bushy trees. Such selected states can be themodes or sub-samplings of the state distribution of the preceding bushytrees.A 3-factor BGWJ LIBOR model has been implemented with thenonexploding bushy tree technique, and accurate results have been obtained for caps/floors, and the European and Bermudan swaption resultsare comparable with those of other techniques. The effects of the number of factors (up to 5 factors) on the Bermudan swaption prices wereobtained through preliminary analyses. Sufficient conditions of converInsThe initial version of the nonexploding bushy tree (NBT) technique was first reportedby Yi Tang and Bin Li at a RISK workshop on February 25, 1999.
294
The Interest Rate Market Model
Chapter 7
gence are also provided.7.9.1
Construction of a Nonexploding Busby Tree
The construction of a conventional bushy tree is to first construct asmaller sub-bushy tree, then from each of the terminal states (or nodes)of the previously constructed sub-bushy tree(s) construct additional subbushy trees, and repeat this process until the tree reaches the desiredsteps, typically out to the maturity or expiration date of the derivatives.The construction of a nonexploding bushy tree is to have the subbushy trees grown only from selected “growth” states (or nodes) of thepreviously constructed sub-bushy tree(s), and repeat this process until thetree reaches the desired steps. This is illustrated in Figure 7.5.
Figure 7.5. Examples of the construction of a conventional bushy tree (a) and anonexploding bushy tree (b). In the former case, additional sub-bushy trees (B)are grown from each of the terminal node or state of the previously constructedsub-bushy tree (A). In the latter case, additional sub-bushy trees (B) are grownonly from selected “growth” nodes or states ( ) of the previously constructedsub-bushy tree (A).
The growth states (or nodes) in a nonexploding bushy tree can bechosen to be the centers (or a few states near the centers) of the modestructures or peaks in the state distributions of the corresponding conventional bushy tree, or to be some sub-samplings of the state distributionsof the corresponding conventional bushy tree, if no clear mode structuresare present. These approaches can reduce the number of tree nodes bymany orders of magnitude, and hence break the computation time barrierof the conventional bushy tree technique. Essentially, the nonexplodingbushy tree technique is actually the sub-sampling of a conventionalbushy tree, but with no state stratification or bundling.The inception of the nonexploding bushy tree technique was inspired
Sec. 7.9
The Nonexploding Bushy Tree Technique
295
by the fact that even though the tree branches in a bushy tree do not recombine, they actually tend to cluster and consequently the bushy tree isnot dramatically different from a recombining tree. Such clusteringtends to result in peaks in the state distributions or the histogram of thetree branches, which we shall term as the mode structures. Particularly,in the 1-factor case, the bushy tree for any given LIBOR under its forward measure can even be reduced to a recombining trinomial lattice(see, e.g., Hull and White, 1994, for the trinomial lattice implementationfor pricing interest rate derivatives).Figure 7.6 shows a histogram of the tree branches of the 5th (quarterly) LIBOR obtained with a 3-factor lognormal BGWJ model implemented by a conventional bushy tree (with 3 tree steps and 20%, 4%, 2%volatilities for the 3 (independent) factors, and a flat initial yield curve).Mode structures (or peaks in the histogram) can be clearly seen.As an example, a nonexploding bushy tree can be constructed bygrowing additional 3 step sub-bushy trees only from the centers of themode structures (marked by Ir in Figure 7.6). In other words, at thisstage, only 7 sub-bushy trees are grown from the 343 (=73) availablestates, which amounts to significant reductions in the number of treestates.
-4
-3.5
-3
-2.5
-2
In(LIBOR)
Figure 7.6. The histogram of the tree branches of a 3-factor lognormal BGM/Jmodel from a conventional bushy tree (with 3 tree steps). Mode structures (orpeaks in the histogram) can be clearly seen. As an example, a nonexplodingbushy tree can be constructed by growing additional 3 step bushy trees only fromthe centers of the mode structures (marked by A ). 1-factor models generally exhibit narrower mode structures.
