A quick refresher
A quanto option is a derivative where the underlying is denominated in one currency, but the option-payoff is settled in a different one (the quanto-currency) at a pre-defined fixed exchange rate \(Q\).
I take the example of gold quoted as \(XAU/USD\) that is quantoed in \(INR\).
Define :
\[ \begin{align*} S_T &:= \text{Price of gold in the underlying currency} \\ X_T &:= \text{Price of the underlying currency in quanto currency terms}\\ K &:= \text{Strike expressed in underlying currency terms}\\ Q &:= \text{Pre-specified exchange rate} \end{align*} \]
Since the payoff is in \(INR\), we take \(INR\) as the base currency or numeraire in the Black-Scholes model.
Then, the payoff of the quanto-call option is:
\[ V_T = Q\left(\frac{S_T}{X_T} - K\right)^{+} \]
\(S_T/X_T\) has units \(\frac{XAU}{USD}\), and \(Q\) - the pre-specified conversion factor has units \(\frac{USD}{INR}\).
The Setup
The domestic risk-neutral measure \(Q^{INR}\) is the probability measure linked to the domestic money-market account \(M_T^{INR}\).
The risk-neutral measure \(Q^{USD}\) is the probability measure linked to the underlying money-market account expressed in quanto currency terms, \((M_T^{USD} \cdot X_T)\). \(M_T^{USD}\) has units \(USD^{-1}\) and \(X_T\) has units \(USD \cdot INR^{-1}\).
Consider the Black-Scholes model:
\[ \begin{align*} {XAU/INR : } \quad dS_t &= r_{INR} S_t dt + \sigma_S S_t dW_{S}^{Q^{INR}}(t) \\ {USD/INR : } \quad dX_t &= (r_{INR} - r_{USD})X_t dt + \sigma_X X(t) dW_X^{Q^{INR}}(t)\\ dW_{S}^{Q^{INR}}(t) \cdot dW_X^{Q^{INR}}(t) &= \rho_{(f,q),(d,q)} dt \end{align*} \]
where \((W^{Q^{INR}}(t),t\geq 0)\) is a \(Q^{INR}\)-standard brownian motion.
The evolution of the underlying XAU-USD
The actual underlying is :
\[ \text{XAU/USD } := \frac{S(t)}{X(t)} \]
Using Ito’s formula, we obtain:
\[ \begin{align*} d\left(\frac{1}{X_t}\right) &= -\frac{1}{X_t^2}dX_t + \frac{1}{2}\cdot\frac{2}{X_t^3} (dX_t)^2 \\ &= -\frac{1}{X_t}[(r_{INR} - r_{USD}) dt + \sigma_X dW_X^{Q^{INR}}(t)] + \frac{1}{X_t}\sigma_X^2 dt\\ &= \frac{1}{X_t} [(\sigma_X^2 + r_{USD} - r_{INR}) dt - \sigma_X dW_X^{Q^{INR}}(t)] \end{align*} \]
and hence:
\[ \begin{align*} d\left(\frac{S_t}{X_t}\right) &= S_t \cdot d\left(\frac{1}{X_t}\right) + \frac{1}{X_t} dS_t + dS_t \cdot d\left(\frac{1}{X_t}\right)\\ &= \frac{S_t}{X_t}[(\sigma_X^2 + r_{USD} - r_{INR}) dt - \sigma_X dW_X^{Q^{INR}}(t)] + \frac{S_t}{X_t}[r_{INR} dt + \sigma_S dW_{S}^{Q^{INR}}(t)] \\ &-\frac{S_t}{X_t} [((\sigma_X^2 + r_{USD} - r_{INR}) dt - \sigma_X dW_X^{Q^{INR}}(t))(r_{INR} dt + \sigma_S dW_{S}^{Q^{INR}}(t))] \\ &=\frac{S_t}{X_t}[(\sigma_X^2 +r_{USD} + \rho \sigma_X \sigma_S ) dt + \sigma_S dW_S^{Q^{INR}(t)} - \sigma_X dW_X^{Q^{INR}}(t)] \end{align*} \]
We can find an orthogonal decomposition of the random vector process \(\begin{bmatrix} W_S^{Q^{INR}}(t) \\ W_X^{Q^{INR}}(t)\end{bmatrix}\).
Define : \[ \begin{bmatrix} W_S^{Q^{INR}}(t) \\ W_X^{Q^{INR}}(t) \end{bmatrix} = \begin{bmatrix} Z_S^{Q^{INR}}(t) \\ \rho Z_S^{Q^{INR}}(t) + \sqrt{1-\rho^2} Z_X^{Q^{INR}}(t) \end{bmatrix} \]
where \(Z_S^{Q^{INR}}(t)\) and \(Z_X^{Q^{INR}}(t)\) are independent standard brownian motions.
So, the SDE can be written as:
\[ \begin{align*} d\left(\frac{S_t}{X_t}\right) &= \frac{S_t}{X_t}[(\sigma_X^2 +r_{USD} + \rho \sigma_X \sigma_S ) dt + \sigma_S dZ_S^{Q^{INR}}(t) - \sigma_X (\rho \cdot dZ_S^{Q^{INR}}(t) + \sqrt{1-\rho^2} \cdot dZ_X^{Q^{INR}}(t))] \\ &=\frac{S_t}{X_t}[(\sigma_X^2 +r_{USD} + \rho \sigma_X \sigma_S ) dt + (\sigma_S - \rho \sigma_X ) dZ_S^{Q^{INR}}(t) - (\sigma_X \sqrt{1-\rho^2}) dZ_X^{Q^{INR}}(t)] \end{align*} \]
Define:
\[ B^{Q^{INR}}(t) = \frac{(\sigma_S - \rho \sigma_X ) Z_S^{Q^{INR}}(t) - (\sigma_X \sqrt{1-\rho^2}) Z_X^{Q^{INR}}(t)}{(\sigma_S - \rho \sigma_X )^2 + (\sigma_X \sqrt{1-\rho^2})^2} \]
It’s easy to see that \(B^{Q^{INR}}(t)\) is Gaussian and has mean and variance given by \(\mathcal{N}(0,t)\).
Consequently, we can re-write the SDE as:
\[ d\left(\frac{S_t}{X_t}\right) = \frac{S_t}{X_t}[(\sigma_X^2 +r_{USD} + \rho \sigma_X \sigma_S ) dt + (\sigma_S^2 - 2\rho \sigma_S \sigma_X + \sigma_X^2)dB^{Q^{INR}}(t)] \]
Thus, \((\frac{S_t}{X_t})_{t\geq 0}\) follows a geometric brownian motion:
\[ \frac{S_t}{X_t} = \frac{S_0}{X_0}\exp\left[\left(\alpha - \frac{\beta^2}{2}\right)T+\beta B^{Q^{INR}}(T)\right] \]
where \(\alpha = \sigma_X^2 +r_{USD} + \rho \sigma_X \sigma_S\), \(\beta = \sigma_S^2 - 2\rho \sigma_S \sigma_X + \sigma_X^2\)
This can easily be plugged into the Black formula to derive analytic expressions for quanto vanilla calls and puts.