for Lj(t).Through changes of probability measures, which will be discussed inmore details in the next section, we can express the LIBOR processes ina common measure of PTM, where M = i, ( N ) . In other words,
dLi ( t ) = (...)dt + Ai (t)Li ( t ) dTTMILi (0 I t Iq),
(7.66)
where W,TM,Lf is a one-dimensional PTM-Brownian motion for Lj(t).Again, we are not concerned with the drift term.Thus, the forward swap rate in Eq. (7.61) can be re-written as
dsi,N(t)=(...)dl+
i C ( W
as,
-Aj(t)Lj(t)dW,TMyLj.j = i c ( i ) aLj
(7.67)
Consequently, the incremental variance-covariance matrix of the forwardswap rate is given by
dSi,N(t)dsit,Nt( t )=
where A4 = max (i, ( N ),i, ( N ' ) ).Let us now find the approximated swap rate volatilities. First, notethat
where (X,Y)denotes the instantaneous correlation between X and Ydefined as'"
lo' Since the instantaneous correlation is measure invariant, so we do not need to specifythe measure under which the expected value is computed.
278
The Interest Rate Market Model
Therefore, the instantaneous normal volatility A$"
Chapter 7
( I ) of the forward
swap rate Si,N(t)is given by
The above equation can be further simplified and approximated by setting L~( t )= L~(0).Therefore, the normal term volatilityswap rate Si,N(t)is given by
&F( t , T ) of the forward
This normal term volatility can be further simplified and approximatedby setting Li (s) = Li ( t ) .The percentage term volatility or the Black volatility
q,,,( t , T ) can
be approximately obtained by
Such term volatility can be used in the Black's formula (Eq. (7.52))for approximately simultaneous consistent pricing of derivatives in capand swap markets. This process is commonly used in the model calibration process to back out the approximated model parameters as the initialguesses for further calibrations as a speed improvement measure.
Sec. 7.6
7.6.2
Overview of Simultaneous and Globally Consistent Pricing and Hedging 279
More on Simultaneous Consistent Pricing
What we have discussed are simple cases for simultaneous consistentpricing of the current spot starting capdfloors and European swaptions.However, the consistency issue can be brought to a bigger scope that includes many additional and more complicated and challenging cases,which we shall discuss below.The first additional case is for the consistent pricing and hedging ofthe current trades, the aged trades, and the future or proposed trades.Typically, the BGM model is calibrated to the current spot startingcaps/floors and European swaptions, as they are more liquid than otheroptions and the reset and the accrual dates for the LIBORs modeled bythe BGM model (approximately) match those of the LIBORs underlyingthe caps/floors and European swaptions. However, the aged trades andthe future or proposed trades can depend on LIBORs with the reset andthe accrual dates very different from what are model by the BGM model.Of course, one can perform interpolations to derive these LIBORs.However, as we pointed out before, interpolations do not automaticallyguarantee the martingale relationships required by the arbitrage pricing.One actually needs the martingale interpolations and Brownian bridgeinterpolations presented in Chapter 4 to achieve the desired martingalerelationships.The second additional case is for the consistent modeling of the interest rate volatility cube. The interest rate implied Black volatilities arefunctions of three variables, i.e., the expiry, term, and strike (or the deltabetween the strike and the break-even forward) of the corresponding options (i.e., caps/floors and European swaptions), hence the name volatility cube. The volatility skew and smile, i.e., the dependency of the volatility on the strike (or the delta between the strike and the break-evenforward), have become increasingly important. The SABR model hasbecome a standard in modeling the volatility skew and smile. While theSABR model can consistently model the volatility skew and smile acrossdifferent strikes at a given expiry and term, it is not consistent across different expiries or terms. This is because that the SABR model treats theunderlying forward LIBOR or swap rate at each expiry and term with aseparate martingale model. In other words, the SABR model is an extension of the Black model, but not a term structure model.A feasible approach to address this inconsistency is to use a termstructure SABR LIBOR market model, i.e., a LIBOR market model withstochastic volatility parameters similar to those of the SABR model and
280
The Interest Rate Market Model
Chapter 7
