Quantum Uncertainty and the Black-Scholes Formula

Abstract

The publication of the Black-Scholes formula in 1973 appeared for the first time to put the pricing of financial options onto a rational and objective basis. While earlier option-pricing models relied on a subjective estimate of the stock’s uncertain future growth rate, the Black-Scholes model replaced this with the risk-free rate, which could be measured objectively. This switch transformed the perception of option pricing from a form of gambling to scientific risk management, and led to a huge increase in options trading. However while the formula revolutionised the world of finance, and remains the industry-standard pricing model today, its proof relies on a number of assumptions about price behaviour which are often contested, such as that log prices follow a random walk with constant volatility, and that one can constantly buy or sell stocks and options without incurring transaction fees. This paper presents an alternative approach to option pricing, based on a quantum oscillator model of stock prices. In the quantum model, the bid/ask spread between buy and sell prices is treated as a fundamental measure of uncertainty which is a main cause of price volatility. It cannot therefore be ignored as a technical detail. Also, the volatility which should be used in the formula is not constant but exhibits a smile-like dependence on strike. Finally, the appropriate rate is not the risk-free rate, but the expected growth rate. We show how the Black-Scholes model and its assumptions lead to a systematic mispricing of commonly-traded options, while results can be improved by adopting the quantum model.

Keywords: financial options, implied volatility, stock markets, quantum economics, quantum finance

JEL Classification: G10, G12