On the Derivation of the Black-Scholes Formula

Seminaire de Probabilites, Vol. 37, pp. 399-414, 2004

12 Pages Posted: 2 May 2005

See all articles by Ioanid Rosu

Ioanid Rosu

HEC Paris - Finance Department

Daniel W. Stroock

Massachusetts Institute of Technology (MIT) - Department of Mathematics

Abstract

Methods of proving the Black-Scholes formula for the price of an European call option fall into two categories: the bond replication method (the original one by Black and Scholes), and the call replication method (originated by Merton). These two methods are not equivalent. While the call replication argument is simple and requires only continuity of the call price C, the bond replication method puts more restrictions on C in order for the argument to work. Moreover, we show that the typical bond replication method fails if the call option delta is equal to one. That implies that at each point either C satisfies the Black-Scholes PDE, or it satisfies the linear PDE given by delta equal to one. We then show that these two PDEs cannot coexist if C is assumed continuously differentiable, which proves that the Black-Scholes PDE holds everywhere.

Keywords: option pricing, bond replication, self-financing strategy

JEL Classification: C6, G1

Suggested Citation

Rosu, Ioanid and Stroock, Daniel W., On the Derivation of the Black-Scholes Formula. Seminaire de Probabilites, Vol. 37, pp. 399-414, 2004. Available at SSRN: https://ssrn.com/abstract=710842

Ioanid Rosu (Contact Author)

HEC Paris - Finance Department ( email )

1 rue de la Liberation
Jouy-en-Josas Cedex, 78351
France

Daniel W. Stroock

Massachusetts Institute of Technology (MIT) - Department of Mathematics ( email )

Cambridge, MA 02139-4307
United States

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