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

33 Pages Posted: 31 Aug 2012 Last revised: 18 Apr 2015

See all articles by Jean-Pierre Fouque

Jean-Pierre Fouque

University of California, Santa Barbara (UCSB) - Statistics & Applied Probablity

Matthew Lorig

University of Washington - Applied Mathematics

Ronnie Sircar

Princeton University - Department of Operations Research and Financial Engineering

Date Written: April 17, 2015

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.

Suggested Citation

Fouque, Jean-Pierre and Lorig, Matthew and Sircar, Ronnie, Second Order Multiscale Stochastic Volatility Asymptotics: Stochastic Terminal Layer Analysis & Calibration (April 17, 2015). Available at SSRN: https://ssrn.com/abstract=2137900 or http://dx.doi.org/10.2139/ssrn.2137900

Jean-Pierre Fouque

University of California, Santa Barbara (UCSB) - Statistics & Applied Probablity ( email )

United States

Matthew Lorig (Contact Author)

University of Washington - Applied Mathematics ( email )

Seattle, WA
United States

Ronnie Sircar

Princeton University - Department of Operations Research and Financial Engineering ( email )

Princeton, NJ 08544
United States