Recovering Risk Neutral Densities from Option Prices: A New Approach

Abstract

In this paper we present a new method of approximating the risk neutral density (RND) from option prices based on the C-type Gram-Charlier series expansion (GCSE) of a probability density function. The exponential form of this type of GCSE guarantees that it will always give positive values of the risk neutral probabilities, and it can allow for stronger deviations from normality, which are two drawbacks of the A-type GCSE used in practice. To evaluate the performance of the suggested expansion of the RND, the paper presents simulation and empirical evidence.

A dynamic equilibrium model for U-shaped pricing kernels

“…The tail of the risk-neutral density of a stock index return is fatter than that of the physical density and the skewness of the risk-neutral return distribution is more negative than that of the physical distribution. See for instance Rubinstein (1984), Jackwerth & Rubinstein (1996) Rompolis & Tzavalis (2008). In particular, our study is closely related to Bakshi et al (2002), which showed how risk aversion induces negative skewness in the risk-neutral distribution of the market return.…”

Retrieving risk neutral densities from European option prices based on the principle of maximum entropy

“…Instead of relying on option price values, it employs the non-central moments of the RND implied by a crosssectional set of European option prices as constraints to solve the MED problem. These moments can be retrieved from option prices in a model-free manner based on Bakshi et al (2003) and Rompolis and Tzavalis (2008) formulas. This new methodology of implementing the principle of maximum entropy has two attractive properties from the empirical point of view compared to the previously mentioned ones.…”

Exploring the role of the realized return distribution in the formation of the implied volatility smile

“…The non-central moments of the risk-neutral density l Q n ðt þ sjtÞ, denoted as the risk-neutral moments, can be directly obtained from out-of-the-money (OTM) European call and put prices employing the formulas suggested by Bakshi et al (2003) for n = 1, 2, 3, 4 and, recently, extended by Rompolis and Tzavalis (2008) to any order n as…”

“…We can view this method as one of efficiently aggregating daily returns so as to extract time-varying higher-order cumulants of the monthly return distribution. Respectively, we obtain riskneutral cumulant estimates directly from out-of-the-money (OTM) European call and put prices in a model-free manner, employing the formulas suggested by Bakshi et al (2003) and recently extended by Rompolis and Tzavalis (2008). We should stress here that our approach makes no assumption about the nature of the stochastic process of the underlying asset price under any probability measure nor assumes investors with specific utility functions or preferences.…”