LIBOR MARKET MODEL AND GAUSSIAN HJM EXPLICIT APPROACHES TO OPTION ON COMPOSITION

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1 LIBOR MARKET MODEL AND GAUSSIAN HJM EXPLICIT APPROACHES TO OPTION ON COMPOSITION MARC HENRARD Absrac. The win brohers Libor Marke and Gaussian HJM models are invesigaed. A simple exoic opion, floor on composiion, is sudied. The same explici approach is used for boh models. Using an approximaion he LLM price is obained wihou Mone Carlo simulaion. The resuls of he approximaion are very good, wih an error well below he uncerainy due o he simulaion. The appendices proves he exisence of he modified normal and shifed log-normal LLM used in he pricing. The link of he laer wih he Ho and Lee coninuous ime model is described.. Inroducion Even if hey are win brohers see [4], he Libor Marke Models LLM and Gaussian HJM models usually lead o very differen numerical implemenaions echniques. Mone Carlo simulaions is he main ool for he former while explici formulas or rees are he sandard for he laer. One simple case where he LLM are no used wih Mone Carlo simulaions is he pricing of vanilla caps and floors. There he explici Black formula is used. This is probably he only case where he almos-always-working-bu-heavy simulaion approach is no used. The insrumen sudied here is an opion on he composiion described below. I is no a plain vanilla insrumen bu a relaively simple exoic one. I involves several Libor raes fixed a differen daes bu only one paymen dae. In he simple one-facor Gaussian HJM, an explici formula can be obained for many insrumens and in paricular for ha one Secion 3. Formulas for similar insrumens, including overnigh-indexed swaps, can be found in [6]. The LLM used is based on raes following an arihmeic Brownian moion by opposiion o a geomeric Brownian moion in he BGM model [] The reason for he choice is ha he normal models on raes seems o represen beer he dynamic of ineres rae producs, a leas for he momen in USD. A comparison beween models ha leads o his conclusion in he case of swapions is proposed in [8]. For he normal LLM an approximaion of he drif and volailiy erms are used o obain an explici formula in Secion 4. The qualiy of he approximaion is analysed hrough examples in Secion 5. The volailiy par is analysed hrough caples. The drif an volailiy approximaions is applied o he opions on composiion. The model used is no exacly based on a pure arihmeic Brownian moion. Appendix A proves ha such equaions for he forward Libor raes can no be embedded ino a HJM framework. The equaions are modified far away from any realisic rae o ensure he heoreical foundaion of he model bu wih no impac on any pracical compuaions. Appendix C is dedicaed o he heoreical analysis of he approximaion used in Secion 4 pricing formula. The approximaed formula can also be considered as an exac formula for a LLM based on shifed log-normal dynamic for he forward rae. In is simples version he same shifed Dae: Firs version: 4 November 5; his version: 9 November 5. Key words and phrases. explici formula, Libor marke model, HJM model, shifed log-normal model, normal model, exisence, opion on composiion. JEL classificaion: G3, E43, C63. AMS mahemaics subjec classificaion: 9B8, 9B4, 9B7, 6G5, 65C5, 65C3. The acronym BGM is an anagram of GBM Geomeric Brownian Moion.

