Tut0S - STAT/ACTSC 446/846 Assignment#1(Solution Problem 1...

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STAT/ACTSC 446/846 Assignment #1 (Solution) Problem 1. Solution: Cov ( Z t ,W u ) = Cov (3 t + 2 W t ,W u ) = 2min( t,u ) = 2 t Problem 2. Solution: (1) X t = μt + σW t X t can become negative. (2) Apply Ito’s lemma: (a) dY t = 0 + 3 X 2 t dX t + 1 2 6 X t ( dX t ) 2 = 3 X 2 t ( μdt + σdW t ) + 3 X t σ 2 dt = (3 Y 2 3 t μ + 3 Y 1 3 t σ 2 ) dt + 3 σY 2 3 t dW t (b) dY t = 2 tdt + 4 e 4 X t dX t + 1 2 16 e 4 X t ( dX t ) 2 = 2 tdt + 4( Y t - 10 - t 2 )( μdt + σdW t ) + 8( Y t - 10 - t 2 ) σ 2 dt = (2 t + ( Y t - 10 - t 2 )(4 μ + 8 σ 2 )) dt + 4 σ ( Y t - 10 - t 2 ) dW t Problem 3. Solution: Let f ( r t ,t ) = r t e αt . So, by It’s formula, we get: df ( r t ,t ) = αr t e αt dt + e αt dr t = αr t e αt dt + e αt ( - α ( r t - r 0 ) dt + σdW t ) = e αt αr 0 dt + σe αt dW t Integrating both sides from 0 to t yields: t integraldisplay 0 df ( r t ,t ) = t integraldisplay 0 e αs αr 0 ds + t integraldisplay 0 σe αs dW s r t e αt - r 0 = r 0 ( e αt - 1) + t integraldisplay 0 σe αs dW s r t = r 0 + e - αt t integraldisplay 0 σe αs dW s Some additional questions: (2) The drift term is equal to α ( r 0 - r t ). How can α and r 0 be interpreted?
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(2) here are many ways to interpret them. Lot’s of answers will be accepted. - α can be interpreted as the rate at which the process reverts to its mean, and r 0 can be interpreted as the mean to which the process reverts. - If you take the expectation of r t it is equal to r 0 since the expectation of integraltext t 0 σe α ( s - t ) dW s is equal to 0 (one easy way to prove it is to say that it is a martingale, so it has constant expectation and thus equal to the expectation at time 0, and the process is integraltext 0 0 σe α ( s - t ) dW s = 0 at time 0.) (3) It is a famous model to model stochastic interest rates, called “Vasicek Model”. A major
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