Hull–White Stochastic Volatility Pricing as an Average of Black Prices
Summary
The note explains a result from the Hull–White stochastic volatility model: when the Brownian motions driving the underlying price and its variance are independent, an option can be priced by averaging Black–Scholes prices over the distribution of time-averaged variance. Conditional on that averaged variance, the log underlying price is normally distributed, so standard Black–Scholes pricing applies at the corresponding volatility. Taking the risk-neutral expectation over the variance distribution then gives the overall price.
The independence condition matters because correlation between returns and variance changes the conditional distribution and can affect option skew and out-of-the-money or in-the-money prices. The discussion therefore does not justify the same simple averaging result when the drivers are correlated. It also notes a limitation of the original model: its geometric Brownian variance process does not mean-revert. It names alternative model families that add mean reversion, while noting a potential negative-variance issue for some Ornstein–Uhlenbeck specifications.
Key ideas
- With independent price and variance Brownian motions, the log price conditional on averaged variance is normal.
- Conditional option values can therefore be computed using Black–Scholes pricing at the implied volatility.
- Averaging those conditional prices over the variance distribution gives the stochastic-volatility option price.
- Correlation between returns and variance changes distribution shape and may affect option prices across strikes.
- The original Hull–White variance process lacks mean reversion, while alternative models introduce other trade-offs.
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Full text
# Caplet price under stochastic volatility is the black price integrated over volatility distribution
# Caplet price under stochastic volatility is the black price integrated over volatility distribution
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?
## Answer by Kevin (score 1)
https://quant.stackexchange.com/a/48667
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).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.