2 M. HENRARD LLM is equivalen o he coninuous ime version of he Ho and Lee model [9]. The equivalence is explained in he same appendix. The shifed log-normal and normal LLM have been used in oher places in paricular [3, Chaper ] and [3] bu he condiions for heir exisence were no discussed. The compounded insrumen pays a floaing rae ypically he Libor compounded on several consecuive periods. The period daes are < < < n. The raes are fixed a he daes s i i = s s < < s. The accrual facors for he periods i i+ are δ i. The composiion is + δ i Ls i, i The paymen is subjec o a floor or a cap. Wihou he floor he value of he insrumen would simply be in, like a floaing rae noe. Wha is special here is ha he floor is on he oal composiion, no on each individual fixing. For a floor wih an amoun K, he paymen a mauriy is max + δ i Ls i, i, K.. Model and hypohesis The wo insances of he HJM framework used in his aricle are described in his secion. They are a one-facor Gaussian version deerminisic volailiy and a muli-facors LLM version wih normal Libor as base equaions. In general, he HJM framework describes he behavior of P, u, he price in of he zerocoupon bond paying in u, u T. When he discoun curve P,. is absoluely coninuous, which is somehing ha is always he case in pracice as he curve is consruced by some kind of inerpolaion, here exiss f, u such ha P, u = exp u f, sds. The idea of Heah-Jarrow-Moron [5] was o exploi his propery by modeling f wih a sochasic differenial equaion df, u = µ, ud + σ, u.dw for some suiable sochasic µ and σ and deducing he behavior of P from here. To ensure he arbirage-free propery of he model, a relaionship beween he drif and he volailiy is required. The volailiy and he Brownian moion are m-dimensional while he drif and he rae are -dimensional. The model echnical deails can be found in he original paper or in he chaper Dynamical erm srucure model of []. The probabiliy space is Ω, {F }, F, P. The filraion F is he augmened filraion of a m-dimensional sandard Brownian moion W T. To simplify he wriing in he res of he paper, we will use he noaion ν, u = u σ, sds. Le N = exp r sds be he cash-accoun numeraire wih r s s T he shor rae given by r = f,. The equaions of he model in he numeraire measure associaed o N are df, u = σ, uν, ud + σ, u.dw or dp N, u = P N, uν, u.dw The noaion P N, s designaes he numeraire rebased value of P, i.e. P N, s = N P, s. The wo following echnical lemmas were presened in [7] for he Gaussian one-facor HJM. Similar formulas can be found in [, 3.3,3.4] in he framework of coheren ineres-rae models.

3 LLM & HJM 3 Lemma. Le u v. In HJM framework he price of he zero coupon bond is P, v u P u, v = P, u exp νs, v νs, u.dw s u νs, v νs, u ds. Lemma. Le u v. In he HJM framework v N u Nv = exp r s ds = P u, v exp u v u νs, vdw s v u ν s, vds... Gaussian HJM. The firs version of he model used is he HJM model wih m = and a deerminisic volailiy funcion σ : [, T ] R +. The simpliciy of he model allows explici formulas for many producs. Secion 3 describes he formulas for he opion on composiion. In one example of Secion 5 caple are used. Le θ i < i+ be he expiry, sar and end daes of he caple. The srike rae is K. The value of he caple a is wih and + δ i KP, i+ N κ α i+ + P, i N κ α i κ = α i = θ νs, i νs, θ ds P, i+ + δ i K ln α i+ α i P, i α i+ αi... Libor Marke Model. The idea behind he Libor Marke model is o embed differen Blacklike equaion for he forward Libor rae beween sandard daes < < < n ino a unique HJM model. The Libor raes L, j are defined by + δ i Ls, i = P s, i P s, i+. The equaions underlying he normal, Gaussian, or Bachelier Libor marke model are 3 dl, j = γ j L, j,.dw j+ in he probabiliy space wih numeraire P, j+. The γ j j n are m-dimensional funcions. To meri he full qualificaion of Bachelier model, γ j should be purely deerminisic no involving L. For fundamenal reasons explained in Appendix A such a model would be ill-defined. In his secion γ is used wih is mos general form. Secion 4 will consider i in is simple deerminisic form. I can be considered also as an affine funcion leading o a displaced log-normal dynamic as described in Appendix C. The Brownian moion change beween he N and he P, j+ numeraires is given by dw j+ The difference ν, j ν, j+ can be wrien as ν, j ν, j+ = Recursively he change of numeraire gives dw j+ = i=j+ = dw + ν, j+ d. L, j + δ j γ j L, j, L, i + δ i γ i L, j, d + dw n. All he raes can be wrien wih respec o he same las numeraire dl, j = L, i + γ i L, j,.γ j L, j, d + γ j L, j,.dw n. δ i i=j+

