We present a new approach to pricing American-style derivatives that is applicable to any Markovian setting

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1 MANAGEMENT SCIENCE Vol. 52, No., Jauary 26, pp. 95 ss ess forms do.287/msc INFORMS Prcg Amerca-Style Dervatves wth Europea Call Optos Scott B. Laprse BAE Systems, Advaced Iformato Techologes, 38 N. Farfax Drve, Arlgto, Vrga 2223, scott.laprse@baesystems.com Mchael C. Fu The Robert H. Smth School of Busess, Uversty of Marylad, College Park, Marylad 2742, mfu@rhsmth.umd.edu Steve I. Marcus Departmet of Electrcal ad Computer Egeerg, Uversty of Marylad, College Park, Marylad 2742, marcus@sr.umd.edu Adrew E. B. Lm Departmet of Idustral Egeerg ad Operatos Research, Uversty of Calfora, Berkeley, Calfora 9472, lm@eor.berkeley.edu Huu Zhag The Robert H. Smth School of Busess, Uversty of Marylad, College Park, Marylad 2742, oyzhag@glue.umd.edu We preset a ew approach to prcg Amerca-style dervatves that s applcable to ay Markova settg.e., ot lmted to geometrc Browa moto) for whch Europea call-opto prces are readly avalable. By approxmatg the value fucto wth a approprately chose terpolato fucto, the prcg of a Amerca-style dervatve wth arbtrary payoff fucto s coverted to the prcg of a portfolo of Europea call optos, leadg to aalytcal expressos for those cases where aalytcal Europea call prces are avalable e.g., the Merto ump-dffuso process). Furthermore, may settgs, the approach yelds upper ad lower aalytcal bouds that provably coverge to the true opto prce. We provde computatoal results to llustrate the covergece ad accuracy of the resultg estmators. Key words: Amerca-style dervatves; Amerca optos; Europea optos; call optos; early exercse; stochastc dyamc programmg Hstory: Accepted by Davd Hseh, face; receved September 9, 23. Ths paper was wth the authors 7 moths ad week for revso.. Itroducto The prcg of Amerca-style dervatves remas oe of the more challegg problems dervatves face. We defe Amerca-style dervatves, sometmes termed Bermuda dervatves, to be dervatve cotracts wth early-exercse opportutes at a fte umber of exercse dates pror to exprato as opposed to cotuously exercsable for a pure Amerca dervatve). The maor dffculty prcg such dervatves wth early-exercse features les the determato of optmal early-exercse polces. Coversely, the prcg of Europea dervatves specfcally, Europea optos s a comparatvely less dffcult task. I partcular, Europea call ad put optos ca be prced aalytcally uder umerous prce processes beyod geometrc Browa moto; otherwse, they geerally ca be determed wthout too much dffculty va a umercal method such as umercal verso or smulato. I ths paper, we reduce the complexty of the prcg of a Amerca-style dervatve to that of prcg Europea call optos. Specfcally, by approxmatg the value fuctos wth approprately chose lear segmets a dyamc programmg recurso, we are able to prce a Amerca-style dervatve by prcg a portfolo of Europea call optos at varyg strke prces, yeldg aalytcal prcg formulas for uderlyg asset prce processes wth aalytcal Europea call-opto prces. We cosder Amerca-style dervatves wrtte o a sgle uderlyg asset that follows a Markova prce process. Oe geeral approach to the prcg of such dervatves s to cast the problem the framework of a stochastc dyamc programmg problem ad employ a backwards ducto algorthm. However, due to the curse of dmesoalty, solvg the dyamc programmg equatos drectly ca become prohbtvely complex. Specfcally, at ay early-exercse date, the payoff from mmedate exercse must be compared to the holdg value, defed 95

2 Laprse et al.: Prcg Amerca-Style Dervatves wth Europea Call Optos 96 Maagemet Scece 52), pp. 95, 26 INFORMS as the dscouted codtoal expectato of the payoff from keepg the dervatve alve. Computg ths codtoal expectato geerally requres the determato of the ext-stage value fucto over ts etre asset space doma. Our approach to approxmatg ths codtoal expectato essetally follows from the deas of umercal quadrature. We frst compute the extstage value fucto at a fte umber of chose terpolatg pots the asset space doma, ad the approxmate ths value fucto wth a pecewse lear fucto over these pots. Ulke the true value fucto, ths approxmate fucto ca be easly determed over the etre asset space doma, ad the computg of ts codtoal expectato s straghtforward. A approxmate holdg value, defed as the dscouted codtoal expectato of the approxmate ext-stage value fucto, s the used the dyamc programmg equatos. Although approxmatg the value fucto wth a pecewse polyomal of hgher order, e.g., a cubc sple, may result a better ft, the resultg approxmate holdg value fucto wll lack the structure that our method explots. We cosder two methods for costructg the pecewse lear approxmato of the value fucto. The frst ad smplest approach, whch we refer to as the secat terpolato, cossts of smply coectg the terpolatg pots wth secat les. The secod approach, the taget terpolato, volves the pecg together of adacet taget les at the terpolatg pots. The key sght to ether terpolato techque s that the resultat terpolato fucto ca be expressed as a fte sum of Europea call-opto payoffs; hece, the holdg value at the prevous early-exercse date ca be approxmated arbtrarly closely by a fte sum of Europea callopto prces. Thus, the far larger arseal of umercal procedures ad approxmatos) that have bee developed for prcg Europea call optos ca be drectly appled to Amerca-style dervatves va our methods. For a broad class of problem settgs, the algorthm utlzg the secat terpolato results a upper boud o the true dervatve prce, the algorthm wth the taget terpolato results a lower boud, ad both methods coverge as the umber of terpolatg pots s creased) to the true prce. Examples of such problem settgs clude Amercastyle put ad call optos wrtte o a uderlyg asset followg a wde rage of stochastc processes, cludg geometrc Browa moto ad the Merto 976) ump-dffuso model. Numercal expermets 3 llustrate the covergece of the two algorthms for these examples. The proposed approach ca hadle pure-ump ad ump-dffuso processes, whch ca sometmes be problematc for the most popular prcg methods, such as partal dfferetato equato PDE) methods, bomal trees, ad other lattce methods. To put our work cotext, we beg by revewg some of the o-smulato-based) lterature o prcg Amerca optos. Carr et al. 992) decompose the value of a Amerca put opto two ways: to the correspodg Europea put prce plus the early-exercse premum, ad also to ts trsc value ad tme value. Ther focus, however, s ot so much o actual mplemetato of umercal procedures. I a closely related work, Km 99) provdes a aalytcal valuato formula for Amerca put-ad-call optos wrtte o a cotuous dvded-payg asset. However, oly uder certa specal cases ca the formulas be solved explctly; otherwse, umercal techques must be used. Ju 998) approxmates the early-exercse boudary wth a multpece expoetal fucto, smlar to the approach of Omberg 987). Broade ad Detemple 996) develop lower ad upper bouds o the prces of stadard Amerca call ad put optos va the use of capped optos. Am 993) develops a exteso of the bomal method to hadle the cluso of umps, makg t applcable to ump-dffuso models. Smlarly, Zhag 997) also develops extesos for the PDE fte-dfferece method to ump-dffuso models, usg varatoal equaltes the prcg of equty optos. Das 997) develops aalogous fte-dfferece methods for prcg bod optos whe the terest rate process follows a ump-dffuso model. Huag et al. 996) approxmate the early-exercse boudary usg Rchardso extrapolato, ad use ths to express the prce of a Amerca opto as the prce of the correspodg Europea opto plus the early-exercse premum. Ther developmet, however, s more problem depedet, that each prce process requres a separate aalyss of the correspodg tegrals used approxmatg the boudary, ad the method was developed for pure Amerca optos. Aother geeral way of extedg beyod geometrc Browa moto s through Mote Carlo smulato. A umber of Mote Carlo based algorthms for prcg Amerca-style dervatves have bee proposed the last decade. May of these algorthms also cast the problem a stochastc dyamc programmg framework ad attempt to approxmate the holdg values,.e., the codtoal expectatos see Tlley 993, Fu et al. 2, Tstskls ad Va Roy 2, Logstaff ad Schwartz 2, Carrere 996). Thus, these algorthms are perhaps the closest sprt to our approach. Aother smulato-based approach approxmates optmal early-exercse polces drectly rather tha the dyamc programmg equatos see Grat et al. 996, Fu ad Hu 995). Fally, Broade ad Glasserma 997a, b) develop algorthms that

