0,15 = d - r Consider modelling the interest rate using the Cox-Ingersoll-Ross model. You are given that: -rebor 09 . dr =0.2(0.025-r) dt +0.3Jr dZ(t) , where Z(t) is a standard Brownian Motion. 158 153 . The sharpe ratio is 0.15. 40 . r(0) =0.05 17 a) Find the yield to maturity on an infinitely lived bond. [05] 79 241 490 P( 1, T, r, ) = A(1, T) e- BUT), 0 , 2 ( 6, 0 35 - 5 - 1 ) = 4 524 32 where 39 72ab A(t, T) = 2 (er(T-1) -1) ( a+ +y) (en(T-1) -1) + 2x/1 , B(1, T ) = ( a+ + y ) (exT -1) - 1)+ 2y y= Va+$)+20? I Lowe The sole, I can sub for eg By & so on With the CIR process, the yield on a long-term bond approaches the value AT =- 2ab as time to maturity goes to infinity. atday Alo ) do all The steps using Ito, pen ance 12 " ( + 1 = 2 0 ( t ) + at 2 5 The following unusual model has been proposed for the (real-world) stochastic behaviour of the short-term interest rate: dr, = ur,dt + o dZ(1), where u > 0 and o are fixed parameters and Z is a standard Brownian motion under the proposed real-world measure P. Under the same measure P, a (zero coupon) bond with maturity Thas price at time -o (Ter B(r,, 1, T) = exp(-(T-1); +0?(1-1)'/6). (a) Derive the SDE satisfied by B(r,,1, T). [05] - Bir,tit )= e (b) Determine the market price of risk and deduce the corresponding SDE for r, under the risk neutral measure Q. [15] CD so I must solve for " V/31 ret) first The sub into SPE You are using the Vasicek one-factor interest-rate model with the short-rate process" calibrated as dr, = 0.6[b-r,]di + odZ, B+ = ert+3 0 4 6 602 + 30 )4 2a [b-r J dt to dalt) For DE drzarat at where {Z(t); is a standard Brownian motion. 1 For t