Risk aversion, fanning preference and volatility smirk on S&P 500 index options

ABSTRACT

This article proposes a novel way of pricing S&P 500 index options in the presence of jump risk. Our analysis is built upon an equilibrium option pricing rule for a representative agent economy. In particular, we use the weighted utility’s certainty equivalent to specify agent’s risk preference, which displays a fanning-out characteristic. We find that the fanning effect captures a remarkably large portion of the total market risk premium implicit in options. As a result, the model with fanning effect generates pronounced volatility smirks.

KEYWORDS:

• Recursive utility
• fanning effect
• jump risk
• option smirk

JEL CLASSIFICATIONS:

• G12
• G13

Acknowledgements

We are grateful to Jingzhi Huang, Xiaoquan Liu, Norvald Instefjord and seminar participants at Xiamen University, the University of Essex, the 2012 XMU-UNCC International Symposium on Risk Management and Derivatives and the 2013 Asian Finance Association annual conference for many helpful comments. We also thank Haomiao Zuo for his assistance in work.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ Liu, Pan and Wang (2005) assume that investors worry about model misspecifications with respect to rare events, or in other words, investors worry if the reference model they used to hedge the extreme downside risk is true or not. This uncertainty about the hedging model is known as model uncertainty.

² Allais (1953) finds that people’s decisions depend on the probability of extreme loss, which violates the independent axiom of expected utility theory. This is known as the common consequence effect of Allais paradox.

³ Hess and Holthausen (1990) name it as eccentricity and show that the risk premium is determined by both the risk aversion and the eccentricity.

⁴ In fact, according to Ma (2011), Equation 4 can be regarded as a limit of the Epstein and Zin’s discrete-time version, when the time lag is close to zero.

⁵ Hess and Holthausen (1990) argue that the probability is weighted by the weighting function as though it has been distorted, and they further demonstrate that the distortion depends on the economic state. In the presence of rare event, this distortion has a significantly large impact on the agent’s behaviour, leading to a choice of the apparently very risky lottery.

⁶ Bates (2008) amplifies the risk aversion to the rare event, Liu, Pan and Wang (2005) add an extra layer of aversion to the model uncertainty and Du (2011) considers a time-varying risk aversion.

⁷ In Equation 16, our setting for the jump size is general and any specific distributions can be assumed. In the literature, Kou (2002) and Kou and Wang (2004) use a double-exponential distribution, and Cai and Kou (2011) extend it to a more general mixed-exponential model. Zhang, Zhao and Chang (2012) also consider a general setting on the jump size, which is a special case of our models (15) and (16) in the absence of fanning effect.

⁸ In our utility generator (5), the elasticity of intertemporal substitution is controlled by $\zeta$. In the special case of additive expected utility, $\phi =0$ as well as $\zeta =\eta$ (see Epstein 1992).

⁹ In this study, the inputs of equilibrium interest rate and dividend yield are from the real market observations. According to Liu, Pan and Wang (2005), this is without much loss of generality.

¹⁰ Here, the interest rate and the dividend yield are set to 5% and 3%, respectively, in line with the setting of Liu, Pan and Wang (2005).

Additional information

Funding

This article is benefitted from the financial support of the National Natural Science Foundation of China [grant numbers 71201136 and 71271058].