Adding and Subtracting Black-Scholes: A New Approach to Approximating Derivative Prices in Continuous-Time Models

Transcription

1 Adding and Subracing Black-Scholes: A New Approach o Approximaing Derivaive Prices in Coninuous-Time Models Dennis Krisensen Columbia Universiy and CREATES Anonio Mele London School of Economics Firs draf: Ocober 28, 28. This version: Sepember 21, 29. Absrac We develop a new approach o approximaing asse prices in he conex of mulifacor coninuous-ime models. For any pricing model ha lacks a closed-form soluion, we provide a soluion, which relies on he approximaion of he inracable model hrough a known, auxiliary one. We derive an expression for he di erence beween he rue (bu unknown) price and he auxiliary one, which we approximae in closed-form, and use o creae increasingly improved re nemens o he iniial mispricing induced by he auxiliary model. The approach is inuiive, simple o implemen and leads o fas and exremely accurae approximaions. We illusrae his mehod in a variey of conexs, including opion pricing wih sochasic volailiy, volailiy conracs and he erm-srucure of ineres raes. Keywords: Asse pricing; sochasic volailiy; closed-form approximaions. JEL-Classificaion: G12, G13. We wish o hank Xavier Gabaix, Albero Mieo, Rolf Poulsen, Erns Schaumburg, Jialin Yu and paricipans a he Inernaional Conference on Finance, Universiy of Copenhagen, 25 for helpful suggesions and commens. Anonio Mele hanks he Briish EPSRC for nancial suppor via gran EP/C522958/1. The usual disclaimer applies. Please send correspondence o Dennis Krisensen a dk2313@columbia.edu and Anonio Mele a a.mele@lse.ac.uk. 1

2 1 Inroducion The las decade has winessed an ever increasing demand for new models addressing a number of empirical puzzles in nancial economics, which relae o pricing, hedging, and spanning derivaives conracs (e.g., Bakshi and Madan, 2; Du e, Pan and Singleon, 2), he erm srucure of ineres raes (e.g., Ahn, Dimar and Gallan, 22; Dai and Singleon, 22), or he aggregae sock marke (e.g., Gabaix, 28; Menzly, Sanos and Veronesi, 24). The vas majoriy of hese models rely on a coninuous-ime framework, which is by now one of he mos celebraed analyical ools in nancial economics. Marke praciioners have also increasingly relied on coninuous-ime models (e.g., Brigo and Mercurio, 26). The reason for his consensus abou he bene s of coninuousime modeling is ha wihin his framework, we are able o provide elegan represenaions for he price of a variey of coningen claims. A he same ime, coninuous-ime models call for an old and well-known pracical issue: how do we go abou dealing wih models no solved in closed-form? As is well-known, closed-form soluions for asse prices consiue he excepion, raher han he norm. This fac has led nancial economiss and praciioners o single ou classes of models for which a soluion could indeed be found, as in he celebraed a ne class (Du e, Pan and Singleon, 2; Heson, 1993), considered o be he benchmark, and in oher classes including quadraic models (Ahn, Dimar and Gallan, 22) or lineariy-generaing processes (Gabaix, 28, 29). However, i is an open quesion as o wheher he assumpions inroduced by hese models clash wih he acual empirical behavior of he sae variables in he economy. Quie ofen, models wih closed-form soluions res on simplifying assumpions ha are ypically unesed, for he sake of analyical racabiliy. This circumsance migh be problemaic, once we move owards a quaniaive assessmen of hese models: how would we know wheher, say, he reason for a model s rejecion would lie in is very economic raionale or, raher, he mere simplifying assumpions underlying i? The role of simplifying assumpions has also been called ino quesion during he 27 subprime crisis, which clearly revealed how a small change in he assumpions underlying a model can hen have dramaic e ecs on he ulimae pricing of derivaive producs (see IMF, 28). In dealing wih models no solved in closed-form, we ypically rely on wo alernaive approaches. The rs approach hinges upon he numerical soluion o a parial di erenial equaion obained hrough, say, nie-di erence or Fourier-inversion mehods (Schwarz, 1978; Hull and Whie, 199; Sco, 1997). The second approach, iniiaed by Boyle (1977), relies on Mone Carlo simulaions, in which a large number of rajecories needs o be generaed for he sae variables underlying he asse pricing model. Boh mehods can be cumbersome o implemen and, compuaionally, quie ime-consuming. This paper develops a new concepual framework o compue asse prices in nonlinear, mulifacor di usion seings. We develop closed-form approximaions o any given coningen claim model, which are easy o implemen and require very lile compuer power. Our main idea is o choose an auxiliary pricing model for which a soluion is available in closed-form. For example, we can choose a ne models o be he auxiliary models, as we shall illusrae hroughou he 2

3 whole paper. Addiional examples of candidae auxiliary models are he quadraic models sudied by Ahn, Dimar and Gallan (22) and he lineariy-generaing processes inroduced by Gabaix (28). For any auxiliary model, we derive an expression for he di erence beween he unknown price of he model of ineres and he auxiliary one. This expression akes he form of a condiional expecaion aken under he risk-neural probabiliy, which, under regulariy condiions, can be cas in erms of a Taylor series expansion. We approximae he unknown price, by reaining a nie number of erms from his series. Our mehod is highly general and herefore applicable in a wide range of seings, which range from he pricing of opions and he compuaion of he associaed Greeks, o he pricing of bonds and variance swaps. We develop several examples o illusrae how o use our general insighs, and provide numerical resuls ha show ha our mehod is quie precise and easily implemened. Our mehod relies, as explained, on Taylor series expansions of condiional expecaions. Similar expansions are widely used in nancial economerics and empirical nance (see, e.g., Aï-Sahalia, 22; Aï-Sahalia and Yu, 26; Aï-Sahalia and Kimmel, 29; Bakshi, Ju and Ou-Yang; 26; Schaumburg, 24). A key feaure in his lieraure is he expansion of a condiional expecaion of a coninuous-ime variable, say some condiional momen relaed o he shor-erm ineres rae expeced o prevail over a small ime-span - e.g., one day or one week a mos. Such small ime expansions are less useful, when he objecive is o approximae opion pricing models, eiher because (i) he presence of opionaliy leads o payo funcions ha are no di ereniable, as for example, in he simple European opion pricing case, or because (ii) he mauriy of he derivaive conracs migh occur a long mauriy daes, as for example, in he erm srucure of ineres raes. For hese reasons, small ime expansions have no been applied o asse pricing models previously, 1 alhough hey are reconsidered in recen work by Kimmel (28), which we discuss in a momen. Our approach sill relies on series expansions of condiional expecaions, bu works di erenly. Raher han being applied direcly o payo funcions, our expansions apply o pricing errors ha summarize he mispricing beween he rue pricing funcion and he auxiliary pricing funcion we choose o approximae he rue model by. These pricing errors are ypically di ereniable even if he payo s are no. In fac, afer compleing his paper, we came across he work of Kimmel (28), who develops a clever mehod o deal wih expansions of he payo funcions, which can be used o address he issues relaed o long mauriy daes. Alhough Kimmel s mehod can no be applied o deal wih payo s ha are no di ereniable, i can be used in e cien conjuncion wih ours, o implemen closed-form approximaions o our pricing errors, which, as noed, are ypically di ereniable. The mehod inroduced in his aricle can be also inerpreed, as an expansion of he risk-neural probabiliy implied by he model of ineres, around ha of some auxiliary model chosen by he user. As such, our approach shares similariies wih he srand of lieraure where opion prices are compued hrough an approximaion of he risk-neural densiy underlying he rue pricing 1 One early and isolaed excepion appears in Chapman, Long and Pearson (1999), which is indeed a special case of our mehod, as we shall explain. However, his special case does no allow one o deal wih derivaives wrien on non-di ereniable payo s. 3

