A Continuous-Time Model for Valuing Foreign Exchange Options

and Applied Analysis 3 Applying Ito’s lemma to the call function c ≡ c(t, T) ≡ c(G, B d , T; X) and the relation dt = −dT, we obtain dc = ∂c ∂G dG + ∂c ∂B d dB d − [ 1 2 ∂ 2 c ∂G 2 σ 2 G G 2 + 1 2 ∂ 2 c ∂B 2 d σ 2 d B 2 d + ∂ 2 c ∂G∂B d ρ Gd σ G σ d GB d − ∂c ∂T ]dT. (10) At this point, we set up a hedge consisting of three assets: G, B d , and c. Let Q G be the number of G, Q d the number of B d , and Q c the number of c in the hedge. Let P h be the value of the hedge portfolio. The hedge is set up in such a way that the value of this hedge portfolio is zero, that is, P h = Q G G + Q d B d + Q c c = 0. Hence, the change in the value of this hedge portfolio is also zero, that is, dP h = Q G dG + Q d dB d + Q c dc = 0. (11) Remark 3. Our hedge is different from that of Black and Scholes [4]. In their case, they create their hedge such that the hedge portfolio is riskless. Hence, its return must equal the riskless rate or the spot rate. In our case, we create our hedge such that the value P h of the hedge portfolio is zero (i.e., the aggregate investment is zero). Hence, we have dP h = 0 in (11). Substituting (5), (9), and (10) into (11) and grouping, (11) becomes dP h = Q c [ 1 2 ∂ 2 c ∂G 2 σ 2 G G 2 + 1 2 ∂ 2 c ∂B 2 d σ 2 d B 2 d + ∂ 2 c ∂G∂B d ρ Gd σ G σ d GB d − ∂c ∂T ]dT + [Q c ∂c ∂G + Q G ] dG + [Q c ∂c ∂B d + Q d ] dB d = 0. (12) Equation (12) suggests that Q c (∂c/∂G) + Q G = 0, Q c (∂c/ ∂B d ) + Q d = 0, and 1 2 ∂ 2 c ∂G 2 σ 2 G G 2 + 1 2 ∂ 2 c ∂B 2 d σ 2 d B 2 d + ∂ 2 c ∂G∂B d ρ Gd σ G σ d GB d − ∂c ∂T = 0. (13) Then the price of a call must satisfy (13) subject to two boundary conditions: (i) the call price is zero if G = 0; (ii) at its expiration, the call has a value of either zero if G ≤ X or G − X if G > X. In notations, the two boundary conditions are c (G = 0, B d , T; X) = 0, c (G, B d = 1, 0; X) = max (0, G − X) . (14) The second-order partial differential equation in (13) has no well-known solution. Hence, to solve for c in (13), we transform (13) to a standard heat equation [14–16] of the form u t (x, t) = α 2 u xx (x, t), where α is some constant. Making use of the linear homogeneity of c in G and XB d , we can carry out such transformation for (13). Consequently, we set θ ≡ θ(G, B d , T) = G/XB d . Remark 4. A function f ≡ f(x 1 , . . . , x n ) is said to be linear homogeneous or homogeneous of degree one in x i , where i = 1, . . . , n, if f(αx 1 , . . . , αx n ) = αf(x 1 , . . . , x n ), where α is some constant. In a competitive and perfect market, the fact that the value of a call is homogeneous of degree one in the asset and exercise pricemeans that the value of the call with exercise price X when the asset value is G will be exactly α times the value of a call on the same asset with exercise priceX/αwhen the asset value is G/α. See [17–19] for more description on linear homogeneity. Using Ito’s lemma and (5) and (9), the total differential of θ is given by dθ = [ ∂θ ∂G μ G G + ∂θ ∂B d r d B d + ∂θ ∂t + 1 2 ∂ 2 θ ∂G 2 σ 2 G G 2 + 1 2 ∂ 2 θ ∂B 2 d σ 2 d B 2 d + ∂ 2 θ ∂G∂B d ρ Gd σ G σ d GB d ]dt + ∂θ ∂G σ G GdW G + ∂θ ∂B d σ d B d dW d . (15) Substituting ∂θ/∂G = 1/XB d , ∂θ/∂B d = −G/XB 2 d , ∂θ/∂t= 0, ∂θ/∂G = 0, ∂θ/∂B d = G/XB 3 d , and ∂θ/∂G∂B d =−1/XB d into (15) and simplifying, (15) becomes dθ θ = μ θ dt + σ θ dW θ , (16) where μ θ = μ G − r d + (σ 2 d /2) − ρ Gd σ G σ d and σ θ = σ 2 G + σ 2 d − 2ρ Gd σ G σ d . To solve (13) for c(G, B d , T; X) subject to the two boundary conditions in (14), we use another variable K such that K ≡ K(θ, T;X) = (c(G, B d , T; X))/XB d . In words, K is the call price expressed in the same units as θ. Expressed in another way, c(G, B d , T; X) = XB d K(θ, T;X). Then ∂ 2 c/∂G 2 = (1/XB d )(∂ 2 K/∂G 2 ), ∂c/∂B d = (G 2 /XB 3 d )(∂ 2 K/ ∂G 2 ), ∂c/∂G∂B d = −(G/XB 2 d )(∂ 2 K/∂G 2 ), and ∂c/∂T = XB d (∂K/∂T). Substituting them into (13) and simplifying, (13) becomes


