Homework Set 1 (Spring2017).pdf - Homework Set 1 STAT...

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Homework Set 1 STAT GU4265 & GR5265 – Stochastic Methods in Finance Irene Hueter 1. (a) Assume that the change of numeraire formula does not hold at time t and we have X Y ( t ) < X Z ( t ) · Z Y ( t ) . Show how to make a risk-free profit by trading in assets X, Y, and Z. (b) Illustrate the trading in (a) when X Y ( t ) = 1 / 4 , X Z ( t ) = 2 , and Z Y ( t ) = 1 / 6 . 2. A discretely rebalanced portfolio P k = N i =1 Δ i ( k ) · X i is self-financing relative to Y if N X i =1 i ( k + 1) - Δ i ( k )] · X i Y ( k + 1) = 0 for each integer k 1 , where Y and { X i } 1 i N are assets. Let X and Y be two no-arbitrage assets. Determine whether the following discretely rebalanced portfolios are self-financing: (a) P k = [max 1 s k X Y ( s )] · Y k , (b) P k = [ 1 k R k 0 X Y ( s ) ds ] · Y k . 3. Assume that Y and { X i } 1 i N are no-arbitrage assets. Show that the price of a discretely rebalanced portfolio P k = N i =1 Δ i ( k ) · X i relative to a reference asset Y is a P Y -martingale if the market is free of arbitrage. Note that a stochastic process { V t } t R
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  • Spring '16
  • Yuchong Zhang

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