3 Laprse et al.: Prcg Amerca-Style Dervatves wth Europea Call Optos Maagemet Scece 52), pp. 95, 26 INFORMS 97 use smulated paths ad backwards recurso to obta upper ad lower bouds. To summarze our work ad put t the cotext of the ust-cted lterature, our approach has two maor cotrbutos: ) It provdes a ew ad terestg way of relatg a Amerca-style dervatve to correspodg Europea call optos; ad ) t leads to computatoal algorthms that may offer computatoal savgs over exstg umercal approaches the case of a relatvely small umber of fte exercse opportutes, provdg bouds may cases. I partcular, t s sgfcatly more computatoally effcet tha smulato-based approaches the settg where aalytcal Europea call prces are avalable. I comparg wth PDE ad lattce-based methods, we ote that both of these requre specfcato of a dscretzato both the tme axs ad state spaces, whch s ot requred our method. As a result, settgs that requre fe dscretzato of the asset prce process especally the tme dmeso) are lkely to be computatoally burdesome for ether the PDE or lattce-based methods, whereas ths s ot the case for our algorthms, whch deped stead o the umber ad placemet of the terpolatg pots ths ca be thought of as aalogous to a dscretzato the state space, as well). Thus, we coecture that for up to fve exercse dates, our method would be superor to lattce/pde methods. Ths s cosstet wth umercal comparsos to the algorthm of Am 993) for a Merto umpdffuso put-opto example, where the computato tme for the lattce creases dramatcally as a fucto of the dscretzato feess. For the pure cotuously exercsable) Amerca case, we use Rchardso extrapolato o the prces obtaed from our terpolato algorthms, ad umercal results dcate that ths s also compettve terms of computato tme ad prcg accuracy. Smlar coclusos hold umercal comparsos wth the method of Huag et al. 996) o Amerca put optos,.e., our method s more effectve the Bermuda settg, but s stll a vable alteratve for prcg the Amerca. The rest of the paper s orgazed as follows. I 2, we preset the backwards recurso algorthm wth the secat or taget terpolato of the value fucto. Also, we establsh crtera for the aalytcal upper ad lower bouds, dscuss covergece wth respect to the umber of terpolatg pots, ad cosder the optmal selecto of the terpolatg pots. I 3, we apply our algorthms to Amercastyle call ad put optos ad provde umercal results. I 4, we offer some coclusos ad dscuss future applcatos of our methods. The appedx cludes proofs of all propostos. 2. Backwards Recurso Algorthms Let S t deote the prce at tme t of a asset whose prce dyamcs follow a tme-homogeeous, Markov process tme homogeety ca be relaxed) govered by S t+t = hz S t t > where s a vector of parameters cludg the rskfree terest rate r ad the cotuous dvded rate, ad Z s some radom vector depedet of S t ad depedece ca be relaxed may cases, but s requred for the accompayg upper- ad lowerboud results). At tme t, wth S t = S, we defe Vt E Sx as the prce of a Europea call opto wrtte o ths asset wth strke prce x ad tme to maturty : V E t Sx= e r E Q S t+ x + S t = S ) where Q deotes the approprate rsk-eutral martgale) measure ad a b + maxa b. Aswe assume tme homogeety the prce process, the Europea call prces are also tme homogeeous; thus, we heceforth drop the tme dexg ) ad wrte V E Sx. Next, gve the asset prce at tme t =, S, the prce of a Amerca-style dervatve wth exprato date T wrtte o the asset ca be expressed as the soluto to the followg optmal stoppg-tme problem: sup E Q e r L S S 2) where L t represets the payoff at tme t we assume the payoff s oly a fucto of the curret asset prce), ad the supremum s over all stoppg tmes t T. Heceforth, for ease of otato, we drop the superscrpt Q o the expectato, but mata that all subsequet expectatos are take wth respect to ths measure. We restrct early-exercse opportutes to dscrete pots t = N, = t <t < <t N = T, where t N s the fal exercse date, ad, to smplfy otato, assume a fxed tmespa betwee exercse dates = t t. Further, wthout loss of geeralty, we assume that the form of the payoff fucto s depedet of the exercse date, whch case we ca drop the subscrpt o L t. Fally, we abuse otato slghtly by wrtg S for S t, = N. We let V S represet the value of the lve dervatve at date t as a fucto of the uderlyg asset prce S. At the exprato date t N, V N S = LS. At prevous dates, we ca express V S as the maxmum of the dervatve s holdg value ad exercse value: V S = maxls H S 3)

4 Laprse et al.: Prcg Amerca-Style Dervatves wth Europea Call Optos 98 Maagemet Scece 52), pp. 95, 26 INFORMS where the holdg value, H S, s the preset value of the expected oe-perod-ahead dervatve value: H S = e r EV + S + S = S 4) As we have assumed that the dervatve caot be exercsed at t, the dervatve prce 2) ca be expressed as V S = H S = e r EV S S. Ideally, backwards recurso could be doe o 3) ad 4) to obta the dervatve prce V S. However, pror to the exprato date, t s geerally mpossble to obta the value fucto V S over the etre asset space doma, yet ths s ecessary to calculate the holdg value at the prevous exercse date. A alteratve approach that we adopt s to replace the curret value fucto at each early-exercse date wth a sutably chose approxmatg value fucto that ca be determed over the etre asset space, ad the calculate the holdg value at the prevous exercse date as the dscouted expectato of ths approxmatg value fucto. The detals of ths geeral approach are as follows. At exercse date t N, H N S = e r EV N S N S N = S = e r ELS N S N = S 5) s the value of a correspodg Europea opto of legth, wth startg asset prce at t N equal to S ad payoff L at t N. Wth V N gve by 3), we costruct a fucto V N that approxmates the value fucto V N, ad defe H N 2 as the approxmate holdg-value fucto obtaed by replacg V N wth V N 4): H N 2 S = e r E V N S N S N 2 = S. Proceedg recursvely, at exercse date t, = N 2, we defe V as the curret value fucto obtaed by replacg H wth H 3): V S = maxls H S 6) costruct a fucto V that approxmates V, ad defe H S = e r E V S S = S 7) The dervatve prce estmate s the V S = H S f exercsg s allowed at t, V S s gve by 6)). We have assumed that the t N holdg value 5) ca be computed wthout dffculty; otherwse, f determg the expectato of a approxmate value fucto V N to L s easer tha fdg the prce of a Europea dervatve wth payoff L, the value fucto approxmato could beg at exprato date t N. Heceforth, we assume the approxmato begs at t N ad, for otatoal coveece, let H N H N ad V N V N. Careful costructo of the approxmatg fuctos ca result upper or lower bouds: Proposto. Let = N 2. If V V for = + N, the V V. Therefore, costructg V as a lower upper) boud to V at all early-exercse dates results lower upper) bouds o the true value fuctos; partcular, V S wll be a lower upper) boud o the true dervatve prce V S. Further, the followg proposto, whch states that errors the approxmato of V wth V oly cotrbute learly to the overall approxmato error, wll be used later sectos to establsh covergece. Proposto 2. Let = N 2. If V S V S < for = + N, the V S V S < N =+. Left uspecfed up to ths pot s the selecto of the approxmatg value fucto V. I our approach, V s a lear terpolato of the curret value fucto V at a selected fte umber of terpolatg pots the asset space. Ths pecewse lear terpolato fucto ca be coveetly expressed as a summato of Europea call-opto payoffs; thus the approxmate holdg value 7), as a expectato of ths terpolato fucto, ca be represeted as a portfolo of Europea call optos. Hece, the valuato of a Amerca-style dervatve s essetally reduced to the prcg of Europea call optos. We cosder two types of lear terpolato fuctos: The secat terpolato fucto terpolates wth secat les ad the taget terpolato fucto terpolates wth taget les. I 2., we descrbe the Secat Algorthm, whch utlzes the secat terpolato fucto the above geeral approach, ad, wth Proposto, show that uder certa codtos, the Secat Algorthm results upper bouds o the true dervatve prce. I 2.2, we provde smlar detals for the Taget Algorthm, whch corporates taget terpolato, ad show that uder certa codtos the Taget Algorthm results lower bouds. 2.. Secat Algorthm At date t, = N, we assume that V s cotuous o ; for cases where ths s ot true, for example, f V = for some fxed terval of the doma space, the costructo, whle smlar, would eed to be updated o a case-by-case bass. We frst choose + terpolatg pots = x <x < < x the asset space of V, where x s a chose large value of the asset space. The the secat terpolato fucto, V, learly terpolates V wth secat les coectg the pots x V x = ; see Fgure. Specfcally, for S x x, V S = m ) ) S x + V x f x = S<x

