a two-factor term structure model under garch volatility
TRANSCRIPT
JUNE 2003 THE JOURNAL OF FIXED INCOME 87
Extending the work of Vasicek [1977],Cox, Ingersoll, and Ross [1985](henceforth CIR), and Heath, Jarrow,and Morton [1992] (henceforth
HJM), researchers have developed manymodels of the term structure of interest ratesthat often involve pricing bonds or interestrate derivatives. One class of models often takesa short-term interest rate to be the variablethat drives the term structure, and then pricesbonds and bond options by deriving the appro-priate risk-neutral or risk-adjusted dynamics ofthe short rate. The HJM approach, however,takes the entire yield curve (or, equivalently,the set of forward rates or bond prices) to bethe state variable and derives prices of optionson bonds. It also matches the current termstructure by default as the term structure (or,equivalently, the set of forward rates) is itselfthe state variable.
It is well known that the dynamics of theterm structure cannot be captured by onefactor (see Litterman and Scheinkman ([1991]).In fact, Dybvig [1997] emphasizes a secondfactor related to the volatility of interest ratesthat may not have any major impact on theprices of the spot bonds, but may be veryimportant for bond options. Andersen andLund [1996] also find that a factor that helpsexplain the curvature of the yield curve is, infact, closely related to the volatility of the shortrate. Unlike the HJM approach, the short rate-based approaches can easily accommodate asecond non-traded state variable such as
volatility and maintain analytical as well asnumerical tractability.
Fong and Vasicek [1991] develop a two-factor model in which one factor is the instan-taneous variance of the short rate, and providesolutions for bond prices. Longstaff andSchwartz [1992] (henceforth LS) and Chenand Scott [1992] (henceforth CS) develop con-tinuous-time two-factor models along the linesof CIR [1985] that can incorporate randomvolatility in the evolution of the short rate andoffer analytical solutions for values of bondsand bond options.
One difficulty of the continuous-timemodels is that the volatility of the short rate isunobservable. Unobservable volatility impliesthat the spot volatility at time t that is neededto price bonds and bond options is not knownat the time the price needs to be calculated. Asa result, one would be constrained to use thespot volatility inferred from a previous period,which may not necessarily reflect the currentinformation in the term structure. Forexample, one can calibrate the continuous-time stochastic volatility models to market data(observed at time t – 1) on options and bonds,and therefore get an estimate of the volatilityat t – 1 that may very well be different fromthe volatility at time t.
Estimating these models on large datasets spanning multiple days also represents acomputational burden. Each day’s volatilitymust be estimated as a separate parameter; thisconsiderably increases the number of param-
A Two-Factor Term StructureModel under GARCH VolatilitySTEVEN HESTON AND SAIKAT NANDI
STEVEN HESTON
is an assistant professor offinance at the R.H. SmithSchool of Business at theUniversity of Maryland,College Park, [email protected]
SAIKAT NANDI
is senior financial engineerin Portfolio Analytics andResearch at Fannie Mae in Washington, [email protected]
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eters over which a criterion function such as squaredpricing errors needs to be minimized. For example, if wehave data for daily prices of bonds of various maturitiesfor four weeks (20 days), we would have to estimate 20extra parameters (other than the time-invariant parame-ters of the model).
We develop a discrete-time model of interest rateswith analytical solutions for values of bonds and bondoptions in which the second state variable is a randomvolatility following a GARCH process, while the firststate variable is the mean-reverting short rate, and thetwo state variables can be correlated. Besides bond andbond futures prices, the model generates analytical solu-tions for values of options on discount bonds and dis-count bond futures as well as other interest rate derivativessuch as caps or floors.
The advantage of this model besides its analyticaltractability is that the time-varying random volatility iseasily filtered from the discrete observations of interestrates. Therefore, the spot volatility at time t that is neededto value bonds and bond options at that time is knownor observable. Furthermore, observability of volatilityimplies that during model calibration, the number ofparameters that needs to be estimated remains finite (equalto the number of time-invariant parameters of the model)instead of increasing proportionately with the number ofdays or time periods. (For expositional clarity, we oftenrefer to the state variables as factors henceforth.)
