Pricing Options on a Forward with Time-Varying Volatility

Summary

The document derives a European call price when the forward price follows a lognormal process with volatility that changes deterministically over time. The option expires at an earlier date than the forward's delivery date. Integrating the instantaneous variance over the option's life gives an equivalent constant volatility, expressed as an average variance rate over that period.

Because the resulting forward price is lognormally distributed at option expiry, the answer applies a Black-style call formula using this effective volatility and discounts the payoff at the risk-free rate. It specifies the normal-distribution terms in terms of the initial forward, strike, effective volatility, and option maturity. The derivation is limited to the stated dynamics and call payoff, with deterministic decay in volatility and the assumed pricing framework; it does not discuss calibration, alternative payoff structures, or empirical validation.

Key ideas

• Integrate the time-varying instantaneous variance over the option's life to obtain total variance.
• The integrated variance can be represented as a constant effective volatility for the pricing horizon.
• The forward price at option expiry is lognormally distributed under the stated dynamics.
• The call value uses a Black-style formula with the effective volatility and discounting to option maturity.
• The result applies to the specified forward process and payoff assumptions.

Tags

# How to derive an option price for an asset with these dynamics?


# How to derive an option price for an asset with these dynamics?












Assuming my underline asset price follows the process:

$$d\ln (F_{t,T})=-(1/2)\sigma ^2e^{-2\lambda(T-t)}dt+\sigma e^{-\lambda(T-t)}dB_t $$

How should I derive an option price formula?

## Answer by Gordon (score 3, accepted)

https://quant.stackexchange.com/a/27654

For $0 < T_0\le T$, consider the option with payoff, at the option maturity $T_0$, of the form \begin{align*} \max(F_{T_0, T}-K, \, 0).\tag{1} \end{align*} Note that \begin{align*} F_{T_0, T} &= F_{0, T}\exp\left(-\frac{\sigma^2}{2}\int_0^{T_0} e^{-2\lambda (T-t)} dt+\sigma \int_0^{T_0}e^{-\lambda (T-t)} dB_t\right). \end{align*} Let \begin{align*} \hat{\sigma}^2 &= \frac{\sigma^2}{T_0}\int_0^{T_0} e^{-2\lambda (T-t)} dt\\ &=\frac{e^{-2\lambda T}\sigma^2}{2\lambda T_0}\left(e^{2\lambda T_0} -1\right). \end{align*} Then, in distribution, \begin{align*} F_{T_0, T} = F_{0, T}\exp\left(-\frac{\hat{\sigma}^2}{2} T_0 + \hat{\sigma} \sqrt{T_0} Z\right), \end{align*} where $Z$ is a standard normal random variable. The value of Payoff $(1)$ is now given by \begin{align*} e^{-r T_0}\Big[F_{0, T}\Phi(d_1) - K\Phi(d_2) \Big], \end{align*} where \begin{align*} d_1 &= \frac{\ln \frac{F_{0, T}}{K} + \frac{\hat{\sigma}^2}{2} T_0}{\hat{\sigma} \sqrt{T_0}},\\ d_2 &= d_1 - \hat{\sigma} \sqrt{T_0}, \end{align*} and $\Phi$ is the cumulative distribution function of a standard normal random variable.

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