hedging - Dynamic Hedge of Quanto Options

Can anybody explain to me step-by-step how can I dynamically hedge and/or replicate a quanto option with the foreign underlying asset, the foreign cash account and the domestic cash account as detailed as possible? And if you could recommend books or articles that would be also great. Thanks

Answer

Your simulation is basically fine, though you need to discount in USD. For hedging purpose, you need to use the instruments available in USD.

Let $S=\{S_t, \, t\ge 0\}$ be the stock price process in EUR, $X=\{X_t, \, t\ge 0\}$ be the exchange rate process from one unit EUR to units USD, $r_f$ and $r_d$ be interest rates in EUR and USD. Moreover, let $B_t^f=e^{r_f t}$ and $B_t^d=e^{r_d t}$ be respectively the money market account values in EUR and USD. Then the available instruments in USD are $XS$, $B^d$, and $B^fX$. Specifically, we assume that $X$ and $S$ satisfy a system of SDEs of the form

$$\begin{align*} dS_t &= S_t\left(\mu_s dt + \sigma_s dW_t^1 \right),\\ dX_t &= X_t\left[\mu_x dt + \sigma_x \left(\rho dW_t^1 + \sqrt{1-\rho^2}dW_t^2\right) \right], \end{align*}$$ where $\mu_s$ and $\mu_x$ are drift terms, $\rho$ is the correlation, $\{W_t^1, t\ge0\}$ and $\{W_t^2, t\ge0\}$ are two independent standard Brownian motions.

Let $C(t, S_t)$ be the quanto option price at time $t$. We seek a self-financing portfolio such that

$$\begin{align*} C(t, S_t) = \Delta_t^1 X_tS_t + \Delta_t^2 X_t B_t^f + \Delta_t^3 B_t^d.\tag{1} \end{align*}$$ Then, $$\begin{align*} dC &= \Delta_t^1 d\left(X_tS_t\right) + \Delta_t^2 d\left(X_t B_t^f\right) + \Delta_t^3 d\left(B_t^d\right)\\ &=\Delta_t^1X_tS_t\left[\left(\mu_s + \mu_x + \rho\sigma_s\sigma_x \right) dt + \sigma_s dW_t^1 + \sigma_x \left(\rho dW_t^1 + \sqrt{1-\rho^2}dW_t^2\right) \right]\\ &\quad + \Delta_t^2 X_t B_t^f\left[(\mu_x + r_f) dt + \sigma_x \left(\rho dW_t^1 + \sqrt{1-\rho^2}dW_t^2\right) \right] + r_d\Delta_t^3 B_t^d dt. \end{align*}$$ On the other hand $$\begin{align*} dC &= \frac{\partial C}{\partial t}dt + \frac{\partial C}{\partial S}S_t \left(\mu_s dt + \sigma_s dW_t^1 \right) + \frac{1}{2}\frac{\partial^2 C}{\partial S^2}S_t^2 \sigma_s^2 dt. \end{align*}$$ That is, $$\begin{align*} \frac{\partial C}{\partial S}S_t\sigma_s dW_t^1 &= \Delta_t^1X_tS_t\left[\sigma_s dW_t^1 + \sigma_x \left(\rho dW_t^1 + \sqrt{1-\rho^2}dW_t^2\right) \right]\\ &\qquad\qquad\qquad + \Delta_t^2 X_t B_t^f \sigma_x \left(\rho dW_t^1 + \sqrt{1-\rho^2}dW_t^2\right),\tag{2} \end{align*}$$ and $$\begin{align*} &\ \frac{\partial C}{\partial t}dt + \frac{\partial C}{\partial S}S_t \mu_s dt + \frac{1}{2}\frac{\partial^2 C}{\partial S^2}S_t^2 \sigma_s^2 dt \\ =&\ \Delta_t^1X_tS_t\left(\mu_s + \mu_x + \rho\sigma_s\sigma_x \right) dt+\Delta_t^2 X_t B_t^f(\mu_x + r_f) dt+r_d\Delta_t^3 B_t^d dt.\tag{3} \end{align*}$$ From $(2)$, $$\begin{align*} &\Delta_t^1X_tS_t \sigma_x + \Delta_t^2 X_t B_t^f \sigma_x =0,\\ &\frac{\partial C}{\partial S}S_t\sigma_s = \Delta_t^1X_tS_t\left(\sigma_s+ \sigma_x \rho\right) + \Delta_t^2 X_t B_t^f \sigma_x \rho. \end{align*}$$ Combining with $(1)$ above, $$\begin{align*} \Delta_t^1 &= \frac{1}{X_t}\frac{\partial C}{\partial S}, \\ \Delta_t^2 &= -\frac{S_t}{B_t^f}\Delta_t^1, \\ \Delta_t^3 &=\frac{C(t, S_t)}{B_t^d}. \end{align*}$$ From $(3)$, we obtain the Black-Scholes type PDE $$\begin{align*} \frac{\partial C}{\partial t} + \left(r_f - \rho\sigma_s\sigma_x \right)S_t \frac{\partial C}{\partial S} + \frac{1}{2}\frac{\partial^2 C}{\partial S^2}S_t^2 \sigma_s^2 = r_d C. \end{align*}$$ See also the notes here.