1. Chapter 24
Interest Rate Models

2. Introduction
We will begin by seeing how the Black model can be used to price bond and interest rate options
The Black model is a version of the Black-Scholes model for which the underlying asset is a futures contract
Next, we see how the Black-Scholes approach to option pricing applies to bonds
Finally, we examine binomial interest rate models, in particular the Black-Derman-Toy model
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3. Bond Pricing Some difficulties of bond pricing are
A casually specified model may give rise to arbitrage opportunities (for example, if the yield curve is assumed to be flat)
In general, hedging a bond portfolio based on duration does not result in a perfect hedge
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4. An Equilibrium Equation for Bonds
When the short-term interest rate is the only source of uncertainty, the following partial differential equation must be satisfied by any zero-coupon bond
(24.18)
where r is the short-term interest rate, which follows the Itô process dr = a(r)dt + (r)dZ, and (r,t) is the Sharp ratio for dZ
This equation characterizes claims that are a function of the interest rate. The risk-neutral process for the interest rate is obtained by subtracting the risk premium from the drift
(24.19)
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5. An Equilibrium Equation for Bonds (cont’d)
Given a zero-coupon bond, Cox et al. (1985) show that the solution to equation (24.18) is
(24.20)
where E* represents the expectation taken with respect to risk-neutral probabilities and R(t,T) is the random variable representing the cumulative interest rate over time:
(24.21)
Thus, to value a zero-coupon bond, we take the expectation over all the discount factors implied by these paths
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6. An Equilibrium Equation for Bonds (cont’d)
In summary, an approach to modeling bond prices is exactly the same procedure used to price options on stock
We begin with a model of the interest rate and then use equation (23.26) to obtain a partial differential equation that describes the bond price
Next, using the PDE together with boundary conditions, we can determine the price of the bond
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7. Delta-Gamma Approximations for Bonds
Equation (24.18) implies that
(24.22)
Using the risk-neutral distribution, bonds are priced to earn the risk-free rate
Thus, the delta-gamma-theta approximation for the change in a bond price holds exactly if the interest rate moves one standard deviation
However, the Greeks for a bond are not exactly the same as duration and convexity
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8. Equilibrium Short-Rate Bond Price Models
We discuss three bond pricing models based on equation (24.18), in which all bond prices are driven by the short-term interest rate, r
The Rendleman-Bartter model
The Vasicek model
The Cox-Ingersoll-Ross model
They differ in their specification of (r), (r), (r)
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9. Equilibrium Short-Rate Bond Price Models (cont’d)
The simplest models of the short-term interest rate are those in which the interest rate follows arithmetic or geometric Brownian motion. For example
dr = adt +dZ (24.23)
In this specification, the short-rate is normally distributed with mean r0 + ar and variance 2t
There are several objections to this model
The short-rate can be negative
The drift in the short-rate is constant
The volatility of the short-rate is the same whether the rate is high or low
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10. The Rendleman-Bartter Model
The Rendleman-Bartter model assumes that the short-rate follows geometric Brownian motion
dr = ardt + rdz (24.24)
An objection to this model is that interest rates can be arbitrarily high. In practice, we would expect rates to exhibit mean reversion
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11. The Vasicek Model The Vasicek model incorporates mean reversion
dr = a(b  r)dt + dz (24.25)
This is an Ornstein-Uhlenbeck process. The a(b  r)dt term induces mean reversion
It is possible for interest rates to become negative and the variability of interest rates is independent of the level of rates
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12. The Cox-Ingersoll-Ross Model
The Cox-Ingersoll-Ross (CIR) model assumes a short-term interest rate model of the form
dr = a(b  r)dt + r dz (24.27)
The model satisfies all the objections to the earlier models
It is impossible for interest rates to be negative
As the short-rate rises, the volatility of the short-rate also rises
The short-rate exhibits mean reversion
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13. Comparing Vasicek and CIR
Yield curves generated by the Vasicek and CIR models
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14. Bond Options, Caps and the Black Model
If we assume that the bond forward price is lognormally distributed with constant volatility , we obtain the Black formula for a bond option
