Answer to Question #190284 in Financial Math for Niroshan

Question

Question 6 [26 marks]. Consider the extension of the Vasicek model for a variable interest rate rt , t > 0, which is described by the following stochastic differential equation: drt = −a(rt − µ)dt + σ(t)dWt , where a > 0, µ > 0 are constants and σ(t), t > 0, is a strictly positive function of t.

(a) Compute the differential of the function U(t) defined by U(t) = e at(rt − µ). [6]

(b) Solve the equation for rt with the initial value r(0) = r0 [7]

(c) State the distribution of rt and provide formulae for E(rt) and Var(rt). [6]

(d) Suppose that a = 2, µ = 0.05, and σ(t) = 0.1e t 4 . In other words, the interest rate is governed by the following stochastic differential equation: drt = −2(rt − 0.05)dt + 0.1e t 4 dWt .

Given that r0 = 0.05, what is the probability that after one year the interest rate will be less than 0.03? 

Answer 1

(a)

U(t) = e at(rt − µ)

µ=0

r=0

t=0

U(t)=0

(b)

drt = −a(rt − µ)dt + σ(t)dWt

r=0

(c)

E(rt)

U(t) = e at(rt − µ)

Var(rt)

drt = −a(rt − µ)dt + σ(t)dWt

(d)

drt = −2(rt − 0.05)dt + 0.1e t 4 dWt

drt = −2(0.01 − 0.05)dt + 0.1e t 4 dWt

drt = −0.12dt + 0.1e t 4 dWt