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MODELLING INTEREST RATES

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Numerical Methods in Finance (Implementing Market Models)

©Finbarr Murphy 2007

Lecture Objectives  Modelling Interest Rates

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 Fitting the BDT Curve

©Finbarr Murphy 2007

Agenda Page

1 Constructing Binomial Trees for the Short 2

Rate Term Structure Consistent Models

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©Finbarr Murphy 2007

Constructing Binomial Models for the Short Rate  Most of the interest rate models discussed have

limited or no analytical solutions for derivatives priced on the SDE’s  Numerical solutions must be used, particularly for

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those derivatives with early exercise opportunities and non-standard terminal payoff profiles  We will look at binomial trees and how these can

be constructed to model the short rate and…  Price interest rate derivatives

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©Finbarr Murphy 2007

Constructing Binomial Models for the Short Rate  For pricing interest rates, the trees are

constructed in a similar manner to those of the equity variety except that the interest rate changes..

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 Obviously!

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©Finbarr Murphy 2007

Constructing Binomial Models for the Short Rate  We break the yield curve into i = 1,…,N segments

of length Δt  We use the following familiar conventions  P(i) = price at time t=0 of a pure discount bond maturing

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

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at iΔt  σR(i) = volatility at time t=0 of R(i)

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©Finbarr Murphy 2007

Constructing Binomial Models for the Short Rate  Although we could build a binomial tree for any of

the short rate processes covered, we will follow C&S and model the BDT90 process. I.e.

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  σ ' (t ) d ln r (t ) = θ ( t ) + ln r ( t )  dt + σ ( t ) dz σ (t )  

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©Finbarr Murphy 2007

Constructing Binomial Models for the Short Rate  The initial short rate, r, is the yield on the bond

that matures at the end of Δt  In the 2-step model here, we need

rUU

to pick rU and rD so that the bond at 2Δt has a value of 1

rU rUD

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r

 In considering how we can fit the

grid to the initial curve, we must also consider the initial volatility curve

rD rDD

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©Finbarr Murphy 2007

Constructing Binomial Models for the Short Rate  Jamshidian (1991) showed that the level of the

short rate at time t in the BDT model is given by:

r(t ) = U (t )e

σ ( t ) z(t )

E.q. 9.2.1

 Where  U(t) is the median of the lognormal distribution for r at

time t

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 σ(t) is the level of the short rate volatility  z(t) is the level of the Brownian motion

 If we plan to fit both the yield curve and the

volatility curve, we must determine U(t) and σ(t) at each time step 9

©Finbarr Murphy 2007

Constructing Binomial Models for the Short Rate  Looking again at the binomial process

j=N

rUU

j=2 rU

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j=1 j=0

rUD

r

j=-1

rD

j=-2

rDD

j=-N i=0

i=1

i=2

i=N-1

i=N

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©Finbarr Murphy 2007

Constructing Binomial Models for the Short Rate  At i = N, j√Δt is distributed with a mean of 0 and a

variance N  Therefore, as Δt→0, the binomial process j√Δt

converges to the Weiner process z(t)

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 Using this result with Eq. 9.2.1, we can say

ri , j = U ( i ) e

σ ( t ) j ∆t

E.q. 9.2.2

 In order to determine U(i) and σ(i) we again use

state prices 11

©Finbarr Murphy 2007

Constructing Binomial Models for the Short Rate  Recall that Qi,j is the value at time = t of a security

that pays 1 if node i,j is reached or zero otherwise  These state prices can be thought of as

discounted probabilities, given a particular state

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 Jamashidian used a process of forward induction

to accumulate state prices as one moves forward through the tree

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©Finbarr Murphy 2007

Constructing Binomial Models for the Short Rate  Clearly Q0,0 = 1  Let di,j represent the discount factor at node i,j

di , j

1 = 1 + ri , j ∆t

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 For simple compounding

Q2,2 Q1,1

Q0,0

Q2,0 Q1,-1

 Q2,0 = ½Q1,-1d1,-1+ ½Q1,1d1,1

Q2,-2

 In general

Qi , j = 12 Qi −1, j −1d i −1, j −1 + 12 Qi −1, j +1d i −1, j +1 13

©Finbarr Murphy 2007

Constructing Binomial Models for the Short Rate  This doesn’t work for the outer nodes so we have

Qi ,i = 12 Qi −1,i −1d i −1,i −1

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

 We will implement a model to fit the yield curve

and the volatility curve but first  We assume that the volatility curve is constant

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©Finbarr Murphy 2007

Constructing Binomial Models for the Short Rate  This means the BDT90 process becomes

d ln r (t ) = θ ( t ) dt + σdz

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 And Eq 9.2.2 becomes

ri , j = U ( i ) e

σj ∆t

E.q. 9.2.3

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©Finbarr Murphy 2007

Constructing Binomial Models for the Short Rate  The price of a pure discount bond maturing at Δt

is given by

Q1,1 d1,1

P(0) = Q1,1d1,1+Q1,-1d1,-1

 And in a general sense

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P( i + 1) = ∑ Qi , j d i , j

Q0,0 d0,0

Δt

Q1,-1 d1,-1

j

 So

1 P( i + 1) = ∑ Qi , j 1 + ri , j ∆t j 16

©Finbarr Murphy 2007

Constructing Binomial Models for the Short Rate  And

P( i + 1) = ∑ Qi , j j

[

1

1 + U (i)e

σj ∆t

]∆t

 If P(i+1) is known, all other factors are known

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except U(i)  We can estimate U(i) using a numerical search

method such as Newton Raphson

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©Finbarr Murphy 2007

Constructing Binomial Models for the Short Rate  The code is listed in BDT90CurveFit.m  I used a home-grown search method as I found

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that the nature of the Σ was difficult to put in a matlab function such as

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©Finbarr Murphy 2007

Constructing Binomial Models for the Short Rate  Once we have U(i) we can estimate the short rate

ri,j and the discount factors di,j by

( σj ri , j = U ( i ) e

)

1 = 1 + ri , j ∆t

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di, j

∆t

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©Finbarr Murphy 2007

Constructing Binomial Models for the Short Rate  I have generated the same results as those in

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Figure 8.4, page 239 in C&S

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©Finbarr Murphy 2007

Constructing Binomial Models for the Short Rate  The next step is to create extend the model to fit

the existing yield curve and volatility curve  Now, the level of the short rate at i,j is given by

ri,j

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( σ (i) j ri , j = U ( i ) e Q0,0 d0,0

∆t

)

Pu(i)

Pd(i)

Δt 21

©Finbarr Murphy 2007

Constructing Binomial Models for the Short Rate  Pu(i) and Pd(i) are up and down discount functions  These must be consistent with the yield curve, so

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1 [ 12 PU ( i ) + 12 PD ( i ) ] P( i ) = 1 + r0 , 0 ∆t  Initial volatilities of the short rate are given by

σR( i )

1 ln PU ( i ) ∆t = ln 2 ln PD ( i )

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©Finbarr Murphy 2007

Constructing Binomial Models for the Short Rate  By working with the existing forward induction

techniques and the two equations from the previous slide

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 We can work to fit the volatility curve

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©Finbarr Murphy 2007

Recommended Texts  Required/Recommended  Clewlow, L. and Strickland, C. (1996) Implementing derivative

models, 1st ed., John Wiley and Sons Ltd. — Chapter 8

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 Additional/Useful

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