ADAPTIVE FINITE DIFFERENCE METHODS FOR VALUING AMERICAN OPTIONS. Duy Minh Dang

Transcription

1 ADAPTIVE FINITE DIFFERENCE METHODS FOR VALUING AMERICAN OPTIONS by Duy Mnh Dang A thess submtted n conformty wth the requrements for the degree of Master of Scence Graduate Department of Computer Scence Unversty of Toronto Copyrght c 2007 by Duy Mnh Dang

2 Abstract Adaptve Fnte Dfference Methods for Valung Amercan Optons Duy Mnh Dang Master of Scence Graduate Department of Computer Scence Unversty of Toronto 2007 We develop space-tme adaptve methods for valung Amercan optons wth strong emphass on Amercan put optons. We examne the applcaton of adaptve technques to the Black-Scholes partal dfferental equaton problem assocated wth an Amercan put opton n the context of non-unform second-order fnte dfferences. At certan tmesteps, we obtan a redstrbuton of the spatal ponts based on a montor functon that attempts to equdstrbute the error. The proposed fnte dfference dscretzaton on non-unform grds and redstrbuton of the spatal ponts lead to lnear complementarty problems wth M-matrces. The Projected Successve Over-relaxaton and a penalty method are consdered to handle the free boundares. We study and compare the accuracy and effcency of the consdered methods. A complete proof of convergence and unqueness of the projected SOR method under certan condtons s also presented.

3 Acknowledgements Ths research was supported by the Natural Scence and Engneerng Research Councl (NSERC) of Canada, the Ontaro Graduate Scholarshp (OGS) Program, and the Department of Computer Scence at the Unversty of Toronto. I would lke to express my specal thanks to my thess supervsor, Professor Chrstna Chrstara, for ntroducng me to the feld of mathematcal fnance, and also for her support, gudance, and encouragement durng my MSc program. I also would lke to thank Professor Kenneth Jackson for hs valuable suggestons and remarks for mprovements. Fnally, I would lke to thank my beloved parents and my lovely Dem for ther endless support, encouragement and love.

4 Contents 1 Introducton 1 2 Adaptve Mesh Methods Spatal error estmaton Crtera for equdstrbuton of error Algorthm Summary Prcng Amercan Optons Prcng Model Dscretzaton wth Fnte Dfferences Dscretzaton for LV Tme Dscretzaton Iteratve Methods Projected SOR Method Penalty Method Adaptve Mesh Methods for Prcng Amercan Optons Algorthm Descrpton Stablty and Convergence Analyss Prelmnares L-matrx Property v

5 5.3 Dagonal Domnance Implementaton Issues Crank-Ncolson Method Numercal Results European Optons Delta and Gamma Valuatons Opton Valuaton Intal Guesses and the SOR method Amercan Optons Unform Mesh Methods Results Adaptve Mesh Methods Results Effcency Comparson Early Exercse Boundary Conclusons and Future Work 99 A Convergence Proof of PSOR 101 v

6 Lst of Tables 6.1 Model parameters for European and Amercan optons Expermental results for the European put opton at S = 100 obtaned by unform mesh methods. The analytcal results of the opton value, delta, and gamma are , , and , respectvely Expermental results for the European put opton at S = 100 obtaned by adaptve mesh methods. The analytcal results of the opton value, delta, and gamma are , , and , respectvely Expermental results for delta and gamma of the European put opton at S = 100. Rannacher smoothng s used. The analytcal values of delta and gamma are and , respectvely Expermental results for the European call opton at S = 100. Rannacher smoothng s used. The analytcal value of the call s Expermental results for the European put opton at S = 100. Rannacher smoothng s used. The analytcal value of the put s Errors of unform and adaptve mesh methods appled to the European call opton. The analytcal value of the call s Approxmate values are from Table Errors of unform and adaptve mesh methods appled to the European put opton. The analytcal value of the put s Approxmate values are from Table v

7 6.9 Expermental results for the European call opton at S = 100 obtaned on unform grds usng SOR-1. The analytcal value of the call s Expermental results for the European call opton at S = 100 obtaned on unform grds usng SOR-2. The analytcal value of the call s Iteraton comparson between SOR-1 and SOR-2 on unform grds for the European call. Numercal results and statstcs are from Tables 6.9 and Expermental results for the Amercan put opton at S = 100 obtaned wth unform mesh methods and constant tmesteps usng PSOR-1. Reference numercal soluton from [14] s Expermental results for the Amercan put opton at S = 100 obtaned wth unform mesh methods and constant tmesteps usng usng PSOR-2. Reference numercal soluton from [14] s Iteraton comparson between PSOR-1 and PSOR-2 for unform mesh methods and constant tmesteps. Numercal results and statstcs are from Tables 6.12 and Expermental results for the Amercan put opton at S = 100 obtaned wth unform mesh methods wth constant tmesteps usng PENALTY-1. Reference numercal soluton from [14] s Expermental results for the Amercan put opton at S = 100 obtaned wth unform mesh methods wth constant tmesteps usng PENALTY-2. Reference numercal soluton from [14] s Expermental results for the Amercan put opton at S = 100 obtaned wth adaptve mesh methods usng PSOR-2. The true value was generated wth accuracy 10 6 based on the results n [14] and extrapolaton v

8 6.18 Comparson of numercal results for the Amercan put opton between unform mesh methods and adaptve mesh methods usng PSOR-2 wth constant tmesteps. Numercal results and statstcs are from Tables 6.13 and The true value was generated wth accuracy 10 6 based on the results n [14] and extrapolaton Expermental results for the Amercan put opton at S = 100 obtaned wth adaptve mesh methods and constant tmesteps usng PENALTY-1. The true value was generated wth accuracy 10 6 based on the results n [14] and extrapolaton Expermental results for the Amercan put opton at S = 100 obtaned wth adaptve mesh methods and constant tmesteps usng PENALTY-2. The true value was generated wth accuracy 10 6 based on the results n [14] and extrapolaton Comparson of numercal results for the Amercan put opton between unform mesh methods and adaptve mesh methods usng PENALTY-2 wth constant tmesteps. Numercal results and statstcs are from Tables 6.16 and The true value was generated wth accuracy 10 6 based on the results n [14] and extrapolaton Expermental results for the Amercan put opton at S = 100 obtaned wth adaptve mesh methods and varable tmesteps usng PENALTY-2. The true value was generated wth accuracy 10 6 based on the results n [14] and extrapolaton Comparson between adaptve mesh methods usng PENALTY-2 wth constant and wth varable tmesteps. Numercal results and statstcs are from Tables 6.20 and The true value was generated wth accuracy 10 6 based on the results n [14] and extrapolaton Model parameters (I) for comparson of early exercse pont n Amercan optons. 95 v

