FX Call under stochastic rates and deterministic volatility

Question

Lets denote $S_t$, $r^d_t$,$r^f_t$ respectively the FX spot, the domestic rate and the foreign rate at time $t$.

Lets $\mathbb{Q}^d$ , $\mathbb{Q}^f$ respectively be the domestic and foreign mesures, and lets suppose that the rates follows Hull and white dynamics and that the FX spot follow a lognormal-like diffusion with constante volatility.

Hence we have the following equations : $$dr_t^d=(a^d+\lambda_dr_t^d)dt+\sigma_ddW_t^d \text{ under }\mathbb{Q}^d$$ $$dr_t^f=(a^f+\lambda_fr_t^f)dt+\sigma_ddW_t^f \text{ under }\mathbb{Q}^f$$ $$ dS_t=(r^d_t-r^f_t)S_tdt+\sigma^{cst}S_tdW_t \text{ under }\mathbb{Q}^d$$

with $$dW_t^ddW_t^f=\rho_1dt$$ $$dW_t^ddW_t=\rho_2dt$$ $$dW_tdW_t^f=\rho_3dt$$

Question: Is there a closed formula solution for the price of a currency call option under this system of equations ? I would like to study the impact of correlations on the price of the option and see how does it impact the closed formula for currency call option obtained under Black-scholes model. One can notice the resemblance with the case of equity call option under black-scholes with stochastic interest rates already answered in a previous post.

Thanks !

Answer 1

Consider the call option with payoff $(S_T-K)^+$ at the option maturity $T$. Note that the forward exchange rate \begin{align*} F(t, T) = S_t \frac{P^f(t, T)}{P^d(t, T)} \end{align*} is a martingale under the domestic $T$-forward probability measure $Q^T$, where $P^d(T, T)$ and $P^f(T, T)$ are prices at time $t$ of respective domestic and foreign zero-coupon bonds with maturity $T$ and unit face values. As in this question, let \begin{align*} B_a(t, T) = \frac{1}{\lambda_a}\Big(e^{\lambda_a(T-t)}-1 \Big), \end{align*} for $a=d$ and $f$. Then, under $Q^T$, \begin{align*} \frac{d F(t, T)}{F(t, T)} &= -\sigma_f B_f(t, T) dW_t^f + \sigma_d B_d(t, T) dW_t^d + \sigma^{cst} dW_t. \end{align*} Here, we use the same notations for the respective Brownian motions. Let $\sigma$ be a quantity defined by \begin{align*} T \sigma^2 &= \int_0^T\Big(\left(\sigma_f B_f(t, T)\right)^2 + \left(\sigma_d B_d(t, T)\right)^2 + (\sigma^{cst})^2 \\ &\qquad\qquad - 2 \sigma_d\sigma_f\rho_1 B_d(t, T)B_f(t, T) - 2 \rho_3 \sigma^{cst} \sigma_f B_f(t, T) + 2\rho_2 \sigma^{cst} \sigma_d B_d(t, T)\Big) dt. \end{align*} Then, the option value is given by \begin{align*} P^d(0, T)E_{Q^T}\left((F(T, T)-K)^+\right) &=P^d(0, T)\Big[F(0, T)N(d_1) - KN(d_2) \Big], \end{align*} where $d_1 = \frac{\ln F(0, T)/K + \frac{1}{2}\sigma^2 T}{\sigma \sqrt{T}}$ and $d_2 = d_1 - \sigma \sqrt{T}$.