Interest Rate Modelling One Factor

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One-factor Interest Rate Modeling
1 In this lecture...
q stochastic models for interest rates
q how to derive the bond pricing equation for many fixed-income products
q the structure of many popular interest rate models
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2 Introduction
In this lecture we see the ideas behind modeling interest rates us-ing a single source of randomness. This isone-factor interest rate modeling.
q The model will allow the short-term interest rate, the spot rate, to follow a random walk.
This model leads to a parabolic partial differential equation for the prices of bonds and other interest rate derivative products. The ‘spot rate’ that we will be modeling is a very loosely-defined quantity, meant to represent the yield on a bond of infinitesimal ma-turity. In practice one should take this rate to be the yield on a liquid finite-maturity bond, say one of one month. Bonds with one day to expiry do exist but their price is not necessarily a guide to other short-term rates.

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3 Stochastic interest rates
Since we cannot realistically forecast the future course of an interest rate, it is natural to model it as a random variable.
q We are going to model the behaviour ofU, the interest rate received by the shortest possible deposit.
From this we will see the development of a model for all other rates. The interest rate for the shortest possible deposit is commonly called thespot interest rate.
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Let us suppose that the interest rateU is governed by a stochastic differential equation of the form
GU X UW GW Z UW G;  (1)
The functional forms ofX UW andZ UW determine the behaviour of the spot rateU. For the present we will not specify any particular choices for these functions.
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4 The bond pricing equation for the
general model
When interest rates are stochastic a bond has a price of the form 9 UW7 .
q Pricing a bond presents new technical problems, and is in a sense harder than pricing an option sincethere is no underlying asset with which to hedge.
We are therefore not modeling atraded asset; the traded asset (the bond, say) is a derivative of our independent variableU.
q The only way to construct a hedged portfolio is by hedging one bond with a bond of a different maturity.
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We set up a portfolio containing two bonds with different maturities 7 and7. The bond with maturity 7 has price9 UW7 and the bond with maturity7 has price9 UW7 . We hold one of the former and a numberb d of the latter. We have

h 9 b d 9 (2)
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The change in this portfolio in a timeGWis given by
Gh
#9
#W
GW
#9
#U
GU


Z

#

9
#U

GWb d
w
#9
#W
GW
#9
#U
GU


Z

#

9
#U

GW
x

(3)
where we have applied Itô’s lemma to functions ofU andW. Which of these terms are random? Once you’ve identified them you’ll see that the choice
d
#9
#U
ƒ
#9
#U
eliminates all randomness inGh. This is because it makes the coef-ficient ofGUzero. 8
We then have
Gh
w
#9
#W



Z

#

9
#U

b
w
#9
#U
ƒ
#9
#U
xw
#9
#W



Z

#

9
#U

xx
GW
Uh GW U
w
9 b
w
#9
#U
ƒ
#9
#U
x
9
x
GW
where we have used arbitrage arguments to set the return on the port-folio equal to the risk-free rate. This risk-free rate is just the spot rate. Collecting all9
terms on the left-hand side and all9 terms on the
right-hand side we find that
#9
#W



Z
#

9
#U b U9
#9
#U
#9
#W



Z
#

9
#U b U9
#9
#U

9
At this point the distinction between the equity and interest-rate worlds starts to become apparent. This isoneequation intwo un-knowns. Fortunately, the left-hand side is a function of7 but not
7 and the right-hand side is a function of7
but not7. The only
way for this to be possible is for both sides to be independent of the maturity date. Dropping the subscript from9 ,we have
#9
#W



Z
#

9
#U b U9
#9
#U
D UW
for some functionD UW . We shall find it...
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