9 6.25 Model parameters (II) for comparson of early exercse pont n Amercan optons Early exercse boundary for set of parameters lsted n Table Unform and adaptve mesh methods are used on a grd wth PENALTY-2 and Rannacher smoothng. S max = Early exercse boundary for set of parameters lsted n Table Unform and adaptve mesh methods are used on a grd wth PENALTY-2 and Rannacher smoothng. S max = x

10 Lst of Fgures 2.1 A non-unform spatal grd at tme t = t ν Molecules of the mplct Euler (a), explct Euler (b), and Crank-Ncolson (c) Detals and notatons of the fnte dfference grd Detals and notatons of an adaptve step European put valued numercally usng Crank-Ncolson tmesteppng on a unform grd European put valued numercally usng Crank-Ncolson tmesteppng wth adaptve mesh methods on a grd Observed error dstrbuton of the European optons on a grd Maxmum resduals of ntal guess for SOR methods on the unform grd Maxmum resduals of ntal guess for PSOR methods on the unform grd The locatons of mesh ponts used by adaptve mesh methods for the Amercan put on a grd Amercan put value at the strke prce versus computatonal cost for unform and adaptve mesh methods wth dfferent teratve solvers Amercan put value at the strke prce versus computaton cost for adaptve mesh methods wth dfferent teratve solvers x

11 6.9 Amercan put value at the strke prce versus computatonal cost for varous methods Profle of the free boundary obtaned by unform and adaptve mesh methods wth set of parameters from Table 6.24 on a grd Profle of the free boundary obtaned by unform and adaptve mesh methods wth set of parameters from Table 6.25 on a grd x

12 Lst of Notaton Symbol S r σ T t τ ν ν max n t ν τ ν t ν or τ ν S ν ν {S ν } n =0 Meanng space varable (asset prce) rsk-free nterest rate volatlty of the asset prce fnal tme (the expry of the opton) the tme varable (forward) τ = T t (backward) ndex for the tmestep the total number of tmesteps the total number of spatal subntervals value of the tme varable t at the νth tmestep value of the tme varable τ at the νth tmestep the νth tmestep sze value of the th spatal grd pont at the tmestep ν the spatal partton at tmestep ν {h ν } n =1 the spatal stepszes at tmestep ν wth h ν = S ν S ν 1 V V ν unknown functon (opton prce) the soluton at tmestep ν V ν value of the soluton at node (S ν, t ν ) or (S ν, τ ν ) V ν k, V ν value of the soluton at S k of spatal partton k at tme τ ν vector of approxmate values to V at the νth tmestep V ν approxmate value to V (S ν, t ν ) or V (S ν, τ ν ) V ν k vector of approxmate values to V at tme τ ν on space partton k V ν k, the th component of V ν k (V ν k, V (Sk, τ ν )) V ν,(j) jth estmate of V ν by an teratve method x

13 V V,ν V,ν ˆV (S, t) payoff functon vector of payoff values on ν the th component of V,ν montor functon ˆV ν vector of approxmate values to ˆV (S, τ ν ) ˆV ν the th component of ˆV ν ( ˆV ν ˆV (S ν, τ ν ) ξ(s, t) gradng functon vector of approxmate values to ξ(s, τ ν ) on partton ν ξ ν vector of approxmate values to ξ S (S, τ ν) on partton ν ξ ν the th component of ξ ν (ξ ν ξ(s ν, τ ν )) ξ ν r ν an estmate of the error for the th subnterval [S ν 1, S ν ] r ν the average of all r ν s ω ν relaxaton factor for the PSOR method for tmestep ν Matrces are denoted by bold upper-case letters wth entres denoted by correspondng bold lower-case letters wth subscrpts. Vectors are denoted by bold lower-case letters wthout subscrpts. A a,j b b a real matrx named A the (, j) entry of A, 1, j n wth n beng the dmenson of A a real vector named b the th component b, 1 n wth n beng the length of b Frequently, gven a matrx A, we defne the followng matrces D U L dagonal part of A strctly upper trangular part of A strctly lower trangular part of A x

14 Chapter 1 Introducton The evaluaton of fnancal opton contracts s of consderable mportance n fnance. An opton s a contract between the holder and the wrter that gves the rght, but not an oblgaton, to the holder to buy or sell a certan asset by a certan tme for a gven prce. In partcular, a call opton gves the holder the rght to buy, whereas a put opton gves the holder the rght to sell ts underlyng asset, or brefly the underlyng, for a prescrbed amount, known as strke prce. An mportant feature of such contracts s the tme when the contract holders can exercse ther rghts. If ths occurs only at the maturty date, the opton s classfed as a European opton. If holders can exercse any tme up to and ncludng the maturty date, the opton s sad to be an Amercan opton. The opton premum s the prce at whch the opton contract s traded. The premum s pad by the potental holder (buyer of the opton) to the wrter of the opton. In return, the wrter of the opton s oblgated to delver the underlyng asset to the opton holder f the call s exercsed or buy the underlyng asset f the put s exercsed. In any case, the wrter keeps the premum whether or not the opton s exercsed. It s then mportant to determne a far prce for an opton accurately. Generally speakng, there are two basc ways to determne the prce of an opton: analytcal methods and numercal methods. The value of a European opton s gven by the soluton of 1

15 CHAPTER 1. INTRODUCTION 2 the Black-Scholes partal dfferental equaton (PDE) (see, e.g. [37]). In some cases, European optons can be prced usng analytcal formulas. In the semnal papers by Black and Scholes ([2]), and Merton ([24]), the authors derve explct formulas for plan European optons, whch are wrtten on a sngle underlyng asset and do not pay dvdends. However, most optons traded on exchanges are Amercan. For Amercan optons, the Black-Scholes model results n a free boundary problem and unfortunately, one can not fnd explct closed-form solutons to the Amercan opton prcng problem n general. Due to the non-exstence of a general closed-form soluton for Amercan optons, researchers and practtoners resort to numercal methods, such as lattce methods, smulatonbased methods, PDE-based methods, etc. We refer the reader to the recent paper by Broade and Detemple [4], and references theren for a revew and comparson of several numercal methods for valung Amercan optons. Here, we would lke to brefly revew some popular numercal methods for Amercan opton prcng problems. Monte-Carlo smulaton and lattce methods such as bnomal and trnomal trees are very popular among fnancal nsttutons. When used to value an opton, Monte-Carlo methods smulate the development of the underlyng asset n a rsk-neutral world to determne many possble path movements. The mean of expected payoffs of each path s obtaned and dscounted at the rsk-free rate to get an estmate of the value of the dervatve. However, prcng Amercan-style optons va Monte-Carlo method stll remans a very challengng problem due to the exstence of the free boundary ([5], [15]). The path smulaton requres a forward algorthm, whereas prcng optons wth early exercse features generally requre backward algorthms from the maturty date (.e. the end of the path rather than the begnnng). The problem arsng from usng smulaton n prcng Amercan-style optons results from usng a forward procedure to a problem that requres a backward algorthm. As a consequence, the methods wll overestmate the true value of the opton ([15]). Ths could be overcome by usng a technque called Least Squares Monte Carlo derved by Longstaff and Schwartz ([22]). The major advantage of Monte-Carlo smulaton s that ts convergence rate s generally ndependent of

