Two Asset Options

12 Two Asset Options Before plunging into the details of specific options, we need to take a broad overview of the principles underlying this chapter. In Appendix A.1 we set out the most important properties of normally distributed variables. Two general results are of particular importance in this chapter and it is worth repeating them here: r The sum of two normally distributed variables is itself normally distributed; the mean of the sum of the variables is equal to the sum of the means of the variables. r The variance of the sum of two normally distributed variables is equal to the sum of the individual variances if the two variances are independently distributed. If they are correlated, the variance of the sum is given by σ 2 = σ 2 1 + σ 2 2 + 2ρσ 1 σ 2 where ρ is the correlation coefficient. Consider two stochastic assets with prices at time t equal to S (1) t and S (2) t . Since ln S (1) t and ln S (2) t are normally distributed, ln S (1) t + ln S (2) t = ln S (1) t S (2) t must also be normally distributed. This means that variables such as S (1) t S (2) t and S (1) t /S (2) t are lognormally distributed and much of the theory developed for a single stochastic asset can be used in analyzing the composite asset. By contrast, S (1) t + S (2) t does not have a simple distribution. This composite asset is therefore extremely difficult to analyze and we have no analytical results, even for apparently simple options such as a call on the sum of two stock prices, with payoff max[0, S (1) T + S (2) T − X ]. 12.1 EXCHANGE OPTIONS (MARGRABE) (i) Consider an option on two assets whose initial prices are S (1) 0 and S (2) 0 , which has a payoff max[0, S (1) T − S (2) T ]. This can be interpreted in three ways: r An option to call a unit of asset 1 in exchange for a unit of asset 2. r An option to put a unit of asset 2 in exchange for a unit of asset 1. r A contract to receive a price differential if this is greater than zero. In general this is referred to as an exchange or a spread or an outperformance option. A very simple way of pricing this option is as follows (Margrabe, 1978): from the form of the payoff, it is clear that the value of the option f (S (1) 0 , S (2) 0 ) is homogeneous in S (1) 0 and S (2) 0 . This condition [see Section 11.1(ii)] can be written f  S (1) 0 , S (2) 0  = S (2) 0 f (Q 0 , 1) where Q t = S (1) t S (2) t We can interpret Q 0 as the price of asset S (1) denominated in units of S (2) . f (Q 0 , 1) is then just a call option with a strike price of unity. Let us make the arguments more concrete by taking 12 Two Asset Options a specific example where S (1) 0 is today’s $ price of a barrel of oil and S (2) 0 is today’s $ price of an ounce of gold. The quantity Q 0 is then today’s oil price expressed as ounces of gold per barrel. f (Q 0 , 1) is the value (expressed in ounces of gold) of a call option to buy a barrel of oil for 1 ounce of gold (probably not worth a lot at present rates!). In order to price this option we need to first make a short detour and re-examine some fundamental principles. (ii) Two concepts underlie the notion of risk neutrality: first, which everybody focuses on, is the no-arbitrage principle. The second is so self-evident that it is easy to overlook: if we borrow or deposit cash, then we pay or receive interest. Taking the simplest case of a forward contract, no-arbitrage tells us that if we buy an asset for S 0 and sell it forward for a price F 0T , then the return on the trade must equal the cost of borrowing the cash to buy the asset: F 0T /S 0 = e rT . Of course, if we were able to borrow money for zero interest rate, then we would simply put r = 0 in all our option formulas. In our current example, prices are denominated in a different form of money: not cash, but ounces of gold. Gold is not like cash: there is no gold-bank where you can deposit 3 ounces of gold and have it grow to 4 ounces a few years later. People hold gold because they expect it to go up in