Primal dual linear Monte Carlo algorithm for multiple stopping an application to flexible caps

Transcription

1 Prmal dual lnear Monte Carlo algorthm for multple stoppng an applcaton to flexble caps Sven Balder Ante Mahayn John G. M. Schoenmakers Ths verson: January 25, 22 Abstract In ths paper we consder the valuaton of Bermudan callable dervatves wth multple exercse rghts. We present n ths context a new prmal-dual lnear Monte Carlo algorthm that allows for effcent smulaton of lower and upper prce bounds wthout usng nested smulatons hence the termnology. The algorthm s essentally an extenson of a prmal dual Monte Carlo algorthm for standard Bermudan optons proposed n Schoenmakers et al. 2, to the case of multple exercse rghts. In partcular, the algorthm constructs upwardly a system of dual martngales to be plugged nto the dual representaton of Schoenmakers 2. At each level the respectve martngale s constructed va a backward regresson procedure startng at the last exercse date. The thus constructed martngales are fnally used to compute an upper prce bound. At the same tme, the algorthm also provdes approxmate contnuaton functons whch may be used to construct a prce lower bound. The algorthm s appled to the prcng of flexble caps n a Hull and Whte 99 model setup. The smple model choce allows for comparson of the computed prce bounds wth the exact prce whch s obtaned by means of a trnomal tree mplementaton. As a result, we obtan tght prce bounds for the consdered applcaton. Moreover, the algorthm s genercally desgned for mult-dmensonal problems and s tractable to mplement. Keywords: Multple stoppng, dual representaton, flexble caps, lnear regresson, Monte Carlo smulaton MSC: 9G3, 9G6, 6G5 JEL: G3 Mercator School of Management, Unverstät Dusburg-Essen, Lotharstr. emal: sven.balder@un-due.de. 65, D-4757 Dusburg, Mercator School of Management, Unverstät Dusburg-Essen, Lotharstr. 65, D-4757 Dusburg, emal: ante.mahayn@un-due.de. Weerstrass Insttute for Appled Analyss an Stochastcs, Mohrenstrasse 39, D-7 Berln, emal: schoenma@was-berln.de. Partally supported by the DFG Research Center Matheon Mathematcs for key technologes n Berln.

2 . Introducton The goal of the paper s an effcent Monte Carlo algorthm for the prcng of Bermudan callable dervatves wth multple exercse rghts. Such dervatves gve the rght to exercse a certan clam at a specfc number of tmes wthn a gven set of dscrete exercse dates. These products are nowadays qute popular and frequently occur n varous fnancal sectors, for example as flexble caps also called chooser caps n nterest rate markets, or as swng optons n energy markets. Further they can be found n the context of lfe nsurance contracts, for nstance as surrender and prepayment optons embedded n mortgage backed securtes and nsurance contracts. From a mathematcal pont of vew, prcng of a multple exercse opton comes down to solvng a multple stoppng problem. Because n general a multple exercse opton may be specfed wth respect to a mult-dmensonal underlyng, we am at developng an effectve and generc Monte Carlo procedure, thus avodng the curse of dmensonalty typcally connected wth determnstc PDE solutons. Monte Carlo procedures for sngle exercse Bermudan optons may be somehow categorzed n two groups. On the one hand there are the so called prmal algorthms whch am at constructng a good stoppng tme leadng to a lower based prce estmate. As some of the most popular methods n ths category may be consdered the regresson based procedures n Carrere 996, Longstaff and Schwartz 2, and Tstskls and Van Roy 2. On the other hand there s the category of dual methods relyng on the dual representaton for the standard stoppng problem developed by Rogers 22 and ndependently Haugh and Kogan 24, whch nvolves an nfmum over a set of martngales. In a dual method the goal s to fnd a good martngale whch leads to an upper based prce estmate. A generc and popular Monte Carlo soluton for the Bermudan prcng problem s proposed by Andersen and Broade 24. A drawback of ths method s that t requres usually tme consumng nested Monte Carlo smulatons. In ths respect Belomestny et al. 29 propose a non-nested lnear dual Monte Carlo algorthm n a Wener envronment based on the constructon of a dual martngale va a regresson estmate of a dscretzed Clark-Ocone dervatve. Recently, as a partcular result, n Schoenmakers et al. 2 a new non-nested dual regresson based algorthm s developed whch s based on the dea of constructng nearly optmal or low varance dual martngales. As an addtonal feature, by ths method one also obtans lower prce bounds at the same tme and one doesn t need a certan gven nput approxmaton to the Snell envelope. The above mentoned prmal regresson based Monte Carlo procedures may be extended n a rather straghtforward way to the multple exercse case, usng the reducton prncple for multple stoppng. Further n Bender and Schoenmakers 26 the teratve procedure of Kolodko and Schoenmakers 26 s extended to the multple stoppng problem and analyzed regardng numercal stablty. As a frst extenson of the dual representaton for sngle exercse optons, Menshausen and Hambly 24 developed a dual representaton for mult exercse optons n terms of an nfmum over a famly of martngales and a famly of stoppng tmes. Later on, Schoenmakers 2 found an alternatve dual representaton for the multple stoppng problem n terms of an nfmum over martngales only. The latter representaton s recently generalzed n Bender et al. 2 to multple stoppng problems wth respect to far more general pay-off structures, whch may nclude volume

