Consider a European gap put option on a stock that pays at expiration t = T the amount K0 S(T) if S(T) K1 and zero otherwise, where K1 = 100 and K0 = 105. Suppose that T = 1/2 (6 months). The current stock price is 120. The dividend rate is = 0.03. The risk-free interest rate is r = 0.08. The volatility is = 0.25.

(a) Give the formula for the Black-Scholes price of this option using only symbols and no numbers, and without using the abbreviations d1 and d2 (or if you use d1 and d2, you must also give the formulas for d1 and d2 in this case, otherwise you get zero points), so that the dependence of the price on K0 and K1 is clearly indicated. Be very careful about the meaning of each of K0 and K1. It suffices to give the answer.

(b) Find the Black-Scholes price of this option expressed as follows, in two ways. (i) Give the answer in a form that involves expressions of the type N(x) for explicitly given numerical values of x, but do not find the numerical values of N(x). Use explicit numbers for all parts of the answer except for the expressions N(x) where only x is an explicit number. In your final answer, use four decimals for each explicit number. Show your work. (ii) Evaluate the expressions of the form N(x) and give your final answer as a single number using two decimals (this number is then the price of the option).