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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 2-4 reinforce our discussion of optimal stopping and American options. 1) Consider the following asset-allocation problem. Two investment opportunities are avail- able. One is risk-free, earning (constant) interest r . The other is lognormal, with (constant) drift and volatility , i.e. it satisfies dp = pds + pdw . You start at time t by investing wealth x . Your control is the weighting of your portofolio between these two assets, i.e. ( s ) = fraction of wealth invested in the risky asset at time s subject to 0 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 ( s ) E y ( t )= x [ h ( y ( T ))] where y ( s ) is the value of the portfolio at time...
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