A Continuous-Time Model for Valuing Foreign Exchange Options

Abstract

This paper makes use of stochastic calculus to develop a continuous-time model for valuing European options on foreign exchange (FX) when both domestic and foreign spot rates follow a generalized Wiener process. Using the dollar/euro exchange rate as input for parameter estimation and employing our FX option model as a yardstick, we find that the traditional Garman-Kohlhagen FX option model, which assumes constant spot rates, values incorrectly calls and puts for different values of the ratio of exchange rate to exercise price. Specifically, it undervalues calls when the ratio is between 0.70 and 1.08, and it overvalues calls when the ratio is between 1.18 and 1.30, whereas it overvalues puts when the ratio is between 0.70 and 0.82, and it undervalues puts when the ratio is between 0.86 and 1.30.

1. Introduction

In this paper, we make use of stochastic calculus [1–3] 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 𝑟(𝑡) and, accordingly, derive a continuous-time model for valuing European call and put FX options as 𝑑𝑟(𝑡)=𝑎𝑑𝑡+𝑏𝑑𝑊. (1) In (1), the generalized Wiener process has a drift rate of 𝑎, a variance rate of 𝑏², and 𝑊 is a standard Wiener process whose increment 𝑑𝑊 has a normal distribution with zero mean and variance 𝑑𝑡.

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 𝑟(𝑡) 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.

2. 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. To proceed, we adopt the following notation:𝑆(𝑡): the spot exchange rate (domestic currency price per unit of foreign currency) at time 𝑡,𝑟_𝑑(𝑡): the domestic spot rate of interest at time 𝑡,𝑟_𝑓(𝑡): the foreign spot rate of interest at time 𝑡,𝐵_𝑑(𝑡,𝑇): the domestic currency price of a pure discount bond at time 𝑡 which pays one unit of domestic currency at time 𝑡+𝑇,𝐵_𝑓(𝑡,𝑇): the foreign currency price of a pure discount bond at time 𝑡 which pays one unit of foreign currency at time 𝑡+𝑇,𝑐(𝑡,𝑇): the domestic currency price at time 𝑡 of a European call on one unit of foreign currency which expires at time 𝑡+𝑇,𝑝(𝑡,𝑇): the domestic currency price at time 𝑡 of a European put on one unit of foreign currency which expires at time 𝑡+𝑇,𝑋: the domestic currency exercise price of a European call or a put on foreign currency.

Using (1) for the domestic and foreign spot rates, their diffusion processes are expressed as follows: 𝑑𝑟_𝑑(𝑡)=𝑎_𝑑𝑑𝑡+𝑏_𝑑𝑑𝑊_𝑑,𝑑𝑟_𝑓(𝑡)=𝑎_𝑓𝑑𝑡+𝑏_𝑓𝑑𝑊_𝑓, (2) with 𝑑𝑊_𝑑𝑑𝑊_𝑓=𝜌_𝑑𝑓𝑑𝑡. Using (2) and applying Ito’s lemma [11–13], we have 𝑑𝐵_𝑑=𝜕𝐵_𝑑𝜕𝑡+𝑎_𝑑𝜕𝐵_𝑑𝜕𝑟_𝑑+12𝑏²_𝑑𝜕²𝐵_𝑑𝜕𝑟²_𝑑𝑑𝑡+𝑏_𝑑𝜕𝐵_𝑑𝜕𝑟_𝑑𝑑𝑊_𝑑,𝑑𝐵_𝑓=𝜕𝐵_𝑓𝜕𝑡+𝑎_𝑓𝜕𝐵_𝑓𝜕𝑟_𝑓+12𝑏²_𝑓𝜕²𝐵_𝑓𝜕𝑟²_𝑓𝑑𝑡+𝑏_𝑓𝜕𝐵_𝑓𝜕𝑟_𝑓𝑑𝑊_𝑓. (3) Letting 𝜇_𝑑=1𝐵_𝑑𝜕𝐵_𝑑𝜕𝑡+𝑎_𝑑𝜕𝐵_𝑑𝜕𝑟_𝑑+12𝑏²_𝑑𝜕²𝐵_𝑑𝜕𝑟²_𝑑,𝜎_𝑑=−1𝐵_𝑑𝑏_𝑑𝜕𝐵_𝑑𝜕𝑟_𝑑=−𝑏_𝑑𝐵_𝑑𝜕𝐵_𝑑𝜕𝑟_𝑑,𝜇_𝑓=1𝐵_𝑓𝜕𝐵_𝑓𝜕𝑡+𝑎_𝑓𝜕𝐵_𝑓𝜕𝑟_𝑓+12𝑏²_𝑓𝜕²𝐵_𝑓𝜕𝑟²_𝑓,𝜎_𝑓=−1𝐵_𝑓𝑏_𝑓𝜕𝐵_𝑓𝜕𝑟_𝑓=−𝑏_𝑓𝐵_𝑓𝜕𝐵_𝑓𝜕𝑟_𝑓, (4) and assuming the local expectations hypothesis holds for the term structure of interest rates (i.e., 𝜇_𝑑=𝑟_𝑑 and 𝜇_𝑓=𝑟_𝑓), we obtain 𝑑𝐵_𝑑𝐵_𝑑=𝑟_𝑑𝑑𝑡+𝜎_𝑑𝑑𝑊_𝑑,𝑑𝐵_𝑓𝐵_𝑓=𝑟_𝑓𝑑𝑡+𝜎_𝑓𝑑𝑊_𝑓. (5)(6)