globally calibrate this model to most of the points on the volatility cube.Monte Carlo simulation with early exercise boundary optimization,Monte Carlo simulation with state stratification, the least squares MonteCarlo simulation, and the nonexploding bushy tree (NBT), all enhancedwith various martingale resampling and interpolation techniques (presented in Chapter 4), are promising numerical techniques for implementing the multi-factor term structure SABR LIBOR market model.A major advantage of the BGM models, particularly the term structure SABR LIBOR market model, as compared with other models, is thatthey can calibrate not only to the entire ATM volatility surface exactly (ifone so desires), but also to most of the points on the volatility cube verywell as a best fit.Other more subtle cases involve the handling various un-hedgeableor difficult-to-hedge parameters, such as correlation (e.g., among interestrates of different terms and different underlying instrument prices orvariables), forward volatilities and forward volatility skews and smiles.These typically require multi-factor models and the secondary modelcalibration that we discussed earlier.As we discussed in Section 2.4, there have been many exotic interestrate derivatives instruments (with the above subtleties) developed forsatisfying the needs of various market participants for, e.g., asset management and liability management to achieve the desired payoffs and riskprofiles (for, e.g., coupon enhancement, trade cheapening, exploring relative value opportunities, and hedging). A lot of these derivatives tradesare motivated by the issuance of structured notes that have these exoticfeatures embedded in the coupon. Typically, a note issuer needs to swapthe exotic coupon with a dealer (unless the issuer itself is a dealer) byreceiving the exotic coupon and paying, e.g., LIBOR plus a fundingspread'02 and, if the note is callable, selling the embedded Bermudan option to the dealer (i.e., making the swap cancelable by the dealer).This essentially allows the note issuer to hedge away almost all therisks associated with the exotic coupon (such as the risks in volatility,volatility skew and smile, correlation, etc.) and the interest rate volatilitylo* The funding spread is explicitly or implicitly determined by the credit quality (or thecredit spread) of the issuer and some liquidity factors. For highly credit rated issuers withliquid issues, the funding spread can actually be negative. This is one way to achievesub-LIBOR funding. For lower credit rated issuers, the funding spread is positive ingeneral, but can vary depending the types of the notes issued. So an issuer has an optionto issue a note with lower funding spread or funding cost.
Sec. 7.6
Overview of Simultaneous and Globally Consistent Pricing and Hedging 281
risk in the callable feature. This also helps the note issuer in terms ofoperations or making the call decisions. Essentially, when the dealercancels the swap is when the note issuer should call the note. The exception is when the dealer does not exercise optimally. It is interesting forthe reader to think about what strategies the note issuer can employ topossibly make a profit should the dealer not exercise the cancel optionoptimally.As usual, the swap transaction involves the counterparty credit risks,which affect both the price and the hedges of the swap. A AAA ratedissuer typically desires to swap with the AAA rated subsidiary of adealer, which is typically A rated.The callable note issuers are major suppliers of the Bermudan optionvolatilities. It is important to point out that by entering a generic interestrate cancelable swap with a dealer, the issuer can hedge away the risk inthe generic interest rate and its volatility, but issuer does not properlyhedge its funding spread or credit spread risk and the volatility of thecredit spread, especially for lower credit rated issuers. For instance, if alower credit rated issuer issues a simple fixed-coupon callable note, itmay want to call the note when its funding rate or par coupon (which isan aggregation of the generic interest rate, such as LIBOR, and its creditspread) declines and issue another one with lower funding rate or parcoupon.Let’s perform a simple scenario analysis. More specifically, let’sconsider the scenario whereby the decline of the finding rate or par coupon can be accompanied by a significant decline of the credit spread anda less significant increase of the generic interest rate. In this case, theissuer may want to call the note, but the dealer with the cancelable generic interest rate swap may not want to cancel the swap. It is interestingfor the reader to think more about this and about what the issuer can doshould this situation happen and what types of trades the issuer can enterinto to prevent this from happening,More specific examples of such exotic derivatives instruments (oftenwith embedded caps/floors) include the following; callable fixed-couponnote or swap, callable inverse floater note or swap, callable or noncallable range accrual note or swap, callable or non-callable zero-couponnote or swap, callable or non-callable CMS spread note or swap, andcallable or non-callable PRDC (Power Reverse Dual Currency) note orswap. All these trades can also have the snow ball feature and/or targetredemption feature embedded in their coupons. The target redemption
282
The Interest Rate Murket Model
Chapter 7
