Consider the following derivative product issued by a financial institution that can be assumed to be credit risk free. The product takes the form of a note paying out continuously a random coupon Ct>0 until default of a reference company, XYZ. The coupon is measurable relative to the information generated by the short rate, rt, which is assumed to evolve as a CIR process. You can assume C to admit the representation C(t,):=f(t,rs();0st) for generic state of the world and some deterministic, bounded function f0. In addition to the coupon payment, the note pays a unitary face value at maturity T>0, should company XYZ not have defaulted by then. If the company's default time, , occurs before maturity, then the face value is replaced by a payment X made at time , whereas the residual coupon payments are cancelled. The random payment X represents a suitably defined moving average of the company's CDS spread, (Rt)t[0,]. In other words, you can assume the following representation: X()():=g(t,Rs();0s () ) for generic state of the world and some deterministic, bounded function g0. You can assume X to be predictable relative to the information generated by the company's default time, so that there is no jump at default. The price of zero-coupon bonds issued by company XYZ are denoted by P(0,s) for any maturity s0 i. Write down the random payoff of the note. (6 marks) ii. Write down an expression for the no arbitrage price of the note by using risk neutral valuation. Assume that coincides with the first jump time of a conditionally Poisson processes with intensity t to make the expression as explicit as possible. Explain every step and assumption supporting your derivation. (6 marks) iii. Explain how the expression found at point ii. would simplify if and and the short rate process (rt)t0 were uncorrelated