16 CHAPTER 1. INTRODUCTION 3 the number of state varables and thus can be easly adapted to accommodate complex payoffs and complex stochastc processes, multple underlyng assets, and path-dependent contracts. However, for low dmenson problems, Monte-Carlo smulaton approaches suffer from low effcency due to hgh smulaton tme. The bnomal model was frst ntroduced by Cox, Ross and Rubnsten [8]. The underlyng assumpton of the model s that the prce of the underlyng asset follows a random walk. At each tmestep untl maturty, t has a certan probablty of movng up and down by a certan amount. The scheme eventually yelds a bnomal tree. To calculate the premum, one could trace the tree backwards, startng at the maturty date where the payoff s known. Durng the tracng process, the prce of the underlyng asset at each node s calculated and compared to determne whether t s more useful to hold or to execute the opton. The process stops when one reaches the root where the desred prce for the opton s obtaned. The trnomal tree approach nvolves a thrd level of prce movements n the tree and s descrbed n [17]. One major dsadvantage of lattce methods s that both bnomal and trnomal trees are equvalent to explct fnte dfference methods, hence suffer from a temporal stepsze restrcton of the form t c S 2, for some constant c. Partal dfferental equaton (PDE) based approaches are very popular for problems n low dmensons because of effcency reasons. In addton, PDE methods allow us to obtan values for all ponts n the spatal doman and hence the term structure of the opton,.e. the development of the opton value functon for each tmestep, can be easly vsualzed. Ths approach s frst ntroduced by Brennan and Schwartz n [3]. They propose a fnte dfference scheme ncorporated wth an teratve projecton method to explctly deal wth the early exercse constrant. In ths thess, we adopt ths approach for the Amercan opton prcng problem on a sngle asset wth constant volatlty and nterest rate. Due to the early exercse possblty, the problem n a PDE approach can be formulated as a tme dependent lnear complementarty problem (LCP). One approach used by many researchers s to dscretze wth e.g. fnte dfferences and

17 CHAPTER 1. INTRODUCTION 4 reduce the problem to a sequence of dscrete LCPs, one per each tmestep. Ths formulaton wll be descrbed n more detal n Chapter 3. Below we wll brefly descrbe two popular ways of solvng the LCP: relaxaton methods and the penalty methods. One common technque for the soluton of the LCPs n ths category s the projected successve over-relaxaton method, also known as PSOR (see, for example [30], [33], [37]). Ths method was frst proposed by Cryer ([9]) under the assumpton that the underlyng matrx s symmetrc postve defnte. In [9], a proof of the unqueness and convergence of the PSOR soluton s gven under the assumpton that the underlyng matrx s symmetrc postve defnte. A refned verson of ths approach s presented n [19], and s based on the observaton that the soluton of the problem at each tmestep can be obtaned as a synthess of the two ndependent components correspondng to the two regons of the spatal doman separated by the free boundary. Penalty methods have been used by several authors. In [40], Zvan, Forsyth and Vetzal ntroduce a penalty formulaton of the dscretzed equatons that enforces the early exercse constrant. In the same paper, a proof of the unqueness and convergence of the penalty soluton s presented under the assumpton that the resultng matrces are M-matrces. A smlar approach s taken by Nelsen, Skavhaug and Tveto n [25] and they ntroduce a penalty term n the contnuous equatons. Adaptve mesh methods are wdely used n the numercal soluton of PDEs (see, for example, [7], [10], [12], [13] [35], [36]). These methods compute the optmal placng of a gven number of dscretzaton ponts so that a chosen norm of the error n the computed approxmaton s mnmzed. The ultmate goal of adaptve mesh methods s to obtan a certan level of accuracy wth a smaller number of dscretzaton ponts, or a hgher level of accuracy wth the same number of ponts, when compared to unform mesh methods. We propose a fnte dfference space dscretzaton on nonunform grds resultng n M- matrces The grd has a fxed number of ponts and the locatons of the grd ponts are determned adaptvely by means of montor functons at selected tmesteps so that the postons

18 CHAPTER 1. INTRODUCTION 5 of grd ponts are well-dstrbuted. The frst-order and the second-order partal dervatves are approxmated usng centered fnte dfferences. In order to obtan an M-matrx for each tmestep, a condton on the grd step szes s enforced by the adaptve procedure. A smple tmestep selector ntroduced n [14] s added to mprove the accuracy and effcency of the proposed method. We ntegrate the adaptve mesh methods wth penalty and PSOR teratve technques for the soluton of LCP at each tmestep. We ntroduce an mproved ntal guess soluton vector for both the penalty and PSOR methods. We gve a proof of the unqueness and convergence of the PSOR soluton under the assumpton that the assocated matrces are M-matrces, not necessarly symmetrc. We present numercal results that demonstrate the performance of the resultng methods. By the numercal experments, t s shown that the adaptve placement of the spatal dscretzaton ponts correctly captures the behavour of the Amercan opton prcng problem, by concentratng many more ponts around the exercse value (.e., the knk pont n ntal condtons) on the frst tmestep and around the free boundary on the subsequent tmesteps. The numercal experments also show that the adaptve mesh methods outperform the unform ones. Moreover, the mproved ntal guess substantally reduces the number of PSOR teratons, whle t only slghtly reduces the number of penalty teratons. The thess s structured as follows. Chapter 2 ntroduces adaptve mesh methods for ntal value problems (IVPs). The dscretzaton of the Amercan opton prcng problem and two teratve methods for solvng the LCPs, namely the PSOR and the penalty methods, are presented n Chapter 3. In Chapter 4, we dscuss an adaptve mesh method for Amercan opton prcng problems. We study the stablty and convergence of the proposed method n Chapter 5. In Chapter 6, we present selected numercal results and study the effcency of several methods. Fnally, we make some concludng remarks and dscuss future work n Chapter 7. A proof of the unqueness and convergence of the PSOR soluton under certan condtons s presented n the Appendx A.