price, not because they can earn interest from it. If you borrow gold, there is no gold-interest charged – merely some handling charge, similar in nature to a stock-borrowing cost. Therefore, if gold is used to denominate the price of a commodity and its derivative, we must set the interest rate equal to zero in our formulas. Two further points should be made: first, we have not abandoned risk neutrality. We under- stand that the underlying growth rate in the price of oil (in barrels per gold ounce) is some unknown quantity whose value we do not need to know. We solve our option problem in the usual risk-neutral way, by setting this growth rate equal to the interest rate and present-valuing the option using the interest rate: it just happens that when the money is not cash, the interest rate equals zero. The second point is that the reader should take care not to confuse the forgoing with the role of dividends. Oil and gold dividends do not make much sense, but these commodities do incur storage charges which as we saw in Section 5.5(v) play a role analogous to dividends. If S (1) and S (2) were company stocks, the usual substitutions S (1) 0 → S (1) 0 e −q 1 T and S (2) 0 → S (2) 0 e −q 2 T can be used to account for continuous dividends. (iii) Margrabe’s Formula: An expression for f (Q 0 , 1) can immediately be written down using the Black Scholes formula for a call option. In the standard notation of equation (5.1), with X = 1 and setting r → 0: f (Q 0 , 1) ={Q 0 N[d 1 ] − N[d 2 ]} One final piece of information is needed: a value for σ Q , the volatility of the composite asset Q t = S (1) t /S (2) t S (2) t . An expression for this is derived in Appendix A.1(xi). Generalizing to allow for dividend-paying assets, Margrabe’s formula can now be written f M arg rabe  S (1) 0 , S (2) 0  = S (1) 0 e −q 1 T N[d 1 ] − S (2) 0 e −q 2 T N[d 2 ] (12.1) d 1 = ln Q 0 + 1 2 σ 2 Q T σ Q √ T ; d 2 = d 1 − σ Q √ T ; σ 2 Q = σ 2 1 + σ 2 2 − 2ρ 12 σ 1 σ 2 (iv) Applying the basic risk-free hedging portfolio arguments of Section 4.2, we would expect to replicate an option on two assets by borrowing cash B(S (1) t , S (2) t , t) and investing this in  (1) t 154 12.2 MAXIMUM OF TWO ASSETS and  (2) t units of each stock, i.e. f  S (1) t , S (2) t  =  (1) t S (1) t +  (2) t S (2) t − B  S (1) t , S (2) t , t  ;  (i) t = ∂ f t ∂ S (i) t Euler’s theorem [see Appendix A.12(i)] states that if f (S (1) t , S (2) t ) is homogeneous, then we must have f  S (1) t , S (2) t  = S (1) t ∂ f t ∂ S (1) t + S (2) t ∂ f t ∂ S (2) t = S (1) t  (1) t + S (2) t  (2) t The last two equations taken together mean that B  S (1) t , S (2) t , t  ≡ 0 always We never need to borrow cash, which is of course why r does not appear in Margrabe’s formula: we merely borrow the right amount of one stock and exchange it at the current rate for the other stock; we are then automatically hedged for small movements in the price of either stock. (v) American Options: The homogeneity arguments that led to the adoption of a modified Black Scholes model apply as much to an American option as to European options. f (Q 0 , 1) can therefore be evaluated using one of the numerical procedures for American options, setting r → 0. 12.2 MAXIMUM OF TWO ASSETS (i) Consider an option whose payoff at time T is max[S (1) T , S (2) T ]. The value of this option today can be written f max  S (1) 0 , S (2) 0  = PV  E  S (1) T : S (2) T < S (1) T  + E  S (2) T : S (1) T < S (2) T  (12.2) = f (1 max) + f (2 max) From the symmetry of the terms, we only need to find an expression for one of these in order to write down the other (Stulz, 1982). Taking the second term and using the fact that the option price must be homogeneous in S (1) 0 and S (2) 0 : f (2 max) = S (2) 0 PV[E[1 | Q T < 1]] = S (2) 0 PV[P[Q T < 1]] (ii) P[Q T < 1] is the probability that the price of oil is less than 1 ounce of gold per barrel. A quick glance back to Section 5.2 will show that this is the first term (the coefficient of the strike