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MODELLING INTEREST RATES

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Numerical Methods in Finance (Implementing Market Models)

©Finbarr Murphy 2007

Lecture Objectives  Two Factor Models  Fong and Vasicek (1992)  Longstaff and Schwartz (1992)

 Term Structure Consistent Models  Ho & Lee 1986  Hull & White 1993

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 Black and and Karasinski (1991)  Heath, Jarrow and Morton (HJM) (1992)

©Finbarr Murphy 2007

Agenda Page

Two Factor Models

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2 2

Term Structure Consistent Models

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Two Factor Models  To recap at this time:  There are two areas of interest rate modelling  Traditional Term structure modelling and  Term Structure Consistent Models

 So far we have looked at two traditional models:

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 Vasicek (1977) and  CIR (1985)

 These 1-factor models are easily programmable

but can only generate curves that are monotonically increasing, monotonically decreasing or slightly humped 4

©Finbarr Murphy 2007

Two Factor Models  More realistic term structures involve those with

two or more sources of uncertainty  Fong and Vasicek (1992) proposed a 2-factor

model of the term structure of interest rates  The two sources of uncertainty are

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 The short rate r and  The variance of the short rate v

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Two Factor Models  These two process are driven by the PDE’s

dr = [α ( r − r ) ] dt + v dz1

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 And

dv = [ γ ( v − r ) ] dt + ξ v dz 2

dz1dz 2 = ρdt

 The solution for the discount bond price and yield

are programmable with a degree of complexity although some analytical shortcuts are available

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©Finbarr Murphy 2007

Two Factor Models  Aside from the relative difficulty of programming

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the VF92 model, the parameters of the PDE must also be estimated through some complex regression testing

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©Finbarr Murphy 2007

Two Factor Models  Longstaff and Schwartz (1992) also developed a 2-

factor model

dx = ( γ − δx ) dt + x dz1 dy = (η − θy ) dt + y dz 2

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 With the short rate defined by

r = αx + β y  And the volatility is given by

v =α x+β y 2

2

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©Finbarr Murphy 2007

Two Factor Models  The LS92 2-factor model is slightly more easily to

programme  Both VF92 and LS92 provide tractable solutions

for the term structure  The resultant curves exhibit many of the

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characteristics of a “real-world” curve  They provide a term-consistent paradigm for

pricing bonds and bond derivatives

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©Finbarr Murphy 2007

Two Factor Models  The problem with traditional term structure of

interest rate models is that they do not correctly price many currently traded bonds  I.e The curve is likely to fit with some traded

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bonds but not will all traded bonds

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©Finbarr Murphy 2007

Agenda Page

Two Factor Models

1

2 2

Term Structure Consistent Models

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3

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©Finbarr Murphy 2007

Term Structure Consistent Models  To recap at this time:  There are two areas of interest rate modelling  Traditional Term structure modelling and  Term Structure Consistent Models

 So far we have looked at two traditional models:

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 Vasicek (1977) and  CIR (1985)

 These 1-factor models are easily programmable

but can only generate curves that are monotonically increasing, monotonically decreasing or slightly humped 12

©Finbarr Murphy 2007

Term Structure Consistent Models  Ho & Lee 1986 were the first to develop a model

consistent with the initial yield curve  The short rate is given by the process

dr = θ (t )dt + σdz  Θ(t), the drift function, represents the slope of

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the initial forward rate curve  And the volatility of the short rate process

∂f (0, t ) 2 θ (t ) = +σ t ∂t

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©Finbarr Murphy 2007

Term Structure Consistent Models  The partial derivative denotes the slope of the

initial forward curve at maturity t  The drift function, Θ(t) allows us to use the

observed market bond prices to “fit” the model

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 we will look at this in more details later

 The HL86 model is a  Single factor  No mean reversion  Negative interest rates are possible

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Term Structure Consistent Models  The drift function Θ(t) does not depend on the

short rate  The volatility structure for spot and forward rates

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is determined by the constant σ

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©Finbarr Murphy 2007

Term Structure Consistent Models  The analytical results relate future bond prices to

the current term structure  The future bond prices are denoted by P(T,s)

where T >= t

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 On a time-line this looks like:

t=0

 Giving

t=t

t=T

P ( T , s ) = A( T , s ) e

t=s

− B (T , s ) r (T )