notes (TARN) have been particularly popular in recent years.The counterparty credit risks involved in these trades, includingcredit mediation, naturally make them credit contingent (or defaultable)or credit hybrid trades. In recent years, explicit credit contingency hasbeen applied to some of these trades, such as the zero-recovery swap orcredit extinguisher in general whereby the trade is terminated at zerovalue or with no recoveries if either trading party is default, and the(callable) real rate swap whereby the floating rate referenced is not thegeneric interest rate, but rather approximately the real funding rate of,e.g., the note issuer. As we pointed out previously, the counterpartycredit risk can have significant impact on the price and hedge ratios of aderivatives instrument. For instance, if one were to trade with a muchlower credit rated counterparty, then in most cases, one’s PV of the tradeis reduced, the absolute value of one’s delta hedge ratio is also reduced, alinear instrument can now have a Vega, an IR instrument can now have asensitivity to the credit spread, and there are also cross hedge ratios, alldue to the counterparty credit risk.Most of these derivatives instruments are highly path-dependent andare sensitive to many (spot and forward) volatility skew and smile pointsand (spot and forward) correlation among various variables (such as forward LIBORs). Many of these variables are not hedgeable (or cannot beeasily hedged with low cost), thus the price of these instruments can dependent on factors that are not handled by the model, such as risk preference, supply and demand, and, therefore the secondary model calibrationis needed, as we discussed previously. Consequently, their valuationsare extremely challenging. Even more challenging are the tasks of accurately computing the sensitivities or hedge ratios.Multi-factor models capable of handling volatility skews and smilesglobally, such as multi-factor term structure SABR LIBOR marketmodels, are good model candidates for the valuation of these derivativesinstruments. As we discussed earlier, Monte Carlo simulation with earlyexercise boundary optimization, Monte Carlo simulation with state stratification, the least squares Monte Carlo simulation, and the nonexplodingbushy tree (NBT), all enhanced with various martingale resampling andinterpolation techniques (presented in Chapter 4), are promising numerical techniques for implementing the multi-factor term structure SABRLIBOR market model.Needless to say, one needs to properly handle the counterparty creditrisk and the secondary model calibration. The latter often requires using
Sec. 7.7 More on the Martingale or Full-dimensional LIBOR Market Model
283
slightly different models for hedging and for pricing. In other words, oneoften needs to price low (e.g., with a low forward volatility) and hedgehigh (e.g., with a high forward volatility) and vice versa, depending onwhether one is long or short the un-hedgeable variables (such as the forward volatility).
7.7
More on the Martingale or Full-dimensional LIBOR MarketModel
The martingale or full-dimensional LIBOR market model presented inthe previous sections is specified by Brownian motions under differentarbitrage-free measures. For general derivatives pricing purposes, models with Brownian motions under the same measure are more convenient.A natural choice of this common measure is the forward arbitrage-freemeasure IPTN for the longest maturity LIBOR LN-,(t)(up to the paymentdate TN). Equivalently, we can also choose the spot arbitrage-free measure Q . The PV of a derivatives instrument does not depend on whichprobability measure we choose.To achieve this purpose, changes of measures are needed. With Radon-Nikodgm derivative dPT /dPT+l, the following change of measurerelationship can be obtained
Define the change of measure multiplier
then the above change of measure relationship can be reduced to
E;
[x,]= Ei+1[ x,rn;+, ( t , T ) ]
(t<T I T ) .
(7.76)
It is also easy to see that
E:" [mi,, ( t , T ) ]= 1
( t < T Iq ) .
(7.77)
The Interest Rate Market Model
284
Chapter 7
An intuitive way of understanding the change of probability measureis as follows. If XT is the price of a simple traded asset (with no couponand dividend payments) and since its price does not depend on whichprobability measure we use, then from the martingale relationship, wehave
X,= P(t,l;)Ef [ X , / P ( T , q ) ]
[
= p(l7 q+1)Ei+' X T / ~T (,q+1)]*
(7.78)
(t<TIq)Thus,m;+l ( t ,T ) =
p ( T , ~ ) / P ( T , ? + , )- 1+ SjLj ( T )p ( t , q ) / p (t , q+l) 1+ SjLj ( t )(tlTlQ
(7.79)
The above change of measure takes care of derivatives pricing (i.e., theexpected value calculation) under different measures.Another aspect of change of measure is that the same stochasticprocess takes different forms under different measures. One way handlethis is to apply Girsanov's change of measure theorem and we shall discuss the details in Section 7.13.3.Here we present a simple approach of formulating the martingale orfull-dimensional LIBOR market model under the common probabilitymeasure of PTN.Recognizing that, in general, Li(t) is not a PT,-martingale, we startby parameterizing the LIBOR process under PTN as follows, i.e.,dLi ( t ) = pidt + uidYTN'Lt
( 0 I t I qR),
(7.80)
where YTN9Liis a one-dimensional PTN-Brownian motion for Li(t). Thekey, as usual, is to determine the drift term p j . Here we focus on thediffusion process. We can also extend to jump processes.Recognizing that
Sec. 7.7
More on the Marfingale or Full-DimensionalLIBOR Market Model
285
(7.81)
n + SiLiN-1
=
(1
(t))
is a PrN -martingale and thus
:!i,?j)),
+(l+SiLi(t))d[+SidLi(t) d
(tI?R,i<N-l
1
must have zero drift, thus we have (by plugging in Eq. (7.80))
Therefore,
(7.82)
The Interest Rate Market Model
286
Chapter 7
.
