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TheLibormarket model (LMM) is commonlyused or pricing exotic interest rate deriv-atives. However, its popularity is limited by the act that its high-dimensionality necessitates (quasi)-Monte Carlo simulation or pric-ing. Such simulation is oten criticised because o its low convergencerate and the diculties o pricing early-exercisable derivatives.

Lattice models do not suer rom these problems, but provideless exibility in terms o correlation structures and ca libration tomarket data. One solution (Piterbarg, 2004) is to use a Markovunctional approximation to the LMM as a control variate. Tusthe part that is able to be priced on a lattice is priced quickly,

while the remaining part has lower variance and thus the MonteCarlo simulation converges quickly. Te key eatures o the con-trol variate are that it can be evaluated using both partial dier-ential equations (PDEs) and simulations, and that the simulationcan be made to be highly correlated with the LMM simulation.

Tere has been little discussion o how to carry out the ftting.In this article, we solve the problem o how to ft a Markov unc-tional approximation to a given calibration o the LMM in such a

way that the control variate varies smoothly with the calibration.One virtue o this approach is that it allows the possibility ousing the control variates or vegas as well as prices and deltas.

Displaced-difusion Libor market model

We briey recall the displaced-diusion LMM. It is based on theidea o evolving discrete market observable orward/Libor rates.

We have a set o tenor dates 0 < T0

< T1

< ... < Tn

and n corre-sponding orward rates f

0, f

1, ... , f

n1, where f

i(t) represents the

orward rate at time tor the interval [Ti, T

i+1).

Let P(t, T) denote the price at time to the zero-coupon bondpaying one at its maturityT. Troughout this article, we will workin the terminal measure, which corresponds to taking P(t, T

n) as

the numeraire.In the displaced-diusion LMM, the orward rates are assumed

to have the ollowing evolution:

dfi t( )fi t( ) + i

= i f, t( )dt + i t( )dW t( )

(1)

where the sis are deterministic D-(1 Dn 1) dimensional

row vectors, the ais are constant displacement coecients, Wis a

standardD-dimensional Brownian motion and the drit term isgiven by no-arbitrage arguments. For more details on the LMM,

we reer the reader to Brace (2007) or Joshi (2003).A calibration procedure results in a covariance matrix o the

rates or each step o the simulat ion. wo procedures that resu ltin the same covariance matrices will yield the same prices. In

what ollows, we will thereore regard a calibration o the LMM

as a vector covariance matrix.Te LMM cannot be represented as a low-dimensional Markov

process or two reasons. Te frst is that the drit is state-depend-ent. Te second is that even i we set all drits to zero, the time-dependence o the volatility loadings will cause the set o attain-able orward-rate curves to be high-dimensional. For thesereasons, it is usual to implement the LMM using Monte Carlosimulation. However, i we modiy the LMM appropriately, wecan obtain a Markov-unctional model.

Markov-unctional model

We take as our starting point the Markov-unctional model oPietersz, Pelsser & Van Regenmortel (2003). However, we canencompass a more general volatility structure. We consider the

case o an F-actor model or generality but it is the case where F= 2 that is important or us.nF-factor model. We require the volatility structure to be sepa-rable, that is:

it( ) = iC t( )

(2)

whereniis anF-dimensional row vector and C(t) is an FFmatrix.

Te volatility structure gets its name because the orward rate andtime dependence are separated out. Separability (see Pietersz,Pelsser & Van Regenmortel, 2003) is oten used to reer to the case

where C(t) is diagonal but this is an overly strong restriction.We illustrate using a two-actor model. Our volatility structure is:

i

t( ) = vi ,1

vi, 2

c11 t( ) c12 t( )

c21

t( ) c22 t( )

(3)

Tis volatility orm diers rom Piterbargs (2004) since he setsc

21(t) = 0. Te extra non-zero term allows better ftting o pseudo-

square roots.We defne an F-dimensional vector o Markov actors via:

dX t( ) = C t( )dW t( )

where W(t) is the projection o the Brownian motion in (1) on tothe frst Fco-ordinates. I we can consider an LMM driven onlyby the frst Factors then its dynamics can be written as:

dfi t( )

fi t( ) + i= i t( )dt + idX t( )

(4)

and so i the drits were dropped the orward rates would beMarkovian inX(t).