X) in the Black Scholes model for a put option. We can therefore lift the formula for this directly from our previous work, remembering that the following points apply in this case: r The volatility of Q t is given by σ 2 Q = σ 2 1 + σ 2 2 − 2ρ 12 σ 1 σ 2 , where σ 1 and σ 2 are the $ price volatilities of oil and gold; ρ 12 (or ρ) is the correlation between them [see Appendix A.1(xi) and (xii)]. r The interest rate in any formula we use is set equal to zero [see Section 12.1(ii) above]. (iii) The two terms in the expression for f max (S (1) 0 , S (2) 0 ) in equation (12.2) are completely sym- metrical and may both be obtained using the first term of the Black Scholes formula for a put 155 12 Two Asset Options option, which is given explicitly in Section 5.2. With a minimal amount of algebra, we get f max  S (1) 0 , S (2) 0  = S (1) 0 N[d 1/2 ] + S (2) 0 N[d 2/1 ] d i/j = − ln S (i) 0  S ( j ) 0 + 1 2 σ 2 Q T σ Q √ T ; d 1/2 + d 2/1 = σ Q √ T If the assets pay continuous dividends, we put S (i) 0 → S (i) 0 e −q i T ; i = 1, 2. (iv) Margrabe Again: This last formula can be used to re-derive Margrabe’s result. Consider the following identity for the payoff: max  0, S (1) T − S (2) T  = max  S (1) T , S (2) T  − S (2) T and find the present value of its expected value: f M arg rabe  S (1) 0 , S (2) 0  = f max  S (1) 0 , S (2) 0  − PV  E  S (2) T  This formula must be homogeneous in S (2) 0 and S (1) 0 . The first term on the right-hand side was evaluated in the last subsection. The second term is simply the forward rate, but remember that we are working in units which imply a zero interest rate [see Section 12.1(iii)]. The last term can therefore simply be written S (2) 0 . Using the properties of the cumulative normal distribution given in Appendix A.1 then gives f M arg rabe  S (1) 0 , S (2) 0  = S (1) 0 N[d 1/2 ] + S (2) 0 N[d 2/1 ] − S (2) 0 = S (1) 0 N[d 1/2 ] − S (2) 0 {1 − N[d 2/1 ]} = S (1) 0 N[d 1/2 ] − S (2) 0 N[d 1/2 − σ Q √ T ] 12.3 MAXIMUM OF THREE ASSETS (i) The method of the last section can be extended to three assets. f max (S (1) 0 , S (2) 0 , S (3) 0 ) is today’s value of an option whose payoff at time T is max[S (1) T , S (2) T , S (3) T ]. The value of this option may be written f max  S (1) 0 , S (2) 0 , S (3) 0  = f (1 max) + f (2 max) + f (3 max) = PV  E  S (1) T : S (2) T < S (1) T ; S (3) T < S (1) T  + E  S (2) T : S (1) T < S (2) T ; S (3) T < S (2) T  + E  S (3) T : S (2) T < S (3) T ; S (1) T < S (3) T  This additive pattern reflects a well-known property of probabilities: if three events are mutually exclusive, the probability of all three happening is equal to the sum of the probabilities of any single one happening. As in the two asset case, the option must be homogeneous in S (1) 0 , S (2) 0 and S (3) 0 , so that the first term can be written f (1 max) = S (1) 0 PV  P  Q 2/1 T < 1; Q 3/1 T < 1  where Q i/j t = S (i) t /S ( j ) t . As in the last two sections, all quantities on the right-hand side (except S (1) 0 ) are measured in units of commodity S (1) . We consequently put r → 0 when we perform our risk-neutral calculations, as explained in Section 12.1(ii). The three terms in the equation for f max are completely symmetrical so only one of them needs to be evaluated. 156 12.3 MAXIMUM OF THREE ASSETS (ii) Setting r → 0, the present value discount factor becomes unity, and we see from Appendix A.1 that z i/j t = ln Q i/j t  Q i/j 0 + 1 2 σ 2 i/j t σ i/j √ t is a standard normal variate. Effecting a change of variables in the manner of equations (A1.7), and using the bivariate normal definitions of equation (A1.12) gives P  Q 2/1 T < 1; Q 3/1 T < 1  =  1 0  1 0 F jo int  Q 2/1 T ,Q 3/1 T  dQ 2/1 T dQ 3/1 T  d 2/1 −∞  d 3/1 −∞ n 2  z 2/1 T , z 3/1 T ; ρ 2/1,3/1  dz 2/1 T dz 3/1 T = N 2 [d 2/1 , d 3/1 ; ρ 2/1,3/1 ] d i/j = − ln Q i/j 0 + 1 2 σ 2 i/j T σ i/j √ T = − ln S i 0  S j 0 + 1 2 σ 2 i/j T σ i/j √ T ; σ 2 i/ k = σ 2 i + σ 2 k − ρ ik σ i σ k ρ i/ k, j/k = 1 σ i/ k σ j/k  σ i σ j ρ ij − σ i σ k ρ ik − σ j σ k ρ jk + σ 2 k  The last expression is demonstrated in equations (A1.24). Taking all three terms, we have by symmetry f max  S (1) 0 , S (2) 0 , S (3) 0  = S (1) 0 N 2 [d 2/1 , d 3/1 ; ρ 2/1,3/1 ] + S (2) 0 N 2 [d 1/2 , d 3/2 ; ρ 1/2,3/2 ] + S (3) 0 N 2 [d 1/3 , d 2/3 ; ρ 1/3,2/3 ] (12.3) As usual, continuous dividends can be accommodated by substituting S (i) 0 → S (i) 0 e −q i T for each asset. An important specific case is an option for the maximum of two stochastic assets or cash. We use equation (12.3) but set S (3) 0 e −q 3 T → X e −rT ; σ X = 0 to give f max  S (1) 0 , S (2) 0 , X  = S (1) 0 N 2 [d 2/1 , d X/1 ; ρ 2/1,X/1 ] + S (2) 0 N 2 [d 1/2 , d X/2 ; ρ 1/2,X/2 ] + X e −rT N 2 [d 1/ X , d 2/ X ; ρ 1/ X,2/ X ] (12.4) where the results of equations (A1.24) give σ i/ X = σ X/i = σ i and ρ 1/ X,2/ X = ρ 2/ X,1/ X = ρ 12 ; ρ X/2,1/2 = σ 2 − σ 1 ρ 12 σ 1/2 ; ρ X/1,2/1 = σ 1 − σ 2 ρ 12 σ 2/1 The adaptations to be made to the d i/j are self-evident. The techniques of this and the last section can be extended to larger numbers of assets (Johnson, 1987); the formula for f max will then involve multivariate normal functions of higher order. In practice, correlations between assets tend to be highly unstable – more so than for example volatility. Any derivative which is a function of a correlation therefore needs to be treated with caution. But a derivative whose price is a complicated function of several correlation coefficients probably has little commercial future. 157 12 Two Asset Options 12.4 RAINBOW OPTIONS These are call or put options on the maximum or minimum of two stochastic assets. Their pricing is obtained directly from equation (12.4) (see also Rubinstein, 1991a). (i) Call on the Maximum: This is by far the most commonly encountered rainbow option, and has payoff max  0, max  S (1) T , S (2) T  − X  = max  S (1) T , S (2) T , X  − X This immediately leads us to the formula C(max) = f max  S (1) 0 , S (2) 0 , X  − X e −rT (ii) Put on the Maximum: Regarding max[S (1) T , S (2) T ] as an asset in its own right, put call parity gives Put  max  S (1) T , S (2) T  + max  S (1) T , S (2) T  = Call  max  S (1) T , S (2) T  + X e −rT which leads directly to the formula P(max) = f max  S (1) 0 , S (2) 0 , X  − f max  S (1) 0 , S (2) 0  (iii) Call and Put on the Minimum: Suppose you have calls on two different assets, but someone else has a call on you for the larger of the two assets. What are you left with? Simply a call on the smaller of the two assets. In the notation of this chapter, this is written C(min) = C  S (1) 0  + C  S (1) 0  − C(max) P(min) = P  S (1) 0  + P  S (1) 0  − P(max) 12.5 BLACK SCHOLES EQUATION FOR TWO ASSETS An extension of the Black Scholes differential equation can be derived, which describes an option on two assets. The steps in the derivation follow those of Section 4.2 precisely, and the reader is advised to return to that section in order to follow the amendments below (i) As in the one asset case, we start with the assumption that a portfolio can be constructed, consisting of the derivative and the underlying stocks, in such quantities that the change in value of the portfolio over a small time interval δt is independent of the stock price movements. Otherwise expressed, we can hedge this option with the underlying stocks. The value of the portfolio is written f t − S (1) t  (1) t − S (2) t  (2) t where the sign conventions of Chapter 4 are used (negative means a short position). In the small time interval δt, the value of this portfolio moves by δ f t − δS (1) t  (1) t − δS (2) t  (2) t − S (1) t  (1) t q 1 δt − S (2) t  (2) t q 2 δt Arbitrage arguments tell us that if the portfolio value movement does not depend on the stock price movement, then the rate of return due to this movement (plus any other predictable cash 158 12.5 BLACK SCHOLES EQUATION FOR TWO ASSETS flows) must equal the risk-free return: δ f t − δS (1) t  (1) t − δS (2) t  (2) t − S (1) t  (1) t q 1 δt − S (2) t  (2) t q 2 