Instruction: Which of the following is most equivalent to writing a straddle?

Response: 580 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Figure 68.6 As with the symmetric butterﬂy spread, an investor can create an asymmetric butterﬂy spread by using only calls, using only puts, or by using a long or short position in the stock, combined with both puts and calls. See Problem 68.12. 68 VOLATILITY SPECULATION: STRADDLES, STRANGLES, AND BUTTERFLY SPREADS581 Practice Problems Problem 68.1 (A) buy stock, write two calls (B) buy stock, buy one put (C) short stock, buy one call (D) short stock, buy one put. Problem 68.2 A butterﬂy spread of an index is created as follows: (1) buying a 35-strike call with premium $6.13 and selling a 40-strike call with premium $2.78 (that is buying a bull call spread). (2) selling a 40-strike call with premium $2.78 and buyin a 45-strike call with premium $0.97 ( that is buying a bear call spread). All options expire in three months from now. The current price of the index is $40. The risk free annual eﬀective rate of interest is 8.33%. Show that the proﬁt diagram of this position coincides with the one in Example 68.3. Problem 68.3 A butterﬂy spread of an index is created by buying a 35-strike call (with premium $6.13), selling two 40-strike calls (with premium $2.78 each) and buying a 45-strike call (with premium $0.97). All options expire in three months from now. The current price of the index is $40. The risk free annual eﬀective rate of interest is 8.33%. Show that the proﬁt diagram of this position coincides with the one in Example 68.3. Problem 68.4 A butterﬂy spread of an index is created by buying a 35-strike put (with premium $0.44), selling two 40-strike puts (with premium $1.99 each) and buying a 45-strike put (with premium $5.08). All options expire in three months from now. The current price of the index is $40. The risk free annual eﬀective rate of interest is 8.33%. Show that the proﬁt diagram of this position coincides with the one in Example 68.3. Problem 68.5 A butterﬂy spread of an index is created by buying the stock, buying a 35-strike put (with premium $0.44), selling two 40-strike calls (with premium $2.78 each) and buying a 45-strike call (with premium $0.97). All options expire in three months from now. The current price of the index is $40. The risk free annual eﬀective rate of interest is 8.33%. Show that the proﬁt diagram of this position coincides with the one in Example 68.3. 582 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Problem 68.6 ‡ You believe that the volatility of a stock is higher than indicated by market prices for options on that stock. You want to speculate on that belief by buying or selling at-the-money options. What should you do? (A) Buy a strangle (B) Buy a straddle (C) Sell a straddle (D) Buy a butterﬂy spread (E) Sell a butterﬂy spread Problem 68.7 ‡ You are given the following information: •The current price to buy one share of ABC stock is 100 •The stock does not pay dividends •The risk-free rate, compounded continuously, is 5% •European options on one share of ABC stock expiring in one year have the following prices: Strike Price Call Premium Put Premium 90 14.63 0.24 100 6.80 1.93 110 2.17 6.81 A butterﬂy spread on this stock has the following proﬁt diagram. Which of the following will NOT produce this proﬁt diagram? (A) Buy a 90 put, buy a 110 put, sell two 100 puts (B) Buy a 90 call, buy a 110 call, sell two 100 calls (C) Buy a 90 put, sell a 100 put, sell a 100 call, buy a 110 call (D) Buy one share of the stock, buy a 90 call, buy a 110 put, sell two 100 puts (E) Buy one share of the stock, buy a 90 put, buy a 110 call, sell two 100 calls. 68 VOLATILITY SPECULATION: STRADDLES, STRANGLES, AND BUTTERFLY SPREADS583 Problem 68.8 Which of the following has the potential to lose the most money? (A) a long butterﬂy spread (B) a short strangle (C) a long strangle (D) an asymetric butterﬂy spread Problem 68.9 Three month European put options with strike prices of $50, $55, and $60 cost $2, $4, and $7, respectively. A butterﬂy is created by buying a 50-strike put, selling two 55-strike put, and buying a 60-strike put. The risk-free 3-month interest rate is 5%. (a) What is the maximum gain when a butterﬂy spread is created from the put options? (b) What is the maximum loss when a butterﬂy spread is created from the put options? (c) For what two values of PT does the holder of the butterﬂy spread breakeven, where PT is the stock price in three months? Problem 68.10 Consider a strangle with the strike price of the put is higher than the strike price of the call. Draw the proﬁt diagram. Problem 68.11 An asymmetric butterﬂy spread has the following payoﬀ diagram: This position was created using calls that are priced as follows: Strike Price Premium 80 4 84 2 90 0.50 Ignoring commissions and bid-ask spreads, what was the cost to establish this asymmetric butterﬂy spread? 584 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Problem 68.12 Show that the asymmetric butterﬂy of Figure 68.6 can be duplicated by buying two 35-strike puts (with premium $0.44 each), selling ten 43-strike puts (with premium $3.674 each ) and buying eight 45-strike puts (with premium $5.08 each). All options expire 3 months from now. The risk free annual eﬀective rate of interest is 8.33%. 69 EQUITY LINKED CDS 585 69 Equity Linked CDs An equity-linked CD (ELCD) (also known as index-linked CD) is an FDIC-insured certiﬁcate of deposit that ties the rate of return to the performance of the market indices such as the S&P 500 Composite Stock Price Index, the Dow Jones Industrial Average, or the NASDAQ 100 composite index. Equity-linked CDs are usually issued by a bank. The CDs may vary by maturity (ﬁve years, six years, etc.), participation rate (some oﬀer 100% of the increase in the market index, others only 90% or 70%), and the method of computing the return of the stock index (some use an averaging method to determine the ending value of the index level, while others use the ﬁnal index value). For example, an equity-linked CD can have the following structure: A CD with a return linked to the S&P 500 index is guaranteed to repay the invested amount plus 70% of the simple appreciation in the S&P 500 at the maturity date of 5.5 years. If the invested amount is $10,000 and the index is 1300 initially then a decline in the index after 5.5 years will result in a return of the $10,000 to the investor. If on the other hand the index rises to 2200 at the maturity date then the return to the investor is the original investment plus 70% of the precentage gain on the index. That is, 10,000 [ 1 + 0.70 (2200 1300 −1 )] = $14,846. More generally, the payoﬀ is given by the formula 10,000 [ 1 + 0.70 max { 0, S5.5 1300 −1 }] where S5.5 is the value of the index after 5.5 years. Example 69.1 A 5.5-year ELCD promises to repay $10,000 and 70% of the gain in S&P 500 index at expiration date. Assume investment occurs when S&P 500 index was 1300. (a) Complete the following table Index after 5.5 years Payoﬀ 500 1000 1500 2000 2500 3000 586 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES (b) Draw the payoﬀ diagram. Solution. (a) Index after 5.5 years Payoﬀ 500 10,000 1000 10,000 1500 11,076.92 2000 13,769.23 2500 16,461.54 3000 19,153.85 (b) The payoﬀ diagram is shown in Figure 69.1 Figure 69.1 Building an ELCD Using a Zero-Coupon Bond and a Call Option We next use ﬁnancial engineering to create the building structure of an ELCD by buying a zero- coupon bond and a call option. Consider an investor who wants to invest $1,000 in a S&P 500 index with no risk of losing principal and the opportunity to earn market-like returns. This investor might purchase a zero-coupon bond and an at-the-money call option on the S&P 500 market index. The investor buys the zero-coupon 69 EQUITY LINKED CDS 587 bond at a discount and receives the $1,000 bond par value at maturity. She then uses the diﬀerence between the $1,000 investment and the price of the zero-coupon bond to pay for the call option premium. The call option has expiration date as the time to maturity of the bond. If, at maturity, the S&P 500 has increased, the investor exercises the call option and earns a return on the capital gains portion of the S&P 500. If the S&P 500 decreases, the investor does not exercise the option and receives nothing for it. However, the investor still receives the face value of the bond thereby insuring the original $1,000 investment. A setback of the ELCD in the above example is that in the case of a decline in the index the investor will get back his $1,000 at maturity date. If, say, the maturity date is 5 years and if the annual eﬀective interest rate during the investment time is 6%, the investor has lost in interest 1,000(1.06)5 −1,000 = $338.22. The present value of this interest is 338.22(1.06)−5 = $252.74.Thus as the investor invests today he has foregone interest with a present value of $252.74, and that is the implied cost today of investing in the CD. Summarizing, an investor is attracted to ELCD’s because it has the potential for market appreciation without risking capital. One disadvantage is the possible loss of interest on the invested principal. In general, the payoﬀ of an ELCD with invested principal P and rate of participation r is given by the formula P ( 1 + rmax{PT P0 −1,0} ) = P + Pr max{PT P0 −1,0}. That is, the payoﬀ of a typical equity-linked CD at expiration is equal to the par amount of the CD plus an equity-linked coupon. In general, the equity-linked coupon is equal to either: (a) zero, if the underlying equity has depreciated from an agreed upon strike level (usually the index level on the issue date of the CD), or (b) the participation rate times the percentage change in the underlying equity times the par amount of the note, if the underlying equity appreciated. Example 69.2 An investor buys a ﬁve-year ELCD with 100% participation in the simple appreciation of the S&P 500 index for $1,000. The starting index-level is 1,400. (a) Find the payoﬀ if after ﬁve years the index-level is 2,100. (b) Find the payoﬀ if after ﬁve years the index-level is below 1,400. (c) Create the payoﬀ diagram. Solution. (a) The payoﬀ is 1,000 [ 1 + (2100 1400 −1 ) ×100% ] = $1,500. 588 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES (b) The total payoﬀ is just the original investment of $,1000. (c) The proﬁt diagram is shown in Figure 69.2 Figure 69.2 Example 69.3 Create a synthetic equivalent to the ELCD in Example 69.2. Solution. A synthetic equivalent can be created as follows: (1) invest in a ﬁve-year 6% discount bond with par value of $1,000 for 1,000(1.06)−5 = $747.26 and (2) buy a ﬁve-year, S&P 500 at-the-money call option with a $1,400 strike price for a premium $252.74 We conclude this section by looking at how the three perspectives the end-user, the market-maker, and the economic observer view an ELCD: •The end-user is interested in the product and whether it meets a ﬁnancial need at a fair cost. •The market-maker (i.e. the bank issuer) is interested in making proﬁt without bearing risk from having issued the CD. Some of the proﬁt come from fees for issuing the CD and from penalities of early withdrawals since ELCDs cannot be withdrawn before maturity date. •The economic observer studies the advantage of having ELCDs. 69 EQUITY LINKED CDS 589 Practice Problems Problem 69.1 The payoﬀ in Example 69.1 can be regarded as the payoﬀ of $10,000 and the payoﬀ of a certain unit of European call options. What is that number of units? Problem 69.2 Suppose that an ELCD promises to repay the principal investment plus 80% of the gain of a linked equity. If the gain on the equity is 50% during the investment period then what is the gain for each dollar invested? Problem 69.3 In a ELCD, an investor will receive a return consisting of his orginal investment only when (a) the underlying index has increased from the starting index value (b) the underlying index has remained static or have fallen. Problem 69.4 An investor has an equity linked CD with a nominal value of $10,000 and a participation of %110. If the positive movement of the equity markets at maturity is 35% then ﬁnd the amount due at maturity to the investor. Problem 69.5 Which of the following statements are true about equity linked CDs: (I) pays a ﬁxed interest rate (II) return depends on the performance of the underlying assets (III) Miniumum return is the initially invested money (IV) can be viewed as a combination of a zero-coupon bond and a call option Problem 69.6 A 5-year CD promises to repay $1,000,000 and 115% of the gain in S&P 500 index at expiration date. Assume investment occurs when S&P 500 index was 500. (a) Complete the following table Index after 5 years Payoﬀ 400 450 500 550 600 700 800 590 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES (b) Draw the payoﬀ diagram. Problem 69.7 What is the minimum value of a call option? An equity-linked CD? Problem 69.8 A 5-year CD promises to repay $1,000,000 and 115% of the gain in S&P 500 index at expiration date. Assume investment occurs when S&P 500 index was 500. Assume annual eﬀective interest rate of 8% during the 5-year period. What is the implied cost today of investing in the equity-linked index ? Problem 69.9 A 5-year CD promises to repay $1,000,000 and 115% of the gain in S&P 500 index at expiration date. Assume investment occurs when S&P 500 index was 500. Create a synthetic ELCD for this investment. 70 PREPAID FORWARD CONTRACTS ON STOCK 591 70 Prepaid Forward Contracts On Stock There are four alternative ways for buying shares of a stock. Namely, •Outright purchase: An investor simultaneously pays S0 dollars in cash at time 0 and owns the stock. •Full leveraged purchase: An investor borrows S0 dollars and receives the ownership of the stock at time 0. Letting r denote the annual continuous compound interest, the investor has to repay S0ert at time T. •Forward contract: An arrangement for an agreed-upon future price (the forward price) and date in which the buyer pays for the stock at time T and the seller transfers the ownership of the stock to the buyer at that time. •Prepaid forward contract or prepay: This is like a forward contract but with the (prepaid) forward price paid at time 0. That is, the contract entails the buyer to pay today for the stock and owns it in the future. The price one pays is not necessarily the stock price S0. In this section, we will discuss the question of pricing a prepaid forward contract. Pricing of Prepaid Forward Contracts with no Dividends Let FP 0,T denote the prepaid forward price for an asset bought at time 0 and delivered at time T. This price depends on the dividends paid to the stockholders. In the absence of dividends, the timing of delivery is irrelevant. Therefore, the prepaid forward price is just the stock price at time 0. That is, FP 0,T = S0. Example 70.1 The current price of a stock is $50. The free-risk interest rate is 4% compounded continuously. Assuming no dividends, what is the prepaid forward price? Solution. The price of a prepaid forward contract on a stock that pays no dividends is the initial stock price. So the answer is $50 Now, arbitrage opportunities (positive cash ﬂows either today or in the future and with no risk) can be achieved with prepaid forward price as illustrated in the following two examples. Example 70.2 Suppose that the prepaid forward price exceeds the stock price. An arbitrageur will buy the stock at the low price of S0 and sell the prepaid forward contract for FP 0,T. Assuming no dividends, show that this transaction has a positive cash ﬂow at time 0 and is risk-free. 592 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Solution. The net cash ﬂow at time 0 is FP 0,T −S0 > 0 since the cash ﬂow (outﬂow) from buying the stock is −S0 and from selling the prepaid contract (inﬂow) is FP 0,T. This transaction is risk-free: Selling the prepaid forward contract obliges the seller to deliver the stock at time T and buying the stock today ensures that the stock will be delivered at that time Example 70.3 Suppose that the stock price exceeds the prepaid forward. An arbitrageur will short the stock at the price of S0 and buy the prepaid forward contract for FP 0,T. Assuming no dividends, show that this transaction has a positive cash ﬂow at time 0 and is risk-free. Solution. The net cash ﬂow at time 0 is S0 −FP 0,T >0 since the cash ﬂow (outﬂow) from buying the prepaid forward is −F0,T and from selling the stock (inﬂow) is S0. This transaction is risk-free: By buying the prepaid forward contract we are entitled to acquire the stock at time T which we use to close the short position Throughout the book all prices are assumed to be free arbitrage prices. Pricing of Prepaid Forward Contracts with Dividends Stocks that pay dividends have prepaid forward price lower than the stock price. Only the owner of the stock receives the dividends but not the holder of the prepaid forward. When dividend is paid the stock price is reduced by the amount of the dividend. Suppose that over the life of a forward contract, a stock receives dividends Di at time ti where i= 1,2,··· ,n. Then the prepaid forward price is lower than the stock price by the present value of those dividends: FP 0,T = S0 − n∑ i=1 PV (Di). (70.1) Example 70.4 Suppose ABC stock costs $75 today and is expected to pay semi-annual dividend of $1 with the ﬁrst coming in 4 months from today and the last just prior to the delivery of the stock. Suppose that the annual continuously compounded risk-free rate is 8%. Find the cost of a 1-year prepaid forward contract. Solution. The cost is FP 0,1 = 75 −e−0.08×4 12 −e−0.08×10 12 = $73.09 70 PREPAID FORWARD CONTRACTS ON STOCK 593 Formula (70.1) applies for the case of discrete dividends. What about a stock index where dividends are given almost daily? In this case, it is common to model the dividends as being continuously paid at a constant continuoulsy compounded dividend yield rate δ(which is deﬁned to be the annualized dividend payment divided by the stock price). Note that δis used here as a continuously compounded dividend rate, rather than a force of interest. Like an interest rate, the dividend rate is a rate per year. Divide the interval [0,T] into n equal subintervals using the mesh point ti = i nT where 1 ≤i≤n. The dollar dividend over each interval is Dividend = δT n S0. Suppose that dividends payments are reinvested into stock. Let Ai be the total number of shares at time ti. Then Ai+1 = Ai ( 1 + δT n ) . Hence, the total amount of shares multiplies by ( 1 + δT n ) so that one share at time 0 grows to ( 1 + δT n )n at time T. Letting n →∞ we get that one share at time 0 grows to eδT shares at time T. Example 70.5 You buy one share of ABC stock and hold it for two years. The dividends are paid at continuously compounded rate of 3% and you reinvest all the dividends when they are paid. The current stock price is $125. (a) Find the daily dollar dividend. (b) How many shares do you have at the end of two years? Solution. (a) The daily dollar dividend is Dividend= 0.03 365 ×125 = $0.01027. (b) At the end of two years, the number of shares increased from one share to e0.03×2 = 1.061836 shares Now, if one share of the index is desired at time T, then one has to buy e−δT shares today. Due to dividend reinvestment, at time T we will have e−δTeδT = 1 share. This process of buying just enough shares today so that when dividends are reinvested we accumulate exactly one share at time T is called tailing the position. The cost per share at time 0 of a tailed position is e−δTS0. This position can be achieved as well by buying a T−year prepaid forward contract on a stock (or 594 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES index) that pays dividends at a continuous rate of δ for delivery of one full share of the index at time T. If there exists no arbitrage, the prepaid forward price must be equal to the cost of the tailing position at time 0. That is, FP 0,T = S0e−δT. Remark 70.1 In either the case of a tailed position or a prepaid forward contratct, the buyer of the share does not receive (in cash) any dividends paid prior to time T. In the case of the tailed position, those dividends are used to buy additional stock to bring the total to one full share; in the case of the prepaid forward contract, dividends are not payable to the holder of a long forward prior to the expiration (delivery) date, time T (at which time the investor receives one full share). Example 70.6 Find the 1-year prepaid forward price of the index in the previous example. Solution. The prepaid forward price is FP 0,1 = 125e−0.03 = $121.306 70 PREPAID FORWARD CONTRACTS ON STOCK 595 Practice Problems Problem 70.1 Complete the following charts that descibe the cash ﬂows and transactions to undertake arbitrage (stocks pay no dividends). F0,T >S0 Cash Flows t= 0 t= T Buy Stock Sell Prepaid Forward Total F0,T <S0 Cash Flows t= 0 t= T Short Stock Buy Prepaid Forward Total Problem 70.2 Suppose that the current price of a stock is $1100. The stock does not pay dividends and the risk-free annual rate is 5%. (a) Suppose that an investor is willing to buy a prepaid forward contract at a prepaid forward price of $1135 on a one year prepaid forward contract. Show how to make an arbitrage gain under these circumstances. (b) Suppose that an investor is willing to sell a prepaid forward contract at a prepaid forward price of $1050 on a one year prepaid forward contract. Show how to make an arbitrage gain under these circumstances. Problem 70.3 Suppose ABC stock costs $X today. It is expected that 4 quarterly dividends of $1.25 each will be paid on the stock with the ﬁrst coming 3 months from now. The 4th dividend will be paid one day before expiration of the forward contract. Suppose the annual continuously compounded risk-free rate is 10%. Find X if the cost of the forward contract is $95.30. Round your answer to the nearest dollar. Problem 70.4 Suppose ABC stock costs $75 today and is expected to pay semi-annual dividend of $X with the 596 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES ﬁrst coming in 4 months from today and the last just prior to the delivery of the stock. Suppose that the annual continuously compounded risk-free rate is 8%. Find X if the cost of a 1-year prepaid forward contract is $73.09. Round your answer to the nearest dollar. Problem 70.5 You buy one share of Ford stock and hold it for 2 years. The dividends are paid at the annualized daily continuously compounded rate of 3.98% and you reinvest all the dividends when they are paid. How many shares do you have at the end of two years? Problem 70.6 Suppose that annual dividend of 30 on the stocks of an index is valued at $1500. What is the continuously compounded dividend yield? Problem 70.7 Suppose XYZ stock costs $50 today and is expected to pay quarterly divident of $1 with the ﬁrst coming in 3 months from today and the last just prior to the delivery of the stock. Suppose that the annual continuously compounded risk-free rate is 6%. What is the price of a prepaid forward contract that expires 1 year from today, immediately after the fourth-quarter dividend? Problem 70.8 Suppose XYZ stock costs $50 today and is expected to pay 8% continuous risk-free dividend. What is the price of a prepaid forward contract that expires 1 year from today, immediately after the fourth-quarter dividend? Problem 70.9 LEPOs (Low Exercised Price Options) exist in order to avoid taxes and transaction fees associated with trading with stocks. These stocks have zero-dividends and are purchased outright. These options are certain to be exercised due to the fact that the strike price K is so low. The payoﬀ at expiration time T is PT −K.Find the cost of this option at time 0. Show that a LEPO is essentially a prepaid forward contract with no dividends. Problem 70.10 An investor is interested in buying XYZ stock. The current price of stock is $45 per share. This stock pays dividends at an annual continuous risk-free rate of 5%. Calculate the price of a prepaid forward contract which expires in 18 months. Problem 70.11 ‡ A non-dividend paying stock currently sells for 100. One year from now the stock sells for 110. The risk-free rate, compounded continuously, is 6%. The stock is purchased in the following manner: 70 PREPAID FORWARD CONTRACTS ON STOCK 597 •You pay 100 today •You take possession of the security in one year Which of the following describes this arrangement? (A) Outright purchase (B) Fully leveraged purchase (C) Prepaid forward contract (D) Forward contract (E) This arrangement is not possible due to arbitrage opportunities 598 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES 71 Forward Contracts on Stock Recall that prepaid forward contracts are forward contracts with payment made at time 0. The forward price is the future value of the prepaid forward price and this is true regardless of whether there are discrete dividends, continuous dividends, or no dividends If the expiration time is T and the continuous risk-free rate is r then for a stock with no dividends (i.e. FP 0,T = S0) the future value of the prepaid forward price is given by the formula F0,T = S0erT. In the case of a stock with discrete dividends D1,D2,··· ,Dn made at time t1,t2,··· ,tn the forward price is F0,T =FV (FP 0,T) = ( S0 − n∑ i=1 Die−rti ) erT =S0erT − n∑ i=1 Dier(T−ti) In the case of a stock with continuous dividends with continuously compounded yield δ we have F0,T = (S0e−δT)erT = S0e(r−δ)T. The forward premium is the ratio of the forward price to the current spot price deﬁned by Forward premium= F0,T S0 . The expression 1 −F0,T S0 is the percentage by which the forward price exceeds (or falls below) the spot price. A number α that satisﬁes the equation F0,T = S0eαT is called the annualized forward premium. Solving this equation for α we ﬁnd α= 1 T ln (F0,T S0 ) . 