Likewise, the payoff of the better-of-two note may be decomposed as follows: Price of gold (US$/oz) Oil price

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1 Exchange Options Consider the Double Index Bull (DIB) note, which is suited to investors who believe that two indices will rally over a given term. The note typically pays no coupons and has a redemption value linked to the lower value of the two indices at maturity. More exactly, the redemption value is gearing min{index 1, index }, where index 1 and index are the dollar values of the two indices on the maturity date of the note and gearing is the degree of leverage of the note. Generally, the two indices are standardized to have identical forward values. This structure is closely related to the better-of-two note, whose redemption value is linked to the higher value of the two indices and defined as gearing max{index 1, index }. The important difference between the two notes lies in the ownership of the embedded option. This may be seen by restating the redemption profile for each. In the case of the DIB note, the redemption payoff may be decomposed as follows: gearing min{index 1, index } = gearing [ index 1 max{0, index 1 index } ]. Likewise, the payoff of the better-of-two note may be decomposed as follows: gearing max{index 1, index } = gearing [ index 1 + max{0, index index 1 } ]. Embedded in the DIB note is the sale, on the part of the investor, of an exchange option. The option premium is used to increase the gearing of the note. On the other hand, the better-of-two note embeds the purchase of an exchange option. The option premium, in this case, decreases the gearing of the note. As a result, the DIB note will pay in excess of 100% of the upside of the minimum of the two indices, whereas the better-of-two note will pay less than 100% of the maximum of the two indices. For example, let us consider an investor who has a bullish view on both oil and gold. In such a case, it is possible to structure a note with US$0 million face value and three-year term, which pays no coupon and has a redemption value of 107% of the minimum value of, ounces of gold and 1,075,350 barrels of oil. The payoff of the note for various gold and oil scenarios is provided below. Price of gold (US$/oz) Oil price (US$/barrel) Redemption value (million)

2 The investor receives full principal repayment if the price of oil per barrel is greater than US$17.38 and the price of gold per ounce is greater than US$0.57. For an investor who is bullish on both indices, this structure offers a higher expected return than an equal-weighted portfolio in the two indices. Specifically, this transaction will provide a return of 17.70% if both indices rally 10% (see table). 1 Payoff Profile An exchange option refers to an option structure whereby the purchaser has the right to exchange one asset for another. In the case of an option to exchange the first asset for the second, the payoff is given by max{0,s (T) S 1 (T)}, where S i (T ) is the spot price of asset i (i =1,) on expiration. As indicated in the opening example, the payoff of an exchange option is related to the payoff of better-of-two or worse-of-two options as follows: The identity max{0,s (T) S 1 (T)} = max{s 1 (T ),S (T)} S 1 (T) is useful for pricing better-of-two options. The identity max{0,s (T) S 1 (T)} = S (T) min{s 1 (T ),S (T)} is useful for pricing worse-of-two options. We return to this point when dealing with outperformance options. Valuation We can price an exchange option by taking the risk-neutral valuation approach. 1 This equates to evaluating the following discounted expectation: C = e rτ E [ max{0,s (T) S 1 (T)} ] {S =e E[ rτ (T) S 1 (T) } ] I {S (T)>S 1 (T )} { [ ] [ ]} = e rτ E S e Y (T) I {S e Y (T ) >S 1 e X(T ) } E S 1 e X(T ) I {S e Y (T ) >S 1 e X(T ) }. (1) We will now demonstrate how to evaluate the first expectation in (1). Our notation will be consistent with the handout Correlation Options: Preliminary Results. First we relate X and Y to Z 1 and Z (Z 1 and Z being constructed from X and Y to have a bivariate standard normal distribution): X = µ X + Z 1 σ X Y =µ Y +Z 1 σ Y +Z σ X, Z σ Y. The event {S e Y (T ) >S 1 e X(T) }can be expressed in the form Z <A+BZ 1, where A = ln(s /S 1 )+µ Y µ X (σ Y +σ X ) (1 ρ)/ and B = σ Y σ X σ Y + σ X. () 1 The value of an exchange option can also be obtained by direct arbitrage-free arguments similar to the original Black-Scholes derivation. This approach would involve constructing a portfolio consisting of the option and the two underlying assets such that the return on the portfolio is riskless. This leads to a partial differential equation (PDE) that is satisfied by the option value; solving the PDE under appropriate boundary conditions yields the option value. Margrabe [1] demonstrates how this is done.

