Solution to Homework Three

# Solution to Homework Three - MAFS 5030 Quantitative...

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Unformatted text preview: MAFS 5030 Quantitative Modeling of Derivatives Securities Solution to Homework Three Course Instructor: Prof. Y.K. Kwok 1. F is generated by the partition P = {{- 3 ,- 2 } , {- 1 , 1 } , { 2 , 3 }} . (i) Since { 2 , 3 } ∈ P and X (2) = 4 negationslash = X (3) = 9, X is not F-measurable. (ii) Since { 2 , 3 } ∈ P and X (2) = 2 negationslash = X (3) = 3, X is not F-measurable. Define the random variable X ( ω ) = max( ω, 3). Now, X ( ω ) = 3 for all ω ∈ Ω, hence X is F-measurable. 2. (a) Suppose F is generated by a partition P . It suffices to show that this property is valid for every B ∈ P . Consider E [ I B E [ X |F ]] = summationdisplay ω ∈ B E [ X | B ] P ( ω ) = E [ X | B ] P ( B ) = summationdisplay ω ∈ B X ( ω )( P ( ω ) /P ( B )) P ( B ) = summationdisplay ω ∈ B X ( ω ) P ( ω ) = E [ XI B ] . (b) Recall E [ X |F ] = J summationdisplay j =1 E [ X | B j ] 1 B j , and consider E [max( X 1 , ··· , X n ) |F ] = J summationdisplay j =1 E [max( X 1 , ···...
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Solution to Homework Three - MAFS 5030 Quantitative...

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