Question

A stock is currently priced at $110, and the volatility is 32% per annum. Within the next one year, a dividend of $1.5 is expected after two months and again after eight months (Hint: There are two dividends). The risk-free rate of interest is 7% per annum with continuous compounding. Keep four decimal places for all calculations.

2) According to Black’s approximation, what is the value of a 10-month American call with a strike price of $105?

Answer 1

T1 = 2/12 = 0.1667; T2 = 8/12 = 0.6667; T = 10/12 = 0.8333; D1 = 1.5; D2 = 1.5; S0 = 110; K = 105; SD = 32%; Rf = 7%

Compare K*[1 - e^(-r(T2 - T1)] with D1: 105*(1-e^(-7%*(0.6667-0.1667)) = 3.6114

This is greater than D1 so option should not be exercised before the first ex-dividend date.

Compare K*[1 - e^(-r(T - T2)] with D2: 105*(1-e^(-7%*(0.8333-0.6667)) = 1.2179

This is less than D2 so option should be exercised before the second ex-dividend date.

Adjustment to be made to the strike price is the present value of the first dividend which is 1.5*e^(-7%*0.1667) = 1.4826

Adjusted strike price = 110 - 1.4826 = 108.5174

Using Black Scholes model, we have:

This option price has to be compared with the value of an equivalent European call as follows:

18.3802 > 15.4967 so the 10-month American call option value is 18.3802.