(b) You put the amount of money I0 into a bank at the interest rate r = const at time t = t0 .

(i) Determine, showing all steps, the amount of money I(t) you will have at time t > t0 .

(ii) Is your investment risk-free? Why?

(c) Suppose that you are the holder of a cash-or-nothing European call option. The pay-off function 入(S) is given by

入(S) = BH(S – E),

where H(S-E) is the Heaviside step function, E is the strike price, S is the underlying price and B is a constant (the amount of cash). Let B = 30 pence, the strike price E = 270 pence and the underlying price S = 280 pence on the expiry date. What is your overall profit/loss on this transaction if you paid 15 pence for this option?

(d) Explain the no arbitrage principle. Use the no arbitrage principle to derive put-call parity for European vanilla options.

(e) (i) Write down the Black-Scholes equation for a European call option C(S, t) if the un- derlying pays out a continuous dividend. Explain carefully the meaning of each constant in the equation.

(ii) Let C(S, t) be

C(S, t) = C1(S, t) + AC2(S, t),

where C1(S, t) and C2(S, t) are solutions of the Black-Scholes equation in part (i) and A is a constant. Demonstrate, showing all steps, that C(S, t) is a solution to the Black-Scholes equation in part (i).

(iii) Consider a European option V (S, t) and let a discrete dividend be paid out on the underlying S at time td. The dividend yield is dy, where 0 s dy < 1. Formulate the jump condition for the underlying price S at time td and the continuity condition for the option price V (S, t) at time td .