strike price can be a cash strike price or a quoted strikeprice. In this case it is a quoted strike price. The bond price shown on the tree is the cash bondprice. The accrued interest at each node is shown below the tree. The cash strike price is calculatedas the quoted strike price plus accrued interest. The quoted bond price is the cash bond price minusaccrued interest. The payoff from the option is the cash bond price minus the cash strike price.Equivalently it is the quoted bond price minus the quoted strike price.
The tree gives the price of the option as 0.668. A much larger tree with 100 time steps gives theprice of the option as 0.699. Two points should be noted about Figure 23.11:
1. The software measures the time to option maturity in days. When an option maturity of1.5 years is input, the life of the option is assumed to be 1.5014 years (or | year and 183 days).
2. The price of the 10-year bond cannot be computed analytically when the lognormal model isassumed. It is computed numerically by rolling back through a much larger tree than thatshown.
Unequal Time Steps
In practice, it is often convenient to construct a tree with unequal time steps. Typically, we wantto choose time steps so that nodes are on certain key dates (e.g., payment dates or dates when a562 CHAPTER 23
 
Bermudan option can be exercised). We can accommodate unequal time steps as follows.Suppose that the nodes are at times tg, t),.... ty. When the x* tree is constructed, the verticalspacing between nodes at time t;,, is set equal to o,/3(t;,; — f;). The branching method is asindicated in Figure 23.12. From any given node at time t;, we branch to one of three adjacentnodes at time ¢;,,;. Suppose that x7 is the value of x* at time t;. The central node to which webranch at time t;,, is chosen to be the node closest to the expected value of x*, that is, it is thenode closest to x} —a(t;,; —1,)x}. The probabilities are determined so that the mean and
At each node:
Upper value = Cash Bond Price
Middle value = Option Price
Lower value = dt-period Rate
Shaded values are as a result of early exercise
Strike price = 105Time step, dt = 0.3753 years, 137.00 days
71.11253}
11.3831%
                
  
79.18324|9.2597%6.85538
|7.4890%
Pu: 14.0101%Pm: 66.3497%Pd: 19.6402%
   
   
87,06608)Lo7.5362%
Pu: 14.8604%Pm: 66.5258%Pd: 18.6138%
Pu: 15.7459%Pm: 66.6314%Pd: 17.6226%
         
0.057585)6.1368%
101.5064
0.468616
4.9633%
107.6932.154657]4.0142%
     
Pu: 16.6667%Pm: 66.6667%Pd: 16.6667%
  
    
   
 
[9.93444ma)105,622)
3.9392%,
} 0.2710314.9296%
107.0144)3.9870%
[112.40913.2246%|
0.097281
              
      
    
    
   
Pu: 17.6226%Pm: 66.6314%Pd: 15.7459%
106.32391.267429
         
Pu: 18.6138%Pm: 66.5258%Pd: 14.8604%
       
116.1785 Pu: 19.6402%| 10,5227} Pm: 66.3497%
  
Pd: 14.0101%
Node Time:
0.0000 0.3753 0.7507 1.1260 1.5014Accrual:
0.0000 1.9028 1.2842 0.6557 0.0275
Figure 23.11 Tree, produced by DerivaGem, for valuing an American bond optionInterest Rate Derivatives: Models of the Short Rate 563
Figure 23.12 Changing the length of the time stepstandard deviation of the change in x* are matched. (This involves solving three simultaneous
linear equations in the probabilities.) The tree for x is constructed from the tree for x” as in thecase where a constant time step is used.
23.13, NONSTATIONARY MODELS
 
