Explanation Given: The given matrix is: X ′ ( t ) = [ 0 1 0 0 0 1 1 − 1 1 ] X ( t ) + [ 0 0 t ] , X ( 0 ) = [ 1 − 1 0 ] and, the result of problem (9) is: e A t = 1 2 [ e t + cos t − sin t 2 sin t e t − cos t − sin t e t − cos t − sin t 2 cos t e t − cos t + sin t e t − cos t + sin t − 2 sin t e t + cos t + sin t ] Formula used: The solution of initial value problem X ′ = AX + f ( t ) , X ( 0 ) = X 0

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Chapter 9.8, Problem 24E

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The solution for the given initial value problem.

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Chapter 9.8, Problem 23E

Chapter 9.8, Problem 25E

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Problem 3. Pricing a multi-stock option the Margrabe formula The purpose of this problem is to price a swap option in a 2-stock model, similarly as what we did in the example in the lectures. We consider a two-dimensional Brownian motion given by W₁ = (W(¹), W(2)) on a probability space (Q, F,P). Two stock prices are modeled by the following equations: dX = dY₁ = X₁ (rdt+ rdt+0₁dW!) (²)), Y₁ (rdt+dW+0zdW!"), with Xo xo and Yo =yo. This corresponds to the multi-stock model studied in class, but with notation (X+, Y₁) instead of (S(1), S(2)). Given the model above, the measure P is already the risk-neutral measure (Both stocks have rate of return r). We write σ = 0₁+0%. We consider a swap option, which gives you the right, at time T, to exchange one share of X for one share of Y. That is, the option has payoff F=(Yr-XT). (a) We first assume that r = 0 (for questions (a)-(f)). Write an explicit expression for the process Xt. Reminder before proceeding to question (b): Girsanov's theorem…

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The solution for the given initial value problem.