Chapter 7Stochastic Volatility

Modeling and Asymptotic Approaches to Option Pricing and Portfolio Selection

Matthew Lorig1 and Ronnie Sircar2

1University of Washington, USA

2Princeton University, USA

7.1 Introduction

Understanding and measuring the inherent uncertainty in market volatility are crucial for portfolio optimization, risk management, and derivatives trading. The problem is made difficult since volatility is not directly observed. Rather, volatility is a statistic of the observable returns of, for example, a stock, and so estimates of it are at best noisy. Among the major empirical challenges have been separating contributions of diffusive and jump components of log returns, typical timescales of fluctuation, and memory effects. Until recently, data were limited to low frequencies, typically daily. The availability of high-frequency data over the past 20 years brings with it issues of deciphering market microstructure effects such as the bid–ask bounce, which contaminate the potential usefulness of such large datasets, and we refer to the recent book by Aït-Sahalia and Jacod (2014) for an overview of the difficulties.

The major problem that has been the driver of stochastic volatility models is the valuation and hedging of derivative securities. This market grew in large part from the landmark paper by Black and Scholes (1973), which showed how to value simple options contracts when volatility is constant. Even at the time of their paper, Black and Scholes realized ...

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