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ma503b-10hm3

# ma503b-10hm3 - W 1 t π 2 t σ 2 d ¯ W 2 t o ii Consider a...

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MATH503b (SPRING 2010) HOMEWORK #3 Due Wednesday, March 10, 2010 1) Exercise 11.1 (of Bj¨ork’s book). 2) Exercise 15.3, 15.4, and 15.5 (of Bj¨ork’s book). 3) Consider a market containing one bond and two stocks, and is described by the following SDEs: ( dB ( t ) = rB ( t ) dt dS i ( t ) = S i ( t )[ rdt + σ i d ¯ W i ( t )] , i = 1 , 2 . (1) where ¯ W 1 and ¯ W 2 are two independent standard Brownian motions. i) Let π ( t ) = ( π 1 ( t ) , π 2 ( t )), t 0 be the portfolio process, where π i ( t ) is the amount of money invested in i -th stock at time t . Show that, for any given self-financing portfolio, the discounted value of the portfolio e V π ( t ) = e - rt V π ( t ) satisfies d [ e V π ( t )] = e - rt n π 1 ( t ) σ 1 d
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Unformatted text preview: W 1 ( t ) + π 2 ( t ) σ 2 d ¯ W 2 ( t ) o . ii) Consider a T-claim X 1 , 2 = ( S 1 ( T )-S 2 ( T )) + . Assume that the price of X 1 , 2 at time t is Π( t, X 1 , 2 ) = F ( t,S 1 ( t ) ,S 2 ( t )), for all t ≥ 0, where F is a smooth deterministic function. Applying the pricing principle to derive the diﬀerential equation that F satisﬁes, as well as a hedging strategy ( π 1 ( t ) ,π 2 ( t )) in terms of the function F . iii) Is it possible to reduce the state space for this problem? If so, carry it out. 1...
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