4703-10-Fall-h5 - IEOR 4703 Homework 5 This assignment will...

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IEOR 4703: Homework 5 This assignment will help you understand how variance reduction works in real applica- tions. For our first problem: You will estimate the price of a European call option, even though we know the price exactly via Black-Scholes option pricing formula. Since we do know the price, we can check it against our estimates, and we can do so for alternate estimate using antithetic variates to see how one estimate can really be better than an- other. For our second problem (see below), we re-consider Homework 3, Problem 4 using antithetic variates as comparison. FIRST PROBLEM: Assume: X ( t ) = σB ( t )+ μt where σ = 0 . 04. S ( t ) = S (0) e X ( t ) , t 0 (Geometric BM). μ = μ * = r - σ 2 / 2 = 0 . 05 - 0 . 008 = 0 . 0492 is the risk-neutral drift and you are to use that since we are only interested in prices of an option for this assignment. We want to estimate the price of a European call option, with payoff ( S ( T ) - K ) + , at time T . We will consider T = 4. Thus the price is given by C 0 ( S (0) , K ) = e - rT E * (( S ( T ) - K ) + ) = e - 0 . 20 E * (( S (4) - K ) + ) . (1) Assuming that S
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