[Paper Review] A theory for Fluctuations in Stock Prices and Valuation of their Options

Gemunu H. Gunaratne, Joseph L. McCauley|ArXiv.org|Sep 19, 2002

Complex Systems and Time Series Analysis3 references4 citations

TL;DR

This paper proposes a new option pricing theory based on a Fokker-Planck equation with a position- and time-dependent diffusion coefficient, leading to an asymmetric exponential distribution of returns. The model explains the volatility smile observed in markets by showing that higher volatility for large price deviations naturally leads to effective volatility adjustments in Black-Scholes pricing.

ABSTRACT

A new theory for pricing options of a stock is presented. It is based on the assumption that while successive variations in return are uncorrelated, the frequency with which a stock is traded depends on the value of the return. The solution to the Fokker-Planck equation is shown to be an asymmetric exponential distribution, similar to those observed in intra-day currency markets. The "volatility smile," used by traders to correct the Black-Scholes pricing is shown to provide an alternative mechanism to implement the new options pricing formulae derived from our theory.

Motivation & Objective

• To develop a theoretical framework for option pricing that accounts for non-Gaussian return distributions observed in financial markets.
• To address the limitations of the Black-Scholes model, which assumes log-normal returns and constant volatility, by incorporating empirically observed return dynamics.
• To explain the empirical phenomenon of the 'volatility smile' not as an ad hoc correction but as a natural outcome of a more realistic stochastic process.
• To derive a solution to the Fokker-Planck equation with a bilinear diffusion coefficient that captures asymmetric fluctuations in stock returns.
• To demonstrate that the effective volatility used in market practice (the volatility smile) can be derived from the underlying distribution of returns in the proposed model.

Proposed method

• Assumes successive return variations are uncorrelated but that the diffusion coefficient D(x,t) depends on the return value x and time t, modeling increased trading frequency during large price deviations.
• Uses a scaling ansatz W(x,t) = t^{-η} F(u), with u = x/t^η, to derive η = 1/2 from the Fokker-Planck equation, consistent with empirical scaling in financial time series.
• Models the diffusion coefficient as a piecewise linear function of u: D(x,t) = (1/γ²)(1 - γu)Θ̄(u) + (1/ν²)(1 + νu)Θ(u), introducing asymmetry via γ ≠ ν.
• Solves the resulting Fokker-Planck equation to obtain a solution W(x,t) = A/(√t) e^{γu} Θ̄(u) + B/(√t) e^{-νu} Θ(u), with A and B determined by normalization and continuity conditions.
• Derives explicit expressions for European call and put option prices under this distribution, showing they deviate from Black-Scholes predictions.
• Demonstrates that the observed 'volatility smile' in option markets can be interpreted as an effective volatility adjustment that matches the model’s predicted option values.

Experimental results

Research questions

• RQ1 How can a non-Gaussian, asymmetric distribution of stock returns be modeled within a stochastic framework that preserves uncorrelated increments?
• RQ2 What form must the diffusion coefficient take to yield a solution matching the empirically observed asymmetric exponential distribution of intra-day returns?
• RQ3 To what extent can the volatility smile—commonly used as a heuristic correction in option pricing—be derived from a physically consistent stochastic process?
• RQ4 How does the inclusion of a consensus price S₀ as a minimum volatility point affect the dynamics and pricing of options compared to the Black-Scholes model?
• RQ5 What are the implications of higher-order corrections (e.g., quadratic terms in the diffusion coefficient) for the tail behavior of return distributions and option valuation?

Key findings

• The solution to the Fokker-Planck equation with a bilinear diffusion coefficient yields an asymmetric exponential distribution for returns, matching empirical data from intra-day currency and bond markets.
• The model predicts that large deviations in returns occur more frequently than in a Gaussian model, leading to higher option prices than predicted by Black-Scholes.
• The effective volatility required to match the model’s option prices with Black-Scholes formulae produces a volatility smile curve, as seen in real market data.
• For γ = 15.0 and ν = 10.0, the annualized volatility of the return distribution is 11.5%, and the resulting effective volatility curve (V_eff(K)) closely resembles observed market volatility smiles.
• The model shows that the consensus price S₀ is a critical point where the diffusion coefficient is minimized, making it a natural reference point for return calculations.
• Higher-order corrections to the diffusion coefficient (e.g., quadratic terms) lead to power-law tails in the distribution, consistent with fat-tailed behavior observed in inter-day returns.