Financial Modeling and Option Theory with the Truncated Levy Process

TL;DR

This paper explores the truncated Levy process (TLP) for financial modeling, demonstrating its advantages over traditional models in capturing market dynamics and deriving a generalized option pricing formula.

Contribution

It introduces the use of TLP in financial modeling, compares hedging strategies, and generalizes the Black-Scholes formula for TLP-based assets.

Findings

TLP captures excess kurtosis and slow convergence to Gaussian.

Optimal hedging strategies differ significantly from delta hedging.

Derived a generalized Black-Scholes formula for TLP models.

Abstract

In recent studies the truncated Levy process (TLP) has been shown to be very promising for the modeling of financial dynamics. In contrast to the Levy process, the TLP has finite moments and can account for both the previously observed excess kurtosis at short timescales, along with the slow convergence to Gaussian at longer timescales. I further test the truncated Levy paradigm using high frequency data from the Australian All Ordinaries share market index. I then consider, for the early Levy dominated regime, the issue of option hedging for two different hedging strategies that are in some sense optimal. These are compared with the usual delta hedging approach and found to differ significantly. I also derive the natural generalization of the Black-Scholes option pricing formula when the underlying security is modeled by a geometric TLP. This generalization would not be possible without the truncation.