Financial economics

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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