Pricing FX Options Under a Quanto Currency Measure

Summary

The response outlines a pricing approach for an FX-linked payoff when payment is made in a third, quanto currency. It defines the exchange rate with the foreign currency as the numerator and gives its drift under the quanto currency measure. The drift includes the domestic and foreign interest rate difference and a correlation adjustment tied to the exchange rate and quanto conversion rate volatilities.

Under constant-rate and constant-volatility assumptions, the terminal log exchange rate is normally distributed, with the stated measure-adjusted drift and variance. The payoff is then valued by discounting its expectation under that distribution at the quanto currency rate. The response illustrates the setup with a call-style payoff and notes its connection to Black–Scholes pricing. It is a simplified framework: the excerpt does not provide the requested derivation in the original notation, and its formula depends on the stated currency convention, measure, and modeling assumptions.

Key ideas

• Currency quotation and payment currency determine the appropriate measure and discounting convention.
• Under a quanto measure, the FX drift includes a correlation adjustment from the conversion rate.
• With constant rates and volatility, the terminal log exchange rate is normally distributed.
• Price the payoff by taking its expectation under the relevant measure and discounting in the payment currency.
• The stated setup relies on simplifying assumptions and consistent exchange-rate conventions.

Tags

# FX pricing replication


# FX pricing replication












Pay in currency : cur

The FX is : $FX^{cur_2/cur_1}$

European options on the FX (and itself) are quoted in currency cur 1.

I'm looking for the price of \begin{equation*} \mathbb{E}^{Q} \left[ e^{-\int_{0}^{T}r_{s}^{cur}ds} f \left( FX_{T_f}^{cur_2/cur_1} \right) | \mathcal{F}_{0} \right] = ? \end{equation*}

If i integrate with respect to the FX_rate density $\phi_{T_{f}}$, is $\frac{B(0,T)^{cur}}{B(0,T_{f})^{cur1}}\int_{0}^{\infty}f(x)\phi_{T_{f}}(x)dx$ the right answer?

## Answer by ir7 (score 1)

https://quant.stackexchange.com/a/55384

If $X$ is FOR-DOM exchange rate (asset always on the left, numeraire on the right), then its dynamics in the QUANTO currency measure (currency different from FOR and DOM; let $Y$ be the DOM-QUANTO exchange rate) is:

$$ dX/X = \left(r_{\rm DOM}-r_{\rm FOR}-\rho_{XY}\sigma_X \sigma_Y \right) dt + \sigma_X dW $$

Terminal distribution:

$$ \ln (X_T/X_0) \sim \phi \left((r_{\rm DOM}-r_{\rm FOR}-\rho_{XY}\sigma_X \sigma_Y - 0.5 \sigma_X^2)T, \sigma^2_X T\right) $$

with $\phi$ normal density.

Price:

$$ {\rm e}^{-r_{\rm QUANTO} T} \int_{-\infty}^{\infty} g(x)\phi(x)dx $$

(we get Black-Scholes formula under my assumptions here and $g(x)=(X_0{\rm e}^x-K)^+$).

(See this resource for further details and proofs on quanto FX options.)

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.