Interest Rate Modelling One Factor
1
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
2
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.
3
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.
4
Let us suppose that the interest rateU is governed by a stochastic differential equation of the form
GU X U W GW Z U W G; (1)
The functional forms ofX U W andZ U W determine the behaviour of the spot rateU. For the present we will not specify any particular choices for these functions.
5
4 The bond pricing equation for the general model
When interest rates are stochastic a bond has a price of the form
9 U W 7 . q Pricing a bond presents new technical problems, and is in a sense harder than pricing an option sincethere is no underlying asset
with
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
2
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.
3
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.
4
Let us suppose that the interest rateU is governed by a stochastic differential equation of the form
GU X U W GW Z U W G; (1)
The functional forms ofX U W andZ U W determine the behaviour of the spot rateU. For the present we will not specify any particular choices for these functions.
5
4 The bond pricing equation for the general model
When interest rates are stochastic a bond has a price of the form
9 U W 7 . q Pricing a bond presents new technical problems, and is in a sense harder than pricing an option sincethere is no underlying asset
with