δt f t − S (1) t  (1) t − S (2) t  (2) t = r δt (12.5) (ii) In order to obtain the generalized Black Scholes equation, we now need to substitute for δS (1) t , δS (2) t and δ f t in the last equation. The two stock prices are assumed to follow the following Wiener processes: δS (1) t = S (1) t (µ 1 − q 1 )δt + S (1) t σ 1 δW (1) t δS (2) t = S (2) t (µ 2 − q 2 )δt + S (2) t σ 2 δW (2) t which immediately gives us two of the terms to substitute. The third term is obtained from Ito’s lemma which needs to be adapted slightly. (iii) Ito’s Lemma for Two Assets: As set out in Section 3.4, Ito’s lemma is based on two elements: 1. The observation that (δW t ) 2 → δt as δt → 0. We use this relationship again, but there is an additional relationship, based on precisely the same reasoning, which states that δW (1) t δW (2) t → ρ 12 δt as δt → 0 where ρ 12 is the correlation between the two Brownian motions. 2. Taylor’s expansion for two assets, making these last substitutions and rejecting all terms of order greater than O[δt] becomes δ f t =  ∂ f t ∂t + (µ 1 − q 1 )S (1) t ∂ f t ∂ S (1) t + (µ 2 − q 2 )S (2) t ∂ f t ∂ S (2) t + 1 2 σ 2 1  S (1) t  2 ∂ 2 f t ∂  S (1) t  2 + ρ 12 σ 1 σ 2 S (1) t S (2) t ∂ 2 f t ∂ S (1) t S (2) t + 1 2 σ 2 2  S (2) t  2 ∂ 2 f t ∂  S (2) t  2  δt + S (1) t σ 1 ∂ f t ∂ S (1) t δW (1) t + S (2) t σ 2 ∂ f t ∂ S (2) t δW (2) t (iv) Having made the necessary substitutions back into equation (12.5), we set the coefficients of δW (1) t and δW (2) t equal to zero, reflecting the fact that the portfolio is perfectly hedged, to give 0 = ∂ f t ∂t + (r − q 1 )S (1) t ∂ f t ∂ S (1) t + (r − q 2 )S (2) t ∂ f t ∂ S (2) t − rf t + 1 2  σ 2 1  S (1) t  2 ∂ 2 f t ∂  S (1) t  2 + σ 2 2  S (2) t  2 ∂ 2 f t ∂  S (2) t  2 + 2ρ 12 σ 1 σ 2 S (1) t S (2) t ∂ 2 f t ∂ S (1) t S (2) t  (12.6) This can be written in more familiar form by making the substitution ∂/∂t =−∂/∂T where T is the time to maturity of an option [see Section 1.1(v)], and setting t = 0: ∂ f 0 ∂T = (r − q 1 )S (1) 0 ∂ f 0 ∂ S (1) 0 + (r − q 2 )S (2) 0 ∂ f 0 ∂ S (2) 0 − rf 0 + 1 2  σ 2 1  S (1) 0  2 ∂ 2 f 0 ∂  S (1) 0  2 + σ 2 2  S (2) 0  2 ∂ 2 f 0 ∂ S (2) 2 0 + 2ρ 12 σ 1 σ 2 S (1) 0 S (2) 0 ∂ 2 f 0 ∂ S (1) 0 S (2) 0  (12.7) 159 12 Two Asset Options 12.6 BINOMIAL MODEL FOR TWO ASSET OPTIONS (i) An extension of the now familiar binomial tree to three dimensions is shown in Figure 12.1. For the sake of simplicity, we use the space variables x t = ln S (1) t /S (1) 0 and y t = ln S (2) t /S (2) 0 , rather than working directly with stock prices. This means that the step sizes (up/down and left/right) are of constant sizes, rather than proportional to the stock values; the first node in the tree has value zero. A tree of this type is described by the basic arithmetic random walk in Appendix A.2. Equation (3.5) shows that if the risk-neutral drift of S (1) t is r − q 1 , then the drift of x t is r − q 1 − 1 2 σ 2 1 = m x , and similarly for y t . t x t t y Figure 12.1 Binomial tree for two assets (ii) Uncorrelated Assets: Consider the first cell in this tree. Figure 12.2 shows this cell looking at the pyramid from the apex. In the middle of the rectangle we have the starting node with x 0 , y 0 = 0. From this point we move a time step of length δt and x 0 and y 0 move to one of the four combinations whose values are given at the corners of the rectangle. In this simple case, x δ t can take two values: x u and x d ; similarly, y δ t takes values y r and y l . This means that the movement in asset 2 is the same whether asset 1 moves up or down, i.e. the two asset prices are uncorrelated. 00 x,y ul x,y 1 4 p= 1 4 p= 1 4 p= 1 4 p= x , y d r x,y dl x,y 00 x,y u r Figure 12.2 Single binomial cell: uncorrelated It is shown in Appendix A.2 and in Chapter 7 that with a binomial model for a single un- derlying asset, we have discretion in choosing nodal values and the probabilities of up- and down-jumps. This is also the case with a three-dimensional tree, and we will choose the transition probability to each corner to be 1 4 (cf. the Cox–Ross–Rubinstein discretiza- tion of the simple binomial tree with p = 1 2 ). Our task now is to find each of the nodal values corresponding to these probabilities (Rubinstein, 1994). 