1. Let {Wt}te[0,1] be a Brownian motion on a probability space (22, F,P), and let Ft, 0 < t

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1 Let Wt Te 0 1 Be A Brownian Motion On A Probability Space 22 F P And Let Ft 0 T T Be The Filtration Generate 1 (106.65 KiB) Viewed 63 times

1. Let {Wt}te[0,1] be a Brownian motion on a probability space (22, F,P), and let Ft, 0 < t<T be the filtration generated by this Brownian motion. In a certain market the risk free rate r = () and the equity index is given by St = S, exp((u – čo?)t+oWt), where , o > 0 are constants. There are no dividends. A derivative contract is to be written at t = 0on the index which has the payoff ST XT = log K T>0, K >0. (i) (1 marks) Write down an expression for the SDE of S. (ii) (2 marks) Use the hedging strategy to derive the partial differential equation (PDE) that the price of derivative contract satisfies. (iii) (1 marks) Use the Feynman-Kac formula to give an conditional expectation representation of the solution to the PDE problem. (iv) (3 marks) Show the explicit pricing formula for the price of the derivative contract. (v) (1 mark) Show the A; in the hedging strategy.