# FX Option with Different Premium Currency

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?

• You need volatilities from two exchange rates and the correlation. That is, you need a measure change from CHF to USD. – Gordon Feb 7 '17 at 21:45
• @Gordon, thx, would you mind providing an example under GBM assumptions? – Tim Feb 7 '17 at 22:00
• @Tim look up quanto options and girsanov. Essentially the result is you change your fwd curve by $e^{-\rho_{AB} \sigma_A \sigma_B t}$. The question then becomes what volatilities do you use? – will Feb 8 '17 at 11:02
• @will, I'm really referring to case (1) given by Gordon's answer below. Sry if my question was confusing. However, I'm also glad for your comment as this may have inspired the comparision between the two payout structures. – Tim Feb 8 '17 at 21:10

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 $X_0^{gbp\rightarrow usd}$.

• That's a gorgeous answer. I'm very thankful for the effort and it's what I was looking for! Just a brief question about (2) above. Shouldn't the payoff be multiplied with a quanto factor, that is, a pre-specified exchange rate? – Tim Feb 8 '17 at 20:58
• @Tim: Thanks for accepting my answer. For payoff (2), there is no quanto factor multiplication. The quanto adjustment has already been factored into the drift term, or a synthetic foreign interest rate. – Gordon Feb 8 '17 at 21:21
• I see that now, thx. Could you just write one more sentence on the dQ/dQ part? I have seen and understood that before, just need a reminder. – Tim Feb 8 '17 at 21:51
• @Tim: That is the measure change based on Radon–Nikodym derivative. A good source is in Chapter 9 of the book by Shreve. – Gordon Feb 8 '17 at 21:56
• I read some parts of the book a while ago. Will read this part again. – Tim Feb 8 '17 at 22:05

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.

• Depends on the exact details of how the option works (what happens at maturity). For a plain vanilla option paid in another currency you are quite right. – noob2 Feb 8 '17 at 16:22
• Thx for your answer, I appreciate. My intuition went into this direction as well, however, I feel more comfortable by the derivation of Gordon. My stochastic calculus skills needed a refesher. – Tim Feb 8 '17 at 21:04