Discrete-time volatility models based on theGARCH dynamics have been very popular in equity andcurrency markets due to their abilities to capture certainsalient features of data such as volatility clustering and thefact that the time-varying volatility is observable (seeBollerslev, Chou, and Kroner [1992]). Brenner, Harjes,and Kroner [1996], Koedjik et al. [1994], and others haveshown that GARCH processes can also capture thevolatility dynamics of interest rates.
Until now, though, there were not explicit solutionsfor bond prices and prices of various interest rate deriva-tives under a GARCH process (to the best of our knowl-edge). Therefore one of our contributions lies in providinganalytical solutions for bond prices and especially bondoption prices in a GARCH framework, thus simplifyingthe computational burden of non-analytical solutions.
Calibrating our model to the yield curve (eight dif-ferent maturities) for several two-week intervals in theperiod 1990-1996, we find that the two-factor versiondoes not improve (statistically and economically) uponthe nested one-factor model (which is a discrete-time
version of the Vasicek model) in terms of pricing thecross-section of spot bonds. This occurs even though theone-factor is rejected in favor of the two-factor model inexplaining the time series behavior of a chosen short rate(computed from the three month T-bill yields). This con-clusion is robust to the length of the interval used tosample the term structure.
Given option prices (on discount bonds) from thetwo-factor model, however, the implied volatilities fromthe Black model exhibit a much more pronounced smirkor skew than the implied volatilities generated from theprices of the nested one-factor model. These option pricesare generated not by conjectured parameter estimates, butby actual parameter estimates obtained though calibratingthe models to the observed yield curve.
Thus our results show that the effects of randomvolatilities of interest rates are visible mostly in bond optionprices rather than in bond prices.
I. MODEL
We assume that the short rate, rt, which is the interestrate on a loan at time t that is to be repaid at time t + 1,follows the processes:
(1)
(2)
where zt is a standard normal. Consequently ht+1 is thevariance of rt+1 – rt, conditional on the information attime t. A non-zero ⁄ allows correlation between the levelof interest rates and the conditional variance in thatCovt–1[h(t + 1), rt] = 2ÿ⁄ht (Covt–1 ( ) is the covarianceconditional on the information at time t – 1). The volatilityof variance or volatility is controlled by the parameter, ÿ.
In particular, note that ht+1 is known as of time t, giventhe history of rt until t and an initial variance h0 as follows:
(3)
In this model, we have a mean-reverting short ratewith GARCH volatility. This is very similar to being thediscrete-time counterpart of the Vasicek [1977] model,augmented with GARCH volatility. In fact, if we restrict
2(r ¡ ¹ ¡ ¹ r ¡ ¸h ¡ °h )t 0 1 t¡1 t th = ! + ¯ h + ®t+1 t
ht
q2h = ! + ¯ h + ®(z ¡ ° h )t+1 t t t
qr = ¹ + ¹ r + ¸h + h zt+1 0 1 t t+1 t+1 t+1
88 A TWO-FACTOR TERM STRUCTURE MODEL UNDER GARCH VOLATILITY JUNE 2003
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ht to be constant, we have exactly the discrete-time coun-terpart of the continuous-time Vasicek model.1
It can be shown following Heston and Nandi [2000]that if rt is taken to be the instantaneous short rate as incontinuous-time models of the term structure, the lawsof motion of rt are:
(4)
(5)
where µ*1, fl, ·, and Ê are related to the parameters of the
discrete-time model (see Heston 1 and Nandi [1999] forthe details).
It can be verified that the continuous-time modellends itself to an affine structure in that the logarithm ofa bond yield is affine (i.e., linear plus a constant) in rt andvt as in Fong and Vasicek [1991]. As a result, one can workout analytical solutions for the continuous-time modelalso. We concentrate only on the discrete-time model, asparameter estimation and therefore model implementa-tion are much easier this way. This is simply because,unlike in the discrete-time GARCH model, one cannotfilter the spot variance, vt, of the continuous-time modelas a function of the path of rt as rt is observed only at dis-crete intervals of time.
