# 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

• That's a very broad question - especially when you insist on the "step-by-step" explanation and "as detailed as possible". I think you would probably attract more answers by explaining what your understanding is and where you get stuck. – LocalVolatility Feb 28 '17 at 0:06
• I have short call option on a VW stock which is denominated in EUR and the payoff /max[S(T)-K); 0]/ is denominated in USD. So I simulated the VW stock process and the EURUSD price process, calculated the the payoff in EUR, converted to USD with the exchange rate at maturity (this is what I simulated), took the average then converted back on the spot rate. I also calculated the delta, the first derivative with respect to the VW stock and I know i have to buy delta stocks to hegde the risk from the stock stock process but I don't how to handle the the risk from the EURUSD process. – Nikola Feb 28 '17 at 18:35

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.
• @Gordon How well off would the replication be if we simply did the usual hedge of investing Delta (of the quanto price) into the underlying foreign stock, and borrowed from the domestic bank? Would it always be completely off, or would it depend on on how much uncertainty there was with respect to exchange rate $X$? My own attempts of using this simpler strategy fails even when $X$ is a constant, which I found surprising. Is that just a mistake on my part, or is it true that this strategy actually does badly even when $X$ is known always? – Jaood Mar 2 '17 at 23:21