The Implied Risk Aversion from Utility Indifference Option Pricing in a Stochastic Volatility Model

International Journal of Applied Mathematics & Statistics, Vol. 16, No. M10, pp. 11-37, March 2010

27 Pages Posted: 11 Oct 2009 Last revised: 25 Nov 2009

See all articles by Fred Espen Benth

Fred Espen Benth

University of Oslo

Martin Groth

University of Oslo

Carl Lindberg

Chalmers University of Technology

Date Written: October 8, 2009

Abstract

The academic interest in utility indifference based approaches to derivative pricing in incomplete markets has grown during the last decades. In lack of arbitrage arguments the fair price of an option can be found through the value function of two optional investment choices, one where an option is issued and one where the wealth is invested directly in the underlying asset. In this paper we consider a stochastic volatility model defined as a positive non-Gaussian Ornstein-Uhlenbeck process, and price call and put options using the indifference methodology in the case of exponential utility. The purpose of the study is to investigate empirically the implied risk aversion for a representative agent in the option market, as a function of time to maturity and strike price. Our study is based on real price data, calibrating the stochastic volatility model using historical price returns. The implied risk aversion is found by numerically inverting the indifference pricing equation, given observed option prices. We find that the option prices in the market are basically set by the issuer, in the sense that it is the issuer’s indifference prices that matches the market prices. Since the stochastic volatility model explains the stylized facts of returns rather well, we expect the implied risk aversion to be close to flat with respect to maturity and strike price of the options. We find on the contrary a clear smile effect for short dated options, which may be explained by the issuer’s fear of a market crash (in the case of the issuance of a put option). Although the stochastic volatility model explains the heavy tails of the returns, the crash risk seems to be unexplained by the stochastic volatility model.

Keywords: stochastic volatility, utility indifference option pricing, risk aversion, Lévy processes

Suggested Citation

Benth, Fred Espen and Groth, Martin and Lindberg, Carl, The Implied Risk Aversion from Utility Indifference Option Pricing in a Stochastic Volatility Model (October 8, 2009). International Journal of Applied Mathematics & Statistics, Vol. 16, No. M10, pp. 11-37, March 2010, Available at SSRN: https://ssrn.com/abstract=1485374

Fred Espen Benth

University of Oslo ( email )

Center of Mathematics for Applications
Oslo, N-0317
Norway

Martin Groth

University of Oslo ( email )

PO Box 6706 St Olavs plass
Oslo, N-0317
Norway

Carl Lindberg (Contact Author)

Chalmers University of Technology ( email )

SE-412 96 Goteborg
Sweden
+46 (0)704 93 63 95 (Phone)

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