4 4 M. HENRARD When δ i is small he rae dependency of drif almos disappear. Le δ i /n one year final rae. If he raes are bounded by L L i L + we obain for he drif erm γ j i=j+ γ i L + n drif γ j i=j+ γ i L + + n If all he γ i are equal and consan, boh bounds are converging o γ. In all cases he raio beween he lower bound and upper bound is L + n L + + n which is rapidly close o when n growhs. So for mos of he raes, he drif is close o is iniial value. This propery of lile dependency of he drif will be used laer. In he pure Bachelier model on rae L., j he caple price is given by [, Secion 3.3.] P, i+ δ i L, i KNκ + γ i θnκ wih κ = L, i K γ i. θ 3. Gaussian one-facor HJM approach o opion on composiion Theorem. Le < < < n, = s s < s < < s wih s i i and K >. In he a HJM one facor model, he price of an insrumen paying in n he maximum of a fixed amoun K and of a principal gross-up by he discree compounding of ineres raes over he periods [ i, i+ ] fixed in s i i.e. Q P s i, i /P s i, i+ is given in by where σ = j= F = P, Nκ + σ + KP, n N κ minsi,s j νs, i+ νs, i νs, j+ νs, j ds. and κ = P, ln σ KP, n σ. The price of an insrumen paying in n he minimum of a fixed amoun K and of a principal gross-up by he discree compounding of ineres raes over he periods [ i, i+ ] fixed in s i is given in by C = P, N κ σ + KP, n Nκ Proof. The price of he insrumen is Using Lemma, we have P s i, i P s i, i+ = P, P, n exp By Lemma, N n F = N E N max si { } P s i, i P s i, i+, K N n. ν s, i+ ν s, i ds + si n = P, n exp νs, n dw s n ν s, n ds. νs, i+ νs, i dw s We denoe his las exponenial by L n. Le W s # = W s + s ντ, ndτ. By he Girsanov s heorem [, Secion 4.., p. 7], W # is a sandard Brownian moion wih respec o he probabiliy P # of densiy L n wih respec o N. This is he n mauriy bond numeraire. Noe ha he

5 LLM & HJM 5 probabiliy P# is no he same as he probabiliy P n used in he LLM as he models are no he same, bu he idea is he same. The sum of he inegrals in he exponenial can be wrien as si = νs, i+ νs, i dw s + si si νs, i+ νs, i dw # s si ν s, i+ ν s, i ds νs, i+ νs, i νs, n νs, i+ νs, i ds Using he ideniy νs, n νs, i = j=i νs, j+ νs, j and rearranging he erms he las sum can be wrien as minsi,s j νs, i+ νs, i νs, j+ νs, j ds. j= The value of he insrumen can now be wrien as P, F = E P #, n max P, n exp σ σx #, K where X # is a random variable wih a sandard normal disribuion wih respec o P #. The firs erm of he maximum operaor is he acual maximum when X # < κ. So we obain F = P, E exp σx # # σ X # < κ + KP, n P # X # κ which by sandard manipulaion on he expecaion and on he normal disribuion lead o he resul. The price of he capped insrumen can be obained by pu-call pariy. 4. Libor Marke model approach o opion on composiion Theorem. Le < < < n, = s s < s < < s wih s i i and K >. In he LLM, he price in of an insrumen paying in n he maximum of a fixed amoun K and of a principal gross-up by he discree compounding of ineres raes over he periods [ i, i+ ] fixed in s i i.e. P s i, i /P s i, i+ is approximaely where T = τ i,j i, j n wih F = P, Nκ + σ + KP, n N κ τ i,j = minsi,s j λ i = L, i + δ i γ i s.γ j sds, σ = λ T T λ and κ = P, ln σ KP, n σ. The price in of an insrumen paying in n he minimum of a fixed amoun K and of a principal gross-up by he discree compounding of ineres raes over he periods [ i, i+ ] fixed in s i is approximaely C = P, N κ σ + KP, n Nκ