296
The Interest Rate Market Model
Chapter 7
For the states (or nodes) where no sub-bushy trees are grown, theirconditional expectations (or values to continue in a Bermudan option orvalues of volatility sensitive underlying instruments) are obtainedthrough interpolations of the conditional expectations on the growthstates. The intrinsic values (or values to exercise a Bermudan option orvalues of volatility insensitive underlying instruments) on all the stateson a nonexploding bushy tree and all the information of the paths leadingto these states are readily available. For better accuracy, more growthnodes per mode structure can be used.It is important to point out that the above processes are performed oneach LIBOR (or state variable) and there is no dimensionality reductionand no state stratification or bundling. Even though the inception of thenonexploding bushy tree technique was inspired by the presence of themode structures, the key idea of the nonexploding bushy tree technique isactually the sub-sampling of a conventional bushy tree. The presence ofthe mode structures can help the convergence and the computation speedof a nonexploding bushy tree, but it is actually not a necessary condition.Sufficient conditions of convergence are provided in the Appendix7.13.2.The number of float operations or computations of a conventionalbushy tree (with n factors and m total tree steps) is proportional to itstotal number of tree nodes that is given by ( n + 1)” , which explodes exponentially, and hence comes the term “dimensionality curse”.Similarly, the memory requirement and the number of float operations or computations of a nonexploding bushy tree are essentially proportional to the total number of nodes in all its sub-bushy trees. For anonexploding bushy tree with n factors, m total tree steps, m’ steps ineach sub-bushy tree, 2n+l branches per node, and one growth state permode, the total number of tree nodes is given by(2n + l)m’[m (m/m‘- 1) + m / d ], which is of the same orders of magnitude as a recombining trinomial tree, if m ’is a constant or a decreasingfunction of n and m. Thus, it is not exponentially exploding, and hencecomes the term nonexploding bushy tree.Let us make a simple comparison between the total number of nodesin a conventional bushy tree and that in a nonexploding bushy tree. For2-factors and 20 or 30 total tree steps (and 3 steps for the sub-bushy tree,in the case of the nonexploding bushy tree), a conventional bushy tree
Sec. 7.9
297
The Nonexploding Bushy Tree Technique
-
-
contains about 3*' 3.5 billion and 330 2 . 1 ~ 1 0 'tree~ nodes, respectively, whereas a nonexploding bushy tree contains only about53x202/3 17,000 and 53x302/3- 38,000 tree nodes, respectively. Thehuge number of nodes of conventional bushy trees imposes significantchallenges to the current computation power.
-
7.9.2
Modeling Stochastic Processes on a Nonexploding Bushy Tree
In general, a nonexploding bushy tree is built on the underlying incremental Brownian motion that governs the stochastic processes in amodel, such as the LIBOR processes in the BGWJ LIBOR marketmodel.A tree is generally determined by the following parameters: thenumber of tree branches emanating from each tree node, the location ofthe end point on each branch (which is also referred to as the stateshock), and the probability for each branch. All these parameters aredetermined by matching the local mean, local variance and co-variancefor all the factors on each tree step to the values determined by the process being modeled on the tree. Optionally, matching the local kurtosiscan also be achieved for better convergence. In a nonexploding bushytree, the local kurtosis is matched for the first 2 factors (when possible).More specifically, in constructing a tree, we need to find the stateshock matrix M and the branching probability vector P such that the incremental Brownian motion dW(t) is modeled asAW (1)' = & M ,
(7.94)
with all the above moment matching conditions satisfied. For an n-factormodel with m tree branches, M i s an m by n matrix and dW(t) is an n bym matrix.It is easy to verify that a binomial tree can match the local mean andvariance (but not kurtosis), as well as all odd moments of a 1-factor (orone-dimensional) incremental Brownian motion. A trinomial tree canmatch the local mean, variance, and kurtosis, as well as all odd momentsof a 1-factor (or one-dimensional) incremental Brownian motion. Whenused with a 2-factor (or 2-dimensional) incremental Brownian motion, atrinomial tree can match the desired local mean, variance and covariance (but not kurtosis).As an example, let us take a look at how the state shock matrix Mand the branching probability vector P are determined in a trinomial tree
The Interest Rate Market Model
298
Chapter 7
for modeling 1 -factor (or one-dimensional) incremental Brownian motion. Let M' = (x, 0, -x) and P' = (pi ,p 2,p i ) , then matching mean (0),variance (At), and kurtosis ( 3 k ) for the incremental Brownian motionAW(t) requires
El [ A W ( t) ] = pix + P I (- x ) = 0K u r t [ A W ( t ) ] = p l x 4+pl(-x) 4 =3At 2
(7.95)
2P, + P2 = I
Solving the above simultaneous equations yields M' = (&,O,-&)
Similarly, for an n-factor model, or, equivalently, an n-dimensionalBrownian motion, generally 2n+l tree branches are needed for matchingthe local mean, variance and co-variance, and all the odd moments for allthe factors on each tree step, the local kurtosis is also matched for thefirst 2 factors. Fewer than 2n+l tree branches may be used when n isgreater than 4, in which case not all the odd moments are matched.The state shock matrix M and the branching probability vector P, fora 3-factor model as an example, can be chosen to be:
See. 7.9
The Nonexploding Bushy Tree Technique
Factor12
3
4 7 0
0
0
0
4
7
O
O
J
0
0
0
0
-m
0
-&
0
-47
0
0
299
Q
0
,P =
(7.96)
' 16
