# Derivation of the Black-Scholes Option-Pricing Model

5 Pages Posted: 12 Jun 2009

See all articles by Robert M. Conroy

## Robert M. Conroy

University of Virginia - Darden School of Business

### Abstract

This note derives the Black-Scholes option-pricing model. It covers the basic assumptions necessary to derive the equation. In addition, it presents the arbitrage conditions used by Black and Scholes in simple terms. The basic stochastic differential equation is derived, and the boundary conditions are specified for the valuation of a European call option.

Excerpt

UVA-F-0945

DERIVATION OF THE BLACK-SCHOLES OPTION-PRICING MODEL

Assume that we have one asset, W, whose price at time t, W(t), depends on the price of another asset at time t, S(t). In this case, we can write the price of W(t) as a function of the price of S at time t, or W(t) = F[S(t), t]. The simplest way of thinking of this is to think of the price of something depending on something else. For example, the prices of bonds depend on interest rates. In the discussion that follows, we focus on call options, where the price of the call option depends on the stock price. The basic definitions we will use are

W(t) = price of options at t, and

W(t) = F[S(t),t], where S(t) is stock price at time t.

Assume dS(t) = asSdt + σsSdz. This stochastic differential equation assumes that the change in the stock price over time, dS(t), follows a log-normal distribution.

. . .

Keywords: capital markets option pricing options

Suggested Citation

Conroy, Robert M., Derivation of the Black-Scholes Option-Pricing Model. Darden Case No. UVA-F-0945, Available at SSRN: https://ssrn.com/abstract=1418303

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