71 FORWARD CONTRACTS ON STOCK 599 If a stock pays no dividends, the annualized forward premium is just the annual risk-free interest rate α= r. For the case of continuous dividends, the annualized forward premium reduces to α= 1 T ln (S0e(r−δ)T S0 ) = r−δ. Remark 71.1 In the literature, the forward premium of an exchange rate forward contract is deﬁned to be the percentage by which today’s forward rate exceeds (falls below) today’s spot rate and is given by the formula ln (F0,T S0 ) . Usually these premia are quoted as annualized rates given by the formula 1 T ln (F0,T S0 ) . Example 71.1 Suppose the stock price is $90 and the continuously compounded interest rate is 10%. (a) What is the 9-month forward price assuming dividends are zero? What is the forward premium? (b) If the 9-month forward price is $92, what is the annualized forward premium? (c) If the 9-month forward price is $92, what is the annualized continuous dividend yield? Solution. (a) F0, 9 12 = 90e0.1×9 12 = $97.01. The forward premium is 97.01 90 = 1.07789. (b) The annualized forward premium is α= 1 T ln (F0,9/12 S0 ) = 12 9 ln (92 90 ) = 0.0293. (c) We have 0.0293 = 0.10 −δ.Solving for δ we ﬁnd δ= 0.0707 = 7.07% In Section 65, we have seen how to create a synthetic forward contract by buying a call and selling a put with both having the same strike price. We next discuss another way for creating a synthetic forward. In this discussion we consider a stock that pays continuous dividends at the rate δ.We buy a tailed position by buying e−δT shares of the stock which cost S0e−δT at time 0 so that at time T we have one share of the stock. We borrow S0e−δT to pay for the shares of stock. In this case, we invest 0 at time 0. At time T we must repay S0e(r−δ)T and we sell the one share of stock at price PT. The payoﬀ for this position is 600 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Payoﬀ at expiration= PT −S0e(r−δ)T = PT −F0,T which is the payoﬀ of a long forward contract with forward price F0,T and expiration time T. In terms of a word equation we have Forward = Stock−Zero coupon bond (71.1) By rearranging the terms of this equation we can derive a synthetic stock Stock = Forward + Zero-coupon bond This position is created by buying a forward contract with forward price F0,T = S0e(r−δ)T and expiration T and lending S0e−δT ( the present value of the forward price) at time 0. The payoﬀ at time T is Payoﬀ at expiration= (PT −F0,T) + F0,T = PT which is the payoﬀ of buying e−δT shares of the stock. We can use (71.1) to create a synthetic bond Zero-coupon bond = Stock −Forward. This position is created by buying e−δT shares of the index at the cost of S0e−δT at time 0 and short one forward contract with forward price F0,T and expiration time T. The payoﬀ of this position at time T is Payoﬀ at expiration= PT + (F0,T −PT) = F0,T which is the payoﬀ of a long risk-free bond with par value of F0,T and expiration time T. Example 71.2 Consider the position of selling e−δT shares for S0e−δT at time 0 and buying a forward with forward price F0,T = S0e(r−δ)T. Complete the following table Cash Flows t= 0 t= T Selling e−δT shares Long one forward Total Solution. 71 FORWARD CONTRACTS ON STOCK 601 Cash Flows t= 0 t= T Selling e−δT shares S0e−δT −PT Long one forward 0 PT −F0,T Total S0e−δT −F0,T Thus going short on a tailed position and buying a forward create cash ﬂows like those of a risk-free short zero-coupon bond (i.e. borrowing) If i is the rate of return on the synthetic bond then i satisﬁes the equation S0e(i−δ)T = F0,T. Solving for i we ﬁnd i= 1 T ln ( F0,T S0e−δT ) = δ+ 1 T ln (F0,T S0 ) . This rate is called the implied repo rate. Remark 71.2 “Repo” is short for “repurchase.” In the synthetic short bond, if we sell the stock short and enter into a long forward contract, then we have sold the stock and agreed to repurchase it. We have in eﬀect taken a loan (shorted a bond), which must be repaid with interest at the implied repo rate. Example 71.3 A $124 index pays a 1.5% continuous dividend. Suppose you observe a 2-year forward price of $135.7. Calculate the annual continuous rate of interest which you earn by buying stock and entering into a short forward contract, both positions for the same nominal amount. Solution. The implies repo rate is i= δ+ 1 T ln (F0,T S0 ) = 0.015 + 1 2 ln (135.7 124 ) = 6% We next discuss how a market-maker uses the synthetic positions strategies to hedge his clients positions. Suppose that a client enters into a long forward position with forward price F0,T and expiration time T. In this case, the conterparty represented by the market-maker is holding a short forward position. To oﬀset this risk, the market-maker borrows S0e−δT and use this amount to buy a tailed position in the index, paying S0e−δT and thus creating a synthetic long forward position. These transactions are summarized in the table below. 602 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Table 71.1 Cash Flows t= 0 t= T Buy tailed position in stock, paying −S0e−δT −S0e−δT PT Borrow S0e−δT S0e−δT −S0e(r−δ)T Short forward 0 F0,T −PT Total 0 F0,T −S0e(r−δ)T A transaction in which you buy the asset and short the forward contract is called cash-and-carry (or cash-and-carry hedge). It is called cash-and-carry, because the cash is used to buy the asset and the asset is kept. A cash-and-carry has no risk: You have obligation to deliver the asset, but you also own the asset. An arbitrage that involves buying the asset and selling it forward is called cash-and-carry arbitrage. Now suppose that a client wishes to enter into a short forward contract with forward price F0,T and expiration time T. In this case, the market-maker is holding a long forward position. To oﬀset this risk, the market-maker short a tailed position in the index receiving S0e−δT a tailed position in the index, paying S0e−δT and thus creating a synthetic long forward position. Example 71.4 A transaction in which you short the asset and long the forward contract is called reverse cash- and-carry (or cash-and-carry hedge). Complete the following table: Cash Flows t= 0 t= T Short tailed position in stock, receiving S0e−δT Lent S0e−δT Long forward Total Solution. We have Table 71.2 Cash Flows t= 0 t= T Short tailed position in stock, receiving S0e−δT S0e−δT −PT Lent S0e−δT −S0e−δT S0e(r−δ)T Long forward 0 PT −F0,T Total 0 S0e(r−δ)T −F0,T Note that with a reverse cash-and-carry the market-maker is long a forward and short a synthetic forward contract 71 FORWARD CONTRACTS ON STOCK 603 Note that if the forward contract is priced as F0,T = S0e(r−δ)T, then proﬁts on a cash-and-carry or reverse cash-and-carry are zero. If F0,T ̸= S0e(r−δ)T, an arbitrageur can do arbitrage. For example, if the forward contract is overpriced, that is F0,T > S0e(r−δ)T, than an arbitrageur can use the strategy described in Table 71.1 to make a risk-free proﬁt. In contrast, if the forward contract is underpriced, that is F0,T < S0e(r−δ)T, than an arbitrageur can use the strategy described in Table 71.2 to make a risk-free proﬁt. Example 71.5 Suppose you are a market-maker in index forward contracts. The current price of an index is $42.50. The index pays no dividends. You observe a 6-month long forward contract with forward price of $43. The risk-free annual continuously compounded interest rate is 5%. Describe a strategy for creating an arbitrage proﬁt and determine the amount of the proﬁt. Solution. The no-arbitrage forward price of the long forward contract is 42.50e0.05(0.5) = $43.58 which is larger than the observed long forward contract. We use the strategy of reverse cash-and-carry: We sell the index for $42.50 and deposit the money into a savings account for six months. We also buy the forward contract. At the expiration date, the money in the bank grew to 42 .50e0.05(0.5) = 43.58 and the cost of the forward is $43. Thus, we make a proﬁt of 43 .58 −43 = $0.58 According to either Table 71.1 or Table 71.2 we have seen that an arbitrageur makes a costless proﬁt only when F0,T ̸= S0e(r−δ)T. This says that there is only one single no arbitrage forward price. In practice that is not the case. That is, there is a range of prices that preclude arbitrage opportunities. In fact, there is a no-arbitrage lower bound and upper bound between which there arbitrage can not be practiced. Transactions costs that determine these bounds are: •Index ask and bid prices denoted by Sa 0 and Sb 0 with Sb 0 <S a 0 . •Forward contract ask and bid prices denoted by Fa 0,T and Fb 0,T with Fb 0,T <F a 0,T. •Index and forward transaction fees (commissions) denoted by CS and CF respectively. •The diﬀerence between borrowing and lending rates rb >rl. Let F+ and F− be the upper and lower bound respecively. Suppose that an arbitrageur believe that the forward price F0,T is too high. Assume that the index pays no dividends and that the arbitrageur applies the cash-and-carry strategy as illustrated in the table below. 604 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Cash Flows t= 0 t= T Buy tailed position in stock −Sa 0 e−δT −CI PT Borrow Sa 0 e−δT + CI + CF [Sa 0 e−δT + CI + CF] −[Sa 0 e−δT + CI + CF]erbT Short forward −CF Fb 0,T −PT Total 0 Fb 0,T −[Sa 0 e−δT + CI + CF]erbT An arbitrageur makes costless proﬁt if Fb 0,T > F+ = [Sa 0 e−δT + CI + CF]erbT. This upper bound reﬂects the fact that we pay a high price for the index (the ask price), pay transaction costs on both the index and the forward, and borrow at a high rate. In our discussion, we assume no transaction fees at time T. In Problem 71.14 we ﬁnd the lower bound to be F−= (Sb 0 −CI −CF)erlT. Example 71.6 Consider an index with current price of $800 and with zero dividends. Suppose that an arbitrageur can borrow at the continuously compounded rate of 5.5% and lend at the continuously compounded rate of 5%. Suppose there is a $1 transaction fee, paid at time 0, only for going either long or short a forward. Find the lower and upper bound between which the arbitrage is not proﬁtable. Solution. We have F−= (800 −1)e0.05 = $839.97 and F+ = (800 + 1)e0.055 = $846.29 Expected Future Price Consider a forward contract with underlying asset a non-dividend paying stock. Suppose that the current price is S0, the risk-free rate is r, and the delivery time is T. A forward price in this case is found to be S0erT. Does this mean that the stock price at time T is S0erT? The answer is no. That is, the forward price does not conveys any information about the future price of the stock. However, there is (conceptually) an expected rate of accumulation for the stock, call it α, that can be used to project an expected price for the stock at time T : S0eαT. Since α= r+ (α−r) the expected rate α has two components: the a risk-free component, r,representing compensation for the passage of time, and a risk component, α−r,which represents compensation for assuming the risk of loss on the stock (risk premium). If α−r >0, then you must expect a positive return from the forward contract. The only way for this to happen is if the forward price predicts too low a stock price. In other words, the forward price a biased predictor of the future stock price. 71 FORWARD CONTRACTS ON STOCK 605 Example 71.7 ‡ The current price of one share of XYZ stock is 100. The forward price for delivery of one share of XYZ stock in one year is 105. Which of the following statements about the expected price of one share of XYZ stock in one year is TRUE? A. It will be less than 100 B. It will be equal to 100 C. It will be strictly between 100 and 105 D. It will be equal to 105 E. It will be greater than 105. Solution. In general, an investor should be compensated for time and risk. A forward contract has no invest- ment, so the extra 5 represents the risk premium. Those who buy the stock expect to earn both the risk premium and the time value of their purchase and thus the expected stock value is greater than 100 + 5 = 105 and the answer is (E) Cost of Carry and Lease Rate Suppose you borrow S0 at the continuously compounded risk-free rate rto purchase a share of stock that pays continuous dividends at the rate δ.At the end of one year, you will repay S0er.However, the dividends will provide an oﬀsetting income of S0eδ.Hence, at the end of one year you will have to pay S0er−δ. This is the net cost of carrying a long position in the stock. We call r−δ the cost of carry. Now suppose you short the index and depositing the proceeds at a savings account paying contin- uously compounded annual rate r. Then at the end of one year you will earn S0er. However, you would have to pay S0eδ to the index lender. Note that the index dividends are what the lender gives up by allowing the stock to be borrowed. We call δ the lease rate of the index: It is the annualized cash payment that the borrower must make to the lender. With the deﬁnitions of the cost of carry and the lease rate, the forward pricing formula F0,T = S0e(r−δ)T has the following word interpretation Forward price = Spot price + Interest to carry the asset - Lease rate or Forward price = Spot price + Cost of carry. 606 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Example 71.8 What is the lease rate for a non-dividend paying stock? What is the cost of carry of a forward contract? Solution. For a non-dividend paying stock the lease rate is δ= 0. A forward contract requires no investment and makes no payouts and therefore has a zero cost of carry 71 FORWARD CONTRACTS ON STOCK 607 Practice Problems Problem 71.1 A stock has a spot price of $35. The continuously compounded risk free rate of interest is 6%. Find the prepaid forward price, the forward price and the annualized forward premium for a one-year forward contract on one share of the stock in each of the following situations. (a) The stock pays no dividends. (b) The stock pays a dividend of 1 in 6 months and 0.5 in one year just before delivery. (c) The stock pays continuous dividends at a rate of 3.5%. Problem 71.2 Suppose the stock price is $55 and the 4-month forward price is $57.50. Assuming the stock pays continuous dividends, ﬁnd (a) the 8-month forward premium (b) the 8-month forward price. Problem 71.3 Suppose the stock price is $55 and the 4-month forward price is $57.50. Assuming the stock pays continuous dividends, ﬁnd (a) Calculate the annualized forward premium. (b) Calculate the 1-year forward price. Problem 71.4 Consider the position of borrowing S0e−δT at time 0 and selling a forward with forward price F0,T = S0e(r−δ)T. Complete the following table Cash Flows t= 0 t= T Short one forward Borrow S0e−δT Total Problem 71.5 Consider the position of selling e−δT shares of the index for S0e−δT at time 0 and lending this amount for a period of T with interest rate r.Complete the following table Cash Flows t= 0 t= T Short tailed position in Stock Lend S0e−δT Total 608 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Problem 71.6 XYZ stock costs $123.118 per share. This stock pays dividends at an annual continuous rate of 2.5%. A 18 month forward has a price of $130.242. You own 10000 shares of XYZ stock. Calculate the annual continuous rate of interest at which you can borrow by shorting your stock. Problem 71.7 You are a market-maker in S&R index forward contracts. The current price of the index is $1100. The index pays no dividends. The risk-free annual continuously compounded interest rate is 5%. (a) What is the no-arbitrage forward price of a 9-month forward contract. (b) Suppose a customer wishes to enter into a long index forward position. If you take the opposite position, describe the strategy you would use to hedge the resulting short position. (c) Suppose a customer wishes to enter into a short index forward position. If you take the opposite position, describe the strategy you would use to hedge the resulting long position. Problem 71.8 Repeat the previous problem that index pays an 1.5% continuous dividend. Problem 71.9 Suppose you are a market-maker in index forward contracts. The current price of an index is $1100. The index pays no dividends. The risk-free annual continuously compounded interest rate is 5%. (a) You observe a 6-month forward contract with forward price of $1135. Describe a strategy for creating an arbitrage proﬁt and determine the amount of the proﬁt. (b) You observe a 6-month forward contract with forward price of $1115. Describe a strategy for creating an arbitrage proﬁt and determine the amount of the proﬁt. Problem 71.10 Suppose you are a market-maker in index forward contracts. The current price of an index is $1100. The index pays continuous dividends at 2%. The risk-free annual continuously compounded interest rate is 5%. (a) You observe a 6-month forward contract with forward price of $1120. Describe a strategy for creating an arbitrage proﬁt and determine the amount of the proﬁt. (b) You observe a 6-month forward contract with forward price of $1110. Describe a strategy for creating an arbitrage proﬁt and determine the amount of the proﬁt. Problem 71.11 An ABC index has a current price of $124 and the continuously compounded risk-free rate is 6%. Suppose you observe a 2-year forward price of $135.7. What dividend yield is implied by this forward price? 71 FORWARD CONTRACTS ON STOCK 609 Problem 71.12 Using the previous problem (a) Suppose you believe that the dividend yield over the next two years will be 0.5%. What arbitrage would you take? (b) Suppose you believe that the dividend yield over the next two years will be 3%. What arbitrage would you take? Problem 71.13 Suppose the current stock price is $45.34 and the continuously compounded intrest rate is 5%. The stock pays dividend of $1.20 in three months. You observe a 9-month forward contract with forward price $47.56. Is there an arbitrage opportunity on the forward contract? If so, describe the strategy to realize proﬁt and ﬁnd the arbitrage proﬁt. Problem 71.14 Suppose that an arbitrageur believes that the observed forward price F0,T is too low. Show that, for the arbitrageur to make costless proﬁt we must have F0,Ta<F −= [Sa 0 e−δT −CI −CF]erlT. Problem 71.15 Suppose that an arbitrageur would like to enter a cash-and-carry for 10000 barrels of oil for delivery in six months. Suppose that he can borrow at an annual eﬀective rate of interest of 4.5%. The current price of a barrel of oil is $55. (a) What is the minimum forward price at which he would make a proﬁt? (b) What is his proﬁt if the forward price is $57? Problem 71.16 Consider an index with current price of $800 and with zero dividends. Suppose that an arbitrageur can borrow at the continuously compounded rate of 5.5% and lend at the continuously compounded rate of 5%. Suppose that there are no transaction fees. Then an arbitrage is not proﬁtable if the forward price is between F− and F+. Determine the values of F− and F+. Problem 71.17 Consider an index with current price of $800 and with zero dividends. Suppose that an arbitrageur can borrow at the continuously compounded rate of 5.5% and lend at the continuously compounded rate of 5%. Suppose the transaction fee for going long or short a forward is $1 and $2.40 for going long or short an index. Find the lower and upper no-arbitrage bounds. 