 
The models discussed in the preceding sections have involved only one function of time, 6(t). Someauthors have suggested extending the models by making a or o (or both) functions of time. Forexample, in 1990, Hull and White produced analytic results for the model in equation (23.17) whenaand o are both functions of time. Also in 1990, Black, Derman, and Toy suggested a procedurefor building a binomial tree that is equivalent to the model!¢
where o’(t) is the partial derivative of o with respect to t. In 1991, Black and Karasinski suggested amore general model where the reversion rate is decoupled from the volatility’
dinr = [6(t) — a(t) In(r)} dt + a(t) dz
'6 See F, Black, KE. Derman, and W. Toy, “A One-Factor Model of Interest Rates and Its Application to TreasuryBond Options,” Financial Analysts Journal, January/February 1990, 33-39.
'” See F. Black and P. Karasinski, “Bond and Option Pricing when Short Rates Are Lognormal,” Financial AnalystsJournal, July/August 1991, 52-59.564 CHAPTER 23
 
The trinomial tree-building procedure in Section 23.12 can be extended to accommodate modelssuch as these where a and o are known functions of time. As in the case of constant a and o, wefirst build a tree for x*, where
dx* = —a(t)x* dt + o(t) dz
The spacing between the nodes at time t;,, is chosen to be
o(ti)/ 3(tin1 — i)
From any given node at time t;, we branch to one of three adjacent nodes at time 1;,,. Suppose thatxj is the value of x* at time t;. The central node we branch to at time t;,; is chosen to be the nodeclosest to the expected value of x*, that is, it is the node closest to x} — a(t;)(t;,; — tj)x7- Theprobabilities are determined similarly to the case where a constant time step is used, so that themean and standard deviation of the change in x” are matched. The tree for x is calculated from thetree for x* as in the case of constant a and o.
The advantage of making o or a (or both) functions of time is that the model can be fitted moreprecisely to the prices of instruments that trade actively in the market. When we use the model toprice deals that are fairly similar to actively traded deals we know that it will provide consistentprices. The disadvantage of making a and o functions of time is that the model has a nonstationaryvolatility structure. The volatility term structure given by the model in the future is liable to bequite different from that existing in the market today.'® For this reason is it usually desirable tochoose o and a as “well-behaved” functions of time that do not change too much from one timestep to the next. We will discuss how this can be done in the next section.
23.14 CALIBRATION
 
 
Up to now, we have assumed that the volatility parameters a and o, whether constant or functionsof time, are known. We now discuss how they are determined. This is known as calibrating themodel.
The volatility parameters are determined from market data on actively traded options (e.g., brokerquotes such as those in Tables 22.1 and 22.2). These will be referred to as the calibrating instruments.The first stage is to choose a “‘goodness-of-fit’” measure. Suppose there are n calibrating instruments.A popular goodness-of-fit measure is
itMu, - viri=I
where U; is the market price of the ith calibrating instrument and V, is the price given by the modelfor this instrument. The objective of calibration is to choose the model parameters so that thisgoodness-of-fit measure is minimized.
When a and a are constant, there are only two volatility parameters. A convenient way of makinga or oa function of time is to assume a step function. Suppose, for example, we elect to make aconstant and o a function of time. We might choose times f), t),..., t, and assume a(t) = 09fort < t, o(t) =o; fort; <t < tj4,(1 <i<n-— 1), ando(t) =o, fort > t,. There would then bea
'8 The point made here is similar to the point we made in connection with the IVF model in Chapter 20. It isdiscussed further in J. Hull and A. White, “Using Hull-White Interest Rate Trees,” Journal of Derivatives, 4,no. 3 (Spring 1996), 26-36.Interest Rate Derivatives: Models of the Short Rate 565
 
