interest rate tree

THE ARBITRAGE-FREE VALUATION FRAMEWORKCHAPTER 8

© 2016 CFA Institute. All rights reserved.

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TABLE OF CONTENTS01 INTRODUCTION

02 THE MEANING OF ARBITRAGE-FREE VALUATION03 INTEREST RATE TREES AND ARBITRAGE-FREE VALUATION

04 MONTE CARLO METHOD05 SUMMARY

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1. INTRODUCTION

• The idea that market prices will adjust until there are no opportunities for arbitrage underpins the valuation of fixed-income securities, derivatives, and other financial assets.

• The purpose of this chapter is to develop a set of valuation tools for bonds that are consistent with this notion.

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2. THE MEANING OF ARBITRAGE-FREE VALUATION

• Arbitrage-free valuation refers to an approach to security valuation that determines security values that are consistent with the absence of arbitrage opportunities.

The traditional approach to valuing bonds is to discount all cash flows with the same discount rate as if the yield curve were flat.

However, the yield curve is rarely flat. Thus, each cash flow of the bond should be discounted at the appropriate spot rate. This leads to arbitrage-free value.• Think of the valuation of a bond as a portfolio of

zeros.

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ARBITRAGE OPPORTUNITIES

• Arbitrage opportunities arise as a result of violations of the law of one price.

There are two types of arbitrage

opportunities.

Value additivity, which means the value of the

whole equals the sum of the values of the parts

Dominance, where a financial asset with a risk-

free payoff in the future must have a positive price

today

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IMPLICATIONS OF ARBITRAGE-FREE VALUATION

• Using the arbitrage-free approach, any fixed-income security should be thought of as a package or portfolio of zero-coupon bonds.

• Dealers in US Treasuries can separate the bond’s individual cash flows and trade them as zero-coupon securities. This process is called “stripping.”

Stripping

• Dealers can recombine the appropriate individual zero-coupon securities and reproduce the underlying coupon Treasury. This process is called “reconstitution.”

Reconstitution

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3. INTEREST RATE TREES AND ARBITRAGE-FREE VALUATION

• For option-free bonds, the simplest approach to arbitrage-free valuation involves determining the arbitrage-free value as the sum of the present values of expected future values using the benchmark spot rates.

General formula:

where z1, z2, zN are the spot rates for period 1, 2, and N.

Benchmark securities are liquid, safe securities whose yields serve as building blocks for other interest rates in a particular country or currency.

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Example. Assume an option-free bond with four years to maturity and an annual coupon of 6.5%.

+ + + +

+

Year Spot Rate One-Year Forward Rate

1 3.5000% 3.500%2 4.215% 4.935%3 4.735% 5.784%4 5.271% 6.893%

OPTION-FREE VALUATION REFRESHER

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DIFFERENT APPROACH FOR BONDS WITH OPTIONS ATTACHED

• 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

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BINOMIAL INTEREST RATE TREE FRAMEWORK

• 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

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HOW TO OBTAIN VALUES FOR AONE-YEAR INTEREST RATE

• To obtain the two possible values for the one-year interest rate one year from today, two assumptions are required.

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 σ is the assumed volatility of the one-year rate.

Interest rate model, which we assume to be lognormal

1Interest rate volatility, represented by a standard deviation measure in our modeling

2

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BINOMIAL INTEREST RATE TREE

i0

i1,H

i1,L

i2,HH

i2,HL

i2,LL

i3,HHH

i3,HLL

i3,LLL

i3,HHL

Time 1Time 0 Time 2 Time 3

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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).

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DETERMINING THE VALUE OF A BOND AT A 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.

Bond value at any node:

Bond value if higher interest rate is realized plus coupon payment

Bond value if lower interest rate is realized plus coupon payment

Forward rateH

Forward rateL

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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.

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PRICING A BOND USING A BINOMIAL TREE

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

2.0%

5.0%

3.0%

8.0%

6.0%

4.0%

Time 0 Time 1 Time 2

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Example (continued).At Time 2, the bond value at each node is equal to the following:

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Example (continued).At Time 1, the bond value at each node is equal to the following:

At Time 0, the bond value of the bond is

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V = 103.0287

C = 5V = 103.2280

C = 5V = 106.9506

C = 5V = 102.2222

C = 5V = 104.0566

C = 5V = 105.9615

C = 5V = 100

C = 5V = 100

C = 5V = 100

C = 5V = 100

2.0%

2.0%

3.0%

3.0%

5.0%

5.0%

8.0%

8.0%6.0%

6.0%4.0%

4.0%

Example (continued).