610 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES 72 Futures Contracts A futures contract is an agreement between two counterparts to buy/sell an asset for a speciﬁed price (the future price) at a speciﬁed date (the delivery date). Like the case of a forward con- tract, the holder of the contract is called the long future and the seller is called the short future. Futures are exchange-traded forward contracts. That is, they are bought and sold in organized future exhanges. Examples of exchanges the Chicago Mercantile Exchange, the Chicago Board of Trade, the International Petroleum Exchange of London, the New York Mercantile Exchange (NYMEX), the London Metal Exchange and the Tokyo Commodity Exchange. In the US, futures transactions are regulated by a government agency, the Commodity Futures Trading Commission (CFTC). Each exchange has a clearinghouse. The role of the clearinghouse is to match the purchases and the sales which take place during the trading day. By matching trades, the clearinghouse never takes market risk because it always has oﬀsetting positions with diﬀerent counterparts. By having the clearinghouse as counterpart, an individual entering a futures contract does not face the possible credit risk of its counterpart. Forwards and futures are similar in many aspects. However, there are still some diﬀerences which we list next. •Forward contracts are privately negotiated and not standardized. They are traded mainly over the counter and can be customized to suit the buyer or the seller. In contrast, futures contracts are standardized and have speciﬁed delivery dates, locations, and procedures. •Forward contracts are settled at the delivery date. Futures contracts are settled daily. As a result of daily settlements, value of a future contract is calculated according with the current value of the the underlying asset. We refer to this process as marking-to-market. Frequent marking-to- market can lead to pricing diﬀerences between futures contracts and identical forward contracts. •Futures contracts are liquid. Unlike a forward contract, which requires settling with the coun- terparty by actually buying or selling the asset on the expiration date, a futures contract can be cancelled by entering into the opposite position. For example, if you are short a futures contract, you can cancel your obligation to sell by entering an oﬀsetting obligation to buy the contract. This is done through a brokerage ﬁrm. •In the presence of clearinghouse, futures contracts are structured so as to minimize the eﬀects of credit risks. In contrast, in over-the-counter forward contracts, each party bears the other’s credit risk. •Because futures contracts are traded on an exchange, trades are subject to the exchange’s rules, which typically include a limit on the amount of daily price changes by imposing temporary halts in trading if the price change exceeds a deﬁned amount. •Futures contracts are commonly cash-settled, rather than requiring delivery of the underlying 72 FUTURES CONTRACTS 611 asset. We next illustrate an example of futures contracts with the S&P 500 index futures contract. The underlying asset for such a contract is the S&P 500 stock index. The contract has the following speciﬁcations: •The futures contract is traded at the Chicago Mercantile Exchange. •Notional Value: One contract has a value that is deﬁned as the level of the index (let’s say it is currently 1300) times $250 (so an S&P 500 futures contract currently has a notional value of $325,000). •The contract has delivery months March, June, September and December. •Trading ends on each business day prior to determination of settlement price. The settlement price is the average of the prices at which the contract traded immediately before the bell signaling the end of trading for the day. •The contracts are cash-settled because it is inconvenient or impossible to deliver the underlying asset which in this case consists of a basket of 500 stocks. The cash settlement is based on the opening price of the index on the third Friday of delivery month. Example 72.1 Suppose you wish to acquire $2.2 million worth of S&P 500 index with futures price 1100. (a) What is the notional value of one contract and how many futures contracts can you long for the invested amount? (b) The open interest of a futures contract is deﬁned to be the number of all futures outstanding contracts. It is the number of long positions. The activities of ﬁve traders are listed below. Complete the following table: Date Trading Activity Open interest Jan 1 A buys 1 contract and B sells 1 contract Jan 2 C buys 5 contracts and D sells 5 contracts Jan 3 A sells his contract and D buys one contract Jan 4 E buys 5 contracts from C who sells his 5 contracts Solution. (a) The notional value of one futures contract is 250 ×1100 = $275,000.With $2.2 million you can long 2200000 275000 = 8 futures contract. (b) 612 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Date Trading Activity Open interest Jan 1 A buys 1 contract and B sells 1 contract 1 Jan 2 C buys 5 contracts and D sells 5 contracts 6 Jan 3 A sells his contract and D buys one contract 6 Jan 4 E buys 5 contracts from C who sells his 5 contracts 6 Margins and Marking to Market One of the advantages of the futures contracts over the forward contracts is the dealing with credit risk. For example, forward contracts can be traded between two parties directly. This creates a credit risk related with the solvency of each party: One of the investors may regret the deal and try to back out or simply might not have the ﬁnancial resources to honor the agreement. In contrast, in the case of futures contracts the exchange tries to organize trading in such a way to avoid credit defaults. This is when margins come in. To avoid credit risk, a brokerage ﬁrm requires investors entering a futures contract to make deposit into an account called the margin account (also known as perfomance bond) which earns interest. The amount to be deposited at the time the contract is entered into is called theinitial margin and is determined by the exchange. It is usually a fraction of the market value of the futures’ underlying asset. The margin account is mainly intended to protect the counterparty of the contract. Margin accounts are adjusted on a daily basis to reﬂect the investor’s gain or loss. This practice is referred to as marking to market the account. A trade is marked to market at the close of the day on which it takes place. It is then marked to market at the close of the trading on each subsequent day. If there exists a loss, the investor’s broker transfers that amount from the investor’s margin account to the clearinghouse. If a proﬁt, the clearinghouse transfers that amount to investor’s broker who then deposits it into the investor’s margin account. We illustrate the role of margins in the following example. Example 72.2 On July 5, 2007, John instructs his broker to buy two gold futures contracts (each contract with nominal amount of 100 ounces of gold) on the New York Commodity Exchange (COMEX) with futures’ price of $400 per ounce. The annual continuously compounded rate of return is 1.5%. (a) How many ounces of gold has John contracted to buy at the price of $400 per ounce? (b) What is the notional value of the two contracts? (c) Suppose that the exchange requires 5% margin with daily settlement. What is the initial margin? (d) Suppose that on July 6,2007, the futures price of gold has dropped from $400 to $397. What is the balance in John’s margin account after settlement? (e) Suppose that on July 7,2007, the futures price of gold has risen to $400. What is the balance in John’s margin account after settlement? 72 FUTURES CONTRACTS 613 Solution. (a) Since each contract consists of 100 ounces of gold, John has contracted to buy 200 ounces at the price of $400 per ounce. (b) The notional value of the two contracts is 2 ×100 ×400 = $80,000. (c) The initial margin is 5% ×80,000 = $4,000. (d) The balance in John’s margin account after settlement is 4000e 0.015 365 + 200(397 −400) = $3400.16. (e) The balance in John’s margin account after settlement is 34,000.16e 0.015 365 + 200(400 −397) = $4,000.30 An investor is entitled any balance in the margin account in excess of the initial margin. However, if the balance in an investors margin account falls, the broker (or clearinghouse) has less protection against default. Investors are required to keep the margin account to a minimum level which is a percentage of the initial margin. This minimum level is called the maintenance margin. When the balance in the margin account drops below the maintenance level the investor receives amargin call and is required to make a deposit to return the balance to the initial margin level. If the investor does not meet the margin call, the position is closed out and the investor receives any remaining balance in the margin account. Example 72.3 A farmer enters into a short futures contract to sell 100,000 pounds of wheat for $1.4 per pound. There is 30% margin and weekly settlement. The maintenance margin is 20% and the annual eﬀective rate of interest is 4.5%. What is the minimum next week price which would lead to a margin call? Solution. The initial margin is (0.30)(100000)(1.4) = $42000. The maintenance margin is (0.20)(100000)(1.4) = $28000. After settlement, next week’s balance is 42000(1.045) 1 52 + 100000(1.4 −P1 52 ). A margin call happens if 28000 >42000(1.045) 1 52 + 100000(1.4 −P1 52 ). 614 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES or P1 52 >1.4 −28000 −42000(1.045) 1 52 100000 = 1.540355672 72 FUTURES CONTRACTS 615 Practice Problems Problem 72.1 The open interest on silver futures at a particular time is the number of (A) all silver futures outstanding contracts (B) outstanding silver futures contracts for a particular delivery month (C) silver futures contracts traded during the day (D) silver futures contracts traded the previous day Problem 72.2 The fact that the exchange is the counter-party to every futures contract issued is important because it eliminates (A) market risk (B) credit risk. (C) interest rate risk Problem 72.3 Margin must be posted by (A) buyers of futures contracts (B) sellers of futures contracts (C) both buyers and sellers of futures contracts (D) speculators only Problem 72.4 The daily settlement of obligations on futures positions is called (A) a margin call (B) marking to market (C) a variation margin check (D) None of the above Problem 72.5 All of the following are false except (A) A margin deposit can only be met by cash (B) All futures contracts require the same margin deposit (C) The maintenance margin is the amount of money you post with your broker when you buy or sell a futures contract (D) The maintenance margin is the value of the margin account below which the holder of a futures contract receives a margin call. 616 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Problem 72.6 You are currently long in a futures contract. You then instruct a broker to enter the short side of a futures contract to close your position. This is called (A) a cross hedge (B) a reversing trade (C) a speculation (D) marking to market Problem 72.7 A silver futures contract requires the seller to deliver 5,000 Troy ounces of silver. An investor sells one July silver futures contract at a price of $8 per ounce, posting a $2,025 initial margin. Neglecting interest, if the required maintenance margin is $1,500, the price per ounce at which the investor would ﬁrst receive a maintenance margin call is closest to: (A) $5.92 (B) $7.89 (C) $8.11 (D) $10.80 Problem 72.8 If the initial margin on a futures contract I have sold increases, it likely means: (A) The market has gotten less liquid. (B) I am not making money in my futures positions. (C) Prices have gotten more volatile. (D) The margins never change once you have a futures position. Problem 72.9 Complete the following statement: At the end of each trading day, the margin account is adjusted to reﬂect the futures trader’s gain or loss. This process is referred to as the account. Problem 72.10 What is the diﬀerence between the initial margin and the maintenance margin? Problem 72.11 Explain some diﬀerences between a futures contract and a forward contract. Problem 72.12 Crude oil futures trade in units of 1,000 U.S. barrels (42,000 gallons). The current value is $50/bar- rel. Find the notional value of this contract. 72 FUTURES CONTRACTS 617 Problem 72.13 Assume today’s settlement price on a CME EUR futures contract is $1.3140/EUR. The nominal amount of one contract is 125,000. You have a short position in one contract. Your margin account currently has a balance of $1,700. The next three days’ settlement prices are $1.3126, $1.3133, and $1.3049. Calculate the changes in the margin account from daily marking-to-market and the balance in the account after the third day. Problem 72.14 Do Problem 72.13 again assuming you have a long position in the futures contract. Assume 40% maintenance margin. Would you receive a margin call? Problem 72.15 Suppose the S&P 500 index futures price is currently 1200. You wish to purchase four futures contracts on margin. (a) What is the notional value of your position? (b) Assuming a 10% initial margin, what is the value of the initial margin? Problem 72.16 Suppose the S&P 500 index futures price is currently 950 and the initial margin is 10%. You wish to enter into 10 S&P 500 futures contracts. (a) What is the notional value of your position? What is the margin? (b) Suppose you earn a continuously compound rate of 6% on your margin balance, your position is marked to market weekly, and the maintenance margin is 80% of the initial margin. What is the greatest S&P 500 index futures price 1 week from today at which you will receive a margin call? 618 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES 73 Understanding the Economy of Swaps: A Simple Com- modity Swap A swap is an agreement to exchange cash ﬂows at speciﬁed future times according to certain spec- iﬁed rules. For example, a forward contract is a single-payment swap. To see this, suppose you enter a forward contract to buy 50 ounces of gold for $300 an ounce in one year. You can sell the gold as soon you receive it. The forward contract acts like a single-payment swap because at the end of the year you will pay 50 ×300 = $15000 and receive 50 ×P1 where P1 is the market price of one ounce of gold on that date. While a forward contract leads to an exchange of cash on a single future dates, swaps typically lead to cash ﬂow exchanges taking place on several future dates. In some sense, a swap is a series of forward contracts combined into one contract or put it diﬀerently it can be regarded as a portfolio of forward contracts. There are various types of swaps, namely, interest rates swap, currency swaps, commodity swaps, and equity swaps. In this section we will try to understand the economy of swaps. Our discussion will be centered around commodity swaps. A commodity swap is a swap in which one party is selling a commodity (or the cash equivalent) and the other is paying cash for the commodity (the buyer). The contract needs to specify the type and quality of the commodity, how to settle the contract, etc. Commodity swaps are predominantly associated with energy-related products such as oil. Suppose that quantity Qi of the commodity is to be delivered at time ti where i = 1 ,2,··· ,n. The buyer can either make one payment of C0 at time 0 for the swap (a prepaid swap) or make a payment of Ci at time ti. Let F0,ti be the forward price of the commodity with delivery at time ti where 1 ≤i≤n.Let ri be the annual interest rate applied over the interval of time [0,ti].Then the present value of the commodity delivered over the life of the swap is n∑ i=1 QiPVi(F0,ti). In a prepaid swap, the buyer makes a single payment today in the amount of n∑ i=1 QiPVi(F0,ti) to obtain multiple deliveries in the future. Example 73.1 A manufacturer is going to buy 100,000 barrels of oil 1 year from today and 2 years from today. 73 UNDERSTANDING THE ECONOMY OF SWAPS: A SIMPLE COMMODITY SWAP619 Suppose that the forward price for delivery in 1 year is $20/barrel and in 2 years $21/barrel. The 1-year spot interest rate is 6% and the 2-year spot interest rate is 6.5%. The manufacturer enters into long forward contracts for 100,000 barrels for each of the next 2 years, committing to pay $20/barrel in 1 year and $21/barrel in 2 years. Alternatively, the manufacturer can make a single payment P now and the supplier would commit to deliver 1 barrel in year 1 and 1 barrel in year 2. Find P. Solution. P is just the cost of a prepaid swap given by 20 1.06 + 21 1.0652 = $37.383 Now, with multiple deliveries and multiple payments, the buyer makes swap payments at the time the commodity is delivered. Usually each swap payment per unit of commodity is a ﬁxed amount say R. That is, at time ti the swap payment is QiR. Thus, the present value of all the payments is given by the sum n∑ i=1 QiPVi(R). We thus obtain the equation n∑ i=1 QiPVi(F0,ti) = n∑ i=1 QiPVi(R) From this equation, one ﬁnds R given by R= ∑n i=1 QiPVi(F0,ti)∑n i=1 QiPVi(1) . Suppose that at time of delivery, the buyer pays a level payment of Rat each of the delivery times. Then, the present value of the cashﬂow of payments is n∑ i=1 PVi(R) and R is given by R= ∑n i=1 QiPVi(F0,ti)∑n i=1 PVi(1) . 620 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Example 73.2 Consider the information of Example 73.1. Suppose a swap calls for equal payments each year. Let x be the payment per year per barrel. This is the swap price. Find x. Solution. We must have x 1.06 + x 1.0652 = 37.383. Solving this equation for xwe ﬁnd x= $20.483.This means, that each year, the manufacturer will pay $20.483 and receives a barrel of oil Remark 73.1 The payments need not be equal. Any payments that have a present value of $37.383 are acceptable. That is, any two payments that satisfy the equation P1 1.06 + P2 1.0652 = 20 1.06 + 21 1.0652 . A swap can be settled either by physical settlement or by cash settlement. If a swap is settled physically, the party with the commodity delivers the stipulated notional amount to the buyer in exchange of cash as shown in the previous example. If a swap is cashed settled, then the commodity is valued at the current spot price. If the current value of the commodity is larger than the value of the cash payment, then the buyer pays the swap counterparty the diﬀerence and then buys the commodity at the spot price. Reciprocally, if the current value of the commodity is smaller than the value of the cash payment, the counterparty pays this diﬀerence to the buyer and the buyer buys the commodity. Example 73.3 Again, we consider the information of both previous examples. Suppose the swap is cash settled. (a) Suppose that the market price is $25. Calculate the payment which the buyer receives. (b) Suppose that the market price is $18. Calculate the payment which the counterparty receives. Solution. (a) The swap counterparty pays the buyer 25 −20.483 = $4.517 per barrel. (b) The buyer pays the swap counterparty 18 −20.483 = −$2.483 per barrel Notice that whatever the market price of oil, the net cost to the buyer is the swap price $20.483: spot price −(spot price −swap price) = swap price. 73 UNDERSTANDING THE ECONOMY OF SWAPS: A SIMPLE COMMODITY SWAP621 This formula describes the payments per one barrel of oil. For a swap agreement involving 100,000 barrels of oil, the values in the formula are multiplied by the notional amount of the swap (i.e. 100,000 barrels). Remark 73.2 Notice that the results for the buyer are the same whether the swap is settled physically or ﬁnancially. In both cases, the net cost to the buyer is $20.483 per barrel. A swap can be regarded as a series of forward contracts, combined with borrowing and/or lending money. In the above oil swap example, the buyer under the swap agreement would pay (and the counterparty would receive) the swap price of $20.483 on each of the two dates. Under a pair of forward contracts, the forward prices of $20 and $21 would have been paid on the ﬁrst and second dates, respectively. Relative to the forward prices, the buyer lends to the counterparty an amount of $0.483 (= 20 .483 −20) on the ﬁrst date by paying that much more than the forward rate; then the counterparty repays the loan by accepting $20.483 on the second date, which is less than the forward price by $0.517 (= 21 −20.483). This loan has the eﬀect of equalizing the net cash ﬂow on the two dates. The interest rate on the loan is 0.517 0.483 −1 = 7% This is the 1-year implied forward rate from year 1 to year 2. A commodity swap allows to lock the price of a sale. It can be used by a producer of a com- modity to hedge by ﬁxing the price that he will get in the future for this commodity. It also can be used by a manufacturer to hedge by ﬁxing the price that he will pay in the future for a commodity. Swap Dealer A dealer is an agent who makes a market buying and selling swaps to parties who have natural exposure to price risk. He earns his keep on the bid/ask spread. The dealer buys a swap from the producer, at ﬁxed price S (the swap price). The seller receives S−PT, where PT is the spot price. The dealer sells a swap to the consumer, at ﬁxed price S+ ∆, where δ is the ask-bid spread. The buyer pays S+ ∆ −PT. The dealer earns diﬀerence between what buyer gives and seller gets, i.e. ∆ . The process of matching a buyer to the seller is called a back-to-back transaction. As you can see from the above discussion, the dealer is subject to each party’s credit risk, but is not exposed to price risk. Figure 73.1 illustrates the role of the swap dealer. 622 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Figure 73.1 We next consider the case when the dealer serves as the counterparty. A dealer enters the swap as the counterparty. Thus, the dealer has the obligation to pay the spot price and receives the swap price. The dealer has a short position in 1- and 2- year oil. The dealer can lose money if the spot price increases. The dealer hedges his short position by going long year 1 and year 2 gold forwards. The table below illustrates how this strategy works. Year Payment from Gold Buyer Long Forward Net 1 $20 .483 −P1 P1 −$20 $0.483 2 $20 .483 −P2 P2 −$21 −$0.517 The net cash ﬂow for the hedged position is a loan where the dealer receives $0.483 in year 1 and repays $0.517 in year two. The dealer has interest rate exposure. In interest rate fall, the dealer will not be able to earn suﬃcient return from investing $0.483 in year 1 to repay $0.517 in year 2. This shows that hedging oil price risk does not fully hedge the position. In summary, the dealer can eliminate its price risk by hedging the swap contract through the use of multiple forward contracts. The dealer must also enter into forward rate agreements in order to hedge the interest rate exposure resulting from the timing diﬀerence between the forward contract’s payments and the payments under the swap agreement. The Market Value of a Swap When a swap agreement is created, its market value is zero. In our oil example, the swap consists of two forward contracts and an agreement to lend money at the implied forward rate of 7%. These forwards have zero values at time 0 so the swap does as well. As time passes the market value of the swap will no longer be zero due to oil prices and interest rates change. So over time, the swap contract develops a non-zero value. Even if prices and interest rates do not change, the swap agreement will have a non-zero value once the ﬁrst payment has been made. Let us ﬁnd the market value of the oil swap example. Suppose that right after entering the swap the oil prices go up by $2 in year 1 and year 2. The original swap in this case has a non-zero market value. Assuming interest rate are the same, the new swap price x satisﬁes the equation x 1.06 + x 1.0652 = 22 1.06 + 23 1.0652 73 UNDERSTANDING THE ECONOMY OF SWAPS: A SIMPLE COMMODITY SWAP623 which gives x= $22.483. In this case, the buyer buys a barrel of oil for $20.483 and can sell it for $22.483 making a net proﬁt of $2 per barrel each year. The present value of this diﬀerence of prices is 2 1.06 + 2 1.0652 = $3.65. The buyer can receive a stream of payments worth $3.65 by oﬀsetting a new swap. Thus, $3.65 is the market value of the swap. If interest rates had changes, we would have used the new interest rate in computing the new swap price. The swap market is very large and has no major governmental oversight. Swap markets have none of the provisions as the futures and options markets do, thus resulting in a default risk and the need to assess the creditworthiness of each counterparty. Traders must carefully assess their trade partners. On the other side of this, swaps have a measure of privacy and can be negotiated for unique quan- tities, qualities, and contract provisions. Example 73.4 Suppose that oil forward prices for 1 year, 2 years, and 3 years are $20, $21, and $22. The eﬀective annual interest rates are shown in the table below Maturity(in years) yield % 1 6 2 6.5 3 7 (a) What is the swap price, assuming level payments? (b) Suppose you are a dealer who is paying the swap price and receiving the spot price. Suppose you enter into the swap and immediately thereafter all interest rates rise by 0.5% with the oil forward prices remain unchanged. What happens to the value of your swap position? (c) What if interest rates fall by 0.5%? (d) What hedging instrument would have protected you against interest rate risk in this position? Solution. (a) The swap price x satisﬁes the equation x 1.06 + x 1.0652 + x 1.073 = 20 1.06 + 21 1.0652 + 22 1.073 . Solving for x we ﬁnd x= $20.952. (b) You should short 3 forward contracts. This strategy is illustrated in the table below. 