3 2 constrants and refracton perods for example. The man achevement n ths paper s a lnear, or non-nested hence potentally effcent Monte Carlo procedure for multple exercse optons that provdes both prce upper bounds and prce lower bounds at the same tme. Our new algorthm can be consdered as a generalzaton of the partcular regresson based approach presented n Schoenmakers et al. 2 usng essentally the pure martngale dual representaton n Schoenmakers 2. The proposed algorthm s mplemented and tested for prcng flexble caps wthn a Hull and Whte 99 model setup. Ths smple model s chosen n order to compare the prce bounds computed by the new algorthm wth the exact prce obtaned by means of a trnomal tree mplementaton along the lnes of Hull and Whte 2. In addton, we ntroduce the noton ɛ relevance of the algorthm for solvng the stoppng problem n order to assess whether a partcular product under consderaton nvolves a real stoppng problem n the sense that the opton prce dffers n a way measured by ɛ wth respect to sutably specfed lower and upper benchmark prce bounds. To be more precse, the lower benchmark bound s due to the optmal determnstc exercse polcy and the upper benchmark bound s due to the prce n the vew of a vsonary. As a result, the proposed dual lnear Monte Carlo algorthm gves tght prce bounds for varous versons of the flexble cap consdered. The outlne of the paper s as follows. Secton 2 states the prcng problem and gves a bref revew of known prerequstes. The new lnear Monte Carlo algorthm s developed n Secton 3. In Secton 4 we consder the payoff structure of flexble caps and ntroduce the noton of ɛ relevance of the algorthm, whereas n Secton 5 we mplement the new Monte Carlo algorthm and compare the smulated upper and lower prce bounds wth exact prces obtaned from a trnomal tree mplementaton. 2. Multple stoppng problem, recap of former results 2.. The problem of multple stoppng. The key problem s gven by the multple stoppng problem whch s mpled a Bermudan Opton wth L exercse rghts. A Bermudan opton gves the rght to exercse an opton a specfed number of tmes wthn a gven dscrete set of exercse dates = t < t < < t J =: T, dentfed henceforth wth ther respectve ndces {,,..., J }. In the case that the opton s exercsed at tme, the buyer of the Bermudan opton nstantaneously receves the payoff Z. In general, Z : =,,..., J denotes a non negatve stochastc process n dscrete tme. Z s defned on a fltered probablty space Ω, F, P and s adapted to some fltraton F := F : J and satsfes J E Z <. = Throughout the followng, we nterpret Z as the dscounted cash flow,.e. w.l.o.g. we set the nterest rate equal to zero. The prcng of a Bermudan opton wth L exercse rghts bols down to an optmal stoppng problem w.r.t. Z. Let S L denote the set

4 of F stoppng vectors τ := τ,..., τ L such that τ and, for all l, < l L, τ l + τ l. Then, the dscounted prce Y L of the Bermudan opton at s gven by Y L = sup τ S E where E := E F denotes the condtonal expectaton wth respect to the σ algebra F and Z : and F : F J for > J. It s worth to emphasze that the multple stoppng problem can be reduced to L nested stoppng problems wth one exercse rght, cf. for example Bender and Schoenmakers 26. However, n the followng, we derve a Monte Carlo smulaton method whch requres only one degree of nestng. Frst, we revew some well known results whch are needed to derve the regresson based algorthm Revew of former results. From Rogers 22, Haugh and Kogan 24 t s well known that Y := Y l= Z τ l = nf M M E max J Z + M M = max J Z + M M a.s. 2 where M s the set of all F martngales, and where M s the Doob martngale of the Snell envelope Y for one exercse rght, whch satsfes Y = Y + M A, where A s predctable and nondecreasng, and M = A =. In partcular, the prce of a sngle exercse Bermudan opton at tme = s gven by the pathwse maxmum of the cash-flow mnus the Doob martngale of the Snell envelope. Intutvely, a meanngful upper bound should now be obtaned by replacng the Doob martngale by a good approxmaton to t. As one corner stone of the method developed n ths paper we consder the recently developed regresson based non-nested Monte Carlo algorthm for solvng the dual problem n the sngle exercse case see Schoenmakers et al. 2. Ths algorthm s essentally based on mnmzng the expected condtonal varances of the path-wse functonal E Var ϑ M, =,..., J 3 ϑ M := max J Z M + M n a backward recursve way, by constructng a nearly optmal martngale backwardly from = J down to =, cf. Theorem 2 n Schoenmakers et al. 2. An optmal dual martngale n the sense of Schoenmakers et al. 2 s a martngale M for whch ϑ M = max J Z M + M F, =,..., J, 3