Introduction
In this paper, we make use of stochastic calculus [1][2][3] to develop a continuous-time model for valuing European options on foreign exchange when both domestic and foreign spot rates follow a generalized Wiener process.
Foreign exchange (FX) options have traded on the Philadelphia Stock Exchange since 1982.An FX option is an agreement between two parties in which one party pays a premium and obtains the right to buy or sell the stated amount of foreign exchange at a later date at the exercise price, where the exercise price is an exchange rate agreed upon at the initial time.Extending the Black-Scholes [4] pricing model for stock options and assuming that both domestic and foreign spot rates are constant during the life of the option, Garman and Kohlhagen [5] developed in 1983 a model for valuing European FX options.
However, the assumption of Garman and Kohlhagen (G-K) that the two spot rates are constant over the life of the option is inappropriate because they are, in actuality, evolving continuously and stochastically through time.In this study, we incorporate the stochastic character of the two spot rates into our FX option model.Specifically, we employ the following stochastic process, often referred to as the generalized Wiener process, to represent the evolution of the spot rate () and, accordingly, derive a continuous-time model for valuing European call and put FX options as () =  + . ( In (1), the generalized Wiener process has a drift rate of , a variance rate of  2 , and  is a standard Wiener process whose increment  has a normal distribution with zero mean and variance .
Remark 1.It is possible that the spot rate under (1) can become negative.But negative spot rate is not probable if the drift rate is positive.More importantly, most FX options traded on an exchange have an expiration of less than one year.Hence, if we use an initial positive value for () and suitable values for the drift rate and variance rate, the expected first-passage time of the spot rate to the origin can easily be made longer than one year.
Remark 2. Our FX option model essentially extends the traditional G-K model to incorporate the stochastic character of the two spot rates.Like the G-K model, our model is for valuing European FX options.To value American options, often we have to resort to numerical procedures because no analytic formulas are available.For some well-articulated numerical procedures for valuing American options, see Hull [6] for pricing FX options with constant interest rates, Ho et al. [7] for pricing stock options with stochastic interest rates, and Zhang and Wang [8] for pricing bond options with a penalty method (see [9,10]).
The rest of this paper is organized as follows.In Section 2, we derive an explicit formula for valuing European call and put FX options when both domestic and foreign spot rates evolve according to (1).In Section 3, we first estimate the various parameters of our FX option model based on the dollar/euro exchange rate; then, we compute the FX option prices for our model and the G-K model using the parameter estimates as inputs, and finally we examine the pricing biases in the G-K model employing our model as a yardstick.A short conclusion is given in Section 4. The derivation of the rather lengthy equation ( 27) in Section 2 is relegated to the Appendix.

Deriving a Formula for Call and Put FX Options
In this paper, we assume that the foreign exchange market is frictionless; that is, there are no trading costs, margin requirements, exchange rate controls, and taxes; trading takes place continuously; borrowing and short-selling are allowed; there exist pure discount bonds at which each currency can be borrowed or lent.Using (1) for the domestic and foreign spot rates, their diffusion processes are expressed as follows: with     =   .Using (2) and applying Ito's lemma [11][12][13], we have Letting and assuming the local expectations hypothesis holds for the term structure of interest rates (i.e.,   =   and   =   ), we obtain Solving ( 5) and (6) for   and   , we obtain explicit formulas for the prices of domestic and foreign pure discount bonds with time to maturity  as Note that  = −( At this point, we set up a hedge consisting of three assets: ,   , and .Let   be the number of ,   the number of   , and   the number of  in the hedge.Let  ℎ be the value of the hedge portfolio.The hedge is set up in such a way that the value of this hedge portfolio is zero, that is,  ℎ =    +     +    = 0. Hence, the change in the value of this hedge portfolio is also zero, that is, Remark 3. Our hedge is different from that of Black and Scholes [4].In their case, they create their hedge such that the hedge portfolio is riskless.Hence, its return must equal the riskless rate or the spot rate.In our case, we create our hedge such that the value  ℎ of the hedge portfolio is zero (i.e., the aggregate investment is zero).Hence, we have  ℎ = 0 in (11).
Remark 6.We equate the two sides of (24) to − 2 so that the two differential equations in (25) have continuous eigenvalues  2 .