6 6 M. HENRARD Proof. The price of he insrumen using P, n as numeraire is { } F = P, n E max n P s i, i P s i, i+, K. In a similar way o he previous heorem, he inegrals appearing in he expeced value can be wrien in he LLM model as si νs, i+ νs, i.dw s + si νs, i+ νs, i ds = si = νs, i+ νs, i.dw n s minsi,s j j= νs, i+ νs, i νs, j+ νs, j ds The sochasic inegrals s i γ isdws n are normally disribued wih mean and covariance T. The sochasic value of he Libor rae is approximaed in he formula by Ls, i = L, i. As noed in Secion, he impac of he rae level is only hrough he raio /L+ δ i and is relaively limied. The iniial raio is λ i. The inegrals can be wrien as The composiion becomes λ i X i λ i λ j τ i,j. P s i, i P s i, i+ = P, P, n exp j= λ i X i λt T λ. Wih he approximaion he large sum in he exponenial is only a consan, denoed α. The sum of he random variables λ i X i is a normally disribued variable wih mean and variance marix λ T T λ. The maximum is he composiion when X < κ. The price can now be wrien as F = P, E exp σx n σ X < κ + KP, n P n X κ. The resul follows easily. The price of he capped insrumen can be found by pu-call pariy. The price of a caple can be deduced from he above formula wih n =. 5. Examples In he firs example a vanilla caple is priced using differen models. The goal of his example is o assess he qualiy of he approximaion used o obain he explici soluion in he LLM. The price is compued wih he exac and approximaed formulas for he normal LLM, he Hull-Whie model, and he log-normal Black model. All he resuls are represened in Figure. The numbers are in erm of normal implied volailiy. The figures used for he example are a simplified curve wih spo equal o oday which schemaically represens he USD curve a he ime of wriing Table. The normal volailiy is consan for all raes and is %. The caple sudied has one year o expiry and he underlying rae is he hree monhs rae. The Hull-Whie and Black model are calibraed o he a-he-money caple. The Hull-Whie mean reversion parameer is %. The relaion beween his approximaion and he shifed log-normal model is described in Appendix C.

7 LLM & HJM Bachelier Black Hull Whie Approximaion Figure. Implied volailiy for he four models sudied. O/N m 3m 6m y y Table. Yield curve. The price of he caple is compued for he four models and he implied volailiy for he normal model is compued from hem. In he price graph all he curves are undisinguishable; he implied volailiy display he differences beer. The wo horizonal doed lines represen a ypical bid-offer spread. The main par of he comparison is beween he approximaed and he exac soluion for he normal LLM. The difference is small. A is maximum i is approximaely one enh of he bid-offer. The Hull-Whie curve is almos equal o he approximaed normal one. From Appendix C equivalence his is no surprising. The Ho and Lee model is equivalen o he Hull-Whie wih a mean reversion. For sandard caples, he correlaion srucure and muli-facor feaures of he normal LLM is no used. The Black curve, ha was included only as a reference poin, is very differen. The similariy beween he resuls for he hree firs model is no due o he fac ha all possible models gives similar resuls bu due o he fac ha he hree models sudied are hem-self very close for insrumens for which no correlaion is used. The second example is relaed o he opion on composiion. The same curve and volailiies are used. For he opion on composiion he full srucure of γ is used, no only is norm. Two differen srucures are used o see is impac. The firs is a one-facor one for which no more deails are required. The second one is a wo facor model. The srucure used is one suggesed in [3, Secion 7.3.] wih γ i = γ i sinθ i, cosθ i. The five θ i are chosen equally spaced beween and π/. The he price for a five hree monh periods composiion is presened in Figure. The firs graph is he inrinsic value F maxp, n K,. In he second graph, o emphasize he differences, he figures are relaive o he Hull-Whie price.

8 8 M. HENRARD.5 3 x 3 Mone Carlo Approximaion Mone Carlo Approximaion Hull Whie x 4 Mone Carlo Approximaion Mone Carlo Approximaion Hull Whie a Inrinsic value b Value relaive o he HJM value Figure. Price using he five differen approaches The prices are compued for a se of srikes raes beween.5% and 7%. The prices are compued five imes for each srike. Once wih he Hull-Whie Gaussian HJM approach and four imes wih he LLM one. There are wo insances of he LLM, one wih one facor and he second wih wo facors. For boh of hem he price is compued wih a Mone Carlo simulaion approach and he explici approximaed formula. The Mone Carlo simulaions are done using he predicor-correcor [3, Secion 5.3.] long-jump echnique and, simulaions. As he number of facors is no he same i was no possible o use he same seed for he wo simulaions. In erm of speed he explici approximaed formula is obviously a lo more efficien. The non-convergence o he same level for low srike is due o he imprecision inheren o he Mone Carlo simulaion. The difference beween he approximaion and he exac figure is smaller han he error coming from he simulaion. Figure 3a repors hose errors. The, simulaions were run imes. The graph repors he difference o he mean of he simulaions. The approximaed resul is represened by he doed line very close o. As can be seen he ypical simulaion error is well above he approximaion error. The error is higher for low srikes. One could ask why his sysemaic error paern appears 3. For lower srikes more simulaed raes and less srikes are used in he expeced value; he raes are subjec o simulaion errors while he fixed srikes are no. The rescaled posiively and negaively exercise probabiliy is also indicaed on he graph o corroborae he inerpreaion. This sugges i is possible o use he pu-call pariy o improve he precision in Mone Carlo simulaions. In our case he pariy is F = P, n K + P, C. The floor on composiion are repriced by Mone Carlo simulaions in Figure 3b using he iniial approach and he pu-call pariy only en simulaions are shown o simplify he graph. The graphs shows he floor and he cap approaches in doed lines. The resuls are relaive o he approximaed formula. The errors are symmerical beween he wo approaches, large on low srikes for he floor and large on high srikes for he cap approach. The a priori bes approach is o ake he cap approach for low srikes and floor approach for high srikes. The cu-off beween he wo approaches can be done a he forward rae. In he example i is around 4.75%. The combine approach is given in solid lines. The verical lines a 4.75% indicae he wo possible choices a he 3 The quesion was acually asked o he auhor by Luis Bengoechea while discussing he paper firs draf.