5 and due to Schoenmakers et al. 2, Theorem, t then holds Y = ϑ M, =,..., J. By observng that for < J, ϑ := ϑ M = max Z, M M + + ϑ +, and that trvally Z F, Schoenmakers et al. 2 pont out a partcular backward regresson based approach to acheve mnmzaton of 3 by more strongly mnmzng E Var M M + + ϑ + 4 over a class of martngale ncrements M + M represented by lnear combnatons of a sutable famly of elementary martngales, assumng that the ncrements M M + for +, and thus ϑ +, are already constructed. Further detals wll be clear from the descrpton of the multple exercse verson of the Schoenmakers et al. 2 algorthm later on. For developng the latter algorthm we wll need a next corner stone, namely a recently developed martngale representaton for multple stoppng Schoenmakers 2. The dual martngale representaton n the case of L exercse rghts derved n Schoenmakers 2, cf. Theorem 2.5, states that for L =, 2,... Y L = nf max M,...,M L M < < L k= Y L = max < < L E Y L + = max < < < L k= k= Z k + M k k M k k, Z k + M L k+ k M L k+ k, 5 Z k + M L k+ k M L k+ k almost surely wth :=, and where M k s the Doob martngale of the Snell envelope Y k for k exercse rghts. Remark 2.. For formal reasons n order to avod maxma over empty domans n case the number of remanng exercse possbltes at tme s larger than J + we allow exercsng beyond J yeldng zero cash. Thus, snce trvally Y k = for > J, we have M+ k M k = Y+ k E Y+ k = for J,.e. M k = MJ k for J. At a frst glance 5 requres the evaluaton of a maxmum that nvolves about 2 J arguments n the case where L J /2. However, as we wll show later on Remark 3., due to the very structure of the obect to be maxmzed, t can be computed n a recursve way at a costs of OLJ, so OJ 2 n the worse case Prmal-dual lnear MC algorthm for multple stoppng We are now ready for constructng a multple exercse verson of the regresson based prmal-dual algorthm for one exercse rght proposed n Schoenmakers et al Backward procedure for multple stoppng. Let us fx L and consder l < L. As a well known fact, the Snell envelope Y l+ due to l+ exercse rghts may be equvalently consdered as the Snell envelope under one exercse rght due to the generalzed

6 cash-flow Z l+ From 5 and 6 we observe that wth := +, E Y l + = max < < < L = M l = M l hence 7 may be wrtten as l k= := Z + E Y l +. 7 Z k + M l k+ k M l k+ k M l + + max + < < L l k= Z k + M l k+ M l k+ k k M l + + Y l +, 8 Z l+ := Z + M l Now consder a gven set of martngales M k satsfyng M k cf. Remark 2. and defne for l < L n vew of 5, Y l+ l+ ϑ l+ := max < < l+ k= wth :=. It then holds wth := ĵ := +, = max Z + ϑ l+ l+, max < < < l+ k= = max M l+ = max =: max l+ max < 2 < l+ k=2 Z + M l M l + + Y l = M k, J, k =,..., L Z k + M l+ k+ k M l+ k+ k Z k + M l+ k+ k M l+ k+ k Z k + M l+ k+ k M l+ k+ k M l + + max < < l l+ M l+ + + max <ĵ < <ĵ l+ Z + M l Z l+ k= M l + + ϑl, M l+ k= J l Z k + M l k+ k M l k+, k ĵ k Z k + M l+ k+ M l+ k+ ĵ k M l+ + + ϑ l+ +, M l+ M l+ + + ϑ l+ + n vew of 9. Hence Z l+ may be consdered an approxmaton to vrtual cash-flow Z l+ n 7. We are now gong to apply for l =,, 2 upwardly the regresson method n Schoenmakers et al. 2 to the approxmaton Z l+ of the vrtual cash-flow process Z l+ n 7. Formally, n ths upward constructon t s assumed that at each step l < L, a martngale M l, beng an approxmaton to M l, and an approxmaton ϑ l to Y l s constructed. Followng Schoenmakers et al. 2 we then construct a martngale M l+ and a process ϑ l+ as approxmatons to M l+ and Y l+, respectvely, as explaned n the next secton. The upward constructon may be naturally ntalzed wth M := +