Let the time t price of a discount bond that maturesat T be P(t, T). It can be shown that under the risk-neu-tral distribution (which is needed for pricing) zt is replacedby zq
t, so that zqt = zt – fi ��ht+1� is a standard normal under
the risk-neutral distribution, where fi is a constant (seeHeston and Nandi [1999]).
In other words, the risk-neutral dynamics of rt andht are given by
(6)
(7)
where ‚* = ‚ + h, and ⁄* = ⁄ – h.
II. BOND PRICES
Consider the price of a zero-coupon bond at t thatexpires at t + 2, i.e., P(t, t + 2). Since P(t, t + 2) is thediscounted expected value of a single-period bond:
qq ¤ 2h = ! + ¯ h + ®(z ¡ ° h )t+1 t tt
qq¤r = ¹ + ¹ r + ¸ h + h zt+1 0 1 t t+1 t+1 t+1
pdv = ·(µ ¡ v )dt+ ¾ v dWt t t t
p¤dr = (¹ + ¹ r + ¸ v ) dt+ v dWt 0 t t t t1
(8)
where Eqt ( ) is the expectation (under the risk-neutral dis-
tribution) conditional on the information set at time t.Thus the yield of the two-period bond is 1⁄2(µ0 +
(µ1 + 1)rt + (‚* + 0.5)ht+1). Note that the bond yield isaffine (i.e., linear plus a constant) in the state variables rtand ht+1. We can similarly calculate the yield on a three-period bond by using iterated conditional expectationand show that the three-period yield is also affine in thestate variables. The affine nature of the bond yield inthe state variables rt and ht+1 suggests that we can write theprice of a bond with T – t periods to maturity in theexponential-affine form as:
(9)
as in CIR [1985], Heston [1990], Duffie and Kan [1996],and many others. As rt and ht+1 are known as of time t, itremains to solve for the coefficients A(t, T), B(t, T), andC(t, T) in terms of the model parameters. These coeffi-cients can be solved recursively from a terminal orboundary condition. For the boundary condition, con-sider the price of a bond (that will mature at T) at T – 2:
(10)
But, following (8), it is also true that P(T – 2, T) =exp[–(µ0 + (µ1 + 1)rT – 2 + (‚* + 0.5)hT – 1)]. Equatingthe two expressions for P(T – 2, T), and matching coef-ficients on the state variables and the constant, we get theboundary conditions:
(11)
(12)
(13)
Now we will derive the backward recursion for-mula for the coefficients A(t, T), B(t, T), and C(t, T),
¤C(T ¡ 2; T ) = ¡(¸ + 0:5)
B(T ¡ 2; T ) = ¡(¹ + 1)1
A(T ¡ 2; T ) = ¡¹0
B(T ¡ 2; T )r + C(T ¡ 2; T )h )T¡2 T¡1
P (T ¡ 2; T ) = exp (A(T ¡ 2; T ) +
P (t; T ) = exp(A(t; T ) +B(t; T )r + C(t; T )h )t t+1
¤(¸ + 0:5)h ))t+1
= exp(¡ (¹ + (¹ + 1)r +0 1 t
qP (t; t+ 2) = exp(¡r )E (P (t+ 1; t+ 2))t t
q= exp(¡r ) E (exp(¡r ))t t t+1t
JUNE 2003 THE JOURNAL OF FIXED INCOME 89
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given the boundary conditions. At each time period, thebond price is the discounted expected value of its next-period price. In other words:
(14)
Substituting the risk-neutralized dynamics for rt + 1and ht+1 , and doing the algebra as shown in the appendix,we get that the coefficients are:
(15)
(16)
+
(17)
In other words, starting from the boundary condi-tions (11), (12), and (13), we have to perform a simplebackward recursion using (15), (16), and (17) to find thecoefficients A(t, T), B(t, T), and C(t, T). Once we havethese, P(t, T) is given by (9).
