108156.pdf

9. Financial Engineering 193

written, that is,z=−1. Then

d

dS

V(S)=x−

d

dS

D(S).

The last termddSD(S), which is thedeltaof the derivative security, can readily
be computed if the model of stock prices is specified, so that an explicit formula
forD(S) is available.

Proposition 9.1

Denote the European call option price in the Black–Scholes model byCE(S).
The delta of the option is given by

d

dS

CE(S)=N(d 1 ),

whereN(x) is the normal distribution function given by (8.10) andd 1 is defined
by (8.9).

Proof

The priceS=S(0) appears in three places in the Black–Scholes formula, see
Theorem 8.6, so the differentiation requires a bit of work, with plenty of nice
cancellations in due course, and is left to the reader. Bear in mind that the
derivativeddSCE(S) is computed at timet=0.

Exercise 9.1

Find a similar expression for the deltaddSPE(S) of a European put option

in the Black–Scholes model.

For the remainder of this section we shall consider a European call option
within the Black–Scholes model. By Proposition 9.1 the portfolio (x, y, z)=
(N(d 1 ),y,−1),where the position in stockN(d 1 ) is computed for the initial
stock priceS=S(0), has delta equal to zero for any money market positiony.
Consequently, its value

V(S)=N(d 1 )S+y−CE(S)

does not vary much under small changes of the stock price about the initial
value. It is convenient to chooseyso that the initial value of the portfolio is
equal to zero. By the Black–Scholes formula forCE(S)thisgives

y=−Xe−TrN(d 2 ),