Pricing American Options using Monte Carlo Method

Transcription

1 Prcng Amercan Optons usng Monte Carlo Method Zhemn Wu St Catherne s College Unversty of Oxford A thess submtted for the degree of Master of Scence n Mathematcal and Computatonal Fnance June 21, 2012

2 Acknowledgements I am grateful to all of those who made ths thess possble. Frst and foremost, I would lke to express my sncere grattude to my supervsor, Dr. Lukas Szpruch, for hs gudance and help wth my thess project. I would also lke to thank Dr. Samuel Cohen and Dr. Lajos Gergely Gyurko for ther valuable dscussons about my thess topcs. Apprecaton also goes to other departmental lecturers and admnstrators, and ths work would not have been possble wthout ther valuable tme and work. The dscussons wth graduate student, Mss. Shu Zhou, s also hghly apprecated.

3 Abstract Ths thess revewed a number of Monte Carlo based methods for prcng Amercan optons. The least-squares regresson based Longstaff-Schwartz method (LSM) for approxmatng lower bounds of opton values and the Dualty approach through martngales for estmatng the upper bounds of opton values were mplemented wth smple examples of Amercan put optons. The effectveness of these technques and the dependences on varous smulaton parameters were tested and dscussed. A computng savng technque was suggested to reduce the computatonal complexty by constructng regresson bass functons whch are orthogonal to each other wth respect to the natural dstrbuton of the underlyng asset prce. The orthogonalty was acheved by usng Hermte polynomals. The technque was tested for both the LSM approach and the Dualty approach. At the last, the Multlevel Mote Carlo (MLMC) technque was employed wth prcng Amercan optons and the effects on varance reducton were dscussed. A smoothng technque usng artfcal probablty weghted payoff functons jontly wth Brownan Brdge nterpolatons was proposed to mprove the Multlevel Monte Carlo performances for prcng Amercan optons.

4 Contents 1 Introducton 1 2 Amercan Optons: the Problem Formulatons Amercan Optons Dynamc Programmng Prncple Optmal Stoppng Rule and Contnuaton Values Longstaff-Schwartz Method: a Lower Bounds Estmator usng Least- Squares Regressons Approxmatng Contnuaton Values Usng Least-Squares Regresson Low-based Estmator Usng Optmal Stoppng Rule Numercal Examples The Dualty through Martngales: an Upper Bounds Estmator The Dualty: a Mnmzaton Problem over Martngales The Dualty Estmaton Usng Martngales from Approxmated Value Functons Numercal Examples Orthogonalty of Hermte Polynomals: A Computng Savng Technque Hermte Polynomals Constructng Orthogonal Bass Functons through Hermte Polynomals Numercal Examples Multlevel Monte Carlo Approach An Analyss of Standard Monte Carlo Multlevel Monte Carlo Numercal Examples Improved MLMC for Lower Bounds Estmaton usng Probablty Weghted Payoff Functons and Brownan Brdge Interpolatons Conclusons 42

5 A Matlab Codes 45 A.1 Sub-codes and Functons A.1.1 Cumulatve Normal Densty Functon A.1.2 Regresson Bass Functons Usng Laguerre Polynomals A.1.3 Orthogonal Bass Functons Based on Hermte Polynomals A.1.4 Dscounted Payoff Functons of Amercan Put Opton A.1.5 Estmatng Regresson Coeffcents Usng Backward Inducton A.1.6 Regresson Coeffcents Usng Smplfed Approach Based on Orthogonalty A.1.7 LSM Estmator A.1.8 Dualty Estmator A.1.9 LSM Estmator Usng Probablty Weghted Payoff Functons A.1.10 FDM penalty method A.2 Man Codes A.2.1 Lower Bounds Estmaton Usng LSM A.2.2 Upper Bounds Estmaton Usng Dualty Method A.2.3 Multlevel Monte Carlo for Prcng Amercan Put Optons A.2.4 Multlevel Monte Carlo Usng Probablty Weghted Payoff Functons jontly wth Brownan Brdge Interpolatons Bblography 65

6 Chapter 1 Introducton Amercan optons are wdely studed n mathematcal fnance. As smple examples, an Amercan call/put opton wrtten on an underlyng asset wth strke K and maturty T gves the opton holder the rght to buy/sell the underlyng asset at prce of K at any tme up to the maturty T. Snce the nvestors can choose an optmal tme to exercse the opton, the valuatons of Amercan type optons are relatvely more complex than prcng European optons, whch only can be exercsed on maturty and are typcally valued usng the wellknown Black Scholes model [1] under the arguments of no-arbtrage and perfect hedgng by tradng the underlyng assets. The value of an Amercan opton s the value acheved by exercsng optmally. Prcng an Amercan opton s essentally an equvalence to a problem of solvng an optmal stoppng problem by defnng the optmal exercse rule. The value of an Amercan opton s thus calculated by computng the expected dscounted payoff under ths rule. It s wdely acknowledged that an analytcal formula does not exst for the value of an Amercan opton. Thus as a result, the valuatons of Amercan optons mostly rely on numercal smulatons and the optmzaton procedure embedded n the problem makes ths a challenge. Smple Amercan type dervatves can be solved by the Fnte Dfference Method [2] (FDM) by statng the problem as a Black-Scholes type PDE wth free boundares assocated wth the early exercse feature. The penalty method [3] s one of the most popular technques of prcng Amercan optons usng FDM. The man drawback of Fnte Dfference Methods s the restrcton to dmensons as the computatonal cost rapdly ncreases wth the dmensons of the underlyng (number of the underlyng assets). In the recent years, Monte Carlo based methods [4] have been ntensvely developed to overcome the current exstng problems wth FDM based approaches. These Monte Carlo based methods can be roughly dvded nto two groups. The frst group drectly employs a recursve scheme for solvng the optmal stoppng problem by usng backward dynamc programmng prncple. Dfferent technques are appled to approxmate the nested condtonal expectatons. The least-squares regresson method of Longstaff and Schwartz [5][6] s one of the most popular approaches n ths group and has attracted the most nterests recently. Ths method uses smple lnear regressons through known bass functons of the current state of underlyng asset prce to approxmate 1

