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equity options, we will sometimes find it useful to convert the price of the underlying foreign stock into the domestic currency. We write ˜S_t^f = Q_tS^f_t to denote the price of a foreign stock S^f expressed in units of domestic currency. Using Itˆo’s formula, and the dynamics under P^∗of the exchange rate Q, which are

dQ_t= Q_t

(r_d− rf) dt + σ_Q· dWt^∗

, (4.21)

one finds that under the domestic martingale measure P^∗, the process ˜S^f satisfies d ˜S_t^f = ˜S^f_t 

rddt + (σ_S^f + σQ)· dW_t^∗

. (4.22)

The last equality shows that the price process ˜S^f behaves as the price process of a domestic stock in the classic Black-Scholes framework; however, the corresponding volatility coefficient is equal to the superposition σ_Sf + σ_Q of two volatilities – the foreign stock price volatility and the exchange rate volatility. By defining in the usual way the class of admissible trading strategies, one may now easily construct a market model in which there is no arbitrage between investments in foreign and domestic bonds and stocks. Since this can be easily done, we leave the details to the reader.

4.2 Currency Forward Contracts and Options

In this section, we consider derivative securities whose value depends exclusively on the fluctuations of exchange rate Q, as opposed to those securities which depend also on some foreign equities. Currency options, forward contracts and futures contracts provide an important financial instrument through which to control the risk exposure induced by the uncertain future exchange rate. The deliverable instrument in a classic foreign exchange option is a fixed amount of underlying foreign currency. The valuation formula that provides the arbitrage price of foreign exchange European-style options was established independently in Biger and Hull (1983) and Garman and Kohlhagen (1983). They have shown that if the domestic and foreign risk-free rates are constant, and the dynamics of the exchange rate are given by (4.2), then a foreign currency option may be valued by means of a suitable variant of the Black-Scholes option valuation formula. More precisely, one may apply formula (3.76), which gives the arbitrage price of a European option written on a stock which pays a constant dividend yield.

4.2.1 Forward Exchange Rate

Let us first consider a foreign exchange forward contract, written at time t, which settles at the future date T. The asset to be delivered by the party assuming a short position in the contract is a prespecified amount of foreign currency, say 1 unit. The party who assumes a long position in a currency forward contract is obliged to pay a certain number of units of a domestic currency, the delivery price. As usual, the delivery price that makes the forward contract worthless at time t≤ T is called the forward price at time t of one unit of the foreign currency to be delivered at the settlement date T. In the present context, it is natural to refer to this forward price as the forward exchange rate. We will write FQ(t, T ) to denote the forward exchange rate.

Proposition 4.2.1 The forward exchange rate FQ(t, T ) at time t for the settlement date T is given by the following formula

FQ(t, T ) = e^{(rd−rf) (T−t)}Qt, ∀ t ∈ [0, T ]. (4.23) Proof. It is easily seen that if (4.23) does not hold, risk-free profitable opportunities arise between

the domestic and the foreign market. ✷

Relationship (4.23), commonly known as the interest rate parity, asserts that the forward ex-change premium must equal, in the market equilibrium, the interest rate differential rd − rf. A

relatively simple version of the interest rate parity still holds even when the domestic and foreign interest rates are no longer deterministic constants, but follow stochastic processes. Under uncer-tain interest rates, we need to to introduce the price processes B_d(t, T ) and B_f(t, T ) of the domestic and foreign zero-coupon bonds with maturity T. A zero-coupon bond with a given maturity T is a financial security which pays one unit of the corresponding currency at the future date T. Suppose that zero-coupon bonds with maturity T are traded in both domestic and foreign markets. Then equality (4.23) may be extended to cover the case of stochastic interest rates. Indeed, it is not hard to show, by means of no-arbitrage arguments, that

F_Q(t, T ) = B_f(t, T )

Bd(t, T )Q_t, ∀ t ∈ [0, T ], (4.24) where Bd(t, T ) and Bf(t, T ) stand for the respective time t prices of the domestic and foreign zero-coupon bonds with maturity T. Notice that in (4.24), both Bd(t, T ) and Bf(t, T ) should be seen as the domestic and foreign discount factors rather than the prices. Indeed, prices should be expressed in units of the corresponding currencies, while discount factors are merely the corresponding real numbers. Finally, it follows immediately from (4.21) that for any fixed settlement date T, the forward price dynamics under the martingale measure (of the domestic economy) P^∗ are

dF_Q(t, T ) = F_Q(t, T ) σ_Q· dWt^∗, (4.25) and F_Q(T, T ) = Q_T.

4.2.2 Currency Option Valuation Formula

As a first example of a currency option, we consider a standard European call option, whose payoff at the expiry date T equals

C_T^{Q def}= N (Q_T − K)⁺,

where QT is the spot price of the deliverable currency (i.e., the spot exchange rate at the option’s expiry date), K is the strike price in units of domestic currency per foreign unit, and N > 0is the nominal value of the option, expressed in units of the underlying foreign currency. It is clear that payoff from the option is expressed in the domestic currency; also, there is no loss of generality if we assume that N = 1. Summarizing, we consider an option to buy one unit of a foreign currency at a prespecified price K, which may be exercised at the date T only.