Solving (5) and (6) for 𝐵_𝑑 and 𝐵_𝑓, we obtain explicit formulas for the prices of domestic and foreign pure discount bonds with time to maturity 𝑇 as 𝐵_𝑑(𝑡,𝑇)=exp−𝑟_𝑑𝑎𝑇−^𝑑𝑇²2+𝑏²_𝑑𝑇³6,𝐵_𝑓(𝑡,𝑇)=exp−𝑟_𝑓𝑎𝑇−^𝑓𝑇²2+𝑏²_𝑓𝑇³6. (7) Note that 𝑇=−(1/𝐵_𝑑)[𝜕𝐵_𝑑/𝜕𝑟_𝑑]=−(1/𝐵_𝑓)[𝜕𝐵_𝑓/𝜕𝑟_𝑓]. Hence, we have 𝜎_𝑑=−(𝑏_𝑑/𝐵_𝑑)[𝜕𝐵_𝑑/𝜕𝑟_𝑑]=𝑏_𝑑𝑇 and 𝜎_𝑓=−(𝑏_𝑓/𝐵_𝑓)[𝜕𝐵_𝑓/𝜕𝑟_𝑓]=𝑏_𝑓𝑇.

Similar to the G-K FX option model, we assume that the spot exchange rate follows the geometric Wiener process 𝑑𝑆𝑆=𝜇_𝑠𝑑𝑡+𝜎_𝑠𝑑𝑊_𝑠, (8) where 𝜇_𝑠 and 𝜎_𝑠 are constant parameters, and 𝑊_𝑠 is a standard Wiener process. In addition, we assume 𝑑𝑊_𝑠𝑑𝑊_𝑑=𝜌_𝑠𝑑𝑑𝑡 and 𝑑𝑊_𝑠𝑑𝑊_𝑓=𝜌_𝑠𝑓𝑑𝑡.

Converting the price of a foreign pure discount bond into domestic currency price, we define a new variable 𝐺=𝑆𝐵_𝑓 such that 𝑑𝐺𝐺=𝜇_𝐺𝑑𝑡+𝜎_𝐺𝑑𝑊_𝐺. (9)

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𝜕𝐺𝑑𝐺+𝜕𝑐𝜕𝐵_𝑑𝑑𝐵_𝑑−12𝜕²𝑐𝜕𝐺²𝜎²_𝐺𝐺²+12𝜕²𝑐𝜕𝐵²_𝑑𝜎²_𝑑𝐵²_𝑑+𝜕²𝑐𝜕𝐺𝜕𝐵_𝑑𝜌_𝐺𝑑𝜎_𝐺𝜎_𝑑𝐺𝐵_𝑑−𝜕𝑐𝜕𝑇𝑑𝑇. (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_𝐺𝐺+𝑄_𝑑𝐵_𝑑+𝑄_𝑐𝑐 = 0. Hence, the change in the value of this hedge portfolio is also zero, that is, 𝑑𝑃_ℎ=𝑄_𝐺𝑑𝐺+𝑄_𝑑𝑑𝐵_𝑑+𝑄_𝑐𝑑𝑐=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_𝑐12𝜕²𝑐𝜕𝐺²𝜎²_𝐺𝐺²+12𝜕²𝑐𝜕𝐵²_𝑑𝜎²_𝑑𝐵²_𝑑+𝜕²𝑐𝜕𝐺𝜕𝐵_𝑑𝜌_𝐺𝑑𝜎_𝐺𝜎_𝑑𝐺𝐵_𝑑−𝜕𝑐+𝑄𝜕𝑇𝑑𝑇_𝑐𝜕𝑐𝜕𝐺+𝑄_𝐺𝑑𝐺+𝑄_𝑐𝜕𝑐𝜕𝐵_𝑑+𝑄_𝑑𝑑𝐵_𝑑=0. (12)