(7.84)
dt(t q R J I N - 1 )
The above equation can be further re-written as
N-l
S Covar, (dLi( t) ,dLj ( t ))
i=i+l&
1+6,Li(t)
V
J
(7.85)
dt
=-cN-1
oiojSjCo"; (dL,(t),dLj( t ) )
j=i+l
1 + 6 j L j( t )
dt
( t I q R , i I N - 1)
Therefore, the above LIBOR process can be rewritten as
dL, ( t ) = -
N-1
OiOjSj
j=i+l1+ 6jLj ( t )
+oidw,'"
I
Li
Con, (dL,(t),dL,(t))dt
.
(7.86)
This is the martingale or full-dimensional LIBOR market model in ageneral form and under the same probability measure ( PTN). The instantaneous correlation among all LIBORs, Con, (dLi ( t ) , dLj ( t ) ) , is an in-
Sec. 7.8
Modeling Interest Rate Volatiliv Skew and Smile
287
put to the model. As we pointed out before, the volatility oi can be timedependent, state dependent, or stochastic.
7.8
Modeling Interest Rate Volatility Skew and Smile
The volatility skew and smile both refer to the strike dependency of theimplied Black volatility that is widely observed in various financial markets, such as the equity, FX, and fixed-income markets.Roughly speaking, the volatility skew refers to the monotonic strikedependency of the implied Black volatility (e.g., higher implied volatilityat lower strike), whereas the volatility smile also refers to the strike dependency of the implied Black volatility, but it is a non-monotonic dependency (e.g., higher implied volatility at both lower and higherstrikes). In other words, the volatility skew and smile differ in the functional form of their strike dependency of the implied Black volatility.Often observed are some combinations of the volatility skew and smile.The volatility skew usually is a more dominant effect than the volatilitysmile. Figure 7.2 shows an illustrative diagram for the volatility skew(the solid line) and a combination of the volatility skew and smile (thedashed line).
50%
100%
150%
200%
Percentage of ATY SWke
Figure 7.2. Illustrative diagram for the volatility skew (the solid line) and a combination of the volatility skew and smile (the dashed line).
Here we start the CEV (Constant Elasticity of Variance) model formodeling the volatility skews. While the CEV (and shifted lognormal)BGM models can model the volatility skews, they cannot model thevolatility smiles. By making the volatility in the BGM model stochastic,one can model a large range of volatility smiles, which we shall discuss
288
The Interest Rate Market Model
Chapter 7
in Section 7.11.
7.8.1
CEV and LCEV Models for Modeling the Volatility Skew
The CEV (Constant Elasticity of Variance) model was first proposed byCox and Ross (1976) for modeling the volatility skew of equities. Intheir model, the process of the price S of a stock paying no dividend, under the spot arbitrage-free measure Q , is given by
dS = rSdt + o S q d K ,
(7.87)
where, as usual, r is the instantaneous continuous compounding shortinterest rate, W,is a one-dimensional Q -Brownian motion, q is the CEVpower, and o is state-independent and non-stochastic.If q=l, then the above process reduces to a regular lognormal processwith CT being the percentage volatility. If 40' , then the above processreduces to a regular Gaussian process. The market implied q is normallybetween 0 and 1.The CEV volatility structure has also been applied to the BGM/Jmodel (Andersen and Andreasen, 1998, Hull and White, 1999, and Blythand Uglum, 1999).As usual, we rewrite the LIBOR volatility term in the following form
o(L;)=A;(t)L;(t),
(7.88)
where Ai(t) is the instantaneous percentage volatility of the LIBOR L;(t).If Ai(t) is state-independent and non-stochastic, it is the regular lognormalvolatility structure.The CEV volatility, on the other hand, has the following state dependence, i.e.,(7.89)where q, as before, is the CEV power. The presence of Li(0) in the aboveequation is for the normalization so that 4' ( t ) can be a certain stateindependent percentage volatility. The state-dependency here is on theoverall volatility, One can also specify different state dependency for thevolatility of each individual factor, which is one way to produce statedependent correlation.