total of n+ 2 volatility parameters: a, og, 67...., 5, The number of volatility parameters mustalways be less than the number of calibrating instruments.
The minimization of the goodness-of-fit measure is accomplished by using the Levenberg-Marquardt procedure.'? When a or o or both are functions of time, a penalty function is oftenadded to the goodness-of-fit measure so that the functions are “well behaved’. In the examplejust mentioned, where a is a step function we would choose the objective function as
n n n-tDOG = WP + YE we) = 1-1) + Yo 2 (oj-1 + 141 — 20,7i=l i=l i=l
The second term provides a penalties for large changes in a between one step and the next. Thethird term provides a penalties for high curvature. Appropriate values for w,; and w»; are basedon experience and are chosen to provide an appropriate level of smoothness in the o function.
The calibrating instruments chosen should be as similar as possible to the instrument beingvalued. Suppose, for example, that we wish to value a Bermudan-style swap option that lasts tenyears and can be exercised on any payment date between year 5 and year 9 into a swap maturingten years from today. The most relevant calibrating instruments are 5 x 5, 6 x 4, 7 x 3, 8 x 2,and 9 x 1 European swap options. (An # x m European swap option is an n-year option to enterinto a swap lasting for m years beyond the maturity of the option.)
A somewhat different approach to calibration is to use all available calibrating instruments tocalculate a “global-best-fit” constant a and o parameters. The parameter a is held fixed at its best-fit value. The model can then be used in the same way as Black-Scholes. There is a one-to-onerelationship between options prices and the o parameter. The model can be used to convert tablessuch as Tables 22.1 and 22.2 into tables of implied o’s.?° These tables can be used to assess the omost appropriate for pricing the instrument under consideration.
23.15 HEDGING USING A ONE-FACTOR MODEL
 
 
We outlined some general approaches to hedging a portfolio of interest rate derivatives inSection 22.9. They can be used with the term structure models discussed in this chapter. Thecalculation of deltas, gammas, and vegas involves making small changes to either the zero curve orthe volatility environment and recomputing the value of the portfolio.
Note that, although we often assume there is one factor when priciig inlerest rate derivatives, wedo not assume only one factor when hedging. For example, the deltas we calculate allow for manydifferent movements in the yield curve, not just those that are possible under the model chosen.The practice of taking account of changes that cannot happen under the model considered, as wellas those that can, is known as outside model hedging and is standard practice for traders?! The
For a good description of this procedure, see W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling,Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press, Cambridge, 1988.
2 Note that in a term structure model the implied o’s are not the same as the implied volatilities calculated fromBlack’s model in Tables 22.1 and 22.2. The procedure for computing implied o's is as follows. The Black volatilitiesare converted to prices using Black’s model. An iterative procedure is then used to imply the o parameter in the termstructure model from the price.
21 A simple example of outside model hedging is in the way that the Black-Scholes model is used. The Black-Scholesmodel assumes that volatility is constant—but traders regularly calculate vega and hedge against volatility changes.566 CHAPTER 23
reality is that relatively simple one-factor models if used carefully usually give reasonable prices forinstruments, but good hedging schemes must explicitly or implicitly assume many factors.
23.16 FORWARD RATES AND FUTURES RATES
 
In Section 5.11 we explained that a convexity adjustment is necessary when a forward interest rateis calculated from a Eurodollar futures price quote.”
Consider a futures contract on the interest rate lasting between times t, and t). As mentioned inSection 5.11, in the Ho—Lee model the continuously compounded forward rate for the periodequals the continuously compounded futures rate minus ott /2. A typical value of o is 0.012. Forthe Eurodollar futures traded on the CME, t2 = t; + 0.25. Typical convexity adjustments that haveto be made to the rates given by these contracts when they have maturities of 2, 4, 6, 8, and 10 yearsare therefore 3, 12, 27, 48, and 74 basis points, respectively.
For the Hull-White model the convexity adjustment that must be subtracted from the con-tinuously compounded futures rate to get the continuously compounded forward rate is
2By, wer, (1 — ety + 2aB(0, yy (23.32)boty —h 4a
where the B function is as defined in equation (23.20) (see Problem 23.21).
Exampie 23.4 Consider the situation where a = 0.05 and o = 0.015 and we wish to caiculate aforward rate when the cight-year Eurodollar futures price is 94. In this case, t; = 8, t = 8.25,B(t;. ts) = 0.2484, B(O, t;) = 6.5936, and the convexity adjustment is
0.0157
Ay ODS4K UNUS
0.2484
O25Ved
= 0.0050
 