Equation (12) suggests that 𝑄_𝑐(𝜕𝑐/𝜕𝐺)+𝑄_𝐺=0, 𝑄_𝑐(𝜕𝑐/𝜕𝐵_𝑑)+𝑄_𝑑=0, and 12𝜕²𝑐𝜕𝐺²𝜎²_𝐺𝐺²+12𝜕²𝑐𝜕𝐵²_𝑑𝜎²_𝑑𝐵²_𝑑+𝜕²𝑐𝜕𝐺𝜕𝐵_𝑑𝜌_𝐺𝑑𝜎_𝐺𝜎_𝑑𝐺𝐵_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 𝑐𝐺=0,𝐵_𝑑𝑐,𝑇;𝑋=0,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)=𝛼²𝑢_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(𝑥₁,…,𝑥_𝑛) is said to be linear homogeneous or homogeneous of degree one in 𝑥_𝑖, where 𝑖=1,…,𝑛, if 𝑓(𝛼𝑥₁,…,𝛼𝑥_𝑛)=𝛼𝑓(𝑥₁,…,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 price 𝑋/𝛼 when 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=𝜕𝜃𝜇𝜕𝐺^𝐺𝐺+𝜕𝜃𝜕𝐵_𝑑𝑟_𝑑𝐵_𝑑+𝜕𝜃+1𝜕𝑡2𝜕²𝜃𝜕𝐺²𝜎²_𝐺𝐺²+12𝜕²𝜃𝜕𝐵²_𝑑𝜎²_𝑑𝐵²_𝑑+𝜕²𝜃𝜕𝐺𝜕𝐵_𝑑𝜌_𝐺𝑑𝜎_𝐺𝜎_𝑑𝐺𝐵_𝑑+𝑑𝑡𝜕𝜃𝜎𝜕𝐺^𝐺𝐺𝑑𝑊_𝐺+𝜕𝜃𝜕𝐵_𝑑𝜎_𝑑𝐵_𝑑𝑑𝑊_𝑑. (15)

Substituting 𝜕𝜃/𝜕𝐺=1/𝑋𝐵_𝑑, 𝜕𝜃/𝜕𝐵_𝑑=−𝐺/𝑋𝐵²_𝑑, 𝜕𝜃/𝜕𝑡=0, 𝜕²𝜃/𝜕𝐺²=0, 𝜕²𝜃/𝜕𝐵²_𝑑=𝐺/𝑋𝐵³_𝑑, and 𝜕²𝜃/𝜕𝐺𝜕𝐵_𝑑=−1/𝑋𝐵²_𝑑 into (15) and simplifying, (15) becomes 𝑑𝜃𝜃=𝜇_𝜃𝑑𝑡+𝜎_𝜃𝑑𝑊_𝜃, (16) where 𝜇_𝜃=𝜇_𝐺−𝑟_𝑑+(𝜎²_𝑑/2)−𝜌_𝐺𝑑𝜎_𝐺𝜎_𝑑 and 𝜎²_𝜃=𝜎²_𝐺+𝜎²_𝑑−2𝜌_𝐺𝑑𝜎_𝐺𝜎_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_𝑑𝐾(𝜃,𝑇;𝑋). Then 𝜕²𝑐/𝜕𝐺²=(1/𝑋𝐵_𝑑)(𝜕²𝐾/𝜕𝐺²), 𝜕²𝑐/𝜕𝐵²_𝑑=(𝐺²/𝑋𝐵³_𝑑)(𝜕²𝐾/𝜕𝐺²), 𝜕²𝑐/𝜕𝐺𝜕𝐵_𝑑=−(𝐺/𝑋𝐵²_𝑑)(𝜕²𝐾/𝜕𝐺²), and 𝜕𝑐/𝜕𝑇=𝑋𝐵_𝑑(𝜕𝐾/𝜕𝑇). Substituting them into (13) and simplifying, (13) becomes 12𝐺²𝑋𝐵_𝑑𝜎²_𝐺+𝜎²_𝑑−2𝜌_𝐺𝑑𝜎_𝐺𝜎_𝑑𝜕²𝐾𝜕𝐺²−𝑋𝐵_𝑑𝜕𝐾𝜕𝑇=0, (17) or 12𝐺𝑋𝐵_𝑑²𝜎²_𝐺+𝜎²_𝑑−2𝜌_𝐺𝑑𝜎_𝐺𝜎_𝑑𝜕²𝐾𝜕𝐺²−𝜕𝐾𝜕𝑇=0. (18)

Since 𝜃²=(𝐺/𝑋𝐵_𝑑)² and 𝜎²_𝜃=𝜎²_𝐺+𝜎²_𝑑−2𝜌_𝐺𝑑𝜎_𝐺𝜎_𝑑, (18) becomes 12𝜎²_𝜃𝜃²𝜕²𝐾𝜕𝜃²−𝜕𝐾𝜕𝑇=0. (19) In other words, 𝐾≡𝐾(𝜃,𝑇;𝑋) has to satisfy (19) subject to the following two boundary conditions: 𝐾(𝜃=0,𝑇;𝑋)=0 and 𝐾(𝜃,0;𝑋)=max(0,𝜃−1).