160 12.6 BINOMIAL MODEL FOR TWO ASSET OPTIONS Using the approach of Section 7.1(iv), the Wiener processes for x t and y t are written δx t = m x δt + σ x √ δtz 1 δy t = m y δt + σ y √ δtz 2 where z 1 and z 2 are uncorrelated standard normal variates. Matching local drifts and volatilities to the tree, and using equation (A2.5) of the Appendix means that we can write E[δx t ] = m x δt = 1 2 (x u + x d ) var[δx t ] = σ 2 x δt = 1 4  x 2 u + x 2 d  − E 2 [δx t ] = 1 4 (x u − x d ) 2 which solves to x u = m x δt + σ x √ δt; x d = m x δt − σ x √ δt Similarly y r = m y δt + σ y √ δt; y l = m y δt − σ y √ δt 00 x,y u x 1 4 p= 1 4 p= 1 4 p= 1 4 p= u a x,y d b x,y d x 0 0 x,y d ,y ,y g Figure 12.3 Single binomial cell: correlated (iii) Correlated Assets: It is much more difficult to find the nodal values in this case since the value of y δ t will depend on the value of x δ t . In graphical terms, the grid becomes squashed so that each cell when viewed from the apex turns into the parallelogram of Figure 12.3. As before, we exercise the discretion we are allowed, first by choosing the transition prob- abilities to each corner to be 1 4 , and second by only allowing x δ t to have two values: x u and x d . This time, y δ t takes different values at each corner. The Wiener processes can be written δx t = m x δt + σ x √ δtz 1 δy t = m y δt + σ y √ δtz 2 = m y δt + σ y √ δt{ρz 1 +  1 − ρ 2 z 3 } where z 1 and z 3 are independently distributed, standard normal variates [see Appendix A.1(vi)]. A heuristic argument might be made that in the first of these two processes, x u and x d are obtained by putting z 1 → 1 and z 1 →−1 respectively. Similarly, putting z 3 equal to ±1 corresponding to each of the values for z 1 gives x u = m x δt + σ x √ δt; x d = m x δt − σ x √ δt and δy α = m y δt + σ y √ δt{ρ +  1 − ρ 2 } δy β = m y δt − σ y √ δt{ρ −  1 − ρ 2 } δy γ = m y δt − σ y √ δt{ρ +  1 − ρ 2 } δy δ = m y δt + σ y √ δt{ρ −  1 − ρ 2 } 161 12 Two Asset Options So much for the flaky argument: a proper confirmation that these are indeed the correct ex- pressions is obtained by substituting them in the following defining equations: Eδy t =m y δt varδy t =σ 2 y δt covδx t δy t =ρσ x σ y δt (iv) Payoff values of the option can be calculated for each final node since each of these contains a value for x T and y T . Discount these back through the tree in the normal way, remembering that the values at four nodes are needed for each step back (rather than two in the single asset tree); probabilities are all set at p = 1 4 . For American options, derivative values at each node are replaced by exercise value if necessary. (v) Alternatives to Trees: It seems that we can extend this tree to higher dimensions to calculate options on three or more assets, but this is not really practical for three reasons: r Correlations in finance are extremely unstable, except for a very special case discussed in Chapter 14. Calculations involving correlation between the prices of two stocks are useful, but must be treated with extreme caution. Three-way correlation just compounds the instability of results to the point where they have little practical use. r The mental agility needed to analyze N-dimensional trees is discouraging. r There are deep theoretical reasons why the efficiency of a tree drops off sharply with an increasing number of dimensions: see Section 10.1(iii). An example is given in Chapter 10 of the pricing of a two asset spread option using quasi-Monte Carlo. This method is very quick and accurate, and can readily be extended to several assets. 162 [...]