A General Control Variate Method

Transcription

1 A General Control Varate Method Chun-Yuan Chu Tan-Shyr Da Hua-Y Ln Aprl 9, 13 Abstract The control varate method s a popular varance reducton technque used n Monte Carlo methods, whch are frequently used to prce complex dervatves. Ths technque s used n much fnancal lterature to explot nformaton about the errors n estmates of known quanttes whch are usually values of the dervatves that can be analytcally prced to reduce the error for estmatng an unknown quantty, whch s usually the prce of a complex dervatve of nterest. Ths paper generalzes the core dea of the control varate method so that t can be appled to reduce the prcng errors ncurred n many numercal prcng methods, such as the tree method, the characterstc-functon-based prcng method, and the convoluton-based prcng method. For example, a numercal method for prcng a complex dervatve, say an Asan opton, may need to calculate the densty functon of the average prce of the underlyng asset n the convoluton-based prcng method. However, ths functon can not be analytcally solved and must be numercally approxmated. Thus we fnd another analytcal functon that s close to the functon of nterest and explot nformaton about the errors n estmates of the analytcal functon to reduce the error for estmatng the functon of nterest. Numercal results shows that our approach can effectvely ncrease the prcng effcency of many numercal prcng methods. JEL classfcaton: C; G13 Keywords: numercal method, prcng, control varate, tree, fast Fourer transform Correspondng author. Department of Mathematcs, Florda State Unversty, 8 Love Buldng, 117 Academc Way, Tallahassee. E-mal: cchu@math.fsu.edu. Insttute of Fnance, Natonal Chao-Tung Unversty, 11 Ta Hsueh oad, Hsnchu 31, Tawan. E-mal: camelda@mal.nctu.edu.tw. Insttute of Fnance, Natonal Chao-Tung Unversty, 11 Ta Hsueh oad, Hsnchu 31, Tawan. E-mal: lnhuay@g.nctu.edu.tw.

2 1 Introducton Due to the boomng of fnancal markets and related academc studes, complex dervatve products, lke Asan optons, are constructed to meet customers need, and sophstcated prcng models, lke the Lévy model, are appled to ft fnancal markets better. Although the presences of these complex dervatves and prcng models mprove the effcency of fnancal markets, they also make the dervaton of analytcal prcng formulae ntractable. Thus we must rely on many dfferent numercal methods, such as Monte Carlo smulatons, tree methods, characterstcfuncton-based prcng methods, convoluton-based methods, and so on. In most numercal methods, contnuous prcng models or equatons are transferred nto dscrete counterparts that are sutable for numercal evaluatons. Take the C tree model proposed by Cox et al. 1979) n Fg. 1 as an example. The tme nterval [, T ] are dscretzed nto several tme steps, each wth the length t. The outgong bnomal structure of each node also dscretze the prce of the underlyng asset at each tme step, say Su and Sd at the tme step 1. The detals of ths fgure wll be dscussed n later sectons. Usually, the prcng results generated by a numercal method converge to the theoretcal dervatve prce as the dscretzaton get fner; for example, the prcng results generated by the C tree converge to the theoretcal dervatve prce as the number of tme steps ncreases see Duffe 1996)). However, ncreasng the resoluton of the dscretzaton would dramatcally) ncreases the computatonal tme complexty; n other words, the trade-off between the prcng accuracy and the computatonal tme s a bg challenge for usng numercal methods. Improvng the performance of the numercal prcng methods s an mportant ssue and s wdely studed n academc lterature. One possble approach s to mprove the effcency for computng numercal methods. For example, evaluatng dervatves under the tree model llustrated n Fg. 1 s tradtonally performed by the backward nducton method. oughly speakng, we frst evaluate the dervatve prce at each node located n the last tme step, say tme step n ths example. Ths nformaton s then used to derve the dervatve prce at each node n tme step 1. The above procedure s repeatedly appled by usng the nformaton at tme step to derve the nformaton at tme step 1 untl the dervatve prce at tme step s obtaned. Lyuu 1998) and Da et al. 7) suggest that some calculaton process of the backward nducton method can be saved by takng advantage of combnatoral propertes. Thus the prcng performance s mproved snce the combnatoral method requres less computatonal tme to acheve the same level of accuracy than the backward nducton method does. Another approach to mprove a numercal method s to mprove the convergence rate of the prcng results. Take the Monte Carlo prcng method MC hereafter) poneered by Boyle 1977) as an example. It randomly samples the changes of economc varables, say the underlyng asset s prce, to evaluate a dervatve s prce, whch s equal to the dscounted expected payoff of that dervatve under the so-called rsk-neutral probablty measure see Harrson and Plska 1981)). Note that the prcng result s also a random varable and a large amounts of samples and hence a large amount of computatonal tme) are requred to obtan a satsfactory prcng result. The control varate method s one of the mportant approaches that mprove the performance of the MC by reducng the varances of the prcng results. Instead of drectly applyng the MC to estmate an unknown quantty whch s usually the prce of a complex dervatve of nterest, the control varate method use MC to estmate the expected dfference between the unknown quantty and the known ones whch are usually the values of dervatves that can be analytcally solved. Thus the unknown quantty can be estmated as the known quantty evaluated by the analytcal formula) plus the expected dfference. The varance of the prcng result can be dramatcally reduced f the sample dfferences are close to zero. Ths approach s wdely adopted n the followng fnancal lterature. For example, Kemna and Vorst 199) take advantage of the analytcal prcng formulae for geometrc Asan optons to evaluate arthmetc Asan optons. It s further mproved by Han and La 1) by usng an optmal control process as a control 1

