2. Consider the simple one factor interest rate modell where the forward rate is given by...

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2. Consider the simple one factor interest rate modell where the forward rate is given by f(t, T) = oWt + f(0, T) + fa(s, T)ds. The tradables are: Cash bond: dBt=rBedt, Bo = 1, T-maturity bond:P(t, T) = exp{- f(t, u)du}. The discounted bond price Zt B¹P(t, T) satisfies the stochastic differential equation = dZtZt-o(T - t)dWt+ [-00 (√²/0² (T − 1)² - T [² a(t, u)du dt (a) Find the measure Q equivalent to P under which Z, is a martingale. (b) Consider another bond with maturity S and let Y₁ = B₁¹P(t, S). (i) Write down the expression for dyt (ii) Show that under no arbitrage the market price of risk is the same for the two different maturity bonds. (c) Use (b) above to determine the function a(t, T). (d) Express B, and P(t, T) in terms of the Q Brownian motion and the market price of risk. 2. Consider the simple one factor interest rate modell where the forward rate is given by f(t, T) = oWt + f(0, T) + fa(s, T)ds. The tradables are: Cash bond: dBt=rBedt, Bo = 1, T-maturity bond:P(t, T) = exp{- f(t, u)du}. The discounted bond price Zt B¹P(t, T) satisfies the stochastic differential equation = dZtZt-o(T - t)dWt+ [-00 (√²/0² (T − 1)² - T [² a(t, u)du dt (a) Find the measure Q equivalent to P under which Z, is a martingale. (b) Consider another bond with maturity S and let Y₁ = B₁¹P(t, S). (i) Write down the expression for dyt (ii) Show that under no arbitrage the market price of risk is the same for the two different maturity bonds. (c) Use (b) above to determine the function a(t, T). (d) Express B, and P(t, T) in terms of the Q Brownian motion and the market price of risk.

Answer rating: 100% (QA)

SOLUTION a To find the equivalent measure Q under which Z is a martingale we need to determine the riskneutral drift rate which is the drift rate adju... View the full answer