Remark 5. Given 𝜃=𝐺/𝑋𝐵_𝑑, 𝐺=0 is equivalent to 𝜃=0. Thus, we have the first boundary condition 𝐾(𝜃=0,𝑇;𝑋)=0. At time 𝑡+𝑇, 𝐵_𝑑(𝑇,𝑇)=1, and we have the second boundary condition 𝐾(𝜃,0;𝑋)=(𝑐(𝐺,𝐵_𝑑(𝑇,𝑇),0;𝑋))/(𝑋𝐵_𝑑(𝑇,𝑇))=(max(0,𝐺−𝑋))/𝑋=max(0,(𝐺/𝑋)−1)=max(0,𝜃−1).

Setting up another variable 𝑉≡∫_𝑡^𝑡+𝑇𝜎²_𝜃(𝑣)𝑑𝑣 and then defining (𝜃,𝑉)≡𝐾(𝜃,𝑇), we have that 𝜕²𝐾/𝜕𝜃²=𝜕²𝑦/𝜕𝜃² and 𝜕𝐾/𝜕𝑇=(𝜕𝑦/𝜕𝑉)(𝜕𝑉/𝜕𝑇)=(𝜕𝑦/𝜕𝑉)𝜎²_𝜃. Substituting them into (19), we obtain 12𝜎²_𝜃𝜃²𝜕²𝑦𝜕𝜃²−𝜕𝑦𝜎𝜕𝑉^2𝜃=𝜎²_𝜃12𝜃²𝜕²𝑦𝜕𝜃²−𝜕𝑦𝜕𝑉=0, (20) or 12𝜃²𝜕²𝑦𝜕𝜃²−𝜕𝑦𝜕𝑉=0. (21) In other words, 𝑦=𝑦(𝜃,𝑉) has to satisfy (21) subject to the following two boundary conditions: 𝑦(0,𝑉)=0 and 𝑦(𝜃,0)=max(0,𝜃−1).

To solve (21) subject to the two boundary conditions, we transform (21) by using the change of variables 𝛿≡log𝜃+(𝑉/2) and 𝑢(𝛿,𝑉)≡𝑦(𝜃,𝑉)/𝜃. Then we have 𝑦(𝜃,𝑉)=𝑢(log𝜃+(1/2)∫_𝑡^𝑡+𝑇𝜎²_𝜃(𝑣)𝑑𝑣,∫_𝑡^𝑡+𝑇𝜎²_𝜃(𝑣)𝑑𝑣)𝜃, 𝜕²𝑦/𝜕𝜃²=(1/𝜃)((𝜕𝑢/𝜕𝛿)+(𝜕²𝑢/𝜕𝛿²)), and 𝜕𝑦/𝜕𝑉=𝜃((1/2)(𝜕𝑢/𝜕𝛿)+(𝜕𝑢/𝜕𝑉)). Substituting them into (21) and simplifying, (21) becomes 𝜕𝑢/𝜕𝑉=(1/2)(𝜕²𝑢/𝜕𝛿²). For the first boundary condition, we have that 𝑢(𝛿,𝑉)=𝑦(𝜃,𝑉)/𝜃=(1/𝜃)𝐾(𝜃,𝑇)=(1/𝜃)(𝑐/𝑋𝐵_𝑑)=(𝑋𝐵_𝑑/𝐺)(𝑐/𝑋𝐵_𝑑)=𝑐/𝐺. As always, 0≤𝑐≤𝐺. Thus, the first boundary condition is |𝑢(𝛿,𝑉)|=(𝑐/𝐺)≤1. For the second boundary condition, we have that 1−(1/𝜃) = 1−exp[−log𝜃]=1−exp[−𝛿], because 𝛿=log𝜃 when 𝑉=0. Hence, 𝑢(𝛿,𝑉=0) = (𝜃−1)/𝜃=1−(1/𝜃)=1−exp[−𝛿] if 1≥exp[−𝛿] and 𝑢(𝛿,𝑉=0)=0 if 1<exp[−𝛿]. In short, our boundary value problem is made up of a boundedness condition |𝑢(𝛿,𝑉)|≤1 and the following two conditions: 𝜕𝑢(𝛿,𝑉)=1𝜕𝑉2𝜕²𝑢(𝛿,𝑉)𝜕𝛿²,𝑢()=[][][].𝛿,01−exp−𝛿if1≥exp−𝛿0if1<exp−𝛿 (22)(23)

Now (22) is in standard heat equation form and hence can be solved by the separation-of-variables method. Let 𝑢(𝛿,𝑉)=𝑓(𝛿)𝑔(𝑉), where 𝑓(𝛿) is some function of 𝛿 and 𝑔(𝑉) is some function of 𝑉. Substituting 𝑢(𝛿,𝑉)=𝑓(𝛿)𝑔(𝑉) into (22) and simplifying, we get 2𝑔(𝑉)𝜕𝑔(𝑉)=1𝜕𝑉𝑓(𝛿)𝜕²𝑓(𝛿)𝜕𝛿². (24) In other words, the right-hand side of (24) depends only on 𝛿 and the left-hand side depends only on 𝑉. Since 𝛿 and 𝑉 are independent variables, we can equate the two sides of (24) to a constant −𝑘², where 𝑘>0.

