Qf Presentation

8/8/2019 Qf Presentation

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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

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A rgentina/Pesos

-2

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A rgentina/Dol lars

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60

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B razil

-6

-4

-2

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Chile

-10

-5

0

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Hong Kong

-20

-10

0

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30

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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

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