## Call Option Formula

Although it is usually impossible to find an analytic solution to the Black-Scholes equation, it is possible to find such a solution for a European call option This analytic solution is of great practical and theoretical use

The formula uses the function N(..v), the standard cumulative normal probability distribution. This is the cumulative distribution of a normal random variable having mean 0 and variance I It can be expressed as

The function N(.x) is illustrated in Figure 13 1 The value N(x) is the area under the familiar bell-shaped curve from —oo to ,.v. Particular values are N(~oo) — 0, N(0) = i and N(oo) = 1.

The function N(x) cannot be expressed in closed form, but there are tables for its values, and there are accurate approximation formulas (See Exercise 1.)

Black-Scholes call option formula Consider a European call option with strike price ~7 K and expiration time T If the underlying stock pays no dividends during the time [0, J ] and if interest is constant and continuously compounded at a rate r, the Black-Scholes solution is f($,t)~C(S,t), defined by

FIGURE 13 J Normal density and cumulative distribution, (a) The curve is the normal density {\}-j2iT)e~*~12 The area under the curve up to the point a gives the value of the cumulative distribution N{x) (b) The cumulative distribution itself rises smoothly from 0 to 1, but it does not have a closed-form representation

FIGURE 13 J Normal density and cumulative distribution, (a) The curve is the normal density {\}-j2iT)e~*~12 The area under the curve up to the point a gives the value of the cumulative distribution N{x) (b) The cumulative distribution itself rises smoothly from 0 to 1, but it does not have a closed-form representation where ln(S/K) + (r + a2/2)(T -1) d] = -

Os/T - t and where N(.x) denotes (he standard cumulative normal probability distribution.

Let us examine some special cases, First suppose t = 7" (meaning the option is at expiration). Then

because the r/'s depend only on the sign of In (S/K). Therefore, since A^(oo) = 1 and N(—oo) =0, we find

which agrees with the known value at T.

Next let us consider T = oo. Then d\ — oo and e~r{1~!) = 0 Thus C(S, oo) = 5, which agrees with the result derived earlier for a perpetual call.

Example 13,2 (A 5-month option) Let us calculate the value of the same option considered in Chapter 12, Example 123, That was a 5-month call option on a stock with a current price of $62 and volatility of 20% per year. The strike price is $60 and the interest rate is 10%. Using S = 62, K = 60, a = ,20, and r = .10, we find

The corresponding values for the cumulative normal distribution are found by the approximation in Exercise 1 to be

Hence the value for the call option is

C = 62 x 7.39332 - 60 x . 95918 x 695740 = $5,798.,

This is close to the value of $5.85 found by the binomial lattice method.

Although a formula exists for a call option on a non-dividend-paying stock, analogous formulas do not generally exist for other options, including an American put option The Black-Scholes equation, incorporating the corresponding boundary conditions, cannot be solved in analytic form

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