3 1 T Tme S P1 = + E E pp11 qp1 E C1 pa11 + qa1 = P1 = max C1, K S ) A Δt p q Su P11 = + E E pp1 qp E C11 pa + qa = P11 = max C11, K Su) A Sd P1 = + E E pp qp3 E C1 pa + qa3 = P1 = max C1, K Sd) A Su E P1 = max K Su,) C 1 = P = P A E 1 1 S E P = max K S,) C = P = P A E Sd E P3 = max K Sd,) C 3 = 1 Δt P = P A E 3 3 Tme step Fgure 1: Prcng Amercan Puts on the Two-tme-Step C Tree S denotes the underlyng asset s value at tme, K denotes the strke prce, T denotes the tme to maturty for the Amercan put, t denotes the length of a tme step, and denotes the dscount factor. p and q denote the upward and the downward branchng probabltes, respectvely. u and d denote the upward and the downward multplcaton factors, respectvely. Each node s represented by a node wth four felds, whch denote the prce of the underlyng asset, the prce of the European put ˆP j E, the contnuaton value ˆV j, and the prce of the Amercan put ˆP j A of that node. Overhead hat symbols are used to dstngush symbols nvolved n the tradtonal backward nducton method n ths fgure from the symbols used n the general control varate method dscussed later). The subscrpt j means the j-th node counted from the uppermost node) at the -th tme step. varate to gan further varance reducton. Besdes, Duan 1995) also apples the control varate technque to effcently evaluate optons under the GACH model. Dngec and Hormann 1) prce path dependent optons under Lévy processes by explotng nformaton about the error to evaluate the same optons under geometrc Brownan moton. Hull and Whte 1988) generalze the core dea of the control varate method by applyng t to prce an Amercan put on the C tree as llustrated n Fg. 1. Specfcally, they take advantage of the error for estmatng the prce of a European put on a C tree to reduce the error for estmatng the prce of an Amercan put on the same tree. They suggest that the Amercan put can be evaluated as the dfference between the estmates of prces of Amercan and European puts.e, ˆP A 1 ˆP E 1) 1 plus the European put prce whch can be analytcally solved by Black and Scholes 1973) formula. Ths paper wll nvestgate the property of the control varate method further and examne how useful t could be to mprove the performance of varous numercal prcng methods. oughly speakng, most numercal prcng methods would evaluate the result for applyng an operaton O on a certan functon X. Usually OX) can not be analytcally evaluated and must be estmated numercally. Our core dea s to suppress the error for estmatng OX) n order to mprove the performance of numercal prcng methods. To nhert the sprt of the control varate, we frst pck another functon Y as a control varate call t the proxy functon for smplcty. That s, 1 The prcng results generated by the C tree for European and Amercan puts are ˆP E 1 and ˆP A 1, respectvely.

4 the proxy functon Y s close to the functon X and OY ) can be analytcally solved. Then a more accurate estmaton for OX) s obtaned by numercally estmatng OX Y ) plus the analytcal result of OY ). Take prcng Amercan puts wth the C tree llustrated n Fg. 1 as an example. The Amercan-put-value functon of the underlyng asset prce at the tme step plays the role of the functon X. Fndng the contnuaton-value functon of the underlyng asset prce) at the tme step 1 based on the functon X plays the role of the operaton O. Operaton O can be numercally mplemented by a 1-tme-step backward nducton; n other words, the contnuaton value of a node s estmated as the expected dscounted opton values of ts two successor nodes as llustrated n the thrd feld of that node. The European-put-value functon at tme step plays the role of proxy functon Y to suppress the error for numercally estmatng OX). Ths s because the functon Y s close to X and each pont n the Europeanput-value functon at tme step 1 or OY )) can be analytcally solved by the Black and Scholes 1973) formula. Note that the Amercan put value for each node at tme step 1 s then evaluated as the maxmum of the contnuaton value and the exercse value at that node as llustrated n the fourth feld of that node. And the aforementoned evaluatons are repeated from the last tme step to the begnnng of the tree to obtan the fnal prcng result. Note that our approach can repeatedly apply the control varate method, one for each tme step, whle Hull and Whte 1988) approach apply once to overall prcng result. Numercal experments suggest that our approach outperforms ther approach. Two more examples are gven to demonstrate how the general control varate method s appled to mprove the performance of numercal prcng methods. Frst, to capture versatle behavors of the underlyng asset prce, varous processes, such as the jump dffuson process proposed by Merton 1976), the stochastc volatlty process proposed by Heston 1993), and so on, are nvented to ft the real world market better. Snce the characterstc functons of most of these processes are smpler to derve analytcally than the probablty densty functon of these processes, Carr and Madan 1999) and Carr and Wu 4) take advantage of the relaton between the Fourer transforms of opton prces and the characterstc functons to prce optons. Specfcally, the opton prces can be evaluated as an nverse Fourer transform of the functon that can be expressed n terms of the characterstc functon, and ths transform can be numercally evaluated by the fast Fourer transform FFT) whch can be vewed as a numercal ntegraton as dscussed later. However, the ntegrand oscllates sgnfcantly, whch make a hgh resoluton of dscretzaton and hence a hgh runnng tme complexty) necessary to obtan a satsfactory prcng result. To mprove the prcng performance, we frst pck a proxy functon; that s, ths proxy functon s close to the characterstc functon and the nverse Fourer transform of the proxy functon can be analytcally solved. Then we can explot nformaton of the error for applyng FFT on ths proxy functon to reduce the prcng error under our general control varate framework. Numercal results suggest that our revsed method provde more accurate prcng results and can avod the negatve-prce problem for prcng deep-out-of money optons. Next, the general control varate method s used to mprove the performance of the convolutonbased Asan opton prcng method. The payoff of an Asan opton depends on the average prce of the underlyng asset. The prcng problem s ntractable snce the densty functon of the average asset prce f A can not be analytcally solved. Carverhll and Clewlow 199) and Benhamou ) suggest that f A can be numercally evaluated by alternatvely applyng convolutons and the Jacoban transformaton 3 on densty functons. Numercal errors propagate and accumulate when these operatons are alternatvely appled numercally. To reduce accumulated numercal errors, we select proxy functons that are close to the nputs to these operatons. In addton, The contnuaton value means the value to hold an Amercan put wthout exercsng t mmedately. 3 It s used to mplement the transformaton of random varables; that s, to fnd the probablty densty of one random varable that s a functon another random varable. 3