Our Markov-unctional model takes this equation and defnesthe orward rates at time tas a unction oX(t) to be the values thatwould be obtained by evolving rom zero to tusing a single-stepdiscretisation. We use the iterative variant o predictor-corrector orstepping (see Hunter, Jckel & Joshi, 2001, and Joshi & Stacey,2008). Te deated price o a derivative, V, satisfes a PDE:

V

t= qij

i, j=1

F

t( )i jV

(5)

with i

= /xi, where x

irepresents the Markov actors. For the

coecients, let cij(t) be the element in the ith row o the jth col-

umn oC(t) in (2), and then:

cuttingedge.inteReStRAtedeRiVAtiVeS

Smooth calibration o Markov unctionalmodels or pricing exotic interest rate derivativesThe Libor market model is widely used but

oten criticised or its slowness. Nick Denson

andMark Joshidevelop an accurate and

stable calibration procedure that allows or

the efective use o a control variate

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cuttingedge.inteReStRAtedeRiVAtiVeS

qij t( ) =1

2cif

f=1

F

t( )cjf t( )

nAnalytic calibration. Te innovation o our article lies in how we ft the Markov-unctional models calibration to that o the

LMM. We make both the evolution o the LMM and that o theftted control variate vary smoothly with the calibration. For theLMM, a cal ibration is a covariance matrix or each step and theevolution is determined by a choice o pseudo-square root oreach step. We use the spectral decomposition: that is, i the cov-ariance matrix or step i has eigenvalues l

jin decreasing order,

with corresponding eigenvectors ej, then we take:

bj = jej

to be columnjo our pseudo-square root,Bi. As long as the eigen-values are distinct, both they and the eigenvectors are analytic(and thereore smooth) unctions o the matrix (Andrew, Chu &Lancaster, 1993). Tis approach also maximises the impact o thefrst Factors.

For ftting the Markov unctional model, we take C(t) to be

constant over each interval [Ti1, Ti). Let C(i) represent the matrixo constant values over that interval, with the convention that T

1

= 0. For an F-dimensional Markov-unctional model and a prod-uct that depends on n orward rates, there are Fn + F2n parame-ters to be ftted: the Fn orward rate specifc scalars and F2npiecewise-constant time-dependent values. We will denote thepseudo-square roots arising rom our separable ft asAi.

One possible methodology, as in Piterbarg (2004), is to use aglobal least-squares numerical optimisation. However, there is noguarantee in that case that the matrixAi varies smoothly withBi.Here, we present a calibration method that is both analytic andstable. It has the eatures o using spectral methods to ft onematrix and inductively ftting the other matrices one by one.

For the rest o this section, we shall address the problem o ft-ting an F-dimensional Markov-unctional model to a D-dimen-sional LMM, where FD. Naturally, we will only attempt to ftthe frst Fcolumns o the matricesB

i. However, the act that we

are using the pseudo-square root defned by spectral decomposi-tion means that these will be the dominant actors.

Consider the evolution o the orward rates over [0, T0), with

the corresponding pseudo-square rootsB0 andA0. By looking atequations (1) and (4), we see that i we have:

dX t( ) = dW t( )

by an appropriate choice oCvalues, then we can exactly cali-brate the frst Fcolumns oA0 toB0, simply by setting:

j = bj,10K bj,F

0

orj= 0, ... , n 1 with:

C 0( ) =

1 0 K 0

0 1 K 0

M M O M

0 0 K 1

(6)

Let Vdenote the matrix with columnsnj

(which is equal toA0).Now that we have set all the orward rate specifc scalars by

exactly ftting to the frst Fcolumns o the frst pseudo-squareroot, we need to ft the remaining n 1 pseudo-square roots. Ourremaining reedom lies in our ability to pick the matrices C(i).