624 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Year Payment from Gold Buyer Short Forward Net 1 P1 −$20.952 $20 −p1 −$0.952 2 P2 −$20.952 $21 −p2 $0.048 3 P3 −$20.952 $22 −p2 $1.048 With the 0.5% increase in rates we ﬁnd that the net present value of the loan is −0.952 1.065 + 0.048 1.072 + 1.048 1.0753 = −$0.0081. The dealer never recovers from the increased interest rate he faces on the overpayment of the ﬁrst swap payment. (c) If interest rates fall by 0.5% then the net present value of the loan is −0.952 1.055 + 0.048 1.062 + 1.048 1.0553 = $0.0083. The dealer makes money, because he gets a favorable interest rate on the loan he needs to take to ﬁnance the ﬁrst overpayment. (d) The dealer could have tried to hedge his exposure with a forward rate agreement or any other derivative protecting against interest rate risk 73 UNDERSTANDING THE ECONOMY OF SWAPS: A SIMPLE COMMODITY SWAP625 Practice Problems Problem 73.1 Match (a)-(d) with (I)-(IV) (a) Notional amount is (b) The contracting parties in a swap are called (c) What the swap dealers receive for their services is (d) The ask-spread commission is (I) ask-spread bid (II) counterparties (III) the diﬀerence between the cash ﬂow one counterparty pays to the dealer and what dealer pays out to the other counterparty (IV) used to calculate the payments that will be exchanged between the two parties. Problem 73.2 ‡ Zero-coupon risk-free bonds are available with the following maturities and yield rates (eﬀective, annual): Maturity(in years) yield % 1 6 2 6.5 3 7 You need to buy corn for producing ethanol. You want to purchase 10,000 bushels one year from now, 15,000 bushels two years from now, and 20,000 bushels three years from now. The current forward prices, per bushel, are $3.89, $4.11, and $4.16 for one, two, and three years respectively. You want to enter into a commodity swap to lock in these prices. Which of the following sequences of payments at times one, two, and three will NOT be acceptable to you and to the corn supplier? (A) $38,900; $61,650; $83,200 (B) $39,083; $61,650; $82,039 (C) $40,777; $61,166; $81,554 (D) $41,892; $62,340; $78,997 (E) $60,184; $60,184; $60,184 Problem 73.3 A manufacturer uses oil in his profession. He needs 10,000 barrels in two months, 12,000 barrels in four months, and 15,000 barrels in six months. The manufacturer enters into a long oil swap. The 626 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES payment of the swap will be made at the delivery times. The current forward prices, per barrel, are $55, $56, and $58 for two, four, and six months re- spectively. Zero-coupon risk-free bonds are available with the following maturities and yield rates (nominal, convertible monthly): Maturity(in months) yield % 2 4.5 4 4.55 6 4.65 (a) Suppose level payments of R are made at each of the delivery times. Calculate R. (b) Suppose that the cash swap payment consists of a unique payment per barrel made at each of the delivery times. Calculate the no arbitrage swap price per barrel. Problem 73.4 Suppose that a transportation company must buy 1000 barrels of oil every six months, for 3 years, starting 6 months from now. Instead of buying six separate long forward contracts, the company enters into a long swap contract. The payment of the swap will be made at the delivery times. The current forward prices, per barrel, are $55, $57, $57, $60, $62, and $64 for 6-, 12-, 18-, 24-, 30-, and 36- months respectively. Zero-coupon risk-free bonds are available with the following maturities and yield rates (nominal, convertible semiannually): Maturity(in months) yield % 6 5.5 12 5.6 18 5.65 24 5.7 30 5.7 36 5.75 (a) With level payments, ﬁnd the price per barrel of oil using the swap. (b) Suppose the swap is settled in cash. Assume that the spot rate for oil in 18 months is $57. Calculate the payment which the counterparty receives. (c) Suppose that immediately after the swap is signed up, the future prices of oil are given by the table Forward Price $55 $58 $59 $61 $62 $63 Expiration (in months) 6 12 18 24 30 36 Find the market value of the swap for the company. 73 UNDERSTANDING THE ECONOMY OF SWAPS: A SIMPLE COMMODITY SWAP627 Problem 73.5 Suppose that a jewelry manufacturer needs to purchase 100 ounces of gold one year from now and another 100 ounces in two years, and that the forward price for gold is $750 per ounce for delivery in one year and $800 for delivery in two years. The manufacturer could enter into two long forward contracts for these two dates at these two prices, and eliminate the risk of price variation. Alternatively, a single swap agreement could be used to ﬁx a single price for the purchases on these two dates. Suppose that the 1-year and 2-year spot rates are 5% and 6% respectively. (a) With level payments, ﬁnd the price per ounce of gold using the swap. (b) Suppose that the payments are not leveled, say with payment P1 per ounce for year 1 and P2 for year 2. Set the equation that P1 and P2 must satisfy. (c) What is the notional amount of the swap? (d) The buyer pays an amount larger than the forward price in 1 year. The diﬀerence is considered as a loan for the counterparty. What is the value of this amount per ounce? (e) The buyer pays an amount smaller than the forward price in 2 years. The diﬀerence is considered as the counterparty repayment of the loan. What is the value of this amount per ounce? (f) What is the implied forward rate from year 1 to year 2 of the loan? Problem 73.6 Consider the information of the previous problem. Suppose the forward prices are $780 and $830. The 1- and 2-year interest rates are 5% and 6% respectively. Find the new swap price. Problem 73.7 Suppose that gold forward prices for 1 year, 2 years, and 3 years are $750, $800, and $850. The eﬀective annual interest rates are shown in the table below Maturity(in years) yield % 1 6 2 6.5 3 7 (a) What is the swap price? (b) What is the price of a 2-year swap with the ﬁrst settlement two years from now and the second in 3 years? Problem 73.8 Consider the 2-year gold swap of Problem 73.5. Suppose a dealer is paying the spot price and receiving the swap price. What position in gold forward contracts will hedge gold price risk in this position? 628 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Problem 73.9 Consider the 2-year gold swap of Problem 73.5. Suppose a dealer is paying the swap price and receiving the spot price. What position in gold forward contracts will hedge gold price risk in this position? 74 INTEREST RATE SWAPS 629 74 Interest Rate Swaps In this section we consider another type of swaps: the interest rate swaps. Companies use interest rate swap as a means of converting a series of future interest payments that vary with changes in interest rates ( ﬂoating payments) to a series of ﬁxed level payments (i.e. swap rate ), or vice-versa. The ﬁxed stream of payments are computed with respect to a rate determined by the contract. The ﬂoating stream of payments are determined using a benchmark, such as the LIBOR 1 The amount in which the interest payments is based is called the notional principal. The life of a swap is called the swap term or swap tenor. Interest rate swaps are used to hedge against or speculate on changes in interest rates: They are often used by ﬁrms to alter their exposure to interest-rate ﬂuctuations, by swapping ﬁxed-rate obligations for ﬂoating rate obligations, or vice versa. By swapping interest rates, a ﬁrm is able to alter its interest rate exposures and bring them in line with management’s appetite for interest rate risk. They are also used speculatively by hedge funds or other investors who expect a change in interest rates or the relationships between them. Interest rate swaps are also very popular due to the arbitrage opportunities they provide. The interest rate swap market is closely linked to the Eurodollar 2 futures market which trades at the Chicago Mercantile Exchange. All interest rate swaps payments are settled in net: If the ﬁxed rate is greater than the ﬂoating rate, the buyer pays the counterparty the diﬀerence. In contrast, if the ﬁxed rate is smaller than the ﬂoating rate, the counterparty pays the buyer the diﬀerence. The interest rate determination date for the ﬂoating interest payment occurs at the beginning of the period with payment due at the end of the period. It is important to keep in mind that there is no exchange of the notional principal in a swap. Example 74.1 On December 31, 2006, ﬁrm ABC enters into an interest rate swap with ﬁrm XYZ. The notional principal is $100 m. Firm ABC pays the ﬁxed rate of 5% compounded annually and receives ﬂoating rate from ﬁrm XYZ at a 12-month LIBOR. The swap term is 5 years. That is, the two parties exchange payments annually on December 31, beginning in 2007 and concluding in 2011. This swap is illustrated in Figure 74.1 1The ( London Interbank oﬀer rate ) LIBOR is the most widely used reference rate for short term interest rates world-wide. The LIBOR is published daily the (British Bankers Association) BBA. It is based on rates that large international banks in London oﬀer each other for interbank deposits. Rates are quoted for 1-month, 3-month, 6-month and 12-month deposits. 2Eurodollars are US dollars deposited at banks outside the United States, primarily in Europe. The interest rate paid on Eurodollars is largely determined by LIBOR. Eurodollar futures provide a way of betting on or hedging against future interest rate changes. 630 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Figure 74.1 (a) On December 31,2006 the 12-month LIBOR rate was 4.5%. Determine the amount paid by ABC to XYZ on December 31,2007. (b) On December 31,2006 the 12-month LIBOR rate was 5.5%. Determine the amount received by ABC from XYZ on December 31,2007. Solution. (a) On December 31,2007 ABC will pay XYZ the amount $100m ×0.05 = $5,000,000 and receives from XYZ the amount $100m ×0.045 = $4,500,000. Thus, on December 31,2007, ABC pays XYZ the net payment of $500,000. (a) On December 31,2007 ABC will pay XYZ the amount $100m ×0.05 = $5,000,000 and receives from XYZ the amount $100m ×0.055 = $5 ,500,000. Thus, on December 31,2007, ABC receives from XYZ the net amount of $500,000. On net, ABC is paying At no point does the principal change hands, which is why it is referred to as a notional amount Remark 74.1 A ﬁxed-for-ﬂoating interest rate swap is often referred to as a plain vanilla swap because it is the most commonly encountered structure. Counterparty Risk of a Swap Unlike future markets, swaps are over-the-counter instruments and so they are not backed by the guarantee of a clearing house or an exchange. If one party defaults, they owe to the other party at most the present value of net swap payments that they are obligated to make at current market prices. Asset Swap An asset swap is an interest rate swap used to convert the cash ﬂows from and underlying security (such a bond) from ﬁxed coupon to ﬂoating coupon. Thus, an asset swap converts a ﬁxed-rate bond 74 INTEREST RATE SWAPS 631 to a synthetic ﬂoating-rate bond. The asset swap transaction is illustrated in Figure 74.2. Figure 74.2 Asset swaps are commonly used by fund managers who own ﬁxed-rate bonds and wish to have ﬂoating-rate exposure while continuing to own the bonds. Computing the Swap Rate Consider an interest rate swap with nsettlements occurring on dates ti,i = 1,2,··· ,n. Let P(0,ti) be the price of a $1 face value zero-coupon bond maturing on date ti. Notice that P(0,ti) is the discount factor from time zero to time ti based on the ti−year spot rate. That is, if it is the t−year spot rate then P(0,t) = (1 + it)−t. Let r0(ti−1,ti) be the implied forward rate between the times ti−1 and ti. From Section 50, we have r0(ti−1,ti) = P(0,ti−1) P(0,ti) −1. An interest rate swap acts much like the oil price swap described in the previous section. When initiated, the swap has a value of zero, i.e., the ﬁxed rate payments and the variable rate payments have equal present values. In terms of a formula, we have: R n∑ i=1 P(0,ti) = n∑ i=1 P(0,ti)r0(ti−1,ti) where R denote the ﬁxed swap rate. Solving this last equation for R we obtain R= ∑n i=1 P(0,ti)r0(ti−1,ti)∑n i=1 P(0,ti) 632 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES where ∑n i=1 P(0,ti)r0(ti−1,ti) is the present value of the variable payments and ∑n i=1 P(0,ti) is the present value of a $1 annuity when interest rates vary over time. Note that the swap rate can be interpreted as a weighted average of the implied forward rates, where zero-coupon bond prices are used to determine the weights. Now, since r0(ti−1,ti) = P(0,ti−1) P(0,ti) −1 we can write n∑ i=1 P(0,ti)r0(ti−1,ti) = n∑ i=1 [P(0,ti−1) −P(0,ti) = 1 −P(0,tn) so that R= 1 −P(0,tn)∑n i=1 P(0,ti) This formula can be rearranged to obtain P(0,tn) + R n∑ i=1 P(0,ti) = 1. This formula states that a payment of 1 that will be received n years from now, plus a payment of R at the end of each of the next n years, has a total present value of 1. This is the valuation equation for a bond priced at par (i.e. face value $1) with a coupon rate R. Thus, the swap rate is the coupon rate for a bond that is selling at par. Example 74.2 The following table lists prices of zerocoupon $1face value bonds with their respective maturities: Years to Maturity Price 1 $0.956938 2 $0.907029 3 $0.863838 (a) Calculate the 1-, 2-, and 3-year spot rates of interest. (b) Calculate the 1- and 2-year forward rates of interest. (c) Calculate the coupon rate Rfor a 3-year bond with annual coupons whose face value, redemption value and price are all one. Solution. (a) Let i1 be the spot rate of interest. Then 1 + i1 = P(0,1)−1 = 0.956938−1 →i1 = 0.956938−1 − 74 INTEREST RATE SWAPS 633 1 = 4 .499967135%. Likewise, i2 = 0 .907029 −1 2 −1 = 5 .000027694% and i3 = 0 .863838 −1 3 −1 = 4.99991613% (b) The 1-year forward rate is given by r0(1,2) = P(0,1) P(0,2) −1 = 0.956938 0.907029 −1 = 5.502470153%. Likewise, the 2-year forward rate is given by r0(2,3) = P(0,2) P(0,3) −1 = 0.907029 0.863838 −1 = 4.999895814%. (c) The coupon rate is given by R= 1 −P(0,3) P(0,1) + P(0,2) + P(0,3) = 1 −0.863838 0.956938 + 0.907029 + 0.863838 = 4.991632466% Deferred Swaps By a k-deferred swap we mean a swap whose exchange of payments start k periods from now. The (deferred) swap rate is applied in the future but it is agreed upon today. It is computed as follows R= ∑n j=kP(0,tj)r0(tj−1,tj)∑n j=kP(0,tj) = ∑n j=kP(0,tj) ( P(0,tj−1) P(0,tj) −1 ) ∑n j=kP(0,tj) = ∑n j=k(P(0,tj−1) −P(0,tj)∑n j=kP(0,tj) =P(0,tk−1) −P(0,tn)∑n j=kP(0,tj) Example 74.3 Given the following zero-coupon bond prices: Quarter 1 2 3 4 5 6 7 8 P(0,t) 0.9852 0.9701 0.9546 0.9388 0.9231 0.9075 0.8919 0.8763 What is the ﬁxed-rate (deferred swap rate) in a 6-quarter interest rate swap with the ﬁrst settlement in quarter 3? 634 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Solution. We have R=P(0,2) −P(0,8)∑8 j=3 P(0,j) = 0.9701 −0.8763 0.9546 + 0.9388 + 0.9231 + 0.9075 + 0.8919 + 0.8763 =1.71% The Value of a Swap At initiation of the swap, the value of the swap is set to be zero to both parties. The ﬁxed rate of the swap is then calculated such that the present value of the ﬁxed payments equals that of the ﬂoating payments. As time evolves, the interest rate may move upward or downward so that the value of the ﬂoating payments changes. Therefore, the value of the swap changes in later times, and its value at each time is given by the diﬀerence in the present values of the remaining cash ﬂows from the two parties. Even in the absence of changes in interest rates, as soon as the ﬁrst swap payment occurs, the swap will have a value. The swap will have a positive value for the party who paid too much (e.g., the ﬁxed rate payer if the ﬁxed rate exceeds the variable rate on the ﬁrst payment date), and a negative value for the party who received the net payment. As with the commodity swap, an interest rate swap is equivalent to entering into a series of forward contracts and also undertaking some borrowing and lending. 74 INTEREST RATE SWAPS 635 Practice Problems Problem 74.1 Interest rate swaps allow one party to exchange a: (A) ﬂoating interest for a ﬁxed rate over the contract term. (B) ﬁxed interest rate for a lower ﬁxed rate over the contract term. (C) ﬂoating interest rate for a lower ﬂoating rate over the contract term. Problem 74.2 All of the following are types of ﬁnancial instruments except for: (A) currency forward contracts (B) currency futures contracts (C) currency options (D) swap agreements (E) money-market hedge Problem 74.3 On March 5,2004, an agreement by Microsoft to receive 6-month LIBOR and pay a ﬁxed rate of 5% per annum every 6 months for 3 years on a notional principal of $100 million. Complete the following table: Date 6-month Fixed Floating Net LIBOR rate Payment Payment Cash Flow March 5,2004 4.2% Sept 5,2004 4.8% March 5, 2005 5.3% Sept 5, 2005 5.5% March 5, 2006 5.6% Sept 5, 2006 5.9% March 5,2007 6.4% Problem 74.4 Firm ABC is paying $750,000 in interest payments a year while Firm XYZ is paying LIBOR plus 75 basis points (i.e. 0.75%) on $10,000,000 loans. The current LIBOR rate is 6.5%. Firm ABC and XYZ have agreed to swap interest payments, how much will paid to which Firm this year? Problem 74.5 Given the following information 636 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES t (in months) P(0,t) 3 0.986923 6 0.973921 9 0.961067 12 0.948242 Calculate the annual nominal interest rate compounded quarterly for a loan with the following maturity dates: 3, 6, 9 and 12 months. Problem 74.6 Given the following information t (in years) P(0,t) 1 0.95785 2 0.91049 3 0.85404 Calculate the 3-year interest rate swap. Problem 74.7 The ﬁxed rate for a 2-year interest rate swap is 6.06%. The 2-year spot rate is 6.10%. What is the price of a 1-year zero-coupon bond with a maturity value of 1,000? Problem 74.8 Given the following zero-coupon bond prices: Quarter 1 2 3 4 5 6 7 8 P(0,t) 0.9852 0.9701 0.9546 0.9388 0.9231 0.9075 0.8919 0.8763 What is the ﬁxed-rate (deferred swap rate) in a 5-quarter interest rate swap with the ﬁrst settlement in quarter 2? Problem 74.9 Given the following zero-coupon bond prices: Quarter 1 2 3 4 5 6 7 8 P(0,t) 0.9852 0.9701 0.9546 0.9388 0.9231 0.9075 0.8919 0.8763 What is the swap rate in an 8-quarter interest rate swap? Problem 74.10 The following table lists prices of zerocoupon $1face value bonds with their respective maturities: 74 INTEREST RATE SWAPS 637 Years to Maturity Price 1 $0.95238 2 $0.89845 3 $0.842.00 Complete the following table: Years to Maturity Price Spot Rate in r0(n−1,n) 1 $0.95238 2 $0.89845 3 $0.842.00 Problem 74.11 Using the data in the previous table, ﬁnd the ﬁxed rate (i.e. the swap rate R). 638 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES 75 Risk Management Risk management consists of identifying the sources of risk, choosing the ones to be hedged, and choosing the way of hedging. A ﬁrm’s risk management consists of the use of ﬁnancial derivatives such as forwards, calls, and puts to alter its exposure to risk and protect its proﬁtability. We will next discuss risk management from both the producer’s perspective as well as the buyer’s perspective. The Producer’s Perspective A producer normally has a long position that consists of its product. In order to protect its future proﬁts from price ﬂuctuations, the producer can oﬀset his long position by using forward contracts, puts options, call options, or some combination of these derivatives. This process is also referred to as hedging or insuring the long position. Some of the risk management strategies for a producer include the following: (1) Hedging with a forward contract: In order for the producer to lock in the price of his product, the producer can enter into a short forward contract that ﬁxes the future sale price at the current forward price. There is no initial cost for the producer. For example, ABC Inc, an oil producing ﬁrm, enters into a short forward contract, agreeing to sell oil at today’s current price of $65 a barrel in one year. (2) Hedging with a put option: Creating a ﬂoor by purchasing a put option which limits the losses if the price declines, but allows unlimited proﬁt if the price rises. Note that the option’s premium is a cost to the producer for creating the ﬂoor. For example, ABC Inc purchases 65-strike put at a premium of $3.50 per barrel. In this case, the put option plays the role of an insurance that guarantees a minimum price of $65 for a barrel of oil. (3) Insuring by selling a call: A written call (selling a cap) sets a maximum price (and therefore limit proﬁt) for the product if the price rises, but does not limit losses if the price declines. The premium received by the producer helps reduce the losses. For example, ABC Inc sells a 65-strike call for a premium of $3.50 per barrel. (4) Creating a collar by purchasing a put at one strike price and writing a call at a higher strike price. A collar sets maximum and minimum prices (the call strike and the put strike, re- spectively) that the producer may realize for its product. The producer is exposed to the risk of variation between these two prices, but is not aﬀected by price variation above or below this range. The put and call options constitute the collar, but in combination with the producer’s long position in the product, they form a bull spread. 75 RISK MANAGEMENT 639 Example 75.1 ABC is a coppermining company with ﬁxed costs $0.75/lb and variable costs of $2.25/lb. (a) If ABC does nothing to hedge price risk, what is its proﬁt 1 year from now, per pound of copper? (b) Suppose that ABC can enter a short forward contract agreeing to sell its copper production one year from now. The 1-year forward price is $3.5/lb. What is the estimated proﬁt one year from now? (c) Suppose that ABC would like to beneﬁt if the price of the copper goes higher than $3.5/lb. It could use options. One year option prices for copper are given below Strike 3.3 3.4 3.5 3.6 3.7 3.8 3.9 Call 0.42567 0.3762 0.33119 0.29048 0.25386 0.22111 0.19196 Put 0.23542 0.28107 0.33119 0.3856 0.44411 0.50648 0.57245 The 1-year continuously compounded interest rate is 5%. Compute the estimated proﬁt in 1 year if ABC buys put options with a strike of $3.7/lb and with a strike of $3.4/lb. (d) Compute the estimated proﬁt in 1 year if ABC sells call options with a strike of $3.7/lb and with a strike of $3.4/lb. (e) Compute the proﬁt in 1 year if ABC buys a 3.6-strike put and sells a 3.9-strike call. Solution. (a) The proﬁt is the one-year spot price minus total cost. That is, P1 −3.A graph of the unhedged position is shown in Figure 75.1. (b) In this case, the proﬁt is P = P1 −3 + (3.5 −P1) = $0 .5. A graph of the hedged position is shown in Figure 75.1. For an uninsured position, the possible losses can be very high. However, by entering the forward, the company has a ﬁxed beneﬁt. Figure 75.1 (c) The proﬁt per pound that results from buying the 3.7-strike put is P1 −3 + max{3.7 −P1,0}−0.44411e0.05. 640 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES The graph of the proﬁt is given in Figure 75.2. Figure 75.2 The proﬁt per pound that results from buying the 3.4-strike put is P1 −3 + max{3.4 −P1,0}−0.28107e0.05. The graph of the proﬁt is given in Figure 75.2. With the 3.4-strike put, the company makes $0.17139924/lb more than the 3.7-strike put if the price of copper is over $3.7/lb. However, its guaranteed proﬁt is $0.104519233, which is smaller than the guaranteed proﬁt under a 3.7-strike put. Buying a put is like buying insurance against small spot prices. The larger the strike price the larger the price of the insurance. A producer company needs to buy a put with a strike large enough to cover low spot prices. If the strike put is too large, the company will be wasting money in insurance which it does not need. (d) The proﬁt per pound that results from selling the 3.7-strike call is P1 −3 −max{P1 −3.7,0}+ 0.25386e0.05. The graph of the proﬁt is given in Figure 75.3.

