Representation of probability distributions with implied volatility and biological rationale

TL;DR

This paper explores how the Black-Scholes-Merton implied volatility can represent various probability distributions, providing insights into both financial modeling and biological uncertainty processing.

Contribution

It introduces a mathematical framework linking implied volatility to different probability distributions, highlighting its potential biological and financial significance.

Findings

Implied volatility can represent a range of probability distributions.

The BSM model's popularity may stem from its intuitive uncertainty representation.

Mathematical relationships between distributions and implied volatility are established.

Abstract

Economic and financial theories and practice essentially deal with uncertain future. Humans encounter uncertainty in different kinds of activity, from sensory-motor control to dynamics in financial markets, what has been subject of extensive studies. Representation of uncertainty with normal or lognormal distribution is a common feature of many of those studies. For example, proposed Bayessian integration of Gaussian multisensory input in the brain or log-normal distribution of future asset price in renowned Black-Scholes-Merton (BSM) model for pricing contingent claims. Standard deviation of log(future asset price) scaled by square root of time in the BSM model is called implied volatility. Actually, log(future asset price) is not normally distributed and traders account for that to avoid losses. Nevertheless the BSM formula derived under the assumption of constant volatility remains a major uniform framework for pricing options in financial markets. I propose that one of the reasons for such a high popularity of the BSM formula could be its ability to translate uncertainty measured with implied volatility into price in a way that is compatible with human intuition for measuring uncertainty. The present study deals with mathematical relationship between uncertainty and the BSM implied volatility. Examples for a number of common probability distributions are presented. Overall, this work proposes that representation of various probability distributions in terms of the BSM implied volatility profile may be meaningful in both biological and financial worlds. Necessary background from financial mathematics is provided in the text.