LECTURE 7: BLACK–SCHOLES THEORY

1. Introduction: The Black–Scholes Model
In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper1 on the pricing and hedging of (European) call and put options. In this paper the famous Black-Scholes formula made its debut, and the Itˆ calculus was unleashed upon the world o 2 of finance. In this lecture we shall explain the Black-Scholes argument in its original setting, the pricing and hedging of European contingent claims. In subsequent lectures, we will see how to use the Black–Scholes model in conjunction with the Itˆ calculus to price and hedge all manner of o exotic derivative securities.
In its simplest form, the Black–Scholes(–Merton) model involves only two underlying assets, a riskless asset Cash Bond and a risky asset Stock.3 The asset Cash Bond appreciates at the short rate, or riskless rate of return rt , which (at least for now) is assumed to be nonrandom, although possibly time–varying. Thus, the price Bt of the Cash Bond at time t is assumed to satisfy the differential equation dBt (1)
= rt Bt , dt whose unique solution for the value B0 = 1 is (as the reader will now check) t (2)

rs ds .

Bt = exp
0

The share price St of the risky asset Stock at time t is assumed to follow a stochastic differential equation (SDE) of the form
(3)

dSt = µt St dt + σSt dWt ,

where {Wt }t≥0 is a standard Brownian motion, µt is a nonrandom (but not necessarily constant) function of t, and σ > 0 is a constant called the volatility of the Stock.
Proposition 1. If the drift coefficient function µt is bounded, then the SDE (3) has a unique solution with initial condition S0 , and it is given by t (4)

St = S0 exp σ Wt − σ 2 (t/2) +

µs ds
0

Moreover, under the risk–neutral measure, it must be the case that
(5)

rt = µt .

1

“The