Physical principles in option pricing: analysis and valuation based on the Black-Scholes model

Abstract

[EN] This project explores an application of physics to the study of financial systems. Particularly, it analyses the physical principles in option pricing. The main objective is to establish a connection between physics and finance. Options are financial instruments whose value depends on the price of an underlying asset (stock, bond,…). One of the purposes of this project is to model the behaviour of the underlying asset price with a Brownian motion so that later it can be applied to build the Black-Scholes model which leads to the partial differential equation used to price options. For European options, the equation can be explicitly solved to find the Black-Scholes formula. This solution can be obtained transforming the Black-Scholes equation into a diffusion equation and treating the problem as a diffusion process. In the case of American options, particularly American put options, the equation requires the use of numerical analysis to find a solution. The problem of valuing American put options is a free boundary problem which can be transformed into a linear complementarity problem to eliminate the dependence on the free boundary. This linear complementarity problem can be written using finite-difference formulation with the Crank-Nicolson scheme and can be solved using an algorithm that includes the LU method. The algorithm needs to be implemented in Python to be able to run simulations to price American put options.