Robust Approximations for Pricing Asian Options and Volatility Swaps Under Stochastic Volatility
Posted: 4 Apr 2012
Date Written: 2011
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: Suggested Citation