Using Lévy Processes to Model Returns and Price Derivatives

Summary

This article introduces Lévy processes as alternatives to geometric Brownian motion for modelling asset prices in derivative-pricing frameworks. Under the standard Black–Scholes assumption, log returns are normally distributed; the article argues that observed returns can depart from normality, including through skewness and heavier tails. It presents summary statistics for several major equity indices over a stated historical period as evidence of such departures.

A Lévy process is characterized by independent, stationary increments and continuity in probability; Brownian motion is one example, while Poisson and Gamma processes are others. The proposed price model expresses the stock price as its initial value multiplied by the exponential of a Lévy process. The article names Variance Gamma and Normal-inverse Gaussian models and points to an option-calibration comparison in which a Lévy-based model fits market prices more closely than Black–Scholes. The evidence is illustrative, not a universal performance test: results depend on the chosen process and calibration, and the article does not provide a full implementation or establish that one model is best in every market.

Key ideas

• Geometric Brownian motion assumes normally distributed log returns, an assumption that may not match observed returns.
• Lévy processes allow non-normal return dynamics while retaining independent and stationary increments.
• Brownian motion, Poisson processes, and Gamma processes are examples of Lévy processes.
• Variance Gamma and Normal-inverse Gaussian processes are candidate models for asset returns.
• The cited option calibration suggests improved fit in its example, but does not establish universal superiority.

Tags

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.