III. BOND OPTIONS
Now we are ready to calculate the price of a Euro-pean option on a discount bond. Let the time to expira-tion of the option be Á and that of the underlying bondbe T, and T > Á. Let the price of a call option (at time t)on the discount bond be Co (t, Á, T). The payoff from theoption at maturity is Co (Á, Á, T) = max[P(t, T) – K, 0].
As in Merton [1973], instead of the money marketaccount, we choose the Á-maturity bond as the numeraire.Deflated by the numeraire, all asset prices are martingalesunder the martingale measure, which we shall refer to asthe forward measure (as in Jamshidian [1989], Brace,Gatarek, and Musiela [1997], and others). Hence,
where EFt ( )is the conditional expecta-
tion under the forward measure.
C (¿;¿;T )oFE ( )t P (¿;¿)C (t;¿;T )o =P (t;¿)
2 32B(t+1;T ) 2¤ ¤¡ 2° B(t+ 1; T ) + °2®C(t+1;T )4 5®C(t+ 1; T )
1¡ 2®C(t+ 1; T )
¤C(t; T ) = ¸ B(t+ 1; T ) + ¯C(t+ 1; T )
B(t; T ) = ¹ B(t+ 1; T )¡ 11
1¡ log(1¡ 2®C(t+ 1; T ))2
A(t; T ) = A(t+ 1; T ) + ¹ B(t+ 1; T ) + !C(t+ 1; T )0
qP (t; T ) = exp(¡r )E [P (t+ 1; T )]t t
q= exp(¡r )E [exp(A(t+ 1; T ) +B(t+ 1; T )rt t+1t
+ C(t+ 1; T )h )]t+2
Noting that P(Á, Á) = 1, we have that
(18)
where x ∫ A(Á, T) + B(Á, T)rÁ + C(Á, T)hÁ + 1. Thus, inorder to calculate the option price, we have to know theconditional density of x under the forward measure, whichin turn is known if we know its corresponding charac-teristic function.
Let f(È) = EFt [exp(Èx)] denote the conditional
moment-generating function of x at time t under the for-ward measure. The function, f(È), also depends on thestate variables and the parameters of the model, althoughwe suppress them for notational convenience. We shallguess the exponential-affine functional form for f(È) as:
(19)
The coefficients A1(t; È, Á), B1(t; È, Á), and C1(t; È,Á) can be obtained from a boundary condition using arecursive procedure. The boundary conditions are:2
A1(Á; È, Á) = ÈA(Á, T) (20)
B1(Á; È, Á) = ÈB(Á, T) (21)
C1(Á; È, Á) = ÈC(Á, T) (22)
where A(Á, T), B(Á, T), and C(Á, T) are known from therecursions needed to calculate the price of the discountbond that expires at T, P(t, T).
Let A*(t + 1; È, Á) ∫ A1(t + 1; È, Á) + A(t + 1; Á);B*(t + 1; Á) ∫ B1(t + 1; È, Á) + B(t + 1; Á); and C*(t + 1;Á) ∫ C1(t + 1; È, Á) + C(t + 1; Á), where we already haveA(t + 1; Á), B(t + 1; Á), and C(t + 1; Á) from the recur-sion needed to compute the price of the discount bondthat expires at Á, P(t, Á). The recursions required for cal-culating f(È) from the boundary conditions are then givenas below (note that these are very similar to the recur-sions for bond prices). The derivations of these recur-sions are similar to those for the recursions for computingthe bond price, and can be found in Heston and Nandi[1999].
f(Á) = exp(A (t;Á; ¿) +B (t;Á; ¿)r1 1 t
C (t; ¿; T )o F= E (max[exp(x)¡K; 0])tP (t; ¿)
90 A TWO-FACTOR TERM STRUCTURE MODEL UNDER GARCH VOLATILITY JUNE 2003
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(23)
(24)
¥
(25)
At time t, now that we have A1 (t; È, Á), B1 (t; È,Á), and C1 (t; È, Á), we know the conditional moment-generating function of x at time t. If f(È) is the moment-generating function, f(iÈ) is the characteristic function.Inverting the characteristic function as in Heston andNandi [2000], one gets the price of the call option to be
¥
(26)
The derivation of (26) is shown in Heston and Nandi[1999].