For each i, we want to choose C(i) to minimise the sum-o-

squares norm o the dierence:

VC i( ) B i

including only the frst Fcolumns oBi. Since the kth column oC(i) only contributes to the kth column oVC(i), we have a sepa-

rate least-squares optimisation problem or each kand i. I we letb denote the target column and let c denote the relevant columnoC(i), we have to minimise:

Vc b

where Vis a nFmatrix, c is an F-vector and b is a n-vector. Tiscan be solved using standard linear algebra techniques or solvingan over-determined system. Since it is low-dimensional linearalgebra, the calibration is perormed eectively instantaneously

within a computer program.We could also perorm the calibration starting with any evolu-

tion interval. Consider the evolution over [Ti1

, Ti). We exactly ft

the frst Fcolumns oBi by setting:

j = bj,1iK bj, F

i

with C(i) equal to the identity matrix (note that some elements onj

will temporarily be set to zero as elements oBi are zero due toreset orward rates). We perorm a least-squares ft to the n i 1pseudo-square roots ater the evolution time T

i.

o ft the pseudo-square roots beore Ti1

, we work backwardsconsidering one evolution interval at a time. We calculate thematrix C(i 1) by solving the system o equations, as above, includ-ingn

jorj= i, ... , n 1. Once C(i 1) has been calculated, we set

ni1

so that the frst Fcolumns o the i 1th row oBi1 are exactlymatched. We repeat, working backwards until all pseudo-squareroots have been ftted. Tis approach gives the exibility o exactlyftting the frst Fcolumns o a particular pseudo-square root.

We see that the choice o pseudo-square root exactly ftted doesaect the accuracy o the overall calibration. However, the cali-bration method is very ast so it is possible to iterate over all pos-sible choices and select the best within a raction o a second.

Pricing

o apply the Markov-unctional model as a control variate, weprice a product Vusing:

Vcontrol

= VLMM

VMC

VPDE( )

where VMC

and VPDE

represent the deated price in the Markov-unctional model implemented via a Monte Carlo simulation andPDE method respectively. Using dierent numerical implementa-tion methods, we should still arrive at the same price. However, thevariance o the price will be reduced because o the correlationbetween V

LMMand V

MC(see section 4.1 o Glasserman, 2003).

Te Markov-unctional Monte Carlo implementation needs to

be as close as possible to that o the LMM. Tis means the samerandom numbers must be used or the frst two actors whenevolving rates. We also use the LMM exercise t imes or the MonteCarlo Markov-unctional model as this leads to the largest vari-ance reduction. For the lattice method, we apply the exercisestrategy rom the LMM. Te slight dierence between MonteCarlo and lattice exercise strategies will result in a bias. However,our results show this bias to be very small.

We consider the prices o at-the-money payer Bermudan swap-tions starting in six months, which can be exercised on any o thereset dates. Forward rates are semiannual and initially at, calcu-lated using a continuously compounded rate o 5%, that is, P(t,

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rsk-maaz. 83

T) = er(Tt). We use the abcdvolatility structure:

i

t( ) = a + b Ti t( )ec T

i t( ) + d

with a = 0.05, b = 0.09, c = 0.44, d= 0.12 and correlation betweenrates:

ij = L + 1 L( )e Ti Tj

withL = 0.5, b = 0.1 as this represents what we consider a airlydicult market scenario. All displacements are equal to zero, thatis, a

i= 0 i.

o calculate the exercise strategy, we use the least-squares algo-rithm with the Andersen method superimposed, as discussed inBender, Kolodko & Schoenmakers (2006). A Sobol quasi-randomnumber generator with 214 paths was used to determine the exercisestrategy and then 10,000 Mersenne wister paths were used to esti-mate the price and standard error. We optimised the ftting proce-dure to get the best overall ft or the Markov-unctional model.

able A shows prices, standard errors and the multiple o stand-ard error reduction using the control variate. Te control variate

works very well. For a fve-year Bermudan swaption, a standarderror reduction o 31 to 183 times is achieved. Te method is alsoeective or longer-dated products. We achieved a standard errorreduction o 10 times or a fve-actor 20-year Bermudan swap-tion, which is an increase in speed o around 100 times.

In line with expectations, the control variate works best orshorter-dated products and when coupled with a low-actor LMM.Somewhat surprising is the control variates eectiveness or longer-dated products, which is due to the accurate cal ibration method.

Vega

o calculate the vega, we use a fnite-dierence approach, where wecompute the price using the control variate method, then perturb theLMM volatility, recalibrate our Markov-unctional model (which isperormed very quickly) and then compute the new price. Te cru-cial act here is that the eigenvalues and eigenvectors o a matrix areanalytic unctions o its entries (see Andrew, Chu & Lancaster,

1993), provided the eigenvalues are distinct. Te constraint o dis-tinctness will generally be ulflled since the frst eigenvalue is usuallydominant when we consider correlation matrices or orward rates.nVega results. We present results or the computed model vegaor the corresponding Bermudan swaption prices in table A. Wecomputed a one-sided dierence, where the change in price isshown in table B.

able B shows the vega estimate when din the abcdvolatilitystructure was perturbed by 1%. Once again we have used 214 Sobolpaths to determine the exercise strategy and 10,000 Mersennewister paths to determine both the unperturbed and perturbedprices. We exactly ft to the pseudo-square root that produces the

best global ft. Focusing on the second row o the table, or exam-ple, we see that or a 10-year Bermudan swaption the standard errorreduction or the vega calculation ranges rom 12 to 21 times.

