2
$\begingroup$

How is the option price of an plain vanilla option (in a Black Scholes setting) derived, which is written on, say XAGGBP but practically hedged with XAGUSD and GBPUSD (because these are more liquid)? Eventually, I am interested in the delta(s) and correlation risk with respect to XAGUSD and GBPUSD.

$\endgroup$
1
$\begingroup$

Let $X_t^{xag\rightarrow gbp}$ be the exchange rate from one unit of XAG to units of GBP. Moreover, let $X_t^{xag\rightarrow usd}$, and $X_t^{gbp\rightarrow usd}$ be the respective exchanges rates from one unit of XAG and GBP to units of USD. We consider an option payoff, in GBP, at maturity $T$ of the form \begin{align*} \left(X_T^{xag\rightarrow gbp} -K\right)^+. \tag{1} \end{align*} Note that \begin{align*} \left(X_T^{xag\rightarrow gbp} -K\right)^+ = \left(\frac{X_T^{xag\rightarrow usd}}{X_T^{gbp\rightarrow usd}} -K\right)^+. \end{align*} We assume that, under the USD risk-neutral probability measure $Q_{usd}$, \begin{align*} dX_t^{xag\rightarrow usd} &= X_t^{xag\rightarrow usd}\left[\left(r^{usd}-r^{xag} \right)dt +\sigma_1 dW_t^1 \right],\\ dX_t^{gbp\rightarrow usd} &= X_t^{gbp\rightarrow usd}\left[\left(r^{usd}-r^{gbp} \right)dt +\sigma_2\left(\rho dW_t^1 +\sqrt{1-\rho^2}dW_t^2\right)\right], \end{align*} where $r^{usd}$, $r^{gbp}$, and $r^{xag}$ are interest rates, $\sigma_1$ and $\sigma_2$ are volatilities, $\rho$ is the correlation, and $\{W_t^1, \, t\ge 0\}$ and $\{W_t^2, \, t\ge 0\}$ are two standard independent Brownian motions.

Let $B_t^{usd}=e^{r^{usd} t}$ and $B_t^{gbp}=e^{r^{gbp} t}$ be the respective USD and GBP money market account values at time $t$. Moreover, let $Q^{gbp}$ be the GBP risk-neutral probability measure. Note that \begin{align*} \frac{dQ^{gbp}}{dQ^{usd}}\big|_t &= \frac{B_t^{gbp}X_t^{gbp\rightarrow USD}}{B_t^{usd}X_0^{gbp\rightarrow 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*} Then, $\{\tilde{W}_t^1, \, t\ge 0\}$ and $\{\tilde{W}_t^2, \, t\ge 0\}$, where \begin{align*} \tilde{W}_t^1 &= W_t^1 - \sigma_2\rho t, \\ \tilde{W}_t^2 &= W_t^2 - \sigma_2\sqrt{1-\rho^2} t, \end{align*} are two standard independent Brownian motions under $Q_{gbp}$. Furthermore, under $Q^{gbp}$, \begin{align*} dX_t^{xag\rightarrow usd} &= X_t^{xag\rightarrow usd}\left[\left(r^{usd}-r^{xag} +\rho\sigma_1\sigma_2\right)dt +\sigma_1 d\tilde{W}_t^1 \right],\\ dX_t^{gbp\rightarrow usd} &= X_t^{gbp\rightarrow usd}\left[\left(r^{usd}-r^{gbp} +\sigma_2^2\right)dt +\sigma_2\left(\rho d\tilde{W}_t^1 +\sqrt{1-\rho^2}d\tilde{W}_t^2\right)\right]. \end{align*} Then \begin{align*} X_t^{xag\rightarrow gbp} &= \frac{X_t^{xag\rightarrow usd}}{X_t^{gbp\rightarrow usd}} \\ &=\frac{X_0^{xag\rightarrow usd}}{X_0^{gbp\rightarrow usd}} e^{\left(r^{gbp}-r^{xag}+\rho\sigma_1\sigma_2-\frac{1}{2}\sigma_1^2 -\frac{1}{2}\sigma_2^2\right)t + (\sigma_1-\rho\sigma_2)\tilde{W}_t^1 -\sigma_2\sqrt{1-\rho^2}\tilde{W}_t^2}\\ &=X_0^{xag\rightarrow gbp} e^{\left(r^{gbp}-r^{xag}-\frac{\sigma_1^2+\sigma_2^2-2\rho\sigma_1\sigma_2}{2}\right)t + \sqrt{\sigma_1^2+\sigma_2^2 - 2\rho\sigma_1\sigma_2 }\frac{(\sigma_1-\rho\sigma_2)\tilde{W}_t^1 -\sigma_2\sqrt{1-\rho^2}\tilde{W}_t^2}{\sqrt{\sigma_1^2+\sigma_2^2 - 2\rho\sigma_1\sigma_2 }} }. \end{align*}