4 model, as in Abadir and Rockinger (23) or in he saddlepoin approximaions considered by Aï-Sahalia and Yu (26), Rogers and Zane (1999), or Xiong, Wong and Salopek (25). In fac, approximaing he risk-neural probabiliy is a special case of our approach, as we shall explain. Our mehod carries some advanages over approximaions of condiional densiies, when applied o asse pricing. Firs, because i relies on a direc expansion of asse prices, our mehod avoids he numerical compuaion of mulidimensional Riemann inegrals agains an approximae condiional densiy. This feaure is aracive in mulifacor models such as hose ha involve sochasic ineres raes, sochasic volailiy or macro- nance deerminans of he yield curve. Second, he expansion we provide, carries new and ineresing economic conen, as we shall illusrae. For example, we shall see ha approximaing sochasic volailiy models hrough our approach leads o errors, which we can inerpre as hedging coss arising hrough he use of misspeci ed Black-Scholes delas. Finally, we provide an explici expression for he di erence beween he pricing funcion of he rue and he auxiliary model, which leads o a more direc analysis of he pricing error and simpler approximaions. The paper is organized as follows. In he nex secion, we illusrae our mehods hrough hree asse pricing examples. In Secion 3, we develop a general framework o approximae asse prices, provide exensions ha allow for he compuaion of sensiiviies of derivaive prices and, nally, relae our approach o hose relying on he expansion of risk-neural probabiliies. In Secion 4, we assess he numerical performance of our mehods in concree applicaions including he yield curve and opion pricing wih sochasic volailiy. Secion 5 concludes. A echnical appendix provides deails omied from he main ex. 2 The Approximaion Mehod in Three Examples We illusrae he basic ideas underlying our mehod by working ou hree examples, ranked in order of increasing complexiy: (i) he pricing of variance conracs, (ii) he pricing of European opions wihin he generalized Black-Scholes model, and (iii) he pricing of bonds in single-facor models of he shor-erm rae. 2.1 Log-conracs and Variance Swaps Our basic example perains o he recen nancial innovaion relaed o variance swaps, which are conracs guaraneeing a payo linked o he realizaion of he fuure variance of some asse price. As is well-known, he forward price of any liquid asse, F () say, is a maringale under he risk-neural probabiliy. Moreover, suppose ha F () exhibis sochasic volailiy, as follows: df () F () = () dw () ; 4

5 where W () is a Brownian moion under he risk-neural probabiliy, and he insananeous variance follows a coninuous-ime ARCH process (Nelson, 199), d 2 () = 2 () d + 2 () dw () ; (1) for some posiive consans, and, and a Brownian moion W () de ned under he risk-neural probabiliy. By enering ino a variance swap a ime, he holder of he conrac will receive, a some ime T, a payo proporional o, R T 2 (u) du 2 srike, for some consan 2 srike. Typically, he variance srike, 2 srike, is se so as o make he conrac worhless a he ime of originaion,, consisenly wih he marke pracice relaed he more familiar ineres rae swaps. Then, if he shor-erm rae is independen of he forward price variance, i mus be ha in he absence of arbirage opporuniies, he variance srike equals he expeced fuure inegraed variance, viz Z T 2 srike = E x; [ 2 (u)]du = 2E x; log F (T ) ; F () where E x; [ 2 (u)] = E[ 2 (u) j 2 () = x] denoes he condiional mean, and he las equaliy follows by a simple applicaion of Iô s lemma. The las erm is he payo of he so-called log-conrac inroduced by Neuberger (1994), which, as shown in many papers (Bakshi and Madan, 2; Brien-Jones and Neuberger, 2; Carr and Madan, 21; Demeer, Derman, Kamal and Zou, 1999), is equal o a cerain porfolio of opions, which is now being used by he CBOE o compue he new VIX index. Alernaively, one may use he parameric model in Eq. (1) o nd 2 srike and, hence, price he conrac. Then, one can calibrae he parameers and o make 2 srike consisen wih he VIX index, and proceed o use he model in Eq. (1) o price oher derivaive asses. Needless o say, o perform hese asks, i is crucial o compue he expecaion of he fuure variance, E x; [ 2 (u)]. In he conex of he model in Eq. (1), i is well-known ha E x; [ 2 (u)] has a closed-form expression, which leads o a closed-form soluion for he variance srike as of any ime, denoed as: w (x; ) = R T E x; [ 2 (u)]du. For he sake of his inroducory example, suppose we are unaware of his soluion, and wish o approximae he variance srike for he model in Eq. (1) wih anoher variance srike ha we can compue. Consider, for insance, he following varian of he Hull and Whie (1987) model, in which he insananeous variance is a maringale under he risk-neural probabiliy, d 2 () = 2 () dw () : For his model, E x;[ 2 (u)] = x, where E x; denoes he condiional expecaion under he Hull and Whie (1987) model, and he variance srike is jus w (x; ) x (T We now illusrae how o use he auxiliary variance srike prediced by he Hull and Whie model, w (x; ), o approximae he supposedly unknown variance srike, w (x; ). ). Firs, noe ha he variance srike is he undiscouned sum of all is fuure dividends, where each of hese 5