5 Laprse et al.: Prcg Amerca-Style Dervatves wth Europea Call Optos Maagemet Scece 52), pp. 95, 26 INFORMS 99 Fgure Secat Iterpolato of Value Fucto ) x ) m ) x ) m 2 ˆ V ) m 3 Notes. x deote terpolatg pots, ad m deote slopes of secat les. where, for =, ) ) V m x V x = 8) x x s the slope of the secat le from x V x to V x. For S>x, we let m S x + x ) x 2 ) x 3 + V x defe the lmtg le wth left edpot x V x, where the lmtg slope, m +, whle ucostraed, should be chose wth regard to the rght-had lmt of V. For may dervatves, the choce of m + s tutve e.g., the Amerca call ad put optos cosdered 3); otherwse, m + could smply be set equal to m from 8). If m + <, the lmtg le tersects the S-axs at x V x /m x, ad we let x + = x V x /m + the S-axs as the ew lmtg le for S>x m +2 =. Otherwse, f m + Thus, for S>x, V + > ad cosder + ;.e.,, we let x + =. m ) ) S x + V x V S = f x S<x = + 9) f S x + We ca the prove the followg smple result. Proposto 3. V S = V x ) + + = m + m ) ) + S x ) where m =, ad, f m +, the last term of the summato s zero. By 7), the approxmate holdg-value fucto ca the be expressed as H S = e r V x ) + + = m + m ) e r E [ S x ) + S = S ] = e r V x ) + + = m + m ) V E Sx ) ) where V E S x, defed ), s the value of a Europea call opto wth tal prce at t equal to S ad maturg at t Thus, ) expresses the approxmate holdg value, H, as a portfolo of Europea call-opto values wth strke prce x. of legth wth varyg strke prces x + =. V E S x ca be evaluated ether a closed-form expresso, as whe the process follows geometrc Browa moto, or va smulato or aother umercal method. The followg summarzes the backwards recurso procedure. Secat Algorthm Step. Let = N. Step. Choose terpolatg pots = x < x < <x ; ad, for =, compute V x va 6), where H N = H N va 5), ad, for <N, H s gve by ). Step 2. Let m =, ad, for =, compute m va 8). Step 3. Choose m +.Ifm + <, let x + = x V x /m +, ad m +2 =. Step 4. =. If >, retur to Step. Otherwse, retur V S = H S va ). If V, = N, s covex over ts etre doma, a careful selecto of the lmtg slope m + wll result a prce estmate V S that s a aalytcal upper boud o the true dervatve prce. To be more precse, as dcated graphcally Fgure, V covex mples V S V S for S x, ad, f the lmtg secat le wth slope m + s chose such that V S V S for S>x, we wll have V V, whch, wth Proposto, mples upper bouds o the true value fuctos. Proposto 4 provdes codtos uder whch the approxmatg value fucto, V, s covex uder the secat algorthm. Proposto 4. Suppose L s covex. If ether L s odecreasg ad hz s covex, or L s ocreasg ad hz s cocave, the H N ad V N are covex. For = N, f V ad hz are covex, m + m, ad hz s lear or m, the H ad V are covex. By Proposto 2, covergece of the secat algorthm wth the umber of terpolatg pots requres beg able to arbtrarly boud V S V S at each early-exercse date. Cosder date t ad assume x = b s the chose rghtmost terpolatg pot;.e., for + terpolatg pots, b = x. The quatty V S V S ca be arbtrarly bouded for S b by

6 Laprse et al.: Prcg Amerca-Style Dervatves wth Europea Call Optos Maagemet Scece 52), pp. 95, 26 INFORMS smply addg terpolatg pots b; however, boudg V S V S for S>bdrectly depeds o the choce of the lmtg slope. For the Amerca putad-call optos cosdered 3, ths tal error ca be bouded ad covergece follows Taget Algorthm The secat terpolato fucto oly requres the evaluato of the curret value fucto at a fte umber of terpolatg pots. The taget terpolato fucto s somewhat more dffcult that costructg taget les at these terpolatg pots also requres the frst dervatve of the curret value fucto. However, for may processes, the structure of the approxmate value fucto that develops throughout the algorthm allows for a relatvely easy calculato of ts frst dervatve. The detals of the taget terpolato closely parallel those for the secat terpolato provded 2.. At date t, = N, we aga assume the curret value fucto, V, to be cotuous o, ad we frst choose + terpolatg pots = x <x < <x. We the costruct the taget terpolato fucto V o the terpolatg pots by pecg together adacet taget les at each of the pots x V x = ; see Fgure 2. Specfcally, for =, m S x + V x s the taget le to V at x, where m = V S/S S=x we defer dscusso of the determato of V S/S for ow). Lettg y z x V x ad y z x V x, for =, we defe y z as the pot of tersecto of the taget les at x ad x +, respectvely;.e., y = m + x + m x V x + m + m ) ) + V x 2) Wthout loss of geeralty, we assume that adacet taget les do deed tersect; the rare stace where we have parallel, otersectg adacet taget les, we could adust the chose terpolatg pots. Smlarly, we ca assume that for =, x y x +. Next, we ote that for =, the taget le to V at x ca also be wrtte as ether m S y + z or m S y + z. Thus, for S x x, the taget terpolato fucto, V, ca be expressed as V S = m ) S y + z f y S<y = y, the costructo of V s detcal up For S>x to otato to that for the secat terpolato fucto. Aga choosg the lmtg slope m + wth respect to the rght-had lmt of V m + = m s the smplest alteratve), we let m +S y + z defe the lmtg le wth left edpot y z = x V x. Ifm + <, we let y + = y z /m + ad cosder the S-axs as the ew lmtg le for S>y + y + ;.e., m +2 =. Thus, for S x, =. Otherwse, f m +, we let m ) S y + z V S = f y S<y = + f S y + The followg result parallels Proposto 3. Fgure 2 Taget Iterpolato of Value Fucto V ) m 2 Vˆ Proposto 5. V S = V x ) + + m = + m ) ) + S y where m =, ad, f m +, the last term of the summato s zero. Notes. x les, ad y ) m ) x ) y ) m ) x deote terpolatg pots, m deote tersecto pots of taget les. ) ) y x 2 deote slopes of taget Thus, aalogous to the secat algorthm, the approxmate holdg value ca be expressed as H S = e r V x ) + + m = + m by 7), where V E Sx s gve by ). ) V E Sy ) 3)

