Option pricing formulas based on a non-Gaussian stock price model

Area of Science:

• Quantitative Finance
• Statistical Mechanics
• Financial Modeling

Background:

• Standard option pricing models like Black-Scholes assume Gaussian distributions for stock returns.
• Empirical data shows stock returns often exhibit non-Gaussian, fat-tailed behavior, leading to phenomena like the volatility smile.
• The volatility smile necessitates multiple volatility inputs in the Black-Scholes model, complicating pricing.

Purpose of the Study:

• To explain non-Gaussian fluctuations in financial options using nonextensive thermodynamics.
• To develop a generalized Black-Scholes equation that accounts for these fluctuations.
• To demonstrate a simplified approach to option pricing using a single volatility parameter.

Main Methods:

• Application of nonextensive thermodynamics, specifically the parameter q, to model stock return distributions.
• Derivation of a generalized Black-Scholes partial differential equation.
• Obtaining closed-form solutions for option pricing under the generalized model.

Main Results:

• A generalized Black-Scholes equation is derived, incorporating the nonextensive parameter q.
• Using q=1.5, which accurately models empirical return distributions, a single volatility value effectively describes option prices.
• This approach successfully models option prices without requiring multiple volatility inputs, addressing the volatility smile.

Conclusions:

• Nonextensive thermodynamics provides a robust framework for understanding non-Gaussian option price dynamics.
• The generalized Black-Scholes model offers a more accurate and simplified method for option pricing compared to the standard model.
• A single, empirically derived volatility parameter can effectively capture option pricing behavior, resolving the volatility smile issue.