7 condtonal expectatons and thus n turn to estmate a lower bound of the value of an Amercan opton followng a sub-optmal exercse rule. Ths s what s called the prmal approach. The second group comes from the Dualty approach developed by Rogers [7] and ndependently by Haugh and Kogan [8]. The approach was mplemented by Andersen and Broade [9]. The basc dea of ths dual approach s to represent the prce of an Amercan opton through a mnmzaton problem, leadng to a hgh-based approxmaton and thus provdng upper bounds on prces. The lower bounds provded by the least-squares regresson based prmal approach and the upper bounds provded by the dual approach contan the true prce. Recently, a very nterestng extenson was presented by Belomestny and Schoenmakers [10]. These authors combned Andersen and Broade s algorthm [9] wth Multlevel Monte Carlo Method (MLMC), developed by Mke Gles [11]. Ths thess presents a revew of above mentoned Monte Carlo based algorthms for prcng Amercan optons and provdes mplementatons wth smple examples of Amercan put optons. The purpose of the study s to test the effectveness of these technques and the dependences on varous smulaton parameters. A potentally promsng computng savng technque usng orthogonal Hermte polynomals s suggested and tested. The Multlevel Mote Carlo method s employed wth the algorthms and the effects on varance reducton are dscussed. A smoothng technque usng artfcal probablty weghted payoff functons jontly wth Brownan Brdge nterpolatons s proposed to mprove the Multlevel Monte Carlo performances for prcng Amercan optons. The thess s organzed n the followng way. Chapter 2 revews the basc theoretcal results of the valuaton problems of Amercan optons. The basc problem formulatons based on dynamc programmng prncple are derved. Chapter 3 provdes an ntroducton to the least-squares regresson based Longstaff- Schwartz method (LSM) for approxmatng lower bounds of the prces of Amercan optons. The algorthms are tested usng smple examples of an Amercan put opton wrtten on sngle non-dvdend payng stock. The dependences on the number of Monte Carlo paths and the number of tme steps (exercse opportuntes) are dscussed. Chapter 4 revews and mplements the Dualty approach for estmatng the upper bounds of the prces of Amercan optons through a martngale technque usng approxmated value functons and nested Monte Carlo smulatons. Varous effects of the smulaton parameters are dscussed. Chapter 5 suggests a computng savng technque for prcng Amercan optons. The technque ams to reduce the computatonal complexty by constructng regresson bass functons whch are orthogonal to each other wth respect to the natural dstrbuton of the underlyng asset prce. The orthogonalty s acheved by usng Hermte polynomals. The technque s tested for the both algorthms of the prmal LSM approach and the Dualty approach. 2

8 Chapter 6 employs the Multlevel Monte Carlo (MLMC) towards the mentoned methods and the effects on varance reducton for both the prmal and the dual approaches are tested. A smoothng technque usng artfcal probablty weghted payoff functons and Brownan Brdge nterpolatons s suggested to mprove MLMC performances wth the lower bounds estmatons through LSM algorthm. 3

9 Chapter 2 Amercan Optons: the Problem Formulatons By defnton, an Amercan opton s a dervatve wrtten on an underlyng asset whch can be exercsed any tme up to ts maturty. The dscretzed verson of Amercan opton, whch s also called Bermudan opton, s an Amercan style dervatve that can be exercsed only on pre-specfed dates up to maturty. The value of an Amercan opton s the maxmum value whch can be acheved by optmal exercsng. Ths value can be estmated by defnng an optmal exercsng rule on the opton and computng the expected dscounted payoff of the opton under ths rule. Ths chapter presents the basc formulatons of the valuaton of Amercan optons. 2.1 Amercan Optons In contrast to an European opton, whch can only be exercsed at maturty T, an Amercan opton can be exercsed at any tme t (0, T ] although t s usually not exercsable at tme 0. For the contnuous tme case, f we denote the payoff of an Amercan opton at any tme t (0, T ] as h(t) = h(s t, t) (2.1) where S t s the prce process of underlyng asset, whch s assumed to be a Markovan process, and gven a class of admssble stoppng tmes T t, whch takes values n [t, T ] (t [0, T ]), the Amercan opton prcng problem can be formulated as [4] V 0 = sup E [e ] τ 0 r(u) du h(sτ, τ) (2.2) τ T 0 where V 0 s the value of opton at tme t = 0, {r(t), 0 t T } s the nstantaneous nterest rate process and the expectaton s taken under the equvalent martngale measure. Wth ths formulaton of prce for Amercan optons, the general no-arbtrage prncple holds [12]. We can also absorb the dscount factor e t 0 r(u) du nto the payoff functon h(s t, t), and rewrte (2.2) as V 0 = sup τ T 0 E [h(s τ, τ)] (2.3) 4

10 where h(s t, t) = e t 0 r(u) du h(st, t) s the dscounted payoff. Also, f the prce process S t s Markovan, t s natural to assume the class of admssble stoppng tmes T t = T (S t ) depends only on the current state of the prce process,.e., nvestors can make an exercse decson at tme t based on the prce process state S t. As the smplest example, an Amercan put opton wrtten on a sngle underlyng asset S t wth strke K and maturty T gves the opton holder the rght to sell the underlyng asset at prce K at any tme up to the maturty T. Assume the asset prce S t follows a general geometrc Brownan moton process gven by ds t = (r q)s t dt + σs t dw t (2.4) where r and q are nterest rate and dvdend rate respectvely, whch are assumed to be constants, σ s the volatlty and W t s a standard Brownan moton process. The value of Amercan put opton at t = 0 s then gven by V 0 = sup τ T 0 E [ e rτ (K S τ ) +] (2.5) Fgure 2.1 shows an example of an Amercan put opton wrtten on a non-dvdends payng stock wth strke K = 100, nterest rate r = 5%, volatlty σ = 0.5 and maturty T = 5 respectvely. European opton. The value of Amercan opton s compared wth that of the correspondng The values of Amercan opton are calculated usng Fnte Dfference Method (FDM) based penalty approach [3] and the values of European opton are calculated usng Black-Scholes formula [1]. Strctly speakng, an Amercan opton s contnuously exercsable,.e. t can be exercsed at any tme up to maturty T. However, ths s not the case for numercal smulatons. To numercally prce an Amercan opton, we need to dvde the entre tme perod [0, T ] nto a dscretzed sequence {0 = t 0 < t 1 < t 2 < < t M = T } and assume the Amercan opton can only be exercsed at ths fxed set of dates, whch s n fact the case wth specal name Bermudan optons. Actually, one may expect the value of the dscretzed verson of Amercan opton s smply the approxmaton of the contnuous one when the number of tme segments M. In ths thess, we wll smply focus on the dscretzed case. Once we restrct our concern wth tme dscretzaton, t s necessary to defne a way to approxmate the underlyng Markovan asset prce process. We denote S = S(t = t ), (0, 1, 2,, M) as the state of asset prce process at the th exercse opportunty. The Euler-Maruyama scheme [13] then can be used to approxmate the asset prce process (2.4), whch s gven by S = S 1 + (r q)s 1 t + σs 1 W, = 1, 2, M (2.6) where t = t t 1 s the sze of tme nterval, whch s assumed to be constant n ths thess, and W = W W 1 s the ndependent Brownan ncrements, whch follows a normal dstrbuton N(0, t). The dscretzed process S gven n ths way s essentally a Markov chan. 5