 
{0.2484(1 — ¢7?*0*8) 4.2 x 0.05 x 6.59367]or 0.50%. The futures rate is 6% per annum with quarterly compounding or 5.96% with continuous
compounding. The forward rate is therefore 5.96 —0.50 = 5.46% per annum with continuouscompounding.
SUMMARY
 
The traditional models of the term structure used in finance are known as equilibrium models.These are useful for understanding potential relationships between variables in the economy buthave the disadvantage that the initial term structure is an output from the model rather than aninput to it. When valuing derivatives, it is important that the model used be consistent with theinitial term structure observed in the market. No-arbitrage models are designed to have thisproperty. They take the initial term structure as given and define how it can evolve.
This chapter has provided a description of a number of one-factor no-arbitrage models of theshort rate. These are very robust and can be used im conjunction with any set of initial zero rates.The simplest model is the Ho—Lee model. This has the advantage that it is analytically tractable.
2 Note that the term convexity adjustment is overworked in derivatives. The convexity adjustment here is quitedifferent from the one talked about in Section 22.6.Interest Rate Derivatives: Models of the Short Rate 567
Its chief disadvantage is that it implies that all rates are equally variable at all times. The Hull-White model is a version of the Ho—Lee model that includes mean reversion. It allows a richerdescription of the volatility environment white preserving its analytic tractability. Lognormal one-factor models have the advantage that they avoid the possibility of negative interest rates but,unfortunately, they have no analytic tractability.
SUGGESTIONS FOR FURTHER READING
 
Equilibrium Approaches to Modeling the Term StructureBrennan, M. J., and E. S. Schwartz, “An Equilibrium Model of Bond Pricing and a Test of MarketEfficiency,” Journal of Financial and Quantitative Analysis, 17, no. 3 (September 1982), 301-29.
Courtadon, G., “The Pricing of Options on Default-free Bonds,” Journal of Financial and QuantitativeAnalysis, 17 (March 1982), 75—100.
Cox, J. C., J. E. Ingersoll, and S. A. Ross, “A Theory of the Term Structure of Interest Rates,”Econometrica, 53 (1985), 385-407.
Jamshidian, F., “An Exact Bond Option Pricing Formula,” Journal of Finance, 44 (March 1989), 205-9.
Longstaff, F. A., and E. S. Schwartz, “Interest Rate Volatility and the Term Structure: A Two FactorGeneral Equilibrium Model,” Journal of Finance, 47, no. 4 (September 1992), 1259-82.
Rendleman, R., and B. Bartter, “The Pricing of Options on Debt Securities,” Journal of Financial andQuantitative Analysis, 15 (March 1980), 11-24.
Schaefer, S. M., and E. S. Schwartz, ‘‘Time-Dependent Variance and the Pricing of Options,” Journal ofFinance, 42 (December 1987), 1113-28.
