Math of finance

1. Consider a non-dividend paying share, whose price at time t, denoted by St, is modelled as:
9, ? s.exp[a(1§T-1§.) + (r- 1/202)(T- n], forT 2t,
where 3‘ is a standard Brownian motion under the risk-neutral probability measure Q and r
is the continuously compounded constant annual risk-free rate of interest.
(a) Determine the distribution of STIF. under measure Q, where F; denotes the filtration up
to time t. [2 marks]
(b) Show that D: ? ??????? ?? a Q-martingale. [2 marks]
Consider a derivative contract which prom’ses to make a payment of:
X2
at time 2.
(c) For time: 1 ? t ? 2, determine the following:
(i) Price of the derivative contract, Vt.
(ii) The share holding, 4),, and bond holding, u’)‘, in the self-financing replicating strategy.
[6 marks]
(d) For time: 0 ? t ? 1, determine the following:
(i) Price of the derivative contract, Vg.
(ii) The share holding, (1);, and bond holding, 117., in the self-financing replicating strategy.
[6 marks]
(e) Amming that the derivative is sold at time 0, d’scuss how the self-financing replicating
strategy over the duration of the contract, 0 ? t ? 2, will ensure that the final maturity
payofi’ at time 2 is met. [4 marks]
(f) Show that the prim obtained in parts (c) and (d) satisfy the Black-Scholes PDE and
the necessary boundary conditions. [6 marks]
[Totalz 26 marks]