On the probability flow in the Stock market I: The Black-Scholes case

TL;DR

This paper examines probability flow in the stock market through the Black-Scholes model, revealing conditions for probability conservation and its implications for stock price stability.

Contribution

It reformulates the Black-Scholes equation in Hamiltonian form and explores the conditions under which probability conservation can occur in the market.

Findings

Probability is generally not conserved in the Black-Scholes model.

Conditions for probability conservation can stabilize stock prices.

Challenges the non-Hermitian assumption of the Black-Scholes Hamiltonian.

Abstract

It is known that the probability is not a conserved quantity in the stock market, given the fact that it corresponds to an open system. In this paper we analyze the flow of probability in this system by expressing the ideal Black-Scholes equation in the Hamiltonian form. We then analyze how the non-conservation of probability affects the stability of the prices of the Stocks. Finally, we find the conditions under which the probability might be conserved in the market, challenging in this way the non-Hermitian nature of the Black-Scholes Hamiltonian.