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

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 Sølna (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.

Robust Numerical Calibration for Implied Volatility Expansion Models

“…In order to address these issues, theoretical results have been proposed that translate model formulations into explicit implied volatility expansions for an alternative faster parameter calibration. Fouque et al [8,9] studied Stochastic Volatility (SV) models with multiple factors driving volatility on different time scales. These multifactor models have been validated by empirical studies that show that the deterministic volatility assumption is inconsistent with real data [2].…”

“…For an extensive review of local-stochastic modelling see [12]. Both multiscale SV and LSV models are based on explicit asymptotic approximations of implied volatility in successive orders of perturbation [9] and Taylor [14] expansions, respectively.…”

“…Both model classes provide successive option price expansions which are translated into successive approximating calibration formulas to the implied volatility surface. We present the theoretical results from [9] and [14] needed in each model class for practical calibration of market parameters within a second order expansion and discuss the calibration challenges of existing approaches and motivate our proposed solution.…”

“…Multiscale SV with Perturbation Expansions. Fouque et al [8,9] develop implied volatility expansions for a class of two-factor two-scale volatility models defined under the risk-neutral pricing measure P by the system of stochastic differential equations:…”

“…In [9], the authors propose the following two-step calibration procedure: 1. Find Θ * that minimizes the least-squares fit error over all maturities i and strikes j:…”