L7. Interest Rate Modeling

Motivation: we are doing interest rating modeling to price fixed income securities whose cash flows depend on future movements of interest rate.

The rate to model: most interest rate models are based on the (hypothetical) instantaneous spot rate — short rate. It can be thought of as a very short-term spot rate (e.g. overnight), though formally, it is for an instantaneous maturity. Some models work with forward rates, such as Heath, Jarrow and Morton.

Stochastic Process

Wiener Process:

, where is a random draw from a standard normal distribution . s are iid, and thus we have the following properties:

where and denote the current and future time, respectively.

A Wiener process has a drift rate of 0 and a variance rate of 1.

In general, if a random variable follows a generalized Wiener process, stochastic process, or Brownian motion:

where the drift (the slope, the deterministic part) and the volatility (random part, diffusion part) can be constants. We have the following properties:

Purpose of Introducing Stochastic Process:

We intend to build a realistic model to describe the underlying source of risk, e.g. interest rate fluctuation (default risk)

The model provides us with the probability distribution of outcomes at an important future time, e.g. expiration date of an option

This probability distribution will allow us to compute the expected value of a security/derivative at the future time and subsequently its present value (price) via discounting

The short rate process

To illustrate the short rate process, we will simulate a discrete-time path of the short rate:

Given , which can be observed at , can be computed as , so

There are also some other kinds of short rate processes

Long-term trend and negative interest rates:

Mean reversion (the Ornstein-Uhlenbeck process):

Mean reversion and non-negative interest rate:

Geometric Brownian motion with time-varying drift and volatility:

Desirable properties of an interest rate model

Economic soundness, e.g. non-negativity of interest rate, mean reversion;

Reasonable distribution of spot rates at a given maturity;

Correlation structure between interest rates of different maturities;

Volatility structure of interest rates, e.g. rate level-dependent volatility;

Tractability (analytical solution versus numerical solution)

The short rate process expressed in a bivariate lattice

Backward Induction: Solving the tree model starting from a bond’s maturity backward for its price

We can use the short rate tree to compute

discount bond prices

longer-term spot rates

forward rates

pricing interest rate derivatives or securities with interest rate-dependent cash flows

The problem then is how to construct the short-rate tree.

The Black-Derman-Toy model

We first need to specify a short rate process. The Black-Derman-Toy model assumes the short rate follows a Geometric Brownian motion:

The BDT model is a popular short rate model used in the pricing of bond options, swaptions, and other interest rate derivatives.

If the short rate follows a Geometric Brownian motion:

It naturally rules out negative interest rates. If the short rate follows a Brownian motion (as in the Vasicek and CIR models):

It is common to fix an up-move and down-move probability to 50% in the bivariate lattice.

Spot rate can be calculated using different start dates. The sequence’s volatility is the spot rate volatility.

There are only two unkown variables in each step of the tree. And thus we only need two equations to solve for the unkowns.

The short rate tree is consistent with the market prices of zero-coupon bonds and zero rate volatilities. The model can be further refined by making the short rate tree consistent with other market data, e.g. derivative prices. Once the tree is calibrated, the valuation of securities is simple, following the no-arbitrage rule. The problem with the bivariate lattice model is, however, recalibration.

Short Rate Tree Application

Consider a European call option written on a 3-year discount bond. The option expires in two years. We obtain the short rate and bond price trees

Suppose the strike price X is 0.9000. Then the option’s payoffs at time is

With the payoffs at the expirations, we compute the option values backward along its price tree

Thus the option to buy the 3-year discount bond with a face value of one million dollars at 90% of its par at time costs 9,160 dollars today.

As for American options which can be exercised before maturity, we should compare the exercise payoff and the continuation values to decide the value of each nodes.

A summary of Some One-Factor Models

(we only model the short rate — one factor)

Multifactor Models

Interest rate models with stochastic volatility process, e.g. Fong and Vasicek, Longstaff and Schwartz

Shortcomings: loss of closed-form solution in general

Arbitrage Models

e.g. The Heath-Jarrow-Morton model

More of a framework than a model

Based on the forward rate rather than the spot rate, it involves the whole term structure of forward rates for automatic calibration

Allows accurate term structure of interest rate volatility