Second Order Multiscale Stochastic Volatility Asymptotics: Stochastic Terminal Layer Analysis & Calibration

TL;DR

This paper advances multiscale stochastic volatility models by deriving second order asymptotics, including a terminal layer analysis, to improve implied volatility surface approximation and calibration to market data.

Contribution

It introduces a new probabilistic approach for second order terminal layer analysis in multiscale stochastic volatility models, enhancing accuracy of implied volatility approximations.

Findings

Second order asymptotics capture implied volatility convexity.

The model fits S&P 500 options data well.

Terminal layer analysis improves approximation accuracy.

Abstract

Multiscale stochastic volatility models have been developed as an efficient way to capture the principle effects on derivative pricing and portfolio optimization of randomly varying volatility. The recent book Fouque, Papanicolaou, Sircar and S{\o}lna (2011, CUP) analyzes models in which the volatility of the underlying is driven by two diffusions -- one fast mean-reverting and one slow-varying, and provides a first order approximation for European option prices and for the implied volatility surface, which is calibrated to market data. Here, we present the full second order asymptotics, which are considerably more complicated due to a terminal layer near the option expiration time. We find that, to second order, the implied volatility approximation depends quadratically on log-moneyness, capturing the convexity of the implied volatility curve seen in data. We introduce a new probabilistic approach to the terminal layer analysis needed for the derivation of the second order singular perturbation term, and calibrate to S&P 500 options data.