By a simple rearrangement, we can also write Equa-tion (26) in the typical Black-Scholes format as Co (t; Á, T)= P(t, T)M1 ( ) – KP(t, Á )M2 ( ) where M1 ( ) and M2 ( ) aretwo probability distribution functions. The two univariateintegrals converge very quickly and are very easy to inte-grate numerically. They can be combined and evaluated asa single univariate integral in fractions of a second using anygood integration routine. We use the Romberg integrationroutine of Press et al. [1992] to generate option prices.
Although we do not have explicit representationsfor the prices of American options on puts, one couldcompute the early exercise premium from the contin-uous-time analogue of the nested one-factor model (theVasicek model) along the lines of Carr, Jarrow, and Myneni[1992], Huang, Subrahmanyam, and Yu [1996], Ju [1998],and others by using as the numeraire the price of the dis-
à " # !Z ¡iÁ log(K)11 1 e f(iÁ)¡ KP (t; ¿) + Re dÁ2 ¼ (iÁ)0
" #¡iÁ log(K)e f(iÁ+ 1)
Re dÁ(iÁ)
Z11 1
C (t; ¿; T ) = P (t; T ) + P (t; ¿)o2 ¼ 0
2 3¤ 2B (t+1;Á;¿) 2¤ ¤ ¤¡ 2° B (t+ 1;Á; ¿) + °¤2®C (t+1;Á;¿)4 5¤1¡ 2®C (t+ 1;Á; ¿)
¤+ ®C (t+ 1;Á; ¿)
¤ ¤ ¤C (t;Á; ¿) = ¡C(t; ¿) + ¸ B (t+ 1;Á; ¿) + ¯C (t+ 1;Á; ¿)1
¤B (t;Á; ¿) = ¡(B(t; ¿) + 1) + ¹ B (t+ 1;Á; ¿)1 1
1¤¡ log(1¡ 2®C (t+ 1;Á; ¿))
2¤!C (t+ 1;Á; ¿)
¤ ¤A (t;Á; ¿) = ¡A(t; ¿) +A (t+ 1;Á; ¿) + ¹ B (t+ 1;Á; ¿) +1 0count bond that matures at the same time as the option.This early exercise value can be added to the Europeanvalue to obtain an approximate American value.
Prices of other types of European interest rateoptions such as caps and floors can be calculated in thesame way as above by noting that these are portfolios ofoptions on discount bonds (see Hull [1999] for the analo-gies). Also, the value of an option on the average of aninterest rate process can be computed as in Bakshi andMadan [2000] as we have an affine model. Options oncoupon bonds cannot be evaluated through a straight-forward analytical procedure, however, contrary to theone-factor model (see Jamshidian [1989]).
IV. OPTIONS ON BOND FUTURES
Now we show how to calculate the value of a Euro-pean option on a discount bond future under our model.Let F(t, T1, T2) denote the current futures price for acontract on a discount bond that expires at T2; the futurescontract expires at T1. Suppose a call option is traded onthe futures contract, and the option expires at Á. Let Á <T1 < T2.
As with options on discount bonds, f(È) is themoment-generating function of the logarithm of thefutures price when the option matures and is given asfollows:
(27)
f(È) can be calculated recursively exactly as with optionson discount bonds, but with a slightly different boundaryor terminal condition as given below:
(28)
(29)
(30)
Note that the terminal coefficients, Af(Á, T1, T2),Bf(Á, T1, T2), and Cf(Á, T1, T2), are known from the recur-sions needed to calculate the futures price.
