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Hull&White 1987 state that when the brownian motion driving the volatility and the brownian motion driving the forward rate are uncorrelated, the caplet price under stochastic volatility is the black price integrated over volatility distribution like this :

$$StochsticVolPrice = \int Black(\sigma) \phi(\sigma) d \sigma$$

Can anyone explain with formulas how to arrive at this result?

Why the condition of the two brownian being uncorrelated? What happens if not the case?

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Consider the model \begin{align*} \mathrm{d}S_t &=rS_t\mathrm{d}t+\sigma_tS_t\mathrm{d}W_{1,t}, \\ \mathrm{d}\sigma^2_t &= \alpha \sigma_t^2\mathrm{d}t+\xi\sigma_t^2\mathrm{d}W_{2,t}, \end{align*} where the Brownian motions $(W_{1,t})$ and $(W_{2,t})$ are independent. Denote the averaged variance by $$\bar{V}=\frac{1}{t}\int_0^t\sigma_s^2\mathrm{d}s.$$

In their paper (page 284), Hull and White prove in a lemma that the conditional distribution of $\ln\left(\frac{S_t}{S_0}\right)$ given the value $\bar{V}$ is normal. Thus, you can price options with the standard Black-Scholes machinery by replacing the constant Black-Scholes parameter $\sigma^2$ with the time averaged $\bar{V}$. This only holds if the conditional variance is driven by an independent Brownian motion!

The equation \begin{align*} \mathrm{Call}=\int_0^\infty \mathrm{BlackScholes}(\sigma) \cdot f(\sigma) \mathrm{d}\sigma \end{align*} follows from risk neutral pricing and can be seen as a risk-neutral expectation of the call price. You take the expectation of the terminal stock price with respect to the stock price (which gives the Black Scholes price) and then the expectation with respect to the conditional variance. Compare their Equation (7) to their Equation (8) where the inner integral equals the Black-Scholes price (if the Brownian motions are independent).

Hull and White also show how the sign of the correlation coefficient (which governs the skewness of the distribution) leads to higher or lower OTM (or ITM) option prices if you incorporate correlation between returns and their variance (leverage effect).

Finally, the model from Hull and White uses a geometric Brownian motion for $(\sigma_t^2)$ which is a non-mean-reverting process which is not what we expect from a sensible volatility model. Mean-reversion is captured by Stein & Stein and Schöbel Zhu (however with potentially negative values for the variance due to the employed Ornstein Uhlenbeck process) and by Heston (whose model has mean-reversion and non-negativity, it is the same process as in the CIR model for the short rate).

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