Provide clearly explained answers.

In a particular two-state intensity-based model with constant transition intensities, the integrated generator matrix M =|A(u)du has the form M = where a = 0.2(r-s) and b = 0. 1(1 - s). Show that, in this case, M= -(a+b)M. Hence deduce an explicit formula for ed in terms of a and b . Hence deduce a set of formulae for the transition probabilities p;(s,f)-To fund an expansion in its operations, a company has just issued 5-year zero-coupon bonds with a total face value of $10 million, taking its total asset value up to $15 million. Explain how the value of the bonds can be expressed in terms of a European put option. [3] Hence calculate the fair price of a holding of the company bonds with a face value of $100 using the Black-Scholes model, given that the price of a 5-year zero-coupon government bond is $77.88. Assume that the annualised volatility of the company's assets over the 5-year period is 25%. [4] (iii) Explain what is meant by a credit spread and calculate its value for the company bonds. [3] Total 101Company X has just issued some 5-year zero-coupon bonds. A continuous-time two-state model is to be used to model the status of the company and to calculate the fair price of the bonds. It is believed that the risk-neutral transition rate for failure of the company is A(1) = 0.002r , where r is the time in years since the issue of the bonds. The 5-year risk-free spot yield is 5.25% expressed as an annual effective rate. Calculate the risk-neutral probability that the company will have failed by the end of 5 years. [2] (ii) In the event of failure of the company, the bonds will make a reduced payment at the maturity date. The recovery rate for a payment due at time / is: 6 (1) =1-0.05 Calculate the fair price to pay for $100 nominal of a Company X bond, taking into account the possibility of company failure. [3] (iii) An analyst is concerned that the estimate of A() may be too simplistic. Explain the possible reasons for his concern and how the model could be developed to deal with this. [3] [Total 8]