7.4 Binomial Interest Rate Tree Framework

For bonds with options attached, changes in future interest rates impact the likelihood that the option will be exercised and, in so doing, impact the cash flows.
The interest rate "tree" framework allows interest rates to take on different potential values in the future based on some assumed level of volatility.
The interest rate tree performs two functions in the valuation process:
Generates the cash flows that are interest rate dependent
Supplies the interest rates used to determine the present value of the cash flows
Because the interest rate tree resembles a lattice, these models are often called "lattice models."
The binomial interest rate tree framework involves building a binomial lattice model, where the short interest rate can take on one of two possible values consistent with the volatility assumption and an interest rate model.
A valuation model involves generating an interest rate tree based on the following:

1. Benchmark interest rates 2. An assumed interest rate model 3. An assumed interest rate volatility

The two possible interest rates next period will be consistent with the following three conditions:

(1) An interest rate model that governs the random process of interest rates, (2) The assumed level of interest rate volatility, and (3) The current benchmark yield curve.

HOW TO OBTAIN VALUES FOR A ONE-YEAR INTEREST RATE To obtain the two possible values for the one-year interest rate one year from today, two assumptions are required.

1. Interest rate model, which we assume to be lognormal 2. Interest rate volatility, represented by a standard deviation measure in our modeling The lognormal random walk posits the following relationship:

where i1, L= the rate lower than the implied forward rate at Time 1; i1,H = the rate higher than the implied forward rate at Time 1; and sigma is the assumed volatility of the one-year rate.

The random possibilities each period are (nearly) centered on the forward rates calculated from the benchmark curve. The intuition of this relationship is deceptively quick and simple.
Think of the one-year forward implied interest rate from the yield curve as the average of possible values for the one-year rate at Time 1. The lower of the two rates, i1,Lis one standard deviation below the mean (one-year implied forward rate) and i1,H is one standard deviation above the mean. Thus, the higher and lower values (i1,Land i1,H) are multiples of each other and the multiplier is .

BINOMIAL INTEREST RATE TREE i0 Time 0 Time 1 Time 2 Time 3 i1,L i1,H i2,HH i2,LL i2,HL i3,LLL i3,HLL i3,HHL i3,HHH

i 1,L = lower one-year forward rate one year from now at Time 1 and i 1,H = higher one- year forward rate one year from now at Time 1.
i2,LL = one-year forward rate at Time 2 assuming the lower rate at Time 1 and the lower rate at Time 2; i2,HH = one-year forward rate at Time 2 assuming the higher rate at Time 1 and the higher rate at Time 2; and i2,HL = one-year forward rate at Time 2 assuming the higher rate at Time 1 and the lower rate at Time 2 or, equivalently, the lower rate at Time 1 and the higher rate at Time 2.
There are four possible values for the one-year forward rate at Time 3. These are represented as follows: i3,HHH, i3,HHL, i3,HLL, and i3,LLL.

ESTIMATING INTEREST RATE VOLATILITY Two methods are commonly used to estimate interest rate volatility.

In estimating historical interest rate volatility, volatility is calculated by using data from the recent past with the assumption that what has happened recently is indicative of the future.

In the implied volatility approach, the estimate interest rate volatility is based on observed market prices of interest rate derivatives (e.g., swaptions, caps, floors).

Volatility is measured relative to the current level of rates. It can be shown that for a lognormal distribution, the standard deviation of the one-year rate is equal to .

7.4.1 The Backward Induction Valuation Methodology and Computation of the Value of

a Fixed-Income Instrument Given Its Cash Flow at Each Node.

To find the value of the bond, the backward induction valuation methodology can be used.

At maturity bonds are valued at par. So, we start at maturity, fill in those values, and work back from right to left to find the bond's value at the desired node.

fH and fL are higher and lower forward rates, respectively.

