sheet06

# sheet06 - J Wissel Financial Engineering with Stochastic...

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J. Wissel Financial Engineering with Stochastic Calculus I Fall 2010 Assignment Sheet 6 1. Let c ( t,S ( t ) ) be the price of a European call option with maturity T and strike K in the Black-Scholes model, and let Δ( t ) be the stock position in the replication portfolio. a) Using the formulas for c ( t,x ) and d ± ( t,x ) in the lecture notes (equation (4.20) and following), show that c x ( t,x ) = Φ ( d + ( t,x ) ) . b) Let X ( t ) = η ( t ) B ( t )+Δ( t ) S ( t ) be the value at time t of the replication portfolio. Compute the bond position η ( t ). In particular, you will ﬁnd that η ( t ) < 0. c) The function c ( t,x ) is given by equation (4.20) only for t < T . For t = T we have c ( T,x ) = ( x - K ) + . Show that c ( t,x ) is continuous at t = T , i.e., lim t T c ( t,x ) = ( x - K ) + for all x > 0 . 2. Let us denote the Black-Scholes model price of a European call option with maturity T and strike K by c ( t,S ( t ) ,T,K,r,σ ) to express the dependence on parameters. a)

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sheet06 - J Wissel Financial Engineering with Stochastic...

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