Effective Media Analysis for Stochastic Volatility Models. Issue 93 (23rd January 2018)

• Record Type: Journal Article
• Authors: Hagan, Patrick S.; Lesniewski, Andrew S.; Woodward, Diana E.
• Is Part Of: Wilmott. Volume 2017:Issue 93(2017)
• Page Start: 46
• Page End: 55
• Publication Date: 2018-01-23
• Subjects: volatility surface -- SABR -- derivatives -- smile -- stochastic volatility -- effective media; Finance -- Periodicals; Financial services industry -- Periodicals; 332
• DOI: 10.1002/wilm.10645
• Languages: English
• ISSNs: 1540-6962

Abstract

Abstract : Analysis of the standard SABR model leads to an effective forward equation which has time‐independent coefficients, and analysis of this reduced‐dimensionality equation leads to explicit asymptotic formulas for the implied normal volatilities of European options. These formulas are accurate to within O (ε 2 ), and are used extensively in practice for pricing and managing the risks of European options. A recent analysis of the dynamic SABR model leads to an identical effective forward equation, except that the coefficients are time‐dependent. Here we use singular perturbation methods to analyze this new equation. For each exercise date Tex, we derive a set of constant coefficients. Replacing the time‐dependent coefficients with the constant coefficients, and solving the effective forward equation, yields the correct probability density function – and thus the correct European option values – to within O (ε 2 ). These constant coefficients now let us apply the existing analysis to obtain explicit asymptotic formulas for the mplied volatilities of European options, accurate to within O (ε 2 ), for the dynamic SABR model. Current analyses of several other stochastic volatility models, including the ‐SABR and generalized Heston models, also yield identical effective forward equations with time‐dependent coefficients. Therefore, the implied volatilities for these models are also given by the SABR implied volatility formulas.