A model was proposed for a country’s sovereign debt as follows: The economy is in one of three states: 1 (good), 2 (bad) and 3 (default). Transition intensities λi,j are constant and as follows: λ1,2 = 1; λ1,3 = 0; λ2,1 = 0.25, λ2,3 = 0.75; λ3j = 0 for all j and λ1,1 = λ2,2 = −1. It follows that if pi(t) is the probability that the economy is in state i at time t then: )(25.0)()(211tptpdttdp+−= and )()()(212tptpdttdp−=. Set h(t) = 2p1 (t) − p2(t). (ii) (a) Show that ()1.5().dhthtdt=− (b) Derive a similar equation for k defined by k(t) = 2p1(t) + p2(t). [2] Suppose that this country’s economy is in state 2 at time 0. (iii) Find the probability that it is in default at time 2. [4] Assume a continuously compounded risk-free interest rate of 2% per annum and a recovery rate of 60%. (iv) (a) Deduce the price under this model for a zero-coupon bond in this country with a redemption value of 100 and a redemption date in two years’ time. (b) Calculate the credit spread. [3] [Total
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