19 Chapter 2 Adaptve Mesh Methods In ths chapter we frst ntroduce a spatal error estmator and then brefly descrbe a space adaptve algorthm for ntal value problems (IVPs) based on ths estmator. Ths algorthm wll be revsted n more detals n the context of Amercan opton prcng n the followng chapter. 2.1 Spatal error estmaton We use the dea of gradng functons ntroduced n [6] to construct the error estmator. The adaptve technques then relocate the nodes to equdstrbute the error n some chosen norm (or sem-norm) among the subntervals of the partton. Ths s called equdstrbuton prncple frst ntroduced n [10] and has been used extensvely by many researchers (for example, see [7], [35], [36]). Both [6] and [10] deal wth two-pont boundary value problems (BVPs) and the dea has been extended to parabolc IVPs ([12], [13], [35]). Here we brefly descrbe the basc deas of gradng functons and the equdstrbuton prncple n the context of parabolc IVPs. Consder a parabolc ntal value problem (IVP) descrbed by the PDE V t p 2 V S 2 q V S sv = f, for a < S < b, 0 < t T (2.1) 6

20 CHAPTER 2. ADAPTIVE MESH METHODS 7 h ν 1 h ν h ν +1 h ν n S ν 0 = a Sν 1 S ν 1 S ν S ν +1 S ν n 1 S ν n = b Fgure 2.1: A non-unform spatal grd at tme t = t ν subject to the boundary condtons V (a, t) = g a (t), V (b, t) = g b (t), for 0 < t T (2.2) and the ntal condton V (S, 0) = γ(s), for a S b (2.3) where the functons p(s, t), q(s, t), s(s, t), f(s, t), g a (t), g b (t), and γ(t) are gven, (a, b) s a gven nterval and V (S, t) s an unknown functon. For convenence, let LV p 2 V S + q V 2 S + sv, then the PDE (2.1) can be rewrtten as V t = LV + f. (2.4) We now ntroduce some notatons that wll be used n subsequent chapters of the thess. Assume that we have a partton ν {S ν } n =0 at tme t = t ν, not necessarly unform, wth step szes {h ν } n =1 as shown n Fgure 2.1. Let V ν denote the vector of approxmate values to V at tme t ν wth V ν beng an approxmaton to V (S ν, t ν ). For convenence, we denote by V ν the soluton at tme t ν, and let V ν represent V (S ν, t ν ). where If the central fnte dfference formulas are employed for the spatal dscretzaton, we have V ν S = aν 1V ν 1 + a ν 2V ν + a ν 3V ν +1 + RM ν 1, (2.5) a ν 1 = h ν +1 h ν (hν + hν +1 ), a ν 2 = hν +1 h ν, a ν 3 = h ν hν +1 h ν h ν +1 (hν + hν +1 ),

21 CHAPTER 2. ADAPTIVE MESH METHODS 8 ν + 1 ν t S (a) (b) (c) Fgure 2.2: Molecules of the mplct Euler (a), explct Euler (b), and Crank-Ncolson (c) and 2 V ν S 2 = bν 1V ν 1 + b ν 2V ν + b ν 3V ν +1 + RM ν 2, (2.6) where b ν 2 1 = h ν (hν + hν +1 ), b ν 2 = 2, b ν 3 = h ν hν +1 2 h ν +1 (hν + hν +1 ). Here, RM1 ν and RM2 ν are truncaton errors whose explct form of the frst non-zero terms are hν +1h ν 3 V ν and hν +1 h ν 3 V ν, respectvely. The tmesteppng s handled by the θ- 3! S 3 3! S3 scheme V ν+1 V ν = θlv ν+1 t ν + (1 θ)lv ν + θf ν+1 + (1 θ)f ν, (2.7) where t ν = t ν+1 t ν s the tme stepsze. Dependng on value of θ, we have dfferent methods as follows. θ = 1: mplct Euler method; θ = 0: explct Euler method; θ = 1 : the Crank-Ncolson (CN) method. 2 The connecton scheme (stencl) of each method s llustrated n Fgure (2.2).

22 CHAPTER 2. ADAPTIVE MESH METHODS 9 The explct scheme s condtonally stable wth truncaton error n O( t ν ). Both the mplct and CN schemes are uncondtonally stable; however the former also has truncaton error n O( t ν ), whle the latter has truncaton error n O( t 2 ν), whch s more appealng. We focus on the CN scheme due to ts second order of convergence. The equdstrbuton prncple attempts to fnd a good placement of the partton ponts such that some measure of the spatal error s equally dstrbuted over the subntervals. Dependng on the norm chosen, a dfferent gradng functon arses, based on whch the poston of the partton ponts s computed. A gradng functon s of the form ξ(s, t) = S a b ˆV ds ˆV ds, a where ˆV (S, t) s an approprate montor functon. The value ξ(s ν, t ν ), = 1,..., n, of the gradng functon at S ν and t ν represents the porton of the approxmate error at tme t = t ν from the left endpont of the spatal doman up to pont S ν. Approprate quadrature rules are used to approxmate the ntegrals. Snce all montor functons nvolve dervatves of hgh order of V, whch are unknown under realstc stuatons, approxmate values V ν, = 0,..., n, are used to approxmate the dervatves at tme t ν. There are several choces for a montor functon. One popular choce s the arclength S 1 + ( V a S functon ([36]), resultng n the gradng functon ξ(s, t) = )2 ds. It s suggested 1 + ( V b a S )2 ds n [10] that, for a method wth error proportonal to h p V ( q), where h s a stepsze and V ( q) s the q-th dervatve of V wth respect to S, a good gradng functon s S a ξ(s, t) = V ( q) 1/ p ds b V a ( q) 1/ p ds. Ignorng hgher order terms, the fnte dfference approxmaton n (2.5) s a second-order approxmaton and that n (2.6) s a frst-order approxmaton. Actually, the truncaton errors RM ν 1 and RM ν 2 can be bounded n terms of max(h ν ) 2 and (h ν +1 h ν ) + max(h ν ) 2, respectvely. Under the assumpton that h ν +1 h ν, the truncaton errors RM ν 1 and RM ν 2 are proportonal to max(h ν ) 2. In ths case, the (spatal) dscretzaton error of LV s second order

