Article2010

MOMENT EXPLOSIONS AND LONG-TERM BEHAVIOR OF AFFINE STOCHASTIC VOLATILITY MODELS

128 citations0/0 files storedMathematical Finance

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Stochastic processes and financial applications Financial Markets and Investment Strategies Financial Risk and Volatility Modeling

Abstract

We consider a class of asset pricing models, where the risk-neutral joint process of log-price and its stochastic variance is an affine process in the sense of Duffie, Filipovic, and Schachermayer. First we obtain conditions for the price process to be conservative and a martingale. Then we present some results on the long-term behavior of the model, including an expression for the invariant distribution of the stochastic variance process. We study moment explosions of the price process, and provide explicit expressions for the time at which a moment of given order becomes infinite. We discuss applications of these results, in particular to the asymptotics of the implied volatility smile, and conclude with some calculations for the Heston model, a model of Bates and the Barndorff-Nielsen–Shephard model.

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References

1993A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options 2000Transform Analysis and Asset Pricing for Affine Jump-diffusions 1997Empirical Performance of Alternative Option Pricing Models 1996Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options 2001Non-Gaussian Ornstein–Uhlenbeck-based Models and Some of Their Uses in Financial Economics 2000Post-'87 crash fears in the S&P 500 futures option market 1997Processes of normal inverse Gaussian type 2003Affine processes and applications in finance

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2009SMALL-TIME ASYMPTOTICS FOR IMPLIED VOLATILITY UNDER THE HESTON MODEL 2012The Small-Time Smile and Term Structure of Implied Volatility under the Heston Model 2014Consistent Modelling of VIX and Equity Derivatives Using a 3/2 plus Jumps Model 2010The large-maturity smile for the Heston model 2015Exponential moments of affine processes

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