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 XUWGW Z UWG; (1)

The functional forms ofXUWandZ UWdetermine 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 price9UW7and the bond with maturity7 has price9UW7. 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

#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

#9

#U

xw

#9

#W

Z

#

9

#U

xx

GW

Uh GW U

w

9 b

w

#9

#U

#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

DUW

for some functionDUW. We shall find it...