Discrete Time vs Continuous Time Stock-price Dynamics and implications for Option Pricing

TL;DR

This paper compares discrete and continuous time stock-price models, showing how different modeling choices between trading times can significantly affect option prices, challenging traditional Black-Scholes assumptions.

Contribution

It constructs stock price processes matching geometric Brownian motion marginals and transition densities, revealing implications for option pricing beyond classical models.

Findings

Option prices can be arbitrarily close to intrinsic value or stock price.

Modeling between trading times critically influences option valuation.

Theoretical analysis of stochastic differential equations with prescribed densities.

Abstract

In the present paper we construct stock price processes with the same marginal log-normal law as that of a geometric Brownian motion and also with the same transition density (and returns' distributions) between any two instants in a given discrete-time grid. We then illustrate how option prices based on such processes differ from Black and Scholes', in that option prices can be either arbitrarily close to the option intrinsic value or arbitrarily close to the underlying stock price. We also explain that this is due to the particular way one models the stock-price process in between the grid time instants which are relevant for trading. The theoretical result concerning scalar stochastic differential equations with prescribed diffusion coefficient whose densities evolve in a prescribed exponential family, on which part of the paper is based, is presented in detail.