23 CHAPTER 2. ADAPTIVE MESH METHODS 10 wth respect to the stepszes, and nvolves V (3), resultng n the montor functon ˆV = V (3) 1/2, and the correspondng gradng functon s ξ(s, t) = S V (3) 1/2 ds a b V a (3) 1/2 ds. However, we encounter some dffculty wth these gradng and montor functons. The value of the Amercan opton often oscllates and does not show convergence as grds are refned. However, f we use the montor and gradng functons ˆV = V (3) 1/3 and ξ(s, t) = S a V (3) 1/3 ds b a V (3) 1/3 ds. (2.8) then the problems are resolved. These are the montor and gradng functons that we use throughout the course of experments. Gven a gradng functon, the equdstrbuton prncple requres that the partton ponts satsfy, for a fxed tme t = t ν, ξ(s ν, t ν ) ξ(s ν 1, t ν ) = S ν a b ˆV ds ˆV ds a S ν 1 a ˆV ds b ˆV ds 1 n, a or equvalently ξ(s ν, t ν ) = S ν a b ˆV ds a ˆV ds n, (2.9) where = 1,..., n 1. Note that the two boundary ponts are fxed, leavng n 1 ponts to dstrbute. In order to solve (2.9), one could apply the teratve scheme S ν,(k+1) = S ν,(k) ) n, (2.10) ξ (S ν,(k) ξ(sν,(k) ) whch s based on Newton s method.

24 CHAPTER 2. ADAPTIVE MESH METHODS Crtera for equdstrbuton of error Based on the montor functon ˆV (S, t), for a fxed tme t = t ν, we evaluate two quanttes r ν = r ν = S ν S ν 1 b a ˆV ds, = 1,..., n, (2.11) ˆV ds n, (2.12) usng quadrature rules. It s noted that r ν represents the estmate of the error for the th subnterval [S ν 1, S ν ], and r ν represents the average of all r ν 1 n 1 { rν } max s. The rato gves an ndcaton of how well-dstrbuted the partton s. Snce we are usng the equdstrbuton prncple for the remeshng, f ths rato s too large, t follows that the maxmum error estmate over the subntervals s consderably larger than the average estmate and thus the current partton s not well-dstrbuted. At each tmestep, the algorthm checks f rdrft max 1 n 1 { rν } r ν α, (2.13) r ν where α s a small number less than 10. Typcal choces for α are α = 2 (see [35]) and α = 4 n our experments. That s the maxmum value of r ν must be roughly at most α tmes as large as the average value r ν. We consder a partton that satsfes ths property to be well-dstrbuted. Numercal experments ndcate that ths crteron works reasonably well for Amercan opton prcng. We also consder another crteron for the equdstrbuton of error, whch s suggested n [6]. We check f drft max 1 n 1 { rν } r ν tol, (2.14) where tol s a user chosen tolerance. In our experments for Amercan opton prcng, the choce tol = 10 1 works well for the set of model parameters n Table 6.1.

25 CHAPTER 2. ADAPTIVE MESH METHODS Algorthm Summary In ths secton, we present a summary of the core code segment of a space adaptve algorthm for IVPs. Ths part of the code s executed when proceedng from tmestep ν to tmestep ν + 1. The algorthm normally works teratvely, wth a stoppng crteron specfed n (2.13) or (2.14). In our experments, we set the maxmum number of teratons maxt = 1, so that at most one re-dstrbuton of the spatal ponts takes place n one tme step, thus the placement of the spatal ponts evolves as the tme steps proceed. We also set the constant smallnum = 6, and explan below how ths constant s used. We now brefly descrbe the algorthm. In Lne 1, we apply the standard tme-steppng (usually Crank-Ncolson), usng the same spatal ponts n steps ν and ν + 1. We then calculate all quanttes necessary to check the crteron (2.13) or (2.14), that decdes whether a re-dstrbuton of the ponts s needed (Lnes 3 and 4). If the ponts are well-dstrbuted, we proceed to the next tme step (Lnes 5 and 6). If not, the new locaton of the spatal ponts s computed usng (2.9) (Lnes 7 and 8). Next, we need to calculate values of the approxmaton at the new spatal ponts at the ν + 1st tme step. There are two ways to do ths: the frst, apples nterpolaton to values of the approxmaton at the current partton ponts at the νth tme step, to compute values of the approxmaton at the new partton ponts, then apples the tme-steppng procedure to compute values of the approxmaton at the new partton ponts, at the ν + 1st tme step; the second, smply apples nterpolaton to values of the approxmaton at the current partton ponts at the ν + 1st tme step, to compute values of the approxmaton at the new partton ponts, at the ν + 1st tme step. The frst technque s used n the frst few (smallnum) tme steps (Lnes 9, 10 and 11), whle the second s used for all other tme steps (Lnes 12 and 13). The choce of nterpolaton technque may be mportant under certan crcumstances. In the case of European opton prcng or other problems wthout specal constrants, one can use standard nterpolaton technques, such as cubc splne nterpolaton, to obtan the nterpolated values. However, we notced that the nterpolaton technques used n valung Amercan op-

26 CHAPTER 2. ADAPTIVE MESH METHODS 13 tons should be chosen carefully, due to the exstence of the free boundary at each tme step. More dscusson on ths wll be provded n Chapter 4. We now gve the space adaptve algorthm for IVPs, whch assumes that an approxmaton to V ν s already computed on partton ν, and computes an approxmaton to V ν+1 on partton ν+1, whch may be dfferent from ν.

27 CHAPTER 2. ADAPTIVE MESH METHODS 14 Algorthm 1: Bref descrpton of space adaptve algorthm for IVPs 1: apply (2.7) to compute approxmaton to V ν+1 on partton ν+1 ν usng the gven approxmaton to V ν ; 2: for k = 1 to maxt do 3: approxmate the approprate dervatves of V ν+1 and estmate the dstrbuton of the error usng (2.8); 4: calculate the quanttes ( r ν+1 ) n 1 =1 and rν+1 usng (2.11) and (2.12); check crteron (2.13) or (2.14) to decde whether a remeshng s needed; 5: f (2.13) or (2.14) s satsfed then 6: ext; 7: else 8: compute a new partton ν+1 ν usng (2.9); 9: f ν smallnum then 10: compute new approxmaton to V ν on ν+1 usng nterpolaton on current approxmaton to V ν ; 11: apply (2.7) to compute new approxmaton to V ν+1 on partton ν+1 usng the 12: else new approxmaton to V ν ; 13: compute new approxmaton to V ν+1 on ν+1 usng nterpolaton on current 14: end f 15: end f 16: end for approxmaton to V ν+1 ;