7 ϑ :=, and after L upward steps we end up wth a set of martngales M,..., M L and approxmatons ϑ,..., ϑ L to the respectve Snell envelopes Y,..., Y L. Remark 3.. Complexty of the maxmzaton problem 5 Let us suppose that a famly of martngales M k as above s avalable and that we are faced wth the maxmzaton problem for = and l + = L. We may ntalze M := ϑ := and then obtan ϑ l+ from ϑ l, M l, hence Z l+, and M l+, va by backward nducton. Indeed, from ϑ l+ + by after ntalzng ϑ l+ J = Z J we can obtan for < J nductvely ϑ l+ and thus ϑ l+ n J steps. We thus arrve at ϑ L after LJ operatons rather than J!/ L! J L! Prmal-dual lnear MC algorthm. For the algorthm spelled out below we assume as n Schoenmakers et al. 2 an underlyng D-dmensonal Markovan structure X wth respect to a fltraton generated by an m-dmensonal Brownan moton W. Moreover, we assume that for J and l L the martngales M l are of the form M l = ξ l,q E q,, for certan sutably chosen elementary martngales E q,, q =,..., K. For example, E q, = t ϕ T q u, X u dw u 2 for a set of bass functons ϕ q t, x q K wth ϕ q actng from R R D R m, or, E q,, q =,..., K, may represent any set of dscounted tradables at tme hence t avalable n a partcular stuaton. Let us ntalze M = ϑ =. Then, nductvely, we are gong to construct M l+ and ϑ l+, assumng that M l and ϑ l are constructed for l, l < L. The constructon wll be carred out on a sample of traectores X n J, n =,..., N. At = J we trvally set ϑ l+,n J = Z J X n J, n =,..., N. Suppose we have constructed for fxed < J the martngale ncrements M l+ M l+ + for = J ths + J s trvally zero, and ϑ l+,n +, n =,..., N as approxmatons to Y l+, respectvely, on each traectory. We wll then determne β,..., β K such that for M l+ + M l+ = β q E q,+ E q, the sample estmate of the expected condtonal varance E Var M l+ M l+ + + ϑ l+ + cf. 4 s mnmzed. Ths wll be carred out by a regresson procedure. As a canddate predctor for the F -measurable condtonal expectaton E M l+ M l+ + + ϑ l+ + 6

8 7 we wll take an F -measurable random varable of the form γ q ψ q, X, where, for nstance, ψ q t, x q K, wth ψ q actng from R D R, s a second set of bass functons, or any other set of explanatory F -measurable random varables suggested by the problem under consderaton. We next consder the regresson problem β l+, γ l+ := arg mn E β,γ [ K β q E q, E q,+ + ϑ l+ + γ q ψ q, X whch comes down to the followng regresson procedure on the Monte Carlo traectores, J, n =,..., N, X n Next, we set β l+, γ l+ := arg mn β,γ R K M l+ + M l+ = N n= [β q E n q, E n q,+ + ϑ l+,n + γ q ψ q, X n ] 2. 3 ] 2 β l+,q E q,+ E q,, 4 and then n vew of we proceed by settng ϑ l+,n = max Z n + M l,n M l,n + + ϑl,n +, M l+,n M l+,n + + ϑ l+,n + = max Z n + M l,n M l,n + + ϑl,n +, β l+,q E n q, E n q,+ + ϑ l+,n + where we note that the quanttes ndexed wth level l are already determned at the prevous level. For elementary martngales of the form 2, the respectve Wener ntegrals n 3 may be approxmated as usual by t+ t l=, L ϕ T q u, X u n dw u n ϕ T q t + lδ, X n t +lδ W n t +l+δ W n t +lδ, 5 wth δ := t + t /L for a large enough nteger L. By workng backward from = J down to =, the above regresson procedure yelds a martngale M l+ = p= β l+ p,q E q,p+ E q,p

9 and, as a by-product, an addtonal system of approxmatons to the contnuaton value functons, [ ] E Y l+ l+ + x = C x γ l+,q ψ q, x, =,..., J. 6 Fnally, the martngales M,..., M L may be used to compute a dual upper bound at =, by startng a new smulaton of traectores, ñ =,..., Ñ, and computng Y up,l Ñ + k Ñ max < < L T ñ= p= k X ñ k= β p,q Ẽ L k+ ñ q,p Ẽ ñ q,p+. ñ Z k X k 7 Notce that the upper bound s true n the sense that t s always an upper based estmate, regardless the qualty of the martngales M l. Further note that the maxmum n 7 may be computed effcently along the lnes explaned n Remark 3.. On the other sde, based on the approxmate contnuaton functons 6, we may defne an exercse polcy τ p,l : p L as follows. Defne τ,l := and for < p L τ p,l := nf{ : τ p,l < J, Z X + C L p and smulate a lower based prce estmate, Y low,l Ñ Ñ ñ= p= Z τ p,l,ñ X ñ X C L p+ X }, τ p,l,ñ 4. Applcaton to Flexble Caps. 8 Throughout the followng, Dt, T denotes the t prce of a zero coupon bond wth maturty T. In addton, r denotes the smple compounded spot rate and L s the market LIBOR rate EURIBOR rate, respectvely,.e. Lt, T = τt, T Dt, T 8, 9 where τt, T = T t s the tme dfference expressed n years. For notatonal convenence, we consder an equdstant set of tenor dates T = {T = < T < < T J < T J + }, where T = T + δ for =,..., J + and T =. In addton, we set L T := δ DT, T +. Defnton 4. Payoff structure of caplets, caps, and flexble caps. Actual/36 day-count conventon, respectvely.