0 0 0 0 0 00 ~6 0 0 0 00 o + o o o oF = <0 0 0 ~ 0 00 o o o j o o0 o o o o + o0 o o o o o +
00
It is easy to see that, at each tree step, the local mean and the covariance of the n-dimensional incremental Brownian motion are given byEt [ A W ( t )] = &M'P
=0
(n-D vector)
COY[AW ( t ) ]= El [ A W ( t )AW (t,'] = AtM'P'M = AtI
7
(7.97)
as they should, where I is an n-dimensional unity matrix. It is also easyto verify that the local kurtosis is matched for the first 2 factors, and allthe odd moments are matched.It is useful to have the state shock at the ( r ~ + l )tree' ~ branch (ie., thecenter branch) to be zero (with total 2n+l tree branches), in which casethe (n+l)Ih tree branch can serve as an approximated center of a mode.This eliminates the necessity of dynamically detecting the center ofmodes. The center branches can also be used as sub-samplings in general when the mode structures are not significant.
300
The Interest Rate Market Model
Chapter 7
So far the nonexploding bushy tree constructed is quite general andnot model specific. All Brownian motion driven models, including thegeneral HJM models, can be mapped on to the nonexploding bushy tree.The general HJM models can be mapped into this framework with eachdiscredited forward yield as a state variable. It can also be applied toCEV volatility structures for handling the volatility skews in caps/floorsand swaptions (see, e.g., Hull and White, 1999).For the BGM/J LIBOR market model in particular, the LIBOR processes can be constructed on each of the nonexploding bushy treebranches as follows.(7.98)which can be readily obtained from Eq. (7.16) through ItB’s lemma andEuler finite difference. This is also valid for CEV and LCEV volatilitystructures. Modeling the log of the LIBOR rates insures positive interestrate.The nonexploding bushy tree can also be applied to the modeling ofa shifted lognormal model (which requires slight modifications of theabove SDE) and a stochastic volatility model (which requires additionalSDE for the stochastic volatility).Essentially, a tree is a discretization of continuous Brownian motions, whereas a nonexploding bushy tree is a further sub-sampling of aconventional bushy tree. The approximation in a conventional bushytree is the state discretization, whereas the approximations in a nonexploding bushy tree are both the interpolations of the conditional expectedpayoffs and the state discretization. The intrinsic values (or values ofvolatility insensitive underlying instruments) on all tree nodes and all theinformation of the paths leading to these nodes in a nonexploding bushytree are readily available.In general, the interpolations of the conditional expected payoffs forderivatives pricing can be performed by using the state values of the underlying asset or the underlying state variable as the independent variables. The available path information can also be used in the interpolations for path dependent derivatives.For instance, for caps and floors, since the payoffs of each caplet andfloorlet depend on only one state variable (e.g., one LIBOR), the interpolation can be easily performed with respect to that state variable. Forswaptions, the interpolation can be performed with respect to the price of
Sec. 7.9
The Nonexploding Bushy Tree Technique
301
the underlying swap or the underlying swap rate.Principal components can also be used as the independent variablesin the interpolation.It seems to be feasible that one growth state per mode can satisfy theaccuracy requirement in many situations. If necessary, more growthstates can be used to further enhance the accuracy. Particularly, forAmerican and Bermudan derivatives, more growth states can be usedwhere the intrinsic values and the early exercise probabilities are high.7.9.3
Application of Martingale Control Variate Technique
In Chapter 4, we have discussed in details various martingale resamplingor control variate techniques and martingale interpolations. The key ideais to have the martingale relationships obtained from the generic arbitrage theory to hold with high accuracy.The fact that the LIBOR BGM/J model models LIBORs with discrete tenors has some implications. For instance, the payoff of a tradebeing priced can be based on LIBORs with tenors not modeled by theBGM/J model. This is particularly true for aged or existing trades. Ofcourse, one can perform interpolations. But interpolations do not automatically guarantee the required martingale relationships. As we pointedout before, this is a criticism of the BGWJ model on a theoretical basis.This has been addressed in details in Chapter 4. Interpolations usingproper martingale quantities and/or combined with martingale resampling techniques, that we termed as martingale interpolation, can addressguarantee the required martingale relationships.Here, we shall present simple (and approximate) applications ofthese techniques in the interest rate market models so that we can demonstrate the feasibility of the nonexploding bushy tree technique (for,e.g., yielding a good prior solution). We can also use more accurate martingale resampling or control variate technique, in which case the put-callparity can be satisfied exactly. If we allow enough degrees of freedom inthe LIBOR market model or in the local volatilities in particular, we cancalibrate it almost exactly to the entire ATM volatility matrix (or all theATM caplet and swaption prices at different expiries and terms) quotedin the market, if we so desire. By adding stochastic volatility terms inthe CEV or shifted lognormal LIBOR market model, we can also globally best fit the volatility skews and smiles at different expiries andterms.Let’s start with some simple martingale relationship. In the case of
The Interest Rate Market Model
302
Chapter 7
the BGM/J LIBOR market model, the LIBOR Lj(t) is a PT.-martingale.1+1In other words,
Li ( 0 )= EF’ [Li ( t) ]
( t Iq ) .