28 Chapter 3 Prcng Amercan Optons In ths chapter, we revew the problem of prcng Amercan optons. We frst ntroduce the prcng model and the formaton of the Amercan opton valuaton as a lnear complementarty problem. We then dscuss the numercal computaton of Amercan optons. In partcular, we present the dscretzaton wth fnte dfferences and two teratve methods for solvng the assocated constraned matrx problem. Although we restrct our attenton to Amercan puts, the approach can be easly appled to Amercan calls on dvdend-payng assets. 3.1 Prcng Model The model ntroduced by Black and Scholes [2] and Merton [24] was the frst, and s stll the most wdely used, for prcng optons. They observed a lognormal behavor of asset prces and derved the followng partal dfferental equaton (PDE) that descrbes the opton s value: V t σ2 S 2 2 V V + rs S2 S rv = 0. (3.1) Here, V = V (S, t) represents the opton prce; S 0 s the underlyng asset prce; r s the rsk-free nterest rate; σ s the volatlty of the underlyng asset prce; and 0 t T where T the expry. In ths thess, we consder only constant volatlty and nterest rate. However, the approach presented here can be extended to cases where volatlty and nterest rate are functons 15

29 CHAPTER 3. PRICING AMERICAN OPTIONS 16 of the underlyng asset prce and tme. Equaton (3.1) s referred to as the Black-Scholes PDE. It s a parabolc PDE and has many solutons. To obtan a unque soluton for a partcular prcng model, the Black-Scholes equaton must be assocated wth addtonal constrants such as fnal condtons, boundary condtons, or free boundary condtons. It can then can be solved backwards n tme from the opton expry tme t = T to the present t = 0. In contrast to European optons whch can only be exercsed at the maturty date T, Amercan optons can be exercsed at any tme up to T. Consequently, dentfyng the optmal exercse strategy s an essental part of the valuaton problem. For Amercan put optons, the possblty of early exercse requres that V (S, t) max(e S, 0), t [0, T ], (3.2) otherwse an arbtrage opportunty would arse ([17], [30]). The evaluaton of an Amercan opton s assocated wth a free boundary value problem ([37]), and the exercse boundary curve S f (t), whch vares wth tme, dvdes the S-t half strp [0, ) [0, T ] nto the contnuaton regon and the stoppng regon. The contnuaton regon {(S, t) [0, ) [0, T ] : V (S, t) > max(e S, 0)} s the set of all ponts (S, t) where the opton should be held, whereas n the stoppng regon {(S, t) [0, ) [0, T ] : V (S, t) = max(e S, 0)} early exercse s advsable. If S f (t) s calculated, the Amercan put opton holder should exercse the opton as early as possble when S < S f (t) and hold the opton otherwse. Under the Black-Scholes framework, the prce V (S, t) of an Amercan put opton satsfes ether V t σ2 S 2 2 V V + rs S2 S rv = 0 V max(e S, 0) 0 (3.3) n the contnuaton regon, or V t σ2 S 2 2 V V + rs S2 S rv < 0 V max(e S, 0) = 0 (3.4)

30 CHAPTER 3. PRICING AMERICAN OPTIONS 17 n the stoppng regon. We also have addtonal boundary condtons at the free boundary S f (t) V (S f (t), t) = max(e S f (t), 0) = E S f (t), V S (S f(t), t) = 1, (3.5) whch are known as smooth boundary condtons ([30]). The fnal condton s V (S, T ) = max(e S, 0). (3.6) The boundary condtons can be obtaned by mposng Drchlet boundary condtons whch are gven by V (0, t) = E, V (S, t) 0 as S. (3.7) These condtons are based on some addtonal knowledge about asymptotc behavor of the solutons, whch may not be avalable n complex contracts. For smplcty, n ths thess we consder only Drchlet boundary condtons for Amercan optons. Defne the operator LV 1 2 σ2 S 2 2 V V + rs S2 S rv (3.8) and let V = max(e S, 0) denote the payoff functon. The Amercan put opton prcng can be reformulated as the lnear complementarty problem (LCP) V t + LV = 0 V V 0 or V t + LV < 0 V V = 0. (3.9) together wth the fnal condton (3.6), boundary condtons (3.7), and smooth boundary condtons (3.5). The optmal free boundary S f (t) s automatcally captured by ths formulaton and can be determned a-posteror. Solutons of lnear complementarty problems can be obtaned by several teratve methods such as the projected successve over-relaxaton method or the penalty methods. We wll dscuss these two methods n a later secton.

31 CHAPTER 3. PRICING AMERICAN OPTIONS 18 τ ν+1 τ ν τ h ν h ν +1 node (S ν +1, τ ν+1 ) τ ν S ν 1 S ν S ν +1 S Fgure 3.1: Detals and notatons of the fnte dfference grd 3.2 Dscretzaton wth Fnte Dfferences In order to wrte the LCP (3.9) n more conventonal form so that we can solve t backwards n tme, defne τ = T t, so that t becomes V τ LV = 0 V V 0 or V τ LV > 0 V V = 0. (3.10) For the dscretzaton of the tme varable, we choose a set of grd ponts forward n tme wth respect to τ {τ 0, τ 1,..., τ νmax 1, τ νmax } τ 0 = 0 < τ 1 < < τ νmax 1 < τ νmax = T (3.11) Defne τ ν = τ ν+1 τ ν, ν = 0, 1,..., ν max 1. Usually, we use unform tme stepsze but we may make use of a non-unform tme stepszes to speed up effcency. The choce of spatal dscretzaton s more complcated. The nfnte doman [0, ) must be truncated down to [0, S max ] and the boundary condton at S = s replaced by the boundary condton at S = S max. More dscusson on how to choose value S max can be found n Chapter 6. The spatal partton on the truncated doman at tme level τ ν s denoted by ν. The Fgure 3.1 llustrates part of the entre (S, τ) grd usng the same notatons ntroduced n prevous chapter.

