Discrete Time vs Continuous Time Stock-Price Dynamics and Implications for Option Pricing

19 Pages Posted: 30 Nov 2001

See all articles by Damiano Brigo

Damiano Brigo

Imperial College London - Department of Mathematics

Fabio Mercurio

Bloomberg L.P.

Date Written: 2000

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.

Keywords: Stochastic Differential Equations, Fokker--Planck Equation, Exponential Families, Stock Price Models, Black and Scholes model, Option Pricing, Trading Time Grid, Delta-Markovianity, Market Incompleteness, Option replication error

JEL Classification: G12

Suggested Citation

Brigo, Damiano and Mercurio, Fabio, Discrete Time vs Continuous Time Stock-Price Dynamics and Implications for Option Pricing (2000). Available at SSRN: https://ssrn.com/abstract=292059 or http://dx.doi.org/10.2139/ssrn.292059

Damiano Brigo (Contact Author)

Imperial College London - Department of Mathematics ( email )

South Kensington Campus
London SW7 2AZ, SW7 2AZ
United Kingdom

HOME PAGE: http://www.imperial.ac.uk/people/damiano.brigo

Fabio Mercurio

Bloomberg L.P. ( email )

731 Lexington Avenue
New York, NY 10022
United States

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