Multi-Asset Options. Mark Davis Department of Mathematics, Imperial College London. February 27, 2003

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1 Multi-Asset Options Mark Davis Department of Mathematics, Imperial College London February 27, 2003 This note covers pricing of options where the exercise value depends on more than one risky asset. Section 1 describes a very useful formula for pricing exchange options, while section 2 gives a model for the FX market, where the option could be directly an FX option or an option on an asset denominated in a foreign currency. 1 The Margrabe Formula This is an expression, originally derived by Margrabe [1], for the value C = E[e rt max(s 1 (T ) S 2 (T ), 0)] of the option to exchange asset 2 for asset 1 at time T. It is assumed that under the risk-neutral measure P, S 1 (t) and S 2 (t) satisfy ds 1 (t) = rs 1 (t)dt + σ 1 S 1 (t)dw 1, S 1 (0) = s 1 (1) ds 2 (t) = rs 2 (t)dt + σ 2 S 2 (t)dw 2, S 2 (0) = s 2, (2) where w 1, w 2 are Brownian motions with E[dw 1 dw 2 ] = ρdt. The riskless rate is r. The Margrabe formula is C(s 1, s 2 ) = s 1 N(d 1 ) s 2 N(d 2 ) (3) where N( ) is the normal distribution function, 1.1 The Probabilistic Method d 1 = ln(s 1/s 2 ) σ2 T σ T d 2 = d 1 σ T σ = σ1 2 + σ2 2 2ρσ 1σ 2 (4) First, C does not depend on the riskless rate r since S i (t) = e rt Si (t), i = 1, 2, where S i (t) is the solution to (1),(2) with r = 0, and hence C = E[max( S 1 (T ) S 2 (T ), 0)] = E[ S 2 (T ) max( S 1 (T )/ S 2 (T ) 1, 0)] (5) 1

2 Henceforth, take r = 0 so that S i (t) = S i (t). By the Ito formula, Y (t) = S 1 (t)/s 2 (t) satisfies and dy = Y (σ 2 2 σ 1 σ 2 ρ)dt + Y (σ 1 dw 1 σ 2 dw 2 ) (6) 1 s 2 S 2 (T ) = exp(σ 2 w 2 (T ) 1 2 σ2 2T ) is a Girsanov exponential defining a measure change Thus from (5) By the Girsanov theorem, under measure P the process d P dp = 1 s 2 S 2 (T ). (7) C = s 2 Ẽ[max(Y (T ) 1, 0)]. (8) d w 2 = dw 2 σ 2 dt is a Brownian motion. We can write w 1 as w 1 (t) = ρw 2 (t) + 1 ρ 2 w (t) where w (t) is a Brownian motion independent of w 2 (t) (under measure P ). You can check that with P defined by (7), w remains a Brownian motion under P, independent of w 2. Hence d w 1 defined by d w 1 = ρd w 2 (t) + 1 ρ 2 dw (t) = dw 1 (t) ρσ 2 dt is a P -Brownian motion. The equation for Y under P turns out miraculously to be which we can write dy = Y (σ 1 d w 1 σ 2 d w 2 ) dy = Y σdw, (9) where w is a standard Brownian motion and σ is given by (4). In view of (8), (9) the exchange option is equivalent to a call option on asset Y with volatility σ, strike 1 and riskless rate 0. By the Black-Scholes formula, this is (3). 1.2 The PDE Method This follows the original Black-Scholes perfect hedging argument. This time we work under the objective probability measure, under which S 1 and S 2 have drifts µ 1, µ 2 (rather than the riskless drift r)in (1),(2). Form a portfolio X t = C α 1 S 1 α 2 S 2 α 3 P, (10) where P (t) = exp( r(t t)) is the zero-coupon bond and the hedging component is selffinancing, i.e. satisfies d(α 1 S 1 + α 2 S 2 + α 3 P ) = α 1 ds 1 + α 2 ds 2 + α 3 dp. 2

