FX Option with Different Premium Currency

Question

When valuing an FX option with some model M, e.g. Garman Kohlhagen for a call on GBPUSD spot, with a different premium currency, say CHF, is it correct to simply multiply the obtained option price (in GBP) with the current spot rate of the premium currency in order to obtain the option value in the premium currency, e.g:

price_in_CHF = GBPCHF * GarmanKohlhagen(GBPUSD,...) 

Would that affect the delta (in terms of GBPUSD) in the same way?

Answer 1

Let $X_t^{gbp\rightarrow usd}$ and $X_t^{chf\rightarrow usd}$ be the respective exchanges rates from one unit of GBP and CHF to units of USD. Depending on the option contractual specification, the payoff in CHF at maturity $T$ can have a form of either \begin{align*} \frac{\left(X_T^{gbp\rightarrow usd} -K\right)^+}{X_T^{chf\rightarrow usd}}, \tag{1} \end{align*} or \begin{align*} \left(X_T^{gbp\rightarrow usd} -K\right)^+. \tag{2} \end{align*} Here, Payoff $(2)$ is of a quanto form.

Let $r^{chf}$, $r^{gbp}$, and $r^{usd}$ be the respective interest rates for currencies CHF, GBP, and USD, and $B_t^{usd}=e^{r^{usd} t}$, $B_t^{gbp}=e^{r^{gbp} t}$, and $B_t^{chf}=e^{r^{chf} t}$ be the corresponding money-market account values at time $t$. Moreover, let $Q^{chf}$ and $Q^{usd}$ be the respective risk-neutral probability measures for currencies CHF and USD, and $E^{chf}$ and $E^{usd}$ be the corresponding expectation operators.

We assume that, under the USD risk-neutral probability measure $Q^{usd}$, \begin{align*} dX_t^{gbp\rightarrow usd}&= X_t^{gbp\rightarrow usd}\left[(r^{usd}-r^{gbp})dt + \sigma_1 dW_t^1 \right],\\ dX_t^{chf\rightarrow usd}&= X_t^{chf\rightarrow usd}\left[(r^{usd}-r^{chf})dt + \sigma_2 \left(\rho dW_t^1 + \sqrt{1-\rho^2}dW_t^2\right) \right], \end{align*} where $\sigma^1$ and $\sigma^2$ are the volatility parameters, $\rho$ is the correlation, $\{W_t^1, t\ge 0\}$ and $\{W_t^2, t\ge 0\}$ are independent standard Brownian motions.

Payoff Form $(1)$

For a payoff of the form $(1)$, note that \begin{align*} \frac{dQ^{chf}}{dQ^{usd}}\big|_t = \frac{X_t^{chf\rightarrow usd}B_t^{chf}}{X_0^{chf\rightarrow usd}B_t^{usd}}. \end{align*} Then \begin{align*} E^{chf}\left(\frac{1}{B_T^{chf}}\frac{\left(X_T^{gbp\rightarrow usd} -K\right)^+}{X_T^{chf\rightarrow usd}} \right) &=E^{usd}\left(\frac{dQ^{chf}}{dQ^{usd}}\big|_T\frac{1}{B_T^{chf}}\frac{\left(X_T^{gbp\rightarrow usd} -K\right)^+}{X_T^{chf\rightarrow usd}} \right)\\ &=\frac{1}{X_0^{chf\rightarrow usd}}E^{usd}\left(\frac{\left(X_T^{gbp\rightarrow usd} -K\right)^+}{B_T^{usd}}\right). \end{align*} That is, the payoff value is indeed the normal option value adjusted by the spot exchange rate.

Payoff Form $(2)$

For a quanto style payoff of the form $(2)$, note that \begin{align*} \frac{dQ^{chf}}{dQ^{usd}}\big|_t &= \frac{X_t^{chf\rightarrow usd}B_t^{chf}}{X_0^{chf\rightarrow usd}B_t^{usd}}\\ &=e^{-\frac{1}{2}\sigma_2^2 t + \sigma_2 \left(\rho W_t^1 + \sqrt{1-\rho^2}W_t^2\right)}. \end{align*} Let $\tilde{W}_t^1 = W_t^1-\rho\sigma_2 t$ and $\tilde{W}_t^2 = W_t^2-\sqrt{1-\rho^2}\sigma_2 t$. Then $\{\tilde{W}_t^1, t\ge 0\}$ and $\{\tilde{W}_t^2, t\ge 0\}$ are two independent standard Brownian motions. Moreover, under the CHF risk-neutral probability measure $Q^{chf}$, \begin{align*} dX_t^{gbp\rightarrow usd}&= X_t^{gbp\rightarrow usd}\left[(r^{usd}-r^{gbp}+\rho\sigma_1\sigma_2)dt + \sigma_1 d\tilde{W}_t^1 \right]\\ &=X_t^{gbp\rightarrow usd}\left[\Big(r^{chf} - (r^{chf} - r^{usd}+r^{gbp}-\rho\sigma_1\sigma_2)\Big)dt + \sigma_1 d\tilde{W}_t^1 \right]. \end{align*} That is, we can treat $X_t^{gbp\rightarrow usd}$ as the exchange rate from a foreign currency to CHF, where the foreign currency has interest rate $r^{chf} - r^{usd}+r^{gbp}-\rho\sigma_1\sigma_2$. The value of Payoff $(2)$ is then given by \begin{align*} E^{chf}\left(\frac{\left(X_T^{gbp\rightarrow usd} -K\right)^+}{B_T^{chf}} \right), \end{align*} which can be computed using the Garman Kohlhagen formula with domestic interest rate $r^{chf}$, foreign interest rate $r^{chf} - r^{usd}+r^{gbp}-\rho\sigma_1\sigma_2$, and spot FX rate $X_0^{gbp\rightarrow usd}$.

Answer 2

I don't see how settling the premium cashflow, whether it's in some currency or other, or paid on time or late, or rolled into the contract, or any other way of paying that you could think of, will affect the value of the option in its base currency.

Yeah so to express the option value in some other currency, then you use the spot rate. If you wanted to delta hedge using some other currency (not sure why?) then likewise for hedging cash flows.