(i) What is an option?

(ii) What is a key difference between a European option and an American option? (iii) What is an American perpetual option?

(iv) What is called an optimal exercise boundary for an American put option?

(v) Consider the American put option with value P(S, t), where S is the asset price and t is time. Let E be the strike price, T be the expiry date, r be the interest rate, o be the volatility, and Sf(t) be an optimal exercise boundary. Write down the partial differential equation on P(S, t) when S > Sf(t), the formula for P(S, t) when S < Sf(t), the inequality which P(S, t) must satisfy for S < Sf(t), the two

conditions at S = Sf(t), the condition at t = T , the boundary condition at S = 0, and the boundary condition at S < &.

(c) Let a discrete dividend be paid out on the underlying S at time td where the dividend yield is dy = 0.2. The value of S immediately after the dividend payment is S = £10.00. What is the value of S immediately before the dividend payment?

(d) (i) Consider smooth function V(F,t) at point (F* , t * ). Write forward finite difference ap- proximation of the first derivative 2V /2t at point (F* , t * ) if variable t has increment △t > 0, and finite difference approximation of the second derivative 22V /2F2 at point (F* , t * ) if variable F has increment △F > 0.

(ii) Consider European vanilla call option C(F, t) without dividends when the underly- ing is a futures contract with forward price F and delivery date T . The value of a European vanilla call option C(F, t) satisfies the following equation

+ o2F2 - rC = 0. (1)

Describe briefly how to generate a computational grid in the (F, t)-plane for numeri- cal solution of equation (1) if the variables are F ∈ [0, &) and t ∈ [0, T].