5 applyng the operatons on the proxy functons must be analytcally calculated. Thus, the nformaton of the error for applyng operatons numercally on proxy functons are appled to suppress accumulaton errors and hence the errors of prcng results. The rest of ths paper s organzed as follows. In Sec., we wll survey the tradtonal control varate method and ntroduce the framework of our general verson. Then we demonstrate how general control varate method reduce the prcng error of the the tree prcng method, the characterstc-functon-based prcng method, and the convoluton-based prcng method n Sec. 3, 4, and 5, respectvely. A smple survey of each numercal method and the related numercal experments wll also be gven n the correspondng secton. Sec. 6 concludes the paper. The Control Varate Method.1 The Tradtonal Verson The tradtonal control varate method s a varance reducton technque used n MC see Glasserman, 4). It uses the estmaton error of known quanttes, say EY ), to mprove the estmaton of unknown quanttes, say EX), where X and Y are random varables. Then the estmaton of EX) can be mproved by estmatng EX Y ) wth MC plus the analytcal value of EY ) nstead of drectly estmatng EX) wth MC. Note that the varances for former and the latter estmaton are VarX) + VarY ) CovX, Y ) and VarX), respectvely. Ths mples that the control varance technque can dramatcally reduce the varance f we can fnd a proper Y that s hghly correlated to X. Note that the dervatve prce s equal to the expected present value of ts future payoff under the rsk neutral probablty measure. Thus much lterature take advantage of the tradtonal control technque to evaluate the value of a complex dervatve wth a smple dervatve that can be analytcally prced. For example, Hull and Whte 1988) prce an Amercan put by usng a European put s payoff as the control varate. Kemna and Vorst 199) prce an arthmetc Asan opton by usng a geometrc Asan opton s payoff as the control varate. Both approaches dramatcally mprove prcng performance snce the payoff of the latter smple) dervatve s hghly correlated to the former complex) dervatve.. The General Verson ecall that evaluatng the expected value of a random varable Z can also be expressed as the ntegral of the random varable Z multpled by ts densty functon. Thus we can extend the control varate technque from MC to the numercal ntegraton descrbed as follows. Note that random varables X and Y can be treated as real-valued functons defned on the sample space Ω.Thus the expected values EX) and EY ) can be expressed n terms of ntegratons Xω)dP ω) and Y ω)dp ω), respectvely, where f denotes the densty functon. If EX) Ω Ω can not be analytcally solved, we can estmate t by numercal ntegraton nstead of MC. To suppress the error ncurred by numercal ntegraton, we can mmc the strateges used n the tradtonal control varate method pckng a random varable Y that s hghly correlated to X and whose expected value can be analytcally solved. Analogously, we pck a functon Y that s close to X. The ntegraton Y ω)dp ω) can be analytcally solved. Thus the estmaton error Ω of EX) can be reduced by estmatng EX Y ) wth numercal ntegraton plus the analytcal value of EY ) nstead of drectly estmatng EX) wth numercal ntegraton. Many numercal prcng methods are desgned to numercally evaluate the result of applyng an operaton O nstead of smply calculatng the expected value) of a functon X. The prcng accuracy can be mproved by suppressng the error for estmatng OX), and our paper wll acheve ths goal by generalzng the aforementoned control varate method. Each numercal prcng method would have ts own operaton and the functon that play the roles of O and 4

6 X. For example, n a characterstc-functon-based prcng method lke Carr and Madan 1999) and Carr and Wu 4), the nverse Fourer transform and the characterstc functon of the underlyng asset s process play the roles of O and X, respectvely. A numercal method may numercally evaluate more than one operatons that can be mproved by the general control varate method. In a convoluton-based prcng method lke Carverhll and Clewlow 199) and Benhamou ), the convoluton and the Jacoban transformaton are alternatvely appled repeatedly. The general control varate method s appled on each operaton to mprove the accuracy of an ntermedate output rather than drectly on the prcng result. In addton, the ntermedate output s not necessary a value; t can be a dscretzed functon, say a dscretzed densty functon n the convoluton-based prcng method. Our numercal experments suggest that the accumulatons of these partal mprovements would result n overall performance mprovement of a numercal prcng method n terms of accuracy and computatonal tme. Applyng the control varate method to reduce the error for numercally estmatng OX) s smlar to the procedure mentoned above. Frst, we fnd a proper proxy functon Y that makes OY ) analytcally solvable and that s close to X. Then the error for estmatng OX) s reduced by estmatng OX Y ) numercally plus the analytcal value of OY ). Note that the effectveness of the general control varate method hghly depends on how the proxy functon Y s close to the functon X. For prcng Amercan puts on the tree, we use the European-put-prce functon of the underlyng asset as the proxy functon snce these two optons are almost dentcal expect the rght to exercse the opton early. For the characterstc-functon-based prcng method, we use the characterstc functon of the jump dffuson process proposed n Merton 1976) to approxmate the characterstc functon of the underlyng asset s prce process. The parameters of the former process s calbrated to make the frst fve cumulants of the former process match the cumulants of the latter process. For the convoluton-based prcng method, we use the normal densty functon to approxmate the densty functon nputted to the convoluton or the Jacoban transformaton. Agan, the parameters of the former densty functon are calbrated to make the frst two moments of the former functon match the estmated moments of the latter one. 3 Prcng Amercan Puts under Tree Methods 3.1 Background Introducton A put opton grants a holder the rght to sell the underlyng asset for a predetermned strke prce K. Whle a European put only allows the opton holder to exercse the rght at the maturty date T, an Amercan put allows the holder to exercse the rght at any tme τ pror to maturty T. Let S t denote the underlyng asset prce process at tme t for convenence. The payoff of a European put at maturty s K S T ) +, and the payoff to exercse an Amercan put at tme τ s K S τ ). By takng advantage of the rsk neutral valuaton method, the value of a European put P E and an Amercan put P A can be expressed as the expected dscounted payoff [ P E = E Q e rt K S T ) +] 1) P A = max E [ Q e rτ K S τ ) ], ) τ Υ where r denotes the rsk-free rate, Q denotes the rsk neutral measure, and Υ denotes the set of all stoppng tmes. Here we follow Hull and Whte 1988) by assumng that S t follow the lognormal dffuson process lns t /S ) = r σ / ) t + σw t, 3) where σ denotes the volatlty of the underlyng asset s prce, and W t denotes a standard Brownan moton. Note that Eq. 1) can be analytcally evaluated by the Black and Scholes 1973) 5