6 dividends, say ha as of ime +u, is he insananeous variance a ime +u. Therefore, he sum of he expeced insananeous capial gain and he insananeous dividend (he variance) mus be zero under he risk-neural probabiliy, in he absence of arbirage, d du E x;[w 2 ( + u) ; + u ] u= + x =. Tha is, he variance srike solves he following parial di erenial equaion, = Lw (x; ) + x; (2) wih boundary condiion w (x; T ) =, where L denoes he in niesimal generaor of 2 (): Lw (x; ) (x; + ( (x; x w (x; 2 : Likewise, w (x; ), he variance srike prediced by he auxiliary Hull and Whie model, sais es, wih boundary condiion w (x; T ) =. (x; + x; (3) Our key idea, now, is o subrac Eq. (3) from Eq. (2). A simple compuaion shows ha he mispricing arising from he use of he wrong model, w (x; ) w (x; ) = Lw (x; ) + (T ) (x) ; w (x; ), sais es, wih boundary condiion w (x; T ) =, and mispricing funcion given by: (x) = ( The soluion o he previous equaion, provided i exiss, can be represened as a condiional expecaion, due o he well-known Feynman-Kac heorem (see, e.g., Karazas and Shreve, 1991). The resul is he following represenaion of he variance mispricing, w (x; ) = Z T x). (T s) E x; [ (x (s))] ds: (4) The expecaion inside he inegral can be wrien explicily, in erms of he in niesimal generaor associaed wih he model of ineres (1), L, as follows, E x; [ (x (s))] = 1X (s ) n L n (x) ; (5) n! where, by a direc compuaion, L n (x) = ( ) n ( x). Our approximaion o he variance mispricing, w (x; ), is obained by replacing he expecaion E x; [ (x (s))] in Eq. (4) wih only he rs N erms of he series expansion in Eq. (5), as follows: w N (x; ) = Z T (T s) NX (s ) n L n (x) ds: (6) n! 6

7 Accordingly, our approximaion o he variance srike w (x; ) is: w N (x; ) = w (x; ) + w N (x; ) = x (T ) + ( x) NX (T ) n+2 (n + 2)! ( ) n ; where he second equaliy follows by he evaluaion of he inegral in Eq. (6). I is easily checked ha as N increases, w N (x; ) approaches he rue variance srike, w (x; ) = (T ) + (x ) 1 e (T ) =. 2.2 The Generalized Black-Scholes Opion Pricing Model Nex, we illusrae our basic ideas in he conex of he pricing of opions wrien on raded asses. Suppose ha he price of a sock, S () say, is he soluion o ds () S () = rd + (S () ; ) dw () ; (7) where W () is a sandard Brownian moion under he risk-neural probabiliy, and r is he shorerm rae, a consan. A European call opion pays b(s (T )) max fs (T ) K; g a mauriy ime T >, where K > is he srike price. Le w (x; ) be he price of he opion a ime < T given he curren sock price is S () = x. Le L be he in niesimal generaor associaed wih Eq. (7), Lw (x; ) (x; ) + rx (x; ) x w (x; 2 : (8) Under sandard regulariy condiions on he volailiy funcion (x; ), he opion price, w (x; ), sais es: = Lw (x; ) rw (x; ) ; (9) wih boundary condiion w (x; T ) = b (x), for all x. In general, he soluion o Eq. (9), provided i exiss, is no known in closed-form. The saring poin of our approximaion mehod is, as in he previous example on variance swaps, he choice of an auxiliary model ha can be solved in closed-form. In his conex, he Black and Scholes (1973) (BS, henceforh) model is a naural candidae. For his model, he volailiy in Eq. (7) is a consan, i.e. (x; ), for all x;. Accordingly, he BS opion price, w bs (x; ; ), say, is soluion o, = L w bs (x; ; ) rw bs (x; ; ) ; (1) where w bs (x; T ; ) = max fx K; g, and he associaed in niesimal operaor, L, is he same as in Eq. (8), bu wih replacing (x; ). Proceeding as we did in Secion 2.1, we now subrac Eq. (1) from Eq. (9). The resul is ha 7

8 he price di erence, w (x; ; ) w (x; ) w bs (x; ; ), sais es, = Lw (x; ; ) rw (x; ; ) + (x; ; ) ; (11) wih boundary condiion w (x; T ; ) = for all x, where our mispricing funcion is now: (x; ; ) 1 2 (x; ) wbs (x; ; ) : (12) Since w bs (x; ; ) is known, we can compue (x; ; ). By relying on he Feynman-Kac represenaion of he soluion o Eq. (11), he unknown pricing funcion can be expressed as he sum of he Black-Scholes price plus a condiional momen, which we shall inerpre in a momen: w (x; ) = w bs (x; ; ) + E x; Z T e r(u ) (S(u); u; ) du : (13) The inerpreaion of he mispricing funcion in Eq. (12) relaes o he hedging cos arising while evaluaing and hedging he opion hrough he BS formula. Precisely, suppose a rader sells he opion and wishes o hedge agains i hrough a self- nancing sraegy, in which he rades he underlying sock using he BS bs (x; ; ) =@x. Then, as shown by El Karoui, Jeanblanc- Picqué and Shreve (1998), and furher elaboraed by Corielli (26), our funcion in Eq. (12) is inerpreed as he insananeous incremen in he oal hedging cos arising from he use of a wrong model (he BS model) o hedge agains he rue model in Eq. (7). The condiional momen in Eq. (13) is aken under he sock price dynamics given by Eq. (7). Therefore, i is in general impossible o obain a closed-form expression for he second erm in Eq. (13). To make his formula operaional, we make use of a series expansion of he condiional momen in Eq. (13) in erms of he corresponding in niesimal generaor. As shown in he Appendix (see Proposiion A.3), Eq. (13) is indeed equivalen o: w (x; ) = w bs (x; ; ) + 1X (T ) n+1 (n + 1)! L n (x; ; ) ; (14) where L = L r. In pracice, his formula needs o be runcaed, yielding: w N (x; ; ) w bs (x; ; ) + NX (T ) n+1 (n + 1)! L n (x; ; ); (15) for some N. For example, a rs order approximaion (N = ) is given by w (x; ; ) w bs (x; ; ) + (T ) (x; ; ). Naurally, he unknown opion price w in Eq. (14) does no depend on, alhough is runcaion w N does. In Secion 4.1, we discuss choices of he nuisance parameer,. In our numerical experimens repored in Secion 4.1, we nd ha he numerical accuracy of w N (x; ; ) does no crucially depend on he choice of. 8