The characteristic function corresponding to f(È) isf(iÈ). Given f(iÈ), it can be shown that the price of theoption, Co(t; Á, T1, T2) is (see Heston and Nandi [1999]for the derivation):
C (t; ¿; T ; T )o 1 2
C (¿ ;Á; ¿) = ÁC (¿; T ; T )1 f 1 2
B (¿ ;Á; ¿) = ÁB (¿; T ; T )1 f 1 2
A (¿ ;Á; ¿) = ÁA (¿; T ; T )1 f 1 2
f(Á) = exp(A (t;Á; ¿) +B (t;Á; ¿)r + C (t;Á; ¿)h )1 1 t 1 t+1
JUNE 2003 THE JOURNAL OF FIXED INCOME 91
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(31)
V. ESTIMATION
How well does the model fit the observed termstructure? And can we account for strike price biases (suchas the smile or smirk in implied volatilities in some interestrate derivatives markets) from a one-factor model such asthe Black model after we have estimated the model param-eters? This exploration will help illustrate the computa-tional feasibility and efficiency of the model. Calibratingthe model to the actual term structure should also gen-erate some basic insights about the basic functionality ofthe model in the real world that may inspire more empir-ical research.
For the zero-coupon bond prices of various matu-rities, we use a modification of the Fisher, Nychka, andZervos [1995] method of constructing zero-coupon yieldcurves constructed from the daily Center for Research inSecurity Prices bond file data due to Waggoner [1996](seeBliss 3 [1997] for the details).3 The criterion function forparameter estimation is minimization of the sum of squarederrors between model and market zero-coupon bondprices for a sample of two weeks in a period spanning1990-1996.4
Specifically, we minimize the criterion function:
(32)
where Pi,t and Mi,t are, respectively, the model and marketprices for bond i on date t. For the short rate, we use thecontinuously compounded overnight rate implicit in thethree month T-bill prices.
Note that in computing this criterion function, oneneeds to know ht+1, which is the conditional variance ofthe rt+1, at time t. For each t, ht+1 is known from the his-tory of the short rate until time t, and need not be esti-mated as a parameter. In continuous-time stochasticvolatility models, however, one would have to estimatethe variance as a separate parameter, making the estima-tion procedure very cumbersome. Given our sample, wewould have to estimate ten extra parameters (for the volatil-ities on ten separate trading days in two weeks) if we wereusing a continuous-time stochastic volatility model.
NT tXX2min (P ¡M )i;t i;t
¹ ;¹ ;!;®;¯;°;¸;´0 1 t=1 i=1
à " # !!Z ¡iÁ log(K)11 1 e f(iÁ)¡ K + Re dÁ2 ¼ (iÁ)0
à " #Z ¡iÁ log(K)11 1 e f(iÁ+ 1)= P (t; ¿) F (t;T ; T ) + Re dÁ1 2
2 ¼ (iÁ)0
The parameter estimates obtained by fitting themodel to market prices for two weeks starting January 3,1994, are as follows: µ0 = 2.13 ¥ 10–8, µ1 = 0.999, ‚ =–3.36, v = 15.58, w = 1.44 ¥ 10–11, Ÿ = 0.256, ÿ = 1.093¥ 10–11, and ⁄ = 12.68 (note that the parameter estimatesare based on the values of a daily (not annualized) yield).According to these parameter estimates, the long-runmean of the annualized volatility of the changes (not thelevels) of the annualized short rate is around 0.041, andthe half life of the volatility process is around 185 days.
Exhibit 1 shows the model yields for various matu-rities on January 3, 1994. Now, if we fit the nested one-factor model (the discrete-time version of the Vasicekmodel) to the same sample, we find that the average abso-lute pricing error is only a little higher than the two-factor model (around 2 basis points).5
A statistical test such as a likelihood-ratio test testingthe nested one-factor model against the two-factor modelcannot reject the one-factor model (see Judge et al.[1988]). In other words, the two-factor model is only atrivial improvement if fitting the entire yield curve overa given period of time is the criterion function.6
In the options markets, volatility is expected to beof much more importance. Using the parameters estimatedfrom the cross-section of bond prices (i.e., from the yieldcurve calibration), we generate prices of options on dis-count bonds of different strike prices under our two-factormodel. Then, using the Black [1976] model, a widely usedone-factor model for pricing these options in the mar-ketplace (see Hull [1998]), we back out the impliedvolatility of different strike prices for an option (with 60
92 A TWO-FACTOR TERM STRUCTURE MODEL UNDER GARCH VOLATILITY JUNE 2003
Maturity (in days) Market Yield Model Yield
90 3.13 3.23
180 3.3 3.36
270 3.46 3.49
360 3.67 3.62
730 4.26 4.11
1095 4.65 4.53
1460 5.04 4.9
3650 6.09 5.95
E X H I B I T 1Market and Model Yields
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days to maturity) on a two-year zero-coupon bond. Exhibit 2 shows that the implied volatilities are
reduced as the strike price increases; the implied volatil-ities display a pronounced smirk or skew, a feature oftenobserved in the bond options market, but the skew fromthe nested one-factor Gaussian model is much more flat.7
Thus our results suggest that volatility as a secondfactor is important primarily for bond options, rather thanfor spot bonds or bond futures.