32 CHAPTER 3. PRICING AMERICAN OPTIONS Dscretzaton for LV We use non-unform central fnte dfference formulas defned n (2.5) and (2.6) for the spatal dscretzaton. Assume that we want to proceed from tme step ν to tme step ν + 1. Recall that LV 1 2 σ2 S 2 2 V V ν + rs rv. Substtutng (2.5) and (2.6) nto LV S2, = 1,..., n 1 and S gnorng the truncaton errors, we obtan LV ν = 1 2 σ2 (S ν ) 2 (b ν 1V ν 1 + b ν 2V ν + b ν 3V ν +1) + rs ν (a ν 1V ν 1 + a ν 2V ν + a ν 3V ν +1) rv ν = ( 1 ) 2 σ2 (S ν ) 2 b ν 1 + rs ν a ν 1 V ν 1 + ( 1 2 σ2 (S ν ) 2 b ν 2 + rs ν a ν 2 r ) V ν + ( 1 ) 2 σ2 (S ν ) 2 b ν 3 + rs ν a ν 3 V ν +1, where a ν 1, a ν 2, a ν 3 and b ν 1, b ν 2, b ν 3 are specfed n (2.5) and (2.6), respectvely. Let m ν, 1 = 1 2 σ2 (S ν ) 2 b ν 1 rs ν a ν 1, (3.12) m ν, = 1 2 σ2 (S ν ) 2 b ν 2 rs ν a ν 2 + r, (3.13) m ν,+1 = 1 2 σ2 (S ν ) 2 b ν 3 rs ν a ν 3, (3.14) then LV ν = m ν, 1V ν 1 m ν,v ν m ν,+1v ν +1, (3.15) = 1, 2,..., n 1. (3.16) Tme Dscretzaton Recall that under the change of varable τ = T t, the Black-Scholes PDE used for Amercan optons becomes V τ LV = 0, or equvalently V τ = LV, (3.17)

33 CHAPTER 3. PRICING AMERICAN OPTIONS 20 where LV 1 2 σ2 S 2 2 V V + rs rv. Applyng the θ-tmesteppng scheme (2.7) to (3.17), S2 S we obtan V ν+1 V ν = θlv ν+1 + (1 θ)lv ν τ ν V ν+1 θ τ ν LV ν+1 = V ν + (1 θ) τ ν LV ν. (3.18) Substtutng the dscretzaton formula (3.15) for LV nto (3.18) gves V ν+1 + θ τ ν (m ν+1, 1 Vν mν+1, V ν+1 + m ν+1,+1 Vν+1 +1 ) = V ν (1 θ) τ ν (m ν, 1V ν 1 + m ν,v ν + m ν,+1v ν +1). (3.19) Note that n equaton (3.19), we have m ν, 1 = m ν+1, 1, mν, = m ν+1, and m ν,+1 = m ν+1,+1. For smplcty, we use m ν, 1, m ν, and m ν,+1 on both sdes of equaton (3.19), resultng n the followng one: V ν+1 + θ τ ν (m ν, 1V ν 1 + m ν,v ν+1 + m ν,+1v ν+1 +1 ) Wrtng ths n matrx form, we have = V ν (1 θ) τ ν (m ν, 1V ν 1 + m ν,v ν + m ν,+1v ν +1). (3.20) (I + θ τ ν M ν )V ν+1 = (I (1 θ) τ ν M ν )V ν, (3.21) where I s the dentty matrx and M ν s a trdagonal matrx n R (n 1) (n 1). Matrx M ν has the followng form: M ν = m ν 1,1 m ν 1, m ν 2,1 m ν 2,2 m ν 2, m ν, 1 m ν, m ν, m ν n 2,n 3 m ν n 2,n 2 m ν n 2,n m ν n 1,n 2 m ν n 1,n 1,

34 CHAPTER 3. PRICING AMERICAN OPTIONS 21 wth m ν, 1, m ν,, m ν,+1, = 1,..., n 1 beng defned n (3.12). Note that besdes constants σ and r, matrx M ν depends also on spatal partton at the tme level τ ν. As we mentoned earler, we would lke to use the CN method due to ts second order of convergence. However, for CN method, spurous oscllatons can be ntroduced nto the soluton. Even though these oscllatons may be small f we look at the opton value, they can be magnfed when computng the opton delta and gamma [42]. In ths thess, we focus on the CN scheme, but we use the mplct Euler scheme for the Rannacher smoothng technque. We wll dscuss the oscllatory behavor of the CN method and ts remedy n Chapter 5. Under the above dscretzaton, the LCP (3.10) s re-formulated as a constraned matrx problem of the form A ν V ν+1 = b ν V ν+1 V,ν+1 0 or A ν V ν+1 > b ν V ν+1 V,ν+1 = 0, (3.22) whch must be solved n order to proceed from tme step ν to tme step ν + 1. Here, A ν = I + θ τ ν M ν, b ν = (I (1 θ) τ ν M ν )V ν, (3.23) are both dependent on the space partton ν and tme step sze τ ν. We denote V,ν as a vector of payoff values on spatal partton ν, where the th component s V,ν = max(e S ν, 0). In the next secton, we wll dscuss some teratve methods for solvng ths type of problem. 3.3 Iteratve Methods Before we descrbe the applcaton of the projected successve over-relaxaton and the penalty method on prcng Amercan optons, we frst revew some background on teratve methods. We are nterested n solvng the lnear system of equatons Ax = b, A R n n, b, x R n. (3.24)

35 CHAPTER 3. PRICING AMERICAN OPTIONS 22 Wth a sutable matrx Q R n n, we can re-wrte (3.24) as Qx = (Q A)x + b x = (I Q 1 A) x + Q 1 b, }{{}}{{} G c resultng n the teraton scheme x (k+1) = Gx (k) + c, (3.25) where k s the teraton ndex and G s the teraton matrx. It has been proved n [34] that teraton scheme (3.25) converges f and only f ρ(g) < 1 where ρ(g) denotes the spectral radus of G. Moreover, to optmze the convergence, we would lke to have ρ(g) to be as small as possble. One way s to splt the matrx A nto three parts A = D + L + U, where D, L and U represent the dagonal, strctly lower trangular and strctly upper trangular parts of A, respectvely. We now ntroduce the relaxaton methods. Let ω > 0 denote the relaxaton parameter. We let Q = D ω + L, whch results n the teraton matrx G = I Q 1 A = I ( D ω + L) 1 A. Ths leads to the teraton scheme x (k+1) = (I ( D ω + L) 1 A)x (k) + ( D ω + L) 1 b. (3.26) Substtutng A = D + L + U nto (3.26) and rearrangng the resultng formula, we obtan the relaxaton scheme x (k+1) = x (k) + ωd 1 (b Lx (k+1) (D + U)x (k) ). (3.27) Note that n formula (3.27), when x (k+1) s beng computed, all prevous components x (k+1) j s, j = 1,..., 1, are already computed and thus they can be used to compute x (k+1) to speed

