stochastic

4. The short term interest rate rt in the state of Claonadh satises the stochastic differential equation: den; = 0:01. radii (77",;th where Wt is a Wiener process. Proposed values for the parameters are: (6) Parameter Value u 0.0/1I a 0.3 oz 0.1 Describe the behaviour of the process rt, commenting on the short and long term features of the paths. [6] A nancial derivative has a price '00", t) that is a function of the short rate rt and of the valuation date if. (b) (C) Applying Ito's formula, write down the Ito derivative of M7315), collecting sep- arately terms in alt and terms in (ii/Vt. [5] The market price of risk in Claonadh is an expected risk premium /\\ per unit standard deviation of a bond price. Write down a partial differential equa tion to be satised by the function '00", t) consistent with this risk premium, assuming that v is decreasing as a function of r. [5] Write down an expression for the risk-neutral drift of the short rate process '1}. [4] Assuming A = 0.2, compare the long run means to which r reverts under the real world law and the risk neutral law. Explain what the difference in the two longrun means implies for expected returns of bonds of different terms. [5] Total marks for this question [25]