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©Finbarr Murphy 2007

Term Structure Consistent Models  Where

B( T , s ) = ( s − T )

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 and

P( t , s ) ∂ ln P( t , T ) ln A(T , s ) = ln − B( T , s ) P( t , T ) ∂T 1 2 2 − σ ( T − t ) B( T , s ) 2

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©Finbarr Murphy 2007

Term Structure Consistent Models  Time to look at a simple example  Let’s assume that the current term structure is a

flat 5% continuously compounded rate  Short rate volatility σ = 1%  We also assume that the short rate in one year,

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r(T) = r(1) = 5%  What is the price of a bond in one year with a 4

year maturity at that time? I.e. it currently has 5 years to maturity

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Term Structure Consistent Models P(0,5) = e-0.05*5 = 0.7788 P(0,1) = e-0.05*1 = 0.9512

 Therefore P(1,5) = A(1,5)e-B(1,5)r(1)

 The A(·, ·) term has a partial derivative that can

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be approximated using the slope. I.e.

∂ ln P( t , T ) ln P( t , T + ∆t ) − ln P( t , T − ∆t ) ≈ ∂T 2∆t 19

©Finbarr Murphy 2007

Term Structure Consistent Models  Letting Δt = 0.1 year, this gives

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P(1,5) = 0.8181

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©Finbarr Murphy 2007

Term Structure Consistent Models  Now we can calibrate the HL86 model to existing

option data  Supposing we have a series of options on discount

bonds  The price of these options is denoted by

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marketi

where i=1,…,M

 One way to calibrate our model is to minimise the

following with respect to σ

 modeli (σ ) − market i    ∑ market i i =1   M

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©Finbarr Murphy 2007

Term Structure Consistent Models  Where modeli(σ)is the price produced using σ  We will need to search through a range of σ’s  If M>1, we can only find σ that best fits the

solution

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 Options on discount bonds are uncommon but we

can use other interest rate derivatives such as cap and floors  These can be thought of as a series of discount

bond options

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©Finbarr Murphy 2007

Term Structure Consistent Models  Hull and White 1993  The SDE for this model is given by

dr = [θ ( t ) − αr ] dt + σdz

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 This model can be thought of as HL86 with mean

reversion

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Term Structure Consistent Models  Hull and White Extensions  One extension to HW93 model is to allow the

reversion rate to be time dependent

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dr = [θ ( t ) − α ( t ) r ] dt + σdz  This automatically introduces time dependent

volatility σ(t,s) which is the volatility of a yield with maturity s as seen at time t  This can be fitted to observed volatility term

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©Finbarr Murphy 2007

Term Structure Consistent Models  Other 1-factor models  Black, Derman and Toy (BDT) 1990

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  σ ' (t ) d ln r (t ) = θ ( t ) + ln r ( t )  dt + σ ( t ) dz σ (t )    Θ(t) and σ(t) are chosen to match current term

structures  But this must be done numerically  Not very tractable 25

©Finbarr Murphy 2007

Term Structure Consistent Models  Black and and Karasinski (1991)

d ln r = [θ ( t ) + α (t ) ln r ] dt + σ ( t ) dz

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 Numerically implemented using trinomial trees

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©Finbarr Murphy 2007

Term Structure Consistent Models  Hull and White (1994b)  2-factor model

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dr = [θ ( t ) + u − αr ] dt + σ 1dz1 du = −budt + σ 2 dz 2

 u is a mean reversion level with a random element  dz1 and dz2 are correlated through ρ

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©Finbarr Murphy 2007

Term Structure Consistent Models  The HW94b model has the advantage of being

partially analytical  The future discount bond price is given as

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P( T , s ) = A( T , s ) e

− r ( T ) B ( T , s ) −u ( T ) C ( T , s )

 The extra stochastic factor allows for a much

richer potential for yield curve shapes and possible volatility term structures

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©Finbarr Murphy 2007

Term Structure Consistent Models  Heath, Jarrow and Morton (HJM) (1992)  This is a multi-factor model  It basically states that the drifts of individual

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yields are a function of that yields volatility and of the correlation between the volatilities across the yields  The Heath-Jarrow-Morton (HJM) model is one of

the most widely used models for pricing interest rate derivatives

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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 7

 Additional/Useful  Hull, J. (2005) Options, futures and other derivatives, 6th ed.,

Prentice Hall

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— Chapters 28

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