36 CHAPTER 3. PRICING AMERICAN OPTIONS 23 up the convergence. In component form, (3.27) can be wrtten as for = 1,..., n do y (k+1) = 1 ( b a,j x (k+1) j ) a,j x (k) j a, j< j> x (k+1) endfor = x (k) + ω(y (k+1) x (k) ) (3.28) Lettng ω = 1, ths results n the Gauss-Sedel method. For 0 < ω < 1, the scheme s called successve under-relaxaton, and for 1 < ω < 2, we obtan the successve over-relaxaton (SOR) method Projected SOR Method The projected SOR (PSOR) method for lnear complementarty problems was frst proposed by Cryer n [9]. The method s well-known and wdely appled for prcng Amercan optons (see [33], [30]). In ths secton, we frst dscuss the PSOR method and then we present an algorthm whch performs PSOR teraton for solvng the constraned matrx problem resultng from Amercan opton prcng. Let us now consder a generc problem of the form Ax = b x g or Ax > b x = g. (3.29) For the soluton of ths problem, Cryer [9] proposes the PSOR based on modfcatons of the teratve scheme of the SOR method by ncludng at each teraton the constrant x g. More specfcally, the PSOR method for solvng problem (3.29) s for = 1,..., n do y (k+1) = 1 ( b a,j x (k+1) j a, j< j> x (k+1) endfor = max ( g, x (k) + ω(y (k+1) x (k) ) ) ) a,j x (k) j (3.30)

37 CHAPTER 3. PRICING AMERICAN OPTIONS 24 The proof of convergence for the PSOR method under the condton that matrx A s symmetrc postve defnte can be found n [9]. In our case, due to adaptvty, the underlyng matrx at each tmestep n our case s hghly non-symmetrc and thus the convergence condton n [9] does not apply. In Appendx A, we present a proof to show that f the matrx A belongs to a specfc class of matrces and the relaxaton parameter ω satsfes certan condtons, then the convergence of the PSOR method s guaranteed. Both SOR and PSOR methods start wth some ntal guess x (0) and compute successve approxmatons x (k) to the soluton for k = 1,... untl some stoppng crteron s satsfed. The stoppng crteron we use s x (k+1) x (k) tol, where. could be 2-norm or -norm and tol s a user-defned tolerance. The relaxaton parameter ω plays an mportant role on the convergence rate of the (P)SOR method. The optmal relaxaton parameter can be determned when A s a postve defnte matrx (see [39]). However, the matrx at each tme step s not symmetrc due to adaptvty, t s not easy to fnd the optmal ω n our case. For the numercal solutons of IVPs, a technque to approxmate dynamcally the optmal relaxaton parameter for (P)SOR method at each tme step s proposed n [37]. To explan the technque descrbed n [37], consder three successve tmesteps ν 1, ν, ν + 1. Let ω ν and t ν denote the value of the relaxaton parameter and the number of teratons requred for convergence at tmestep ν, respectvely. Frst, we compare the number of teratons the method requred n tmesteps ν 1 and ν, whch use ω ν 1 and ω ν, respectvely. Here, ω ν = ω ν 1 + ω wth ω beng a very small, constant number ether greater or smaller than zero. Typcally, the value of ω n absolute value s between 0.01 and Dependng on the result of that comparson we decde the value of ω ν to be used n the next tmestep by adjustng ω as follows. (a) If t ν 1 < t ν, set ω = ω; (b) Set ω ν+1 = ω ν + ω; We adopt ths technque n our experments.

38 CHAPTER 3. PRICING AMERICAN OPTIONS 25 The ntal guess also plays a role n the convergence of the (P)SOR method. The most common ntal guess s x (0) = 0. In the context of IVPs, a better ntal guess for the current tme step s the approxmate soluton for the prevous tme step. However, usng extrapolaton on the approxmatons at the prevous tme steps could produce even a better ntal guess. More dscusson on ths wll be provded next. We now gve an mplementaton of the PSOR method for solvng the LCP resultng from Amercan optons. Recall that the LCP s descrbed by (3.9) and the dscretzaton of the partal dfferental operator gves rse to the constraned matrx problem (3.22). We notce the smlartes n form between the constraned matrx problem (3.22) and problem (3.29) and thus the PSOR teratve scheme can be used to solve the non-lnear problem assocated wth the Amercan opton prcng. For each teraton and each component of the soluton vector, the valuaton nvolves a comparson between the value of the opton that would be obtaned f the holder does not exercse and the value of the opton that would be obtaned f the holder does exercse the opton. Snce t s assumed that the holder would act optmally, the larger of these two values would be chosen as the value of the opton at that pont. We ntroduce some addtonal notatons used for the algorthm and for the rest of the thess. V ν+1,(k) : kth estmate of V ν+1 ; a ν,j: (, j)th entry of matrx A ν ; b ν : th entry of vector b ν ; A bref descrpton of PSOR teraton for valung an Amercan put s presented n Algorthm 2. It has been noted n the lterature that the convergence rate of the PSOR method deterorates when the dscretzaton s refned whch makes PSOR a very slow method on fner grds. The convergence rate of the PSOR method depends on the ntal guess and relaxaton factor ω. As we mentoned earler, we follow a technque n [37] to dynamcally determne ω at each tme step. Wth respect to the ntal guess, one popular choce for the ntal guess s V ν+1,(0) = V ν, (3.31)

39 CHAPTER 3. PRICING AMERICAN OPTIONS 26 whch s an approxmate soluton at the prevous tme step τ ν. Our expermental results show that ths ntal guess stll results n deteroratng behavors of the convergence rate. However, deteroraton of the convergence rate could be reduced f we had a better ntal guess. In partcular, we use lnear extrapolaton on V ν and V ν 1, resultng n the ntal guess V ν+1,(0) = ( τ ν + τ ν 1 ) τ ν 1 V ν τ ν τ ν 1 V ν 1, ν = 2,..., ν max. (3.32) In case of constant tmesteps, that s τ ν = τ ν 1, ν = 2,..., ν max, the above formula reduces to V ν+1,(0) = 2V ν V ν 1, ν = 2,..., ν max. Experment results show that for opton prcng, ths choce of ntal guess produce much faster convergence and the deteroraton of convergence rate on fner grds s less serous than those obtaned from the choce (3.31). We provde numercal results related to ths ssue n Chapter 6. Algorthm 2: Bref descrpton of PSOR Amercan put opton constrant teraton 1: ntalze V ν+1,(0) ; 2: for k = 0,..., untl convergence do 3: for = 1,..., n 1 do 4: calculate Ṽν+1,(k+1) = 1 a ν, 5: end for V ν+1,(k+1) 6: f V ν+1,(k+1) V ν+1,(k) tol then 7: break; 8: end f 9: end for 10: V ν+1 = V ν+1,(k+1) ; ( b ν a ν,jv ν+1,(k+1) j j< j> ) a ν,jv ν+1,(k) j ; = max ( V,ν+1, V ν+1,(k) ν+1 + ω (Ṽν+1,(k+1) V ν+1,(k) ) ) ;