Market Complete Option Valuation using a Jarrow-Rudd Pricing Tree with Skewness and Kurtosis

TL;DR

This paper introduces a generalized Jarrow-Rudd option pricing model that incorporates skewness and kurtosis through a skew random walk, extending traditional models to better capture market asymmetries and tail risks.

Contribution

It develops a GJR model based on the Cherny-Shiryaev-Yor invariance principle, incorporating skewness, kurtosis, and transaction costs, with methods to fit it to market data.

Findings

The GJR model captures skewness and kurtosis in option prices.

Implied surfaces for GJR parameters are constructed.

Numerical examples demonstrate the model's applicability.

Abstract

Applying the Cherny-Shiryaev-Yor invariance principle, we introduce a generalized Jarrow-Rudd (GJR) option pricing model with uncertainty driven by a skew random walk. The GJR pricing tree exhibits skewness and kurtosis in both the natural and risk-neutral world. We construct implied surfaces for the parameters determining the GJR tree. Motivated by Merton's pricing tree incorporating transaction costs, we extend the GJR pricing model to include a hedging cost. We demonstrate ways to fit the GJR pricing model to a market driver that influences the price dynamics of the underlying asset. We supplement our findings with numerical examples.