Asymptotic Methods for Computing Implied Volatilities Under Stochastic Volatility
NCCR-Finrisk Working Paper No. 204
38 Pages Posted: 15 Feb 2005
Date Written: October 2004
In this paper we propose an analytical formula for computing implied volatilities of European options based on their short term asymptotics. The analysis is performed in a general framework with local and stochastic volatility. Assuming CEV volatility of volatility we first obtain a quasi-analytical solution for the limit of implied volatilities as time-to-maturity goes to zero (instanteneous implied volatility). Then we develop our analytical formula in the form of a local transformation of the instanteneous implied volatility. Numerical experiments suggests that this approximation is extremely accurate at short maturities (one or two month). We further introduce a class of models under which this method is accurate even for long maturity options. In the particular case of SABR model we improve the formula derived in Hagan et al. (2002).
Keywords: Option pricing, stochastic volatility, local volatility, implied volatility, short term asymptotics
JEL Classification: G12
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