7 Laprse et al.: Prcg Amerca-Style Dervatves wth Europea Call Optos Maagemet Scece 52), pp. 95, 26 INFORMS We ow cosder the calculato of the taget le slopes. For = N, by 6), H V S S S f H S > LS S = 4) S LS f H S <LS at all pots of dfferetablty of V. Geerally, the oly pots of odfferetablty of V wll be whe H S = LS; thus, for most cases, we would oly select terpolatg pots for V where the holdg ad exercsg values do ot equal. Assumg LS/S s easly calculated, we cosder H S/S. Frst, at t N, for a smooth prce process where the domated covergece theorem could be appled, we have by 5), [ S H N S = e r E S LS N Next, at t, = N 2, by 3), S H S = + + = + m + m + ) S N = S S V E Sy + ] 5) ) 6) Thus, the determato of the taget le slopes over the etre recurso oly requres H N S/S ad V E Sx/S. I geeral, for a smooth prce process where Europea call-opto values ca be determed aalytcally, H N S/S ad V E Sx/S ca also be determed aalytcally, or at least easly approxmated by a fte dfferece estmate. Fally, f computg H N S/S s more dffcult tha computg V E Sx/S, t may be beefcal to beg the taget terpolato,.e., the value fucto approxmato, at the exprato date t N, rather tha at t N. I ths case, H N S/S ca be computed va 6). We aga provde a step-by-step summary of the backwards recurso algorthm. Taget Algorthm Step. Let = N. Step. Choose terpolatg pots = x < x < <x ; ad, for =, compute V x va 6), where H N = H N va 5), ad, for <N, H s gve by 3). Step 2. Let m =, ad, for =, calculate m = V S/S S=x va 4), where H N S/S = H N S/S va 5) ad/or 5), ad, for <N, H S/S s gve by 6). Step 3. Let y = x, y = x, ad, for =, compute y va 2). Step 4. Choose m +.Ifm V x /m + ad m +2 =. + <, let y + = x Step 5. =. If >, retur to Step. Otherwse, retur V S = H S va 3). Aalogous to the secat algorthm where covexty of the approxmate value fucto may lead to upper bouds, covexty may lead to lower bouds for the taget algorthm. Specfcally, f for = N, V s covex over ts etre doma, the V S V S s cho- for S x see Fgure 2) ad, f slope m + se such that V S V S for S>x, we wll have that V V, whch, wth Proposto, mples lower bouds o the true value fuctos. I partcular, V S wll be a lower boud o the true dervatve prce. Proposto 6 parallels Proposto 4 ad provdes codtos for whch V s covex uder the taget algorthm. Proposto 6. Suppose L s covex. If ether L s odecreasg ad hz s covex, or L s ocreasg ad hz s cocave, the H N ad V N are covex. For = N, f V ad hz are covex, m + m, ad hz s lear or m, the H ad V are covex. I partcular, by Propostos 4 ad 6, f hz s lear ad L s covex ad mootoe, a careful choce of the lmtg slopes at each early-exercse date wll result upper ad lower bouds o the true dervatve prce va the secat ad taget algorthms, respectvely. Covergece of the taget algorthm wth the umber of terpolatg pots ca be establshed va Proposto 6 usg a smlar argumet as for the secat algorthm, aga depedg o the choce of the lmtg slopes. The accuracy of the secat or taget algorthm s clearly depedet o the selecto of the terpolatg pots. If possble, the terpolatg pots should be chose so as to mmze the error replacg the curret value fucto V wth the terpolato fucto V. I ths regard, more terpolatg pots should be cocetrated o the areas of the state space where V s most covex or cocave. Oe smple heurstc where the terpolatg pots are chose teratvely s as follows: Gve a curret set of pots ad the correspodg secat or taget les, addtoal terpolatg pots are serted to those areas where the absolute dfferece betwee the slopes of adacet secat or taget les s large, as these areas should correspod to areas of hgher covexty or cocavty. Further dscusso of the selecto of terpolatg pots ca be foud Laprse 22), whch cludes alteratve heurstcs for teratvely selectg the terpolatg pots, dscusses the possble accumulato of errors as a result of poorly chose terpolatg pots, ad, for the secat terpolato algorthm, shows that usg the same set of terpolatg pots at each early-exercse date may result a reducto of overall computatoal tme.

8 Laprse et al.: Prcg Amerca-Style Dervatves wth Europea Call Optos 2 Maagemet Scece 52), pp. 95, 26 INFORMS 3. Amerca-Style Call ad Put Optos We llustrate the umercal propertes of the algorthms by prcg two commo Amerca-style dervatves: dvded-payg) call ad put optos. By followg a specfc approach to the selecto of terpolatg pots ad lmtg slopes, we prove that for a wde rage of uderlyg stochastc processes, the secat ad taget terpolato algorthms result aalytcal upper ad lower bouds, respectvely, ad argue that both algorthms coverge to the true opto prce wth the umber of terpolatg pots. We also prove that for each opto, optmal early-exercse polces are threshold polces ad the algorthms provde aalytcal bouds o the true thresholds. Icluded ths class of stochastc processes are geometrc Browa moto ad the Merto 976) ump dffuso, whch we cosder as examples 3.3 ad 3.4, respectvely. The put-ad-call optos are assumed to be wrtte o a uderlyg asset that possbly pays cotous dvdeds at a rate of. For the call opto, we assume >; otherwse, as s well kow, the optmal exercse polcy s to ever early exercse ad the Amerca-style opto s essetally a Europea opto. For the propostos ths secto, we cosder stochastc processes that satsfy the followg assumptos: Assumptos. Assumpto. V E Sx/S e. Assumpto 2. V E x=. Assumpto 3. For the call opto, hz s covex; for the put opto, hz s lear. We ote that hz lear wll also satsfy Assumpto 3 for the call opto. Assumpto s a mld codto equvalet to oegatve ad opostve deltas for the Europea call ad put optos, respectvely. Further, for a multplcatve process,.e., S t+t = S t X tt+t where X tt+t > s a radom varable depedet of all S u u t, Assumptos 2 ad 3 are trvally satsfed. The stock prce models we cosder are multplcatve. The results ths secto mplctly assume that exact Europea call values ad Europea call delta values are avalable, whereas oly the call values are eeded for the secat algorthm. However, we ote that oly the clams ths secto, e.g., the upper ad lower bouds, are depedet upo these exact values ad the satsfacto of the above assumptos; the specfc procedures detaled are applcable for a wde rage of uderlyg stock prce models. 3.. Call Opto I ths case, LS = S K +, ad we assume >. The holdg value at the latest early-exercse date s smply the value of a Europea call opto: H N S = e r ES K + S N = S = V E SK 7) Thus, applyg the secat or taget algorthm to the Amerca call opto, the Europea call opto s the oly opto that eeds prcg. For the taget algorthm, because H N S = V E SK, the dscusso 2.2 mples that the determato of the taget le slopes oly requres the delta of the Europea call opto, V E Sx/S. We ow show that the optmal exercse polcy at t N s a threshold polcy. Frst, V N S = H N S for S K, ad by 7) ad Assumpto, H N S/S e <. Thus, as S K/S =, there exsts a fte sn >K such that H N sn = Ls N ad H N S > <LS for S <>sn ;.e., the opto should oly be exercsed f S sn ad H N S f S<sN V N S = 8) LS = S K f S sn For both algorthms, we take advatage of learty the value fuctos ad choose terpolatg pots accordgly. For ease of otato, we assume a fxed umber of terpolatg pots at each early-exercse date ad omt the subscrpt o. Secat Algorthm. We apply the secat algorthm va the geeral summary provded 2. wth some added specfcatos. As V N S s lear wth slope. for S sn, we let xn = sn ad N m = as H N s gve by 7), we assume sn ca be determed umercally va some rootfdg method). After choosg terpolatg pots betwee N N x = ad x, H N 2 ad V N 2 ca be determed va ) ad 6), respectvely. Moreover, H N 2 also vokes a threshold polcy for the approxmate value fucto at t N 2 ;.e., H N 2 S f S< s N 2 V N 2 S = LS = S K f S s N 2 for some fte s N 2 >K. Proceedg recursvely, at t, gve that V admts a threshold polcy wth threshold s,fweletx = s ad m =, the resultat H vokes a threshold polcy at t. The followg proposto formalzes ths result, ad va Proposto 4, states that the approxmate value fuctos provde upper bouds o the true value fuctos; partcular, V S V S. For Propostos 7, 8, ad 9, H N H N s gve by 7), s N s N s gve by 8), ad s ad s are take to be fty.