VI. CONCLUSION
We have developed a discrete-time two-factor modelof interest rates, in which the second factor is a time-varying volatility following a GARCH process that canbe correlated with the level of the short rate. The modelcan be used to price spot bonds, bond futures, and bondoptions using easily computable analytical solutions.Unlike continuous-time two-factor models with randomvolatility, volatility in our model is observable on the basis
of the history of interest rates. This gives our model a dis-tinct computational advantage in terms of estimation andimplementation over continuous-time stochastic volatilitymodels of interest rates.
We find that the second factor (volatility) does notmatter as much for valuing spot bonds as for valuing bondoptions. In fact, we cannot reject the nested one-factorversion of the model for the entire cross-section of spotbonds on the basis of parameters estimated from the U.S.Treasury yield curve. Option values from the two- factormodel produce a much more pronounced smirk or skewin implied volatilities (using the one-factor Black model)for bond options than for option values from the one-factor model. This suggests that the effects of randomvolatilities of interest rates would be visible mostly in bondoption prices rather than in bond prices.
JUNE 2003 THE JOURNAL OF FIXED INCOME 93
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.94 0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 1.03
Moneyness
Imp
lied
Vo
lati
lity
(Bla
ck)
E X H I B I T 2Implied Volatility Smile
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APPENDIX
Bond Valuation Equation
Here we show how to derive Equations (15), (16),and (17), the coefficients needed for bond pricing. Sub-stituting the dynamics of rt+1 and ht+1 in (14):
Completing the square in the portion to which theexpectation applies, using the fact that for a standardnormal z, E(a(z+b)2) = exp(– 1⁄2 log(1 – 2a) + ab2⁄1-2a) andmatching the coefficients on rt,ht+1, and the constant, weget (15), (16), and (17).
ENDNOTES
The views expressed here are those of the authors andnot those of Fannie Mae.
1Note that we can easily drop ‚ from Equation (1) andstill generate all our results. Similarly, we can include r in theright-hand side of Equation (2) to reflect that the level of thevariance depends on the level of interest rates, and still get theanalytical solutions.
2The boundary conditions can be derived as in Heston[1993] or Heston and Nandi [2000].
3We thank Daniel Waggoner for constructing the zero-coupon yield curves.
4The data for daily coupon bond prices that we haveaccess to ended in 1996.
5The discount bond prices for the one-factor or the con-stant-volatility model are easily derived by noting that zt
* = zt= fi, instead of zt
* = zt – fi �ht+1�� as in the two-factor model.6When we repeat this estimation exercise for other ran-
domly chosen two-week periods between 1990 and 1996, ourconclusions regarding the mispricing remain essentiallyunchanged, although the parameter estimates are somewhatdifferent, depending on the period.
7It is possible that a different one-factor model such as thatof CIR [1985] would also exhibit a smirk or smile because the
E exp(B(t+ 1; T ) h zt+1 t+1t ¸q 2q ¤+ ®C(t+ 1; T )(z ¡ ° h ) )t+1t+1
= exp(¡r +A(t+ 1; T )t
¤+ B(t+ 1; T )(¹ + ¹ r + ¸ h )0 1 t t+1
+ C(t+ 1; T )(! + ¯h )):t+1
exp(A(t; T ) +B(t; T )r + C(t; T )h )t t+1
Black formula is non-linear in volatility for out-of-the-moneyoptions.
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