Hull–White Mixing for Stochastic Volatility and Jump Option Pricing

Summary

The document asks whether a stock price in a stochastic volatility with jumps (SVJ) model can be factored into a component driven by Brownian risk independent of volatility, a correlated Brownian component, and an independent jump multiplier. It then proposes pricing a vanilla option by averaging Black–Scholes prices over the correlated and jump components, using integrated variance to set the remaining volatility.

The central idea is a mixing representation associated with Hull and White, with Lewis’s 2002 discussion cited as background. The document does not provide an accepted answer, derivation, or empirical evidence, so it does not establish that the proposed factorization or option-pricing expectation is correct. In particular, the risk-neutral drift, jump compensation, dependence assumptions, and conditioning behind the proposed average would need careful specification before applying the formula. Treat this as a question that motivates further study rather than a validated pricing recipe.

Key ideas

• The document asks whether an SVJ stock price can be split into independent Brownian, correlated Brownian, and jump factors.
• It proposes averaging Black–Scholes prices over the correlated and jump components.
• The residual Black–Scholes volatility is tied to integrated variance and the Brownian correlation.
• No derivation or answer is supplied, so the proposed pricing representation remains unverified.

Tags

# Mixing formula for SVJ models


# Mixing formula for SVJ models












I am trying to understand the mixing formula (Hull and White formula) for stochastic volatility models with jumps in the asset price. One article which discusses this is Lewis, The mixing approach to stochastic volatility and jump models, Wilmott, 2002.

Is it correct to write the general stock price process in a SVJ model as follows: $$ S_T = \tilde S_T X_T Y_T $$ with $$ \tilde S_T = S_t \exp \left\{ -\frac{1-\rho^2}{2} \int_t^T \sigma^2_u du + \sqrt{1-\rho^2} \int_t^T \sigma_u dZ_u \right\} $$ $$ X_T = \exp \left\{ -\frac{\rho^2}{2} \int_t^T \sigma^2_u du + \rho\int_t^T \sigma_u dW_u \right\} $$ where $\sigma_t$ is the stochastic instantaneous volatility and adapted $W$, $dW dZ = 0$, $\rho$ the correlation between the vol and the stock price, and $$ Y_T = \exp \left \{ \int_t^T dJ_u \right\} $$ is the (compensated) jump process (eg a compensated compound Poisson process)? Also, for simplicity I assume that $J$ is independent of the standard Brownian motions $W$ and $Z$ and the instantaneous volatility $\sigma$

If so (if what I wrote above is correct), then I can write the price of a vanilla option as follows, correct? $$ C^{SVJ}(S_t,K,T) = E_t \left[ C^{BS}\left(S_tX_TY_T,K,T; \bar\sigma \sqrt{1-\rho^2} \right) \right] $$ Here $C^{SVJ}$ is the SVJ model option price, and $C^{BS}$ the Black-Scholes formula, which in the expectation has implied volatility $\bar\sigma \sqrt{1-\rho^2}$ and $$ \bar\sigma = \left( \frac{1}{T-t} \int_t^T \sigma^2_u du \right)^{1/2} $$

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.