9 2.3 Bond Pricing in a Single-Facor Model For our hird example, we consider he pricing of bonds in a single-facor model of he shor-erm ineres rae. Suppose ha he shor-erm rae r is soluion o dr() = (r(); )d + (r(); )dw (); (16) for some drif and di usion funcions (x; ) and (x; ), and a sandard Brownian moion W () de ned under he risk-neural probabiliy. Le w (x; ) be he price as of ime of a defaul-free bond mauring a ime T >, when r () = x. Under sandard regulariy condiions on (x; ) and (x; ), w (x; ) is soluion o, wih boundary condiion w (x; T ) = 1. = Lw (x; ) xw (x; ) ; @x ; (17) Nex, le us inroduce, as usual, an auxiliary model, dr() = (r(); )d + (r(); )dw (); where (x; ) and (x; ) are some drif and di usion funcions. Associaed wih his model is a bond pricing funcion, w (x; ; ), which solves Eq. (17), wih boundary condiion w (x; T ) = 1, bu wih and replacing and. In w (x; ; ), he vecor is noaion we use o denoe he nuisance parameer vecor in he wo auxiliary funcions (x; ) and (x; ). I parallels he BS of he previous secion. I is easy o show ha he price di erence, w (x; ; ) w (x; ) w (x; ; ), sais es: = Lw (x; ; ) xw (x; ; ) + (x; ; ) ; (18) where w (x; T ) =, and he mispricing funcion is, (x; ; ) = ( (x; ) (x; (x; (x; ) 2 (x; 2 w (x; 2 : (19) Noe, he funcion summarizing he mispricing arising from he use of he auxiliary model, (x; ; ), has now a more complex srucure han ha we nd in Secion 2.2 in he opion pricing case. Is second componen, he convexiy adjusmen, is now familiar, by he resuls in Secion 2.2. Is rs erm, which is new, arises because he shor-erm rae is obviously no a raded risk, which makes he wo drifs under he risk-neural probabiliy, and, di er. In he opion example deal wih in Secion 2.2, insead, he asse underlying he conrac is radable, and is expeced o appreciae a an insananeous rae of rd, under he risk-neural probabiliy, independenly of he evaluaion model. A choice ha simpli es he funcion in Eq. (19) is =, which we shall use in our numerical experimens of Secion

10 By he Feynman-Kac represenaion, he soluion o Eq. (18) is, w (x; ; ) = Z T E x; exp Z u r(s)ds (r (u) ; u; ) du: Using he same ype of power series expansions as in Secions 2.1 and 2.2, we obain he following approximaing formula for he bond price funcion w (x; ): w N (x; ; ) = w (x; ; ) + NX (T ) n+1 (n + 1)! where now L is de ned as L (x; ; ) = L (x; ; ) x (x; ; ). L n (x; ; ) ; (2) As for he opion pricing model discussed in Secion 2.2, he approximaing bond price, w N (x; ), depends on some nuisance parameers, which arise hrough he use of he auxiliary model. For example, assume ha he drif of he auxiliary model is chosen so as o mach he drif of he model we approximae, =. Suppose, also, ha he di usion funcion of he auxiliary model is a consan. This consan, hen, is he nuisance parameer. Secion 4.2 provides examples based on his case of a consan nuisance parameer, and discusses simple choices of i, which sill lead o quie reliable approximaion oucomes. 3 A General Approximaing Pricing Formula In his secion, we derive a general approximaion formula for asse prices in models no solved in closed-form, obained following he same lead as ha for he hree examples discussed in Secion 2. In Secion 3.1, we inroduce noaion for he model we approximae and is auxiliary counerpar, and provide our approximaing formula. In Secion 3.2, we discuss approximaions for derivaives of he pricing funcions of ineres, which can be useful for hedging purposes. In Secion 3.3, we explain how our approach relaes o mehods ha rely on he expansion of risk-neural densiies. 3.1 The model and is approximaion We consider a mulifacor model in which a d-dimensional vecor of sae variables x () a ecs all asse prices in he economy. We assume ha under he risk-neural probabiliy x () sais es: dx () = (x () ; ) d + (x () ; ) dw () ; (21) where W () is a d-dimensional sandard Brownian moion under he risk-neural probabiliy, and (x; ) and (x; ) are some drif and di usion funcions. Le w (x; ) be he price of a derivaive wrien on he realizaion of x (T ), for some T >, when he curren sae is x () = x. The derivaive price is characerized by hree componens: Firs, is payo funcion a T as given by b (x (T )), for some funcion b (x). Second, le R (x; ) denoe he insananeous shor-erm ineres rae a ime, when he sae vecor is x () = x. Finally, le c (x; ) be he insananeous coupon 1

11 rae promised by he asse a ime. De ne he in niesimal generaor operaor L associaed o Eq. (21), Lw (x; ) (x; + dx i (x; ) (x; i dx i;j=1 2 ij (x; w (x; j : (22) The derivaive price, w (x; ), is hen he soluion o he following parial di erenial equaion: Lw (x; ) + c (x; ) = R (x; ) w (x; ) ; (23) wih boundary condiion w (x; T ) = b (x) for all x. In words, an invesmen ino his asse mus be such ha he expeced insananeous capial gain, Lw (x; ), plus he insananeous coupon rae, c (x; ), mus equal he insananeous yield on a safe asse, under he risk-neural probabiliy. To approximae he unknown price w (x; ), we inroduce an auxiliary model, dx () = (x () ; ) d + (x () ; ) dw () ; (24) for some drif and di usion funcions (x; ) and (x; ). Our objecive is a suiable expansion of he iniial model abou such an auxiliary model. We assume ha he dimension of he auxiliary model is he same as ha of he iniial model, i.e. x is a d-dimensional vecor. This assumpion does no enail any loss of generaliy, since we can always add consan componens, as we now explain. Suppose, for example, ha we wish o consider an auxiliary model wih a lower dimension, where he sae vecor y () 2 R m, wih m < d, solves, for some drif and di usion funcions Y and Y : dy () = Y (y () ; ) d + Y (y () ; ) dw 1 () ; and W 1 () is a m-dimensional sandard Brownian moion. The vecor process y > >, x m+1 x d where he las d m componens remain consan over ime, is hen a soluion o Eq. (24) wih: ( ( ;i (x; ) = Y;i (y; ) ; 1 i m Y;ij (y; ) ; 1 i; j m ;ij (x; ) = ; oherwise ; oherwise In Secion 4.3, we use his modeling rick o approximae he price of opions in models wih sochasic volailiy, using he Black-Scholes model as an auxiliary device. As for he derivaive associaed wih he auxiliary marke, we assume ha he derivaive is worh b (x (T )) a ime T, for some funcion b (). This complicaion helps illusrae a few properies of our approximaion mehods arising wihin he pricing of bonds, as we shall explain in Secion 4.2. However, in mos cases, one will choose b (x) = b (x) such ha he auxiliary pricing funcion, w (x; ), mimics w (x; ). Finally, and crucially, we assume ha we have a closed-form soluion w (x; ) for he pricing funcion in he markes where he sae vecor sais es Eq. (24). To save on noaion, we do no make explici ha he pricing funcion w (x; ) depends on nuisance parameers, as we did in our inroducory examples of he previous secion. 11