9 Laprse et al.: Prcg Amerca-Style Dervatves wth Europea Call Optos Maagemet Scece 52), pp. 95, 26 INFORMS 3 Proposto 7 Secat Algorthm for Amerca Call). Let = N. Ifx = s ad m = for = + N, there exsts a fte s >K such that H S f S< s V S = LS = S K f S s Further, V V ad H H. By Proposto 7, for = N, V S = V S for S> s = x. Hece, by Proposto 2 ad the argumet 2., the secat algorthm coverges wth the umber of terpolatg pots. Next, Proposto 8 states that the true optmal early-exercse polces are threshold polces ad the approxmate thresholds provde upper bouds o the true thresholds. Proposto 8. For = N, there exsts a s, where K s s, such that H S f S<s V S = LS = S K f S s Taget Algorthm. We also clude some added specfcatos the applcato of the taget algorthm. At t N, we aga take advatage of the learty of V N ad let x = sn N ad mn =. However, because V N s odfferetable at sn, N we let m = H N S/S N S=x, where H N s gve by 7). Further, as H N = by 7) ad N Assumpto 2, we set m = to esure lower bouds. The, as for the secat algorthm, H N 2, gve by 3), vokes a threshold polcy for V N 2, ad proceedg recursvely, costructg the taget terpolato fucto as above at each early-exercse date results threshold polces throughout. Further, the approxmate value fuctos ad thresholds provde lower bouds o the true value fuctos ad thresholds, respectvely. Proposto 9 Taget Algorthm for Amerca Call). Let = N. If x = s, m =, m = H S/S S=x, ad m = for = + N, there exsts a fte s >K such that H S f S< s V S = LS = S K f S s Further, V V, H H, ad s s. I partcular, V S V S. As prevously argued for the secat algorthm, Proposto 9 mples the covergece of the taget algorthm wth the umber of terpolatg pots Put Opto I ths case, LS = K S +. The holdg value at t N s the value of a Europea put opto, ad by put-call party, we have: H N S = Ke r Se + V E SK 9) Therefore, as s the case for the Amerca call opto, the Europea call opto s the oly opto that eeds prcg whe applyg the secat or taget algorthm to the Amerca put prcg problem. Further, 9) mples that the oly requremet for determg the taget le slopes s V E Sx/S. As for the call opto, the optmal early-exercse polcy at t N s a threshold polcy. I partcular, for S K, H N S > LS =, ad, by 9) ad Assumpto 2, H N = Ke r K = L. Further, by 9) ad Assumpto, H N S/S. Therefore, there exsts a sn <K such that H N S <LS for S<sN ad H N S LS for S sn ;.e., LS = K S f S<sN V N S = 2) H N S f S sn Thus, the opto should oly be exercsed f S sn ; partcular, sn s postve uless r =. I applyg our algorthms, we aga take advatage of learty the value fuctos. Secat Algorthm. As V N S s lear wth slope for S sn, we let xn = sn ad mn =. Next, by 9) ad 2), V N S approaches zero wth creasg S; thus, gve some small tolerace, we choose x >K large eough such that N N H N x <, ad, to esure upper bouds, let N m =. The, after choosg terpolatg pots N N betwee x ad x, H N 2 ca be determed N va ). I partcular, as m, the last term N the summato of ) s zero, ad, as x =, N V N x = L = K, ad V E S = Se, H N 2 ca be wrtte as: H N 2 S = Ke r Se + N m = N + m ) V E N Sx ) 2) Comparg 9) ad 2), we ote that the Europea call opto compoet of the holdg value at t N cossts of oe call opto, whle the respectve compoet for the approxmate holdg value at t N 2 cossts of a portfolo of call optos. Next, t ca be show that H N 2 vokes a threshold polcy the approxmate value fucto at t N 2,.e., LS = K S f S< s N 2 V N 2 S = H N 2 S f S s N 2

10 Laprse et al.: Prcg Amerca-Style Dervatves wth Europea Call Optos 4 Maagemet Scece 52), pp. 95, 26 INFORMS for some s N 2 < K. Also, as V E Sx approaches Se xe r for S x ad, 2), N =mn + m =, lm S H N 2 S = ; hece lm S V N 2 S =. Proceedg recursvely, at t, gve that V admts a threshold polcy wth threshold s ad lm S V S =, we let x = s, m =, ad m =, ad choose terpolatg pots x 2 x, where x >K s arbtrarly large. The, the resultat H aga trggers a threshold polcy at t, has a form detcal to that see 2), ad thus approaches zero wth creasg S. The followg proposto formalzes ths result, ad va Proposto 4, states that the approxmate value fuctos provde upper bouds o the true value fuctos; partcular, V S V S. For Propostos,, ad 2, H N H N s gve by 9), s N s N s gve by 2), ad s ad s are take to be zero. Proposto Secat Algorthm for Amerca Put). Let = N. Ifx = s, m =, x >K, ad m = for = +N, there exsts a s <K such that H S < LS for S< s ;.e., LS = K S f S< s V S = H S f S s where, for <N, H ca be wrtte the followg form: H S = Ke r Se + + m p= p+ m+ p ) V E Sx + p ) 22) Further, V V ad H H. We ext cosder covergece wth the umber of terpolatg pots. For = N 2, by Proposto, lm S V S =. Thus, as prevously argued for = N, for arbtrary tolerace, we ca choose x large eough such that V x <; whch case, gve that m =, V S V S < for S>x. So, for suffcetly large x ad eough terpolatg pots, V S V S ca be arbtrarly bouded whch mples covergece. Next, Proposto states that the true optmal early-exercse polces are threshold polces ad the approxmate thresholds provde lower bouds o the true thresholds. Proposto. For = N, there exsts a s, where s s <K, such that LS = K S f S<s V S = H S f S s Taget Algorthm. As for the secat algorthm, N we set x = sn ; however, wth the odfferetablty of V N S at S = sn, we set mn = N H N S/S N S=x. Also, we let m = ad N N N N y z = x V N x. Further, we N choose x arbtrarly large ad let the lmtg slope m = m. As H N, hece V N, N N N s a decreasg fucto, m <, whch mples N N y < ad m+2 =. The, H N 2 ca be determed va 3), ad, partcular, as y = x = N N N, V N x = L = K, ad V E S = Se, H N 2 ca be wrtte the followg form: H N 2 S = Ke r Se N + m + m = N ) V E Sy N ) Thus, we have a smlar form for the approxmate holdg value at t N 2 usg the taget terpolato as whe usg the secat terpolato 2). Recursvely costructg V ths same maer, we fd that the approxmate holdg values admt threshold polces, the approxmate value fuctos provde lower bouds o the true value fuctos, ad the approxmate thresholds provde upper bouds o the true thresholds. Proposto 2 Taget Algorthm for Amerca Put). Let = N. Ifx = s, y = x, m =, m = H S/S S=x, ad m = m for = + N, there exsts a s <K such that H S < LS for S< s ;.e., { LS = K S f S< s V S = H S f S s where H ca be wrtte the followg form: H S = Ke r Se + + m p= p+ m+ p ) V E Sy + ) Further, V V, H H, ad s s. I partcular, V S V S. As argued for the secat algorthm, V s a decreasg fucto approachg zero; hece, wth m = m <, for ay where V x <, V S V S < for S>x, mplyg covergece wth umber of terpolatg pots. I the ext two sectos, we prce call ad put optos va the procedures detaled 3. ad 3.2 o assets followg geometrc Browa moto 3.3) ad Merto ump dffuso 3.4) Geometrc Browa Moto We assume the uderlyg stock prce process follows geometrc Browa moto: ds t = S t r dt + dw t, where W t s a stadard Browa moto p

