An Elementary Introduction to Stochastic Interest Rate Modeling

The following sections are included:

• Preface

• Contents

We include a review of Brownian motion and stochastic integrals since they are a key tool to the modeling of interest rate processes. For simplicity, our presentation of the stochastic integral is restricted to square-integrable processes and we refer the reader to more advanced texts such as e.g. [Protter (2005)] for a comprehensive introduction.

The Black-Scholes formula can be considered as a building block for the pricing of financial derivatives, and its importance is not restricted to the pricing of options on stocks. Indeed, the complexity of the interest rate models makes it in general difficult to obtain closed form expressions, and in many situations one has to rely on the Black-Scholes framework in order to find pricing formulas for interest rate derivatives, in particular in the BGM model, cf. Chapter 10.

This chapter is a short introduction to some common short term interest rate models. Here we do not aim at completeness as the study of these models has already been extensively developed in the literature, see e.g. [Brigo and Mercurio (2006)], [Carmona and Tehranchi (2006)], [James and Webber (2001)], [Kijima (2003)], [Rebonato (1996)], [Yolcu (2005)]. In the next chapters we will mainly use the Vasicek mean-reverting model in our examples as it allows for explicit calculations.

In this chapter we describe the basics of bond pricing in the absence of arbitrage opportunities. Explicit calculations are carried out for the Vasicek model, using both the probabilistic and PDE approaches. The definition of zero-coupon bounds will be used in Chapter 5 in order to construct the forward rate processes.

In this chapter we define the forward and instantaneous forward rates from absence of arbitrage arguments, and using the construction of zero-coupon presented in Chapter 4. We also consider the problem of parametrization of forward rates.

In this chapter we present the general framework of Heath, Jarrow and Morton [Heath et al. (1992)] in which the evolution of forward rates is infinite dimensional and viewed as a stochastic process valued in a function space. We also expose the HJM absence of arbitrage condition. Under this condition we recover some time dependent short term interest rate models such as the Hull-White model.

In this chapter we introduce the notion of forward measure for the pricing of interest rate derivatives. We use the Girsanov theorem to obtain the dynamics of the short rate processes under forward measures, with explicit calculations in the case of the Vasicek model.

The short rate models considered in the previous chapters are one-factor models in the sense that their stochastic evolution is driven by a single Brownian motion. Such models pose several restriction to the fitting of forward curves, and they induce undesirable correlations in the price of zero-coupon bonds with different maturities. Our goal in this chapter is to study two-factor models which use two sources of randomness and allow for a wider choice of parameters for the fitting of forward curves.

The bond models studied in the previous chapters rely on the hypothesis that the bond payoff at maturity is always equal to $1, thus excluding the possibility of default. In this chapter we study a bond model in which default can occur at a random time τ. We also consider the associated options (credit default swaps) that are designed as a protection against default.

In this chapter we consider the pricing of caps and swaptions using forward measures on the London Interbank Offered Rates (LIBOR), in which forward rates are defined using different compounding conventions. We also introduce the swap rates to be used in the next Chapter 11 for the Brace-Gatarek-Musiela (BGM) model.

This chapter is devoted to the BGM model, a nonlinear model for LIBOR rates which, unlike the HJM model, ensures the positivity of interest rates, which can be built according to a drifted geometric Brownian motion. After constructing the BGM model we give a quick outlook on its calibration following the approach of [Schoenmakers (2005)].

This appendix surveys some basic results in probability and measure theory used in the lectures. It does not aim to completeness and the reader is referred to standard texts in probability such as [Jacod and Protter (2000)], [Protter (2005)] for more details.

In this appendix we list some recent issues investigated in the literature.

Exercise 1.1. We need to check the five properties of the Brownian motion:

(i) starts at 0 at time 0,

(ii) independence of increments,

(iii) almost sure continuity of trajectories,

(iv) stationarity of the increments,

(v) Gaussianity of the increments…

The following sections are included:

• Bibliography

• Index

• Author Index