Human: Which of the following is most equivalent to writing a straddle?

Actuarial AI: 580 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Figure 68.6 As with the symmetric butterﬂy spread, an investor can create an asymmetric butterﬂy spread by using only calls, using only puts, or by using a long or short position in the stock, combined with both puts and calls. See Problem 68.12. 68 VOLATILITY SPECULATION: STRADDLES, STRANGLES, AND BUTTERFLY SPREADS581 Practice Problems Problem 68.1 (A) buy stock, write two calls (B) buy stock, buy one put (C) short stock, buy one call (D) short stock, buy one put. Problem 68.2 A butterﬂy spread of an index is created as follows: (1) buying a 35-strike call with premium $6.13 and selling a 40-strike call with premium $2.78 (that is buying a bull call spread). (2) selling a 40-strike call with premium $2.78 and buyin a 45-strike call with premium $0.97 ( that is buying a bear call spread). All options expire in three months from now. The current price of the index is $40. The risk free annual eﬀective rate of interest is 8.33%. Show that the proﬁt diagram of this position coincides with the one in Example 68.3. Problem 68.3 A butterﬂy spread of an index is created by buying a 35-strike call (with premium $6.13), selling two 40-strike calls (with premium $2.78 each) and buying a 45-strike call (with premium $0.97). All options expire in three months from now. The current price of the index is $40. The risk free annual eﬀective rate of interest is 8.33%. Show that the proﬁt diagram of this position coincides with the one in Example 68.3. Problem 68.4 A butterﬂy spread of an index is created by buying a 35-strike put (with premium $0.44), selling two 40-strike puts (with premium $1.99 each) and buying a 45-strike put (with premium $5.08). All options expire in three months from now. The current price of the index is $40. The risk free annual eﬀective rate of interest is 8.33%. Show that the proﬁt diagram of this position coincides with the one in Example 68.3. Problem 68.5 A butterﬂy spread of an index is created by buying the stock, buying a 35-strike put (with premium $0.44), selling two 40-strike calls (with premium $2.78 each) and buying a 45-strike call (with premium $0.97). All options expire in three months from now. The current price of the index is $40. The risk free annual eﬀective rate of interest is 8.33%. Show that the proﬁt diagram of this position coincides with the one in Example 68.3. 582 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Problem 68.6 ‡ You believe that the volatility of a stock is higher than indicated by market prices for options on that stock. You want to speculate on that belief by buying or selling at-the-money options. What should you do? (A) Buy a strangle (B) Buy a straddle (C) Sell a straddle (D) Buy a butterﬂy spread (E) Sell a butterﬂy spread Problem 68.7 ‡ You are given the following information: •The current price to buy one share of ABC stock is 100 •The stock does not pay dividends •The risk-free rate, compounded continuously, is 5% •European options on one share of ABC stock expiring in one year have the following prices: Strike Price Call Premium Put Premium 90 14.63 0.24 100 6.80 1.93 110 2.17 6.81 A butterﬂy spread on this stock has the following proﬁt diagram. Which of the following will NOT produce this proﬁt diagram? (A) Buy a 90 put, buy a 110 put, sell two 100 puts (B) Buy a 90 call, buy a 110 call, sell two 100 calls (C) Buy a 90 put, sell a 100 put, sell a 100 call, buy a 110 call (D) Buy one share of the stock, buy a 90 call, buy a 110 put, sell two 100 puts (E) Buy one share of the stock, buy a 90 put, buy a 110 call, sell two 100 calls. 68 VOLATILITY SPECULATION: STRADDLES, STRANGLES, AND BUTTERFLY SPREADS583 Problem 68.8 Which of the following has the potential to lose the most money? (A) a long butterﬂy spread (B) a short strangle (C) a long strangle (D) an asymetric butterﬂy spread Problem 68.9 Three month European put options with strike prices of $50, $55, and $60 cost $2, $4, and $7, respectively. A butterﬂy is created by buying a 50-strike put, selling two 55-strike put, and buying a 60-strike put. The risk-free 3-month interest rate is 5%. (a) What is the maximum gain when a butterﬂy spread is created from the put options? (b) What is the maximum loss when a butterﬂy spread is created from the put options? (c) For what two values of PT does the holder of the butterﬂy spread breakeven, where PT is the stock price in three months? Problem 68.10 Consider a strangle with the strike price of the put is higher than the strike price of the call. Draw the proﬁt diagram. Problem 68.11 An asymmetric butterﬂy spread has the following payoﬀ diagram: This position was created using calls that are priced as follows: Strike Price Premium 80 4 84 2 90 0.50 Ignoring commissions and bid-ask spreads, what was the cost to establish this asymmetric butterﬂy spread? 584 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Problem 68.12 Show that the asymmetric butterﬂy of Figure 68.6 can be duplicated by buying two 35-strike puts (with premium $0.44 each), selling ten 43-strike puts (with premium $3.674 each ) and buying eight 45-strike puts (with premium $5.08 each). All options expire 3 months from now. The risk free annual eﬀective rate of interest is 8.33%. 69 EQUITY LINKED CDS 585 69 Equity Linked CDs An equity-linked CD (ELCD) (also known as index-linked CD) is an FDIC-insured certiﬁcate of deposit that ties the rate of return to the performance of the market indices such as the S&P 500 Composite Stock Price Index, the Dow Jones Industrial Average, or the NASDAQ 100 composite index. Equity-linked CDs are usually issued by a bank. The CDs may vary by maturity (ﬁve years, six years, etc.), participation rate (some oﬀer 100% of the increase in the market index, others only 90% or 70%), and the method of computing the return of the stock index (some use an averaging method to determine the ending value of the index level, while others use the ﬁnal index value). For example, an equity-linked CD can have the following structure: A CD with a return linked to the S&P 500 index is guaranteed to repay the invested amount plus 70% of the simple appreciation in the S&P 500 at the maturity date of 5.5 years. If the invested amount is $10,000 and the index is 1300 initially then a decline in the index after 5.5 years will result in a return of the $10,000 to the investor. If on the other hand the index rises to 2200 at the maturity date then the return to the investor is the original investment plus 70% of the precentage gain on the index. That is, 10,000 [ 1 + 0.70 (2200 1300 −1 )] = $14,846. More generally, the payoﬀ is given by the formula 10,000 [ 1 + 0.70 max { 0, S5.5 1300 −1 }] where S5.5 is the value of the index after 5.5 years. Example 69.1 A 5.5-year ELCD promises to repay $10,000 and 70% of the gain in S&P 500 index at expiration date. Assume investment occurs when S&P 500 index was 1300. (a) Complete the following table Index after 5.5 years Payoﬀ 500 1000 1500 2000 2500 3000 586 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES (b) Draw the payoﬀ diagram. Solution. (a) Index after 5.5 years Payoﬀ 500 10,000 1000 10,000 1500 11,076.92 2000 13,769.23 2500 16,461.54 3000 19,153.85 (b) The payoﬀ diagram is shown in Figure 69.1 Figure 69.1 Building an ELCD Using a Zero-Coupon Bond and a Call Option We next use ﬁnancial engineering to create the building structure of an ELCD by buying a zero- coupon bond and a call option. Consider an investor who wants to invest $1,000 in a S&P 500 index with no risk of losing principal and the opportunity to earn market-like returns. This investor might purchase a zero-coupon bond and an at-the-money call option on the S&P 500 market index. The investor buys the zero-coupon 69 EQUITY LINKED CDS 587 bond at a discount and receives the $1,000 bond par value at maturity. She then uses the diﬀerence between the $1,000 investment and the price of the zero-coupon bond to pay for the call option premium. The call option has expiration date as the time to maturity of the bond. If, at maturity, the S&P 500 has increased, the investor exercises the call option and earns a return on the capital gains portion of the S&P 500. If the S&P 500 decreases, the investor does not exercise the option and receives nothing for it. However, the investor still receives the face value of the bond thereby insuring the original $1,000 investment. A setback of the ELCD in the above example is that in the case of a decline in the index the investor will get back his $1,000 at maturity date. If, say, the maturity date is 5 years and if the annual eﬀective interest rate during the investment time is 6%, the investor has lost in interest 1,000(1.06)5 −1,000 = $338.22. The present value of this interest is 338.22(1.06)−5 = $252.74.Thus as the investor invests today he has foregone interest with a present value of $252.74, and that is the implied cost today of investing in the CD. Summarizing, an investor is attracted to ELCD’s because it has the potential for market appreciation without risking capital. One disadvantage is the possible loss of interest on the invested principal. In general, the payoﬀ of an ELCD with invested principal P and rate of participation r is given by the formula P ( 1 + rmax{PT P0 −1,0} ) = P + Pr max{PT P0 −1,0}. That is, the payoﬀ of a typical equity-linked CD at expiration is equal to the par amount of the CD plus an equity-linked coupon. In general, the equity-linked coupon is equal to either: (a) zero, if the underlying equity has depreciated from an agreed upon strike level (usually the index level on the issue date of the CD), or (b) the participation rate times the percentage change in the underlying equity times the par amount of the note, if the underlying equity appreciated. Example 69.2 An investor buys a ﬁve-year ELCD with 100% participation in the simple appreciation of the S&P 500 index for $1,000. The starting index-level is 1,400. (a) Find the payoﬀ if after ﬁve years the index-level is 2,100. (b) Find the payoﬀ if after ﬁve years the index-level is below 1,400. (c) Create the payoﬀ diagram. Solution. (a) The payoﬀ is 1,000 [ 1 + (2100 1400 −1 ) ×100% ] = $1,500. 588 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES (b) The total payoﬀ is just the original investment of $,1000. (c) The proﬁt diagram is shown in Figure 69.2 Figure 69.2 Example 69.3 Create a synthetic equivalent to the ELCD in Example 69.2. Solution. A synthetic equivalent can be created as follows: (1) invest in a ﬁve-year 6% discount bond with par value of $1,000 for 1,000(1.06)−5 = $747.26 and (2) buy a ﬁve-year, S&P 500 at-the-money call option with a $1,400 strike price for a premium $252.74 We conclude this section by looking at how the three perspectives the end-user, the market-maker, and the economic observer view an ELCD: •The end-user is interested in the product and whether it meets a ﬁnancial need at a fair cost. •The market-maker (i.e. the bank issuer) is interested in making proﬁt without bearing risk from having issued the CD. Some of the proﬁt come from fees for issuing the CD and from penalities of early withdrawals since ELCDs cannot be withdrawn before maturity date. •The economic observer studies the advantage of having ELCDs. 69 EQUITY LINKED CDS 589 Practice Problems Problem 69.1 The payoﬀ in Example 69.1 can be regarded as the payoﬀ of $10,000 and the payoﬀ of a certain unit of European call options. What is that number of units? Problem 69.2 Suppose that an ELCD promises to repay the principal investment plus 80% of the gain of a linked equity. If the gain on the equity is 50% during the investment period then what is the gain for each dollar invested? Problem 69.3 In a ELCD, an investor will receive a return consisting of his orginal investment only when (a) the underlying index has increased from the starting index value (b) the underlying index has remained static or have fallen. Problem 69.4 An investor has an equity linked CD with a nominal value of $10,000 and a participation of %110. If the positive movement of the equity markets at maturity is 35% then ﬁnd the amount due at maturity to the investor. Problem 69.5 Which of the following statements are true about equity linked CDs: (I) pays a ﬁxed interest rate (II) return depends on the performance of the underlying assets (III) Miniumum return is the initially invested money (IV) can be viewed as a combination of a zero-coupon bond and a call option Problem 69.6 A 5-year CD promises to repay $1,000,000 and 115% of the gain in S&P 500 index at expiration date. Assume investment occurs when S&P 500 index was 500. (a) Complete the following table Index after 5 years Payoﬀ 400 450 500 550 600 700 800 590 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES (b) Draw the payoﬀ diagram. Problem 69.7 What is the minimum value of a call option? An equity-linked CD? Problem 69.8 A 5-year CD promises to repay $1,000,000 and 115% of the gain in S&P 500 index at expiration date. Assume investment occurs when S&P 500 index was 500. Assume annual eﬀective interest rate of 8% during the 5-year period. What is the implied cost today of investing in the equity-linked index ? Problem 69.9 A 5-year CD promises to repay $1,000,000 and 115% of the gain in S&P 500 index at expiration date. Assume investment occurs when S&P 500 index was 500. Create a synthetic ELCD for this investment. 70 PREPAID FORWARD CONTRACTS ON STOCK 591 70 Prepaid Forward Contracts On Stock There are four alternative ways for buying shares of a stock. Namely, •Outright purchase: An investor simultaneously pays S0 dollars in cash at time 0 and owns the stock. •Full leveraged purchase: An investor borrows S0 dollars and receives the ownership of the stock at time 0. Letting r denote the annual continuous compound interest, the investor has to repay S0ert at time T. •Forward contract: An arrangement for an agreed-upon future price (the forward price) and date in which the buyer pays for the stock at time T and the seller transfers the ownership of the stock to the buyer at that time. •Prepaid forward contract or prepay: This is like a forward contract but with the (prepaid) forward price paid at time 0. That is, the contract entails the buyer to pay today for the stock and owns it in the future. The price one pays is not necessarily the stock price S0. In this section, we will discuss the question of pricing a prepaid forward contract. Pricing of Prepaid Forward Contracts with no Dividends Let FP 0,T denote the prepaid forward price for an asset bought at time 0 and delivered at time T. This price depends on the dividends paid to the stockholders. In the absence of dividends, the timing of delivery is irrelevant. Therefore, the prepaid forward price is just the stock price at time 0. That is, FP 0,T = S0. Example 70.1 The current price of a stock is $50. The free-risk interest rate is 4% compounded continuously. Assuming no dividends, what is the prepaid forward price? Solution. The price of a prepaid forward contract on a stock that pays no dividends is the initial stock price. So the answer is $50 Now, arbitrage opportunities (positive cash ﬂows either today or in the future and with no risk) can be achieved with prepaid forward price as illustrated in the following two examples. Example 70.2 Suppose that the prepaid forward price exceeds the stock price. An arbitrageur will buy the stock at the low price of S0 and sell the prepaid forward contract for FP 0,T. Assuming no dividends, show that this transaction has a positive cash ﬂow at time 0 and is risk-free. 592 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Solution. The net cash ﬂow at time 0 is FP 0,T −S0 > 0 since the cash ﬂow (outﬂow) from buying the stock is −S0 and from selling the prepaid contract (inﬂow) is FP 0,T. This transaction is risk-free: Selling the prepaid forward contract obliges the seller to deliver the stock at time T and buying the stock today ensures that the stock will be delivered at that time Example 70.3 Suppose that the stock price exceeds the prepaid forward. An arbitrageur will short the stock at the price of S0 and buy the prepaid forward contract for FP 0,T. Assuming no dividends, show that this transaction has a positive cash ﬂow at time 0 and is risk-free. Solution. The net cash ﬂow at time 0 is S0 −FP 0,T >0 since the cash ﬂow (outﬂow) from buying the prepaid forward is −F0,T and from selling the stock (inﬂow) is S0. This transaction is risk-free: By buying the prepaid forward contract we are entitled to acquire the stock at time T which we use to close the short position Throughout the book all prices are assumed to be free arbitrage prices. Pricing of Prepaid Forward Contracts with Dividends Stocks that pay dividends have prepaid forward price lower than the stock price. Only the owner of the stock receives the dividends but not the holder of the prepaid forward. When dividend is paid the stock price is reduced by the amount of the dividend. Suppose that over the life of a forward contract, a stock receives dividends Di at time ti where i= 1,2,··· ,n. Then the prepaid forward price is lower than the stock price by the present value of those dividends: FP 0,T = S0 − n∑ i=1 PV (Di). (70.1) Example 70.4 Suppose ABC stock costs $75 today and is expected to pay semi-annual dividend of $1 with the ﬁrst coming in 4 months from today and the last just prior to the delivery of the stock. Suppose that the annual continuously compounded risk-free rate is 8%. Find the cost of a 1-year prepaid forward contract. Solution. The cost is FP 0,1 = 75 −e−0.08×4 12 −e−0.08×10 12 = $73.09 70 PREPAID FORWARD CONTRACTS ON STOCK 593 Formula (70.1) applies for the case of discrete dividends. What about a stock index where dividends are given almost daily? In this case, it is common to model the dividends as being continuously paid at a constant continuoulsy compounded dividend yield rate δ(which is deﬁned to be the annualized dividend payment divided by the stock price). Note that δis used here as a continuously compounded dividend rate, rather than a force of interest. Like an interest rate, the dividend rate is a rate per year. Divide the interval [0,T] into n equal subintervals using the mesh point ti = i nT where 1 ≤i≤n. The dollar dividend over each interval is Dividend = δT n S0. Suppose that dividends payments are reinvested into stock. Let Ai be the total number of shares at time ti. Then Ai+1 = Ai ( 1 + δT n ) . Hence, the total amount of shares multiplies by ( 1 + δT n ) so that one share at time 0 grows to ( 1 + δT n )n at time T. Letting n →∞ we get that one share at time 0 grows to eδT shares at time T. Example 70.5 You buy one share of ABC stock and hold it for two years. The dividends are paid at continuously compounded rate of 3% and you reinvest all the dividends when they are paid. The current stock price is $125. (a) Find the daily dollar dividend. (b) How many shares do you have at the end of two years? Solution. (a) The daily dollar dividend is Dividend= 0.03 365 ×125 = $0.01027. (b) At the end of two years, the number of shares increased from one share to e0.03×2 = 1.061836 shares Now, if one share of the index is desired at time T, then one has to buy e−δT shares today. Due to dividend reinvestment, at time T we will have e−δTeδT = 1 share. This process of buying just enough shares today so that when dividends are reinvested we accumulate exactly one share at time T is called tailing the position. The cost per share at time 0 of a tailed position is e−δTS0. This position can be achieved as well by buying a T−year prepaid forward contract on a stock (or 594 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES index) that pays dividends at a continuous rate of δ for delivery of one full share of the index at time T. If there exists no arbitrage, the prepaid forward price must be equal to the cost of the tailing position at time 0. That is, FP 0,T = S0e−δT. Remark 70.1 In either the case of a tailed position or a prepaid forward contratct, the buyer of the share does not receive (in cash) any dividends paid prior to time T. In the case of the tailed position, those dividends are used to buy additional stock to bring the total to one full share; in the case of the prepaid forward contract, dividends are not payable to the holder of a long forward prior to the expiration (delivery) date, time T (at which time the investor receives one full share). Example 70.6 Find the 1-year prepaid forward price of the index in the previous example. Solution. The prepaid forward price is FP 0,1 = 125e−0.03 = $121.306 70 PREPAID FORWARD CONTRACTS ON STOCK 595 Practice Problems Problem 70.1 Complete the following charts that descibe the cash ﬂows and transactions to undertake arbitrage (stocks pay no dividends). F0,T >S0 Cash Flows t= 0 t= T Buy Stock Sell Prepaid Forward Total F0,T <S0 Cash Flows t= 0 t= T Short Stock Buy Prepaid Forward Total Problem 70.2 Suppose that the current price of a stock is $1100. The stock does not pay dividends and the risk-free annual rate is 5%. (a) Suppose that an investor is willing to buy a prepaid forward contract at a prepaid forward price of $1135 on a one year prepaid forward contract. Show how to make an arbitrage gain under these circumstances. (b) Suppose that an investor is willing to sell a prepaid forward contract at a prepaid forward price of $1050 on a one year prepaid forward contract. Show how to make an arbitrage gain under these circumstances. Problem 70.3 Suppose ABC stock costs $X today. It is expected that 4 quarterly dividends of $1.25 each will be paid on the stock with the ﬁrst coming 3 months from now. The 4th dividend will be paid one day before expiration of the forward contract. Suppose the annual continuously compounded risk-free rate is 10%. Find X if the cost of the forward contract is $95.30. Round your answer to the nearest dollar. Problem 70.4 Suppose ABC stock costs $75 today and is expected to pay semi-annual dividend of $X with the 596 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES ﬁrst coming in 4 months from today and the last just prior to the delivery of the stock. Suppose that the annual continuously compounded risk-free rate is 8%. Find X if the cost of a 1-year prepaid forward contract is $73.09. Round your answer to the nearest dollar. Problem 70.5 You buy one share of Ford stock and hold it for 2 years. The dividends are paid at the annualized daily continuously compounded rate of 3.98% and you reinvest all the dividends when they are paid. How many shares do you have at the end of two years? Problem 70.6 Suppose that annual dividend of 30 on the stocks of an index is valued at $1500. What is the continuously compounded dividend yield? Problem 70.7 Suppose XYZ stock costs $50 today and is expected to pay quarterly divident of $1 with the ﬁrst coming in 3 months from today and the last just prior to the delivery of the stock. Suppose that the annual continuously compounded risk-free rate is 6%. What is the price of a prepaid forward contract that expires 1 year from today, immediately after the fourth-quarter dividend? Problem 70.8 Suppose XYZ stock costs $50 today and is expected to pay 8% continuous risk-free dividend. What is the price of a prepaid forward contract that expires 1 year from today, immediately after the fourth-quarter dividend? Problem 70.9 LEPOs (Low Exercised Price Options) exist in order to avoid taxes and transaction fees associated with trading with stocks. These stocks have zero-dividends and are purchased outright. These options are certain to be exercised due to the fact that the strike price K is so low. The payoﬀ at expiration time T is PT −K.Find the cost of this option at time 0. Show that a LEPO is essentially a prepaid forward contract with no dividends. Problem 70.10 An investor is interested in buying XYZ stock. The current price of stock is $45 per share. This stock pays dividends at an annual continuous risk-free rate of 5%. Calculate the price of a prepaid forward contract which expires in 18 months. Problem 70.11 ‡ A non-dividend paying stock currently sells for 100. One year from now the stock sells for 110. The risk-free rate, compounded continuously, is 6%. The stock is purchased in the following manner: 70 PREPAID FORWARD CONTRACTS ON STOCK 597 •You pay 100 today •You take possession of the security in one year Which of the following describes this arrangement? (A) Outright purchase (B) Fully leveraged purchase (C) Prepaid forward contract (D) Forward contract (E) This arrangement is not possible due to arbitrage opportunities 598 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES 71 Forward Contracts on Stock Recall that prepaid forward contracts are forward contracts with payment made at time 0. The forward price is the future value of the prepaid forward price and this is true regardless of whether there are discrete dividends, continuous dividends, or no dividends If the expiration time is T and the continuous risk-free rate is r then for a stock with no dividends (i.e. FP 0,T = S0) the future value of the prepaid forward price is given by the formula F0,T = S0erT. In the case of a stock with discrete dividends D1,D2,··· ,Dn made at time t1,t2,··· ,tn the forward price is F0,T =FV (FP 0,T) = ( S0 − n∑ i=1 Die−rti ) erT =S0erT − n∑ i=1 Dier(T−ti) In the case of a stock with continuous dividends with continuously compounded yield δ we have F0,T = (S0e−δT)erT = S0e(r−δ)T. The forward premium is the ratio of the forward price to the current spot price deﬁned by Forward premium= F0,T S0 . The expression 1 −F0,T S0 is the percentage by which the forward price exceeds (or falls below) the spot price. A number α that satisﬁes the equation F0,T = S0eαT is called the annualized forward premium. Solving this equation for α we ﬁnd α= 1 T ln (F0,T S0 ) . 