11 Laprse et al.: Prcg Amerca-Style Dervatves wth Europea Call Optos Maagemet Scece 52), pp. 95, 26 INFORMS 5 process ad s the volatlty. Ths leads to the multplcatve logormal prce process: S t+t = hz S t = S t e r 2 /2t+ tz 23) where Z N ad = r. The Europea call-opto prce s gve by the Black-Scholes formula: V E Sx= Se a + xe r a where a = logs/x + r 2 /2 24) ad s the stadard ormal cumulatve dstrbuto fucto. Thus, the Europea call-opto delta s gve by S V E Sx= e a + 25) where a s gve 24). By 25), Assumpto s satsfed, ad as the prce process 23) s multplcatve, Assumptos 2 ad 3 are satsfed. Hece, Propostos 7 2 apply; partcular, the secat ad taget algorthms provde aalytcal upper ad lower bouds, respectvely, o the Amerca-style call ad put opto prces. Table shows the results of applyg the algorthms to a three-year call opto, exercsable every.5 years, wth strke prce K =, = 2, r = 5, ad = 4. For each choce of # terpolatg pots), Table dsplays opto prce estmates for a rage of startg asset prces, the correspodg threshold estmates threshold values are depedet of the startg prces, ad the t 5 = 25 threshold s omtted, as t s obtaed depedetly), ad CPU tmes secods). The fal row, labeled Eur, dsplays the correspodg Europea call prces,.e., o early-exercse opportutes, gve by 24) wth = 3. Our results show that the upper ad lower bouds va the secat ad taget algorthms, respectvely, tghte quckly wth the umber of terpolatg pots. For example, wth ust terpolatg pots, the upper ad lower bouds are wth a pey of the true prce. The covergece s eve faster for the threshold values. To get a dea of the relatve computatoal burde wth respect to other methods, we also estmated the prces usg a bomal lattce o the same computatoal platform. The results dcate comparable prce accuracycomputato trade-offs. For example, the S = case, a dscretzato of daly cremets led to a lattce prce of $3.552 takg.36 CPU secods, fallg somewhere betwee the = 5 ad = results of the taget algorthm terms of both accuracy ad computato. A thrce fer dscretzato yelds a more accurate prce of $3.554 takg 3.95 CPU secods, correspodg roughly to the = 2 taget algorthm. Also, other umercal expermets reported Laprse 22) dcate oly lear growth the computato tme of our algorthm wth the umber of exercse dates. Aalogous to Table, Table 2 shows the results of applyg the secat ad taget algorthms to a put opto wth the same parameters as for the call, except for the dvded rate set to zero. The row labeled Eur dsplays the correspodg Europea put prces gve by 9) ad 24) wth = 3. Aga, the bouds o the opto prce tghte quckly wth the umber of terpolatg pots; see also Fgure 3 llustratg the covergece for the S = case. The -the-moey prces are the most accurate; e.g., for S = 6, oly terpolatg pots are eeded for the upper ad lower bouds to bracket the true prce to wth a pey, whereas over 2 pots are requred for comparable accuracy for S = 4. The threshold bouds for the put opto are eve tghter tha for the call Table ). Table Bermuda Call Opto o Asset Uder Geometrc Browa Moto Opto prce Thresholds Alg type S = 6 S = 9 S = S = S = 4 t = 5 t 2 = t 3 = 5 t 4 = 2 CPU Ta Sec Eur Notes. K =, t = yrs, : umber of terpolatg pots, Eur : Europea prce.

12 Laprse et al.: Prcg Amerca-Style Dervatves wth Europea Call Optos 6 Maagemet Scece 52), pp. 95, 26 INFORMS Table 2 Bermuda Put Opto o Asset Uder Geometrc Browa Moto Opto prce Thresholds Alg type S = 6 S = 9 S = S = S = 4 t = 5 t 2 = t 3 = 5 t 4 = 2 CPU Ta Sec Pure Am Eur Notes. K =, t = yrs, pure Am : estmated pure Amerca prce, : umber of terpolatg pots for Ta/Sec ad umber of tervals for pure Am, Eur : Europea prce. The pure Am etres dcate a prce estmate for the cotuously exercsable Amerca put usg the method of Huag et al. 996), where # tervals dcates the umber of tervals used approxmatg the early-exercse boudary. The results dcate that the computatoal burde s comparable, so the Bermuda case, the precso cludg bouds) of our method makes t very hghly recommeded, but f the pure Amerca prce s the goal, the the choce s ot so clear, as applyg our method would requre Rchardso extrapolato. Computatoal comparsos wth prcg va a bomal lattce were smlar to those of the prevous example; however, terms of practcal mplemetato, our algorthms have two mportat advatages. Frst, lattce methods geerally do ot provde ay formato o the level of precso of ther Fgure 3 Prce Covergece of Bouds for Amerca Put Opto S = Secat terpolato Taget terpolato : Number of terpolatg pots estmates, whle the secat ad taget algorthms provde bouds. Secod, our algorthms ca provde prce estmates for ay startg asset prce oe pass of the algorthm;.e., each row Tables ad 2 s computed from a sgle recurso, whereas a lattce method would requre computato over a dfferet lattce for each startg prce Merto Jump Dffuso The ump-dffuso model from Merto 976) ca be wrtte as follows: S t+t = hz S t = S t e r 2 /2t+ N t tz + = Z 2 /2 26) where Z N..d., N t Possot, the ump szes are..d. logormally dstrbuted: LN 2 /2 2, ad r,, ad reta ther deftos from the geometrc Browa moto process 23); here = r. Merto 976) derves the followg closed-form soluto for the prce of a Europea call opto wrtte o a asset followg 26): V E Sx= e V E Sx 27)! = where V E Sx s the modfed Black-Scholes formula 24) wth 2 replaced by 2 = /. Thus, V E Sx s a weghted sum of Black-Scholes prces. Further, by 27), the Europea call delta s gve by S V E Sx = = e =! S V E Sx e a! + 28) =

