Probabilities of Positive Returns and Values of Call Options

TL;DR

This paper examines the probability of positive returns for European call options under the Black-Scholes model, proposing an alternative valuation method based on positive return probability and analyzing its implications.

Contribution

It introduces a new method for pricing European call options using positive return probability, addressing biases in the Black-Scholes formula.

Findings

Black-Scholes probability depends on market factors and growth rate.

Proposed method yields different prices for in/out-of-the-money options.

Implied volatility exhibits a typical smile shape in numerical results.

Abstract

The true probability of a European call option to achieve positive return is investigated under the Black-Scholes model. It is found that the probability is determined by those market factors appearing in the BS formula, besides the growth rate of stock price. Our numerical investigations indicate that the biases of BS formula is correlated with the growth rate of stock price. An alternative method to price European call option is proposed, which adopts an equilibrium argument to determine option price through the probability of positive return. It is found that the BS values are on average larger than the values of proposed method for out-of-the-money options, and smaller than the values of proposed method for in-the-money options. A typical smile shape of implied volatility is also observed in our numerical investigation. These theoretical observations are similar to the empirical anomalies of BS values, which indicates that the proposed valuation method may have some merit.