11 Fgure 2.1: The value of Amercan put opton compared wth the correspondng European opton. The value of Amercan opton s calculated wth Fnte Dfference Method (FDM). The value of European opton s calculated wth Black-Scholes formula. Parameters used are K = 100, r = 5%, q = 0%, σ = 0.5, T = Dynamc Programmng Prncple Gven the dscretzaton settng n Secton 2.1, the dynamc programmng prncple ntroduced by Rchard Bellman [14] provdes us an effectve tool to prce Amercan optons. Let s assume at tme t = t, the Amercan opton under the queston has not been excsed before. Thus, accordng to (2.2), assumng the constant nterest rate, we can wrte the value of an Amercan opton at t = t as [ ] Ṽ (s) = sup E e r(τ t ) h(sτ, τ) S = s τ T (2.7) whch s essentally equvalent to a newly ssued opton at t wth the state of asset prce startng from S = s. Now, f we work recursvely from t = t M = T, we are able to defne a recurson formula for the prce of Amercan optons. Frstly, let s consder the termnal tme at opton s maturty T. We know at maturty, the opton holder wll ether choose to exercse the opton or let the opton expre worthless. Thus, at t = t M, the value of an Amercan opton s smply the opton s payoff, gven by Ṽ M (s) = h(s, T ), S M = s (2.8) 6

12 where h(x, t) s the payoff functon. Now, consder the tme t = t M 1 wth the state of asset prce S M 1 = s, an nvestor wll choose to exercse the opton f and only f ts current payoff s greater than the dscounted expected value to be receved f the nvestor choose not to exercse, thus accordng to (2.7), the value of opton at t M 1 s gven by [ ]} Ṽ M 1 (s) = max { h(s, tm 1 ), E D M 1,M Ṽ M (S M ) S M 1 = s (2.9) where D M 1,M = e r(t M t M 1 ) s the dscount factor from t M 1 to t M. Thus, by dynamc programmng prncple, we can derve a recursve formulaton for the value of Amercan optons as blow {ṼM (s) = h(s, T ), S M = s Ṽ 1 (s) = max { h(s, t 1 ), E [ ]} D 1, Ṽ (S ) S 1 = s, = 1 M (2.10) Ths formulaton essentally provdes the basc prncple of optmally exercsng an Amercan opton,.e., the exercse descson s made at t = t 1 by comparng the mmedate exercse payoff h(s, t 1 ) aganst the expected present value of contnung. Also, an Amercan opton s usually not exercsable at tme 0, and ths can be done by smply settng h(s, 0) = 0. In (2.10), all the value functons Ṽ(s) are gven n tme-t currency. However, we are essentally nterested n prcng an Amercan opton at tme 0. So, t s natural to dscount all the values to tme 0, so that the dscount factors D 1, can be dscarded. Now let h (s) = D 0, h(s, t ), V (s) = D 0, Ṽ (s) = 0,, M where D 0, = e r(t t 0 ) s the dscount factor from t 0 to t. Then, accordng to (2.10), we have V 0 (s) =Ṽ0(s), V M (s) =h M (s), V 1 (s) =D 0, 1 Ṽ 1 (s) [ ]} =D 0, 1 max { h(s, t 1 ), E D 1, Ṽ (S ) S 1 =s { [ ]} = max h 1 (s), E D 0, 1 D 1, Ṽ (S ) S 1 =s = max {h 1 (s), E [V (S ) S 1 =s]} Thus, we can absorb the dscount factors nto the payoff and value functons, and (2.10) s then smplfed as { V M (s) = h M (s), S M = s V 1 (s) = max {h 1 (s), E [V (S ) S 1 =s]}, = 1 M (2.11) 7

13 2.3 Optmal Stoppng Rule and Contnuaton Values The dynamc programmng recursons (2.11) gve a way to choose optmal exercse strategy for the Amercan opton. Assume for any stoppng tme τ {t 1, t 2,, t M }, a sub-optmal value of Amercan opton s defned as V τ 0 (S 0 ) = E [h τ (S τ )] (2.12) The queston here s to choose an optmal stoppng tme. We know that at any tme t, a dscounted value functon V (s) s assgned to each state of asset prce process S =s by the dynamc programmng recursons (2.11). Thus, t s natural to choose our optmal stoppng tme τ as the frst tme when the payoff exceeds the value functon, τ s thus gven by [4] τ = mn {τ {t 1,, t M } : h (S ) V (S )} (2.13) And on the other hand, at any tme t, an nvestor wll exercse the opton when the state of asset prce S gves a payoff larger than the value functon, gven that the opton has not been exercsed before. And we call the set of asset prce states {s : h (s) V (s)} the exercse regon at tme t and the complement to ths set as contnuaton regon. Wth these defntons, the optmal stoppng rule (2.13) can be vewed as the frst tme the underlyng asset prce process S enters an exercse regon. We can now defne the contnuaton value of an Amercan opton as the value of holdng rather than exercsng the opton. The contnuaton value s smply defned as [4] { C M (s) = 0 (2.14) C (s) = E [V +1 (S +1 ) S = s], = 0,, M 1 where V (x) s defned by dynamc programmng recursons (2.11). Thus, accordng to (2.11) we have V (s) = max {h (s), C (s)}, = 1,, M (2.15) and the optmal stoppng rule n (2.13) can be re-wrtten as τ = mn {τ {t 1,, t M } : h (S ) C (S )} (2.16) and the value of the opton s then determned by ths rule as V τ 0 (S 0 ) = E [h τ (S τ )] (2.17) 8