Proposition 4.2.2 The arbitrage price, in units of domestic currency, of a currency European call option is given by the risk-neutral valuation formula

C_t^Q= e^−rd(T−t)E_P∗

(Q_T − K)⁺Ft

, ∀ t ∈ [0, T ]. (4.26)

Moreover, the price C_t^Q is given by the following expression C_t^Q= Qte^−rf(T−t)N

h1(Qt, T − t)

− Ke^−rd(T−t)N

h2(Qt, T− t) , where N is the standard Gaussian cumulative distribution function, and

h1,2(q, t) =ln(q/K) + (rd− rf±¹₂σ²_Q)t σ_Q√

t .

Proof. Let us first examine a trading strategy in risk-free domestic and foreign bonds, which we call a currency trading strategy in what follows. Formally, by a currency trading strategy we mean an adapted stochastic process φ = (φ¹, φ²). In financial interpretation, φ¹B˜_t^f and φ²B_t^d represent the amounts of money invested at time t in foreign and domestic bonds. It is important to note that

4.2. CURRENCY FORWARD CONTRACTS AND OPTIONS 81 both amounts are expressed in units of domestic currency (see, in particular, (4.4)). A currency trading strategy φ is said to be self-financing if its wealth process V (φ), which equals

Vt(φ) = φ¹_tB˜_t^f+ φ²_tB^d_t, ∀ t ∈ [0, T ], where ˜B_t^f = B_t^fQ_t, B_t^d= e^rdt, satisfies the following relationship

dV_t(φ) = φ¹_td ˜B_t^f+ φ²_tdB_t^d.

For the discounted wealth process V_t^∗(φ) = e^−rdtV_t(φ) of a self-financing currency trading strategy, we easily get

dV_t^∗(φ) = φ¹_td(e^−rdtB˜_t^f) = φ¹_tdQ^∗_t.

On the other hand, by virtue of (4.21), the dynamics of the process Q^∗, under the domestic mar-tingale measure P^∗, are given by the expression

dQ^∗_t = σQQ^∗_tdW_t^∗.

Therefore, the discounted wealth V^∗(φ) of any self-financing currency trading strategy φ follows a martingale under P^∗. This justifies the risk-neutral valuation formula (4.26). Taking into account the equality Q_T = ˜B^f_Te^−rfT, one gets also

C_t^Q = e^−rd(T−t)E_P∗

(Q_T − K)⁺| Ft



= e^−rfTe^−rd(T−t)EP^∗

( ˜B^f_T− Ke^rfT)⁺| Ft



= e^−rfTC( ˜B^f_t, T − t, Ke^rfT, rd, σQ),

where C stands for the standard Black-Scholes call option price. More explicitly, we have C_t^Q = e^−rfT

B˜_t^fN

d1( ˜B_t^f, T − t)

− Ke^rfTe^−rd(T−t)N

d2( ˜B_t^f, T − t)

= Qte^−rf(T−t)N

d1( ˜B_t^f, T− t)

− Ke^−rd(T−t)N

d2( ˜B^f_t, T − t) . This proves the formula we wish to show, since

d_i( ˜B_t^f, T − t, Ke^rfT, r_d, σ_Q) = h_i(Q_t, T − t)

for i = 1, 2. Finally, one finds immediately that the first component of the self-financing currency trading strategy that replicates the option equals

φ¹_t = e^−rfTN

d1( ˜B_t^f, T− t)

= e^−rfTN

h1(Qt, T − t) .

Therefore, to hedge a short position, the writer of the currency call should invest at time t≤ T the amount (expressed in units of foreign currency)

φ¹_tB_t^f= e^−rf(T−t)N

h1(Qt, T− t)

in foreign market risk-free bonds (or equivalently, in the foreign savings account). On the other hand, she should also invest the amount (denominated in domestic currency)

C_t^Q− Qte^−rf(T−t)N

h1(Qt, T− t)

in the domestic savings account. ✷

Remarks. (a) As mentioned earlier, a comparison of the currency option valuation formula established in Proposition 4.2.2 with expression (3.76) shows that the exchange rate Q can be formally seen as the price of a fictitious domestic “stock”. Under such a convention, the foreign interest rate rf can be interpreted as a dividend yield that is continuously paid by this fictitious stock.

(b) It is easy to derive the put-call relationship for currency options. Indeed, the payoff in domestic currency of a portfolio composed of one long call option and one short put option is

C_T^Q− P_T^Q= (Q_T− K)⁺− (K − QT)⁺= Q_T− K,

where we assume, as before, that the options are written on one unit of foreign currency. Conse-quently, for any t∈ [0, T ], we have

C_t^Q− Pt^Q= e^−rf(T−t)Qt− e^−rd(T−t)K. (4.27) (c) We may also rewrite the currency option valuation formula of Proposition 4.2.2 in the following way

C_t^Q= e^−rd(T−t)



FtNd˜1(Ft, T − t)

− KNd˜2(Ft, T − t)

, (4.28)

where Ft= FQ(t, T ) and

d˜1,2(F, t) = ln(F/K)±¹₂σ_Q² t σQ

√t , ∀ (F, t) ∈ R+× (0, T ].

This shows that the currency option valuation formula can be seen as a variant of the Black futures formula (3.91) of Sect. 3.3. Furthermore, it is possible to re-express the replicating strategy of the option in terms of domestic bonds and currency forward contracts. Let us mention that under the present assumptions of deterministic domestic and foreign interest rates, the distinction between the currency futures price and forward exchange rate is not essential. In market practice, currency options are frequently hedged by taking positions in forward and futures contracts, rather than by investing in foreign risk-free bonds.