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

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.

Keywords: Black-Scholes; Lie groups; affine; cotangent bundle; diffusion; unimodular.

Figure 1

Convergence of second order propagated $\mathsf{\Sigma }$ with reducing time-step, relative to the sample standard deviation ${\mathsf{\Sigma }}_s(t^{\prime })$. The horizontal axis shows the reversed time $t^{\prime }$ from 0.1 to 0.7 units.

Figure 2

Relative error comparing $\mathsf{\Sigma }(t^{\prime })$ at a given time-step with the sampled covariance ${\mathsf{\Sigma }}_s(t^{\prime })$ for parameter values ${\sigma }_1=1$ and ${\sigma }_2=0.5$, representing a scenario with large diffusion.

Figure 3

Contour plots at 300 time steps into the simulation ($t^{\prime }=0.30$) for the small diffusion scenario, showing the close match between the first order and second order propagation against the ground truth (kernel estimated probability density) but a worse match for the finite difference solutions.

Figure 4

Relative error in covariance: Comparison between first order propagation, second order propagation and explicit and implicit finite difference (inset) for the small diffusion scenario. The propagation results nearly coincide and therefore cannot be distinguished in the plot.

Figure 5

Finite difference solution (explicit) at 450 time steps ($t^{\prime }=0.45$) showing spurious oscillations due to the small covariances in the small diffusion scenario.

Figure 6

Contour plots at 300 time steps into the simulation ($t^{\prime }=0.30$) for the large diffusion scenario, with the kernel density estimated probability distribution used as the ground truth.

Figure 7

Relative error in mean: Comparison between first order, second order propagation and finite difference methods (explicit and implicit) for the large diffusion scenario. The propagation results nearly coincide and therefore cannot be distinguished in the plot.

Figure 8

Relative error in covariance: Comparison between first order, second order propagation and finite difference methods (explicit and implicit) for the large diffusion scenario.

Figure 9

Plot of the analytical and propagated solution to the converted 1D Black-Scholes equation in (130), ${\tilde{u}}_f(a,t^{\prime })$, for the small diffusion case ($\sigma =0.5$, $r=3$ and $t^{\prime }=0.3$). Both first order and second order propagation results coincide.

Figure 10

Plot of the analytical and propagated solution to the converted 1D Black-Scholes equation in (130), ${\tilde{u}}_f(a,t^{\prime })$, for the large diffusion case ($\sigma =1$, $r=3$ and $t^{\prime }=0.3$). Both first order and second order propagation results coincide.