3 Assuming C(t, s 1, s 2 ) is a smooth function and writing C 1 = C/ S 1 etc we have by the Ito formula, dx t = C t + C 1 ds C 11σ 2 1S 2 dt + C 2 ds C 22σ 2 2S 2 2dt +C 12 ρσ 1 σ 2 S 1 S 2 dt α 1 ds 1 α 2 ds 2 α 3 rp dt. If we choose this reduces to α 1 = C 1, α 2 = C 2 dx t = (C t C 11σ 2 1S C 22σ 2 2S C 12 ρσ 1 σ 2 S 1 S 2 α 3 rp )dt (11) and by standard no-arbitrage arguments X t must grow at the riskless rate, i.e. satisfy We see that (10),(11),(12), are satisfied if C satisfies the PDE dx t = rx t dt. (12) C t C 11σ 2 1s C 22σ 2 2s C 12 ρσ 1 σ 2 s 1 s 2 = rc rc 1 s 1 rc 2 s 2 (13) (The terms involving α 3 cancel.) Now from the definition of C and the price processes (1),(2) it is clear that C satisfies λc(s 1, s 2 ) = C(λs 1, λs 2 ), for any λ > 0, so taking λ = 1/s 2 we have C(t, s 1, s 2 ) = s 2 f(t, s 1 /s 2 ) (14) where f(t, y) = C(t, y, 1). We can now calculate derivatives of C in terms of those of f; for example C 1 = f y, C 2 = f s 1 f s 2 y. From these expressions we see that s 1 C 1 + s 2 C 2 = C, so that the right-hand side of (13) is equal to zero for any C satisfying (14). This removes the dependence on r in (13). Substituting the remaining expressions into (13) we find that (13) is equivalent to the following PDE for f: where σ is as defined above. The boundary condition is f t y2 σ 2 2 f = 0, (15) y2 f(t, y) = C(T, y, 1) = max(y 1, 0). (16) But (15) (16) is just the Black-Scholes PDE whose solution is (3). Since the left-hand side of (13) is equal to zero, we see from (11) that X t 0 if and only if X 0 = C(0, S 0 ) and α 3 0. The hedging strategy places no funds in the riskless asset. 3

4 1.3 Exercise Probability As in Section 1.1, define Y (t) = S 1 (t)/s 2 (t). We see from (5) that exercise takes place when Y (T ) > 1, and under the risk-neutral measure Y (t) satisfies (6). Hence the forward is F = (s 1 /s 0 ) exp((σ 2 2 σ 1σ 2 ρ)t ), and by standard calculations where Risk-neutral probability of exercise = P [Y (T ) > 1] = N( ˆd 2 ), ˆd 2 = ln(f ) 1 2 σ2 T σ T = ln(s 1/s 2 ) (σ2 2 σ2 1 )T σ T Note that the exercise probability is not N(d 2 ), which is the exercise probability under the transformed measure P. 2 Cross-Currency Options This section concerns valuation of an option on an asset S t denominated in currency F (for foreign ) which pays off in currency D (for domestic ). We denote by f t the exchange rate at time t, interpreted as the domestic currency price of one unit of foreign currency. Thus the currency D value of the asset S t is f t S t at time t. In this note we ignore interest-rate volatility and take the foreign and domestic interest rates as constants r F, r D respectively, so that the corresponding zero-coupon bonds have values 2.1 Forward FX rates P F (t, T ) = e r F (T t) P D (t, T ) = e r D(T t). To deliver one unit of currency F at time T, we can borrow f 0 P F (0, T ) units of domestic currency at time 0 and buy a foreign zero-coupon bond maturing at time T. At that time the value of our short position in domestic currency is f 0 P F (0, T )/P D (0, T ). By standard arguments, an agreement to exchange K units of domestic currency for one unit of currency F at time T is arbitrage-free if and only if K = f 0 P F (0, T )/P D (0, T ). In summary: Forward price = f 0 e (r D r F )T. This coincides with the formula for the forward price of a domestic asset with dividend yield r F. 2.2 The domestic risk-neutral measure The traded assets in the domestic economy are the domestic zero-coupon bond, value Z t = P D (t, T ), the foreign zero-coupon bond, value Y t = f t P F (t, T ), and the foreign asset, value X t = f t S t. An analogous set of assets is traded in the foreign economy. It is important to realize that there are two risk-neutral measures, depending on which economy we regard as home. 4.