13 Laprse et al.: Prcg Amerca-Style Dervatves wth Europea Call Optos Maagemet Scece 52), pp. 95, 26 INFORMS 7 Table 3 Bermuda Put Opto o Asset Uder Merto Jump Dffuso N = 2, = /4 N = 3, = /6 N = 4, = /8 N = 6, = /2 Alg Prce CPUPrce CPUPrce CPUPrce CPU Ta Sec Sm 8597.) ) ) ) 694 Notes. S =, K =, T = 5, r =, = 2828, = 2, = 2, N = umber of exercse dates, Sm : smulato method, stadard error paretheses, Europea prce N = : $ where a s gve by the expresso for a 24) wth 2 replaced by 2 = /. Gve 28) ad the multplcty of the prce process 26), the three assumptos are satsfed ad Propostos 7 2 apply; partcular, the secat ad taget algorthms wll provde upper ad lower bouds, respectvely, o the true opto prces. Table 3 shows the results of applyg our algorthms to a sx-moth T = 5 put opto wrtte o the ump-dffuso model wthout dvdeds =, r =, = 2828, = 2, = 2 usg terpolatg pots per early-exercse date for each algorthm ad eough terms the summatos 27) ad 28) to boud the trucato errors to less tha 4 ad 6, respectvely. For comparso, show the row labeled Sm stadard errors paretheses), are based low) prces obtaed va a smulato method Grat et al. 996), dcatg that for ths example, smulato-based methods requre sgfcatly more computato for a comparable level of umercal precso. The algorthms ca also be used to prce pure cotuously exercsable) Amerca optos by usg Rchardso extrapolato. Deotg P as the prce of the opto wth exercse opportutes at T / = P would correspod to the Europea opto), the three-pot Rchardso extrapolato s gve by P P 3 P 2 5P 2 P, ad the four-pot Rchardso extrapolato s gve by P /3P 4 P 3 23/6P 3 P 2 + /6P 2 P. Table 4 Table 4 Estmatg Pure ) Amerca Put Opto Prce Uder Merto Jump Dffuso Prce CPU Ta Sec Lattce Am) Notes. S =, K =, T = 5, r =, = 2828, = 2, = 2, : umber of tme steps Am lattce or # pots Rchardso extrapolato for Ta/Sec alg. shows the results for the prevous ump-dffuso example, comparg wth the lattce method of Am 993). The algorthms are qute compettve terms of accuracy/computato trade-offs. The upper ad lower bouds also provde a gudele for determg a approprate umber of terpolato pots, whereas determg whe the umber of dscretzato tme steps s suffcetly large applyg lattce methods s ofte arbtrary. 4. Coclusos We preseted a ew approach to prcg Amercastyle dervatves through approxmatg the value fucto wth lear terpolato fuctos, whch coverts the prcg of a Amerca-style dervatve to that of prcg a portfolo of Europea call optos of varyg strkes ad maturtes), ad may cases obta tght aalytcal upper ad lower bouds for the true prce. Implemetato s qute modular, that the same code ca be used for essetally ay asset prce model, smply by substtutg the ew Europea opto prce fucto where t s called by the terpolato model. To llustrate the approach, we appled our algorthms to Amerca call ad put optos wrtte o uderlyg assets followg geometrc Browa moto ad ump-dffuso processes. Relatve to the exstg approaches of Am 993) ad Huag et al. 996), our methods demostrate that they are partcularly effectve the Amerca-style settg for whch they are desged, but also offer a vable alteratve for cotuously exercsable optos whe used wth Rchardso extrapolato. Our methods ca also easly hadle pure ump processes Laprse 22). Laprse 22) also descrbes applcato to Amerca-Asa optos, where the path-depedet payoff depeds o the curret average of the stock prce over a specfed durato cf., Be Ameur et al. 22, Wu ad Fu 23). However, extesos to dervatves o more tha oe uderlyg radom process, e.g., multple assets or stochastc volatlty, s a ope research problem. Ackowledgmets Ths work was supported part by the Natoal Scece Foudato uder Grats DMI-97372, DMI , ad DMI-32322, ad by the Ar Force of Scetfc

14 Laprse et al.: Prcg Amerca-Style Dervatves wth Europea Call Optos 8 Maagemet Scece 52), pp. 95, 26 INFORMS Research uder Grats F49626 ad FA The authors thak Dlp B. Mada for may useful dscussos ad the referees ad assocate edtor for ther helpful commets ad suggestos that have led to a much mproved exposto. Prelmary portos of ths work were reported Laprse et al. 2). Appedx The followg s used the proofs of Propostos 4 ad 6: Lemma A.. Let Z be a radom varable. a) Let g be covex. If ether g s odecreasg ad hz s covex z, org s ocreasg ad hz s cocave z, the EghZ s a covex fucto. b) Let g be cocave. If ether g s odecreasg ad hz s cocave z, org s ocreasg ad hz s covex z, the EghZ s a cocave fucto. I partcular, f both g ad hz are lear, the EghZ s also lear.) Proof. We prove a); the proof of b) s early detcal. For ad ay x, y, z, ghz x + ghz y ghz x + hz y ghz x + y where the frst equalty follows from the covexty of g ad the secod equalty follows from ether hz covex ad g odecreasg, or hz cocave ad g ocreasg. Combg the result wth the learty of the expectato operator gves a). Proof of Proposto. We prove the case the case s detcal) by ducto. Let = N 2. V N V N = V N mples H N 2 S = e r E V N S N S N 2 = S e r EV N S N S N 2 = S = H N 2 S so V N 2 V N 2. By ducto, V + V +.As V + V + V +, we ca smlarly show H H ; hece V V. Proof of Proposto 2. Frst we establsh the followg clam: For = N 2, f V + S V + S <, the V S V S <. V S V S =maxls H S maxls H S H S H S =e r E V + S + V + S + S = S e r E V + S + V + S + S = S Jese s equalty) <e r E S = S Now we prove the proposto by ducto. At = N 2, V N S V N S < N, whch, wth the clam, mples V N 2 S V N 2 S < N. Assume the result holds for t + ;.e., V + S V + S < N =+2. The, wth V + S V + S < +, we have V + S V + S = V + S V + S+ V + S V + S V + S V + S+V + S V + S < + + N =+2 = N =+ Thus, by the clam, V S V S < N =+ Proof of Proposto 3. For ease of otato, we omt superscrpts ad subscrpts. By 9), for S x, VS= m S x + Vx x S<x = where s the dcator fucto. As x S<x = S x S x, VS = m S x + Vx S x = m S x + Vx S x = = m S x + Vx S x m S x + Vx S x + =2 [ m S x + Vx S x ] + m S x + Vx S x = = m m S x + Vx S x m S x + Vx S x + m + S x + Vx S x = m S x + Vx S x = = V m S x + Vx S x + m + m S x + 29) = where the thrd equalty, m = ad m S x + Vx = m S x + Vx. Now for x <,.e., x = x Vx /m ad m +2 =, m S x + Vx = m S x = m +2 m S x ad for x =, S x =. Thus, by 29), VS= V + m + m S x + = where, f x =, the last term the summato s zero. Proof of Proposto 4. Frst, as V H, H covex mples V covex. Further, for = N, as V = maxl H ad L s covex, H covex mples V covex. Cosder H N x = e r ELS N S N = x = e r ELhZ x By Lemma A.a), H N s covex f ether L s odecreasg ad hz s covex, or L s ocreasg ad hz s cocave. For = N, assume V ad hz are covex, ad m + m.by), H s covex f a) m ES S = x s covex; ad b) for =, m + m V E x x s covex, where = + fx + <, ad = otherwse.