Let \begin{align*} \sigma = \sqrt{\sigma_1^2+\sigma_2^2 - 2\rho\sigma_1\sigma_2 }, \end{align*} and \begin{align*} W_t^3 = \frac{(\sigma_1-\rho\sigma_2)\tilde{W}_t^1 -\sigma_2\sqrt{1-\rho^2}\tilde{W}_t^2}{\sqrt{\sigma_1^2+\sigma_2^2 - 2\rho\sigma_1\sigma_2 }}. \end{align*} Then $\{W_t^3, \, t \ge 0\}$ is a standard Brownian motion under $Q^{gbp}$, by Levy's characterization. Moreover, \begin{align*} d X_t^{xag\rightarrow gbp} = X_t^{xag\rightarrow gbp}\left[\left(r^{gbp}-r^{xag} \right)dt +\sigma dW_t^3 \right]. \end{align*} Therefore, the option payoff $(1)$ can be valued using the Garman Kohlhagen formula, while replace the initial exchange rate $X_0^{xag\rightarrow gbp}$ by $\frac{X_0^{xag\rightarrow usd}}{X_0^{gbp\rightarrow usd}}$. The respective hedge ratios can be computed subsequently.

$\endgroup$
  • $\begingroup$ Great answer, thank you! In order to compute the vega(s), there's basically three choices (w.r.t $σ$, $σ_1$, $σ_2$). The correlation risk could be computed w.r.t. $ρ$ and "using" this model would be a question about choosing proper values for $ρ$, $σ_1$ and $σ_2$, right? $\endgroup$ – Tim Feb 12 '17 at 16:27
  • $\begingroup$ The parameter $\sigma_1$ and $\sigma_2$ can be obtained from market quotes as they are more liquid, while $\rho$ is usually from historical estimation. $\endgroup$ – Gordon Feb 12 '17 at 16:49
  • $\begingroup$ Sry for the bothering, I've another question about the derivation above. What I find hard to understand is why are we performing the measure change from the USD risk neutral proabalility measure to the GBP risk neutral proabalility measure? Could we not have derived an answer already under the USD risk neutral proabalility measure? I assume the answer lies in the Radon-Nikodym derivative. $\endgroup$ – Tim Feb 13 '17 at 9:58
  • $\begingroup$ What may help, would be a brief verbal explanation of the right side of the Radon-Nikodym derivative. I'm still reading the chapter of Shreve. $\endgroup$ – Tim Feb 13 '17 at 10:08
  • $\begingroup$ The dynamics of the exchange rate from Cur to Cur1 has a standard form as shown in the answer. However, for the payoff in Cur2, we first find the Radon-Nikodym derivative and then use Girsanov theorem to find the dynamics under the Cur2 risk-neutral measure. The Radon-Nikodym derivative is basically the ratio of the normalized numeraires, converted to the same currencies. $\endgroup$ – Gordon Feb 13 '17 at 14:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.