# LECTURE 9: A MODEL FOR FOREIGN EXCHANGE

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2 investor; the choice of numeraire (dollar or pound sterling) determines which asset is riskless. Let s take the point of view of the pound-sterling investor. Let Y t be the rate of exchange at time t (that is, Y t is the number of British pounds that one dollar will buy at time t). In the simplest model 3, Y t behaves like a geometric Brownian motion, that is, it follows a stochastic differential equation of the form (1) dy t = µy t dt + σy t dw t, where W t is a Wiener process. Let A t and B t denote the share prices of the assets US Money- Market and UK Money Market, reported in units of dollars and British pounds, respectively, and normalized so that the time-zero share prices are both 1. Then (2) (3) A t = exp{r A t} B t = exp{r B t}. The share price of US MoneyMarket at time t in pounds sterling is A t Y t. Solving the stochastic differential equation (1) gives the explicit formula and (4) A t Y t = Y 0 exp{r A t + µt σ 2 t/2 + σw t }. Proposition 1. Let Q B be a risk-neutral probability measure for the pound-sterling investor. If the dollar/pound sterling exchange rate obeys a stochastic differential equation of the form (1), and if the riskless rates of return for dollar investors and pound-sterling investors are r A and r B, respectively, then under Q B it must be the case that (5) µ = r B r A. Therefore, exchange rate Y t is given by (6) Y t = Y 0 exp{(r B r A )t σ 2 t/2 + σw t }, where under the measure Q B the process W t is a standard Wiener process. Proof. Under Q B, the discounted share price of the asset US MoneyMarket in pound sterling must be a martingale. But the discounted share price is exp{ r B t}a t Y t, which by equation (4) equals Y 0 exp{(r A r B )t + µt} exp{ σ 2 t/2 + σw t }. The second exponential is by itself a martingale (see Lecture 8), and the first exponential is nonrandom. Thus, in order that the product of the two exponentials be a martingale it must be that r A r B + µ = 0. Exercise: Show that if M t is a martingale and f(t) is a continuous, nonrandom function of t, then f(t)m t is a martingale if and only if f(t) is constant or M t is identically zero. Proposition 1 is fairly straightforward, given the results from stochastic calculus that we have obtained earlier in the course. Nevertheless, there is something puzzling about it. What if we had taken the point of view of the dollar investor? Then the roles of r A and r B would be reversed, and the share price of UK Money Market in dollars at time t would be B t /Y t. Reasoning as above, we would be led to the following result: 3 probably due to Merton

3 Proposition 2. Let Q A be a risk-neutral probability measure for the dollar investor. If the dollar/pound sterling exchange rate obeys a stochastic differential equation of the form (1), where W t is a standard Brownian motion under Q A, and if the riskless rates of return for dollar investors and pound-sterling investors are r A and r B, respectively, then under Q A it must be the case that (7) µ = r B r A + σ 2. Proof. Homework. Thus, unless the riskless rates of return for dollar and pound-sterling investors are equal, they will necessarily disagree about the drift coefficient µ in the stochastic differential equation (1)! Put this way, the situation seems somewhat paradoxical: one s stochastic model for an observable process, the exchange rate, must depend on the currency in which he or she trades. But this is the wrong way to put it. The risk-neutral probability measure is not (necessarily) an accurate model for the price processes of traded assets; rather, it is imposed by the market, and dependent on the numeraire. It is useful only insofar as it allows the computation of arbitrage prices of derivative securities. 3. The Likelihood Ratio The risk-neutral probability measures Q A and Q B for dollar and pound-sterling investors are both measures on the same space of market scenarios, specifically, the set of all possible paths {Y t } 0 t T of the exchange rate. Proposition 3. The measures Q A and Q B are mutually absolutely continuous, that is, they are related by the likelihood ratio ( ) dqa (8) = exp{σw T σ 2 T/2}. dq B F T Proof. Consider a contingent claim whose value at time t in dollars is V t. The price V 0 of this claim at time t = 0, in dollars, is the discounted expected value of its price, in dollars, at time T, where the expectation is computed under Q A, the risk-neutral measure for dollar investors: (9) V 0 = e r AT E A V T. Let U t be the time-t price of the claim in pounds sterling. Then U t = V t Y t, where Y t is the rate of exchange between dollars and pounds sterling. Since the claim is a tradeable asset, its price in pounds sterling must be a martingale under Q B, the risk-neutral measure for pound-sterling investors. In particular, the time-zero price must be the discounted expected value of the time-t price: (10) U 0 = e r BT E B U T = V 0 Y 0 = e r BT E B (V T Y T ) = V 0 = e r AT E B (V T (Y T /Y 0 )e (r A r B )T ). Comparing equations (9) and (10) shows that the expectation operators E A and E B are related as follows: E A V T = E B (V T (Y T /Y 0 )e (r A r B )T ).