(7.99)
However, due to various numerical errors, such as those introducedby finite difference and interpolations, the above equation would nothold exactly. This is equivalent to the drift term of Lj(t)being modeledwith a slight error.Even though the relative errorlLi ( 0 )- EF’ [Li (t)]l/Li( 0 ) is normally only about 10” or less, it hassignificant effect on the put-call parity.In the case of the lognormal BGWJ LIBOR market model, let(7.100)Then, it is easy to see that
Li(0)=L,I(O)=EF’[L:(t)]
(t<q),
(7.101)
and In L,!( t ) and In Li ( t ) have the (approximately) same second andhigher order central moments. Using L,!(t) (in the place of Li ( t ) ) inderivatives pricing can result in better accuracy and precision.It is important to point out that even though the above equation appears to hold exactly, it is actually only an approximation of the martingale relationship. This is because that the numeraire ( P ( t , q + l ) )of themeasure depends on Li ( t ) and when we change Li ( t ), the PT+lmeasure changes accordingly. This approximated martingale resamplingis what we used to obtain all the numerical results.For accurate martingale resampling, we can utilize the fact that%+I
Li ( t ) is a PT -martingale, i.e.,l+siLi(t)(7.102)With the above martingale relationship, we can easily perform mar-
Sec. 7.9 The Nonexploding Bushy Tree Technique
303
tingale resampling for lognormal, shifted lognormal, and normal LIBORmarket models through linear transformations of the LIBORs and theNewton-Ralphson method (conditioned on future time s, if such conditional expectations are available). This has been discussed in details insection 4.5.1. Such martingale relationship needs to be satisfied by anyLIBOR market model, including those with stochastic volatilities.This martingale resampling or control variate technique is quite necessary in Monte Carlo simulations. It is also beneficial for the conventional bushy trees and the recombining bushy trees when the size of thetree step is coarse.For better accuracy and precision, analytical or numerical expectedvalues can be used at the penultimate tree steps. Such techniques havebeen applied to derivatives valuations under the Black-Scholes framework (Broadie and Detemple, 1996). For the BGM/J model, such expected values can be obtained with the Black’s formula for caps/floorsand the Blacks or the BGM rank one approximations for swaptions.This technique can also be readily applied to Monte Carlo simulations.This technique is equivalent to generating a large number of branches orpaths at the penultimate tree steps near the strike (or the barrier, or, ingeneral, the states of significance).Similar expected value techniques can also be applied to the interiortree nodes with certain assumed functional forms for the payoffs or interpolation and extrapolations (Broadie and Detemple, 1996, Benhenniand Li, 1997). These can further improve the accuracy and convergenceof American, Bermudan, as well as some exotic (such as barrier) derivatives.7.9.4
Numerical Results
Lognormal BGM/J LIBOR market models with 1 through 3 factors havebeen implemented with the nonexploding bushy tree technique with onegrowth state per mode and 1 to 7 steps for the sub-bushy trees. The approximated martingale resampling or control variate technique describedin the previous section was used and the expected value technique wasused at the penultimate tree steps for caps/floors. Unless otherwise indicated, all the results were obtained with flat initial yield curve (5% for allinitial LIBORs), quarterly LIBORs and swaps, and the estimated piecewise constant market correlation (as discussed in the following section).
