Question

Solve the following options employing the Black-Scholes model:

A: European call option on XYZ: stock value = $13. Strike price = $12, r = 6%. strand deviation = 20%. T = 6 months, 1st dividend $1 in 2 months, 2nd dividend of $1 in 5 months.

B: American call option for XYZ (Black's Approximation).

Answer 1

B)

The Black–Scholes formula (hereinafter, "BS Formula") provides an explicit equation for the value of a call option on a non-dividend paying stock. In case the stock pays one or more discrete dividend(s) no closed formula is known, but several approximations can be used, or else the Black–Scholes PDE will have to be solved numerically. One such approximation is described here

The method essentially entails using the BS formula to compute the value of two European call options:
(1) A European call with the same maturity as the American call being valued, but with the stock price reduced by the present value of the dividend, and
(2) A European call that expires on the day before the dividend is to be paid.

The largest of (1) and (2) is taken as the approximate value for the American call.The resulting value is sometimes called the "pseudo American" value of the call.

Here (1) is the same as the european call option value derived above=0.37963

(2)

This is the second method calculation, which states:

A European call that expires on the day before the dividend is to be paid.

This method begins just like the previous method except that this options maturity is set to the last maturity before the last dividend (meaning the second dividend in the fifth month)

Therefore (2) gives a value of 0.775512

since this is greater than (1)

american call option value = 0.775512