Option Returns, Positive-Return Probabilities, and Black–Scholes Pricing Bias

Summary

This document studies the probability that a European call option earns a positive return under the Black–Scholes framework. It says this probability depends on the market inputs in the model as well as the stock’s growth rate, and reports that numerical pricing biases are associated with that growth rate.

The authors propose an alternative call valuation based on an equilibrium argument and the probability of a positive return. In their numerical investigation, Black–Scholes prices tend to be higher for out-of-the-money options and lower for in-the-money options than the alternative values. The analysis also produces a familiar implied-volatility smile. These theoretical patterns are presented as resembling observed Black–Scholes anomalies, but the description provides no empirical dataset, estimation procedure, or out-of-sample validation. The proposed valuation method should therefore be read as a theoretical alternative rather than an established replacement for standard pricing.

Key ideas

• The study examines positive-return probabilities for European calls under Black–Scholes assumptions.
• The reported probability depends on model market inputs and the stock growth rate.
• A proposed equilibrium method values calls using their probability of positive return.
• Numerical comparisons show different relative prices for out-of-the-money and in-the-money calls.
• The observed implied-volatility smile and pricing patterns are theoretical findings in the provided description.

Tags

# Probabilities of Positive Returns and Values of Call Options


# Probabilities of Positive Returns and Values of Call Options









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.

Shown in full with attribution under the source's licence. Licence: abstract CC0

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