(24.32)
where
and F is an abbreviation for the bond forward price F0,T [P(T,T + s)], where Pt(T,T + s) denotes the time-t price of a zero-coupon bond purchased at T and paying $1 at time T + s
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15. Bond Options, Caps, and the Black Model (cont’d)
One way for a borrower to hedge interest rate risk is by entering into a call option on a forward rate agreement (FRA), with strike price KR
This option, called a caplet, at time T + s pays
Payoff to caplet = max [0, RT(T, T + s) – KR] (24.34)
An interest rate cap is a collection of caplets
Suppose a borrower has a floating rate loan with interest payments at times ti, i = 1, . . ., n. A cap would make the series of payments
Cap payment at time ti+1 = max [0, Rti(ti, ti+1) – KR] (24.37)
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16. A Binomial Interest Rate Model
Binomial interest rate models permit the interest rate to move randomly over time
Notation
h is the length of the binomial period. Here a period is 1 year, i.e., h=1
rt0(t, T) is the forward interest rate at time t0 for time t to time T
rt0(t, T; j) is the interest rate prevailing from t to T, where the rate is quoted at time t0 < t and the state is j
At any point in time t0, there is a set of both spot and implied forward zero-coupon bond prices, Pt0(t, T; j)
p is the risk-neutral probability of an up move
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17. A Binomial Interest Rate Model (cont’d)
Three-period interest rate tree of the 1-year rate
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18. Zero-Coupon Bond Prices
Because the tree can be used at any node to value zero-coupon bonds of any maturity, the tree also generates implied forward interest rates of all maturities, as well as volatilities of implied forward rates
Thus, we can equivalently specify a binomial interest rate tree in terms of
Interest rates
Zero-coupon bond prices, or
Volatilities of implied forward interest rates
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19. Zero-Coupon Bond Prices (cont’d)
We can value a bond by considering separately each path the interest rate can take
Each path implies a realized discount factor
We then compute the expected discount factor, using risk-neutral probabilities
All bond valuation models implicitly calculate the following equation
(24.44)
where E* is the expected discount factor and ri represents the time-i rate
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20. Zero-Coupon Bond Prices (cont’d)
The volatility of the bond price is different from the behavior of a stock
With a stock, uncertainty about the future stock price increases with horizon
With a bond, volatility of the bond price initially grows with time. However, as the bond approaches maturity, volatility declines because of the boundary condition that the bond price approached $1
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21. Yields and Expected Interest Rates
Uncertainty causes bond yields to be lower than the expected average interest rate
The discrepancy between yields and average interest rates increases with volatility
Therefore, we cannot price a bond by using the expected interest rate
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22. Option Pricing
Using the binomial tree to price a bond option works the same way as bond pricing
Suppose we have a call option with strike price K on a (T  t)- year zero-coupon bond, with the option expiring in t  t0 periods. The expiration value of the option is
O (t, j) = max [0, Pt(t, T; j) – K] (24.46)
To price the option we can work recursively backward through the tree using risk-neutral pricing, as with an option on a stock. The value one period earlier at the node j’ is
O (t  h, j’) = Pt  h (t  h, t; j’)
 [p  O (t, 2  j’ + 1) + (1  p)  O (m, 2  j’)] (24.47)
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23. Option Pricing (cont’d)
In the same way, we can value an option on a yield, or an option on any instrument that is a function of the interest rate
Delta-hedging works for the bond option just as for a stock option
The underlying asset is a zero-coupon bond maturing at T, since that will be a (T  t)-period bond in period t
Each period, the delta-hedged portfolio of the option and underlying asset is financed by the short-term bond, paying whatever one-period interest rate prevails at that node
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24. The Black-Derman-Toy Model
The models we have examined are arbitrage-free in a world consistent with their assumptions. In the real world, however, they will generate apparent arbitrage opportunities, i.e., observed prices will not match theoretical prices
Matching a model to fit the data is called calibration
This calibration ensures that it matches observed yields and volatilities, but not necessarily the evolution of the yield curve
The Black-Derman-Toy (BDT) tree is a binomial interest rate tree calibrated to match zero-coupon yields and a particular set of volatilities