14 Chapter 3 Longstaff-Schwartz Method: a Lower Bounds Estmator usng Least-Squares Regressons As we have seen n Chapter 2, prcng Amercan optons s essentally reduced to a dynamc programmng problem wth the recursve formulaton (2.11). The man dffculty les n the estmatons of the condtonal expectatons,.e. the contnuaton values C (s) gven by C (s) = E [V +1 (S +1 ) S = s], = 0,, M 1 (3.1) The most popular approach to estmate the contnuaton values are based on regresson methods, suggested by Longstaff and Schwartz [5]. The contnuaton value C (s) n (3.1), that essentally s a condtonal expectaton, s approxmated by a lnear combnaton of known functons (bass functons) of the current state of asset prce s. Regresson method s then used to estmate the optmal coeffcents for the approxmaton. In ths thess, the least-squares based regresson s employed. 3.1 Approxmatng Contnuaton Values Usng Least-Squares Regresson Our am s to approxmate the contnuaton values C (s) by regresson,.e., we want to fnd an expresson of the form C (s) = E [V +1 (S +1 ) S =s] = J β j ψ j (s) = β ψ(s) (3.2) where ψ(s) = (ψ 1 (s), ψ 2 (s),, ψ J (s)) s the vector of some bass functons and β = (β 1, β 2,, β J ) s the vector of the regresson coeffcents, whch depends on tme t. If we use the least-squares rule, then the regresson coeffcents s determned by the followng theorem. j=1 9

15 Theorem 3.1. Least-squares based regresson coeffcents If there exsts a relatonshp between the contnuaton value C (s) and current state of asset prce s n the form (3.2), the least-squares regresson wll gve the coeffcents β n the form ( [ β = E ψ(s )ψ(s ) ]) 1 E [ψ(s )V +1 (S +1 )] B 1 ψ B ψv (3.3) where B ψ =E [ ψ(s )ψ(s ) ] s a J J matrx and B ψv =E [ψ(s )V +1 (S +1 )] s a vector of length J. Proof. The regresson coeffcents β s determned by the least-squares optmzaton of [ ( ) ] 2 E ψ (S ) β E [V +1 (S +1 ) S ] mn By takng the dervatve wth respect to β and equatng to 0, we get [ ( )] E ψ (S ) ψ (S ) β E [V +1 (S +1 ) S ] = 0 By rearrangng, we get [ E ψ (S ) ψ (S ) ] β =E [ψ (S ) E [V +1 (S +1 ) S ]] =E [ψ (S ) V +1 (S +1 )] Solvng for β, we fnally get ( [ β = E ψ(s )ψ(s ) ]) 1 E [ψ(s )V +1 (S +1 )] (3.4) The theoretcal value of regresson coeffcents β s gven by (3.3), whch can be estmated n ( practce by Monte Carlo ) smulaton. Gven N ndependent smulated paths of asset prce S (n) 1, S (n) 2,, S (n) M, n = 1,, N, whch can be obtaned by Euler-Maruyama ( ) scheme (2.6), and assumng that at t the value functons V +1 S (n) are known, the leastsquares estmaton of the regresson coeffcents β s then gven by +1 ˆβ = ˆB 1 ψ, ˆB ψv, (3.5) where ˆB ψ, = 1 N ˆB ψv, = 1 N N ψ(s (n) n=1 N ψ(s (n) n=1 )ψ(s (n) ) (3.6) )V +1 (S (n) +1 ) (3.7) and S (n) need to be replaced by the estmated values ˆV +1, gven by and S (n) +1 correspond to the same Monte Carlo trajectory. Also, n practce, V +1 ˆV +1 (S +1 ) = max { h +1 (S +1 ), Ĉ+1(S +1 ) } (3.8) 10

16 Then the contnuaton value C (S ) can be estmated by Ĉ (S ) = ˆβ ψ(s ) (3.9) Now we can summarze the procedure for the regresson coeffcents estmatons as n the followng algorthm. Algorthm 3.2. Backward nducton for least-squares regresson coeffcents estmaton 1. Smulate N ndependent trajectores of the asset prce process {S (n) 1, S (n) 2,, S (n) M }, n = 1,, N. 2. At termnal tme t M, set ˆV M (S (n) M ) = h M(S (n) M ). 3. Apply backward nducton for = M 1,, 1 (a) estmate the regresson coeffcents by ˆβ = gven by (3.6) and (3.7) respectvely. ˆB 1 ψ, ˆB ψv,, where ˆB ψ, and ˆB ψv, are (n) (b) calculate the contnuaton values Ĉ(S ) = ˆβ ψ(s(n) ), n = 1,, N. { } (n) (c) set ˆV = max h (S (n) (n) ), Ĉ(S ), n = 1,, N. 4. Store vectors β ready to use. Once we have the estmatons of the regresson coeffcents β, the value of the opton can be estmated wth the stoppng rule defned n (2.16) usng the 2nd set of ndependent Mote Carlo paths. 3.2 Low-based Estmator Usng Optmal Stoppng Rule The estmated regresson coeffcents ˆβ determnes the approxmatons of the contnuaton values Ĉ(S ) as n (3.9) at tme step t, gven the state of asset prce S, whch n turn defne an optmal stoppng strategy gven by { } ˆτ = mn τ {t 1,, t M } : h (S ) Ĉ(S ) (3.10) Thus, by smulatng a 2nd set of Monte Carlo paths and applyng ths stoppng strategy, the value of the Amercan opton can be estmated as ˆV 0 (S 0 ) = E [hˆτ (Sˆτ )] (3.11) As we can see, the approxmated stoppng rule gven n (3.10) s nevtably sub-optmal,.e., t departs from the true value gven by (2.16). Thus, we must have V 0 (S 0 ) = sup τ T 0 E [h τ (S τ )] E [hˆτ (Sˆτ )] = ˆV 0 (S 0 ) (3.12) whch ndcates that the estmator defned n (3.11) s a low-based estmator whch provdes a lower bound of the true value. We now summarze the procedure n the followng algorthm. 11

17 Algorthm 3.3. Low estmator usng optmal stoppng rule 1. Load regresson coeffcents β, = 1,, M 2. Smulate N ndependent paths of the asset prce process {S (n) 1, S (n) 2,, S (n) M }, n = 1,, N. (Ths should be a 2nd set of paths ndependent from the 1st one used to estmate the regresson coeffcents β gen n Algorthm 3.2) 3. Apply forward nducton for = 1,, M 1, and for each n = 1,, N (n) (a) calculate contnuaton values Ĉ(S ) = ˆβ ψ(s(n) ) (b) calculate the payoff functons h (S (n) ) 4. For termnal tme = M, set C M (S (n) M ) = 0 and calculate h M(S (n) M ) { } (n) 5. Set ˆV 0 = h (S (n) ), =mn {1,, M} : h (S (n) (n) ) Ĉ(S ) 6. Calculate the estmated value of opton as ˆV 0 = 1 N 3.3 Numercal Examples In ths secton, the Longstaff-Schwartz method s tested usng a smple numercal example of an Amercan put opton wrtten on a sngle non-dvdend payng stock wth the srke K = 20, nterest rate r = 5%, volatlty σ = 0.4 and maturty T = 1 respectvely. For the bass regresson functons ψ (x), we choose the weghted Laguerre polynomals suggested by Longstaff and Schwartz [5], whch are defned as where L k (x) s the Laguerre polynomals defned as N n=1 ˆV (n) 0 ψ j (x) = e x/2 L k (x), k = j 1 (3.13) L k (x) = ex d k ( k! dx k e x x k) 1, k = 0 = 1 x, k = 1 1 k ((2k 1 x) L k 1(x)) (k 1)L k 2 (x), for k 1 For the numercal testng, we fxed the number of bass functons J = 5. (3.14) We ntally set the number of Monte Carlo paths to be N = and fxed the number of tme steps M = 2 5 = 32. Fgure 3.1 shows the numercal results of Longstaff-Schwartz method (LSM) compared wth those gven by the penalty approach usng Fnte Dfference Method (FDM) [3]. For further nvestgatons, we change the number of ndependent Monte Carlo paths from 5e3 to 1e5. The results are shown n Table 3.1 and compared wth those of the Fnte Dfference Method. 12