Bond value at any node:

Bond value if lower interest rate is realized plus coupon payment Bond value if higher interest rate is realized plus coupon payment Forward rate L Forward rate H CALCULATING THE BOND VALUE AT ANY NODE The bond value at a node is equal to the following:

( )

where VH = the bond's value if the higher forward rate is realized one year hence; VL = the bond's value if the lower forward rate is realized one year hence; i= the one-year forward rate at a particular node; and C = the coupon payment that is not dependent on interest rates.

Because there are two possible interest rates one year from today, there are two present values to calculate. The two states of the world are whether chance selects the higher or lower one-year forward rate one year hence. Because it is assumed that either outcome is equally likely, the average of the two present values is computed.
This same procedure holds for any node with forward rates discounting cash flows moving from node to node (right to left).

Example. Using the interest rate tree below, find the correct price for a three-year, annual-pay bond with a coupon rate of 5%.

A three-year bond pays coupons and returns principal at the end of each year.
When we state an annual interest rate, that rate is effective as of the beginning of that year.

At Time 2, the bond value at each node is equal to the following:

V ( )

No matter what level interest rates move to at Time 3, the cash flow from a three-year bond at Time 3 will be the same: par plus a final coupon payment.

2.0% 5.0% 3.0% 4.0% 6.0% 8.0% Time 0 Time 1 Time 2 AtTime 1, the bond value at each node is equal to the following:

V ( )

AtTime 0, the bond value of the bond is V ( )

Time 1 values will be the average of Time 2 discounted plus the coupon payment.
Because no time has elapsed, there is no coupon payment at Time 0, making the Time 0 value the average of the Time 1 values discounted to today

7.4.2 Process of Calibrating a Binomial Interest Rate Tree to Match a Specific Term

Structure

The construction of a binomial interest rate tree requires multiple steps.
There are two potential changes in the forward rate at each node of the binominal tree: higher rate and lower rate.
One of the forward rates (typically lower) can be found iteratively or by solving simultaneous equations subject to using the following:
Known (shorter) spot and/or forward rates
Features of a coupon bond of given maturity
The relationship between a lower and higher rate and their volatility V = 103.0287 C = 5 V =103.2280 C = 5 V = 106.9506 C = 5 V = 102.2222 C = 5 V = 104.0566 C = 5 V = 105.9615 Time 0 Time 1 Time 2 Time 3 C = 5 V = 100 C = 5 V = 100 C = 5 V = 100 C = 5 V = 100 5.0% 5.0% 3.0% 3.0% 2.0% 8.0% 8.0% 6.0% 6.0% 4.0% 4.0% 2.0%
Analytical tools, such as Solver in Excel, can help with the calculations.
For example, assuming a one-year spot rate is 1%, two-year par bond yielding 1.2% and volatility model of with sigma= 0.15, we would need to solve the following simultaneous equations:
{ ( ( ) ( )

( ) ( ))

d CALIBRATING A BINOMIAL TREE To calibrate a binominal tree to match a specific term structure, the following steps should be applied:
For a known par value yield curve, appropriate spot and forward rates can be estimated.
Then, for an assumed interest rate volatility and model, the interest rate tree is built.
Finally, using the backward induction method, the values of the appropriate zero-coupon bonds at each node are calculated.
Once completed, the tree is calibrated to be arbitrage free.
The approximate interest rate in the interest rate tree can be found from forward-rate curve.
For example, assume the one-year forward rate one year from now is 4.04% and the model specification is We approximate the lower rate as d The real rates are 3.442% and 4.646%, respectively.

Lesson Wrap-Up

This lesson should leave you able to explain the binomial interest rate tree framework in a fixed-income context and connect it to the decisions made by issuers, investors, or analysts.

Review Prompts

Explain the backward induction valuation methodology and computation of the value of in your own words.
Explain process of calibrating a binomial interest rate tree to match a specific term in your own words.
State one exam-style risk, valuation, or market implication of the binomial interest rate tree framework.