71 FORWARD CONTRACTS ON STOCK 599 If a stock pays no dividends, the annualized forward premium is just the annual risk-free interest rate α= r. For the case of continuous dividends, the annualized forward premium reduces to α= 1 T ln (S0e(r−δ)T S0 ) = r−δ. Remark 71.1 In the literature, the forward premium of an exchange rate forward contract is deﬁned to be the percentage by which today’s forward rate exceeds (falls below) today’s spot rate and is given by the formula ln (F0,T S0 ) . Usually these premia are quoted as annualized rates given by the formula 1 T ln (F0,T S0 ) . Example 71.1 Suppose the stock price is $90 and the continuously compounded interest rate is 10%. (a) What is the 9-month forward price assuming dividends are zero? What is the forward premium? (b) If the 9-month forward price is $92, what is the annualized forward premium? (c) If the 9-month forward price is $92, what is the annualized continuous dividend yield? Solution. (a) F0, 9 12 = 90e0.1×9 12 = $97.01. The forward premium is 97.01 90 = 1.07789. (b) The annualized forward premium is α= 1 T ln (F0,9/12 S0 ) = 12 9 ln (92 90 ) = 0.0293. (c) We have 0.0293 = 0.10 −δ.Solving for δ we ﬁnd δ= 0.0707 = 7.07% In Section 65, we have seen how to create a synthetic forward contract by buying a call and selling a put with both having the same strike price. We next discuss another way for creating a synthetic forward. In this discussion we consider a stock that pays continuous dividends at the rate δ.We buy a tailed position by buying e−δT shares of the stock which cost S0e−δT at time 0 so that at time T we have one share of the stock. We borrow S0e−δT to pay for the shares of stock. In this case, we invest 0 at time 0. At time T we must repay S0e(r−δ)T and we sell the one share of stock at price PT. The payoﬀ for this position is 600 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Payoﬀ at expiration= PT −S0e(r−δ)T = PT −F0,T which is the payoﬀ of a long forward contract with forward price F0,T and expiration time T. In terms of a word equation we have Forward = Stock−Zero coupon bond (71.1) By rearranging the terms of this equation we can derive a synthetic stock Stock = Forward + Zero-coupon bond This position is created by buying a forward contract with forward price F0,T = S0e(r−δ)T and expiration T and lending S0e−δT ( the present value of the forward price) at time 0. The payoﬀ at time T is Payoﬀ at expiration= (PT −F0,T) + F0,T = PT which is the payoﬀ of buying e−δT shares of the stock. We can use (71.1) to create a synthetic bond Zero-coupon bond = Stock −Forward. This position is created by buying e−δT shares of the index at the cost of S0e−δT at time 0 and short one forward contract with forward price F0,T and expiration time T. The payoﬀ of this position at time T is Payoﬀ at expiration= PT + (F0,T −PT) = F0,T which is the payoﬀ of a long risk-free bond with par value of F0,T and expiration time T. Example 71.2 Consider the position of selling e−δT shares for S0e−δT at time 0 and buying a forward with forward price F0,T = S0e(r−δ)T. Complete the following table Cash Flows t= 0 t= T Selling e−δT shares Long one forward Total Solution. 71 FORWARD CONTRACTS ON STOCK 601 Cash Flows t= 0 t= T Selling e−δT shares S0e−δT −PT Long one forward 0 PT −F0,T Total S0e−δT −F0,T Thus going short on a tailed position and buying a forward create cash ﬂows like those of a risk-free short zero-coupon bond (i.e. borrowing) If i is the rate of return on the synthetic bond then i satisﬁes the equation S0e(i−δ)T = F0,T. Solving for i we ﬁnd i= 1 T ln ( F0,T S0e−δT ) = δ+ 1 T ln (F0,T S0 ) . This rate is called the implied repo rate. Remark 71.2 “Repo” is short for “repurchase.” In the synthetic short bond, if we sell the stock short and enter into a long forward contract, then we have sold the stock and agreed to repurchase it. We have in eﬀect taken a loan (shorted a bond), which must be repaid with interest at the implied repo rate. Example 71.3 A $124 index pays a 1.5% continuous dividend. Suppose you observe a 2-year forward price of $135.7. Calculate the annual continuous rate of interest which you earn by buying stock and entering into a short forward contract, both positions for the same nominal amount. Solution. The implies repo rate is i= δ+ 1 T ln (F0,T S0 ) = 0.015 + 1 2 ln (135.7 124 ) = 6% We next discuss how a market-maker uses the synthetic positions strategies to hedge his clients positions. Suppose that a client enters into a long forward position with forward price F0,T and expiration time T. In this case, the conterparty represented by the market-maker is holding a short forward position. To oﬀset this risk, the market-maker borrows S0e−δT and use this amount to buy a tailed position in the index, paying S0e−δT and thus creating a synthetic long forward position. These transactions are summarized in the table below. 602 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Table 71.1 Cash Flows t= 0 t= T Buy tailed position in stock, paying −S0e−δT −S0e−δT PT Borrow S0e−δT S0e−δT −S0e(r−δ)T Short forward 0 F0,T −PT Total 0 F0,T −S0e(r−δ)T A transaction in which you buy the asset and short the forward contract is called cash-and-carry (or cash-and-carry hedge). It is called cash-and-carry, because the cash is used to buy the asset and the asset is kept. A cash-and-carry has no risk: You have obligation to deliver the asset, but you also own the asset. An arbitrage that involves buying the asset and selling it forward is called cash-and-carry arbitrage. Now suppose that a client wishes to enter into a short forward contract with forward price F0,T and expiration time T. In this case, the market-maker is holding a long forward position. To oﬀset this risk, the market-maker short a tailed position in the index receiving S0e−δT a tailed position in the index, paying S0e−δT and thus creating a synthetic long forward position. Example 71.4 A transaction in which you short the asset and long the forward contract is called reverse cash- and-carry (or cash-and-carry hedge). Complete the following table: Cash Flows t= 0 t= T Short tailed position in stock, receiving S0e−δT Lent S0e−δT Long forward Total Solution. We have Table 71.2 Cash Flows t= 0 t= T Short tailed position in stock, receiving S0e−δT S0e−δT −PT Lent S0e−δT −S0e−δT S0e(r−δ)T Long forward 0 PT −F0,T Total 0 S0e(r−δ)T −F0,T Note that with a reverse cash-and-carry the market-maker is long a forward and short a synthetic forward contract 71 FORWARD CONTRACTS ON STOCK 603 Note that if the forward contract is priced as F0,T = S0e(r−δ)T, then proﬁts on a cash-and-carry or reverse cash-and-carry are zero. If F0,T ̸= S0e(r−δ)T, an arbitrageur can do arbitrage. For example, if the forward contract is overpriced, that is F0,T > S0e(r−δ)T, than an arbitrageur can use the strategy described in Table 71.1 to make a risk-free proﬁt. In contrast, if the forward contract is underpriced, that is F0,T < S0e(r−δ)T, than an arbitrageur can use the strategy described in Table 71.2 to make a risk-free proﬁt. Example 71.5 Suppose you are a market-maker in index forward contracts. The current price of an index is $42.50. The index pays no dividends. You observe a 6-month long forward contract with forward price of $43. The risk-free annual continuously compounded interest rate is 5%. Describe a strategy for creating an arbitrage proﬁt and determine the amount of the proﬁt. Solution. The no-arbitrage forward price of the long forward contract is 42.50e0.05(0.5) = $43.58 which is larger than the observed long forward contract. We use the strategy of reverse cash-and-carry: We sell the index for $42.50 and deposit the money into a savings account for six months. We also buy the forward contract. At the expiration date, the money in the bank grew to 42 .50e0.05(0.5) = 43.58 and the cost of the forward is $43. Thus, we make a proﬁt of 43 .58 −43 = $0.58 According to either Table 71.1 or Table 71.2 we have seen that an arbitrageur makes a costless proﬁt only when F0,T ̸= S0e(r−δ)T. This says that there is only one single no arbitrage forward price. In practice that is not the case. That is, there is a range of prices that preclude arbitrage opportunities. In fact, there is a no-arbitrage lower bound and upper bound between which there arbitrage can not be practiced. Transactions costs that determine these bounds are: •Index ask and bid prices denoted by Sa 0 and Sb 0 with Sb 0 <S a 0 . •Forward contract ask and bid prices denoted by Fa 0,T and Fb 0,T with Fb 0,T <F a 0,T. •Index and forward transaction fees (commissions) denoted by CS and CF respectively. •The diﬀerence between borrowing and lending rates rb >rl. Let F+ and F− be the upper and lower bound respecively. Suppose that an arbitrageur believe that the forward price F0,T is too high. Assume that the index pays no dividends and that the arbitrageur applies the cash-and-carry strategy as illustrated in the table below. 604 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Cash Flows t= 0 t= T Buy tailed position in stock −Sa 0 e−δT −CI PT Borrow Sa 0 e−δT + CI + CF [Sa 0 e−δT + CI + CF] −[Sa 0 e−δT + CI + CF]erbT Short forward −CF Fb 0,T −PT Total 0 Fb 0,T −[Sa 0 e−δT + CI + CF]erbT An arbitrageur makes costless proﬁt if Fb 0,T > F+ = [Sa 0 e−δT + CI + CF]erbT. This upper bound reﬂects the fact that we pay a high price for the index (the ask price), pay transaction costs on both the index and the forward, and borrow at a high rate. In our discussion, we assume no transaction fees at time T. In Problem 71.14 we ﬁnd the lower bound to be F−= (Sb 0 −CI −CF)erlT. Example 71.6 Consider an index with current price of $800 and with zero dividends. Suppose that an arbitrageur can borrow at the continuously compounded rate of 5.5% and lend at the continuously compounded rate of 5%. Suppose there is a $1 transaction fee, paid at time 0, only for going either long or short a forward. Find the lower and upper bound between which the arbitrage is not proﬁtable. Solution. We have F−= (800 −1)e0.05 = $839.97 and F+ = (800 + 1)e0.055 = $846.29 Expected Future Price Consider a forward contract with underlying asset a non-dividend paying stock. Suppose that the current price is S0, the risk-free rate is r, and the delivery time is T. A forward price in this case is found to be S0erT. Does this mean that the stock price at time T is S0erT? The answer is no. That is, the forward price does not conveys any information about the future price of the stock. However, there is (conceptually) an expected rate of accumulation for the stock, call it α, that can be used to project an expected price for the stock at time T : S0eαT. Since α= r+ (α−r) the expected rate α has two components: the a risk-free component, r,representing compensation for the passage of time, and a risk component, α−r,which represents compensation for assuming the risk of loss on the stock (risk premium). If α−r >0, then you must expect a positive return from the forward contract. The only way for this to happen is if the forward price predicts too low a stock price. In other words, the forward price a biased predictor of the future stock price. 71 FORWARD CONTRACTS ON STOCK 605 Example 71.7 ‡ The current price of one share of XYZ stock is 100. The forward price for delivery of one share of XYZ stock in one year is 105. Which of the following statements about the expected price of one share of XYZ stock in one year is TRUE? A. It will be less than 100 B. It will be equal to 100 C. It will be strictly between 100 and 105 D. It will be equal to 105 E. It will be greater than 105. Solution. In general, an investor should be compensated for time and risk. A forward contract has no invest- ment, so the extra 5 represents the risk premium. Those who buy the stock expect to earn both the risk premium and the time value of their purchase and thus the expected stock value is greater than 100 + 5 = 105 and the answer is (E) Cost of Carry and Lease Rate Suppose you borrow S0 at the continuously compounded risk-free rate rto purchase a share of stock that pays continuous dividends at the rate δ.At the end of one year, you will repay S0er.However, the dividends will provide an oﬀsetting income of S0eδ.Hence, at the end of one year you will have to pay S0er−δ. This is the net cost of carrying a long position in the stock. We call r−δ the cost of carry. Now suppose you short the index and depositing the proceeds at a savings account paying contin- uously compounded annual rate r. Then at the end of one year you will earn S0er. However, you would have to pay S0eδ to the index lender. Note that the index dividends are what the lender gives up by allowing the stock to be borrowed. We call δ the lease rate of the index: It is the annualized cash payment that the borrower must make to the lender. With the deﬁnitions of the cost of carry and the lease rate, the forward pricing formula F0,T = S0e(r−δ)T has the following word interpretation Forward price = Spot price + Interest to carry the asset - Lease rate or Forward price = Spot price + Cost of carry. 606 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Example 71.8 What is the lease rate for a non-dividend paying stock? What is the cost of carry of a forward contract? Solution. For a non-dividend paying stock the lease rate is δ= 0. A forward contract requires no investment and makes no payouts and therefore has a zero cost of carry 71 FORWARD CONTRACTS ON STOCK 607 Practice Problems Problem 71.1 A stock has a spot price of $35. The continuously compounded risk free rate of interest is 6%. Find the prepaid forward price, the forward price and the annualized forward premium for a one-year forward contract on one share of the stock in each of the following situations. (a) The stock pays no dividends. (b) The stock pays a dividend of 1 in 6 months and 0.5 in one year just before delivery. (c) The stock pays continuous dividends at a rate of 3.5%. Problem 71.2 Suppose the stock price is $55 and the 4-month forward price is $57.50. Assuming the stock pays continuous dividends, ﬁnd (a) the 8-month forward premium (b) the 8-month forward price. Problem 71.3 Suppose the stock price is $55 and the 4-month forward price is $57.50. Assuming the stock pays continuous dividends, ﬁnd (a) Calculate the annualized forward premium. (b) Calculate the 1-year forward price. Problem 71.4 Consider the position of borrowing S0e−δT at time 0 and selling a forward with forward price F0,T = S0e(r−δ)T. Complete the following table Cash Flows t= 0 t= T Short one forward Borrow S0e−δT Total Problem 71.5 Consider the position of selling e−δT shares of the index for S0e−δT at time 0 and lending this amount for a period of T with interest rate r.Complete the following table Cash Flows t= 0 t= T Short tailed position in Stock Lend S0e−δT Total 608 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Problem 71.6 XYZ stock costs $123.118 per share. This stock pays dividends at an annual continuous rate of 2.5%. A 18 month forward has a price of $130.242. You own 10000 shares of XYZ stock. Calculate the annual continuous rate of interest at which you can borrow by shorting your stock. Problem 71.7 You are a market-maker in S&R index forward contracts. The current price of the index is $1100. The index pays no dividends. The risk-free annual continuously compounded interest rate is 5%. (a) What is the no-arbitrage forward price of a 9-month forward contract. (b) Suppose a customer wishes to enter into a long index forward position. If you take the opposite position, describe the strategy you would use to hedge the resulting short position. (c) Suppose a customer wishes to enter into a short index forward position. If you take the opposite position, describe the strategy you would use to hedge the resulting long position. Problem 71.8 Repeat the previous problem that index pays an 1.5% continuous dividend. Problem 71.9 Suppose you are a market-maker in index forward contracts. The current price of an index is $1100. The index pays no dividends. The risk-free annual continuously compounded interest rate is 5%. (a) You observe a 6-month forward contract with forward price of $1135. Describe a strategy for creating an arbitrage proﬁt and determine the amount of the proﬁt. (b) You observe a 6-month forward contract with forward price of $1115. Describe a strategy for creating an arbitrage proﬁt and determine the amount of the proﬁt. Problem 71.10 Suppose you are a market-maker in index forward contracts. The current price of an index is $1100. The index pays continuous dividends at 2%. The risk-free annual continuously compounded interest rate is 5%. (a) You observe a 6-month forward contract with forward price of $1120. Describe a strategy for creating an arbitrage proﬁt and determine the amount of the proﬁt. (b) You observe a 6-month forward contract with forward price of $1110. Describe a strategy for creating an arbitrage proﬁt and determine the amount of the proﬁt. Problem 71.11 An ABC index has a current price of $124 and the continuously compounded risk-free rate is 6%. Suppose you observe a 2-year forward price of $135.7. What dividend yield is implied by this forward price? 71 FORWARD CONTRACTS ON STOCK 609 Problem 71.12 Using the previous problem (a) Suppose you believe that the dividend yield over the next two years will be 0.5%. What arbitrage would you take? (b) Suppose you believe that the dividend yield over the next two years will be 3%. What arbitrage would you take? Problem 71.13 Suppose the current stock price is $45.34 and the continuously compounded intrest rate is 5%. The stock pays dividend of $1.20 in three months. You observe a 9-month forward contract with forward price $47.56. Is there an arbitrage opportunity on the forward contract? If so, describe the strategy to realize proﬁt and ﬁnd the arbitrage proﬁt. Problem 71.14 Suppose that an arbitrageur believes that the observed forward price F0,T is too low. Show that, for the arbitrageur to make costless proﬁt we must have F0,Ta<F −= [Sa 0 e−δT −CI −CF]erlT. Problem 71.15 Suppose that an arbitrageur would like to enter a cash-and-carry for 10000 barrels of oil for delivery in six months. Suppose that he can borrow at an annual eﬀective rate of interest of 4.5%. The current price of a barrel of oil is $55. (a) What is the minimum forward price at which he would make a proﬁt? (b) What is his proﬁt if the forward price is $57? Problem 71.16 Consider an index with current price of $800 and with zero dividends. Suppose that an arbitrageur can borrow at the continuously compounded rate of 5.5% and lend at the continuously compounded rate of 5%. Suppose that there are no transaction fees. Then an arbitrage is not proﬁtable if the forward price is between F− and F+. Determine the values of F− and F+. Problem 71.17 Consider an index with current price of $800 and with zero dividends. Suppose that an arbitrageur can borrow at the continuously compounded rate of 5.5% and lend at the continuously compounded rate of 5%. Suppose the transaction fee for going long or short a forward is $1 and $2.40 for going long or short an index. Find the lower and upper no-arbitrage bounds. 