Sec. 7.8
Modeling Interest Rate Volatility Skew and Smile
289
Depending on the value of the CEV power q, the CEV volatilitystructure is known to result in diverging and non-unique solutions. Tocircumvent these technical difficulties, Andersen and Andreasen (1998)proposed a limited CEV (LCEV) volatility structure, which is equivalently given by(7.90)where M i s a positive constant.The basic idea of LCEV is to have an upper bound set for the percentage volatility. As such, the LCEV volatility structure satisfies thetechnical conditions (Lee, local Lipschitz continuity and growth condition), and thus ensures bounded and unique solution. The exact value ofA4 is not critical. M=5 or greater was found to be satisfactory.On good feature of CEV and LCEV models is that closed-form solutions and/or approximations are available for simple European options,including caplets/floorlets and European swaptions (Schroder, 1989, Andersen and Andreasen, 1998).Another good feature is that a simple approximation for the impliedpercentage volatility as a function of the strike R and the CEV power q isavailable (Blacher, 1998, Blyth and Uglum, 1999). Applying this approximation to the implied percentage LIBOR volatility ( t ,q ,R ) (asused in Eq. (7.3 I)), we obtain
7
(7.91)
qTM
~ ~ is the at-the-money (ATM) strike and(t,?) iswhere R A(=L;(O))the implied percentage volatility for ATM strike.The above approximation can be used to determine the CEV powerq. The basic procedure is as follows.Obtain the implied percentage LIBOR volatility /zi ( t ,q , R ) from themarket quoted cap/floor volatilities as a function of the strike R.Perform a regression on ~n (t,q,R ) / -ATMA ~ ( t ,q )) versus
(4
h ( R / R A T M (enforcing)zero intercept) and the slope is (q-1)/2.It is important to emphasize the differences between the impliedvolatility and the instantaneous (or local) volatility. The implied volatility is a term volatility (a certain average volatility) and is normally used
290
The Interest Rate Market Model
Chapter 7
to back out the market quoted prices, from which the instantaneous (orlocal) volatility can be determined through calibration processes. Theimplied volatility may not be directly used in an SDE for modeling,whereas the instantaneous (or local) volatility is exactly what is neededin an SDE for modeling. In terms of the CEV and LCEV models, theimplied percentage volatility and instantaneous percentage volatility appear to have similar functional forms. Subtle differences are that the former is a function of the strike and the latter is a function of the statevariable, and the former has an exponent of (q-1)/2 and the latter has anexponent of q- I.7.8.2
Examples of Volatility Skew for Caplets and Swaptions
In this section, we will show some numerical results for the volatilityskew for caplets and European swaptions, as well as, the effect of thevolatility skew on Bermudan swaptions. These results were obtained bya 3-factor LCEV BGWJ model (Jamshidian’s approach) implementedwith the nonexploding bushy tree (NBT) technique. The detailed numerical implementation will be discussed in Section 7.9.The results were obtained with 5% initial yield and 19.2% total ATMinstantaneous volatility, i.e.,
Li (0) = 5%
)I
12.1; ( t = 19.2%
(7.92)
Also used was the estimated instantaneous (historical) market correlation(between Li(t)and L,(t) ) given by Rebonato (1999) as follows.
Figure 7.3 shows the volatility skew for the last caplet in a 5 yearquarterly cap. The points were obtained by the 3-factor LCEV BGWJmodel and the solid lines by the approximation according to Eq. (7.91).From the figure, the approximation appears to be quite good.Hull and White (1999) show that, for the USD cap/floor market, q0.7 based on all available caplets/floorlets. Andersen and Andreasen(1998) show that, for the JPY cap/floor market, q 0.6. In recent years,and in the USD market, in particular, q became much smaller due to thelower interest rates.