12 The price di erence, w (x; ) w (x; ) w (x; ), sais es, Lw (x; ) + (x; ) = R (x; ) w (x; ) ; (25) wih boundary condiion w (x; T ) = d (x). The wo adjusmen erms are given by d (x) = b (x) b (x) ; (26) and where (x; ) = dx i=1 i (x; (x; i dx i;j=1 2 ij (x; w (x; j ; (27) i (x; ) = i (x; ) ;i (x; ) ; 2 ij (x; ) = 2 ij (x; ) 2 ;ij (x; ) : Under sandard regulariy condiions reviewed in he Appendix, we can apply he Feynman- Kac represenaion of he soluion o he derivaive mispricing in Eq. (25), w (x; ), o obain he following represenaion of he asse price, w (x; ), in erms of ha arising wihin he auxiliary model, w (x; ): Theorem 1 (Asse Price Represenaion) Assume ha he wo soluions, w (x; ) and w (x; ) o Eq. (23) and (25) exis. Then he following ideniy holds: w (x; ) = w (x; ) + E x; exp Z T + E x; exp Z s Z T where x () sais es Eq. (21), and d; are given in Eq. (26)-(27). R (x (s) ; s) ds d (x (T )) R (x (u) ; u) du (x (s) ; s) ds; (28) The above represenaion formula holds under quie weak assumpions. 2 The righ hand side delivers an exac expression for he error arising from he use of he auxiliary model o price he claim, insead of he rue model. This represenaion is useful in is own righ, as i shows precisely how he pricing error is relaed o he auxiliary model. Ye our main goal is o look for an approximaion of he error erm in order o adjus he price w (x; ) for he error involved. Accordingly, our nex sep is o approximae he wo expecaions on he righ hand side of Eq. (28) using a series expansion. For he series expansion o hold, we have o impose sronger assumpions. For an N-h order approximaion o be valid, we need o assume ha d (x; ) and (x; ) are 2N imes di ereniable wih respec o x and N imes di ereniable wih respec o. Under his assumpion, we consider he following de niion: 2 The only condiion ha could no possibly hold in sandard asse pricing models is he linear growh condiion ha we impose on he drif and di usion erms. However, his condiion is only needed o ensure ha he soluions x () and x () o he wo sochasic di erenial equaions exis. Oher condiions oher han he linear growh condiions can be used o ensure hese soluions do acually exis. 12

13 De niion 1 (Asse Price Approximaion) The N-h order approximaion w N (x; ) o he unknown price w (x; ) in Eq. (28), a ime and sae x, is given by: w N (x; ) = w (x; ) + NX (T ) n d n (x; ) + n! where d (x; ) = d (x), (x; ) = (x; ) and NX (T ) n+1 n (x; ) ; (29) (n + 1)! d n (x; ) = Ld n 1 (x; ) R (x; ) d n 1 (x; ) ; n (x; ) = L n 1 (x; ) R (x; ) n 1 (x; ) : In Appendix A, we provide addiional regulariy condiions under which our asse price approximaion formula is valid, asympoically, in ha w N (x; )! w(x; ) as N! 1. I also provides error bounds applying o any xed approximaion order, N 1. Noe, nally, ha he approximaion in De niion 1 is only a means o esimae he righ hand side of Eq. (28) in Theorem 1. Oher mehods migh be available. For example, one could approximae he wo condiional expecaions appearing in he righ hand side of Eq. (28) hrough simulaions. However, one migh hen jus use simulaions, and direcly compue he condiional expecaion appearing in he Feynman-Kac represenaion of w (x; ). he power expansion in Eq. compuaion ime. 3.2 Approximaing Greeks The aracive feaure of (29) is, naurally, ha, once implemened, i requires virually no We ouline how our approximaion mehods can be used o obain closed-form approximaions o he parial derivaives of asse prices, which can be useful o esimae Greeks. The approximaion o hese parial derivaives are readily obained indeed, by di ereniaing he approximaing formula in Eq. (29) of De niion 1 wih respec o he variables of ineres. where The approximaion of he k-h order derivaive of w (x; ) is given k w N (x; k w (x; k + The wo sequences, d (k) n NX (T ) n d (k) n (x; ) + n! NX d (k) n (x; ) d n (x; k ; (k) n (x; ) n n (x; k : (T ) n+1 n (k) (x; ) ; (n + 1)! (x; ) and (k) n (x; ), can be evaluaed eiher numerically (using, say, nie-di erence mehods) or analyically. For example, o compue he approximaion o he rsorder derivaives, k = 1, we use he following recursion: d (1) (x; ) (x) =@x, (1) (x; ) (x; ) =@x and, d (1) n (x; ) = Ld (1) n 1 (x; ) R(x; )d(1) n 1 (x; ) + L(1) d n 1 (x; ) d n 1 (x; ) 13

14 where (1) n (x; ) = L (1) n 1 (x; ) R(x; )(1) n 1 (x; ) + L(1) n 1 (x; ) L (1) (x; ) = dx i (x; (x; ) i 2 dx 2 ij ) n 1 (x; 2 (x; j : In Appendix B, we provide full deails abou he recursive scheme needed o compue he erms d (2) n (x; ) and (2) n (x; ) relaed o he approximaion o he second-order parial derivaives of he asse price. 3.3 Risk-Neural Probabiliies Asse prices are condiional expecaions aken under he risk-neural probabiliy. Approximaing asse prices, hen, does necessarily enail approximaing risk-neural probabiliies. How do our approximaion mehods precisely relae o hose approximaing risk-neural probabiliies? In his secion, we link he expansion in Theorem 1 of he price w (x; ) abou he auxiliary price w (x; ), o he expansion of he risk-neural probabiliy of he asse pricing model around ha of he auxiliary pricing model. In Appendix C, we provide a few more echnical deails, such as hose peraining o he evaluaion of risk-neural densiies based on saddlepoin approximaions, which we show o be special cases of our approximaion mehods. For he purpose of simplifying he presenaion, le he insananeous shor-erm rae R and he coupon c in Eq. (23) be idenically zero, R (x; ) = c (x; ). Then, he wo prices, w (x; ) and w (x; ), are simply: Z Z w (x; ) = b (y) p (y; T jx; ) dy; R d w (x; ) = b (y) p (y; T jx; ) dy; R d where p and p are he risk-neural condiional densiies underlying he wo models: he rue, p, and he auxiliary, p. Clearly, we have: where p p Z w (x; ) = w (x; ) + b (y) p (y; T jx; ) dy R d (3) p is he di erence beween he wo condiional densiies, he risk-neural ransiion discrepancy, using a erminology due o Aï-Sahalia (1996). I is easy o see ha he asse price represenaion in Theorem 1 implies ha he following ideniy holds rue: Z R d b (y) p (y; T jx; ) dy = Z T E x; [ (x (s) ; s)] ds; (31) where is as in Eq. (27) (see Appendix C). Therefore, our expansion of R T E x; [ (x (s) ; s)] ds in De niion 1, is relaed o a corresponding expansion of he risk-neural ransiion discrepancy, p. In fac, in Appendix C, we derive an explici represenaion of p (y; T jx; ) in erms of a condiional expecaion (see Eq. (C1)), which highlighs he fac ha he represenaion and approximaions 14

