(e)  Suppose that the solution V = V (S) is a perpetual solution (meaning it is independent of

t) of the Black-Scholes equation. Show that V satisfies the differential equation o2S2 +rS - rV = 0

and by considering solutions of the form V = Sk deduce the general solution of the diffe-         rential equation.

2. (a)  Give the definition of an American Vanilla Put option.

(b)  Suppose that P represents the value of an American Vanilla Put with expiry T .  Explain why for any time t ● T there is a maximal value of S, Sf(t), such that the holder of the option should exercise the option if S ● Sf(t) and otherwise should sell or hold on to the option.

Give an informal explanation why = -1 at Sf.

(c)  Suppose that V is an American cash-or-nothing call option with pay-off V (S﹐ T ) = B > 0

if S exceeds the strike price and 0 otherwise. For a fixed time t ● T , describe an optimal         strategy for the holder of V dependent on the value of S.

(d)  Suppose that CA and PA are the values of an American Vanilla Call and an American Vanilla Put respectively.  Let CE and PE be the values of the corresponding European options.  Assume that both the put and the call have underlying S (which does not pay dividends), strike price E and expiry T ; the risk free interest rate is r. Mark the following statements as either true or false and in each case justify your conclusion.

(i) CA(S﹐ t) ● max(S - E ﹐ 0); (ii) CA(S﹐ t) < S - Ee-r(T -t); (iii) CE CA;

(iv) S - E ● CA(S﹐ t) - PA(S﹐ t); and

(v) CA(S﹐ t) - PA(S﹐ t) ● S - Ee-r(T -t) .

3. Suppose that V (S﹐ t) represents the value of a European option with strike price E and expiry date T on an underlying S which does not pay dividends. Then the Black-Scholes equation can be transformed into the diffusion equation

2u 22u

2 r     2x2

using the transformations S = Eex , t = T - 2r/o2 and

V = Eu(x﹐ r)exp ╱ - - ← ﹐

where the constant k = 2r/o2  is determined by the volatility o and the risk free interest rate r. Assume that V is the value of a European Vanilla Put.

(a)  Write down an expression for u(x﹐ 0) for x e (-&﹐ &).

(b)  What is the value of limx>& u(x﹐ t)?

(c)  What is the value of V (0﹐ t)? Explain your answer.

(d)  What is the value of limx>-& u(x﹐ r)?

(e)  Assume that r = . Use a numerical grid and the explicit finite difference method with x+ = 4, x- = - , 8x = = 8 r to approximate u(x﹐ r) at (x﹐ r) = (-1 ﹐ 1).                        [13]

Hint: At some stages you might like to use the iteration rule :

un﹐ m+1 = un﹐ m+ (un+1﹐ m+un-1 ﹐ m - 2un﹐ m)．