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Let $X_t^{gbp\rightarrow usd}$ and $X_t^{chf\rightarrow usd}$ be respectively the 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 $X_0^{gbp\rightarrow usd}$.

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 $X_0^{gbp\rightarrow usd}$.

Let $X_t^{gbp\rightarrow usd}$ and $X_t^{chf\rightarrow usd}$ be respectively the 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*} \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 for a quanto style option.

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 the foreign currency interest rate given by $r^{chf} - r^{usd}+r^{gbp}-\rho\sigma_1\sigma_2$.

Let $X_t^{gbp\rightarrow usd}$ and $X_t^{chf\rightarrow usd}$ be respectively the 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 $X_0^{gbp\rightarrow usd}$.

Let $X_t^{gbp\rightarrow usd}$ and $X_t^{chf\rightarrow usd}$ be respectively the 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*} \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 for a quanto style option.

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

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]. \end{align*} 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 also be expressed by a Black-Scholes' type formula. Note, however, the correlation $\rho$ and both volatilities $\sigma_1$ and $\sigma_2$ are involved.

Let $X_t^{gbp\rightarrow usd}$ and $X_t^{chf\rightarrow usd}$ be respectively the 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*} \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 for a quanto style option.

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 the foreign currency interest rate given by $r^{chf} - r^{usd}+r^{gbp}-\rho\sigma_1\sigma_2$.