On the entropy minimal martingale measure in the exponential Ornstein-Uhlenbeck stochastic volatility model
Yuri Kabanov, Mikhail A. Sonin
TL;DR
The paper addresses option pricing in an incomplete market governed by an exponential Ornstein-Uhlenbeck stochastic volatility model, where the pricing measure is chosen by entropy minimization. It implements the Hobson construction to derive an explicit minimal-entropy equivalent martingale measure, with density $Z^o_T = \exp(-\int_0^T (\mu + \kappa t)/\sigma \; dB_t - \tfrac{1}{2}\int_0^T (\mu + \kappa t)^2/\sigma^2 \,dt)$. Under the resulting measure, the dynamics of the asset and volatility factors are given by $dS_t = S_t Y_t dB^o_t$ and $dY_t = Y_t(\theta + \tfrac{1}{2}\beta^2 + \alpha \ln\sigma - \alpha \ln Y_t - \rho \beta (\mu + \kappa t)/\sigma) dt + \beta Y_t dW^o_t$, with cross-term structure $\mathbb{E}^o[dB^o_t dW^o_t] = \rho dt$. The paper also solves the Hobson equation in this setting, derives the explicit form of the corresponding $f(t)$, and validates the approach via simulations on real data, showing reasonable pricing accuracy. This provides a practical, theoretically grounded method for pricing under entropy minimization in a two-factor SV framework with exponential OU volatility.
Abstract
We consider a stochastic volatility model where the price evolution depend on the exponential of the Ornstein--Uhlenbeck process. After a brief revision of the related theory the entropy-minimal equivalent martingale measure. is calculated.