610 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES 72 Futures Contracts A futures contract is an agreement between two counterparts to buy/sell an asset for a speciﬁed price (the future price) at a speciﬁed date (the delivery date). Like the case of a forward con- tract, the holder of the contract is called the long future and the seller is called the short future. Futures are exchange-traded forward contracts. That is, they are bought and sold in organized future exhanges. Examples of exchanges the Chicago Mercantile Exchange, the Chicago Board of Trade, the International Petroleum Exchange of London, the New York Mercantile Exchange (NYMEX), the London Metal Exchange and the Tokyo Commodity Exchange. In the US, futures transactions are regulated by a government agency, the Commodity Futures Trading Commission (CFTC). Each exchange has a clearinghouse. The role of the clearinghouse is to match the purchases and the sales which take place during the trading day. By matching trades, the clearinghouse never takes market risk because it always has oﬀsetting positions with diﬀerent counterparts. By having the clearinghouse as counterpart, an individual entering a futures contract does not face the possible credit risk of its counterpart. Forwards and futures are similar in many aspects. However, there are still some diﬀerences which we list next. •Forward contracts are privately negotiated and not standardized. They are traded mainly over the counter and can be customized to suit the buyer or the seller. In contrast, futures contracts are standardized and have speciﬁed delivery dates, locations, and procedures. •Forward contracts are settled at the delivery date. Futures contracts are settled daily. As a result of daily settlements, value of a future contract is calculated according with the current value of the the underlying asset. We refer to this process as marking-to-market. Frequent marking-to- market can lead to pricing diﬀerences between futures contracts and identical forward contracts. •Futures contracts are liquid. Unlike a forward contract, which requires settling with the coun- terparty by actually buying or selling the asset on the expiration date, a futures contract can be cancelled by entering into the opposite position. For example, if you are short a futures contract, you can cancel your obligation to sell by entering an oﬀsetting obligation to buy the contract. This is done through a brokerage ﬁrm. •In the presence of clearinghouse, futures contracts are structured so as to minimize the eﬀects of credit risks. In contrast, in over-the-counter forward contracts, each party bears the other’s credit risk. •Because futures contracts are traded on an exchange, trades are subject to the exchange’s rules, which typically include a limit on the amount of daily price changes by imposing temporary halts in trading if the price change exceeds a deﬁned amount. •Futures contracts are commonly cash-settled, rather than requiring delivery of the underlying 72 FUTURES CONTRACTS 611 asset. We next illustrate an example of futures contracts with the S&P 500 index futures contract. The underlying asset for such a contract is the S&P 500 stock index. The contract has the following speciﬁcations: •The futures contract is traded at the Chicago Mercantile Exchange. •Notional Value: One contract has a value that is deﬁned as the level of the index (let’s say it is currently 1300) times $250 (so an S&P 500 futures contract currently has a notional value of $325,000). •The contract has delivery months March, June, September and December. •Trading ends on each business day prior to determination of settlement price. The settlement price is the average of the prices at which the contract traded immediately before the bell signaling the end of trading for the day. •The contracts are cash-settled because it is inconvenient or impossible to deliver the underlying asset which in this case consists of a basket of 500 stocks. The cash settlement is based on the opening price of the index on the third Friday of delivery month. Example 72.1 Suppose you wish to acquire $2.2 million worth of S&P 500 index with futures price 1100. (a) What is the notional value of one contract and how many futures contracts can you long for the invested amount? (b) The open interest of a futures contract is deﬁned to be the number of all futures outstanding contracts. It is the number of long positions. The activities of ﬁve traders are listed below. Complete the following table: Date Trading Activity Open interest Jan 1 A buys 1 contract and B sells 1 contract Jan 2 C buys 5 contracts and D sells 5 contracts Jan 3 A sells his contract and D buys one contract Jan 4 E buys 5 contracts from C who sells his 5 contracts Solution. (a) The notional value of one futures contract is 250 ×1100 = $275,000.With $2.2 million you can long 2200000 275000 = 8 futures contract. (b) 612 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Date Trading Activity Open interest Jan 1 A buys 1 contract and B sells 1 contract 1 Jan 2 C buys 5 contracts and D sells 5 contracts 6 Jan 3 A sells his contract and D buys one contract 6 Jan 4 E buys 5 contracts from C who sells his 5 contracts 6 Margins and Marking to Market One of the advantages of the futures contracts over the forward contracts is the dealing with credit risk. For example, forward contracts can be traded between two parties directly. This creates a credit risk related with the solvency of each party: One of the investors may regret the deal and try to back out or simply might not have the ﬁnancial resources to honor the agreement. In contrast, in the case of futures contracts the exchange tries to organize trading in such a way to avoid credit defaults. This is when margins come in. To avoid credit risk, a brokerage ﬁrm requires investors entering a futures contract to make deposit into an account called the margin account (also known as perfomance bond) which earns interest. The amount to be deposited at the time the contract is entered into is called theinitial margin and is determined by the exchange. It is usually a fraction of the market value of the futures’ underlying asset. The margin account is mainly intended to protect the counterparty of the contract. Margin accounts are adjusted on a daily basis to reﬂect the investor’s gain or loss. This practice is referred to as marking to market the account. A trade is marked to market at the close of the day on which it takes place. It is then marked to market at the close of the trading on each subsequent day. If there exists a loss, the investor’s broker transfers that amount from the investor’s margin account to the clearinghouse. If a proﬁt, the clearinghouse transfers that amount to investor’s broker who then deposits it into the investor’s margin account. We illustrate the role of margins in the following example. Example 72.2 On July 5, 2007, John instructs his broker to buy two gold futures contracts (each contract with nominal amount of 100 ounces of gold) on the New York Commodity Exchange (COMEX) with futures’ price of $400 per ounce. The annual continuously compounded rate of return is 1.5%. (a) How many ounces of gold has John contracted to buy at the price of $400 per ounce? (b) What is the notional value of the two contracts? (c) Suppose that the exchange requires 5% margin with daily settlement. What is the initial margin? (d) Suppose that on July 6,2007, the futures price of gold has dropped from $400 to $397. What is the balance in John’s margin account after settlement? (e) Suppose that on July 7,2007, the futures price of gold has risen to $400. What is the balance in John’s margin account after settlement? 72 FUTURES CONTRACTS 613 Solution. (a) Since each contract consists of 100 ounces of gold, John has contracted to buy 200 ounces at the price of $400 per ounce. (b) The notional value of the two contracts is 2 ×100 ×400 = $80,000. (c) The initial margin is 5% ×80,000 = $4,000. (d) The balance in John’s margin account after settlement is 4000e 0.015 365 + 200(397 −400) = $3400.16. (e) The balance in John’s margin account after settlement is 34,000.16e 0.015 365 + 200(400 −397) = $4,000.30 An investor is entitled any balance in the margin account in excess of the initial margin. However, if the balance in an investors margin account falls, the broker (or clearinghouse) has less protection against default. Investors are required to keep the margin account to a minimum level which is a percentage of the initial margin. This minimum level is called the maintenance margin. When the balance in the margin account drops below the maintenance level the investor receives amargin call and is required to make a deposit to return the balance to the initial margin level. If the investor does not meet the margin call, the position is closed out and the investor receives any remaining balance in the margin account. Example 72.3 A farmer enters into a short futures contract to sell 100,000 pounds of wheat for $1.4 per pound. There is 30% margin and weekly settlement. The maintenance margin is 20% and the annual eﬀective rate of interest is 4.5%. What is the minimum next week price which would lead to a margin call? Solution. The initial margin is (0.30)(100000)(1.4) = $42000. The maintenance margin is (0.20)(100000)(1.4) = $28000. After settlement, next week’s balance is 42000(1.045) 1 52 + 100000(1.4 −P1 52 ). A margin call happens if 28000 >42000(1.045) 1 52 + 100000(1.4 −P1 52 ). 614 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES or P1 52 >1.4 −28000 −42000(1.045) 1 52 100000 = 1.540355672 72 FUTURES CONTRACTS 615 Practice Problems Problem 72.1 The open interest on silver futures at a particular time is the number of (A) all silver futures outstanding contracts (B) outstanding silver futures contracts for a particular delivery month (C) silver futures contracts traded during the day (D) silver futures contracts traded the previous day Problem 72.2 The fact that the exchange is the counter-party to every futures contract issued is important because it eliminates (A) market risk (B) credit risk. (C) interest rate risk Problem 72.3 Margin must be posted by (A) buyers of futures contracts (B) sellers of futures contracts (C) both buyers and sellers of futures contracts (D) speculators only Problem 72.4 The daily settlement of obligations on futures positions is called (A) a margin call (B) marking to market (C) a variation margin check (D) None of the above Problem 72.5 All of the following are false except (A) A margin deposit can only be met by cash (B) All futures contracts require the same margin deposit (C) The maintenance margin is the amount of money you post with your broker when you buy or sell a futures contract (D) The maintenance margin is the value of the margin account below which the holder of a futures contract receives a margin call. 616 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Problem 72.6 You are currently long in a futures contract. You then instruct a broker to enter the short side of a futures contract to close your position. This is called (A) a cross hedge (B) a reversing trade (C) a speculation (D) marking to market Problem 72.7 A silver futures contract requires the seller to deliver 5,000 Troy ounces of silver. An investor sells one July silver futures contract at a price of $8 per ounce, posting a $2,025 initial margin. Neglecting interest, if the required maintenance margin is $1,500, the price per ounce at which the investor would ﬁrst receive a maintenance margin call is closest to: (A) $5.92 (B) $7.89 (C) $8.11 (D) $10.80 Problem 72.8 If the initial margin on a futures contract I have sold increases, it likely means: (A) The market has gotten less liquid. (B) I am not making money in my futures positions. (C) Prices have gotten more volatile. (D) The margins never change once you have a futures position. Problem 72.9 Complete the following statement: At the end of each trading day, the margin account is adjusted to reﬂect the futures trader’s gain or loss. This process is referred to as the account. Problem 72.10 What is the diﬀerence between the initial margin and the maintenance margin? Problem 72.11 Explain some diﬀerences between a futures contract and a forward contract. Problem 72.12 Crude oil futures trade in units of 1,000 U.S. barrels (42,000 gallons). The current value is $50/bar- rel. Find the notional value of this contract. 72 FUTURES CONTRACTS 617 Problem 72.13 Assume today’s settlement price on a CME EUR futures contract is $1.3140/EUR. The nominal amount of one contract is 125,000. You have a short position in one contract. Your margin account currently has a balance of $1,700. The next three days’ settlement prices are $1.3126, $1.3133, and $1.3049. Calculate the changes in the margin account from daily marking-to-market and the balance in the account after the third day. Problem 72.14 Do Problem 72.13 again assuming you have a long position in the futures contract. Assume 40% maintenance margin. Would you receive a margin call? Problem 72.15 Suppose the S&P 500 index futures price is currently 1200. You wish to purchase four futures contracts on margin. (a) What is the notional value of your position? (b) Assuming a 10% initial margin, what is the value of the initial margin? Problem 72.16 Suppose the S&P 500 index futures price is currently 950 and the initial margin is 10%. You wish to enter into 10 S&P 500 futures contracts. (a) What is the notional value of your position? What is the margin? (b) Suppose you earn a continuously compound rate of 6% on your margin balance, your position is marked to market weekly, and the maintenance margin is 80% of the initial margin. What is the greatest S&P 500 index futures price 1 week from today at which you will receive a margin call? 618 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES 73 Understanding the Economy of Swaps: A Simple Com- modity Swap A swap is an agreement to exchange cash ﬂows at speciﬁed future times according to certain spec- iﬁed rules. For example, a forward contract is a single-payment swap. To see this, suppose you enter a forward contract to buy 50 ounces of gold for $300 an ounce in one year. You can sell the gold as soon you receive it. The forward contract acts like a single-payment swap because at the end of the year you will pay 50 ×300 = $15000 and receive 50 ×P1 where P1 is the market price of one ounce of gold on that date. While a forward contract leads to an exchange of cash on a single future dates, swaps typically lead to cash ﬂow exchanges taking place on several future dates. In some sense, a swap is a series of forward contracts combined into one contract or put it diﬀerently it can be regarded as a portfolio of forward contracts. There are various types of swaps, namely, interest rates swap, currency swaps, commodity swaps, and equity swaps. In this section we will try to understand the economy of swaps. Our discussion will be centered around commodity swaps. A commodity swap is a swap in which one party is selling a commodity (or the cash equivalent) and the other is paying cash for the commodity (the buyer). The contract needs to specify the type and quality of the commodity, how to settle the contract, etc. Commodity swaps are predominantly associated with energy-related products such as oil. Suppose that quantity Qi of the commodity is to be delivered at time ti where i = 1 ,2,··· ,n. The buyer can either make one payment of C0 at time 0 for the swap (a prepaid swap) or make a payment of Ci at time ti. Let F0,ti be the forward price of the commodity with delivery at time ti where 1 ≤i≤n.Let ri be the annual interest rate applied over the interval of time [0,ti].Then the present value of the commodity delivered over the life of the swap is n∑ i=1 QiPVi(F0,ti). In a prepaid swap, the buyer makes a single payment today in the amount of n∑ i=1 QiPVi(F0,ti) to obtain multiple deliveries in the future. Example 73.1 A manufacturer is going to buy 100,000 barrels of oil 1 year from today and 2 years from today. 73 UNDERSTANDING THE ECONOMY OF SWAPS: A SIMPLE COMMODITY SWAP619 Suppose that the forward price for delivery in 1 year is $20/barrel and in 2 years $21/barrel. The 1-year spot interest rate is 6% and the 2-year spot interest rate is 6.5%. The manufacturer enters into long forward contracts for 100,000 barrels for each of the next 2 years, committing to pay $20/barrel in 1 year and $21/barrel in 2 years. Alternatively, the manufacturer can make a single payment P now and the supplier would commit to deliver 1 barrel in year 1 and 1 barrel in year 2. Find P. Solution. P is just the cost of a prepaid swap given by 20 1.06 + 21 1.0652 = $37.383 Now, with multiple deliveries and multiple payments, the buyer makes swap payments at the time the commodity is delivered. Usually each swap payment per unit of commodity is a ﬁxed amount say R. That is, at time ti the swap payment is QiR. Thus, the present value of all the payments is given by the sum n∑ i=1 QiPVi(R). We thus obtain the equation n∑ i=1 QiPVi(F0,ti) = n∑ i=1 QiPVi(R) From this equation, one ﬁnds R given by R= ∑n i=1 QiPVi(F0,ti)∑n i=1 QiPVi(1) . Suppose that at time of delivery, the buyer pays a level payment of Rat each of the delivery times. Then, the present value of the cashﬂow of payments is n∑ i=1 PVi(R) and R is given by R= ∑n i=1 QiPVi(F0,ti)∑n i=1 PVi(1) . 620 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Example 73.2 Consider the information of Example 73.1. Suppose a swap calls for equal payments each year. Let x be the payment per year per barrel. This is the swap price. Find x. Solution. We must have x 1.06 + x 1.0652 = 37.383. Solving this equation for xwe ﬁnd x= $20.483.This means, that each year, the manufacturer will pay $20.483 and receives a barrel of oil Remark 73.1 The payments need not be equal. Any payments that have a present value of $37.383 are acceptable. That is, any two payments that satisfy the equation P1 1.06 + P2 1.0652 = 20 1.06 + 21 1.0652 . A swap can be settled either by physical settlement or by cash settlement. If a swap is settled physically, the party with the commodity delivers the stipulated notional amount to the buyer in exchange of cash as shown in the previous example. If a swap is cashed settled, then the commodity is valued at the current spot price. If the current value of the commodity is larger than the value of the cash payment, then the buyer pays the swap counterparty the diﬀerence and then buys the commodity at the spot price. Reciprocally, if the current value of the commodity is smaller than the value of the cash payment, the counterparty pays this diﬀerence to the buyer and the buyer buys the commodity. Example 73.3 Again, we consider the information of both previous examples. Suppose the swap is cash settled. (a) Suppose that the market price is $25. Calculate the payment which the buyer receives. (b) Suppose that the market price is $18. Calculate the payment which the counterparty receives. Solution. (a) The swap counterparty pays the buyer 25 −20.483 = $4.517 per barrel. (b) The buyer pays the swap counterparty 18 −20.483 = −$2.483 per barrel Notice that whatever the market price of oil, the net cost to the buyer is the swap price $20.483: spot price −(spot price −swap price) = swap price. 73 UNDERSTANDING THE ECONOMY OF SWAPS: A SIMPLE COMMODITY SWAP621 This formula describes the payments per one barrel of oil. For a swap agreement involving 100,000 barrels of oil, the values in the formula are multiplied by the notional amount of the swap (i.e. 100,000 barrels). Remark 73.2 Notice that the results for the buyer are the same whether the swap is settled physically or ﬁnancially. In both cases, the net cost to the buyer is $20.483 per barrel. A swap can be regarded as a series of forward contracts, combined with borrowing and/or lending money. In the above oil swap example, the buyer under the swap agreement would pay (and the counterparty would receive) the swap price of $20.483 on each of the two dates. Under a pair of forward contracts, the forward prices of $20 and $21 would have been paid on the ﬁrst and second dates, respectively. Relative to the forward prices, the buyer lends to the counterparty an amount of $0.483 (= 20 .483 −20) on the ﬁrst date by paying that much more than the forward rate; then the counterparty repays the loan by accepting $20.483 on the second date, which is less than the forward price by $0.517 (= 21 −20.483). This loan has the eﬀect of equalizing the net cash ﬂow on the two dates. The interest rate on the loan is 0.517 0.483 −1 = 7% This is the 1-year implied forward rate from year 1 to year 2. A commodity swap allows to lock the price of a sale. It can be used by a producer of a com- modity to hedge by ﬁxing the price that he will get in the future for this commodity. It also can be used by a manufacturer to hedge by ﬁxing the price that he will pay in the future for a commodity. Swap Dealer A dealer is an agent who makes a market buying and selling swaps to parties who have natural exposure to price risk. He earns his keep on the bid/ask spread. The dealer buys a swap from the producer, at ﬁxed price S (the swap price). The seller receives S−PT, where PT is the spot price. The dealer sells a swap to the consumer, at ﬁxed price S+ ∆, where δ is the ask-bid spread. The buyer pays S+ ∆ −PT. The dealer earns diﬀerence between what buyer gives and seller gets, i.e. ∆ . The process of matching a buyer to the seller is called a back-to-back transaction. As you can see from the above discussion, the dealer is subject to each party’s credit risk, but is not exposed to price risk. Figure 73.1 illustrates the role of the swap dealer. 622 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Figure 73.1 We next consider the case when the dealer serves as the counterparty. A dealer enters the swap as the counterparty. Thus, the dealer has the obligation to pay the spot price and receives the swap price. The dealer has a short position in 1- and 2- year oil. The dealer can lose money if the spot price increases. The dealer hedges his short position by going long year 1 and year 2 gold forwards. The table below illustrates how this strategy works. Year Payment from Gold Buyer Long Forward Net 1 $20 .483 −P1 P1 −$20 $0.483 2 $20 .483 −P2 P2 −$21 −$0.517 The net cash ﬂow for the hedged position is a loan where the dealer receives $0.483 in year 1 and repays $0.517 in year two. The dealer has interest rate exposure. In interest rate fall, the dealer will not be able to earn suﬃcient return from investing $0.483 in year 1 to repay $0.517 in year 2. This shows that hedging oil price risk does not fully hedge the position. In summary, the dealer can eliminate its price risk by hedging the swap contract through the use of multiple forward contracts. The dealer must also enter into forward rate agreements in order to hedge the interest rate exposure resulting from the timing diﬀerence between the forward contract’s payments and the payments under the swap agreement. The Market Value of a Swap When a swap agreement is created, its market value is zero. In our oil example, the swap consists of two forward contracts and an agreement to lend money at the implied forward rate of 7%. These forwards have zero values at time 0 so the swap does as well. As time passes the market value of the swap will no longer be zero due to oil prices and interest rates change. So over time, the swap contract develops a non-zero value. Even if prices and interest rates do not change, the swap agreement will have a non-zero value once the ﬁrst payment has been made. Let us ﬁnd the market value of the oil swap example. Suppose that right after entering the swap the oil prices go up by $2 in year 1 and year 2. The original swap in this case has a non-zero market value. Assuming interest rate are the same, the new swap price x satisﬁes the equation x 1.06 + x 1.0652 = 22 1.06 + 23 1.0652 73 UNDERSTANDING THE ECONOMY OF SWAPS: A SIMPLE COMMODITY SWAP623 which gives x= $22.483. In this case, the buyer buys a barrel of oil for $20.483 and can sell it for $22.483 making a net proﬁt of $2 per barrel each year. The present value of this diﬀerence of prices is 2 1.06 + 2 1.0652 = $3.65. The buyer can receive a stream of payments worth $3.65 by oﬀsetting a new swap. Thus, $3.65 is the market value of the swap. If interest rates had changes, we would have used the new interest rate in computing the new swap price. The swap market is very large and has no major governmental oversight. Swap markets have none of the provisions as the futures and options markets do, thus resulting in a default risk and the need to assess the creditworthiness of each counterparty. Traders must carefully assess their trade partners. On the other side of this, swaps have a measure of privacy and can be negotiated for unique quan- tities, qualities, and contract provisions. Example 73.4 Suppose that oil forward prices for 1 year, 2 years, and 3 years are $20, $21, and $22. The eﬀective annual interest rates are shown in the table below Maturity(in years) yield % 1 6 2 6.5 3 7 (a) What is the swap price, assuming level payments? (b) Suppose you are a dealer who is paying the swap price and receiving the spot price. Suppose you enter into the swap and immediately thereafter all interest rates rise by 0.5% with the oil forward prices remain unchanged. What happens to the value of your swap position? (c) What if interest rates fall by 0.5%? (d) What hedging instrument would have protected you against interest rate risk in this position? Solution. (a) The swap price x satisﬁes the equation x 1.06 + x 1.0652 + x 1.073 = 20 1.06 + 21 1.0652 + 22 1.073 . Solving for x we ﬁnd x= $20.952. (b) You should short 3 forward contracts. This strategy is illustrated in the table below. 