Te contribution o the third and above actors appears to belarger or the vega than the price and so we obtain smaller variancereductions. However, a actor o fve translates into an increase inspeed o around 25 times, so these are stil l signifcant.

Conclusion

We have presented a Libor Markov-unctional model as a controlvariate or the displaced-diusion LMM and introduced a newanalytic cal ibration method. Te calibration method is eectiveor vegas as it a llows or small perturbations in the LMM volatil-ity structure to be translated into small perturbations in the sepa-rable volatility structure. Te control variate produces largestandard error reductions or pricing, ranging rom 10 to 20 timesor the 20-year Bermudan swaptions. It also produces useulreductions or vega calculations, ranging rom fve to eight timesor the 20-year product. Tese standard error reductions are sig-nifcant as they dramatically reduce the number o required sim-ulations or a desired level o accuracy. n

nkdsosaPhdsaMarkJoshsaassoaprossora

hcrorAaralSsahuvrsyoMlbor.e-mal:

so@mal.om,mark@markjosh.om

Andrew A, K-W Chu and

P Lancaster, 1993

Derivatives o eigenvalues and

eigenvectors o matrix unctionsSIAMJournalonMatrixAnalysisand

Applications14(4),pages903926

Bender C, A Kolodko and

J Schoenmakers, 2006

Iterating cancellable snowballs and

related exotics

RiskSeptember,pages126130

Brace A, 2007

Engineering BGMChapman&Hall/CRC

Glasserman P, 2003

Monte Carlo methods in fnancial

engineering

Springer

Hunter C, P Jckel and M Joshi, 2001

Getting the dritRiskJuly,pages8184

Joshi M, 2003

The concepts and practice o

mathematical fnanceCambridgeUniversityPress

Joshi M and A Stacey, 2008

New and robust drit approximations

or the Libor market modelQuantitativeFinance8(4),pages

427434

Pietersz R, A Pelsser and M VanRegenmortel, 2003

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BGM modelJournaloComputationalFinance8(1),

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References

A.Prsasaarrrorsbassposorfr

orBrmaswapos

two-aor thr-aor Fv-aor

5Y LMM 226 (2.66) 220 (2.62) 219 (2.60)

corol 225 (0.01) 183x 221 (0.07) 39x 220 (0.08) 31x

10Y LMM 479 (5.63) 480 (5.73) 476 (5.44)

corol 481 (0.10) 53x 482 (0.25) 23x 477 (0.29) 19x

20Y LMM 852 (9.58) 850 (9.65) 839 (9.80)

corol 853 (0.48) 20x 853 (0.86) 11x 838 (0.97) 10x

Note:two-,three-andfve-actorLMMswereusedwiththecontrolvariate

B.Molvasasaarrrorsbassposor

hBrmaswaposprablA

two-aor thr-aor Fv-aor

5Y LMM 10.8 (1.5E-01) 10.6 (1. 5E-01) 10.5 (1.4E-01)

corol 1 0.8 (1.9E -0 3) 8 0x 1 0.5 (9.3E -0 3) 1 6x 1 0.7 (5.6E -0 3) 2 6x

1 0Y L MM 24.4 (3.7E-01) 24.9 (3. 8E-01) 24.2 (3.6E-01)

corol 2 4.5 (1.7E -0 2) 2 1x 2 5.0 (2.9E -0 2) 1 3x 2 4.4 (3.0E -0 2) 1 2x

2 0Y L MM 46.1 (7.7E-01) 46.2 (7. 8E-01) 45.5 (8.2E-01)

corol 46 .2 ( 9. 6E-02 ) 8 x 4 6. 3 ( 1. 1E- 01) 7x 45. 6 ( 1. 7E- 01) 5x

Note:allcapletvolatilitieswereincreasedby1%