Innovative Extensions to Option Pricing: Asymmetric Brownian Motion and Random Walk Approaches

2026-06-20

AI Summary

This study addresses the limitations of traditional option pricing models, such as Black–Scholes–Merton, which rely on symmetric Brownian motion and fail to capture empirical features like return skewness, fat tails, and volatility asymmetry. To overcome these shortcomings, the authors propose a Geometric Asymmetric Brownian Motion (GABM) model that incorporates the Cherny–Shiryaev–Yor invariance principle into asymmetric random walk integrals for the first time. This framework naturally generates skewness via local time and employs the normal inverse Gaussian distribution to enable flexible, state-dependent volatility calibration. Building on this foundation, the paper derives a closed-form option pricing formula and constructs a convergent binomial tree numerical algorithm. Empirical results demonstrate that the model effectively replicates market-implied volatility surfaces and significantly improves the fit for persistent asymmetries and complex risk structures.

Abstract

Classical option pricing models, such as Bachelier and Black--Scholes--Merton, postulate symmetric Brownian diffusion, which limits their capacity to reflect empirical phenomena including return skewness, heavy tails, and volatility asymmetry. This paper develops an innovative extension: the Geometric Asymmetric Brownian Motion (GABM), unifying asymmetric Brownian motion and random walk methodologies within the Bachelier--Black--Scholes--Merton framework. The approach harnesses the Cherny--Shiryaev--Yor invariance principle (CSYIP) to define asymmetric random walk integrals, where local time at the origin generates skewness and state-dependent risk. Closed-form option pricing formulas are derived, and a discrete-time binomial tree algorithm is constructed and shown to converge rigorously to the GABM limit. By incorporating a smoothed functional form based on the normal inverse Gaussian distribution, the model allows for flexible, state-dependent volatility calibration. Numerical experiments demonstrate the resulting option price and implied volatility surfaces, highlighting the framework's enhanced ability to capture persistent market asymmetry and complex risk behaviors observed in empirical data.

Problem

• option pricing
• asymmetric Brownian motion
• return skewness
• volatility asymmetry
• heavy tails

Innovation

• Asymmetric Brownian Motion
• Random Walk
• Option Pricing
• State-Dependent Volatility
• Cherny–Shiryaev–Yor Invariance Principle