Question1 solve clearly.

If A(t) is a strictly positive supermartingale, then zero-coupon bond prices can be modelled using the formula B(, T) = =PLA() | F,] where P is a suitably-chosen A(1) probability measure. (i) (a) Express mathematically the fact that A(t) is a strictly positive supermartingale. (b) Verify that the function A(1) =e-0.05+0.02"(), where W() denotes standard Brownian motion, satisfies the properties in (i)(a). (c) State why the supermartingale property is required. (d) Write down the name given to this type of process. [7] (ii) By writing A(t) in the form A(() =e""), or otherwise, show that A() satisfies a stochastic differential equation of the form: dA(!) = A(D)Lu(D)di + (1)dW()] State the forms of the functions #(t) and GA(!). [4] (iii) (a) Write down or derive a formula for B(t,7) based on the process A(!) specified in (i)(b). (b) Write down expressions for the instantaneous forward rate f(t,7) and the spot rate R(t, 7) based on this model. (c) State one problem that this model of interest rates has. [4] (iv) Calculate the prices at time 5 according to the model in (ii) of the following risk- free bonds: (a) a 10-year zero-coupon bond (b) a 10-year bond that pays a coupon of 5% at the end of each year. [4] [Total 19](i) Explain what is meant by a default-free bond. [1] (ii) State the possible outcomes of a default. [2] (iii) List four types of credit event. [2] (iv) Explain what is meant by the recovery rate for a bond. [1] [Total 6]