Constructing Binomial Models For The Short Rate

Constructing Binomial Models for the Short Rate

©Finbarr Murphy 2007

 P(i) = price at time t=0 of a pure discount bond maturing at iΔt

Qi ,i = 12 Qi −1,i −1d i −1,i −1 Qi , − i = 12 Qi −1, − i +1d i −1, − i +1 Qi , j = 12 Qi −1, j −1d i −1, j −1 + 12 Qi −1, j +1d i −1, j +1

MSc

COMPUTATIONAL FINANCE

 the BDT90 process becomes (const volatility)

d ln r (t ) = θ ( t ) dt + σdz σj ∆t ri , j = U ( i ) e

1 di, j = 1 + ri , j ∆t P( i + 1) = ∑ Q d i, j

j

i, j

1

Constructing Binomial Models for the Short Rate

©Finbarr Murphy 2007

P(3) = 0.8638

P(2) = 0.9070

P(1) = 0.9524 U(1) = 0.498

Q0,0 = 1 r0,0 = 0.05 d0,0 = 0.9524 P(0) = 1

Q1,1 = 0.4762 r1,1 = 0.055 d1,1 = 0.9479

Q1,1 = 0.4762 r1,1 = 0.045 d1,1 = 0.9569

MSc

COMPUTATIONAL FINANCE

 Now We can can repeat say that now values. calculate to Calculate Q1,1for andsubsequent Q1,-1  Q Q = 0.055 1 r1,11,10,0 == and r = 0.045  1/2Q 0,0d0,0 =1,-10.4762

 Remember that the outer states are treated a little  Q rd0,01,1 ===0.05 0.9479 d1,-1 = 0.569  1/2Q0,0dand 1,-1 0,0 = 0.4762

differently from the inner states.  d0,0 = 1/(1+0.05∆t) = 0.9524 1  Solve P( i + 1) = P(2) = = 0.9070 = 2 (1 + 0.05) 1 1 = ∑ Q1, j 1 + U (i ) exp σj ∆t ∆t j = −1

(

)

1

∑Q j = −1

d

1, j 1, j

2