7 formula, whle Eq. ) can not be analytcally evaluated due to the dffculty to estmate the early exercse premum. The evoluton of a lognormal dffuson process n Eq. 3) can be dscretely smulated by a tree, say the C tree see Cox et al., 1979), as llustrated n Fg. 1. It dvdes the tme nterval [, T ] nto several equal tme steps, each wth length t. For convenence, the number xy n the subscrpt of ˆP E and ˆP A denote the values of the European put and the Amercan put at node,j) the j-th node counted from the uppermost node) at the -th tme step. The underlyng asset prce S at an arbtrary node,j) would move to Su node + 1,j)) wth probablty p or Sd node + 1,j + 1)) wth probablty q at the next tme step, where multplcaton factors u and d are set to e σ t and 1/u, respectvely, and branchng probabltes p and q are set to e r t d and 1 p, respectvely, to match the frst two moments of the lognormal dffuson process. u d Both European and Amercan puts can be numercally prced on a C tree by the backward nducton method; that s, we evaluate the opton values of the nodes) from the end of the tree to ts begnnng. Specfcally, the value of a European put for an arbtrary node pror to maturty, say ˆP E j, s evaluated as the expected dscounted value of ts successor node ˆP E j = p ˆP E +1,j +q ˆP E +1,j+1, where denotes the 1-tme-step dscount factor e r t. An Amercan put grants holder the rght to exercse the put at node,j) for the proft K S, j), where S, j) denotes the asset value at that node. The holder can also choose not to exercse the opton at node,j) mmedately and the value of the opton, called the contnuaton value for smplcty, can be estmated by ˆV j = p ˆP A +1,j + q ˆP A +1,j+1. 4) The holder wll decde whether he/she exercses the opton or not to maxmze the beneft; that s, the value of Amercan put at node,j) s ) max ˆVj, K S, j). 5) The prcng results for European and Amercan puts are ˆP E 1 and ˆP A 1, respectvely. The accuracy of the prcng results can be mproved as the dscretzaton of the tree get fner. 3. Applyng the General Control Varate Method The operaton that calculates the expected dscounted payoff as n Eqs. 1) and ) can be numercally estmated by the tree method. Hull and Whte 1988) extend the control varate method by applyng t on the tree method nstead of MC. They take advantage of the error for estmatng P E by the tree method to reduce the error for estmatng P A. The estmaton result ˆP 1 A ˆP 1 E + BSS, T ) s much more accurate than ˆP 1 A gven the same dscretzaton level of the tree), where BSa, b) denotes the Black and Scholes 1973) put opton prcng formula wth the underlyng asset prce a and the tme to maturty b. We generalze the control varate method further by applyng t to mprove the accuracy of certan operatons n the tree method. Here we detal how the contnuous prcng model n Eq. ) s transferred nto ts dscrete counterpart snce smlar concepts and notatons wll be used n the followng sectons. ecall that a n-tme-step tree method dscretzes the tme nterval [, T ] nto n equal-dstance tme steps. Thus, prcng the Amercan put on the tree can be vewed to repeat the operaton O B, to evaluate the contnuaton value functon of the underlyng asset s value at tme step 1 based on the opton-value functon at tme step, n tmes from the last step back to the begnnng of the tree. Take Fg. 1 as an example. Let P A, V, and P E denote the Amercan put value functon, contnuaton value functon, and the European put value functons of the underlyng asset at tme step, respectvely. P A can be dscretely approxmated by keepng the opton values at certan ponts, say S u, S, and S d, n a lst ˆP A ; that s, 6

8 T Tme S O1 = BS S, T) pe11 + qe1 C1 = O1 + A P = max C, K S ) 1 1 A E = P O Δt p q Su O11 = BS Su, Δt) C11 = O11 + pe 1 + qe A P = max C, K S u ) A E = P O Sd O1 = BS Sd, Δt) pe + qe3 C1 = O1 + A P = max C, K S d ) 1 1 A E = P O Δt Su O1 = max K Su,) C 1 = A P = O 1 1 E 1 = S O = max K S,) C = A P = O E = Sd O3 = max K Sd,) C 3 = A P = O 3 3 E 3 = Tme step Fgure : Prcng Amercan Puts wth the General Control Varate Method The meanngs of S, u, d, p, q, and K are the same as those n Fg. 1. Each node,j) s represented by a rectangle wth fve felds, whch denote the prce of the underlyng asset, the analytcal value of the European put Pj E, the contnuaton value V j, the value of the Amercan put Pj A, and the early exercse premum E j. ˆP A { ˆP 1, A ˆP, A ˆP 3}. A The operaton O B s numercally evaluated by applyng the 1-tme-step backward nducton formula lke Eq. 4) on ˆP A to obtan the dscretzaton of the contnuaton value functon at tme step 1, ˆV 1 { ˆV 11, ˆV 1 }. The above numercal estmaton can be expressed as ˆV 1 = O N ˆP A ), where OB N denotes that the operaton O B s evaluated numercally. Then the lst of the Amercan put values at tme step 1, ˆP A 1 s obtaned by judgng whether exercsng the put s benefcal or not by Eq. 5) at each node n the tme step 1. OB N and the early exercse judgement can be alternatvely appled to obtan the fnal prcng result ˆP 1. A Now we mprove the accuracy for estmatng the contnuaton value functon and as a consequence the Amercan put value functon by the general control varate method as llustrated n Fg.. The Amercan and the European-put-value functons play the roles of the functon of nterest X and the proxy functon Y, respectvely. Unlke Eq. 4), the contnuaton value for a node,j) pror to maturty s now estmated as V j pe +1,j+qE +1,j+1 + P E j, where E,j denotes the early exercse premum, or the dfference of the values of the Amercan and the European puts at node, j). Pj E denotes the European put value at node, j) that s analytcally evaluated by the Black and Scholes 1973) formula 4. The above formula for evaluatng V j s appled to evaluate the value of each element n the lst V that dscretely approxmate the functon V. Indeed, the above approxmaton can be expressed n terms of the general control varate method as V = OB N P A +1 P E +1) + O B P+1), E 6) where the dfference of two lsts P A +1 P E +1 s the lst of the early exercse premum or the dscretzaton of the dfference between the functons X and Y. Note that applyng O B on the European put value functon at tme step + 1, P+1, E wll obtan the functon P E. Ths can be 4 Note that P E j and P E are dfferent symbols dstngushed by the number of elements n the subscrpt. The latter symbol denotes the European put value functon at the tme step. 7

