Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

MATH 375

Stochastic modelling in insurance and finance

1. Let α．β．γ．λ．α0．be given positive constants, and a standard Brownian motion under the risk-neutral probability measure be the solution to the following equation:

Consider the following interest rate model:

i) Can the process (r(t)．t > 0) take a negative value for some t > 0? Justify your answer.                                                                                                      [6 marks]

ii) Derive the price p(t．T) of the zero-coupon bond at time t ∈ [0, T].  [9 marks]

iii) Let α = 1, β = ,3, γ = 1, λ = 1, α0  = 0．03.  Consider a forward contract on the zero-coupon bond of part (ii) with maturity T = 1, the delivery date of which is T1  = 0．5 and the delivery price is K = 0．5. Find the value of this forward contract at time ≠ = 0 for the holder with a short position.                  [10 marks]

2. State the three types of recovery rules for the intensity based credit risk models.  If the intensity is constant, i.e. γ(t) = λ > 0 for all t > 0, then what is the expected value and the variance of default time ≠ under the risk-neutral probability measure?     [8 marks]