The Special Elliptic Option Pricing Model and Volatility Smile

Abstract

A special version of option pricing model based on elliptic distribution is developed to explain the volatility smile for short-maturity options. A skew fractional exponential distribution, called λ distribution, is formulated to facilitate the so-called λ transformation for option pricing. The distribution has a single shape parameter that covers the normal distribution, Laplace distribution, and the elliptic cusp distribution. The option prices are hypothesized to have two regimes: The local regime is the high-kurtosis regime where the prices can be calculated, but not observed directly. Mathematical procedures are developed to handle moment explosion in this regime. The global regime is based on the normal distribution and Black-Scholes model where the prices can be observed by the market. The λ transformation is a 4-step transformation projecting between the local regime and the global regime. The implied volatility curve is hypothesized to be movable by the expectation of market momentum during the regime transformation. This model captures the major features of the volatility smile for short-maturity SPX options from a few days to a few weeks.

Keywords: distribution, option pricing, volatility smile, high frequency, fat tail, leptokurtic

JEL Classification: G12