15 of w in Theorem 1 and De niion 1 rely on equivalen represenaions and approximaions of he risk-neural condiional densiy. In spie of his equivalence, our mehods o er greaer exibiliy, as hey lead o closed-form approximaions for pricing errors ha are easily implemened. To illusrae, he righ hand side of Eq. (31), which is he pricing error arising from he use of an auxiliary asse price, can be easily compued hrough a power series expansion, as ha in De niion 1. In conras, he lef hand side of Eq. (31), which is he pricing error arising from he use of an auxiliary risk-neural densiy, requires he compuaion of a Riemann inegral. This compuaion can be cumbersome, especially when he dimension of he model, d, is large. Finally, he previous equivalence holds when he shor-erm ineres rae and he coupon are consan. In general, i is unclear as o how o use approximaions of risk-neural probabiliies o deal wih condiional expecaions such as, E x; exp Z T R (x (s) ; s) ds b (x (T )) : These cases need o be deal wih in many insances, especially hose including he pricing of xed income producs, or derivaives in he presence of sochasic ineres raes. Our mehods, which rely on approximaions direcly obained hrough auxiliary pricing funcions (no auxiliary risk-neural probabiliies), do handle hese cases in a quie naural manner. 4 Numerical Accuracy of Approximaion We assess he performance of our asse price approximaions in hree applicaions: (i) opion pricing in models wih CEV volailiy, such as ha in Secion 2.2; (ii) he erm-srucure of ineres raes; (iii) opion pricing wih sochasic volailiy. 4.1 Opion Pricing wih CEV Volailiy Consider he generalized BS model in Secion 2.2, which, as explained, we wish o approximae hrough our mehods, using he BS model as an auxiliary pricing device. The use of an auxiliary model ineviably leads o a nuisance parameer a parameer ha does no a ec he unknown price, bu does ener he pricing formula for he auxiliary model. In he BS case, he nuisance parameer is he insananeous volailiy in Eq. (1). There are several alernaives o deal wih his parameer. For example, le ^ be some esimae of. Then, we may approximae w(x; ) wih w N (x; ; ^ ). Alernaively, we may consider, ^ N (x; ) = arg min (w N (x; ; ) w (x; ; )) 2 ; where w (x; ; ) w bs (x; ; ), in erms of he noaion in Secion 2.2. As a simple example, we have ha for N =, ^ (x; ) = (x; ). Clearly, lim N!1 ^ N (x; ) = IV(x; ), where IV(x; ) denoes he Black-Scholes implied volailiy, de ned by w (x; ) = w bs (x; ; IV(x; )). For xed N, 15

16 hen, he unknown opion price can be approximaed by w N (x; ; N (x; )), or more generally, w N (x; ; M (x; )), where M N, as we do in he numerical experimens repored below. cev x To gauge he performance of he approximaion, we consider he CEV model, for which (x; ) = 1, where cev is consan and >. For his model, he opion price is known in closedform (see Schroder, 1989), which allows us o achieve a precise quaniaive assessmen of he approximaion. In Figure 1, we depic he approximaion errors resuling from our mehod, arising for di eren levels of he asse price, when he parameer values are cev = 1%, = 1 2, r = 5% and, nally, he srike price is K = 1 and ime-o-mauriy is hree monhs. The approximaing price is obained as w N (x; ; ^ (x)), where (x) = cev x 1, which explains why he percenage errors for N = and N = 1 coincide. More fundamenally, he errors are several orders of magniude lower han one percenage poin, wih only a very small number of correcion erms (N = 3). Figure 2 depics he errors arising in pricing he opion wih a larger mauriy (one year): our approximaion is sill quie accurae in his case, even for he more exreme far-in and far-ou of he money opions. 4.2 The Term Srucure of Ineres Raes The framework developed in Secion 3 allows us o approximae asse prices hrough quie general auxiliary models. This secion illusraes our mehods and analyzes he numerical performance of wo auxiliary models in approximaing he (supposedly unknown) soluion o he Cox, Ingersoll and Ross (1985) (CIR, henceforh) model of he yield curve, where he shor-erm rae is soluion o: dr() = ( r())d + p r ()dw (); (32) where >, > and > are consans. In erms of Eq. (23), herefore, he shor-erm rae is R (r; ) = r, where r is soluion o Eq. (32). The rs auxiliary model we analyze is jus one for which: (i) he payo paid by he bond is zero, raher han one; and (ii) he shor-erm rae is he same as in he rue daa generaing mechanism, ha in Eq. (32). We argue ha he approximaing formula we shall come up wih is ha provided by Chapman, Long and Pearson (1999). We sudy his case in Secion In Secion 4.2.2, we invesigae he performance of our mehods when he auxiliary model is such ha: (i) he payo of he bond equals he rue payo, one; and (ii) he auxiliary model is he Vasicek (1977) model, where he shor-erm rae is soluion o: dr () = ( r ())d + dw (); (33) for hree consans, and A Simple Power Expansion We consider a quie sraigh forward auxiliary marke, one where drif and di usion erms coincide wih he drif and di usion of he CIR shor-erm rae in Eq. (32), i.e. =, =. We 16

