Math of finance1. 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 ris the continuously compounded constant annual risk-free rate of interest.(a) Determine the distribution of STIF. under measure Q, where F; denotes the filtration upto 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:X2at 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 replicatingstrategy over the duration of the contract, 0 ? t ? 2, will ensure that the final maturitypayofi? at time 2 is met. [4 marks](f) Show that the prim obtained in parts (c) and (d) satisfy the Black-Scholes PDE andthe necessary boundary conditions. [6 marks][Totalz 26 marks]
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