sheet07 - J Wissel Financial Engineering with Stochastic...

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J. Wissel Financial Engineering with Stochastic Calculus I Fall 2010 Assignment Sheet 7 1. Let Z 0 be a random variable on a space (Ω , F ,P ) with E [ Z ] = 1. a) Show that ˜ P [ A ] = E [ ZI A ] , A ∈ F defines a probability measure ˜ P on (Ω , F ). b) Suppose that Z > 0 P -a.s. Show that P and ˜ P are equivalent, and that 1 Z is a Radon-Nikod´ym derivative of P w.r.t. ˜ P . 2. Consider the Black-Scholes model for a stock price S ( t ) with interest rate r and vola- tility σ . Use the risk-neutral pricing formula to compute the price at time t [0 ,T ] of a digital option which pays 1 unit of cash at time T if S ( T ) K , and zero otherwise. 3. In the Black-Scholes model with time-varying non-random coefficients, the bank ac- count and stock price are given by dB ( t ) = B ( t ) r ( t ) dt, B (0) = 1 , dS ( t ) = S ( t ) r ( t ) dt + S ( t ) σ ( t ) d f W ( t ) , S (0) = S 0 for a Brownian motion f W ( t ) under the risk-neutral measure ˜ P , and r ( t ) ( t ) are
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sheet07 - J Wissel Financial Engineering with Stochastic...

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