624 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Year Payment from Gold Buyer Short Forward Net 1 P1 −$20.952 $20 −p1 −$0.952 2 P2 −$20.952 $21 −p2 $0.048 3 P3 −$20.952 $22 −p2 $1.048 With the 0.5% increase in rates we ﬁnd that the net present value of the loan is −0.952 1.065 + 0.048 1.072 + 1.048 1.0753 = −$0.0081. The dealer never recovers from the increased interest rate he faces on the overpayment of the ﬁrst swap payment. (c) If interest rates fall by 0.5% then the net present value of the loan is −0.952 1.055 + 0.048 1.062 + 1.048 1.0553 = $0.0083. The dealer makes money, because he gets a favorable interest rate on the loan he needs to take to ﬁnance the ﬁrst overpayment. (d) The dealer could have tried to hedge his exposure with a forward rate agreement or any other derivative protecting against interest rate risk 73 UNDERSTANDING THE ECONOMY OF SWAPS: A SIMPLE COMMODITY SWAP625 Practice Problems Problem 73.1 Match (a)-(d) with (I)-(IV) (a) Notional amount is (b) The contracting parties in a swap are called (c) What the swap dealers receive for their services is (d) The ask-spread commission is (I) ask-spread bid (II) counterparties (III) the diﬀerence between the cash ﬂow one counterparty pays to the dealer and what dealer pays out to the other counterparty (IV) used to calculate the payments that will be exchanged between the two parties. Problem 73.2 ‡ Zero-coupon risk-free bonds are available with the following maturities and yield rates (eﬀective, annual): Maturity(in years) yield % 1 6 2 6.5 3 7 You need to buy corn for producing ethanol. You want to purchase 10,000 bushels one year from now, 15,000 bushels two years from now, and 20,000 bushels three years from now. The current forward prices, per bushel, are $3.89, $4.11, and $4.16 for one, two, and three years respectively. You want to enter into a commodity swap to lock in these prices. Which of the following sequences of payments at times one, two, and three will NOT be acceptable to you and to the corn supplier? (A) $38,900; $61,650; $83,200 (B) $39,083; $61,650; $82,039 (C) $40,777; $61,166; $81,554 (D) $41,892; $62,340; $78,997 (E) $60,184; $60,184; $60,184 Problem 73.3 A manufacturer uses oil in his profession. He needs 10,000 barrels in two months, 12,000 barrels in four months, and 15,000 barrels in six months. The manufacturer enters into a long oil swap. The 626 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES payment of the swap will be made at the delivery times. The current forward prices, per barrel, are $55, $56, and $58 for two, four, and six months re- spectively. Zero-coupon risk-free bonds are available with the following maturities and yield rates (nominal, convertible monthly): Maturity(in months) yield % 2 4.5 4 4.55 6 4.65 (a) Suppose level payments of R are made at each of the delivery times. Calculate R. (b) Suppose that the cash swap payment consists of a unique payment per barrel made at each of the delivery times. Calculate the no arbitrage swap price per barrel. Problem 73.4 Suppose that a transportation company must buy 1000 barrels of oil every six months, for 3 years, starting 6 months from now. Instead of buying six separate long forward contracts, the company enters into a long swap contract. The payment of the swap will be made at the delivery times. The current forward prices, per barrel, are $55, $57, $57, $60, $62, and $64 for 6-, 12-, 18-, 24-, 30-, and 36- months respectively. Zero-coupon risk-free bonds are available with the following maturities and yield rates (nominal, convertible semiannually): Maturity(in months) yield % 6 5.5 12 5.6 18 5.65 24 5.7 30 5.7 36 5.75 (a) With level payments, ﬁnd the price per barrel of oil using the swap. (b) Suppose the swap is settled in cash. Assume that the spot rate for oil in 18 months is $57. Calculate the payment which the counterparty receives. (c) Suppose that immediately after the swap is signed up, the future prices of oil are given by the table Forward Price $55 $58 $59 $61 $62 $63 Expiration (in months) 6 12 18 24 30 36 Find the market value of the swap for the company. 73 UNDERSTANDING THE ECONOMY OF SWAPS: A SIMPLE COMMODITY SWAP627 Problem 73.5 Suppose that a jewelry manufacturer needs to purchase 100 ounces of gold one year from now and another 100 ounces in two years, and that the forward price for gold is $750 per ounce for delivery in one year and $800 for delivery in two years. The manufacturer could enter into two long forward contracts for these two dates at these two prices, and eliminate the risk of price variation. Alternatively, a single swap agreement could be used to ﬁx a single price for the purchases on these two dates. Suppose that the 1-year and 2-year spot rates are 5% and 6% respectively. (a) With level payments, ﬁnd the price per ounce of gold using the swap. (b) Suppose that the payments are not leveled, say with payment P1 per ounce for year 1 and P2 for year 2. Set the equation that P1 and P2 must satisfy. (c) What is the notional amount of the swap? (d) The buyer pays an amount larger than the forward price in 1 year. The diﬀerence is considered as a loan for the counterparty. What is the value of this amount per ounce? (e) The buyer pays an amount smaller than the forward price in 2 years. The diﬀerence is considered as the counterparty repayment of the loan. What is the value of this amount per ounce? (f) What is the implied forward rate from year 1 to year 2 of the loan? Problem 73.6 Consider the information of the previous problem. Suppose the forward prices are $780 and $830. The 1- and 2-year interest rates are 5% and 6% respectively. Find the new swap price. Problem 73.7 Suppose that gold forward prices for 1 year, 2 years, and 3 years are $750, $800, and $850. The eﬀective annual interest rates are shown in the table below Maturity(in years) yield % 1 6 2 6.5 3 7 (a) What is the swap price? (b) What is the price of a 2-year swap with the ﬁrst settlement two years from now and the second in 3 years? Problem 73.8 Consider the 2-year gold swap of Problem 73.5. Suppose a dealer is paying the spot price and receiving the swap price. What position in gold forward contracts will hedge gold price risk in this position? 628 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Problem 73.9 Consider the 2-year gold swap of Problem 73.5. Suppose a dealer is paying the swap price and receiving the spot price. What position in gold forward contracts will hedge gold price risk in this position? 74 INTEREST RATE SWAPS 629 74 Interest Rate Swaps In this section we consider another type of swaps: the interest rate swaps. Companies use interest rate swap as a means of converting a series of future interest payments that vary with changes in interest rates ( ﬂoating payments) to a series of ﬁxed level payments (i.e. swap rate ), or vice-versa. The ﬁxed stream of payments are computed with respect to a rate determined by the contract. The ﬂoating stream of payments are determined using a benchmark, such as the LIBOR 1 The amount in which the interest payments is based is called the notional principal. The life of a swap is called the swap term or swap tenor. Interest rate swaps are used to hedge against or speculate on changes in interest rates: They are often used by ﬁrms to alter their exposure to interest-rate ﬂuctuations, by swapping ﬁxed-rate obligations for ﬂoating rate obligations, or vice versa. By swapping interest rates, a ﬁrm is able to alter its interest rate exposures and bring them in line with management’s appetite for interest rate risk. They are also used speculatively by hedge funds or other investors who expect a change in interest rates or the relationships between them. Interest rate swaps are also very popular due to the arbitrage opportunities they provide. The interest rate swap market is closely linked to the Eurodollar 2 futures market which trades at the Chicago Mercantile Exchange. All interest rate swaps payments are settled in net: If the ﬁxed rate is greater than the ﬂoating rate, the buyer pays the counterparty the diﬀerence. In contrast, if the ﬁxed rate is smaller than the ﬂoating rate, the counterparty pays the buyer the diﬀerence. The interest rate determination date for the ﬂoating interest payment occurs at the beginning of the period with payment due at the end of the period. It is important to keep in mind that there is no exchange of the notional principal in a swap. Example 74.1 On December 31, 2006, ﬁrm ABC enters into an interest rate swap with ﬁrm XYZ. The notional principal is $100 m. Firm ABC pays the ﬁxed rate of 5% compounded annually and receives ﬂoating rate from ﬁrm XYZ at a 12-month LIBOR. The swap term is 5 years. That is, the two parties exchange payments annually on December 31, beginning in 2007 and concluding in 2011. This swap is illustrated in Figure 74.1 1The ( London Interbank oﬀer rate ) LIBOR is the most widely used reference rate for short term interest rates world-wide. The LIBOR is published daily the (British Bankers Association) BBA. It is based on rates that large international banks in London oﬀer each other for interbank deposits. Rates are quoted for 1-month, 3-month, 6-month and 12-month deposits. 2Eurodollars are US dollars deposited at banks outside the United States, primarily in Europe. The interest rate paid on Eurodollars is largely determined by LIBOR. Eurodollar futures provide a way of betting on or hedging against future interest rate changes. 630 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Figure 74.1 (a) On December 31,2006 the 12-month LIBOR rate was 4.5%. Determine the amount paid by ABC to XYZ on December 31,2007. (b) On December 31,2006 the 12-month LIBOR rate was 5.5%. Determine the amount received by ABC from XYZ on December 31,2007. Solution. (a) On December 31,2007 ABC will pay XYZ the amount $100m ×0.05 = $5,000,000 and receives from XYZ the amount $100m ×0.045 = $4,500,000. Thus, on December 31,2007, ABC pays XYZ the net payment of $500,000. (a) On December 31,2007 ABC will pay XYZ the amount $100m ×0.05 = $5,000,000 and receives from XYZ the amount $100m ×0.055 = $5 ,500,000. Thus, on December 31,2007, ABC receives from XYZ the net amount of $500,000. On net, ABC is paying At no point does the principal change hands, which is why it is referred to as a notional amount Remark 74.1 A ﬁxed-for-ﬂoating interest rate swap is often referred to as a plain vanilla swap because it is the most commonly encountered structure. Counterparty Risk of a Swap Unlike future markets, swaps are over-the-counter instruments and so they are not backed by the guarantee of a clearing house or an exchange. If one party defaults, they owe to the other party at most the present value of net swap payments that they are obligated to make at current market prices. Asset Swap An asset swap is an interest rate swap used to convert the cash ﬂows from and underlying security (such a bond) from ﬁxed coupon to ﬂoating coupon. Thus, an asset swap converts a ﬁxed-rate bond 74 INTEREST RATE SWAPS 631 to a synthetic ﬂoating-rate bond. The asset swap transaction is illustrated in Figure 74.2. Figure 74.2 Asset swaps are commonly used by fund managers who own ﬁxed-rate bonds and wish to have ﬂoating-rate exposure while continuing to own the bonds. Computing the Swap Rate Consider an interest rate swap with nsettlements occurring on dates ti,i = 1,2,··· ,n. Let P(0,ti) be the price of a $1 face value zero-coupon bond maturing on date ti. Notice that P(0,ti) is the discount factor from time zero to time ti based on the ti−year spot rate. That is, if it is the t−year spot rate then P(0,t) = (1 + it)−t. Let r0(ti−1,ti) be the implied forward rate between the times ti−1 and ti. From Section 50, we have r0(ti−1,ti) = P(0,ti−1) P(0,ti) −1. An interest rate swap acts much like the oil price swap described in the previous section. When initiated, the swap has a value of zero, i.e., the ﬁxed rate payments and the variable rate payments have equal present values. In terms of a formula, we have: R n∑ i=1 P(0,ti) = n∑ i=1 P(0,ti)r0(ti−1,ti) where R denote the ﬁxed swap rate. Solving this last equation for R we obtain R= ∑n i=1 P(0,ti)r0(ti−1,ti)∑n i=1 P(0,ti) 632 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES where ∑n i=1 P(0,ti)r0(ti−1,ti) is the present value of the variable payments and ∑n i=1 P(0,ti) is the present value of a $1 annuity when interest rates vary over time. Note that the swap rate can be interpreted as a weighted average of the implied forward rates, where zero-coupon bond prices are used to determine the weights. Now, since r0(ti−1,ti) = P(0,ti−1) P(0,ti) −1 we can write n∑ i=1 P(0,ti)r0(ti−1,ti) = n∑ i=1 [P(0,ti−1) −P(0,ti) = 1 −P(0,tn) so that R= 1 −P(0,tn)∑n i=1 P(0,ti) This formula can be rearranged to obtain P(0,tn) + R n∑ i=1 P(0,ti) = 1. This formula states that a payment of 1 that will be received n years from now, plus a payment of R at the end of each of the next n years, has a total present value of 1. This is the valuation equation for a bond priced at par (i.e. face value $1) with a coupon rate R. Thus, the swap rate is the coupon rate for a bond that is selling at par. Example 74.2 The following table lists prices of zerocoupon $1face value bonds with their respective maturities: Years to Maturity Price 1 $0.956938 2 $0.907029 3 $0.863838 (a) Calculate the 1-, 2-, and 3-year spot rates of interest. (b) Calculate the 1- and 2-year forward rates of interest. (c) Calculate the coupon rate Rfor a 3-year bond with annual coupons whose face value, redemption value and price are all one. Solution. (a) Let i1 be the spot rate of interest. Then 1 + i1 = P(0,1)−1 = 0.956938−1 →i1 = 0.956938−1 − 74 INTEREST RATE SWAPS 633 1 = 4 .499967135%. Likewise, i2 = 0 .907029 −1 2 −1 = 5 .000027694% and i3 = 0 .863838 −1 3 −1 = 4.99991613% (b) The 1-year forward rate is given by r0(1,2) = P(0,1) P(0,2) −1 = 0.956938 0.907029 −1 = 5.502470153%. Likewise, the 2-year forward rate is given by r0(2,3) = P(0,2) P(0,3) −1 = 0.907029 0.863838 −1 = 4.999895814%. (c) The coupon rate is given by R= 1 −P(0,3) P(0,1) + P(0,2) + P(0,3) = 1 −0.863838 0.956938 + 0.907029 + 0.863838 = 4.991632466% Deferred Swaps By a k-deferred swap we mean a swap whose exchange of payments start k periods from now. The (deferred) swap rate is applied in the future but it is agreed upon today. It is computed as follows R= ∑n j=kP(0,tj)r0(tj−1,tj)∑n j=kP(0,tj) = ∑n j=kP(0,tj) ( P(0,tj−1) P(0,tj) −1 ) ∑n j=kP(0,tj) = ∑n j=k(P(0,tj−1) −P(0,tj)∑n j=kP(0,tj) =P(0,tk−1) −P(0,tn)∑n j=kP(0,tj) Example 74.3 Given the following zero-coupon bond prices: Quarter 1 2 3 4 5 6 7 8 P(0,t) 0.9852 0.9701 0.9546 0.9388 0.9231 0.9075 0.8919 0.8763 What is the ﬁxed-rate (deferred swap rate) in a 6-quarter interest rate swap with the ﬁrst settlement in quarter 3? 634 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Solution. We have R=P(0,2) −P(0,8)∑8 j=3 P(0,j) = 0.9701 −0.8763 0.9546 + 0.9388 + 0.9231 + 0.9075 + 0.8919 + 0.8763 =1.71% The Value of a Swap At initiation of the swap, the value of the swap is set to be zero to both parties. The ﬁxed rate of the swap is then calculated such that the present value of the ﬁxed payments equals that of the ﬂoating payments. As time evolves, the interest rate may move upward or downward so that the value of the ﬂoating payments changes. Therefore, the value of the swap changes in later times, and its value at each time is given by the diﬀerence in the present values of the remaining cash ﬂows from the two parties. Even in the absence of changes in interest rates, as soon as the ﬁrst swap payment occurs, the swap will have a value. The swap will have a positive value for the party who paid too much (e.g., the ﬁxed rate payer if the ﬁxed rate exceeds the variable rate on the ﬁrst payment date), and a negative value for the party who received the net payment. As with the commodity swap, an interest rate swap is equivalent to entering into a series of forward contracts and also undertaking some borrowing and lending. 74 INTEREST RATE SWAPS 635 Practice Problems Problem 74.1 Interest rate swaps allow one party to exchange a: (A) ﬂoating interest for a ﬁxed rate over the contract term. (B) ﬁxed interest rate for a lower ﬁxed rate over the contract term. (C) ﬂoating interest rate for a lower ﬂoating rate over the contract term. Problem 74.2 All of the following are types of ﬁnancial instruments except for: (A) currency forward contracts (B) currency futures contracts (C) currency options (D) swap agreements (E) money-market hedge Problem 74.3 On March 5,2004, an agreement by Microsoft to receive 6-month LIBOR and pay a ﬁxed rate of 5% per annum every 6 months for 3 years on a notional principal of $100 million. Complete the following table: Date 6-month Fixed Floating Net LIBOR rate Payment Payment Cash Flow March 5,2004 4.2% Sept 5,2004 4.8% March 5, 2005 5.3% Sept 5, 2005 5.5% March 5, 2006 5.6% Sept 5, 2006 5.9% March 5,2007 6.4% Problem 74.4 Firm ABC is paying $750,000 in interest payments a year while Firm XYZ is paying LIBOR plus 75 basis points (i.e. 0.75%) on $10,000,000 loans. The current LIBOR rate is 6.5%. Firm ABC and XYZ have agreed to swap interest payments, how much will paid to which Firm this year? Problem 74.5 Given the following information 636 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES t (in months) P(0,t) 3 0.986923 6 0.973921 9 0.961067 12 0.948242 Calculate the annual nominal interest rate compounded quarterly for a loan with the following maturity dates: 3, 6, 9 and 12 months. Problem 74.6 Given the following information t (in years) P(0,t) 1 0.95785 2 0.91049 3 0.85404 Calculate the 3-year interest rate swap. Problem 74.7 The ﬁxed rate for a 2-year interest rate swap is 6.06%. The 2-year spot rate is 6.10%. What is the price of a 1-year zero-coupon bond with a maturity value of 1,000? Problem 74.8 Given the following zero-coupon bond prices: Quarter 1 2 3 4 5 6 7 8 P(0,t) 0.9852 0.9701 0.9546 0.9388 0.9231 0.9075 0.8919 0.8763 What is the ﬁxed-rate (deferred swap rate) in a 5-quarter interest rate swap with the ﬁrst settlement in quarter 2? Problem 74.9 Given the following zero-coupon bond prices: Quarter 1 2 3 4 5 6 7 8 P(0,t) 0.9852 0.9701 0.9546 0.9388 0.9231 0.9075 0.8919 0.8763 What is the swap rate in an 8-quarter interest rate swap? Problem 74.10 The following table lists prices of zerocoupon $1face value bonds with their respective maturities: 74 INTEREST RATE SWAPS 637 Years to Maturity Price 1 $0.95238 2 $0.89845 3 $0.842.00 Complete the following table: Years to Maturity Price Spot Rate in r0(n−1,n) 1 $0.95238 2 $0.89845 3 $0.842.00 Problem 74.11 Using the data in the previous table, ﬁnd the ﬁxed rate (i.e. the swap rate R). 638 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES 75 Risk Management Risk management consists of identifying the sources of risk, choosing the ones to be hedged, and choosing the way of hedging. A ﬁrm’s risk management consists of the use of ﬁnancial derivatives such as forwards, calls, and puts to alter its exposure to risk and protect its proﬁtability. We will next discuss risk management from both the producer’s perspective as well as the buyer’s perspective. The Producer’s Perspective A producer normally has a long position that consists of its product. In order to protect its future proﬁts from price ﬂuctuations, the producer can oﬀset his long position by using forward contracts, puts options, call options, or some combination of these derivatives. This process is also referred to as hedging or insuring the long position. Some of the risk management strategies for a producer include the following: (1) Hedging with a forward contract: In order for the producer to lock in the price of his product, the producer can enter into a short forward contract that ﬁxes the future sale price at the current forward price. There is no initial cost for the producer. For example, ABC Inc, an oil producing ﬁrm, enters into a short forward contract, agreeing to sell oil at today’s current price of $65 a barrel in one year. (2) Hedging with a put option: Creating a ﬂoor by purchasing a put option which limits the losses if the price declines, but allows unlimited proﬁt if the price rises. Note that the option’s premium is a cost to the producer for creating the ﬂoor. For example, ABC Inc purchases 65-strike put at a premium of $3.50 per barrel. In this case, the put option plays the role of an insurance that guarantees a minimum price of $65 for a barrel of oil. (3) Insuring by selling a call: A written call (selling a cap) sets a maximum price (and therefore limit proﬁt) for the product if the price rises, but does not limit losses if the price declines. The premium received by the producer helps reduce the losses. For example, ABC Inc sells a 65-strike call for a premium of $3.50 per barrel. (4) Creating a collar by purchasing a put at one strike price and writing a call at a higher strike price. A collar sets maximum and minimum prices (the call strike and the put strike, re- spectively) that the producer may realize for its product. The producer is exposed to the risk of variation between these two prices, but is not aﬀected by price variation above or below this range. The put and call options constitute the collar, but in combination with the producer’s long position in the product, they form a bull spread. 75 RISK MANAGEMENT 639 Example 75.1 ABC is a coppermining company with ﬁxed costs $0.75/lb and variable costs of $2.25/lb. (a) If ABC does nothing to hedge price risk, what is its proﬁt 1 year from now, per pound of copper? (b) Suppose that ABC can enter a short forward contract agreeing to sell its copper production one year from now. The 1-year forward price is $3.5/lb. What is the estimated proﬁt one year from now? (c) Suppose that ABC would like to beneﬁt if the price of the copper goes higher than $3.5/lb. It could use options. One year option prices for copper are given below Strike 3.3 3.4 3.5 3.6 3.7 3.8 3.9 Call 0.42567 0.3762 0.33119 0.29048 0.25386 0.22111 0.19196 Put 0.23542 0.28107 0.33119 0.3856 0.44411 0.50648 0.57245 The 1-year continuously compounded interest rate is 5%. Compute the estimated proﬁt in 1 year if ABC buys put options with a strike of $3.7/lb and with a strike of $3.4/lb. (d) Compute the estimated proﬁt in 1 year if ABC sells call options with a strike of $3.7/lb and with a strike of $3.4/lb. (e) Compute the proﬁt in 1 year if ABC buys a 3.6-strike put and sells a 3.9-strike call. Solution. (a) The proﬁt is the one-year spot price minus total cost. That is, P1 −3.A graph of the unhedged position is shown in Figure 75.1. (b) In this case, the proﬁt is P = P1 −3 + (3.5 −P1) = $0 .5. A graph of the hedged position is shown in Figure 75.1. For an uninsured position, the possible losses can be very high. However, by entering the forward, the company has a ﬁxed beneﬁt. Figure 75.1 (c) The proﬁt per pound that results from buying the 3.7-strike put is P1 −3 + max{3.7 −P1,0}−0.44411e0.05. 640 AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES The graph of the proﬁt is given in Figure 75.2. Figure 75.2 The proﬁt per pound that results from buying the 3.4-strike put is P1 −3 + max{3.4 −P1,0}−0.28107e0.05. The graph of the proﬁt is given in Figure 75.2. With the 3.4-strike put, the company makes $0.17139924/lb more than the 3.7-strike put if the price of copper is over $3.7/lb. However, its guaranteed proﬁt is $0.104519233, which is smaller than the guaranteed proﬁt under a 3.7-strike put. Buying a put is like buying insurance against small spot prices. The larger the strike price the larger the price of the insurance. A producer company needs to buy a put with a strike large enough to cover low spot prices. If the strike put is too large, the company will be wasting money in insurance which it does not need. (d) The proﬁt per pound that results from selling the 3.7-strike call is P1 −3 −max{P1 −3.7,0}+ 0.25386e0.05. The graph of the proﬁt is given in Figure 75.3.