Remark 6. We equate the two sides of (24) to −𝑘² so that the two differential equations in (25) have continuous eigenvalues 𝑘².

Hence, by setting the two sides of (24) equal to −𝑘², we obtain the following two ordinary differential equations: 𝜕𝑔(𝑉)+1𝜕𝑉2𝑘²𝑔(𝑉)𝜕=0,₂𝑓(𝛿)𝜕𝛿²+𝑘²𝑓(𝛿)=0. (25)

Rewriting (25), we have 𝑔(𝑉)=exp(−(1/2)𝑘²𝑉) and 𝑓(𝛿)=𝐴(𝛿)cos(𝑘𝛿)+𝐵(𝛿)sin(𝑘𝛿). The generalized linear combination of functions 𝑢(𝛿,𝑉)=𝑓(𝛿)𝑔(𝑉) becomes 𝑢(𝛿,𝑉)=^∞₀[𝐴(𝑘)()(𝑘)(−1cos𝑘𝛿+𝐵sin𝑘𝛿)]×exp2𝑘²𝑉𝑑𝑘. (26)

By the Fourier integral theorem, the expression in (26) is legitimate if 𝐴(𝑘)=(1/𝜋)∫^∞_−∞𝑓(𝜔)cos(𝑘𝜔)𝑑𝜔 and 𝐵(𝑘)=(1/𝜋)∫^∞_−∞𝑓(𝜔)sin(𝑘𝜔)𝑑𝜔. Substituting 𝐴(𝑘) and 𝐵(𝑘) into (26), we obtain (see the Appendix for derivation of (27)) 𝑢()=1𝛿,𝑉𝜋^∞₀^∞_−∞[()()()((𝜔)−1cos𝑘𝜔cos𝑘𝛿+sin𝑘𝜔sin𝑘𝛿)]×𝑓𝑑𝜔exp2𝑘²𝑉=1𝑑𝑘√2𝑉𝜋^∞−∞𝑓(𝜔)−()exp𝜔−𝛿²2𝑉𝑑𝜔. (27)

Setting 𝑞=(𝜔−𝛿)/√2𝑉, we have 𝜔=𝛿+𝑞√2𝑉 and 𝑑𝜔=√2𝑉𝑑𝑞. Substituting 𝑞=(𝜔−𝛿)/√2𝑉 and (24) into (27), we get 𝑢()=1𝛿,𝑉√𝜋^∞_−∞𝑓𝛿+𝑞√2𝑉exp−𝑞²=1𝑑𝑞√𝜋^∞√_{−𝛿/2𝑉}1−exp−𝛿−𝑞√2𝑉exp−𝑞²=1𝑑𝑞√𝜋^∞√_{−𝛿/2𝑉}exp−𝑞²−1𝑑𝑞√𝜋^∞√_{−𝛿/2𝑉}exp−𝛿−𝑞√2𝑉−𝑞²𝑑𝑞. (28)

In order to solve (28), we make another change of variables by setting 𝑎=√2𝑞. Then the first term of (28) becomes 1√𝜋^∞√_{−𝛿/2𝑉}exp−𝑞²1𝑑𝑞=√2𝜋^{∞√−𝛿/𝑉}−𝑎exp²2=1𝑑𝑎√2𝜋^{√𝛿/𝑉−∞}−𝑎exp²2𝛿𝑑𝑎=𝑁√𝑉=𝑁𝑎₁, (29) where 𝑎₁=𝛿/√𝑉 and 𝑁(⋅) is the cumulative probability distribution function for a standardized normal random variable; that is, 𝑁(𝑎₁) is the probability that such a variable will be less than 𝑎₁.

