Robust Approximations for Pricing Asian Options and Volatility Swaps Under Stochastic Volatility

Posted: 4 Apr 2012

See all articles by Martin Forde

Martin Forde

Dublin City University - Department of Mathematical Sciences

Antoine (Jack) Jacquier

Imperial College London; The Alan Turing Institute

Date Written: 2011

Abstract

We show that if the discounted Stock price process is a continuous martingale, then there is a simple relationship linking the variance of the terminal Stock price and the variance of its arithmetic average. We use this to establish a model-independent upper bound for the price of a continuously sampled fixed-strike Arithmetic Asian call option, in the presence of non-zero time-dependent interest rates. We also propose a model-independent lognormal moment-matching procedure for approximating the price of an Asian call, and we show how to apply these approximations under the Black-Scholes and Heston models. We then apply a similar analysis to a time-dependent Heston stochastic volatility model, and we show to construct a time-dependent mean reversion and volatility-of-variance function, so as to be consistent with the observed variance swap curve and a pre-specified term structure for the variance of the integrated variance. We characterize the small-time asymptotics of the first and second moments of the integrated variance, and derive an approximation for the price of a volatility swap under the time-dependent Heston model, using the Brockhaus-Long approximation. We also outline a bootstrapping procedure for calibrating a piecewise-linear mean reversion level and volatility of volatility function.

Keywords: volatility swaps, Heston model, Asian options, calibration

JEL Classification: G12, G13, C6

Suggested Citation

Forde, Martin and Jacquier, Antoine, Robust Approximations for Pricing Asian Options and Volatility Swaps Under Stochastic Volatility (2011). Applied Mathematical Finance 17 (3): 241-259, Available at SSRN: https://ssrn.com/abstract=1514079

Martin Forde

Dublin City University - Department of Mathematical Sciences ( email )

Dublin
Ireland

Antoine Jacquier (Contact Author)

Imperial College London ( email )

South Kensington Campus
London SW7 2AZ, SW7 2AZ
United Kingdom

HOME PAGE: http://wwwf.imperial.ac.uk/~ajacquie/

The Alan Turing Institute ( email )

British Library, 96 Euston Road
London, NW12DB
United Kingdom

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