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# HO4s - Math 238 Financial Mathematics Problem set 4 These...

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Math 238, Financial Mathematics Problem set 4 February 17, 2009 These problems are due on Tuesday February 24. You can give them to me in class, drop them in my office, or put them in my mailbox outside the mathematics department. 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-financing 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( integraltext 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-financing implies the second condition above.

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