Financial Risk Management

Term Structure Models

Financial Risk Management

Term Structure Models

Jan AnnaertGhent University

Hull, Chapter 23

23-2

What is the problem?What is the problem?

• The standard model implies little for the interest rate process and its time path

• It is therefore difficult to handle American interest rate options, callable bonds, …

• This chapter deals with these problems in an internally consistent framework

• Two groups:– equilibrium models– no-arbitrage models

23-3

Model illustrationModel illustration

TTr feE

TreE)T,0(P

)T,0(PlnT1

)T,0(R

TreElnT1

)T,0(R

Start with a process for the short term rate

23-4

• stochastic model for r

• Expected value using model• Discount at risk-free rate• Estimate model

Principle TSIR modelsPrinciple TSIR models

23-5

Parameter estimation

Compute prices

Compare tomarket prices

Parameter adjustment

Estimation TSIR modelEstimation TSIR model

23-6

• geometric Brownian motion

• binomial tree– build interest rate tree– build bond tree– build “derivative” tree

Rendleman & BartterRendleman & Bartter

rdrrdrdr

23-7

Mean ReversionMean Reversion

• Interest rate = stock price ???• Interest rates tend to a LT-equilibrium

– high r: tendency to interest rate decreases– low r: tendency to interest rate increases

• volatility LT rate < volatility ST rate• bond volatility is not proportional with

duration

23-8

Mean reversion: VasicekMean reversion: Vasicek

• Interest rate model:

• Intuition:

23-9

Vasicek: interpretationVasicek: interpretation

• b– LT-equilibrium

• a– speed with which disequilibria are

“corrected”

23-10

• formula:

• Analytical formula for European options on zero coupon bonds exist

Vasicek (II)Vasicek (II)

23-11

Vasicek: Coupon bondsVasicek: Coupon bonds

• Idea: option on a coupon bond is the sum of options on zero coupon bonds

• Define:

23-12

JamshidianJamshidian

• Exercise call:

n

1iiii Xs,T,rP,0maxc

23-13

• CIR-Process

• New: : the higher r, the higher its volatility

• Comparable formula available

Cox Ingersoll & Ross ModelCox Ingersoll & Ross Model

23-14

Two factor modelsTwo factor models

• Brennan & Schwartz– long rate and short rate

• Longstaff & Schwartz– short rate and volatility

23-15

No-Arbitrage modelsNo-Arbitrage models

• Problem in previous models is that often the prices of existing assets are not replicated, e.g. present term structure

• NA-models: start from the present term structure

• Here: only one factor models

23-16

Principle NA ModelsPrinciple NA Models

• Assume a process for bond returns

• Derive the process for forward rates

• Derive the process for interest rates

23-17

Bond return processBond return process

23-18

f t T TP t T P t T

T T

d P t T r tv t T

dt v t T dz t

df t T Tv t T v t T

T Tdt

v t T v t T

T Tdz t

( , , )ln ( , ) ln ( , )

ln ( , ) ( )( , )

( , ) ( )

( , , )( , ) ( , )

( , ) ( , )( )

1 21 2

2 12

1 222

12

2 1

1 2

2 1

2

2

Forward rate processForward rate process

23-19

T T T T T T

dF t Tv t T

Tdt v t T dz t

dF t T v t T v t T dt v t T dz t

T

T T

1 22

0

1

2

; ; lim

( , )( , )

( , ) ( )

( , ) ( , ) ( , ) ( , ) ( )

Instantaneous forward rate processInstantaneous forward rate process

23-20

r t F t t F t dF t

r t F t v t v t d v t dz t

dr t F t dt v t v t v t d dt

v t dz dt v t dz t

t

tt

tt

t tt tt

tt t t

( ) ( , ) ( , ) ( , )

( ) ( , ) ( , ) ( , ) ( , ) ( )

( ) ( , ) ( , ) ( , ) ( , )

( , ) ( ) ( , ) ( )

0

0

0

0

0 0

20

RN short term interest rate processRN short term interest rate process

23-21

Heath Jarrow & MortonHeath Jarrow & Morton

• Specify volatility for the instantaneous forward rates at each moment

• The implied binomial tree may grow very large (exponential growth)

• Non-Markovian

23-22

• Process:

• Markov-model• Analytical expressions for bonds and

European options are available

Ho and Lee ModelHo and Lee Model

23-23

Ho & Lee model (II)Ho & Lee model (II)

Disadvantages:• all spot and forward rates share the

same volatility

• no mean reversion

23-24

Hull & White modelHull & White model

• Extension of Vasicek’s model, but is able to replicate the initial TSIR

• Also the Ho & Lee model is a special case

• Process:

23-25

Hull & White model (II)Hull & White model (II)

• Analytical formula available

• A wide(r) range of volatility structures are available

• Equivalent trinomial tree is availableProblem: (t) has to be determined simultaneously

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