5 We will assume that in the domestic risk-neutral (DRN) measure the asset S t is log-normal, i.e. satisfies ds t = S t µdt + S t σ S dw S (t) (17) for some drift µ and volatility σ S. The asset is assumed to have a dividend yield q. Similarly the FX rate f t is log-normal: df t = f t γdt + f t σ f dw f (t), (18) with drift γ and volatility σ f. w S and w f are Brownian motions with Edw S dw f = ρ dt. The discounted domestic value of the foreign zero-coupon bond is e r Dt f t P F (t, T ) = e r F T f t e (r D r F )t. This is a martingale in the DRN measure, which is true if and only if γ = r D r F. (19) Now consider a self-financing portfolio of foreign assets in which we hold φ t units of asset S t and keep the remaining value in foreign zero-coupon bonds. The portfolio value process V t then satisfies dv t = φ t ds t + qφ t S t dt + (V t φ t S t )r F dt. = V t r F dt + φ t S t (µ + q r F )dt + φ t S t σ S dw S t. Using (18),(19) and the Ito formula we find that the domestic value U t = f t V t of this portfolio satisfies du t = r D U t dt + σ f U t dw f t + φ tf t S t σ S dw S t + φ t f t S t (µ + q r F + ρσ S σ f )dt. Again, the discounted value e r Dt U t is a martingale in the DRN measure, and this holds if and only if µ = r F q ρσ S σ f. (20) In summary, under the DRN measure the FX rate and asset value satisfy the following equations df t = f t (r D r F )dt + f t σ f dw f (t) (21) ds t = S t (r F q ρσ S σ f )dt + S t σ S dw S (t) (22) By applying the Ito formula to (21),(22) we find that X t := S t f t, the asset price expressed in domestic currency, satisfies By computing variances we find that dx t = X t (r D q)dt + X t (σ S dw S (t) + σ f dw f (t)). (23) σ S w S (t) + σ f w f (t) = σw(t), (24) where w(t) is a standard Brownian motion and σ = σs 2 + σ2 f + 2ρσ Sσ f (25) E[dwdw f ] = 1 σ (σ f + ρσ S ). (26) Thus (23) becomes dx t = X t (r D q)dt + X t σdw(t). (27) 5

6 2.3 Option Valuation Options on Foreign Assets This refers to, for example, a call option with value at maturity time T max[x T K, 0], i.e. the foreign asset value is converted to domestic currency at the spot FX rate f T and compared to a domestically-quoted strike K. Since X t satisfies (27) we see that the option value is just the Black-Scholes value for a domestic asset with volatility σ given by (26) Currency-Protected (Quanto) Options Here the option value at maturity is A 0 max[s T K, 0] units of domestic currency, where A 0 is an arbitrary exchange factor, for example the time-zero exchange rate. The option value at time zero is A 0 e r DT E(max[S T K, 0]). The expectation is taken under the DRN measure, in which S t satisfies (22). Note that the volatility is σ S and the drift is r F q ρσ S σ f = r D (q + r D r F + ρσ S σ f ). We can therefore calculate the option value in two equivalent ways: (i) Use the forward form of the BS formula with forward F T = S 0 exp((r F q ρσ S σ f )T ) and discount factor exp( r D T ). (ii) Use the stock form of BS with riskless rate r D and dividend yield q + r D r F + ρσ S σ f. 2.4 Hedging Quanto Options Deriving the Hedge The value of the quanto option given above has the usual interpretation as the initial endowment of a perfect hedging portfolio, but the formula does not indicate how the hedging takes place. To discover this, we re-derive the formula using the traditional Black-Scholes perfect hedging argument. For this we use the objective probability measure - not the risk-neutral measure - under which X t = S t f t and f t are log-normal processes satisfying dx t = λx t dt + σx t d w(t) (28) df t = υf t dt + σ f f t dw f (t) (29) for some drift coefficients λ, υ the value of which, it turns out, we do not need to know. The point about the hedging argument is that from the perspective of a domestic investor, S t itself is not a traded asset: the traded assets are X t (the domestic value of S t ) and the foreign and domestic bonds Y t, Z t. From (29), the equation satisfied by Y t is dy t = (υ + r F )Y t dt + σ f Y t dw f. (30) 1 Note that the sign of ρ would be reversed if we had written the FX model in terms of 1/f t rather than f t. 6