18 Fgure 3.1: Values of Amercan opton estmated by Longstaff-Schwartz method compared wth the Fnte Dfference Method (FDM). The parameters used are, K = 20, r = 5%, σ = 0.4 and T = 1. Number of paths smulated N = 10000, number of tme steps M = 32. LSM estmated values Varances of LSM estmator S 0 FDM N=5e3 N=1e4 N=1e5 N=5e3 N=1e4 N=1e E E E E E E E E E E E E E E E E E E E E E-05 Table 3.1: Results of Longstaff-Schwartz based Monte Carlo method usng dfferent number of smulated paths N, compared wth Fnte Dfferent Method. The parameters used are, K = 20, r = 5%, σ = 0.4, T = 1 and number of tme steps M = 32. As we can see n the table, all the three sets of results gve the smlar estmated values of the opton, whch have the smlar accuracy compared wth those gven by FDM. However, fewer smulated paths wll gve much hgher varances. Another notceable fact s that the varances of the LSM estmators tend to be larger when the opton s close to at-themoney. When the ntal stock prce S 0 = 20, the MC estmator gves the largest varances. It s worth mentonng that, some estmated values usng Monte Carlo method are lower than those of the Fnte Dfference Method whle others are hgher, especally for the deepout-the-money optons. However, we shouldn t consder ths as a contradcton wth the 13

19 argument that the Longstaff-Schwartz method provdes a low-based estmator due to the fact that Monte Carlo method comes wth two sources of errors. One s the bas from the method tself and another s from the varance of MC estmators. For deep-out-the-money optons, the opton values are relatvely low, thus the same level varance wll gve a hgh based value even wth a low-based estmator. Next, to nvestgate the effects of number of exercse opportuntes, we set the number of tme steps M=8, 16, 32, 64 and 128 respectvely and fx the number of Monte Carlo paths to be N= The results are shown n Tabel 3.2. We notce that, when the tme steps ncreases, the LSM estmated values are generally gettng closer to the results gven by FDM, whch uses a very fne tme steppng (500 steps). Ths s n agreements wth the argument that the value of a dscretzed verson of Amercan opton wll be gettng to an approxmaton of the value of contnuous verson Amercan opton when the number of tme steps M. The varances of the MC estmator are not notceably affected for most cases. But for deep-n-the-money and deep-out-of-money optons, hgher number of tme steps gves lower varances. LSM estmated values Varances of LSM estmator S0 FDM M=8 M=16 M=32 M=64 M=128 M=8 M=16 M=32 M=64 M= E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-04 Table 3.2: Results of Longstaff-Schwartz based Monte Carlo method usng dfferent number of tme steps M, compared wth Fnte Dfference Method. The parameters used are K = 20, r = 5%, σ = 0.4, T = 1 and number of smulated MC parths N =

20 Chapter 4 The Dualty through Martngales: an Upper Bounds Estmator As we have seen n Chapter 3, the Longstaff-Schwartz based Monte Carlo method (LSM) provdes a low-based estmator, gvng a lower bound of the true value of the opton. We call ths the prmal problem. A new drecton n Monte Carlo smulaton of Amercan optons s the dual approach, developed by Rogers [7] and ndependently by Haugh and Kogan [8]. One of the most popular mplementatons of the method was presented by Andersen and Broade [9]. The basc dea of ths dual approach s to represent the prce of an Amercan opton through a mnmzaton problem. The dualty mnmzes over a class of supermartngales or martngales and leads to a hgh-based approxmaton, provdng upper bounds on prces. The lower bounds provded by LSM and the upper bounds provded by the dual approach contan the true prce. 4.1 The Dualty: a Mnmzaton Problem over Martngales As we have seen n Chapter 2, the dscounted value functons V (S ) satsfes the recursve formulaton gven by (2.11), whch s re-wrtten below { V M (s) = h M (s), S M = s Thus, we have the nequalty: V 1 (s) = max {h 1 (s), E [V (S ) S 1 =s]}, = 1 M (4.1) V (S ) E [V (S +1 ) S ], = 0,, M 1 (4.2) whch n fact ndcates that the value functon V (S ) s a supermartngale. And also, we have V (S ) h (S ), = 0,, M (4.3) The value functon process V (S ), = 0,, M s n fact the mnmal supermartngale domnatng the dscounted payoff functons h (S ) at all exercse tmes t, whch s a wellknown characterzaton of Amercan opton prce [15]. Ths characterzaton was extended 15

21 by Haugh and Kogan [8], who proposed a mnmzaton problem to formulate the prcng of Amercan optons. In ths thess, we wll employ the major results from Haugh and Kogan wth specalzatons to martngales. Let M = {M, = 0,, M} be a martngale wth M 0 = 0, then the followng theorem holds. Theorem 4.1. Dualty through martngales For any martngales M = {M, = 0,, M} satsfyng M 0 = 0, the prce of Amercan opton V 0 (S 0 ) satsfes the nequalty V 0 (S 0 ) nf M [ ] E max {h (S ) M } =1,,M (4.4) The equalty holds wth the optmal martngale M defned by M0 = 0, M = k, for = 1,, M k=1 (4.5) where s gven by = V (S ) E [V (S ) S 1 ] (4.6) Proof. We frst prove the nequalty (4.4) holds. Accordng to the optonal samplng theorem of martngales, for any stoppng tme τ {t 1, t 2,, t M }, E [M τ ] = M 0 = 0, and thus we have [ ] E [h τ (S τ )] = E [h τ (S τ ) M τ ] E max {h (S ) M } =1,,M By takng the nfmum over the martngales M, we get [ ] E [h τ (S τ )] nf E max {h (S ) M } M =1,,M (4.7) The nequalty (4.7) holds for any stoppng tmes τ, thus t also holds for the supremum over τ. Thus we have proved the nequalty V 0 (S 0 ) = sup τ E [h τ (S τ )] nf M [ ] E max {h (S ) M } =1,,M Next, we prove the equalty n (4.4) holds for M defned by (4.5) and (4.6). We frst show that M s actually a martngale. Gven the defnton of the, we have E [ S 1 ] = E [V (S ) E [V (S ) S 1 ] S 1 ] = 0 (4.8) Thus, we have E [M [ ] 1 S 1 ] = E k S 1 = k = M k=1 k=1 1 16