9 proved by applyng the tower rule on Eq. 1) as follows: [ ] P E S t ) = E Q e rt t) K S T ) + S t = E Q [ e r t E Q [ e rt +1) t) K S T ) + S +1) t ] S t ] = E Q [ e r t P E +1S +1) t S t ]. By takng advantage of the general control varate method n Eq. 6), we get the lst V that s more accurate than the lst ˆV evaluated by drectly applyng the backward nducton. The lst of Amercan put values P A s then obtaned by substtutng V nto Eq. 5). Note that P A tends to be more accurate than ˆP A snce V s more accurate than ˆV Our accurate prcng method for Amercan put s constructed by repeatedly applyng the above procedure as descrbed n Algorthm 1. Note that other numercal methods n the followng sectons wll also be descrbed n ths format wthout gvng a specfczed example lke Fg. for brevty. Algorthm 1 The General Control Varate Method for Prcng Amercan Puts on a n-tme-step Tree. 1: Evaluate the lst of the early exercse premum E n P A n P E n ). : for = n 1 down to do 3: Evaluate the lst of the contnuaton value V by substtutng E +1 nto Eq. 6). 4: Evaluate the lst of the Amercan put value P A by substtutng V nto Eq. 5). 5: Evaluate the lst of the early exercse premum E P A P E ). 6: end for 7: The prcng result s the Amercan put value at the begnnng node of the tree;.e, P A Numercal Experments ecall that the prcng results of a tree method converge to the theoretcal value about n ths example) as the number of tme steps n ncreases. Compared to the results generated by drectly applyng the backward nducton method to the tree model denoted by the gray dotted lne), the Hull and Whte 1988) method apples the control varate method once on the overall prcng results denoted by black dashed lne) and slghtly mprove the accuracy for prcng Amercan puts as llustrated n Fg. 3 a). That s, gven the same number of tme steps of the tree method), the prcng result generated by the Hull and Whte 1988) method tends to be closer to than the result generated by the drect backward nducton. In addton, our approach repeatedly apples the general control varate method and the prcng results denoted by the black sold lne) sgnfcantly mprove the accuracy and reduce the oscllaton problem. Note that a sophstcated prcng method lke our approach would requre extra computatons lke evaluatng the Black-Scholes prcng formula many tmes) and hence long computatonal tme. Thus evaluatng the performances among dfferent prcng methods should compare the convergence rates of these methods n terms of computatonal tme as llustrated n Fg. 3 b). Indeed, most of extra computatons n Algorthm 1 can be skpped by takng advantage of the property of the early exercse boundary see Curran, 1994). For convenence, we call a node,j) an early-exercse node f the opton holder decdes to exercse the Amercan put mmedately at that node.e, V j < K S, j)); otherwse, we call t a non-early-exercse node. There exsts a default boundary, lke the gray curve n Fg., that dvde the tree nodes nto two groups: the group of non-early-exercse nodes n the upper part of the tree and the group of early-exercse nodes n the lower part. We mmc the Curran 1994) method to estmate the default boundary and use the general control varate method to mprove the estmaton. For each non-early-exercse node,j), we drectly evaluate the early exercse premum E j wthout calculatng Pj A ; that s, we evaluate E j as pe +1,j+qE +1,j+1 wthout nvolvng the Black-Scholes 8

10 Value Number of Tme Steps n) Value Tme Second) a) b) Fgure 3: Convergence Comparsons for Prcng Amercan Puts The x axes n panel a) and b) denote the number of tme steps n thousands) and the computatonal tme, respectvely. The y axes n both panels denote the Amercan put prces generated by dfferent numercal methods. The gray dotted lnes denote the results generated by drectly applyng the backward nducton method on the C tree. The black dashed lnes denote the results generated by Hull and Whte 1988) method. The black sold lnes denote the results generated by our method. The ntal underlyng asset s prce S s 4, the tme to maturty T s 3 year, the strke prce K s 35, the rsk-free rate r s.5, and the volatlty σ s.. prcng formula. For early-exercse nodes, we only calculate the values for the node rght below the default boundary. The evaluaton for other early-exercse nodes can be skpped wthout nfluencng the prcng results as mentoned n Curran 1994). Indeed, the above concept can be used to mprove all tree-based prcng algorthms analyzed n ths numercal experment. The experment suggests that our method stll converges much faster than the other two methods n terms of the computatonal tme as n Fg. 3 b). 9

11 4 Characterstc-Functon-Based Prcng Methods 4.1 Background Introducton Many prce processes, say the varance gamma process studed by? and the stochastc volatlty model studed by Heston 1993), are proposed to ft the market phenomena better. However, ths makes dervatve valuaton become ntractable snce under those underlyng assumptons, the analytcal formulae for even the European optons no longer exst. Fortunately, the return characterstc functon for a vast class of underlyng processes are stll avalable. Therefore, many scholars have proposed plenty of lterature to take advantage of the return characterstc functons to prce dervatves see e.g. Baksh and Chen 1997), Bates 1996), Chen and Scott 199)). Among all characterstc-functon-based prcng methods, the method studed by Carr and Madan 1999) s the most computatonally favorable one snce t further ncorporates the computatonal power of the FFT nto the contngent valuaton regon. Consder a target underlyng dynamcs G by whch we are nterested n the European call prce mpled. Let CG EU and ψ G respectvely denote the European call prce and the Fourer transform of the damped call prce mpled by the dynamcs G. Denote by S t the underlyng asset s prce at tme t. Carr and Madan 1999) propose that the value of an European call opton wth strke prce e k and tme to maturty T s: C E Ge k ) = E Q [ e rt S T e k ) +] = k e s e k )qs) ds = e dk π e vk ψ G v) dv, 7) where q s the rsk-neutral densty of ln S T mpled by the dynamcs G; ϕ G denotes the characterstc functon of q: ϕ G v) e vs qs) ds; ψ s the Fourer transform of the damped call prce c E e k ) e dk C E e k ) for some dampng constant d > 5, whch can be further expressed n terms of ϕ: ψ G v) = e rt ϕ G v d + 1)) d + d v + d + 1)v. 8) As a result, the call value n Eq. 7) can be approxmated by the Trapezod rule numercally: C E Ge k ) e dk π N e ηjk ψ G jη) η [ δ j δ N j ], 9) j= where η s the dstance between quadrature ponts; N s the number of quadrature ponts; δ j s the Kronecker delta whch equals 1 when j s zero and otherwse. Note that ths method can also be used to smultaneously evaluate N otherwse dentcal European optons wth dfferent strke prces va the FFT. 4. The Use of Our Generalzed Control Varate Technque on the Characterstc-Functon-Based Method In ths secton, we wll demonstrate the applcaton of our general control varate technque on the characterstc-functon-based method. 5 The Fourer transform of C E e k ) doesn t exst snce C E e k ) tends to S as k tends to, and therefore s not a square ntegrable functon. Carr and Madan 1999) nvoke the dampng constant d so that the Fourer transform of c E e k ) exsts. 1

