Chapter 24 Interest Rate Models

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 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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) Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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) Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

Comparing Vasicek and CIR Yield curves generated by the Vasicek and CIR models Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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) Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

A Binomial Interest Rate Model (cont’d) Three-period interest rate tree of the 1-year rate Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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) Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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) Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

The Black-Derman-Toy Model (cont’d) The BDT tree, which depicts 1-year effective annual rates, is constructed using the market data: Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

The Black-Derman-Toy Model (cont’d) The tree of 1-year bond prices implied by the BDT interest rate tree Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

The Black-Derman-Toy Model (cont’d) Verifying yields Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 $0.9091 = 1 / (1+R0) (24.49) Thus, R0 = 0.10 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 = 0.1082 In the same way, it is possible to solve for the parameters for each subsequent period Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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) Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

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 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.