22 whch ndcates that M satsfes the martngale property. Next, we use backward nducton to prove that V (S ) =E[max{h (S ), h +1 (S +1 ) +1, h +2 (S +2 ) +2 +1, h M (S M ) M +1 } S ], for = 1,, M (4.9) Ths holds for the termnal tme step t M, snce we have V M (S M )=h M (S M )=E[h M (S M ) S M ]. Now, assume (4.9) holds for the tme step t, then accordng to (4.1) and (4.6), we have V 1 (S 1 ) = max {h 1 (S 1 ), E [V (S ) S 1 ]} =E [max {h 1 (S 1 ), E [V (S ) S 1 ]} S 1 ] =E [max {h 1 (S 1 ), V (S ) } S 1 ] =E[max{h 1 (S 1 ), h (S ), h +1 (S +1 ) +1, h M (S M ) M } S 1 ] whch ndcates (4.9) also holds for t 1. Fnally, at t = t 0, recall that we have assumed the opton s not exercsable mmedately and h 0 (S 0 ) = 0, the opton value at t 0 s thus gven by V 0 (S 0 ) = E [V 1 (S 1 ) S 0 ] = E [V 1 (S 1 ) 1 S 0 ] and accordng to (4.9) V 1 (S 1 ) =E[max{h 1 (S 1 ), h 2 (S 2 ) 2, h 3 (S 3 ) 3 2, h M (S M ) M 2 } S 1 ] We then fnally get [ [ ] V 0 (S 0 ) = E E max {h (S ) M } S 1 =1,,M [ ] = E max {h (S ) M } =1,,M S 0 ] (4.10) Ths verfes that equalty n (4.4) holds for M defned by (4.5) and (4.6), whch s ndeed an optmal martngale. Equaton (4.10) provdes an estmator for prcng Amercan optons. If we can fnd a martngale Mˆ that s close to the optmal martngale M, then we can estmate the value of an Amercan opton by [ ˆV 0 (S 0 ) = E max =1,,M whch s ndeed our dualty estmator. Notce that the martngale { h (S ) M ˆ } ] (4.11) ˆ M s nevtably suboptmal and thus (4.11) provdes a hgh-based estmator, provdng an upper bound for the opton prce. Ths, conjunct wth the lower bound gven by LSM approach, gves a range contanng the true prce of the Amercan opton. 17

23 4.2 The Dualty Estmaton Usng Martngales from Approxmated Value Functons As we have seen n the prevous secton, we can use (4.11) to estmate the dualty by constructng a martngale Mˆ that s close to the optmal martngale M. Ths can be done by constructng the martngales based on the approxmated value functons ˆV (S ). Gven the defnton of the optmal martngale M n (4.5) and (4.6), t s natural to defne the approxmaton Mˆ as Mˆ 0 = 0, Mˆ = (4.12) ˆ k, for = 1,, M where ˆ s gven by It easy to verfy that ˆ M defned by (4.12) and (4.13) satsfes the general martngale property. Recall that ˆV (S ) s gven by k=1 ˆ = ˆV [ ] (S ) E ˆV (S ) S 1 (4.13) ˆV (S ) = max { } h (S ), Ĉ(S ) and Ĉ(S ) s the estmaton of the contnuaton value, gven by (4.14) Ĉ (S ) = Ê [V +1(S +1 ) S ] = ˆβ ψ(s ) (4.15) where ˆβ s the vector of LSM regresson coeffcents and ψ(x) s the vector [ of our bass ] functons. One mght thnk of replacng the condtonal expectaton E ˆV (S ) S 1 n (4.13) wth Ĉ 1(S 1 ). However, ths wll not gve a vald constructon of martngales. The reason s that Ĉ 1(S 1 ) s the estmaton of the condtonal expectaton of true value functons,.e. Ĉ 1 (S 1 )=Ê [V (S ) S 1 ], whch s generally not equvalent to an estmaton of E[ ˆV (S ) S 1 ], whch s the condtonal expectaton of approxmated value functons. Thus, to construct a vald martngale, we need to use a nested smulaton [4]. Assume we have smulated the man Monte Carlo paths {S (n) : n = 1,, N}, at each step S 1, we smulate m sub-successors { expectaton E[ ˆV (S ) S 1 ] by S (k) Ê[ ˆV (S ) S 1 ] = 1 m : k = 1,, m} and estmate the condtonal m k=1 (k) ˆV ( S ) (4.16) where ˆV (k) ( S ) s calculated n the same way as gven n (4.14). Equaton (4.16) gves a condtonally unbased estmator of E[ ˆV (S ) S 1 ] gven S 1, thus provdng a vald way to construct the martngales as n (4.12) and (4.13), whch n turn gves a vald dualty estmator as n (4.11). We summarze the procedure n the followng algorthm. 18

