Closed-Form Option Pricing Formulas with Extreme Events

Abstract

This paper explores the effect of extreme events or big jumps downwards and upwards on the jump-diffusion option pricing model of Merton (1976). It starts by obtaining a special case of the jump-diffusion model where there is a positive probability of a big jump downwards. Then, it obtains a new limiting case where there is an asymptotically big jump upwards. The paper extends these models to allow both types of jumps. In some cases these formulas nest Samuelson's (1965) formulas. This simple analysis leads to several closed-form solutions for calls and puts, which are able to generate smiles, and skews with similar shapes to those observed in the marketplace.