Adapting Heston Pricing to FX Options with Two Interest Rates

Summary

The document asks how to adapt a Heston European call formula for an FX pair, where pricing involves both a domestic rate and a foreign rate. It supplies a stock-option integral formula, a characteristic function, and proposed asset and variance dynamics, then asks how to incorporate the foreign yield. The relevant modeling idea is that the foreign rate acts like a continuous yield on the exchange rate: under the domestic risk-neutral measure, the FX rate’s drift is the domestic rate less the foreign rate, while the option payoff is discounted at the domestic rate.

The document does not include an answer or validate its equations. In particular, the displayed characteristic function’s drift term and the variance dynamics appear questionable as written, so they should not be treated as a reliable derivation. A sound implementation needs a consistent risk-neutral FX model, the correct forward level, and conventions for which currency is domestic. The text provides no numerical example, comparison, calibration, or evidence that its proposed formula has been checked.

Key ideas

• For FX options, the domestic rate discounts the payoff and the foreign rate acts as a yield on the exchange rate.
• Under the domestic risk-neutral measure, the FX rate drift is the domestic rate minus the foreign rate.
• The document poses a formula adaptation question but does not provide a validated solution.
• Its stated characteristic function and variance dynamics require checking before implementation.

Tags

# Heston formulae and characteristic function for FX options (or dividend paying yield)


# Heston formulae and characteristic function for FX options (or dividend paying yield)












I've seen the formulae for Call valuation with the Heston model for non-dividend paying Stocks.

How should I modify it to use it in an FX pair (which has two risk free rates: local currancy rate $r$ and foreign currency rate $q$)?

Formulas for Stock (only $r$) i'd like with foreign currency rate $q$:

$$ C(\theta;K,T)=\frac{1}{2}(S_0-e^{-rT}K)+\frac{e^{-rT}}{\pi}\left[\int_0^{\infty}Re\left(\frac{e^{-iu\log(K)}}{iu}\phi(\theta;u-i,T)\right)du - K\int_0^{\infty}Re\left(\frac{e^{-iu\log(K)}}{iu}\phi(\theta;u,T)\right)du \right] $$ Where $\theta$ is the calibration parameters vector $\theta:=[\nu_0, \overline{\nu}, \rho, \kappa, \sigma]^T$ and $\phi$ is the characteristic function given by: $$ \phi(\theta;u,t)=exp\left( iu\log(S_0 + rt) + \frac{\kappa\overline{\nu}}{\sigma^2}\left[ (\xi + d)t -2\log\frac{1-g_1e^{dt}}{1-g_1} \right] + \frac{\nu_0}{\sigma^2}(\xi+d)\frac{1-e^{dt}}{1-g_1e^{dt}} \right) $$ Where $$ \xi := \kappa - \sigma\rho iu \\ d := \sqrt{\xi+\sigma^2(u^2+íu)} \\ g1 := \frac{\xi+d}{\xi-d} $$

Dynamics: $$ dS_t = \mu S_tdt + \sqrt{\nu_t}S_tdW_t^1 \\ d\nu_t = \kappa(\overline{\nu} - \nu_t)dt + \sigma\sqrt{\nu_t}S_tdW_t^2 \\ dW_t^1dW_t^2=\rho dt $$

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