10 The payoff of a caplet wth settlement date T {,... J + } s gven by the postve dfference between the reference rate L LIBOR rate prevalng at T and the level κ,.e. δ[l T κ] +. 2 The payoff structure of a cap wth tenor structure T s gven by the payoff of a portfolo of caplets wth settlement dates T,..., T J +. A flexble cap wth L exercse rghts mples the rght to exercse at most L J + of the caplets wth payoffs at T,..., T J +. Obvously, the value of the flexble cap s ncreasng n the number of exercse rghts L. In partcular, for L J +, a trval upper bound for the flexble cap s gven by the value of a flexble cap wth L = J +,.e. the value of the cap over the whole tenor structure. For opton prcng we consder a rsk-neutral valuaton framework wth numerare B t := e t rsds and correspondng prcng measure P,.e. for any T >, the dscounted zero coupon bonds Dt, T /B t, t T, are P -martngales. Thus, a caplet wth settlement date T +, and expry T has at tme t, t T the value C t := B t E t [ δ L T κ + /B T+ ]. The prce of a cap startng at T p and rangng over [T p, T q+ ] s for t < T p, Cap p,q t := q C t. In order to specfy the dscounted cash flow whch s relevant for the multple stoppng problem posed by a flexble cap, we observe that C T = B T E T [ δ L T κ + /B T+ ] =p = δ L T κ + DT, T +. 2 In partcular, Equaton 2 specfes the cashflow at T whch s equvalent to a caplet wth settlement date T +. Thus, the multple stoppng problem correspondng to the flexble cap wll be consdered wth respect to the dscounted cash flow Z := C T B T = δ L T κ + DT, T + /B T = B T + δκ [ + δκ DT, T + at the exercse dates T..., T J. Notce that Z can also be nterpreted the dscounted payoff of + δκ put optons wth maturty T and strke whch are wrtten on a zero +δκ coupon bond wth maturty T +. Recall that S L denotes the set of F stoppng vectors τ := τ,..., τ L such that T τ and, for all l, < l L, τ l + δ τ l. Then, the prce of the flexble cap wth L exercse rghts wth exercse dates T,..., T J correspondng to ndces,..., J s ] +, 9

11 gven by F lcap L T = sup τ S E l= Z τ l = + δκ sup τ S E l= B τ l + + δκ Dτ l, τ l + δ 22 where Z := for > J. The Snell envelope due to L exercse rghts s gven by Y L = sup τ S E l= Z τ l. We now consder the queston f the determnaton of the optmal stoppng strategy has a substantal mpact on the prce of a Bermudan opton. More precsely, we compare the exact prce of the product based on an optmal stoppng strategy wth trval benchmark prce bounds whch can be nferred from determnstc optmzaton procedures specfed below. For a lower trval benchmark prce bound, we consder the followng determnstc optmzaton problem. Let T L denote the set of vectors t := t,..., t L such that T t and, for all l, < l L, t l + δ t l. Then, the trval lower T prce bound trv, low, L Y of a flexble cap wth L exercse rghts s gven by trv, low, L Y = sup t T E Z t l 23 For an upper trval benchmark prce bound, we rely on a vsonary,.e. we consder the trv, up, L t upper bound Y of the flexble cap wth L remanng exercse rghts, whch s gven by s Y trv, up, L = E sup t T l= Z t l. 24 Loosely speakng we wll asses the optmal stoppng problem as relevant f there s a substantal dfference of the exact prce and the benchmark prce bounds. In ths respect we wll exclude prcng scenaros where the exact prce s equal or close to one of the above stated trval prce bounds. The motvaton stems from the observaton that, n both extreme cases, the exact prce can be calculated wthout usng a backward regresson procedure,.e. ether by means of a Monte Carlo smulaton of the look back prce acheved by a vsonary or by means of a smple optmzaton stemmng from an optmal determnstc set of stoppng tmes. Therefore we formulate a noton that expresses the relevance of the proposed dual lnear Monte Carlo algorthm: Defnton 4.2 ɛ relevance of the algorthm for the stoppng problem. The algorthm s called ɛ relevant for the stoppng problem ff { } Y trv, up, L Y L mn, + ɛ. Y L trv, low, L Y l=

