Please Solve these ones for me.

1 Explain the similarities and differences in the following three interest rate models: the Hull-White model the Cox-Ingersoll-Ross model the Vasicek model 2 Explain the following formulae as they are used in interest rate modelling: (a) Pit,T) = EQ exp[ -[ r(uldu ) rit> (b) A(t) P(t,T) = - Ep[A()IF] 3 The stochastic differential equations defining the short-rate process assumed in three commonly used models for the term structure of interest rates are shown below: Model 1: dr(t) = alu-r(t)]dt + odw(t) Model 2: dr(t)= alu-r(t)]dt + avr(t)ow(t) Model 3: dr(t) = alu(t)-r(t)ldt + adw(t) In each case, Wit) denotes a standard Brownian motion under the risk-neutral probability measure. (i) Identify these three models. (W) Outline the key statistical properties of the short-rate processes for each of these models. The dynamics of a fourth model are defined by: Model 4: dr(t) = 0dt + odw(t) where o and of are constants. (iii) State the limitations of this model.4 Company X has the following financial structure at time 0: Debt E3m (current book value) Equity E6m (issued share capital) The debt is a zero-coupon bond with face value 65m that is repayable at par at time 10. There are 400,000 shares in circulation. (i) Explain how the Merton model could be used to value shares in Company X. (ii) Assuming that the debt is repaid directly from the company's funds at that time, state the share price at time 10 if the total value of Company X at that time is: (a) E15m (b) E4m 5 A two-state model is to be used to model the probability that a bond defaults: A(t) No default, N Default, D where A(t) =2 5+20t -+2 Osts20. 500 (i) Calculate the probability that the bond does not default between times 5 and 10. (ii) Explain how the model may be modified to allow the default intensity _ (t) to depend on future unforeseen events such as a sudden downturn in the economy. 6 A company has just issued 4-year zero-coupon bonds with a nominal value of f4 million. The total tyle value of the company now stands at $7.5 million. A constant risk-free rate of return of 2% pa continuously-compounded is available in the market. (1) Use the Merton model to calculate the theoretical price of f100 nominal of the company's bonds, assuming that the annual volatility of the value of the company's assets is 30%. [4] (ii) Estimate the risk-neutral probability of default on the company's bonds. [3] [Total 7]