Revisiting the Black-Scholes equation

TL;DR

This paper derives the Black-Scholes option pricing PDE within a general equilibrium framework considering uncertain production and technology change, extending classical models with stochastic processes.

Contribution

It introduces a novel derivation of the Black-Scholes PDE using Cox-Ingersoll-Ross type assumptions for production and technology change.

Findings

Black-Scholes PDE derived from stochastic production and technology models

Inclusion of uncertain production processes in option pricing

Extension of equilibrium asset market models

Abstract

In common finance literature, Black-Scholes partial differential equation of option pricing is usually derived with no-arbitrage principle. Considering an asset market, Merton applied the Hamilton-Jacobi-Bellman techniques of his continuous-time consumption-portfolio problem, deriving general equilibrium relationships among the securities in the asset market. In special case where the interest rate is constant, he rederived the Black-Scholes partial differential equation from the general equilibrium asset market. In this work, I follow Cox-Ingersoll-Ross formulation to consider an economy which includes (1) uncertain production processes, and (2) the random technology change. Assuming a random production stochastic process of constant drift and variance, and assuming a random technology change to follow a log normal process, the equilibrium point of this economy will lead to the Black-Scholes partial differential equation for option pricing.