Vasicek, O. A., “An Equilibrium Characterization of the Term Structure,” Journal of FinancialEconomics, 5 (1977), 177-88.
Black, F., E. Derman, and W. Toy, “A One-Factor Model of Interest Rates and Its Application toTreasury Bond Options,” Financial Analysts Journal, January/February 1990, 33-39.
Black, F., and P. Karasinski, “Bond and Option Pricing When Short Rates Are Lognormal,” FinancialAnalysts Journal, July/August 1991, 52-59.
Burghardt, G., and B. Hoskins, “A Question of Bias,” R7SK, March 1995, pp. 63-67.
Ho, T. S. Y., and S.-B. Lee, ‘““Term Structure Movements and Pricing Interest Rate ContingentClaims,” Journal of Finance, 41 (December 1986), 1011-29.
Hull, J., and A. White, “Bond Option Pricing Based on a Model for the Evolution of Bond Prices,”Advances in Futures and Options Research, 6 (1993), 1-13.
Hull, J., and A. White, “Branching Out,” RZSK, January 1994, pp. 34-37.Hull, J., and A. White, “In the Common Interest,” RISK, March 1992, pp. 64-68.
Hull, J., and A. White, ‘Numerical Procedures for Implementing Term Structure Models I: Single-Factor Models,’ Journal of Derivatives, 2, no. 1 (Fall 1994), 7-16.
Hull, J., and A. White, “One-Factor Interest Rate Models and the Valuation of Interest RateDerivative Securities,” Journal of Financial and Quantitative Analysis, 28 (June 1993), 235-54.568 CHAPTER 23
Hull, J., and A. White, ‘Pricing Interest Rate Derivative Securities,” The Review of Financial Studies, 3,no. 4 (1990), 573-92.
Hull, J., and A. White, “Taking Rates to the Limit,” RZSK, December 1997, pp. 168-69.
Hull, J., and A. White, “ The Pricing of Options on Interest Rate Caps and Floors Using the Hull-White Model,” Journal of Financial Engineering, 2, no. 3 (1993), 287-96.
Hull, J., and A. White, “Using Hull-White Interest Rate Trees,” Journal of Derivatives, Spring 1996,26-36.
Hull, J., and A. White, ‘‘Valuing Derivative Securities Using the Explicit Finite Difference Method,”Journal of Financial and Quantitative Analysis, 25 (March 1990), 87-100.
Kijima, M., and I. Nagayama, “Efficient Numerical Procedures for the Hull-White Extended VasicekModel,” Journal of Financial Engineering, 3 (September/December 1994), 275-92.
Kijima, M., and I. Nagayama, ‘‘A Numerical Procedure for the General One-Factor Interest RateModel,” Journal of Financial Engineering, 5 (December 1996), 317-37.
Li, A., P. Ritchken, and L. Sankarasubramanian, “Lattice Models for Pricing American Interest RateClaims,” Journal of Finance, 50, no. 2, June 1995, 719-37.
Pelsser, A. A. J., “Efficient Methods for Valuing and Managing Interest Rate and Other DerivativeSecurities,” Ph.D. Dissertation, Erasmus University, Rotterdam, 1996.
Rebonato, R., Interest Rate Option Models, Wiley, Chichester, 1996.
QUESTIONS AND PROBLEMS(ANSWERS IN SOLUTIONS MANUAL)
 