9 LLM & HJM 9 4 x 5 3 Mean Approximaion Exercise probabiliy 4 x a Mone Carlo simulaions repored o heir average compared o he approximaed formula. b Pu-call pariy improvemens Figure 3. Mone Carlo simulaions cu-off srike. From he picure i is clear ha mos of he wors performing cases are removed. The sandard deviaion of error simulaions and 5 srikes was. bps for he floors,. bps for he caps and.5 bps for he combined approach. 6. Conclusion The normal version of he Libor marke model is used o price simple exoic opions. The non- exisence of such a model and is link o he Ho and Lee model is described. The pricing is done hrough an approximaed explici formula. The resul of he approximaion on boh he sandard caples and floor on composiion is found o be very good. This approach combine he flexibiliy of he LLM in erm of number of facors and correlaion wih he explici resuls usually available only in he less flexible model like he Gaussian one-facor HJM. Appendix A. Non-arbirage free normal LLM This appendix is devoed o analysing if he arihmeic Brownian equaions for he forward raes 3 can be embedded in a HJM framework for γ deerminisic. Unforunaely he answer is no and his can be seen easily. In he HJM framework, he bond prices are given by Equaion and are always posiive. On he oher side he link beween forward rae and prices is + δ i Ls, i = P s, i P s, i+. If L is modelled by a pure arihmeic Brownian moion, i can become very negaive wih a posiive probabiliy. When Ls, i < /δ i, he raio of he prices is negaive. A conradicion wih he previous asserion. The dynamic of he forward rae has o be modified arificially o ensure ha he model can be embedded in an well behaved HJM framework. Condiions sufficien o ensure he exisence of such a framework are presened in Appendix B. The impac of he funcion modificaion far away from he curren rae level is very small. For a hree monhs rae saring in one year, a curren rae of 5% and a volailiy of %, he probabiliy o have a negaive rae is N L /σ θ = N 5 = 3. 7.

