FI-3 Lecture 3

Term Structure Models

Multiple Interest Rate Factors

Key Rates & Perturbations

We assume that

where are the 2-year, 5-year, 10-year and 30-year par coupon rates.

For some security with deterministic cash flows, the first-order approximation can be expressed as

The quantities

can be thought of as the DV01s corresponding to the i-th key rate.

The quantities

can be thought of as the durations corresponding to the i-th key rate.

The second-order approximation takes the form

with multiple factors, duration is described by a vector and convexity is described by a matrix.

Key Rate Perturbations

Let’s assume that a change of one basis point in each individual key rate leads to a change in the par-coupon yield curve of basis points where the function are as shown in the graphs below (Note that all the impacts in the graphs are our assumptions.)

Par Coupon yields and Discount Factors

Note that we can derive the discount factors from the par-coupon yields.

Recall that the par coupon yield corresponding to maturity is given by

starting with a par-coupon yield curve , the discount factors can be computed recursively by

Construct a Hedge using Key Rates

Generalization of the Method

Suppose we have sold a security whose price is given by

and we wish to hedge the sale of this security using a portfolio of basic securities. (Typically we will want securities in the hedging portfolio). If the price of the hedging portfolio is given by

then we choose the amounts of each of the securities in the hedging portfolio so that

This is analogous to matching all of the key rate DV01s. (note that for corresponding to the original yield curve, i.e., the perturbation is zero).

Risk-Neutral Measure for Binomial Framework

Suppose that there are two trading times . Assume that there is a zero-coupon bond that will pay $1 at time 1 (i.e. the bond has face value and maturity .) Denote the price of this bond at time 0 by . Assume that there is another basic (traded) security that has initial price and price at time 1 taking one of two possible different values, each with prob > 0.

Besides, assume .

Consider a strategy described by , which means that the total initial capital is and that the portfolio will hold shares of the risky asset.

Denote the capital of the strategy at time 1 by , which is a random variable. The initial capital invested in the risky asset is and the initial capital invested in the bond is . The terminal capital is given by

for all .

An arbitrage strategy means

That is, a strategy is an arbitrage iff

Absence of Arbitrage

If we put

then

Replication and Pricing

Consider a derivative security that pays the amount at time 1. Use it the replicate the payoff of the security by a portfolio with initial capital and shares of the risky asset. Thus satisfies

we can solve for

thus

where

and .

We can define another probability measure on by

which is called the risk-neutral measure or pricing measure.

Since

thus the arbitrage-free time-0 price of is given by

Remark:

It is customary to define

thus the no aribitrage condition is

and the risk-neutral probabilities are given by

For simplicity, questions in this course all assume that . But in more complex situations, we may calculate different .

N-Period Binomial Models

In the N-period binomial model, there will be N+1 trading dates 0, 1, 2, …, N. Denote for the price at time n of a ZCB that pays $1 at time . Denote for the spot rate that will prevail at time for loans to be settled at time (one-period compounding).

Note that the interest rates are known at time , but not earlier.

Short Rate Models v.s. Whole-Yield Models

Short rate models, also called equilibrium models, have no arbitrage (although they may give rise to negative interest rates). The spot rate curve is an output of the mode, rather than an input. Typically there are parameters in the model that are chosen to try to match the spot-rate curve as closely as possible.

Whole-Yield models, also called arbitrage-free models, take the current spot-rate as an input, and model the random evolution of the spot-rate curve. A key idea is to use forward rates. Prices are computed using a risk-neutrail measure.

Binomial Short Rate Models

Notations:

Dates:

Sample Sapce:

Interest Rate Process:

adapted process (values of the process at time n depend only on the information available at time n).

Pricing Measure: (risk-neutral measure)

Discount Process:

predictable (values of the process at time n are determined by the information available at time n-1)

Risk-Neutral Pricing Formula

For a security that makes a single payment of amount

at a given date , the prices of the security at the dates are given by

where

In particular, the time-0 price is given by

Remark:

Note , because is known at time 0, but as for , are not known at time 0.

the discount factor for time is given by

Given an interest rate process and a pricing measure, we can compute the zero-coupon yield curve (spot-rate curve).

Four Binomial Models

In all 4 of these models, we take the risk-neutral measure to be a binomial product measure (be H or T with prob of p & q), with probability of heads equal to 0.5. We put

Ho-Lee Model

Remark:

are called drift parameters and is called the volatility parameter.

The Ho-Lee Model can be written in the form

where

Ho-Lee with Constant Drift:

Vasicek Model

Binomial Verision of Vasicek Model:

The solution of this difference equation is given by

where is a random walk.

Remarks:

This model exhibits mean reversion

is called the long-term mean, and is called the speed of mean reversion

Hull-White Model

Binomial Version of Hull-White Model:

Black-Derman-Toy Model

which can be rewritten as

Pricing

Zero Coupon Bonds

For each with , let denote the price at time of a zero coupon bond with maturity and face value $1. Then

where is the discount process. In particular,

Coupon Bonds

Given and , let denote the price at time of a coupon bond with maturity , face value $1 and one-period coupon rate . If , we make the convention that is the price after the coupon payment is received at time . Then

provided that .

General Security with Adapted Cash Flows

Let be an adapted process. The price at time 0 of a security that pays the amount at each of the times is given by

One-Period Forward Rates

For , denote as the interest rate agreed upon at time for a loan initiated at time and settled at time . If $1 is the amount borrowed at time then the amount to be repaid at time is . Then

note .

Backward Induction

Sinlge Payment

Consider a security that makes a single payment of amount at time . Start at time and work backward by using

for

Multiple Payments

Consider a security that makes payments at each of the times . We can determine the price at time 0 by viewing this as a string of securties each making one payment and pasting together the results.

Let be the price of the security at time after the payment has been made. The backward induction is

for .

Notice that with multiple payments, the discounted price process will not be a martingale under .

Number of Heads as a State Variable

Simplification is possible when the short spot rate at time depends only on only through the number of heads up to time , i.e.

and the pricing measure is a binomial product measure ().

For this scenario, for a security that makes a single payment at time and the payment can be expressed as function of

The functions can be computed by backward induction:

where are the risk-neutral probabilities of heads and tails, is the number of heads up to time .

For a security that pays at each of the times where each can be expressed as a function of either or , i.e.

Then the function can be computed by backward induction: