Black-Scholes Theory and Diffusion Processes on the Cotangent Bundle of the Affine Group.

Gregory S Chirikjian,Domenico Campolo,Amitesh S Jayaraman

doi:10.3390/e22040455

Abstract

The Black-Scholes partial differential equation (PDE) from mathematical finance has been analysed extensively and it is well known that the equation can be reduced to a heat equation on Euclidean space by a logarithmic transformation of variables. However, an alternative interpretation is proposed in this paper by reframing the PDE as evolving on a Lie group. This equation can be transformed into a diffusion process and solved using mean and covariance propagation techniques developed previously in the context of solving Fokker–Planck equations on Lie groups. An extension of the Black-Scholes theory with coupled asset dynamics produces a diffusion equation on the affine group, which is not a unimodular group. In this paper, we show that the cotangent bundle of a Lie group endowed with a semidirect product group operation, constructed in this paper for the case of groups with trivial centers, is always unimodular and considering PDEs as diffusion processes on the unimodular cotangent bundle group allows a direct application of previously developed mean and covariance propagation techniques, thereby offering an alternative means of solution of the PDEs. Ultimately these results, provided here in the context of PDEs in mathematical finance may be applied to PDEs arising in a variety of different fields and inform new methods of solution.

Highlights

The Nobel Prize-winning Black-Scholes Equation [1] is arguably the most well-known partial differential equation in mathematical finance
We extend the regime of applicability of mean and covariance propagation techniques by first considering the groups GL+ (1) and GL+ (1) × GL+ (1) that arise in the one-asset and two-asset Black-Scholes equations, respectively, and the affine group, A f f + (1) that arises in a coupled-asset dynamics extension of the Black-Scholes theory
In the case of asset dynamics from mathematical finance, the method yields the exact solution for the one-asset and two-asset problems by matching the Lie derivatives of the one-asset and two-asset Black-Scholes equations with the Lie derivatives of GL+ (1) and GL+ (1) × GL+ (1), respectively; this trivially reduces to the logarithmic coordinate transformation that converts these equations to heat equations

Summary

Introduction

The Nobel Prize-winning Black-Scholes Equation [1] is arguably the most well-known partial differential equation in mathematical finance. We offer a new Lie group-theoretic interpretation of the Black-Scholes equation by reformulating the original equation and extensions of it as diffusion processes on Lie groups. Group-theoretic approaches have been used extensively in the analysis of symmetries of partial differential equations (PDEs) in mathematical finance [2,3,4]. One of the central questions there is to identify the group of transformations of variables that can be applied to the equations that preserve the structure of the equations while reducing it to a simpler form for analysis and solution. The one-asset [2] and in general, the multi-asset Black Scholes equation [5] can be reduced to a heat equation through a logarithmic transformation of variables

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Conclusion