10 M. HENRARD Appendix B. Exisence resuls The firs par is an exisence resul. Le γ i L, = p i Lγ i. The funcions p i are globally Lipschiz, p i L, i > 4 and p i have zeros z i wih /δ i z i < L, i. Like in Wih hose condiions i is possible o prove he exisence of a HJM model ha conains he Equaions 3. The argumen is he same as in [, Secion 8..] wih. The firs seps is o prove he exisence of he soluion of 3. This follows from he global Lipschiz condiion wih Iô s heorem [, Theorem 6.7]. Noe ha because of he condiion on he zero of p i and he Lipschiz propery, he soluions have a lower non-aainable barrier a z i δ i. The second main sep is o prove ha he P., n numeraire rebased asses are maringales. For his i is useful o prove ha he inegrals δ j p j Ls, j + δ j Ls, j γ js.dw j s are of bounded quadraic variaion. Or by he ideniy beween he quadraic variaion of an Iô inegral and a Lebesgue inegral [, Theorem 4.8] i is sufficien o prove ha δj p j Ls, j γ j s ds + δ j Ls, j is bounded. The boundedness resul comes from he global Lipschiz condiion in paricular a z i and a infiniy, he lower barrier on he soluion L in z i and he fac ha z i /δ i. LMM wih displaced log-normal or normal raes have been described in oher places, in paricular in [3, Chaper ], bu he quesion of exisence of such a model was no discussed and he condiion on he displacemen no menioned. The firs se of funcions p i o which he resul is applied in his noe is p i L = modified close o L = /δ i. The modificaion is done by keeping p coninuous, seing p i L = for L < /δ i, p i L affine beween /δ i and /δ i + ɛ and leaving p i L = unchanged for L > δ + ɛ. Appendix C. Shifed log-normal Libor marke model The second se of funcions is + δ i L p i L = + δ i L, i. Wih ha choice, he volailiy differences simplify o δ j ν, j ν, j+ = + δ j L, j γ j. This is exacly he value ha was used as an approximaion in Theorem. The same framework can be linked o he coninuous version of he Ho and Lee model [9]. For his ake he simplified version of he shifed log-normal model were all volailiies are consan and such ha γ j + δ j L, j = σ for a cerain σ R +. Then he bond volailiy funcion is given by ν, j ν, j+ = δ j σ = σ j+ j which is exacly he Ho and Lee volailiy srucure wih shor rae volailiy σ. The old fashioned Ho and Lee model can now be renamed wih he more fashionable name of one-facor shifed log-normal Libor marke model. This emphasize once more he broherhood beween LLM and HJM models quoed in he firs senence of his noe, a designaion borrowed from G aarek [4]. Disclaimer: The views expressed here are hose of he auhor and no necessarily hose of he Bank for Inernaional Selemens. 4 The choice of pi being posiive a he curren level of rae is arbirary and wihou loss of generaliy. If p i L, i <, by changing he Brownian moion o W j, he condiion is saisfied. The sandardizaion o a posiive volailiy is more a radiion han a mahemaical consrain.

11 LLM & HJM References [] A. Brace, D. Gaarek, and M. Musiela. The marke model of ineres rae dynamics. Mahemaical Finance, 7:7 54, 997. [] D. C. Brody and L. P. Hughson. Chaos and coherence: a new framework for ineres-rae modelling. Proc. R. Soc. Lond. A., 46:85, 4. [3] E. Errais and F. Mercurio. Yes, libor models can capure ineres rae derivaives skew: a simple modelling approach. Technical repor,???, 5. [4] D. Gaarek. Nonparameric calibraion of forward rae models. Technical repor, NumeriX, 5., [5] D. Heah, R. Jarrow, and A. Moron. Bond pricing and he erm srucure of ineres raes: a new mehodology for coningen claims valuaion. Economerica, 6:77 5, January 99. [6] M. Henrard. Overnigh indexed swaps and floored compounded insrumen in HJM one-facor model. Ewp-fin 48, Economics Working Paper Archive, 4. [7] M. Henrard. A semi-explici approach o Canary swapions in HJM one-facor model. Applied Mahemaical Finance, To appear 5. Preprin in Economics Working Paper Archive, Ewp-fin 38, 3. [8] M. Henrard. Swapions: price, delas, and... 6 / gammas. Wilmo Magazine, pages??????, November 5. [9] T. S. Y. Ho and S.-B. Lee. Term srucure movemens and pricing of ineres rae coningen claims. Journal of Finance, 4: 9, 986., [] P. J. Hun and J. E. Kennedy. Financial Derivaives in Theory and Pracice. Wiley series in probabiliy and saisics. Wiley, second ediion, 4., [] D. Lamberon and B. Lapeyre. Inroducion au calcul sochasique appliqué à la finance. Ellipses, [] M. Musiela and M. Rukowski. Maringale Mehods in Financial Modelling, volume 36. Springer, second ediion, 5. 4 [3] R. Rebonao. Modern pricing of ineres-rae derivaives: he LIBOR marke model and Beyond. Princeon Universiy Press, Princeon and Oxford,., 7, 8, Conens. Inroducion. Model and hypohesis.. Gaussian HJM 3.. Libor Marke Model 3 3. Gaussian one-facor HJM approach o opion on composiion 4 4. Libor Marke model approach o opion on composiion 5 5. Examples 6 6. Conclusion 9 Appendix A. Non-arbirage free normal LLM 9 Appendix B. Exisence resuls Appendix C. Shifed log-normal Libor marke model References Derivaives Group, Banking Deparmen, Bank for Inernaional Selemens, CH-4 Basel Swizerland address: Marc.Henrard@bis.org