financial risk management term structure models jan annaert ghent university hull, chapter 23
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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
23-9
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:
23-13
• 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
23-18
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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