The Interest Rate Market Model
304
Chapter 7
7.9.4.1 Market Correlation and Principal Component AnalysisFor our numerical implementation, we use an estimated piecewise constant market correlation that is given by (Rebonato, 1999)Pij =Pm+(l-Pm)exP(-BIT
-q(),
(7.103)
where pg is the time homogeneous instantaneous correlation betweenLIBORs Li and Lj with accrual start dates of(in years),andpm=0.3, p=po+plmax(T,Tj), and Po~ 0 . 1 2 ,PI =0.005. Let pdenote the correlation matrix.Examples of such estimated market correlation is shown in Figure7.7
5
-.2
S
10.8
0
c.
2
0.6
b
0 0.40
2
4
6
8 1 0 1 2
Ti (Years)Figure 7.7. Examples of the estimated instantaneous market correlation used inthe numerical implementation of the nonexploding bushy tree.
Principal component analysis can be applied to the correlation matrixp such that
p=UfU,
(7.104)
and we call the rows of U the (piecewise constant and time homogeneous) volatility vectors (for unit variance). Such volatility vectors for upto the 5" factor (ranked by the eigenvalues) are shown in Figure 7.8.
305
The Nonexploding Bushy Tree Technique
Sec. 7.9A
8E
'
.2 0.8m2.- 0.65 0.4
o 1st factoro 2nd factor
L
0
A 3rd factor
k5 0.2
b
4th factor
zQ o
> -0.2
.-0-0.4
-m
-0.60
2
4
6
8
10
12
Time to Maturity (Years)
Figure 7.8. Piecewise constant and time homogeneous volatility vectors (for thefirst 5 factors) for unit variance obtained through the principal component analysis.
A volatility multiplier can be used to multiply to all the volatilityvectors to achieve the desired overall volatility. More generally, thevolatility vectors need to be multiplied by a diagonal matrix Z to achievethe desired overall VolatiIity, i.e.,(7.105)
V=UC,
where the diagonal element Z;; can be obtained from the implied volatility of dLi ( t ) / L j( t ) . In this case, the covariance matrix of dLj ( t ) / L i( t )is given byCovar(dLj ( t ) / L , (t),dLj( t ) / L j( t ) )= C'pCdt = C'U'UCdt
. (7.106)
= V'Vdt
As with usual factor analysis, to reduce to an n-factor model (with nbeing no greater than the number of LIBORs), we keep only n rows in Vcorresponding to the first n largest eigenvalues. Let V, denote thejfhcolumn in V, then the (piecewise constant and time homogeneous) lognormal LIBOR volatilities Aj(t) (1 by n vectors) in Eq. (7.16) are given
306
The Interest Rate Market Model
Chapter 7
by(7.107)
In general, Z(and other parameters) can be derived from calibrationprocedures whereby the market prices for caps/floors and Europeanswaptions are reproduced by the model preferably to the accuracy of thebid-ask spread. For more discussions on the calibration of the BGMmodel, see, e.g., Rebonato (1999). If all the diagonal elements of Zarethe same, then Z i s the same as a volatility multiplier, which is what weuse in the derivatives pricing for this chapter for illustrations and comparison with the results of other researchers.
7.9.4.2 Results for Caps/Floors, and European and BermudanS waptionsBenchmark tests have been performed by comparing closed-form results(the Black's formula for caps/floors, and the BGM rank one approximation for European swaptions in 1-factor case) to those of the nonexploding bushy tree, which reveals good accuracy of the nonexploding bushytree technique. The put-call parity checking has also been performed forat-the-money (ATM)lo6caps/floors and European swaptions.The accuracy for caps/floors obtained by the nonexploding bushytree technique as compared with Black's results is mostly better than0.2% and often better than 0.1% (Table 7.2 and Table 7.3). For 1-factorEuropean swaptions, the results obtained by the nonexploding bushy treetechnique are mostly within 0.5% (Table 7.5) as compared with those ofthe BGM rank one approximations. For more than one factors, the BGMrank one approximations may not give accurate results.As usual, all the numerical results, unless otherwise indicated, areobtained without the consideration of (specific) counterparty default'06 Here ATM (at-the-money) means that the strikes equal the corresponding presentlyobserved forward LIBOR for caps/floors and the corresponding presently observed forward swap rates for European swaptions. This is also referred to as at-the-money forward. Under these conditions, a cap and a floor having otherwise identical parametersshould have the same present values. This is also true for an European payer swaptionand an European receiver swaption. This is part of the so-called put-call parity, which isa generic arbitrage result.
