Post on 29-May-2018
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One Factor and Two FactorInterest Rate modeling
Presented By :-
Pravesh SuranaRavi Somani
Raounak J
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Interest rate model or process
It is important to recognize that the interest rate model or process that underliesfixed income securities is quite different from equity or FX process such as
lognormal diffusion
Interest rates display mean reversion and may have volatility dependent on the
interest level
Bill or bond prices that depend on the underlying interest rates also converge to
(or pull to) par at maturity
An Interest Rate Tree or Interest Rate Lattice is a Numerical Representation of theInterest Rate Model. It facilitates the numerical computations of derivative prices
based on the interest rates. One common representation is the binomial tree.
The lattice is a time-discrete process. It may be a discrete model in its own right,
or it may serve as a discrete approximation to a continuous time model.
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-6
-4
-2
0
2
4
6
2/08/94 1/09/96 12/09/97
A rgentina/Pesos
-2
-1
0
1
2
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A rgentina/Dol lars
-60
-40
-20
0
20
40
60
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B razil
-6
-4
-2
0
2
4
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Chile
-10
-5
0
5
10
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Hong Kong
-20
-10
0
10
20
30
40
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Mexico
Changes in interest rates: Argentina, Brazil, Chile, Mexico and HK
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Terminologies It is a model where the stochastic changes depend on one
random factor or noise in the economy. this implies all theinterest rates of different maturities will be perfectlycorrelated.
A one-factor model is conveniently represented by a binomialtree.
ONE-FACTOR INTEREST
RATE MODEL
Examining the drift term, we see that for large r the (risk-neutral) interest rate will tend to decrease towards the mean,which may be a function of time.
When the rate is small it will move up on average.
Mean Reversion
A spot rate, for maturity T is the rate of interest earned on aninvestment that provides a payoff only at time T .Spot Rates
The forward rate is the future spot rate implied by todaysterm structure of interest rates.Forward Rates
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Mean Reversion(Hull: Figure 28.1, page 651)
Interestrate
HIGH interest rate has negative trend
LOW interest rate has positive trend
ReversionLevel
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Models
Vasicek Model
Cox, Ingersoll & Ross Model
Ho & Lee Model
Hull & White Model
ARCH Model
EWMA Model
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Vasicek
The Vasicek model is a particularly simple form:
dzdtidi tt WUO !
Controls Persistence
Controls Mean
Controls Variance
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Using the Vasicek Model
Choose parameter values
Choose a starting value
Generate a set of random numbers with mean 0 and variance 1
%6
262.
0!
!i
dzdtiditt
t=0t=0 t=1t=1 t=2t=2 t=3t=3 t=4t=4
i 6% 6.8% 6.84% 4.202% 5.5616%
.2(6.2(6--i)i) 00 --.16.16 --.168.168 .3596.3596
dzdz .4.4 .1.1 --1.11.1 .5.5
didi .8.8 .04.04 --2.3682.368 1.35961.3596 --.9.9
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Vasicek (sigma = 2, kappa = .17)
0
0.020.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 816 24 32 40 48 56 64 72 80 88 96
Path1
Path2
Path 3
Path 4
Path 5
Average
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Cox, Ingersoll, Ross (CIR) Model
The CIR framework allows for volatility that depends on the currentlevel of the interest rate (higher volatilities are associated withhigher rates)
dr = k(-r)dt + r dZ
is the long run equilibrium rate of interest towards which theshort rate reverts
k is a measure of speed at which the gap is reduced
This formulation is sometimes referred to as a mean reversionmodel
is a partial measure of interest rate volatility and is assumed to
be constant, the full measure of volatility rwill depend on thelevel of interest rates
The model does not allow interest rates to be negative
Does not allow interest rates to explode to levels without bounds
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Cox, Ingersoll, Ross (CIR) Model
This models of the term structure was solved inclosed form by CIR. The price of a zero couponbond is given by:
P(r,) = A()eB()rwhere
A() = [2e(k- ) /2/g()]2k/2
B() = -2(1-e-)/g()
g() = 2 + (k-)(1-e-) = k2 + 22
= T- t
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Alternative Term Structures in CIR
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ARCH MODEL
Few measures of volatility:
Actual: at particular time.
Historical/realized: for some period in the
past.
Implied: used in Black-Scholes model for
option pricing
Forward: for some period in future.
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Assumptions Of ARCH Model
In developing an ARCH model, you will have to
provide three distinct assumptions
one for the conditional mean equation
one for the conditional variance
one for the conditional error distribution.
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Why ARCH Model is used??
The ARCH-M model is often used in financial
applications where the expected return on an
asset is related to the expected asset risk. The
estimated coefficient on the expected risk is a
measure of the risk-return tradeoff.
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ARCH Model
Moving-window volatility; observe the plateauing.
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Auto-Regressive Conditional Heteroskedasticity
Engle (1982)
This model cleverly estimates the unobservable (latent) variance.
The model is not efficient when q is large
ARCH
Generalized ARCH model Bollerslev (1986)
Glosten, L.R., R. Jagannathan and D. Runkle (1993), "Relationshipbetween the Expected Value and the Volatility of the Nominal ExcessReturn on Stocks,"
GARCH Non-linear ARCH model
Higgins and Bera (1992) and Hentschel (1995) These models apply the Box-Cox transformation to the conditional
variance.
The variance depends on both the size and the sign of the variancewhich helps to capture leverage type (asymmetric) effects.
NARCH
Threshold ARCH Model
Rabemananjara, R. and J.M. Zakoian (1993)
Large events to have an effect but no effect from small eventsTARCH
Switching ARCH
Hamilton, J. D. and R. Susmel (1994)
In SWARCH models, the states refer to the states of volatility. For a 2-stateexample, we have high, or low volatility states.
SWARCH
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EWMA Model
In an exponentially weighted moving average
model, the weights assigned to the u2 decline
exponentially as we move back through time,
and the decline rate is
This leads to2
1
2
1
2
)1( PPW!W
nnn
P
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Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate
of the variance rate and the most recentobservation on the market variable
Tracks volatility changes
Risk Metrics usesP
= 0.94 for daily volatilityforecasting
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EWMA Model
Exponentially weighted volatility; no plateauing.
g
!
!
1i
1in21i2
n R)(1
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Ho and Lee
dr =U(t )dt + Wdz
Many analytic results for bond prices
and option prices Interest rates normally distributed
One volatility parameter, W
All forward rates have the samestandard deviation
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Initial ForwardCurve
Short
Rate
r
r
r
r
Time
Diagrammatic Representation of
Ho and Lee (Figure 28.3, page 655)
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Hull and White Model
dr = [U(t) ar]dt + Wdz
Many analytic results for bond prices and
option prices Two volatility parameters, a and W
Interest rates normally distributed
Standard deviation of a forward rate is adeclining function of its maturity
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Diagrammatic Representation of Hull and
White (Figure 28.4, page 656)
Short
Rate
r
r
r
r
Time
Forward Rate
Curve