3 Thus, the first expectation in (1) can be written as [ A+Bz1 ] I 1 = S e µ Y exp z 1 σ Y z σ Y n(z 1 )n(z )dz dz 1. (3) By completing the squares, we have [ ] ( ) exp z 1 σ Y n(z 1 )=e σ Y ()/ n z 1 σ Y, [ ] ( ) 1 ρ 1 ρ exp z σ Y n(z )=e σ Y ()/ n z + σ Y. It now follows from (3) through a change of variable to w 1 = z 1 σ Y (1 + ρ)/ and w = z + σ Y (1 ρ)/ that a+bw1 ( ) I 1 = S e µ Y +σy / a n(w 1 ) n(w ) dw dw 1 = N, 1+b where a = A + Bσ Y +σ Y and b = B. Referring to () to work out the ratio a/ 1+b leads to this final expression: I 1 = S e (r q )τ N(d 1 ), where d 1 = ln(s /S 1 )+(q 1 q +σ a/)τ σ a τ and σ a = σ 1 + σ ρσ 1 σ. It can be shown by a similar argument that the second expectation in (1) equates to I = S 1 e (r q 1)τ N(d ) where d = d 1 σ a τ. Therefore, collecting terms, the price of the exchange option is C = S e q τ N(d 1 ) S 1 e q 1τ N(d ), () where S i is the spot price of asset i, τ the time to expiration, q i the dividend yield of asset i, σ i the volatility of asset i, and ρ the instantaneous correlation between the asset returns. Note that the option price given in () can be written in the following manner: C = S e qτ N(d 1 ) e q1τ N(d ). S 1 S 1 This expression admits an interpretation in terms of a change of numeraire from cash (rate of return r) to asset 1 (rate of return q 1 ). That is, if prices are denominated by units of asset 1, the price of the exchange option is given by the familiar Black-Scholes formula with spot price S /S 1, unit strike price, riskless rate q 1, dividend rate q, and volatility σ a. This last observation is consistent with the following restatements of the exchange option: A call (resp. put) option on asset (resp. asset 1) with a strike price equal to the future value of asset 1 (resp. asset ). 3

4 Example: Suppose that there are two stocks with the spot prices S 1 = $100, S = $100, the volatilities σ 1 = 15%, σ = 0%, the yields q 1 = %, q = 5%, and the two stock returns are correlated with correlation coefficient ρ = 75%. What is the price of an option to exchange the first stock for the second stock in one year? For the given market values, we find that σ a = 0.133, d 1 = 0.006, d = 0.117, N(d 1 )=0.96, and N(d )=0.36. Therefore, the price of the exchange option is 100e e = $ Hedging As a direct consequence of our discussion in Section 1, a short position in an exchange option to exchange the first asset for the second can be hedged statically by using one of the following portfolios of securities: Long a better-of-two-assets option and short the first asset. Long the second asset and short a worse-of-two-assets option. Alternatively, we can hedge such a short position dynamically. Specifically, we need to be long 1 units of asset 1 and units of asset for every unit of option short. Here, 1 and are the partial deltas, given by 1 = C S 1 = e q 1τ N(d ) and = C S = e q τ N(d 1 ). We have used the identity S e qτ n(d 1 )=S 1 e q1τ n(d ). It is easy to see that 1 < 1 < 0 and 0 < < 1. The analysis of price sensitivities for correlation options encompasses sensitivity to the correlation coefficient ρ. Intuitively, a higher correlation coefficient reduces the aggregate volatility σ a which is positively related to the value of an exchange option and this leads to a lower option premium. We demonstrate this observation formally: C ρ = S e qτ n(d 1 ) d 1 ρ S 1e q1τ n(d ) d ρ = S e qτ n(d 1 ) σ 1σ τ, σ a noting that σ a / ρ = σ 1 σ /σ a. Variations In the basic exchange option the assets are given the same weightage, hence the payoff is the larger of zero and S (T ) S 1 (T ). The first variation allows for different weights on the two assets, so the payoff can be modified to max{0,a S (T) a 1 S 1 (T)} where a 1 and a are positive constants. By an adaptation of the valuation in Section, such an option can be priced relatively easily. We can also incorporate a strike price into the structure of the option, giving rise to the spread option (to be discussed separately). The payoffs of spread options are given below for reference: Spread call option: max { 0, [ a S (T ) a 1 S 1 (T ) ] K }. Spread put option: max { 0,K [ a S (T) a 1 S 1 (T) ]}.

5 There is yet another variation of the basic exchange option that involves the exchange of multiassets. We give an example to illustrate the structure. In 1999, Argentina issued a bond with an embedded option. This option allowed the investor to swap one of two assets A or B for one for two assets C or D. Naturally, the investor would choose to trade the lower-priced asset between A and B for the higher-priced asset between C and D. The investor would only do so if there is a positive difference between them. Thus, the investor s payoff from the option could be written as 5 Summary max { 0, max{c, D} min{a, B} }. Exchange options are the basic correlation options which can be used to analyze and price many other correlation options. They are also the simplest correlation options because their payoff patterns are the simplest. Because of that, their prices can be expressed in terms of univariate normal cumulative distribution functions. To a certain degree, exchange options have been superceded by more complex multifactor option structures such as spread options, which have similar characteristics. 6 Problems 1. Margrabe [1] discussed four applications of exchange options. As an example consider the following situation. An adviser receives a performance incentive fee R m R s multiplied by a fixed percentage of the total managed portfolio, where R m and R s stand for the returns of the managed portfolio and a standard portfolio against which the performance is measured, respectively. If the adviser has the protection of limited liability in case the fee became negative, calculate the management fee if the current managed portfolio and the standard portfolio returns are 10% and 5%, respectively, the volatilities of the two returns are both 10%, the fee arrangement lasts for one year, the two portfolios are correlated with a correlation coefficient 50%, and the fixed percentage is 15% of the total managed portfolio of 10 million dollars.. Refer to the example in Section. (a) What is the price of an option to exchange the second stock for the first stock in one year? Comment. (b) What are the highest and lowest prices of the option to exchange the first stock for the second in one year as the correlation coefficient varies? 3. Derive the pricing formula for a general exchange option whose payoff is given by max{0,a S (T) a 1 S 1 (T)} where a 1 and a are constants. References [1] Margrabe, W. (1978). The Value of an Option to Exchange One Asset for Another. Journal of Finance, 3 (Mar 78),