4 Since this formula holds for any nonnegative, F T measurable random variable V T, it follows from equation (6) above that ( ) dqa = (Y T /Y 0 )e (r A r B )T = exp{σw T σ 2 T/2}. dq B F T It is noteworthy that the likelihood ratio (8) is of the same type as occurs in the Cameron- Martin theorem (Lecture 8). Thus, under the risk-neutral measure Q A for dollar investors, the process {W t } 0 t T appearing in the stochastic differential equation (1) is not a standard Wiener process but rather a Wiener process with drift σ. Equivalently, if we define (11) Wt = W t σt, then under Q A the process { W t } 0 t T is a standard Wiener process. Substituting this equation into formula (6) gives the following alternative form for the exchange rate process Y t : (12) Y t = Y 0 exp{(r B r A )t σ 2 t/2 + σw t } = Y 0 exp{(r B r A )t + σ 2 t/2 + σ W t }. It follows by Itô s formula that the exchange rate process obeys the stochastic differential equation (13) dy t = (r B r A + σ 2 )Y t dt + σy t d W t. Observe that this is a second proof of Proposition 2 (the first proof was a homework exercise). 4. Options on Foreign Exchange Consider an option Call that gives the owner the right (but no obligation) to exchange 1 dollar for K pounds sterling at time T. What is the time t = 0 arbitrage price, in dollars, of this option? Let s take the viewpoint of a dollar investor. (In the homework you will be asked to do the entire problem again from the viewpoint of the pound-sterling investor, and to verify that this leads to the same arbitrage price.) The value in dollars of K pounds sterling at time T will be K/Y T, and so the value V T (in dollars) of the option at termination t = T will be V T = (K/Y T 1) +. Consequently, the value at time zero is (14) V 0 = e r AT E A (K/Y T 1) + = e r AT E A (KY 0 exp{(r A r B + σ 2 /2)t σw t } 1) +. This last expectation is identical in form to the expectation that we encountered in pricing a European call option on a stock in the Black-Scholes theory, and thus may be evaluated explicitly in terms of the cumulative normal distribution function Φ. Exercise: Do this. 5. Exercises Problem 1: Show that if M t is a martingale and f(t) is a continuous, nonrandom function of t, then f(t)m t is a martingale if and only if f(t) is constant or M t is identically zero. Problem 2: Prove Proposition 2. Problem 3: Compute the arbitrage price at time t = 0 of the call option discussed in section 4 by calculating the discounted risk-neutral expectation under the risk-neutral measure Q B for

5 pound-sterling investors. Verify that the price agrees with that computed using the risk-neutral measure Q A for dollar investors. Problem 4: In a forward exchange contract, two parties A and B agree at time t = 0 to an exchange of currency at a future date T. In particular, A agrees to pay 1 dollar at time T to B in exchange for K pounds sterling. What is the arbitrage forward price K? Do the problem in two different ways: (i) using the information about the risk-neutral measure Q A for dollar investors given in Proposition 1, and (ii) by a direct arbitrage argument.

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