We can now calculate the Black-Scholes formula (see [8]) for the price of a call option. As we have seen in equations (5.8) and (5.45), this formula can be derived in two ways. Equation (5.45) derives the formula for pricing call options as the expected value of the future payoff based on the risk-neutral probability, discounted by the risk-free rate of interest. Equation (5.9) expresses the price of a call option as the present value of the payoff based on the stochastic discount factor. In either approach, the formula is obtained by the evaluation of (essentially) the same integral.

If we approach the call option pricing formula from the perspective of the discounted expected value we must use the risk-neutral probability which we defined as the MAPLE function given below.

> Lnrn(s,t,P,r,sigma);

The price of the call option based on equations (5.45) and (5.44) will thus be

> CallPrice:='exp(-r*t)*int(Call(s,K)*Lnrn(s,t,P,r,sigma),s=0..infinity)';

We have seen in equation (5.16) the relationship between the risk-neutral probability and the stochastic discount facotor . The function is defined as below.

> psi:=unapply(exp(-r*t)*Lnrn(s,t,P,r,sigma),s,t,P,r,sigma);

The price of the call option can also be calculated based on equation (5.45), as the expression CallAlt:

> CallAlt:=Int((s - K)*psi(s,t,P,r,sigma),s=K..infinity);

where was subsituted for , and thus the integral is as above. Clearly these two expressions are the same. We shall proceed now with the calculation of CallPrice . Some readers may wish to skip these calculations. The result is reported in equation (6.2) toward the end of this section. We begin by informing MAPLE that some of our variables must be positive:

> assume (P>0,t>0,r>0,K>0,sigma>0);

We then ask MAPLE to attempt an evaluation of the integral. To understand MAPLE's evaluation we need to introduce the definition of the function erf , a built-in MAPLE function. We previously defined the cumulative Normal probability function in MAPLE as Normalcdf . erf is a built-in MAPLE function which is closely related to the function Normalcdf . erf is defined as

> erf(x)=2/sqrt(Pi)*Int((exp(-v^2), v=0..x));

Similarly, the cumulative normal probability function is defined by

> Normalcdf:=(x,mu,sigma)->int(exp(-(v-mu)^2/2/(sigma)^2)/sqrt(2*Pi*(sigma)^2), v=-infinity..x);

To demonstrate the relationship between erf and the normal probability function we evaluate Normalcdf at for and .

> Normalcdf(x*sqrt(2),0,1);

MAPLE recognizes that Normalcdf can be defined in terms of the MAPLE function erf . Conversely, we can express the function erf in terms of the function Normalcdf . Mutliplying Normalcdf by two, evaluating it at the point is equivalent to the erf function plus one. This is demonstrated as follows:

> 2*Normalcdf(x*sqrt(2),0,1)-1;

To facilitate the MAPLE calculation the integral defining CallPrice is rewritten in terms of the density function of -- the continuously compounded rate of return. As you recall, we assumed it to be the normal distribution with expected value of and a standard deviation of . This could be achieved by simply replacing the variable of integration in the above integral using the relation in equation, (5.24) i.e., .

Alternatively, one can apply the fact that given a random variable the expected value of may be calculated based on the density of as . Thus in our case plays the rule of and is the normal density function. Consequently we can equivalently define CallPrice as below.

> CallPrice:='exp(-r*t)*int(Call(P*exp(y),K)*Normalpdf(y,(r-sigma^2/2)*t, sigma*sqrt(t)),y=0..infinity)';

Since when the value of vanishes we can change the lower limit of the integral to be and replace in the integrand with . Thus, the integral to be calculated is

> CallPrice:='exp(-r*t)*int((P*exp(y)-K)*Normalpdf(y,(r-sigma^2/2)*t, sigma*sqrt(t)),y=ln(K/P)..infinity)';

and as we can see below, the results of the calculation are written in terms of the MAPLE function erf .

> CallPrice:=expand(CallPrice);

Let us ask MAPLE to simplify the above expression.

> CallPrice:=simplify(CallPrice);

Now the resemblance between the CallPrice and the erf functions and consequently the Normalcdf function is more apparent.

Given the relationship between Normalcdf and erf we can convert this expression in terms of Normalcdf . Indeed, the classical way of stating this expression is not in terms of erf . However, at this time it is more efficient to use erf to perform calculations using MAPLE. We shall return to this point soon, but first we define a function in MAPLE that calculates the price of a call option. We name this function Eqbs . The following command in MAPLE defines the function Eqbs based on the results of our calculation:

> Eqbs:=unapply(CallPrice,K,t,P,sigma,r);

The reader can now calculate the price of a call option given specific parameter values. For example, an option that expires in one month ( of a year) where and are given per annum will have the following value:

> evalf(Eqbs(90,1/12,95,0.18,0.15));

Given the relation between erf and the normal distribution it is possible to write the pricing formula in terms of the normal distribution.

We can also ask MAPLE to express the pricing formula in terms of the cumulative standard normal distribution, i.e., in terms of Normlacdf ( ). For simplicity let us call this function . Let us define a function called newerf ; this function is simply the erf function in terms of the function. As discussed earlier, this relationship is

> newerf:=t->2*N(t*sqrt(2))-1;

We can now substitute newerf for erf in the CallPrice expression. The pricing formula is now expressed in terms of the function:

> NewCallPrice:= simplify(subs(erf=newerf,CallPrice));

which equals

(6.1)

Recognizing that , we restate equation (6.1) as

.

(6.2)

The above is the Black-Scholes option pricing formula. The formula is classically restated as

,

(6.3)

where is the cumulative normal distribution with and , i.e.,

(6.4)

(6.5)

and

(6.6)

The procedure BstCall is a slight modification of the built-in MAPLE procedure which calculates the call price. Using the above example, the parameters for this procedure are as follows: exercise price (90); time to expiration of the option (1 month); current price of the stock (95); standard deviation of the continuous return on the stock per unit of time (0.18); and continuously compounded risk-free interest rate per unit of time (0.15).

Let us value an option with the above specifications. As discussed earlier it is common practice to use 1 year as the basic unit of time. Hence a one month time period is inputted as . We assume the interest rates and standard deviation are given in per annum values. We can now value such a call option as follows:

> BstCall(90,1/12,95,0.18,0.15);

Footnotes

The reader may have skipped this discussion as suggested.