Homework Three

Homework Three - p, < p < 1. We let N...

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MAFS 5030 Quantitative Modeling of Derivatives Securities Homework Three Course Instructor: Prof. Y.K. Kwok 1. Consider the sample space Ω = {- 3 , - 2 , - 1 , 1 , 2 , 3 } and the algebra F = { φ, {- 3 , - 2 } , {- 1 , 1 } , { 2 , 3 } , {- 3 , - 2 , - 1 , 1 } , {- 3 , - 2 , 2 , 3 } , {- 1, 1 , 2, 3 } , Ω } . For each of the following random variables, determine whether it is F -measurable: (i) X ( ω ) = ω 2 , (ii) X ( ω ) = max( ω, 2). Find a random variable that is F -measurable. 2. Let X, X 1 , ··· , X n be random variables de±ned on (Ω , F , P ). Prove the following prop- erties on conditional expectations: (a) E [ XI B ] = E [ I B E [ X |F ]] for all B ∈ F , (b) E [max( X 1 , ··· , X n ) |F ] max( E [ X 1 |F ] , ··· , E [ X n |F ]). 3. Let X = { X t ; t = 0 , 1 , ··· , T } be a stochastic process adapted to the ±ltration F = {F t ; t = 0 , 1 , ··· , T } . Does the property: E [ X t +1 - X t |F t ] = 0 , t = 0 , 1 , ··· , T - 1 imply that X is a martingale? 4. Consider the binomial experiment with probability of success
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Unformatted text preview: p, < p < 1. We let N k denote the number of successes after k independent trials. De±ne the discrete process Y k by N k-kp , the excess number of successes above the mean kp . Show that Y k is a martingale. 5. Consider the two-period securities model in the lecture note of Topic 3, p.16. Suppose the riskless interest rate r violates the restriction r < . 2, say, r = 0 . 3. Construct an arbitrage opportunity associated with the securities model. 6. Deduce the price formula for a European put option with terminal payo² max( X-S, 0) for the n-period binomial model. 1...
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