soln1 - Stochastic Calculus in Finance Assignment 1 -...

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Unformatted text preview: Stochastic Calculus in Finance Assignment 1 - SOLUTIONS (February 2007) 1. (a) = { ( H,H,H ) , ( H,H,T ) , ( H,T,H ) , ( H,T,T ) , ( T,H,H ) , ( T,H,T ), ( T,T,H ) , ( T,T,T ) } , F consists of all subsets of , and P ( A ) = #( A ) / 8. In other words, P makes each outcome equally likely. (b) G is a -field with 4 atoms, namely { ( T,H,H ) , ( H,T,H ) , ( H,H,T ) } , { ( H,H,H ) } , { ( H,T,T ) , ( T,H,T ) , ( T,T,H ) } , and { ( T,T,T ) } . The - field G itself consists of all possible unions of these four atoms, so the events in G consist of all possible unions of 0, 1, 2, 3, or 4 of the atoms. That is, # G = 2 4 = 16. Put another way, each element of G is deter- mined by making 4 choices: whether to include the first atom, whether to include the second atom, etc. and there are 16 ways to make all these choices. To see that G actually is the desired -field, observe that X is G measurable, since any event of the form { | X ( ) x } can be written as a union of atoms, so belongs to G . Conversely, if H is any -field with respect to which X is measurable, then { | X ( ) = x } H for every x . In particular, all the atoms of G belong to H , so G H too. In other words, G really is the smallest such -field. (c) We use the formula from class, that E [1 A | G ]( ) = P ( A | B ), which is valid whenever B is the atom of G containing . Thus E [1 A | G ]( ) = , if = ( T,T,T ) 1 / 3 , if = ( H,T,T ) , ( T,H,T ), or ( T,T,H ) 2 / 3 , if = ( T,H,H ) , ( H,T,H ), or ( H,H,T ) 1 , if = ( H,H,H ) . 2. G has 4 atoms, namely A B , A \ B , B \ A , and A c B c . Our assumptions imply that none of these events is empty. Putting them together in all possible ways yields the following 16 events: G = { ,A B,A \ B,B \ A,A c B c ,A,B,A c ,B c , ( A B ) ( A c B c ) , ( A \ B ) ( B \ A ) ,A B,A B c ,B A c ,A c B c , } 3. To show that max( X,Y ) (which I denote X Y ) is G-measurable, we need to show that { X Y z } G for every any real number z . But { X Y z } = { X z } { Y z } G 1 2 since the latter two sets belong to G (as X and Y are G-measurable), and G is closed under intersections (as it is a -field). Similarly, { X Y z } = { X z } { Y z } G , so X Y is G-measurable. 4. Let X be the number of days with trades, and let A i denote the event that there is a trade on the i th day. Then E [ X ] = E [ 10 X i =1 1 A i ] = 10 X i =1 E [1 A i ] = 10 P ( A 1 ) . Let B j be the event that the j th manager doesnt trade on the first day. Therefore P ( A 1 ) = 1- P ( A c 1 ) = 1- P ( B 1 B 12 ) = 1- P ( B 1 ) 12 = 1- ( . 9) 12 ....
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soln1 - Stochastic Calculus in Finance Assignment 1 -...

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