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25. The Black-Derman-Toy Model (cont’d)
The basic idea of the BDT model is to compute a binomial tree of short-term interest rates, with a flexible enough structure to match the data
Consider market information about bonds that we would like to match
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26. The Black-Derman-Toy Model (cont’d)
Black, Derman, and Toy describe their tree as driven by the short-term rate, which they assume is lognormally distributed
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27. The Black-Derman-Toy Model (cont’d)
For each period in the tree, there are two parameters
Rih can be though of as a rate level parameter at a given time and
i can be though of as a volatility parameter
These parameters can be used to match the tree with the data
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28. The Black-Derman-Toy Model (cont’d)
The volatilities are measured in the tree as follows
Let the time-h price of a zero-coupon bond maturing at T when the time-t short-term rate is r(h) be P[h, T, r(h)]
The yield of the bond is
y[h, T, r(h)] = P[h, T, r(h)]1/(T  h)  1
At time h, the short-term rate can take on the two values ru and rd. The annualized lognormal yield volatility is then
(24.48)
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29. The Black-Derman-Toy Model (cont’d)
The BDT tree, which depicts 1-year effective annual rates, is constructed using the market data:
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30. The Black-Derman-Toy Model (cont’d)
The tree behaves differently from binomial trees
Unlike a stock-price tree, the nodes are not necessarily centered on the previous period’s nodes
This is because the tree is constructed to match the data
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31. The Black-Derman-Toy Model (cont’d)
Verifying that the tree is consistent with the data
To verify that the tree matches the yield curve, we need to compute the prices of zero-coupon bonds with maturities of 1, 2, 3, and 4 years
To verify the volatilities, we need to compute the prices of 1-, 2-, and 3-year zero-coupon bonds at year 1, and then compute the yield volatilities of those bonds
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32. The Black-Derman-Toy Model (cont’d)
The tree of 1-year bond prices implied by the BDT interest rate tree
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33. The Black-Derman-Toy Model (cont’d)
Verifying yields
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34. The Black-Derman-Toy Model (cont’d)
Verifying volatilities
For each bond, we need to compute implied bond yields in year 1 and then compute the volatility
Using equation (24.48), the yield volatility
for the 2-year bond (1-year bond in year 1) is
for the 3-year bond in year 1 is
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35. Constructing a Black-Derman-Toy Tree
Given the data, how do we generate the tree?
We start at early nodes and work to the later nodes, building the tree outward
The first node is given by the prevailing 1-year rate.
The 1-year bond price is
$ = 1 / (1+R0) (24.49)
Thus, R0 = 0.10
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36. Constructing a Black-Derman- Toy Tree (cont’d)
For the second node, the year-1 price of a 1-year bond is P(1, 2, ru) or P(1, 2, rd). We require that two conditions be satisfied
(24.50)
(24.51)
The second equation gives us  = 0.1, which enables us to solve the first equation to obtain R1 =
In the same way, it is possible to solve for the parameters for each subsequent period
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37. Black-Derman-Toy Examples
Caplets and caps
An interest rate cap pays the difference between the realized interest rate in a period and the interest cap rate, if the difference is positive
The payments in each node in a tree are the present value of the cap payments for the interest rate at that node
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38. Black-Derman-Toy Examples (cont’d)
Forward rate agreements (FRA)
The standard FRA calls for settlement at maturity of the loan, when the interest payment is made
If r(3, 4) is the 1-year rate in year 3, the payoff to a standard FRA 4 years from today is
(24.54)
We can compute by taking the discounted expectation along a binomial tree of r(3, 4) paid in year 4 and dividing by P(0, 4)
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39. Black-Derman-Toy Examples (cont’d)
Forward rate agreements (FRA)
Eurodollar-style settlement, calls for payment at the time the loan is made
If a FRA settles on the borrowing date in year 3, then
(24.55)
We can compute by taking the discounted expectation along a binomial tree of r(3, 4) paid in year 3, and dividing by P(0, 3).
There is no marking-to-market prior to settlement, but the timing of settlement is mismatched with the timing of interest payments
Thus, the two settlement procedures generate different fair forward interest rates
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