12 Assume that there s a sutable proxy functon ψ proxy such that a closed-form soluton for the European call opton on the underlyng dynamcs mpled by ψ proxy, whch s denoted by Cproxy, EU s avalable. Let ψ resdual v) ψ G v) ψ proxy v) be the dfference between ψ G and ψ proxy. Based on the decomposton above, CG EU can be decomposed nto a proxy part and a resdual part: C EU G e k ) = e dk π = e dk π e vk ψ G v) dv e vk [ψ proxy v) + ψ resdual v)] dv C EU proxye k ) + C EU resduale k ). 1) Snce we choose a proxy functon ψ proxy whch closely approxmates ψ so that the ampltude of ψ resdual becomes smaller and smoother than ψ, the ampltude of the second order dervatve of ψ resdual wll have a better chance to be smaller than ψ. Because the Trapezod rule s used for numercal approxmaton, the quadrature errors ncurred are bounded above by the supremum of the second order dervatve of the ntegrand. Consequently, our method wll have a better chance to generate less quadrature errors because the upper bound of quadrature error has been sgnfcantly mproved. In ths paper we choose ψ proxy based on the Merton s jump-dffuson model. Merton 1976) assumes that the underlyng asset s dynamcs follows the stochastc dfferental equaton: ds t = µ + σ ) S t dt + σs t dw t + e J t 1)S t dm t, 11) where µ and σ are respectvely the nstantaneous mean and varance of the return condtonal on the absence of the Posson events; W t s a standard Brownan moton; M t s a Posson process wth ntensty λ, whch s ndependent of W t ; J t s a sequence of ndependent normal random varables wth mean µ J and standard dervaton σ J. In Merton s jump-dffuson model, the characterstc functon of the logarthm of underlyng asset s prce can be expressed as an nfnte seres: ϕ MJD v) = exp λt + vµt + ln S ) 1 ) σ T v λt expvµ J σ J v ))h. h! Let H be a postve nteger. We choose the frst H terms of ϕ MJD to play the role of ψ proxy : where ϕ MJDH v) exp ψ proxy v) e µ+ σ +λeµ J + σ J h= 1))T ϕ MJDH v d + 1)), d + d v + d + 1)v λt + vµt + ln S ) 1 ) H 1 σ T v Consequently, the proxy part call prce wll be: C E proxye k ) = H 1 h= h= λt expvµ J σ J v ))h h! e λ T λ T ) h BS C S, e k, σ h, r h ) for u =, 1,..., m 1, h! where λ λe µ J + σ J ; BS C S, e k, σ h, r h ) denotes a lst of Black-Scholes call opton prces wth underlyng asset s prce S, strke prce e k, tme to maturty T, rsk-free rate r h µ + σ hµ J + σ J ), and volatlty rate σ T h σ + hσ J T. 11 +

13 Defne by ψ G {ψ G ), ψ G η),, ψ G Nη)} the lst of ψ s whch s used to estmate C E G ek ) n Eq. 9)and ψ POXY {ψ proxy ), ψ proxy η),, ψ proxy Nη)}. Let O IF symbolze the Fourer nverson operaton. The orgnal Carr and Madan s method whch apples the Fourer nverson numercally to ψ G to prce optons: C E Ge k ) = O N IFψ G ). In contrast, our generalzed control varate method explots the nformaton of ψ G ψ POXY to reduce the numercal errors generated by estmatng C E G ek ): C E Ge k ) = O N IFψ G ψ POXY ) + O IF ψ proxy ). Next we calbrate the parameters for the proxy characterstc functon to approxmate the characterstc functon mpled by the dynamcs G. We solve for the correspondng parameter values through cumulant matchng method. Snce the characterstc functons for a vast class of process) admt nfnte dfferentablty, we assume the frst fve cumulants of ln S T dynamcs G exsts and equal to m 1 T + ln S, m T, m 3 T, m 4 T, and m 5 T, respectvely. The key dea s to match the frst fve cumulants of ln S T for dynamcs G wth those for Merton s jump-dffuson model, whch can be obtaned through the dfferentaton of ϕ MJD. Ths yelds µ + λµ J ) T + ln S = m 1 T + ln S, 1) σ + λ )) µ J + σj T = m T, 13) λµ 3 J + 3µ J σj)t = m 3 T, 14) λµ 4 J + 6µ Jσ J + 3σ 4 J)T = m 4 T, 15) λµ 5 J + 1µ 3 Jσ J + 15µ J σ 4 J)T = m 5 T. 16) Note that the left hand sdes of the above equatons are the frst fve cumulants of ln S T Merton s jump-dffuson model. We frst rewrte Eq. 14) as for σ J = m 3 λµ 3 J 3λµ J 17) and then substtute all the σ J n Eq. 15) and Eq. 16) nto Eq. 17). Ths yelds µ 6 Jλ + 4m 3 µ 3 J 3µ Jm 4 )λ + m 3 =, µ 6 Jλ 3µ J m 5 λ + 5m 3 =. By equatng the above two equatons, we get λ = 4m 3 3m 5 µ J 3m 4 µ J + 4m. 18) 3µ 3 J Now we can replace the λ and σj n Eq. 14) by Eq. 17) and Eq. 18) respectvely and get a polynomal equaton of µ J as follows: 48µ 4 Jm 4 3 1µ 3 Jm 3 3m 4 + 9µ Jm 3 8m3 m 5 + 5m 4) 54µJ m 3m 4 m 5 + 9m 3m 5 =. 19) Clearly, the varable µ J n Eq. 19) can be easly solved snce t s a polynomal equaton of µ J wth degree 4. If there are two or more roots among the real numbers, we pck the one mples smallest λ because n ths case, Cproxy EU wll play a major role n C EU6. However, t s possble 6 Note that we nvoke the frst H terms of the call prce mpled by Merton s model as the the proxy part call prce. Therefore, we choose the answer wth smaller λ snce n ths case, the probablty mass of the Posson dstrbuton wll be more concentrated on the frst H terms. 1 G

