Reference no: EM13951859
1. Consider a market with one stock and one money market account, with the same assumptions on parameters (such as r, µ and σ 2 ) as in the derivation of the Black-Scholes formula given in class for European call option. Consider a contingent claim whose payoff depends on two dates T1 and T2, with T2 > T1 > 0 and two numbers K1 > 0 and K2 > 0.
The buyer of this contingent claim has the right to buy 1 share of the stock at time T1 at price K1. If the price of the stock at time T1 is lower than or equal to K1, she then has a right to by the stock at price K2 at time T2. (She does not have the buying opportunity at time T2 if the price of the stock at time T1 is strictly larger than K1.)
Find a formula for the current price (i.e. at time t = 0) of this contingent claim. The formula has to be explicit in the sense that one could feed into a computer program just like the Black-Scholes formula.
The formula may contain integrals. It is not necessary to simplify the formula.
2. Let Bt be the standard Brownian motion and Ft = σ(Bs, s ≤ t).
Deduce from the fact that Zt = exp(θBt - 1 2 θ 2 t) is a martingale with respect to Ft to show that both B2 t -t and B3 t -3tBt are martingales with respect to Ft.
(Hint: Use Taylor expansion of Z as a function of θ.)
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