24 Algorthm 4.2. Dualty estmaton usng martngales based on approxmated value functons 1. Load regresson coeffcents β, = 1,, M, whch are gven by Algorthm Smulate N ndependent paths of the asset prce process {S (n) 1, S (n) 2,, S (n) M }, n = 1,, N. (Ths should be a 2nd set of paths ndependent from the 1st one used to estmate the regresson coeffcents β gen n Algorthm 3.2) 3. Set the ntal martngale ˆ M 0 = 0 4. Apply forward nducton for = 1,, M 1, and for each n = 1,, N (n) (a) calculate contnuaton values Ĉ(S ) = ˆβ ψ(s(n) ) (b) calculate the payoff functons h (S (n) ) (c) calculate the approxmated { value functons } ˆV (S (n) ) = max h (S (n) (n) ), Ĉ(S ) (1) (d) smulate m ndependent sub Monte Carlo successors { S, the prevous tme step S (n) 1 (e) calculate the estmaton of martngale dfferental by ˆ (n) = ˆV (S (n) m ) 1 (k) m ˆV ( S ) where ˆV ( S (k) k=1 (f) calculate the martngales ) are calculated smlarly usng regresson. M ˆ (n) = ˆ M (n) 1 5. For termnal tme = M, for each n = 1,, N 6. Set (a) set C M (S (n) M ) = 0 and calculate h M(S (n) M ) (b) set ˆV M (S (n) M ) = h M(S (n) M ) + ˆ (n) (c) smulate m ndependent sub Monte Carlo successors { the prevous tme step S (n) M 1 (d) calculate the estmaton of martngale dfferental by ˆ (n) M = ˆV M (S (n) M ) m 1 (k) m ˆV M ( S M ) k=1 where ˆV (k) M ( S M ) = h (k) M( S M ). (e) calculate the martngale ( h (S (n) ˆV (n) 0 = max =1,,M M ˆ (n) M = ˆ ) M (n) ) ˆ M (n) M 1 + ˆ (n) M 7. Calculate the estmated value of dualty as ˆV 0 = 1 N N n=1 ˆV (n) 0 S (1) M (2) S,,, S (2) M (m) S } from (m),, S M } from 19

25 4.3 Numercal Examples We now test the algorthm 4.2 usng the same example provded n Secton 3.3. The Amercan put opton s wrtten on a sngle non-dvdend payng stock wth the srke K = 20, nterest rate r = 5%, volatlty σ = 0.4 and maturty T = 1 respectvely. The weghted Laguerre polynomals defned n (3.13) and (3.14) are used as our regresson bass functons. For the numercal testng, we fxed the number of bass functons J =5. Agan, we fxed the number of tme steps M = 32 and change the number of man Monte Carlo paths from 5e3 to 1e5. The number of smulated sub-successors for the constructon of martngales are fxed wth m = 100. The results are shown n Table 4.1. As we can see, the dualty estmatons are generally hgher than the results ether by the LSM or by the Fnte Dfference Method, whch confrms the argument that the dualty approach provdes a hgh-based estmator, thus gvng a upper bound of the true prce. When we change the number of man Monte Carlo paths, the fewer smulated paths agan gve hgher varances and the hghest varances tend to be wth the at-the-money optons. Another notceable fact s that the varances of dualty estmators are generally much smaller than those of the LSM, whch provdes the lower bounds. MC estmated values N=5e3 N=1e4 N=1e5 S 0 FDM LSM DUAL LSM DUAL LSM DUAL Varances of MC estmators N=5e3 N=1e4 N=1e5 S0 LSM DUAL LSM DUAL LSM DUAL E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-07 Table 4.1: Results of dualty approach usng dfferent number of smulated paths N, compared wth Fnte Dfferent Method. The parameters used are, K = 20, r = 5%, σ = 0.4, T = 1, number of tme steps M = 32 and number of sub-successors m =

26 To nvestgate the effects of the number of sub-successors m for the constructon of martngales, we fxed the number of man Monte Carlo paths N = 1e4 and the number of tme steps M = 32. We set the number of sub-successors as m = 50, 100, and 200 respectvely. The results are shown n Table 4.2. As we can see, when the number of subsuccessors ncreases, the estmated dualty values decrease, gvng smaller upper bounds and thus tghter ranges contanng the true prce, conjunct wth the lower bounds gven by LSM. Ths s reasonable because larger number of sub-successors gves better estmaton of the condtonal expectaton E[ ˆV (S ) S 1 ], whch n turn gves hgher accuracy n the estmatons of the martngales M ˆ. Another nterestng fact s that hgher number of subsuccessors also gves lower overall varances of the estmator, whch agan s the outcome of the better estmatons of the condtonal expectatons. MC estmated values S 0 FDM LSM DUAL/m=50 DUAL/m=100 DUAL/m= Varances of MC estmator S 0 LSM DUAL/m=50 DUAL/m=100 DUAL/m= E E E E E E E E E E E E E E E E E E E E E E E E E E E E-06 Table 4.2: Results of dualty approach usng dfferent number of sub-successors m for the constructon of martngales, compared wth Fnte Dfferent Method. The parameters used are, K = 20, r = 5%, σ = 0.4, T = 1, number of man Monte Carlo paths N = 1e4, number of tme steps M = 32. Fnally, we nvestgated the effects of number of tme steps whle fxng the number of man Monte Carlo paths N = 1e4 and the number of sub-successors m = 100. set M = 32, 64 and 128 respectvely. The results are shown n Table 4.3. As shown n the table, whle the LSM estmates are gettng closer to the results gven by the Fnte Dfference Method, the estmates of dualty have a trend of ncreasng, whch means the ranges provded by the upper-lower bounds are gettng wder. We The reason behnd ths 21

27 phenomena s stll unknown. Agan, the varances of the MC estmators are not notceably affected by the number of tme steps for most cases. MC estmated values M=32 M=64 M=128 S 0 FDM LSM DUAL LSM DUAL LSM DUAL Varances of MC estmator M=32 M=64 M=128 S 0 LSM DUAL LSM DUAL LSM DUAL E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-06 Table 4.3: Results of dualty approach usng dfferent tme steps M, compared wth Fnte Dfferent Method. The parameters used are, K = 20, r = 5%, σ = 0.4, T = 1, number of man Monte Carlo paths N = 1e4, number of sub-successors m =

28 Chapter 5 Orthogonalty of Hermte Polynomals: A Computng Savng Technque As we have seen n the prevous chapters. For prcng Amercan optons, both the Longstaff- Schwartz method estmatng lower bounds (Algorthm 3.3) and the dualty method estmatng upper bounds (Algorthm 4.2) rely on the regresson based estmatons of contnuaton values C(S), whch n turn depend on the qualty of estmatons of the regresson coeffcents β. Thus, for both methods, certan efforts are made towards the computng of regresson coeffcents, whch s fulflled by a pre-processng procedure (Algorthm 3.2). Recall that the the regresson coeffcents β are estmated by backward nducton usng ˆβ = ˆB 1 ψ, ˆB ψv,, = M 1,, 1 (5.1) where [ ˆB ψ, = Ê ψ(s )ψ(s ) ] = 1 N [ ˆB ψv, = Ê ψ(s ) ˆV ] +1 (S +1 ) = 1 N N ψ(s (n) n=1 N ψ(s (n) n=1 )ψ(s (n) ) (5.2) ) ˆV +1 (S (n) +1 ) (5.3) and ˆV { } +1 (S +1 )=max h +1 (S +1 ), Ĉ+1(S +1 ) are estmated value functons and Ĉ+1(S +1 ) = ˆβ +1 ψ(s +1) are estmated contnuaton functons. As we can see, for good estmatons of β, a lot effort s made towards the estmatons of ˆBψ, gven by (5.2), whch requres computng the sum of N J J matrces, where J s the number of bass functons. of man Monte Carlo paths N s requred to be large. Usually, to acheve suffcent accuracy n β, the number Ths wll cause extremely slow computng wth respect to Equaton (5.2). However, f we can fnd a way to construct our bass functons ψ(x) such that the matrx B ψ, = E [ ψ(s )ψ(s ) ] s a J J dentty matrx 23