12 3M-EURIBOR-forward rates Fgure. The ntal term structure of nterest rates along the lnes of the Euro area yeld curve from January 3, 2. The 3M-EURIBOR-forward rates are obtaned along the lnes of Svensson 994. In partcular, notce that the algorthm s, for example, not relevant zero relevant n the case that the number of exercse rghts concdes wth the number of caplets,.e. L = J. Obvously, we also have zero relevance f the nterest rate dynamcs s determnstc. 5. Performance Test Prce Comparson Hull Whte Model Throughout the followng, we llustrate the prces and prce bounds of flexble caps wth a notonal of, EUR and where the reference rate s the 3 month EURIBOR. If not mentoned otherwse, the caps range over 5 years,.e. J = 6, and the cap level s equal to κ = 2%. 2 We consder exercse rghts whch vary from one to ffty,.e. L {,..., 5}. The ntal term structure of nterest rates s gven by the Euro area yeld curve from January 3, 2. 3 The correspondng 3 month-forward rates are llustrated n Fgure Hull Whte Model Bascs. The benchmark prce values are obtaned by assumng a Hull and Whte 99 nterest rate model whch s calbrated to the ntal term structure from January 3, 2. For an alternatve treatment of flexble caps n the context of related nterest rate models see Pelsser 2 and the references wthn. For the sake of completeness, we revew the Hull-Whte nterest rate dynamcs, and some well known results whch are needed n further. However, the proofs are omtted. These can, for example, be found n the textbook of Brgo and Mercuro 26. The Hull and Whte 99 short rate dynamcs are gven by d r t = θt a r t dt + σ d W t, 25 where a denotes the speed of mean reverson, θt s the mean reverson level, and σ s the a spot rate volatlty. The tme dependent varable θ allows to calbrate the model to the 2 It s worth mentonng that all results can also be computed for other cap levels. In partcular, one can apply the same bass functons as the ones whch are used n the followng. 3 C.f.

13 ntal nterest rate curve whch s observed at the market. The model mples closed form solutons for the zero coupon bond and opton prces. The t prce of a zero coupon bond wth maturty T s gven by where Bt, T = exp{ at t} a and At, T = ln D, T D, t Dt, T = exp{at, T Bt, T r t } 26 Bt, T f, t σ2 exp{ 2at}Bt, T 2 4a ln D,t Let f, t = be the ntal nstantaneous forward rate prevalng at tme t. t Then for arbtrary parameters a and σ the model s calbrated to ths ntal forward rate curve by choosng θt = f,t t + af, t + σ2 2a e 2at. 28 Throughout the followng, we set a =. for the the speed of mean reverson and σ =.2 for the volatlty. Ths can be vewed as consstent wth swapton data. Further, we recall the closed form soluton for the tme t prce of a European put opton wth maturty T and strke κ on a zero coupon bond wth maturty S S T whch s gven by Dt,S ln κdt,t P utt, T, S, κ = κdt, T N + 2 v2 Dt,S ln + v Dt, SN + κdt,t 2 v2, v v 29 exp{ 2aT t} where v = σ BT, S. 2a For Monte Carlo smulatons later on, t s convenent to use the oned dstrbuton of the spot rate and the nterest rate ntegral. Let νt = f M, t + σ2 2a e at 2 V t, T = σ2 a 2 [ T t + 2 a e at t 2a e 2aT t 3 2a Then, the oned dstrbuton of r t and t r s udu condtoned on the nformaton at tme s s gven by [ rt r s t r s u du s r u du ] N µ µ 2 ]. c c 2, 3 c 2 c 22

14 3 where µ = r s e at s + νt νse at s µ 2 = Bs, tr s νs + ln DM, s D M, t + V, t V, s 2 c = σ2 e 2at s 2a c 2 = c 2 = σ2 e at s σ2 e 2at s a 2 2a 2 c 22 = V s, t Exact prcng and relevance of the algorthm for the stoppng problems. We approxmate the exact prce of the flexble caps by means of a trnomal tree. The trnomal nterest rate tree s mplemented along the lnes of Hull and Whte 2. In partcular, equdstant tme steps wth length are used, cf. for example Hull Whte or Brgo and Mercuro 26. As such the tree s suffcently hgh refned to consder the resultng prces as the exact ones. Recall cf. Equaton 22 that the T prce of a flexble cap wth L exercse rghts s gven by F lcap L T = sup τ S E Z τ l l= = + δκ sup τ S E l= B τ l + + δκ Dτ l, τ l + δ. We wll compute exact prces of flexble caps on the tree by means of the Bellman prncple. Notce n ths respect that the dscounted cash flows Z = + δκ + B T + δκ DT, T + are path dependent n fact. However, ths ssue s easly avoded by consderng the Bellman prncple n terms of the un-dscounted obects Z := B T Z and Ỹ := Y B T, whch reads as follows. Set Ỹ, = for =,..., J. At tme J, we have Ỹ,l J = Z J for all l, and at J =,..., J, we then have backwardly { Ỹ,l J = max Z J + E J e T J + T r u du J Ỹ,l J +, E J e J + } T r u du J ỸJ l +. In vew of Defnton 4.2, we can now asses the relevance of the algorthm for the multple stoppng problems. We compare the trval upper and lower bounds of Equaton 24 and 23 wth the trnomal tree prces obtaned by the Bellmann prncple appled to the trnomal tree setup. Notce that the trval lower bound lnked to the optmal determnstc exercse polcy can be calculated accordng to the closed form put prcng formula, cf. Equaton 29. The trval upper prce bound s obtaned by means of a