Time 0 Time 1 Time 2 Time 3C = cash flow (% of par); R = 1-year interest rate (%); V = value of bond’s future cash flows (% of par)

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CONSTRUCTING THE BINOMIAL INTEREST RATE TREE

• 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:

• Analytical tools, such as Solver in Excel, can help with the calculations.

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

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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.

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VALUING AN OPTION-FREE BOND WITH THE TREE

• If two valuation methods are arbitrage free, they should provide the same valuation result. Check using this process:

First:

• Calculate the arbitrage-free value of an option-free, fixed-rate coupon bond.

Second:

• Compare the pricing using the zero-coupon yield curve with the pricing using an arbitrage-free binomial lattice.

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Example. Consider an option-free bond with three-years remaining to maturity and a coupon rate of 5%. Spot rates are the following:

Maturity (years)

Spot Rates

1 2.000%2 3.015%3 4.055%

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Example (continued).The interest rate tree is as follows:

2.0%

4.646%

3.442%

8.167%

6.051%

4.482%

Time 0 Time 1 Time 2

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V = 102.8101

C = 5V = 103.4658

C = 5V = 106.2668

C = 5V = 102.0721

C = 5V = 104.0090

C = 5V = 105.4958

C = 5V = 100

C = 5V = 100

C = 5V = 100

C = 5V = 100

2.0%

2.0%

3.442%

3.442%

4.646%

4.646%

8.167%

8.167%6.051%

6.051%4.482%

4.482%

Example (continued).

Time 0 Time 1 Time 2 Time 3

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PATHWISE VALUATION

• An alternative approach to backward induction in a binomial tree is called “pathwise valuation.”

• Pathwise valuation calculates the present value of a bond for each possible interest rate path and takes the average of these values across paths.

• Pathwise valuation involves the following steps:

Specify a list of all potential paths through the tree.

Determine the present value of a bond along each potential path.

Calculate the average across all possible paths.

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PATHWISE VALUATION

Example. Consider the same option-free bond as on slide 23, with three years remaining to maturity and a coupon rate of 5%.

• There are four potential paths of interest rates: HH, HL, LH, and LL. Using actual interest rates results in the following::

Path Time 0 Time 1 Time 21 2% 4.602% 8.167%

2 2% 4.602% 6.051%

3 2% 3.409% 6.051%

4 2% 3.409% 4.482%

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PATHWISE VALUATIONExample (continued).

Present values:

The result is the same as calculated using the binominal tree and spot rates.

Path Time 0

1 100.5296

2 102.3449

3 103.4792

4 104.8876

Average 102.8103

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4. MONTE CARLO METHOD

The Monte Carlo method is an alternative method for simulating a sufficiently large number of

potential interest rate paths in an effort to discover how a value of a security is affected.

This method involves randomly selecting paths in an effort to approximate the results of a complete

pathwise valuation.

Monte Carlo methods are often used when a security’s cash flows are path dependent (e.g.,

asset-backed securities).

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5. SUMMARY

• Using the arbitrage-free approach, viewing a security as a package of zero-coupon bonds means that two bonds with the same maturity and different coupon rates are viewed as different packages of zero-coupon bonds and valued accordingly.

• For bonds that are option free, an arbitrage-free value is simply the present value of expected future values using the benchmark spot rates.

Arbitrage-free valuation of a fixed-income instrument

• A binomial interest rate tree permits the short interest rate to take on one of two possible values consistent with the volatility assumption and an interest rate model.

Binomial interest rate tree framework

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SUMMARY

Backward induction

• Backward induction valuation methodology involves starting at maturity, filling in those values, and working back from right to left to find the bond’s value at the desired node.

Calibrating a binomial interest rate tree

• An interest rate tree is a visual representation of the possible values of interest rates (forward rates) based on an interest rate model and an assumption about interest rate volatility.

• The interest rate tree is fit to the current yield curve by choosing interest rates that result in the benchmark bond value.

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SUMMARY

Comparison of arbitrage-free pricing methods

• An option-free bond that is valued by using the binomial interest rate tree should have the same value as discounting by the spot rates.

Pathwise valuation in a binomial interest rate framework

• Pathwise valuation calculates the present value of a bond for each possible interest rate path and takes the average of these values across paths.

Monte Carlo forward-rate simulation • The Monte Carlo method is an alternative method for simulating a

sufficiently large number of potential interest rate paths in an effort to discover how the value of a security is affected and involves randomly selecting paths in an effort to approximate the results of a complete pathwise valuation.

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