29 I and the estmaton ˆB ψ, s close to an dentty matrx. Then we can smply dscard ˆB ψ, from (5.1) and construct a smplfed estmator as whch wll gve sgnfcant computng savng. ˆβ = ˆB ψv, (5.4) Notce that the qr entry of the matrx B ψ, s gven by [ ( (B ψ, ) qr =E ψ(s )ψ(s ) ) ] = E [ψ q (S )ψ r (S )], qr q, r {1,, J} (5.5) Thus, f we can construct our bass functons wth the orthogonalty such that { + 1 q = r E [ψ q (S )ψ r (S )] = ψ q (S )ψ r (S )p(s ) ds = δ qr = 0 q r (5.6) where p(s ) s the uncondtonal probablty densty functon of asset prce state S, the smplfed estmator (5.4) wll then be vald. Hermte polynomals are promsng canddates for the constructons of our bass functons due to the natural orthogonalty nherent. 5.1 Hermte Polynomals The probablsts Hermte Polynomals are a classcal orthogonal polynomal sequence named after Charles Hermte [16], defned by 1, k = 0 H k (x)=( 1) k e x2 /2 dk /2 dx k e x2 = x, k = 1 xh k 1 (x) (k 1)H k 2 (x), for k 1 (5.7) The The most notceable property of Hermte Polynomals s that they are orthogonal wth respect to the weght functon w(x) = e x2 /2 (5.8) And the followng equaton holds for all probablsts Hermte Polynomals [17]. + H q (x)h r (x)e x2 /2 dx = 2πq!δ qr (5.9) By rearrangng, we get Notce that φ(x) = e x2 /2 2π let + H q (x) q! H r (x) r! e x2 /2 2π dx = δ qr (5.10) s the probablty densty of a standard normal dstrbuton and ψ k (x) = H k(x) k! (5.11) 24

30 we get [ E ψq (x) ψ ] + r (x) = ψ q (x) ψ r (x)φ(x) dx = δ qr (5.12) where the expectaton s takng wth respect to random a varable x N(0, 1). Comparng (5.12) whth (5.6), we fnd that ψ k (x) s a good canddate for our bass functons. The only thng we need to do s to convert our asset prce state S to a random varable followng a standard normal dstrbuton. 5.2 Constructng Orthogonal Bass Functons through Hermte Polynomals As we have seen n the prevous secton, f we can convert our state of the asset prce S nto a standard normal random varable, we are able to use (5.11) to construct a set of bass functons whch are orthogonal to each other. For ths purpose, nstead of usng Euler- Maruyama scheme as gven n (2.6), we change our method to smulate the Monte Carlo paths usng the exact solutons to SDE (2.4), gven by ) S = S 1 exp ((r q σ2 2 ) t + σ W, = 1,, M (5.13) Here, we assume the entre tme horzon s equally dvded. Under ths method, the prce state S has an explct relaton wth the ntal value S 0, gven by ( ) S = S 0 exp (r q σ2 2 ) t + σ W k k=1 (5.14) Notce that the Brownan ncrements W k n (5.14) are ndependent to each other, S thus has an explct log-normal dstrbuton and log(s ) follows a normal dstrbuton ) log(s ) N (log(s 0 )+(r q σ2 2 ) t, σ2 t Now, we can easly convert S nto a standard normal random varable Y by (5.15) Y = log(s ) log(s 0 ) (r q σ2 2 ) t σ N(0, 1) (5.16) t Fnally, we construct our bass functons through ψ j (S ) = ψ k (Y ) = H k(y ) k!, k =j 1, j = 1,, J (5.17) where H k (x) s the Hermte Polynomals gven by (5.7). By dong ths, t s easy to verfy that [ E [ψ q (S )ψ r (S )] = E ψq 1 (Y ) ψ ] r 1 (Y ) = δ qr (5.18) Thus, we essentally construct a set of bass functons who are orthogonal to each other wth respect to the natural dstrbuton of the underlyng asset prce sate S. To estmate the regresson coeffcents wth these orthogonal bass functons, the smplfed estmator (5.4) can be used, provdng large amounts of computng savng. 25

31 5.3 Numercal Examples To test the dea, we use the same example of an Amercan put opton wrtten on a sngle non-dvdend payng stock wth the ntal prce S 0 = 20, the srke K = 20, nterest rate r = 5%, volatlty σ = 0.4 and maturty T = 1 respectvely. As mentoned earler, we do not use regresson on the termnal tme step t M snce the value functons are smply set as equal to the payoff functons at t M. So, we start from the smplest case wth 2 tme steps and focus on the mddle tme step t 1. We tested the Algorthm 3.2 usng the orthogonal bass functons gven n (5.17). For comparson, we tested both the usual estmator gven by (5.1) and the smplfed estmator gven by (5.4). We frst computed the estmated matrx ˆB ψ,1 = Ê [ ψ(s 1 )ψ(s 1 ) ] and set the number of Monte Carlo paths as N=1e4, 1e5 and 1e6 respectvely. We also fxed the number of bass functons as J=5. The results are shown n Table 5.1 below. As we can see, by usng the orthogonal bass functons, the estmated matrx ˆB ψ,1 =Ê [ ψ(s 1 )ψ(s 1 ) ] s close to an dentty matrx I, as expected. The qualty of ths smlarty wth dentty matrx mproves wth larger number of Monte Carlo paths N. N=1e N=1e N=1e Table 5.1: The estmated matrx ˆB ψ = Ê [ ψ(s)ψ(s) ] usng orthogonal bass functons. The parameters used are S 0 = 20, K = 20, r = 5%, σ = 0.4, T = 1, number of tme steps M=2 and number of bass functons J=5. Table 5.2 gves the estmated regresson coeffcents ˆβ 1 and the correspondng computng tme usng the smplfed estmator (5.4) compared wth those gven by the standard estmator (5.1). Agan we set the number of Monte Carlo paths as N=1e4, 1e5 and 1e6 26