Encyclopedia of Financial Models I

Black-Scholes Option Pricing Model

SVETLOZAR T. RACHEV, PhD, DrSci

Frey Family Foundation Chair Professor, Department of Applied Mathematics and Statistics, Stony Brook University; and Chief Scientist, FinAnalytica

CHRISTIAN MENN, Dr Rer Pol,

Managing Partner, RIVACON

FRANK J. FABOZZI, PhD, CFA, CPA

Professor of Finance, EDHEC Business School

Abstract: The most popular continuous-time model for option valuation is based on the Black-Scholes theory. Although certain drawbacks and pitfalls of the Black-Scholes option pricing model have been understood shortly after its publication in the early 1970s, it is still by far the most popular model for option valuation. The Black-Scholes model is based on the assumption that the underlying follows a stochastic process called geometric Brownian motion. Besides pricing, every option pricing model can be used to calculate sensitivity measures describing the influence of a change in the underlying risk factors on the option price. These risk measures are called the “Greeks” and their use will be explained and described.

In this entry, we look at the most popular model for pricing options, the Black-Scholes model, and look at the assumptions and their importance. We also explain the various Greeks that provide information about the sensitivity of the option price to changes in the factors that the model tells us affects the value of an option.

MOTIVATION

Let us assume that the price of a certain stock in June of Year 0 (t = 0) is given ...