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Unformatted text preview: PDE for Finance, Spring 2011 – Homework 5 Distributed 4/4/11, due 4/18/11 . Problem 1 is a classic example (due to Merton) of optimal asset allocation. Problems 24 reinforce our discussion of optimal stopping and American options. 1) Consider the following assetallocation problem. Two investment opportunities are avail able. One is riskfree, earning (constant) interest r . The other is lognormal, with (constant) drift μ and volatility σ , i.e. it satisfies dp = μpds + σpdw . You start at time t by investing wealth x . Your control is the weighting of your portofolio between these two assets, i.e. α ( s ) = fraction of wealth invested in the risky asset at time s subject to 0 ≤ α ≤ 1. You never withdraw from or add to the portfolio, and you have a fixed horizon T . Your goal is to maximize the utility of your portfolio value at time T ; in other words, your value function is u ( x,t ) = max α ( s ) E y ( t )= x [ h ( y ( T ))] where y ( s ) is the value of the portfolio at time...
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 Fall '11
 Bayou
 Finance, Options, optimal investment strategy, max Ey

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