For the second term of (28), note that −𝛿−𝑞√2𝑉−𝑞²=−log𝜃−(𝑉/2)−𝑎√𝑉−(𝑎²/2) = log(1/𝜃)−(1/2)[𝑎²+2𝑎√𝑉+(√𝑉)²]=log(1/𝜃)−(1/2)(𝑎+√𝑉)². Hence, the integrand of the second term is exp(−𝛿−𝑞√2𝑉−𝑞²)=exp[log(1/𝜃)−(1/2)(𝑎+√𝑉)²]=(1/𝜃)exp[−(1/2)(𝑎+√𝑉)²]. Substituting it into the second term and simplifying, (28) becomes 1√𝜋^∞√_{−𝛿/2𝑉}exp−𝛿−𝑞√2𝑉−𝑞²=1𝑑𝑞𝜃√2𝜋^𝑎1−∞−1exp2𝑎+√𝑉²=1𝑑𝑎𝜃√2𝜋^𝑎2−∞−𝑏exp²21𝑑𝑏=𝜃𝑁𝑎₂, (30) where 𝑏=𝑎+√𝑉 and 𝑎₂=𝑎₁−√𝑉. Combining (29) and (30), we obtain 𝑢(𝛿,𝑉)=𝑁𝑎₁−1𝜃𝑁𝑎₂, (31) or 𝑦(𝜃,𝑉)=𝜃𝑢(𝛿,𝑉)=𝜃𝑁𝑎₁−𝑁𝑎₂. (32)

As stated earlier, 𝐺=𝑆𝐵_𝑓, 𝜃=𝐺/𝑋𝐵_𝑑, and 𝑐(𝐺,𝐵_𝑑,𝑇;𝑋)=𝑋𝐵_𝑑𝐾(𝜃,𝑇;𝑋)=𝑋𝐵_𝑑𝑦(𝜃,𝑉). By the linear homogeneity property of 𝑐≡𝑐(𝑡,𝑇)≡𝑐(𝐺,𝐵_𝑑,𝑇;𝑋), the price of a European call FX option is 𝑐=𝑋𝐵_𝑑𝜃𝑁𝑎₁−𝑁𝑎₂=𝑆𝐵_𝑓𝑁𝑎₁−𝑋𝐵_𝑑𝑁𝑎₂, (33) where 𝑎₁=𝛿/√𝑉=(log𝜃+(𝑉/2))/√𝑉 and 𝑎₂=𝑎₁−√𝑉=(log𝜃−(𝑉/2))/√𝑉. In addition, 𝑉 is given by 𝑉≡_𝑡^𝑡+𝑇𝜎²_𝜃(𝑣)=𝑑𝑣_{𝑡𝑡+𝑇}𝜎²_𝑠−2𝜌_𝑠𝑑𝜎_𝑠𝑏_𝑑−𝜌_𝑠𝑓𝜎_𝑠𝑏_𝑓𝑣+𝑏²_𝑑+𝑏²_𝑓−2𝜌_𝑑𝑓𝑏_𝑑𝑏_𝑓𝑣²𝑑𝑣=𝜎²_𝑠𝑇−𝜌_𝑠𝑑𝜎_𝑠𝑏_𝑑−𝜌_𝑠𝑓𝜎_𝑠𝑏_𝑓𝑇²+13𝑏²_𝑑+𝑏²_𝑓−2𝜌_𝑑𝑓𝑏_𝑑𝑏_𝑓𝑇³. (34)

According to put-call parity for European options, the price 𝑝≡𝑝(𝐺,𝐵_𝑑,𝑇;𝑋) of a European put FX option is 𝑝=𝑐−𝑆𝐵_𝑓+𝑋𝐵_𝑑=𝑋𝐵_𝑑𝑁−𝑎₂−𝑆𝐵_𝑓𝑁−𝑎₁. (35)

Equations (33) through (35), as a whole, constitute our FX option model. If both domestic and foreign spot rates are constant (i.e., 𝑎_𝑑=𝑎_𝑓=𝑏_𝑑=𝑏_𝑓=0 in (2)), then the domestic and foreign bond prices in (7) become 𝐵_𝑑(𝑡,𝑇)=exp(−𝑟_𝑑𝑇) and 𝐵_𝑓(𝑡,𝑇)=exp(−𝑟_𝑓𝑇), and 𝑉≡∫_𝑡^𝑡+𝑇𝜎²_𝜃(𝑣)𝑑𝑣=𝜎²_𝑠𝑇. Substituting them into (33) and (35), our FX model then reduces to the following G-K FX model for valuing European call and put options: 𝑐_𝐺𝐾=𝑆exp−𝑟_𝑓𝑇𝑁𝑑₁−𝑋exp−𝑟_𝑑𝑇𝑁𝑑₂,𝑝_𝐺𝐾=𝑋exp−𝑟_𝑑𝑇𝑁−𝑑₂−𝑆exp−𝑟_𝑓𝑇𝑁−𝑑₁, (36) where 𝑑₁=(log(𝑆/𝑋)+(𝑟_𝑑−𝑟_𝑓+𝜎²_𝑠/2)𝑇)/𝜎_𝑠√𝑇 and 𝑑₂=𝑑₁−𝜎_𝑠√𝑇.

