HO4s-sol - Math 238, Financial Mathematics Problem Set 4...

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Math 238, Financial Mathematics Problem Set 4 Solutions February 28, 2009 Problem 1 Consider the Vasicek (or Ornstein-Uhlenbeck) model for the observed short rate (money market rate) r t which is dr t = a ( r - r t ) dt + σdW t Here r is the observed equilibrium level, α is the rate of mean reversion and σ the volatility. We want to price a zero coupon bond by constructing a replicating, self-Fnancing portfolio using a bond with a longer maturity and a money market account. Let P ( t, r ; T ) be the price of the zero coupon bond to be priced, maturing at time T , with t and r the current time and short interest rate, respectively. Let P l ( t, r ; T l ) be the bond with the longer maturity T l > T . Let I ( t ) = exp( R t 0 r s ds ) be the short rate growth factor, with dI ( t ) = r t I ( t ) dt . The portfolio for P is P = Δ P l + bI, with dP = Δ dP l + bdI (a) Explain why self-Fnancing implies the second condition above. In discrete time this condition is P n +1 - P n = Δ n ( P l n +1 - P l n ) + b n ( I n +1 - I n ). In this form we see clearly that the instantaneous change in the value of the portfolio is due to the change in the price of the bond and in the accumulation of interest. After the new value P n +1 is calculated then we go to the replication relation and calculate the new hedge ratio Δ n +1 . No new funds are added to the portfolio. (b) Use Ito’s formula in dP = Δ dP l + bdI and then choose Δ so that the coe±cients of the dW terms cancel out. Show that this gives Δ = ∂P ∂r ∂P l ∂r and then equate the dt terms and rearrange them to get 1 ∂P ∂r ± ∂P ∂t + 1 2 σ 2 2 P ∂r 2 - rP ² = 1 ∂P l ∂r ± ∂P l ∂t + 1 2 σ 2 2 P l ∂r 2 - rP l ² This is done by direct calculation with Ito’s formula and use of the SDE. (c) Explain what the essential conclusion from the above relation is. Show that if the relevant
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HO4s-sol - Math 238, Financial Mathematics Problem Set 4...

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