14 that Eq. 19) does not have any roots among the real numbers. If so, we smply pck a proper µ J whch mnmzes the absolute value of the left hand sde of Eq. 19). Once µ J s determned, the other 4 parameters, λ, σ J, µ, and σ, can also be determned sequentally by Eqs. 18), 17), 1), and 13). Algorthm sums up the use of our generalzed control varate method on Carr and Madan s characterstc-functon-based prcng method. Algorthm The General Control Varate Method for the Characterstc-Functon-Based Method. 1: Fnd a sutable proxy functon ψ proxy to approxmate ψ G ; : Decompose the call prce lst C EU G eku ) nto C EU proxye ku ) and C EU resdual eku ); 3: Evaluate the prce lst C EU proxye k u ) by analytcal formulae and C EU resdual ek u ) va the FFT; 4: Add them up and get C EU G ek u ). 4.3 Numercal esults 13

15 LogError LogError Tme a) LogError c) e) Tme Tme LogError LogError LogError b) d) Fgure 4: Convergence of prcng results. The prcng errors are computed from, n panel a) and b), the VG model wth S = 1, K = 1, r =, T = 4 months, ς =.113, ν =.1686, ϑ =.1436, n panel c) and d), the Heston model wth S = 1, K = 1, r =, T = 4 months, κ = 1.49, θ =.671, ϵ =.74, ρ =.571, V =.6 and, n panel e) and f), Kou s jump-dffuson model wth S = 1, K = 98, r =.5, T = 6 months, ω 1 = 1, ω = 5, λ = 1, p =.4, q =.6, σ =.16. In panel a), c), and e), lnes are computatonal tme used plotted aganst the logarthm of the absolute prcng errors. In panel b), d), and f), lnes are the logarthm of the grd spacng plotted aganst the logarthm of the absolute prcng errors. The squares denote the prcng results for the orgnal Carr and Madan s method; the stars, trangles, and crcles denote the prcng results for the Carr and Madan s method n combnaton wth our generalzed control varate technque wth H = 1, 3, and 7, respectvely. The dampng coeffcent, the number of quadrature ponts, and the grd spacng are set to be 1.5, 496, and.5, respectvely, as n Carr and Madan 1999). The prcng results mpled by the Carr and Madan s method wth N = and η = 1 are chosen as the benchmarks f) LogΗ LogΗ LogΗ 14

16 5 Convoluton-Based Asan Opton Prcng Method 5.1 Background Introducton Consder an arthmetc Asan call opton wth strke prce K ntated at tme whch matures at tme T. Denote by S t ts underlyng asset s prce at tme t. The payoff of an Asan call opton s dependent on the average underlyng asset s prce sampled at tmes t, t 1, t,..., t n, where = t < t 1 < t < < t n = T. Defne the average underlyng asset s prce as A 1 n + 1 n S t. Accordng to Harrson and Plska 1981), the value of an arthmetc Asan call opton can be expressed as the expectaton of the dscounted payoff : = C AS = E Q [ e rt A K) +]. ) Snce the expectaton n ) s ntractable, Carverhll and Clewlow 199) and Benhamou ) dscuss an useful convoluton-based method for prcng arthmetc Asan optons. The underlyng asset s return from tme t 1 to tme t, denoted as lns t /S t 1 ), s assumed to follow the same Brownan moton n Eq. 3). Defne U e = S t /S t 1. Snce S t = S t e 1+ +, Eq. ) can be rewrtten as: A = 1 n + 1 n = S t = S t n U 1 + U 1 U + + U 1 U U n ) = S t n U U U n U n )))) = S t 1 + e e e n e n )))) n + 1 = S t ) 1 + e 1 +ln1+exp +ln1+exp n)))) n + 1 = S t 1 + e D +µ ), 1) n + 1 where µ s defned by the followng recurrence relaton: µ n 1 = E[ n ] µ 1 = E[ ] + ln1 + expµ )), = n 1, n,..., 1; the sequence { } s defned recursvely as follows: where D n 1 = n µ n 1, ) 1 = + Z, = n 1, n,..., 1, 3) Z ln1 + exp + µ )) µ 1. 4) Let f U represent the densty functon of a random varable U for convenence. Snce Z s a ncreasng functon of, f Z can be expressed by a functon of f D accordng to the Jacoban transformaton method: f Z x) = { f D lne x+µ 1 1) µ ) ex+µ 1 f x > µ e x+µ 1 1 1, otherwse. 15 5)