3. 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 3-month 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 𝑟_𝑑 and the euro-currency rate to represent the foreign spot rate 𝑟_𝑓. Accordingly, the estimates for the six parameters of our FX option model are as follows: 𝜎_𝑠=0.198428, 𝑏_𝑑=0.018534, 𝑏_𝑓=0.011398, 𝜌_𝑠𝑑=−0.347350, 𝜌_𝑠𝑓=−0.297520, and 𝜌_𝑑𝑓=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 the G-K model (36) by setting the initial time 𝑡=1 January 2010 when 𝑟_𝑑(𝑡)=0.0008, 𝑟_𝑓(𝑡)=0.0049, and 𝑆_𝑡=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 𝑆_𝑡/𝑋, where 𝑆_𝑡 is the exchange rate on 1 January 2010 and 𝑋 is the exercise price. Given that 𝑆_𝑡 is fixed at 1.4389, we vary 𝑋 such that 𝑆_𝑡/𝑋 ranges from 0.70 to 1.30—a range large enough to include even extreme values of 𝑋 not commonly used in practice. Tables 1, 2, 3, and 4 show the FX option prices and pricing biases in the G-K model when time to maturity 𝑇 is 1, 3, 6, and 9 months, respectively.

Table 1 

Prices for European call and put FX options when time to maturity 𝑇=1 month.

Table 2 

Prices for European call and put FX options when time to maturity 𝑇=3 months.

Table 3 

Prices for European call and put FX options when time to maturity 𝑇=6 months.

Table 4 

Prices for European call and put FX options when time to maturity 𝑇=9 months.

We first focus on call options. For 𝑇=1,3,6, and 9 months, call prices increase as 𝑆_𝑡/𝑋 increases from 0.70 to 1.30 under both our model and the G-K model. For example, in Table 1 where 𝑇=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 𝑆_𝑡/𝑋 increases from 0.90 to 1.00 and to 1.10, respectively. For each of the four 𝑇s, the G-K model incorrectly values FX calls for different values of 𝑆_𝑡/𝑋. Specifically, it undervalues calls when 𝑆_𝑡/𝑋 ranges from 0.70 to 1.08, and it overvalues calls when 𝑆_𝑡/𝑋 ranges from 1.18 to 1.30. For example, in Table 2 where 𝑇=3 months, call price is 0.000666 under our model and 0.000657 under the G-K model for 𝑆_𝑡/𝑋=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 𝑆_𝑡/𝑋=1.30, which amounts to a negative pricing bias of −0.0187%.

The situation is completely opposite for put options. For 𝑇=1,3,6, and 9 months, put prices decrease as 𝑆_𝑡/𝑋 increases from 0.70 to 1.30 under both models. For example, in Table 3 where 𝑇=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 𝑆_𝑡/𝑋 increases from 0.80 to 1.00 and to 1.20, respectively. For each of the four 𝑇s, the G-K model incorrectly values FX puts for different values of 𝑆_𝑡/𝑋. Specifically, it overvalues puts when 𝑆_𝑡/𝑋 ranges from 0.70 to 0.82, and it undervalues puts when 𝑆_𝑡/𝑋 ranges from 0.86 to 1.30. For example, in Table 4 where 𝑇=9 months, put price is 0.375783 under our model and 0.376021 under the G-K model for 𝑆_𝑡/𝑋=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 𝑆_𝑡/𝑋=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 euro-currency option is €62,500. For example, when 𝑇=9 months and 𝑆_𝑡/𝑋=0.90, call premium is ($0.042077 × 62,500) = $2,629.81 under our model and ($0.041614 × 62,500) = $2,600.88 under the G-K model—a difference of $28.93 underpaid by a call option buyer; similarly, when 𝑇=9 months and 𝑆_𝑡/𝑋=1.10, put premium is ($0.044294 × 62,500) = $2,768.38 under our model and ($0.043837 × 62,500) = $2,739.81 under the G-K model—a difference of $28.57 underpaid 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.

4. Conclusion

Prior research often assumes constant spot rates when valuing FX options. The traditional G-K FX option model assumes that both domestic and foreign spot rates remain unchanged over the life of the FX option. The fact of the matter is that both spot rates are changing continuously and stochastically through time. In this paper, we utilize stochastic calculus to develop a continuous-time model for valuing European call and put options on foreign exchange when both spot rates are assumed to follow a generalized Wiener process. Using the dollar/euro exchange rate as input for parameter estimation 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 𝑆_𝑡/𝑋. Specifically, it undervalues calls when 𝑆_𝑡/𝑋 is between 0.70 and 1.08, and it overvalues calls when 𝑆_𝑡/𝑋 is between 1.18 and 1.30, whereas it overvalues puts when 𝑆_𝑡/𝑋 is between 0.70 and 0.82, and it undervalues puts when 𝑆_𝑡/𝑋 is between 0.86 and 1.30.

