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

Martin Forde and Antoine Jacquier ()

Applied Mathematical Finance, 2010, vol. 17, issue 3, 241-259

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 (Theorem 1.2). 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 (subsection 1.3). We then apply a similar analysis to a time-dependent Heston stochastic volatility model, and we show how 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 (Theorem 2.1). We characterize the small-time asymptotics of the first and second moments of the integrated variance (Proposition 2.2) and derive an approximation for the price of a volatility swap under the time-dependent Heston model ( Equation (52)), using the Brockhaus-Long approximation (Brockhaus, and Long, 2000). We also outline a bootstrapping procedure for calibrating a piecewise-linear mean reversion level and volatility-of-volatility function (Subsection 2.3.2).

Keywords: Asian options; Heston; stochastic volatility; calibration; volatility swaps (search for similar items in EconPapers)
Date: 2010
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Citations: View citations in EconPapers (9)

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DOI: 10.1080/13504860903335348

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