 
23.1. What is the difference between an equilibrium model and a no-arbitrage model?
23.2. If a stock price were mean reverting or followed a path-dependent process, then there would bemarket inefficiency. Why is there not a market inefficiency when the short-term interest rate does80?
23.3. Suppose that the short rate is currently 4% and its standard deviation is 1% per annum. Whathappens to the standard deviation when the short rate increases to 8% in (a) Vasicek’s model;(b) Rendleman and Bartter’s model: and (c) the Cox, Ingersoll, and Ross model?
23.4. Explain the difference between a one-factor and a two-factor interest rate model.
23.5. Can the approach described in Section 23.4 for decomposing an option on a coupon-bearingbond into a portfolio of options on zero-coupon bonds be used in conjunction with a two-factormodel? Explain your answer.
23.6. Suppose that a = 0.1 and b=0.1 in both the Vasicek and the Cox, Ingersoll, Ross model. Inboth models, the initial short rate is 10% and the initial standard deviation of the short ratechange in a short time 4¢ is 0.02./8t. Compare the prices given by the models for a zero-couponbond that matures in year 10.
23.7. Suppose that a = 0.1, b = 0.08, and o = 0.015 in Vasicek’s model with the initial value of theshort rate being 5%. Calculate the price of a one-year European call option on a zero-couponbond with a principal of $100 that matures in three years when the strike price is $87.interest Rate Derivatives: Models of the Short Rate 569
23.12.
23.13.
23.14.
23.15,
23.16.23.17.
23.18.
23.19,23.20.
23.21.
23.22.23.23.
. Repeat Problem 23,7 valuing a European put option with a strike of $87. What is the put-call
parity relationship between the prices of European call and put options? Show that the put andcall option prices satisfy put-call parity in this case.
. Suppose that a = 0.05, b = 0.08, and o = 0.015 in Vasicek’s model with the initial short-term
interest rate being 6%. Calculate the price of a 2.1-year European call option on a bond that willmature in three years. Suppose that the bond pays a coupon of 5% semiannually. The principalof the bond is 100 and the strike price of the option is 99. The strike price is the cash price (notthe quoted price) that will be paid for the bond.
Tee the an 929 and witeeca
. ewer tra Proablen an YW oaette aroiments. iq calenlate the atice ofa ad. Use the answer to Problem 23.9 and pui-cau parity arguments to calculate the price of a pui
option that has the same terms as the call option in Problem 23.9.
. In the Hull -White model, a = 0.08 and o = 0.01. Calculate the price of a one-year European call
option on a zero-coupon bond that will mature in five years when the term structure is flat at10%, the principal of the bond is $100, and the strike price is $68.
Suppose that a = 0.05 and o = 0.015 in the Hull-White model with the initial term structurebeing flat at 6% with semiannual compounding. Calculate the price of a 2.1-year European calloption on a bond that will mature in three years. Suppose that the bond pays a coupon of 5% perannum semiannually. The principal of the bond is 100 and the strike price of the option is 99.The strike price is the cash price (not the quoted price) that will be paid for the bond.
Using equation (23.13) and the results in Section 23.16, show that the drift rate of the short rateat time ¢ in the Ho and Lee model is G,(0, 1), where G(t, T) is the instantaneous futures rate asseen at time t for a contract maturing at time 7 and the subscript denotes a partial derivative.Using equation (23.18) and the results in Section 23.16, show that the drift rate of the short rateat time ¢ in the Hull-White model is G,(0, t) + a[G(0, t) — r], where G(t, T) is the instantaneousfutures rate as seen at time t for a contract maturing at time T and the subscript denotes a partialderivative.
Suppose that a = 0.05, o = 0.015, and the term structure is flat at 10%. Construct a trinomialtrec for the Hull-White model whcre there are two time steps, each one year in length.
Calculate the price of a two-year zero-coupon bond from the tree in Figure 23.6.
Calculate the price of a two-year zero-coupon bond from the tree in Figure 23.9 and verify that itagrees with the initial term structure.
Calculate the price of an 18-month zero-coupon bond from the tree in Figure 23.10 and verifythat it agrees with the initial term structure.
What does the calibration of a one-factor term structure model involve?
Use the DerivaGem software to value 1 x 4, 2 x 3, 3 x 2, and 4 x | European swap options toreceive fixed and pay floating. Assume that the 1-, 2-, 3-, 4-, and S-year interest rates are 6%,5.5%, 6%, 6.5%, and 7%, respectively. The payment frequency on the swap is semiannualand the fixed rate is 6% per annum with semiannual compounding. Use the Hull-Whitemodel with a= 3% and o = 1%. Calculate the volatility implied by Black’s model for eachoption.
Use equation (23.32) to calculate an expression for the difference between the instantaneousforward rate and the instantaneous futures rate. Show that it is consistent with equation (23.27).
Prove equations (23.15) and (23.16).Prove equations (23.22), (23.23), and (23.24).570
CHAPTER 23
ASSIGNMENT QUESTIONS
 
 
23.24.
23.25.
23.26.
23.27,
23.28,
23.29.
Construct a trinomial tree for the Ho and Lee model where o = 0.02. Suppose that the the initialzero-coupon interest rate for maturities of 0.5, 1.0, and 1.5 years are 7.5%, 8%, and 8.5%. Usetwo time steps, cach six months long. Calculate the value of a zero-coupon bond with a facevalue of $100 and a remaining life of six months at the ends of the final nodes of the tree. Use thetree to value a one-year European put option with a strike price of 95 on the bond. Compare theprice given by your tree with the analytic price given by DerivaGem.
Suppose that a = 0.1, 0 = 0.02 in the Hull-White model, and a 10-year Eurodollar futures quoteis 92. What is