15 Laprse et al.: Prcg Amerca-Style Dervatves wth Europea Call Optos Maagemet Scece 52), pp. 95, 26 INFORMS 9 We frst cosder b). The covexty of V mples that m + m for =, whch, wth the prevous assumpto, mples the equalty for =. If x + <, the m + < ad m +2 =, whch mples that m +2 >m +. Therefore, m + m for =. Next, f we let gx = x x + we have V E x = e r EghZ. As g s covex ad odecreasg ad hz s covex, by Lemma A.a), V E x, hece m + m V E x, for =, s covex. For a), f we let gx = x, the ES S = x = EghZ x. By the covexty of h, Lemma A. mples that ES S = x s covex. However, a) requres covexty of the product m ES S = x, whch s acheved f we further requre m or learty of h, whch would make ES S = x lear, ad hece the product lear. Proof of Proposto 5. The proof s early detcal to that for Proposto 3. Proof of Proposto 6. The proof s early detcal to that for Proposto 4. Proof of Proposto 7. Note that LS = S K + s covex ad odecreasg. The proof s va ducto, where we also prove V ad H are covex, V =, ad H S/S e. For = N, the clam s gve by 8). Further, L covex ad odecreasg ad hz covex by Assumpto 3 mply V N ad H N are covex by Proposto 4. Also, V N = H N = V E K= by Assumpto 2, ad S H N S = S V E SK e by Assumpto. By ducto, assume { H V + S = + S f S< s + S K f S s + for some fte s +, where V + V + ad H + H +. Further, assume V + ad H + are covex, V + =, ad H + S/S e. Let x + = s + ad m+ =. We cosder the propertes of V +. Frst, V + ad V + = mply m +. Further, H + S/S e mples m + e. Thus, m + m+. Hece, by Proposto 4, V ad H are covex. Next, as x + = s + ad m+ =, V + S = V + S = S K for S>x +. Thus, by Proposto ad the argumet 2., V V ad H H. Next, as x + =, V + =, ad m +, by ), + H S = m + m + ) V E Sx + ) = I partcular, by Assumpto 2, V = H =. Now, V + covex mples m + + m + for =, ad as m + = m + ad m + m+, the coeffcets the summato are all postve. Further, by Assumpto, H S + S = m + m + ) S V E Sx + ) = e = + m + m + ) = e Next, we ote that for S<K, LS = mples V S = H S. Thus, as H S/S s bouded above by a value strctly smaller tha because >) ad S K/S =, there exsts a uque, fte s >K such that L s = H s ad LS <> H S for S<> s ;.e., { H V S = S f S< s S K f S s Ths completes the proof of the ducto step ad thus the proposto. Proof of Proposto 8. Proposto 7 ad the covergece of H to H mply that the true optmal polcy s a threshold polcy. Notg that H S S K for S s ad H s = s K, H s s K from Proposto 7, whch mples s s. Proof of Proposto 9. The proof s va ducto where we also prove V ad H are covex, V =, ad H S/S e. For = N, the clam of the proposto s gve by 8), ad V N ad H N covex, V N =, ad H N S/S e are show the proof of Proposto 7. By ducto, assume H + S f S< s + V + S = S K f S s + for some fte s + s +, where V + V + ad H + H +. Further, assume V + ad H + are covex, V + =, ad H + S/S e. Let x + = s +, m+ = H + S/S + S=x, ad m + =. We cosder the propertes of V +. Frst, H + S/ mples m + e. Thus, m + m +. As = ad hz s covex by Assumpto 3, V ad S e m + H are covex by Proposto 6. Next, as x + = s + ad m + =, V + S = V + S for S>x +. Thus, by Proposto ad the argumet 2.2, V V ad H H. We ow cosder H gve by 3). Frst, as m + = ad, the frst ad last terms of the summato 3) are zero. Thus, as x + = ad V + =, we have m + H S = + m + m + ) V E Sy + ) = I partcular, by Assumpto 2, V = H =. Now, H + covex mples m + + m + for =, ad as m + m+, the coeffcets the summato are all postve. Further, by Assumpto, S H S = + m + m + ) S V E Sy + ) = e = + m + m + ) = e Next, we ote that for S<K, LS = mples V S = H S. Therefore, as H S/S s bouded above by a value strctly smaller tha ad S K/S =, there exsts a uque, fte s >K such that L s = H s ad LS <> H S

16 Laprse et al.: Prcg Amerca-Style Dervatves wth Europea Call Optos Maagemet Scece 52), pp. 95, 26 INFORMS for S<> s ;.e., H S V S = S K f S< s f S s Now cosder s. H S S K for S s ad H s = s K. Thus, H s s K, mplyg s s. Ths completes the proof of the ducto step. Proof of Proposto. We frst ote that LS = K S + s covex ad ocreasg, ad hz s both covex ad cocave. The proof s va ducto where we also prove V ad H are covex, V = K, ad e H S/S. For = N, the proposto clam s gve by 2). Further, L covex ad ocreasg ad hz cocave mply V N ad H N covex by Proposto 4. By 9), S H N S = e + S V E SK Thus, by Assumpto, e H N S/S. Fally, by 2), V N = L = K. By ducto, assume K S f S< s + V + S = H + S f S s + for some s +, where V + V + ad H + H +. Further, assume V + ad H + are covex, V + = K, ad e H + S/S. Let x + = s +, m+ =, ad m + =. We cosder the propertes of V +. Frst, H + S/S mples m + for = 2. Thus, m + m+. Hece, as hz s lear, V ad H are covex by Proposto 4. Next, for S>x +, V + S = H + S mples V + S/S. Thus, as m + =, V + S V + S for S>x +. Thus, by Proposto ad the argumet 2., V V ad H H. As x + =, V + = K, m + =, m + =, m + =, V E S = Se, H S = Ke r Se + + m + m + ) V E Sx + ) = by ). I partcular, by Assumpto 2, H = Ke r K = L, whch mples that V = K. Now, V + covex mples m + + m + for =, ad as m + m+, the coeffcets the summato are all postve. Further, H S S = e + + m + m + ) S V E Sx + ) = Notg that m + = ad m + =, Assumpto mples that e H S/S. Next, for S K, H S > LS = ;.e., V S = H S. Thus, as H L ad H S/S, there exsts a s <K such that H S < LS for S< s ;.e., K S f S< s V S = H S f S s where s > uless r =. Ths completes the proof of the ducto step. Proof of Proposto. Proposto ad the covergece of H to H mply that the true optmal polcy s a threshold polcy. Notg that H S K S for S s ad H s = K s, H s K s from Proposto, whch mples s s. Proof of Proposto 2. The proof s very smlar to that of Propostos 9 ad. Refereces Am, K. I Jump dffuso opto valuato dscrete tme. J. Face 485) Be Ameur, H., M. Breto, P. L Ecuyer. 22. A umercal procedure for prcg Amerca-style Asa optos. Maagemet Sc. 485) Broade, M., J. Detemple Amerca opto valuato: New bouds, approxmatos, ad a comparso of exstg methods. Rev. Facal Stud. 94) Broade, M., P. Glasserma. 997a. Prcg Amerca-style securtes usg smulato. J. Ecoom. Dyam. Cotrol 28/9) Broade, M., P. Glasserma. 997b. Mote Carlo methods for prcg hgh-dmesoal Amerca optos: A overvew. Net Exposure Carr, P., R. Jarrow, R. Myem Alteratve characterzatos of Amerca put optos. Math. Face 22) Carrere, J. F Valuato of the early-exercse prce for dervatve securtes usg smulatos ad sples. Isurace: Math. Ecoom Das, S Dscrete-tme bod ad opto prcg for umpdffuso processes. Rev. Dervatves Res. 3) Fu, M. C., J. Q. Hu Sestvty aalyss for Mote Carlo smulato of opto prcg. Probab. Egrg. Iform. Sc Fu, M. C., S. B. Laprse, D. B. Mada, Y. Su, R. Wu. 2. Prcg Amerca optos: A comparso of Mote Carlo smulato approaches. J. Comput. Face 43) Grat, D., G. Vora, D. Weeks Smulato ad the earlyexercse opto problem. J. Facal Egrg. 53) Huag, J., M. Subrahmayam, G. Yu Prcg ad hedgg Amerca optos: A recursve tegrato method. Rev. Facal Stud. 9) Ju, N Prcg a Amerca opto by approxmatg ts early exercse boudary as a multpece expoetal fucto. Rev. Facal Stud. 3) Km, I. J. 99. The aalytcal valuato of Amerca optos. Rev. Facal Stud. 34) Laprse, S. B. 22. Stochastc dyamc programmg: Smulato based approaches ad applcatos to face. Doctoral thess, Uversty of Marylad at College Park, College Park, MD. Laprse, S. B., M. C. Fu, A. E. B. Lm, S. I. Marcus. 2. A ew approach to prcg Amerca-style dervatves. Proc. 2 Wter Smulato Cof Logstaff, F. A., E. S. Schwartz. 2. Valug Amerca optos by smulato: A smple least-squares approach. Rev. Facal Stud Merto, R. C Opto prcg whe uderlyg stock returs are dscotuous. J. Facal Ecoom Omberg, E The valuato of Amerca puts wth expoetal exercse polces. Adv. Futures Optos Res Tlley, J Valug Amerca optos a path smulato model. Tras. Soc. Actuares Tstskls, J. N., B. Va Roy. 2. Regresso methods for prcg complex Amerca-style optos. IEEE Tras. Neural Networks 44) Wu, R., M. C. Fu. 23. Optmal exercse polces ad smulatobased valuato for Amerca-Asa optos. Oper. Res. 5) Zhag, X Numercal aalyss of Amerca opto prcg a ump-dffuso model. Math. Oper. Res. 223)