Testing Option Pricing with the Edgeworth Expansion

TL;DR

This paper extends the Black-Scholes option pricing model by incorporating Edgeworth expansions to account for skewness and kurtosis, reducing the volatility smile and improving hedging strategies.

Contribution

It provides an explicit formula for option prices using Edgeworth expansion, allowing for non-gaussian return distributions in a practical setting.

Findings

Reduced volatility smile in Brazilian and American markets

More efficient hedging strategies achieved

Improved accuracy over Gaussian-based models

Abstract

There is a well developed framework, the Black-Scholes theory, for the pricing of contracts based on the future prices of certain assets, called options. This theory assumes that the probability distribution of the returns of the underlying asset is a gaussian distribution. However, it is observed in the market that this hypothesis is flawed, leading to the introduction of a fudge factor, the so-called volatility smile. Therefore, it would be interesting to explore extensions of the Black-Scholes theory to non-gaussian distributions. In this contribution we provide an explicit formula for the price of an option when the distributions of the returns of the underlying asset is parametrized by an Edgeworth expansion, which allows for the introduction of higher independent moments of the probability distribution, namely skewness and kurtosis. We test our formula with options in the brazilian and american markets, showing that the volatility smile can be reduced. We also check whether our approach leads to more efficient hedging strategies of these instruments.