7 We know from section that the quanto call option value at time t is a function C(t, S t ) = C(t, X t /f t ) but we need to regard it as a function of X t, f t separately for hedging purposes. Note that if we define g(t, x, f) := C(t, x/f) then with C = C/ S we have g = C t t (31) g x = 1 f C (32) g f = x f 2 C (33) 2 g x 2 = 1 f 2 C (34) 2 g f 2 = 2x f 3 C + x2 f 4 C (35) 2 g x f Let C(t, S t ) be the call value and consider the portfolio = 1 f 2 C x f 3 C (36) V t = C(t, X t /f t ) φ t X t ψ t Y t χ t Z t, (37) where φ t, ψ t, χ t are the number of units of X t, Y t, Z t respectively in the putative hedging portfolio. Recall that S (and hence X) pays dividends at rate q. Applying the Ito formula using (31) - (36) and (28),(29) and then substituting S = X/f we eventually obtain ( C dv t = t SC (v + ρσ S σ f ) + 1 ) 2 σ2 SS 2 C ψ(υ + r F )Y t χr D Z t dt +( 1 f C φ)dx φqxdt (ψy t + SC )σ f dw f Taking φ = 1 f C (38) this becomes dv t = ψ = 1 Y SC (39) ( C t + SC (r F y ρσ S σ f ) + 1 ) 2 σ2 Ss 2 C χr D Z t dt (40) The usual no-arbitrage argument implies that V t must grow at the domestic riskless rate, i.e. dv t = V t r D dt and, from (40) and (41), this equality is satisfied if C satisfies = (C C f X + SC Y Y χz)r Ddt = (C χz)r D dt (41) C t + SC (r F q ρσ S σ f ) σ2 SS 2 C r D C = 0. (42) 7

8 The boundary condition is If we write C(T, s) = A 0 [s K] + (43) r F q ρσ S σ f = r D (q + r D r F + ρσ S σ f ), we can see that (42),(43) is just A 0 times the Black-Scholes PDE with volatility σ S, riskless rate r D and dividend yield (q + r D r F + ρσ S σ f ). This agrees with the valuation obtained in Section We still have to check the two key properties of the hedging portfolio, namely perfect replication and self-financing. The former is obtained by suitably defining χ t ; from (41), V t 0 if χ t = C(t, S t) Z t. (44) To check the latter, note that the hedging portfolio value is W = φx + ψy + χz and we now know that this is equal to the option value C(t, S t ). Hence dw = ( dc C = t + 1 ) 2 σ2 SS 2 C dt + C ds = r D Cdt SC (r F q ρσ S σ f )dt + C ds. (45) (The second line is just an application of the Ito formula and the third uses the Black-Scholes PDE (42).) Now S = X/f, so using (28), (29) we get from the Ito formula Thus ds = 1 f dx S df f ρσ f σ S Sdt. dw = r D Cdt SC (r F q ρσ S σ f )dt + C f dx C S f df ρσ Sσ f C Sdt ( = r D Cdt SC r F dt + df ) + qsc dt + C dx (46) f f Using the definitions of φ, ψ and χ at (38),(39),(44) we see that the first, third and fourth terms of (46) are equal to χdz, φqxdt and φdx respectively. Now Y t = e r F (T t) f t, so dy t = r F Y t dt + e r D(T t) df t = df r F Y t dt + Y t f, showing that the second term in (46) is equal to ψdy. Thus (46) is equivalent to which is the self-financing property. dw = φdx + qφxdt + ψdy + χdz, 8

9 2.4.2 Interpretation of the Hedging Strategy Recall that the hedging portfolio is where φx + ψy + χz φ = 1 f C ψ = SC Y χ = C Z The net value of the first two terms is zero, and this is what eliminates the FX exposure: φ represents a conventional delta-hedge in Currency F, financed by Currency F borrowing (this is ψ). All increments in the hedge value are immediately repatriated and deposited in the home currency riskless bond Z. The value in this domestic account is χz = C so that, in particular, the value at the exercise time T is (for a call option) A 0 [S T K] +, the exercise value of the option. References [1] W. Margrabe, The value of an option to exchange one asset for another, Journal of Finance 33 (1978)