Appendix

To derive (27), we first prove the following:^∞₀[𝑘(−1cos𝜔−𝛿)]exp2𝑘²𝑉=𝑑𝑘𝜋−()2𝑉exp𝜔−𝛿².2𝑉 (A.1)

Proof. Since cos[𝑘(𝜔−𝛿)]=(1/2)exp[𝑖𝑘(𝜔−𝛿)]+(1/2)exp[−𝑖𝑘(𝜔−𝛿)], we obtain [𝑘(−1cos𝜔−𝛿)]exp2𝑘²𝑉=12−1exp2𝑘²(+1𝑉+𝑖𝑘𝜔−𝛿)2−1exp2𝑘²𝑉−𝑖𝑘(𝜔−𝛿), (A.2) where 𝑖=√−1. Since −(1/2)𝑘²𝑉+𝑖𝑘(𝜔−𝛿)=−((𝜔−𝛿)²/2𝑉)−[(𝑘𝑉−𝑖(𝜔−𝛿))/√2𝑉]², we have 12^∞₀−1exp2𝑘²(=1𝑉+𝑖𝑘𝜔−𝛿)𝑑𝑘2^∞₀−()exp𝜔−𝛿²−()2𝑉𝑘𝑉−𝑖𝜔−𝛿√2𝑉²=1𝑑𝑘2−()exp𝜔−𝛿²×2𝑉_∞0−()exp𝑘𝑉−𝑖𝜔−𝛿√2𝑉²𝑑𝑘. (A.3)
Letting 𝑥=(𝑘𝑉−𝑖(𝜔−𝛿))/√2𝑉, which implies 𝑑𝑘=(2/𝑉)𝑑𝑥, we have 12^∞₀−1exp2𝑘²(=1𝑉+𝑖𝑘𝜔−𝛿)𝑑𝑘2−()exp𝜔−𝛿²2𝑉^∞₀exp−𝑥²2𝑉=1𝑑𝑥2𝜋−()2𝑉exp𝜔−𝛿².2𝑉 (A.4)
Similarly, since −(1/2)𝑘²𝑉−𝑖𝑘(𝜔−𝛿)=−((𝜔−𝛿)²/2𝑉)−[(𝑘𝑉+𝑖(𝜔−𝛿))/√2𝑉]², we have 12^∞₀−1exp2𝑘²(=1𝑉−𝑖𝑘𝜔−𝛿)𝑑𝑘2−()exp𝜔−𝛿²×2𝑉_∞0−()exp𝑘𝑉+𝑖𝜔−𝛿√2𝑉²𝑑𝑘. (A.5)
Letting 𝑧=(𝑘𝑉+𝑖(𝜔−𝛿))/√2𝑉, which implies 𝑑𝑘=(2/𝑉)𝑑𝑧, we have 12^∞₀−1exp2𝑘²(=1𝑉−𝑖𝑘𝜔−𝛿)𝑑𝑘2−()exp𝜔−𝛿²2𝑉^∞₀exp−𝑧²2𝑉=1𝑑𝑧2𝜋−()2𝑉exp𝜔−𝛿².2𝑉 (A.6)
Hence, combining (A.4) and (A.6), we obtain (A.1). Now substituting 𝐴(𝑘)=(1/𝜋)∫^∞_−∞𝑓(𝜔)cos(𝑘𝜔)𝑑𝜔 and 𝐵(𝑘)=(1/𝜋)∫^∞_−∞𝑓(𝜔)sin(𝑘𝜔)𝑑𝜔 into (26), we obtain 𝑢()=1𝛿,𝑉𝜋^∞₀^∞_−∞[()()()((𝜔)−1cos𝑘𝜔cos𝑘𝛿+sin𝑘𝜔sin𝑘𝛿)]×𝑓𝑑𝜔exp2𝑘²𝑉=1𝑑𝑘𝜋^∞_−∞𝑓(𝜔)^∞₀[𝑘(−1cos𝜔−𝛿)]×exp2𝑘²𝑉𝑑𝑘𝑑𝜔. (A.7)
Substituting (A.1) into (A.7), we get (27) as follows: 𝑢()=1𝛿,𝑉𝜋^∞_−∞𝑓(𝜔)𝜋−()2𝑉exp𝜔−𝛿²=12𝑉𝑑𝜔√2𝑉𝜋^∞−∞𝑓(𝜔)−()exp𝜔−𝛿²2𝑉𝑑𝜔. (A.8)

Acknowledgments

The author would like to thank two anonymous referees whose valuable suggestions have greatly improved the quality of this paper. This research was supported by the National Science Council of Taiwan (ROC) under Grant NSC-98-2410-H-130-028.