Sec. 7.9
The Nonexploding Bushy Tree Technique
307
risks. Pricing of (specific) counterparty default risks or the default premiums in OTC derivatives is the subject of the next chapter.It is also important to point out that good accuracy for put-call parity(including cap-floor and payer-receiver parities) has also been achieved.Such accuracy is mostly about 0.1% or better for caps/floors (Table 7.2and Table 7.3) and mostly better than 0.5% for European swaptions(Table 7.5, Table 7.6, and Table 7.7). From the theoretical perspective,this ensures the no-arbitrage condition (to that accuracy), whereas fromthe practical perspective, this is very helpful for yielding good modelcalibration accuracy. Better accuracy can be obtained with more thanone growth states per mode.Here we only used simple and approximated martingale resamplingor control variate technique for the purpose of demonstrate the feasibilityof the nonexploding bushy tree technique (for, e.g., yielding a good priorsolution). We can also use more accurate martingale resampling or control variate technique, in which case the put-call parity can be satisfiedexactly. If we allow enough degrees of freedom in the LIBOR marketmodel or in the local volatilities in particular, we can calibrate it almostexactly to the entire ATM volatility matrix (or all the ATM caplet andswaption prices at different expiries and terms) quoted in the market, ifwe so desire. By adding stochastic volatility terms in the CEV or shiftedlognormal LIBOR market model, we can also globally best fit the volatility skews and smiles at different expiries and terms.For Bermudan swaptions, the nonexploding bushy tree results areclose to those from MCA Monte Carlo simulations. In all the casestested, the Bermudan swaption prices are always no less than those of thecorresponding European swaptions (Table 7.4).A European swaption is usually referred to as an NxM swaption withN denoting the option expiry and the swap start, and M the term of theunderlying swap. Such swap is also referred to as an NxM swap (or forward starting swap). For instance, a 5x10 swaption denotes a swaptionwith the expiry date in 5 years and an underlying 5x10 swap (with thestart date in 5 years and the end date in 10 year relative to the start).In practice, the most popular Bermudan swaptions are those of N NCM with N denoting the initial non-call or lock-out period and M the termof the underlying swap relative to today. For instance, a 10 NC 2 Bermudan swaption is one that can be exercised starting in 2 years and ending in 10 years on a swap with the end date in 10 years relative to today.Commonly, Bermudan swaptions are in the embedded form in callable
308
The Interest Rate Market Model
Chapter 7
bonds and cancelable swaps. The cancelable swaps are commonly usedto hedge the callable bonds.In the literature, some authors report the prices of a different type ofBermudan swaptions that are similar to NxM European swaptions exceptthey can be early exercised up to the European option expiry N at theswap frequency. In order to compare our Bermudan swaption priceswith those in the literature, we also compute the prices of the Bermudanswaptions of the different type.As we pointed out before, due to the existence un-hedgeable variables, the risk preference of market participants on these variables, supply-demand, etc. come into the picture for the derivatives pricing. Thisis particularly true for exotic derivatives, but also true for simple Bermudan swaptions. One approach to handle this situation is through the secondary model calibration on these un-hedgeable variables and possibleseparation of model for pricing and model for hedging. This is one ofthe authors’ current research areas.All the prices shown are in basis points (1 bp = 1%/100, or $100 per$1 million notional).Table 7.2 shows 5 year quarterly ATM cap/floor prices (with volatility multiplier = 20%) obtained with a nonexploding bushy tree (compared with analytical Black’s results). Table 7.3 shows 10 year quarterlyATM cap/floor prices (with volatility multiplier = 10%).Table 7.2. 5 year quarterly ATM cap/floor prices (with volatility multiplier =20% and the estimated market correlation) obtained with a nonexploding bushytree (compared with analytical Black’s results).
Table 7.3. 10 year quarterly ATM cap/floor prices (with volatility multiplier =10% and the estimated market correlation).
Black
Nonexploding bushy treeCap%Error Floor % Error % Parity ErrorFactor (s)0.09%1 259.46 259.76 0.11% 259.31 -0.06%2 283.82 284.52 0.24% 283.78 -0.02%0.13%
I
31 291.991 292.871 0.30%)292.15) 0.05%)
0.12%1