Chapter 28: Mathematical Applications 
Theoretical option price= pN(d 1) se-rtN(d2) 
p v2 
ln(8 )+ (r +2 )t 
where d1 = _ r. 
V-4 t 
d2 = d1 - v--ft 
The variables are: 
p = stock price 
s = striking price 
t = time remaining until expiration, expressed as a percent of a year 
r = current risk-free interest rate 
v = volatility measured by annual standard deviation 
ln = natural logarithm 
N(x) = cumulative normal density function 
457 
An important by-product of the model is the exact calculation of the delta - that 
is, the amount by which the option price can be expected to change for a small 
change in the stock price. The delta was described in Chapter 3 on call buying, and 
is more formally known as the hedge ratio. 
Delta= N(d1) 
The formula is so simple to use that it can fit quite easily on most programmable cal­
culators. In fact, some of these calculators can be observed on the exchange floors as 
the more theoretical floor traders attempt to monitor the present value of option pre­
miums. Of course, a computer can handle the calculations easily and with great 
speed. A large number of Black-Scholes computations can be performed in a very 
short period of time. 
The cumulative normal distribution function can be found in tabular form in 
most statistical books. However, for computation purposes, it would be wasteful to 
repeatedly look up values in a table. Since the normal curve is a smooth curve (it is 
the "bell-shaped" curve used most commonly to describe population distributions), 
the cumulative distribution can be approximated by a formula: 
x = l-z(l.330274y 5 - l.821256y 4 + l.781478y 3 - .356538y 2 + .3193815y) 
where y 1 and z = .3989423e-<r212 
= 1 + .2316419lcrl 
Then N(cr) = x if cr > 0 or N(cr) = 1- x if cr < 0