17 assume, however, ha in his auxiliary marke, he nal payo is idenically zero, b : The price of he conrac in he auxiliary marke is, naurally, zero, w (x; ) =, and we also have d (x) = 1, (x; ) =. By simple compuaions, hen, we obain ha he approximaion in Eq. (29) of De niion 1 collapses o: where: NX (T ) n w N (x; ) = d n (x; ) ; (34) n! d (x; ) = 1; d 1 (x; ) = x; d 2 (x; ) = (x; ) x 2 ; d 3 (x; ) = + (x; ) 2x (x; (x; 2 (x; 2 + x (x; ) x 2 : Eq. (34) is a sligh generalizaion o he power series expansion appearing in Chapman, Long and Pearson (1999, Proposiion 3), and Wilmo (23, p. 572). We assess he accuracy of his expansion o approximae he bond prices prediced by he CIR model, using he following parameer values: = = :6, = = :1 and = :12, and xing he iniial shor-erm rae level a x = 1%. The gures for, and roughly mach he average, sandard deviaion and rsorder auocorrelaion of he US overnigh rae, using pos-war daa. Figure 3 plos he percenage pricing error arising for N = 2; 4; 6; 8; and 1. A runcaion of Eq. (34) based on a few erms provides a quie accurae approximaion o shor mauriy bond prices. Insead, many more erms are needed for he resuling approximaion o work a longer mauriies, as also documened by Kimmel (28). As an example, he approximaion based on only he rs hree erms works quie poorly for T 3. We now urn o an expansion based on a richer auxiliary marke, i.e. one for which he nal payo is no zero A Beer Expansion: The Vasicek Model as Auxiliary Pricing Device The resuls peraining o he previous example can be improved, once we use a more informaive auxiliary marke, where he payo of he bond is one, i.e. b (x) = 1, such ha d (x) =. Consider, hen, an auxiliary marke, where he shor-erm rae is as in Vasicek (1977), and is soluion o Eq. (33). The soluion for he bond price, denoed wih w, is well-known as his is he simples example of an exponenial a ne model. The mispricing funcion,, is now, (x; ; ) = ( (x; ) ( (x; (x; ) 2 w (x; 2 ; where now = [ ] > is he nuisance parameer vecor arising from he use of he misspeci ed Vasicek model. Accordingly, he approximaion o he CIR model is given, formally, by Eq. (2). As 17

18 in he opion pricing problem of he previous secion, we have a nuisance parameer vecor o choose. In analogy wih he Secion 4.1, we could use: N (x; ) = arg min (w N (x; ; ) w (x; ; )) 2. Moreover, he wo models, Vasicek and CIR, have a linear drif, which we perfecly mach, by seing and equal o he numerical values we use for he CIR and, i.e. = = :6 and = = :1. The mispricing funcion, hen, simpli es o: (x; ; ) = x 2 w (x; 2 : Figure 4 plos he approximaion error agains ime-o-mauriy for di eren values of N, and he same parameer values of he CIR model used for he numerical analysis summarized in Figure 3. We se he nuisance parameer so as o equal 2 = 2 x, for a given level of he shor-erm rae, which corresponds o choosing N (x; ) = arg min 2 (x; ; ) for all N. Finally, we se he curren shor-erm rae o x = 1%, as in he previous secion. Compared o he approximaion error of he simple expansion, he approximaion based on he Vasicek model works considerably beer, as i only needs a few erms o achieve a quie high level of precision. Numerical resuls no repored here con rm ha he approximaion works equally well for oher iniial values of he shor-erm rae, x. 4.3 Opion Pricing wih Sochasic Volailiy Finally, we sudy he numerical performance of our mehods, by gauging he abiliy of he BS model, where sock volailiy is consan, o approximae he soluion o an European opion price prediced by a model, where volailiy is sochasic. Noe, even if he BS model has consan volailiy, our mehod, which in his secion we choose o be relying on he BS model, does allow o feed informaion abou sochasic volailiy. The reason is ha our power series expansions hinge upon he iniial mispricing arising from he use of he BS model, and his mispricing is a funcion of he iniial sae, price and volailiy. The expansions, hen, deliver re nemens ha are increasingly more informaive abou sochasic volailiy, as we shall illusrae. Naurally, o approximae he unknown price in he marke of ineres, we migh have wished o rely on an auxiliary marke where sock volailiy is random, raher han consan, as in he BS case. Our experimen o approximae a marke wih a given sae space (ha wih sochasic volailiy) hrough a marke wih a lower sae space (ha wih consan volailiy) serves he purpose o make a srong case for our mehods. All in all, he numerical issue we wish o invesigae here links o how many erms in he expansion are needed, in pracice, o feed informaion abou sochasic volailiy so as o make our approximaion reasonable. Our benchmark model is ha of Heson (1993), where, under he risk-neural probabiliy, he sock price, fs ()g, is soluion o, 8 >< >: ds () S () = rd + p v ()dw 1 () dv () = ( v) d + p v () dw 1 () + p 1 (35) 2 dw 2 () 18

19 for four consan,, and, and wo sandard Brownian moions, W 1 and W 2. In Eq. (35), fv ()g is he volailiy process. Le w (S; v; ) be he price of he European opion as of ime 2 [; T ], when he sock price is S and volailiy is v. Come ime T, w (S; v; T ) = max fs K; g, where K is he srike price, as usual. Subjec o his boundary condiion, he pricing funcion, hen, sais es, Lw (x; v; ) rw (x; v; ) =, where L is he in niesimal generaor operaor associaed o Eqs. (35): : (36) As explained, we expand he price w (x; v; ) in he sochasic volailiy marke around he auxiliary BS model, using he expansion se forh in Secion 3 (Theorem 1 and De niion 1). Noe ha for our auxiliary model, he sock price is soluion o: 8 >< >: ds () = rd + p v ()dw 1 () S () dv () = d + dw 1 () + p 1 2 dw 2 () where he iniial condiion for he volailiy process is v () =, and is consan. While his way of re-wriing he BS model appears more complicaed han needed, i acually furher illusraes our approach. Le us denoe he BS pricing funcion as we did in Secion 2.2, wih w bs (x; ; ), which is hen he opion price in our auxiliary marke. The price di erence, w (x; v; ; ) w (x; v; ) w bs (x; ; ), sais es, Lw (x; v; ; ) rw (x; v; ; ) = (x; v; ; ) ; (x; v; ; ) = 1 2 v 2 x w bs (x; ; 2 : where w (x; v; ; ) =, for all x and v. Appealing o Theorem 1 and De niion 1, our N-order approximaion o he rue price w (x; v; ), Heson s, is: w N (x; v; ; ) = w bs (x; ; ) + NX (T ) n+1 n (x; v; ; ) ; (37) (n + 1)! where n sais es n+1 (x; v; ; ) = L n (x; v; ; ) r n (x; v; ; ), wih. The approximaion in Eq. (37) is seemingly idenical o ha in he exended BS-model of Secion 2.2 (see Eq. (15)). In paricular, he mispricing funcion,, has he same funcional form as ha in Eq. (12), and sill bears he inerpreaion of an insananeous hedging cos arising from he use of a wrong model, he BS model. However, he in niesimal operaor L in Eq. (36), which n (x; ; ) ieraes upon, provides increasingly precise informaion abou random volailiy, as he ieraions develop. To illusrae, consider he rs-order approximaion o he Heson s price, w (x; v; ), viz w 1 (x; v; ; ) = w bs (x; ; ) + (x; v; ; ) (T ) (x; v; ; ) (T ) 2 : (38) 19

