Affine Approximation for Moment Generating Function of Log-Normal Stochastic Volatility Model

Abstract

While empirical studies have established that the log-normal stochastic volatility (SV) model is superior to its alternatives, the model does not allow for the analytical solutions available for affine models. To circumvent this, we show that the joint moment generating function (MGF) of the log-price and the quadratic variance (QV) under the log-normal SV model can be decomposed into a leading term, which is given by an exponential-affine form, and a residual term, whose estimate depends on the higher order moments of the volatility process. We prove that the second-order leading term is theoretically consistent with the expected values and covariance matrix of the log-price and the quadratic variance. We further extend this approach to the log-normal SV model with jumps. We use Fourier inversion techniques to value vanilla options on the equity and the QV and, by comparison to Monte Carlo simulations, we show that the second-order leading term is precise for the valuation of vanilla options. We generalize the affine decomposition to other non-affine stochastic volatility models with polynomial drift and volatility functions, and with jumps in the volatility process.