1    PDE option pricing

A “supershare option” is an option that provides payoff:

f(ST ) = {0(S)T /XL

if XL  ≤ ST  < XH

otherwise.

The price of such an option is given by:

wt =       [N(d1 ) − N(d2 )] ,

where

d1                 log(St /XL ) + (r + σ2 /2)(T − t)

σ ^T − t

2      =

1. Sketch the payoff of a supershare option (as a function of ST ).

2. Confirm that the supershare option pricing formula satisfies the boundary condition for the option’s payoff.

3. Show that

∂w            1                                        1

∂S           XL                                             σXL ^T − t

=                         (N\ (d1 ) − N\ (d2 )) +                        (d2 N\ (d2 ) − d1 N\ (d1 ))

    =    id1 N\ (d1 ) − d2 N\ (d2 ) + 2 (  + ) ^T − t[N\ (d2 ) − N\ (d1 )]] . Hint: for , it may help to know that N\\ (d) = −dN\ (d) .

4. Hence, confirm that the supershare option pricing formula satisfies the Black-Scholes PDE.

2    Martingale option pricing

A stock price follows:

dSt = µSt dt + σSt dW.

1. A digital call option provides payoff (at time T):

f(ST ) =

Use risk-neutral valuation to write down a formula for the option’s price. Hint:  You may find it useful to write down what the option’s payoff is in terms of the log stock price at maturity.

2. A “forward start option”does not initially have a known strike price.  Instead, part way through its life, the strike price is set equal to the current stock price.  Mathematically, an option which matures at time T has its

strike price set at time τ to be equal to Sτ  where 0 < τ < T. We consider (here) a forward start call option.

(a) What is the option’s value at time τ (as a function of Sτ )? .