-
-
Sec. 7.8
291
Modeling Interest Rate Volatiliv Skew and Smile
300h
+ q=18 q=0.75
A q4.5
q=0.25x q=o
10%
50%
100%
150%
200%
Percentage of ATM Strike
Figure 7.3. Volatility skew for the last caplet in a 5 year quarterly cap. Thepoints were obtained by the 3-factor LCEV BGWJ model and the solid lines bythe approximation according to Eq. (7.91).
Figure 7.4. shows the volatility skew for a 5 year x 5 year Europeanpayer swaption with, for simplicity, quarterly swap.'03 The points wereobtained by the 3-factor LCEV BGM/J model implemented with theNBT technique and the solid lines by the approximation according to Eq.(7.91). From the figure, the results from the approximation and the NBTappear to be consistent.With the current simplified setting, the CEV power for caplets/floorlets is almost the same as that for European swaptions, i.e., qcapq~,,,ap,ion.In the real market, the CEV power each caplet and Europeanswaption are all different, which can be handled SABR like models,which we shall discuss more later on.
-
103A European swaption is usually referred to as an NXM swaption with N denoting theoption expiry and the swap start, and M the term of the underlying swap. Such swap isalso referred to as an NxM swap (or forward starting swap). For instance, a 5x10 swaption denotes a swaption with the expiry date in 5 years and an underlying 5x10 swap(with the start date in 5 years and the end date in 10 year relative to the start).
292
The Interest Rate Market Model
300h
10%
Chapter 7
A
I
50%
100%
150%
200%
Percentage of ATM Strike
Figure 7.4. Volatility skew for a 5 year X 5 year European payer swaption withquarterly swap. The points were obtained by the 3-factor LCEV BGM/J modeland the solid lines by the approximation according to Eq. (7.91).
7.9
The Nonexploding Bushy Tree Technique104
Implementing a LIBOR market model (which we shall also term it as theBGM/J model), even a multi-factor one with stochastic volatilities, withMonte Carlo simulation is not difficult. However, one needs to use various martingale resampling and interpolation techniques (presented inChapter 4) together with the simulation. Otherwise, it is difficult toachieve the desired convergence or computation speed, especially for amulti-factor model with stochastic volatilities. These martingale resampling techniques can also be used as preliminary or incremental modelcalibrations and can significantly improve the calibration speed withoutthe need of closed-form solutions or approximations. This is quite important, as one is no longer limited to special models just because theyhave closed-form solutions or approximations.As we know, a straight forward Monte Carlo simulation does notprovide conditional expectations and, thus, in general, cannot accuratelyprice derivatives instruments whose values depend on the conditionalexpectations especially when the volatilities are stochastic. These derivatives instruments are mainly the callable instruments, which typically‘04 This section includes, with more details added, a reprint of the paper entitled “A Nonexploding Bushy Technique and Its Application to the Multifactor Interest MarketModel” (Tang and Lange, 2001) published in the Journal of Computational Finance (Vol.4, NO.4, pp 5-31).
Sec. 7.9
The Nonexploding Bushy Tree Technique
293
contain highly compounded options, and, to a lesser extent, other compound options and forward starting options. Various other techniques forestimating the conditional expectations or the exercise boundaries, suchas closed-form solutions or approximations, exercise boundary optimization, state stratification, least squares regression, need to be combinedwith the Monte Carlo simulation for pricing these derivatives instruments.Alternatively, one can consider the lattice or tree based techniques.The first challenge is that the BGWJ models in general (and the generalHJM model) are non-Markovian with respect to the underlying drivingfactors (but can be Markovian with respect to all state variables) and,especially for multi-factor models, cannot be solved with the recombining lattice or tree techniques. The conventional bushy tree, while capable of solving non-Markovian models in principle, suffers from a severepractical limitation - its computation time grows exponentially as thenumber of tree time steps increases (see Heath, Jarrow, and Morton,1990 and Heath, Jarrow, Morton, and Spindel, 1993 for applications ofthe bushy tree technique to the HJM model). A 2-factor conventionalbushy tree is typically limited to 18 tree steps. This renders the conventional bushy tree technique inaccurate for modeling derivatives withmany event monitoring dates, such as cash flow dates, early exercisedates (for American and Bermudan style derivatives), and barrier monitoring dates.In this section, we present a multi-factor nonexploding bushy tree(NBT) te~hnique,''~which breaks the computation time barrier of theconventional bushy tree, and easily allows over 100 tree steps. A nonexploding bushy tree typically consists of a number of small sub-bushy treeunits with subsequent sub-bushy trees grown only from selected nodes orstates of the preceding