FX Option Prices and Pricing Biases in the G-K Model
Retrieved from the Datastream database, three sets of daily data (a total of 2,869 observations from 4 January 1999 to 31 December 2009) are used for parameter estimation.One set is the dollar/euro exchange rate and the other two sets are the 3month US.Treasury Bill Rate and the 1-month euro-currency rate; that is, we use the Treasury Bill Rate to represent the domestic spot rate   and the euro-currency rate to represent the foreign spot rate   .Accordingly, the estimates for the six parameters of our FX option model are as follows: σ = 0.198428, b = 0.018534, b = 0.011398, ρ = −0.347350,ρ = −0.297520,and ρ = 0.524996.
Remark 7. The euro was introduced as an accounting currency on 1 January 1999.Euro coins and banknotes have been in circulation since 1 January 2002.
Using the above parameter estimates as inputs, we compute the option prices for our FX model ((33)-( 35)) and We first focus on call options.For  = 1, 3, 6, and 9 months, call prices increase as   / increases from 0.70 to 1.30 under both our model and the G-K model.For example, in Table 1 where  = 1 month, call price increases from 0.001109 to 0.032643 and to 0.131898 under our model, and from 0.001105 to 0.032624 and to 0.131898 under the G-K model as   / increases from 0.90 to 1.00 and to 1.10, respectively.For each of the four s, the G-K model incorrectly values FX calls for different values of   /.Specifically, it undervalues calls when   / ranges from 0.70 to 1.08, and it overvalues calls when   / ranges from 1.18 to 1.30.For example, in Table 2 where  = 3 months, call price is 0.000666 under our model and 0.000657 under the G-K model for   / = 0.80, which amounts to a positive pricing bias of 1.3699%, whereas call price is 0.330616 under our model and 0.330678 under the G-K model for   / = 1.30, which amounts to a negative pricing bias of −0.0187%.
The situation is completely opposite for put options.For  = 1, 3, 6, and 9 months, put prices decrease as   / increases from 0.70 to 1.30 under both models.For example, in Table 3 where  = 6 months, put price decreases from 0.367578 to 0.082131 and to 0.008767 under our model, and from 0.367695 to 0.081844 and to 0.008639 under the G-K model as   / increases from 0.80 to 1.00  4 where  = 9 months, put price is 0.375783 under our model and 0.376021 under the G-K model for   / = 0.80, which amounts to a negative pricing bias of −0.0633%, whereas put price is 0.006367 under our model and 0.006198 under the G-K model for   / = 1.30, which amounts to a positive pricing bias of 2.7267%.Foreign exchange options are actively traded on the Philadelphia Stock Exchange.The contract size of a eurocurrency option is C62,500.For example, when  = 9 months and   / = 0.90, call premium is ($0.042077 × 62,500) = $2,629.81under our model and ($0.041614 × 62,500) = $2,600.88under the G-K model-a difference of $28.93 underpaid by a call option buyer; similarly, when  = 9 months and   / = 1.10, put premium is ($0.044294 × 62,500) = $2,768.38under our model and ($0.043837 × 62,500) = $2,739.81under the G-K model-a difference of $28.57underpaid by a put option buyer.In other words, option sellers are at an obvious disadvantage if FX option valuation is based on the G-K model.

Table 1 :
Prices for European call and put FX options when time to maturity  = 1 month. is the exchange rate at initial time ,  is the exercise price, Call  and Put  are call and put prices for our model, Call GK and Put GK are call and put prices for the G-K model, Bias  = 100(Call  − Call GK )/Call GK , and Bias  = 100(Put  − Put GK )/Put GK .January 2010 when   () = 0.0008,   () = 0.0049, and   = 1.4389.In addition, employing our FX option model as a yardstick, we examine whether or not the G-K model values correctly FX call and put options for different values of   /, where   is the exchange rate on 1 January 2010 and  is the exercise price.Given that   is fixed at 1.4389, we vary  such that   / ranges from 0.70 to 1.30-a range large enough to include even extreme values of  not commonly used in practice.
the G-K model (36) by setting the initial time  = 1

Table 2 :
Prices for European call and put FX options when time to maturity  = 3 months. is the exchange rate at initial time t,  is the exercise price, Call  and Put  are call and put prices for our model, Call GK and Put GK are call and put prices for the G-K model, Bias  = 100(Call  − Call GK )/Call GK , and Bias  = 100(Put  − Put GK )/Put GK .and to 1.20, respectively.For each of the four s, the G-K model incorrectly values FX puts for different values of   /.Specifically, it overvalues puts when   / ranges from 0.70 to 0.82, and it undervalues puts when   / ranges from 0.86 to 1.30.For example, in Table

Table 3 :
Prices for European call and put FX options when time to maturity  = 6 months. is the exchange rate at initial time ,  is the exercise price, Call  and Put  are call and put prices for our model, Call GK and Put GK are call and put prices for the G-K model, Bias  = 100(Call  − Call GK )/Call GK , and Bias  = 100(Put  − Put GK )/Put GK .and employing our FX option model as a yardstick, our numerical results show that the G-K model values incorrectly both call and put options for different values of   /.Specifically, it undervalues calls when   / is between 0.70 and 1.08, and it overvalues calls when   / is between 1.18 and 1.30, whereas it overvalues puts when   / is between 0.70 and 0.82, and it undervalues puts when   / is between 0.86 and 1.30.

Table 4 :
Prices for European call and put FX options when time to maturity  = 9 months. is the exchange rate at initial time t,  is the exercise price, Call  and Put  are call and put prices for our model, Call GK and Put GK are call and put prices for the G-K model, Bias  = 100(Call  − Call GK )/Call GK , and Bias  = 100(Put  − Put GK )/Put GK .