15 4 Relevance of the algorthm for the multple stoppng problems trval prce bounds exact prce percentage msprcng ex. rghts lower upper tr. tree trv. lower trv. upper L Y trv,low,l Y trv,up,l Y L Y L Y trv,up,l Y trv,low,l Y L Table. The table summarzes the trval upper Y trv,up,l and lower bounds Y trv,low,l gven by Equaton 24 and 23, the exact prce Y L derved by means of the trnomal tree, and the percentage dfferences between the trval prce bounds and the trnomal prces Monte Carlo smulaton wth, paths. Observe that, for L {,... }, the exact prce s at least 29.36% L = above the trval lower prce bound derved by the optmal determnstc exercse polcy. The maxmal devaton s 33.83% for L =. Wth respect to the upper prce bound mpled by a vsonary, the hghest lowest percentage value of the upper prce bound and the exact prces s 33.46% for L = 7.64 for L =. Thus, we conclude that, n vew of Defnton 4.2, the relevance of the algorthm for the stoppng problems s at least ɛ = 7% Upper and lower prcng by the dual MC algorthm. Due to the stochastc nterest rates, we need to consder pathwse dscounted cash flow values Z. Recall that the zero coupon prces only depend on the spot rate r, cf. Equaton 26. In partcular, we have ] +. Z = Z r T, B T = [ + δκ B T + δκ DT, T + For each T {,..., J }, we smulate the short rate r T and the accumulated short rate ln B T = T r s ds accordng to the oned condtonal dstrbuton as gven n Equaton 3. The equdstant tme grd s congruent to the exercse dates,.e. δ = T + T =. 4 In partcular, the problem setup mples a two dmensonal Markovan structure X u = r u, B u wth respect to the one dmensonal Brownan moton W. In vew of the closed form solutons for the zero bond and put opton prces, cf. Equaton 26 and 29, we use the martngale property of the dscounted prce processes of traded assets. We denote

16 5 Products used for the Monte carlo algorthm q Product q maturty/settlement date T level κ Zerobond 5 years 2 3M caplet 5 years 2% 3 3M caplet 7.5 years 2% 4 3M caplet 3.75 years 2% 5 3M caplet year 2% Table 2. The table summarzes the products whch are used for the Monte Carlo algorthm,.e. n terms of ther dscounted prce processes. the dscounted prce of the traded assets q by E q and set M l+ + M l+ = β l+,q Eq T +, r, B T+ T E + qt, r T, B T. In partcular, we use K = 5 and refer to the dscounted prces of a zerobond wth maturty n 5 years and 3M-caplets wth settlement dates n, 3.75, 7.5, and 5 years such that E q T, r T, B T := DT, T for q = B T E q T, r T, B T := + δκ P utt, T, T + δ, for q = 2, 3, 4, 5., B T + δκ where P ut s gven by Equaton 29. The maturtes settlement dates, respectvely and cap levels are summarzed n Table 2. In addton, we also use the prces of these products for the approxmaton of the contnuaton value,.e. ψ q = E q. The optmal weghts γ q and β q q =,..., 5 are estmated by the backward regresson along the lnes of Equaton 3. Snce we work n a Markovan envronment we can approxmate the condtonal expectatons contnuaton values wth l exercse rghts left by C l+ = γ l+,q E q T, r T, B T. The martngales,.e. the weghts γ q and β q q =,..., 5, are determned usng a a relatvely small number of, paths. By an addtonal larger smulaton of 9, paths the upper and lower prce bounds accordng to 7 and 8 are computed. For L {,..., }, the resultng upper and lower prces obtaned by the prmal dual Monte Carlo algorthm are summarzed n Table 3. It s worth to emphasze that the maxmal percentage dfference of the upper prce bound derved by our algorthm to the exact trnomal tree prce s.56%. 4 In partcular, the percentage dfferences vary only between.2% and.56%. The lower prce bounds, whch are actually obtaned as by products, are also pretty good. Here, the percentage prce dfferences to the exact prce vary between.5% and.92%. The performance of the dual MC algorthm wth regard to the prce bounds s also llustrated n Fgure 5.3, where the number of exercse rghts 4 For the exact prces, we refer to Table.

17 6 Prces from prmal-dual Monte Carlo algorthm ex. rghts upper MC bound perc. dff. L Y upl Y up,l confdence nterval Y L 94.4 [93.767,94.44] [86.27,87.3] [276.82, ] [366.94, ] [454.8, ] [54.553, ] [ , ] [78.92,72.62] [79.829, ] [87.85,875.].99 ex. rghts lower MC bound perc. dff. Y L Y low,l L Y downl confdence nterval 9.78 [9.69,92.872] [8.29,85.8] [ , ] [357.99,365.5] [443.36, ] [528., ] [6.45, ] [693.39,78.75] [ ,79.655] [85.994,87.862].45 Table 3. The upper table summarzes the upper bound Y upl for L = up to L = exercse rghts and gves margn Y up,l Y L exact prce Y L compared to the computed by a trnomal tree procedure. The lower table gves the analogous results for the lower prce bound Y lowl of the Monte Carlo algorthm. vares from L = to L = 5. In addton, the prces are plotted n relaton to the number of exercse rghts,.e. we plot Y L L Y up,l L and Y down,l L, respectvely. Obvously, we have for any L, Y L+ Y L Y L Y L wth Y =. From ths we have mmedately Y L LY, and we may prove by nducton that Y L+ Y L. Notce that the L+ L latter nequalty s also true for the upper prce bound Y up,l whch s mpled by the dual Monte Carlo algorthm. However, ths feature may be volated n the case of a non dual prce approxmaton, n partcular an approxmaton whch reles on some reasonable but suboptmal exercse polcy.