20 The rs erm on he righ-hand side is he BS price. The second, is he rs adjusmen, which is proporional o ime-o-mauriy, T, wih proporionaliy facor equal o he mispricing funcion,. Inuiively, consider Eq. (28) in Theorem 1. The payo s in boh Heson s and BS markes are obviously he same and, hence, d =. Therefore, in he conex of his secion, i is only he hird erm on he righ hand side of Eq. (28) ha maers. By approximaing he inegrand of his hird erm wih is value aken a, we obain he second erm in Eq. (38). This approximaion is quie rough: for example, he coe ciens of he sochasic volailiy process,,, and, do no ener. These coe ciens ener he hird, and nal, erm on he righ hand side of Eq. (38). This erm is produc of a quadraic adjusmen for ime-o-mauriy and he funcion 1, obained hrough one ieraion upon he mispricing funcion. I equals: 1 (x; v; ; ) = v 2 x 2 ' (x; v; ; ) ( v) w bs (x; ; 2 + vx w bs (x; ; w bs (x; ; 3 r (x; v; ; ) ; where he funcion ' is de ned as: ' (x; v; ; ) 1 w bs (x; ; ) w bs (x; ; ) 2 + w bs (x; 2 2 w bs (x; ; 3 w bs (x; ; ) + v w bs (x; ; ) 4 4 : The rs-order approximaion in Eq. (38) is no expeced o be accurae. I is only useful o illusrae, analyically, how he approximaing price becomes more informaive as we add new erms. For example, he volailiy of volailiy parameer,, eners 1 only hrough a correlaion channel: if =, does no ener w 1 (x; v; ; ) anymore. Ye one ieraion is su cien o feed he approximaing price wih informaion abou he parameers of he drif funcion of volailiy, and. Figure 5 depics he percenage approximaion error, as a funcion of he curren sock price, x, when: he curren value of he volailiy is such ha v = :5, he srike price K = 1, ime-omauriy is one year, T = 1, he shor-erm rae r = 1%, and he parameer values are roughly ha same as hose in Heson (1993): = 2, = :4, = :1, and = :5. Finally, we se he nuisance parameer, he BS, equal o p v. Figure 5 con rms ha he rs-order approximaion, while improving over ha obained for N = (i.e., ha semming from he use of he rs wo erms in Eq. (38)), sill produces signi can pricing errors. A he same ime, he approximaion in Eq. (37) considerably improves, and quie quickly, as we add new erms. Wih N = 3, for example, Eq. (38) provides a reasonable approximaion o he Heson s price, wih pricing errors amouning o less han one percen from he ruh, over a realisic range of variaion for he sock price. Wih N = 4, our approximaion produces pricing errors as small as :2%, even for far-ou-of-he money opions. 2

21 5 Conclusion We have developed a novel mehod o approximae he price of derivaive asses in he conex of mulifacor coninuous-ime models. The idea underlying our approach is quie simple: given a model wih no closed-form soluion, selec an auxiliary model, which has a closed-form soluion, and expand he unknown price around he auxiliary one. We apply his mehod o a variey of asse pricing problems, spanning from he yield curve o sochasic volailiy opion pricing, and show ha a runcaion of his expansion up o a few erms is quie accurae. Naurally, our approach does no require any simulaion, and once implemened, requires a rivial amoun of compuaional ime. Our mehod can be used in a variey of relaed conexs, such as, for example, hose peraining o pricing exoic conracs hrough calibraed volailiy surfaces. Calibraed volailiy surfaces, even when smoohed, are unlikely o lead o closed-form expressions for he price of exoic derivaives, say a far-ou-of-he money opion. These prices, hen, are ypically solved hrough simulaions. Our approach is a viable alernaive. A second example where our approach has a poenial is he esimaion and calibraion of asse pricing models. Esimaion of coninuous-ime models ypically ceners around a se of condiional momens for asse reurns, which can be readily obained hrough simulaion of our approximaing pricing formulae. 21

22 References Abadir, K. M. and M. Rockinger (23): Densiy Funcionals, wih an Opion-Pricing Applicaion, Economeric Theory 19, Ahn, D.-H., R. F. Dimar, A. R. Gallan (22): Quadraic Term Srucure Models: Theory and Evidence, Review of Financial Sudies 15, Aï-Sahalia, Y. (1996): Tesing Coninuous-Time Models of he Spo Ineres Rae, Review of Financial Sudies 9, Aï-Sahalia, Y. (22): Maximum-Likelihood Esimaion of Discreely-Sampled Di usions: A Closed-Form Approximaion Approach, Economerica 7, Aï-Sahalia, Y. and J. Yu (26): Saddlepoin Approximaions for Coninuous-Time Markov Processes, Journal of Economerics 134, Aï-Sahalia, Y. and R. L. Kimmel (29): Esimaing A ne Mulifacor Term Srucure Models Using Closed-Form Likelihood Expansions, forhcoming in he Journal of Financial Economics. Bakshi, G. S. and D. Madan (2): Spanning and Derivaive-Securiy Valuaion, Journal of Financial Economics 55, Bakshi, G. S., N. Ju and H. Ou-Yang (26): Esimaion of Coninuous-Time Models wih an Applicaion o Equiy Volailiy, Journal of Financial Economics 82, Barone-Adesi, G. and R. E. Whaley (1987): E cien Analyic Approximaion of American Opion Values, Journal of Finance 42, Black, F. and M. Scholes (1973): The Pricing of Opions and Corporae Liabiliies, Journal of Poliical Economy 81, Boyle, P. P. (1977): Opions: A Mone Carlo Approach, Journal of Financial Economics 4, Brennan, M. and E. Schwarz (1978): Finie Di erence Mehods and Jump Processes Arising in he Pricing of Coningen Claims: A Synhesis, Journal of Financial and Quaniaive Analysis 13, Brigo, D. and F. Mercurio (26): Ineres Rae Models: Theory and Pracice (2nd ediion). Heidelberg: Springer Finance. Brien-Jones, M. and A. Neuberger (2): Opion Prices, Implied Price Processes and Sochasic Volailiy, Journal of Finance 55,