[Paper Review] Finance Market Dynamics with Option Pricing

Joseph L. McCauley, Gemunu H. Gunaratne+1 more|arXiv (Cornell University)|Jun 1, 2006

Complex Systems and Time Series Analysis6 references2 citations

TL;DR

This paper proposes a generalized option pricing model using scaling solutions of Markov processes with time- and state-dependent diffusion coefficients, extending the Black-Scholes PDE. It proves that the resulting model remains a martingale under risk-neutral pricing for a broad class of return distributions, while explaining why option prices diverge when fat tails are included in the dynamics.

ABSTRACT

We show how finance markets can be modeled empirically faithfully by using scaling solutions for Markov processes. Classes of exact scaling solutions are presented. We then show that our generalization of the Black-Scholes partial differential equation (pde) for nontrivial diffusion coefficients is equivalent to a Martingale in the risk neutral discounted stock price. Previously, this was proven for the case of the Gaussian logarithmic returns model by Harrison and Kreps, but we prove it for much a much larger class of returns models where the diffusion coefficient depends on both returns x and time t. That option prices blow up if fat tails in logarithmic returns x are included in the market dynamics is also explained.

Motivation & Objective

• To develop a more general framework for modeling finance market dynamics beyond the Gaussian assumption.
• To extend the Black-Scholes PDE to include diffusion coefficients that depend on both stock returns and time.
• To prove that the generalized model preserves the martingale property under risk-neutral measure for a broad class of return processes.
• To explain the mathematical origin of option price divergence when fat tails are introduced in logarithmic returns.

Proposed method

• Derives exact scaling solutions for Markov processes to model financial market dynamics.
• Generalizes the Black-Scholes partial differential equation to allow diffusion coefficients dependent on both returns x and time t.
• Applies risk-neutral pricing theory to show equivalence to a martingale in the discounted stock price process.
• Uses stochastic calculus to establish the martingale condition for nontrivial diffusion coefficients.
• Analyzes the behavior of option prices under fat-tailed logarithmic returns using the generalized PDE framework.

Experimental results

Research questions

• RQ1 How can Markov processes with time- and state-dependent diffusion coefficients be used to model realistic finance market dynamics?
• RQ2 In what class of return models does the generalized Black-Scholes PDE preserve the martingale property under risk-neutral measure?
• RQ3 Why do option prices become unbounded when fat tails are included in the logarithmic return distribution?
• RQ4 What is the mathematical mechanism behind the blowup of option prices under non-Gaussian return dynamics?

Key findings

• The generalized Black-Scholes PDE with time- and state-dependent diffusion coefficients remains equivalent to a martingale in the risk-neutral discounted stock price process.
• The martingale equivalence is proven for a significantly broader class of return models than previously established, including non-Gaussian distributions.
• Option prices diverge when fat tails are present in the logarithmic returns, due to the unbounded growth of the diffusion coefficient's influence in the PDE.
• The scaling solution framework provides an empirically faithful representation of financial market dynamics beyond the Gaussian assumption.