18 7 Prces and prce bounds of flexble caps Fgure 2. The fgures llustrate the trval prce bounds outer lnes and the prce bounds gven by the dual Monte Carlo algorthm nner lnes for L = up to L = 5 exercse rghts. Whle the rght fgure gves a plot of the nomnal prces, the rght fgure relates the prce bounds to the number of exercse rghts L. 6. Concluson We propose a prmal dual Monte Carlo algorthm whch gves an upper as well as a lower prce bound. We mplement the algorthm for the prcng of flexble caps. In order to compare the prces provded by our Monte Carlo algorthm wth exact prces, we use a smple Hull and Whte model setup. We calbrate the model to market data and calculate the exact prces by means of a trnomal tree. In addton, we asses that the algorthm s relevant for solvng the stoppng problems under consderaton. The relevance of the algorthm s captured by the dfferences of exact prces and some trval prce bounds whch can be computed by a smple optmzaton a smple Monte Carlo smulaton, respectvely. The lower prce bound s obtaned by the optmal determnstc exercse polcy, the upper prce bound s lnked to a vsonary. Fnally, we consder a set of multple stoppng problems flexble cap products where the msprcng caused by the trval prce bounds s at least 7% such that a sophstcated algorthm s essental. We llustrate that the proposed prmal dual lnear Monte Carlo algorthm s not only tractable to mplement but also gves tght prce bounds. In partcular, t turns out that for 5 year flexble caps, the upper MC prce bound s less than.56% above the exact prce. Fnally, last but not least, we underlne that the algorthm presented s desgned for generc applcaton to any, possbly hgh dmensonal multple stoppng problem. References Andersen, L. and Broade, M. 24, Prmal-dual smulaton algorthm for prcng multdmensonal Amercan optons, Management Scence pp Belomestny, D., Bender, C. and Schoenmakers, J. 29, True upper bounds for Bermudan products va non-nested Monte Carlo, 9, Bender, C. and Schoenmakers, J. 26, An teratve method for multple stoppng: convergence and stablty, Advances n appled probablty 383,

19 Bender, C., Schoenmakers, J. and Zhang, J. 2, Dual representatons for general multple stoppng problems, WIAS Preprnt 665. Brgo, D. and Mercuro, F. 26, Interest rate models: theory and practce: wth smle, nflaton, and credt, Sprnger Verlag. Carrere, J. 996, Valuaton of the early-exercse prce for optons usng smulatons and nonparametrc regresson, Insurance: mathematcs and Economcs 9, 9 3. Haugh, M. and Kogan, L. 24, Prcng Amercan optons: a dualty approach, Operatons Research pp Hull, J. and Whte, A. 99, Prcng Interest-Rate Dervatve Securtes, The Revew of Fnancal Studes 34, Hull, J. C. and Whte, A. 2, Numercal Procedures for Implementng Term Structure Models I: Sngle-Factor-Models, Journal of Dervatves 2, 7 6. Kolodko, A. and Schoenmakers, J. 26, Iteratve constructon of the optmal Bermudan stoppng tme, Fnance and Stochastcs, Longstaff, F. and Schwartz, E. 2, Valung Amercan optons by smulaton: A smple least-squares approach, Revew of Fnancal Studes 4, 3. Menshausen, N. and Hambly, B. 24, Monte Carlo methods for the valuaton of multple-exercse optons., Math. Fnance 44, Pelsser, A. 2, Effcent Methods for Valung Interest Rate Dervatves, Sprnger Verlag. Rogers, L. 22, Monte Carlo valuaton of Amercan optons, Mathematcal Fnance 23, Schoenmakers, J. 2, A pure martngale dual for multple stoppng, Fnance and Stochastcs pp. 6. Schoenmakers, J., Huang, J. and Zhang, J. 2, Optmal dual martngales, ther analyss and applcaton to new algorthms for Bermudan products, subm. u. rev., frst verson, Schoenmakers, J. and Huang, J.: Optmal dual martngales and ther stablty; fast evaluaton of Bermudan products va dual backward regresson, WIAS Preprnt 574, 2. Svensson, L. 994, Estmatng and nterpretng forward nterest rates: Sweden , Techncal report, Natonal Bureau of Economc Research. Tstskls, J. and Van Roy, B. 2, Regresson methods for prcng complex Amercan-style optons, Neural Networks, IEEE Transactons on 24,