17 Besdes, accordng to Eq. 3), the densty functon f D 1 equals the convoluton of f and f Z. Thus f D 1 can be effcently calculated by applyng the nverse Fourer transform on the product of the characterstc functons of f and f Z. Snce the analytcal soluton for the nverse transform s not avalable, Benhamou ) harnesses the fast Fourer transform to evaluate f D 1 on a grd lst b = x m < < x = < < x m = b between [ b, b], a wndow whch contans the bulk of the probablty mass of all denstes nvolved n hs algorthm, and stores t n 1. Moreover, when estmatng f Z wth Eq.5), the Jacoban transformaton method has to be mplemented n combnaton wth numercal nterpolaton. It s because only the densty lst but the analytcal formula of f D s avalable. Therefore Benhamou ) numercally estmates a lst of densty ponts on f Z by the Jacoban transformaton method and stores t n Z. Algorthm 3 sums up the Carverhll-Clewlow and Benhamou algorthm, where the lst of densty estmate for f s stored n. Algorthm 3 Carverhll-Clewlow and Benhamou algorthm. 1: Evaluate the denstes values n the lst D n 1. : for = n 1 down to 1 do 3: Calculate Z by applyng the Jacoban transformaton method on n combnaton wth numercal nterpolaton see Eq. 4)). 4: Calculate 1, the dscrete convoluton of and Z, va the FFT. 5: end for 6: Evaluate the Asan opton value C AS by the densty functon D and Eqs. ) and 1). 5. The Use of Control Varate Technque on the Convoluton-Based Prcng Method Two operatons, convoluton and the Jacoban transformaton, are alternatvely appled repeatedly to help construct the forward probablty densty of an Asan opton for prcng. In ths secton we are gong to demonstrate the applcaton of our generalzed control varate method on the convoluton-based prcng method studed by Carverhll and Clewlow 199) and Benhamou ). Frst of all, n the thrd step of Algorthm 3, we mplement the the Jacoban transformaton operaton n combnaton wth the lnear nterpolaton wth a vew to estmatng Z. It s because only a densty lst but the analytcal formula of f D s avalable; therefore the numercal nterpolaton s used to estmate {f D lne x+µ 1 1)) µ } xm x=x m see Eq. 5)). Clearly, the accuracy of Z depends sgnfcantly on the estmaton qualty of {f D lne x+µ 1 1) µ )} x m x=x m. When more accurate prcng results are requred, usually plenty of computatonal grds have to be used to mprove the nterpolaton qualty. Nevertheless, dong ths wll deterorate the prcng speed dramatcally. It provdes us a motvaton to apply our generalzed control varate technque to speed up the Carverhll-Clewlow and Benhamou algorthm. Assume that there s a proxy functon f proxy the densty lst {f D lne x+µ 1 1) µ )} x m x=x m. We decompose f D f resdual, where f resdual whch serves as a control varate for estmatng nto two parts: f proxy and f D f proxy be the dfference between f D and f proxy. Accordng to the and D resdual, where D proxy on the same grds as ; D resdual densty decomposton above, can also be decomposed nto D proxy denotes the lst of densty values calculated from f proxy D resdual denotes the dfference between and D resdual 7. To estmate Z more accurately, nstead of drectly applyng the Jacoban transformaton numercally to, we frst calculate 7 If we vew and D resdual vector subtracton. as m-dmensonal Eucldean vectors, then the mnus operator can be treated as 16

18 the denstes lst {f proxy lne x+µ 1 1) µ ) ex+µ 1 e x+µ 1 1 }xm x=x m then apply the Jacoban transformaton numercally to D resdual {fd resdual lne x+µ 1 1) µ ) ex+µ 1 by the control varate f proxy, and n order to estmate the denstes e x+µ 1 1 }x m x=x m. Fnally, the two estmatons above wll add up to Z. In ths way the evaluaton of Z wll become much more effcent snce the use of our generalzed control varate technque sgnfcantly suppress the numercal errors generated by estmatng {f D lne x+µ 1 1) µ )} x m x=x m ; therefore fewer grd ponts hence less computatonal tme) are needed to attan a gven accuracy level. Eq.6) llustrates the frst applcaton of our generalzed control varate technque on the convoluton-based prcng method: The operaton J symbolzes the Jacoban transformaton, whch can be used to estmate a lst of probablty denstes of a transformed random varable; {f proxy control varate term. lne x+µ 1 1) µ ) ex+µ 1 e x+µ 1 1 }xm x=x m s the densty lst whch can be calculated from the Z = J CV ) = J N D resdual ) + {f proxy lne x+µ 1 ) e x+µ 1 1) µ } x m e x+µ 1 1 x=x m. 6) Moreover, n the fourth step of Algorthm 3, dscrete convolutons are employed to estmate 1. Our generalzed control varate method can be harnessed to speed up ths step as well. Assume that there s a proxy functon f proxy Z whch possesses the followng two crtcal propertes to play the role of a control varate when estmatng 1 : One s that f proxy Z should closely approxmates f Z. The other s that there should be an analytcal soluton for the convoluton of f proxy Z and f. Let fz resdual f Z f proxy Z be the dfference between f Z and f proxy Z. Accordng to the densty decomposton above, Z can also be dvded nto Z proxy and Z resdual, where Z proxy denotes the lst of densty values calculated from f proxy Z on the same grd ponts as Z ; Z resdual Z Z proxy denotes the dfferences between Z and Z proxy. Based on the set-ups above, the dscrete convoluton step n Algorthm 3 can be decomposed nto two parts: The dscrete convoluton of Z proxy and plus the dscrete convoluton of Z resdual and. The frst part can be evaluated analytcally by the closed-form soluton for the convoluton of f proxy Z and f wthout nvokng any numercal technques. Only the second part needs to be evaluated va the FFT. Comparng to the orgnal the Carverhll-Clewlow and Benhamou algorthm whch only takes advantage of the nformaton on the densty lst, our algorthm explots more nformaton on the analytcal formula f. Ths could be one of the explanatons why the use of our generalzed control varate technque can suppress numercal errors generated by dscrete convoluton. Eq.7) llustrates the second applcaton of our generalzed control varate technque on the convoluton-based prcng method: Besde nvokng the FFT to evaluate the dscrete convoluton of Z and, we addtonally take advantage of the closed-form soluton for the convoluton of f proxy Z and f to make better estmates for the convoluton of Z and. The notaton denotes the convoluton. 1 = CV Z, ) = N Z resdual, ) + f proxy Z, f ). 7) Algorthm 4 sums up the use of control varate method on Benhamou algorthm One of the most crtcal problems s that what functons could be chosen as sutable control varates, namely, f proxy and f proxy Z. Snce we assume the underlyng assets dynamcs follows the geometrc Brownan moton, one natural canddate for the control varates could be the probablty densty functon of the normal dstrbuton functon. Here comes another problem: what parameters, say mean and varance, should we choose for the control varates? Now we wll ntroduce an dea to tackle ths problem. Sutable parameters for the control varates can be determned by moment matchng method. The frst step s to fnd the frst few moments of and Z. Comparng to determnng the frst n th order moments of and Z, t s much easer to 17