... euros There are two issues to be decided: first, should the New York or the Frankfurt office handle the business, i.e is the option better regarded as a euro option or as a dollar option? Second, how is it priced? We will see that some options can be regarded as either € or $ options, although the analysis is different depending on the approach But one thing is certain: if there are two approaches, they... be the same as they would be for f t Note that the payoff FT (= f T ) is completely independent of ψt so that FT is independent of ψT f quanto (t) is a € asset dependent on two stochastic prices, and so satisfies the Black Scholes equation for two assets in the form given by equation (12.6), with r → r€ ; St(1) → St ; q1 → q; St(2) → ψt ; q2 → r S , i.e 0= ∂ f quanto (t) ∂ f quanto (t) ∂ f quanto (t)... payoff is to be in € or $; k is just a scaling constant These are clearly Margrabe options where one of the assets is an ADR type security If the fund manager were neither in Germany nor the US, but in the UK, the payoff might be required in sterling: (£/$) £ max 0, (S&P)T φT £/€ − (CAC)T φT Again, Margrabe is used, but between two ADR type securities r Alternatively, the payoff may be defined as max 0, k... payoff may be defined as max 0, k (S&P)T φT − X (DAX)T Here we use a modification of the Black Scholes equation for the quotient of two $-securities, (S&P)t , and (DAX)t /φt The latter must in turn be decomposed into two lognormal random variables Note that each of the above options involves correlations between three or four random variables Each of these correlations is likely to be unstable over time...13 Currency Translated Options 13.1 INTRODUCTION (i) These options do not require much new theory, yet newcomers tend to find them fairly tricky It is therefore best to make the analysis as concrete as possible Consider an international investment... a large part of its assets in the US will have a stock price which is strongly correlated with the €/$ exchange rate But apart from such obvious cases, the observed correlation coefficient usually flops about, often changing sign as well as magnitude We have developed nice pricing formulas containing a constant correlation coefficient, while in fact it is a random variable There are two ways in which this... quantoed into pesos, the Mexican buyer can tell himself that he is not only eliminating FX risk but getting his option cheaper as well! (iii) Index Outperformance Options: These are fairly popular in the investment community and are related to the options analyzed above Consider an option to take advantage of the outperformance of the US S&P index over the German DAX r The payoff might be fixed as €max[0,... Translated Options mixed up in a non-trivial way so it is not clear whether we should regard this as a $ or a € option; nor is it apparent whether the transaction is better handled in Frankfurt or New York We will examine it from both viewpoints (i) A € View: The Frankfurt office of our bank sees an option whose payoff is max[0, € ST − € K ψT ] This is an option to exchange one € stochastic asset (K times... expressed in € at maturity is a pure € quantity: ¯ ¯ ¯ €ψT max[0, ST − X ] φ = max[0, ψT ST − X ψT ] φ = max[0, UT − X ψT ]φ Once again, we have a Margrabe’s option: the exchange of stochastic asset UT for stochastic asset X ψT The price of this option is ¯ € f quanto = € U0 e−qU T N[d1 ] − X ψ 0 e−r$ T N[d2 ] φ 2 ln U0 e−qU T /X ψ 0 e−r$ T + 1 σU/ X ψ T 2 d1 = √ σU/ X ψ T √ 2 2 2 σU/ X ψ = σU + σ X ψ... forms, it is actually insoluble analytically The problem is that the difference of two lognormal random variables is not itself lognormal; by contrast, the product is lognormal The reader is referred to Chapter 10 on Monte Carlo (particularly quasi-Monte Carlo) for quick and efficient numerical methods for pricing these options 168 . commercial future. 157 12 Two Asset Options 12.4 RAINBOW OPTIONS These are call or put options on the maximum or minimum of two stochastic assets. Their pricing. (2) 0 ∂ 2 f 0 ∂ S (1) 0 S (2) 0  (12.7) 159 12 Two Asset Options 12.6 BINOMIAL MODEL FOR TWO ASSET OPTIONS (i) An extension of the now familiar binomial