Centre for Internatonal Captal Markets Dscusson Papers ISSN 1749-3412 Valung Amercan Style Dervatves by Least Squares Methods Maro Cerrato No 2007-13
Valung Amercan Style Dervatves by Least Squares Methods Maro Cerrato London Metropoltan Busness School London Metropoltan Unversty October 2007 Correspondng author: M.Cerrato; E-mal:m.cerrato@londonmet.ac.uk; I am grateful to Guglelmo Mara Caporale and L. Stentoft for valuable comments. I am also grateful to Kan Kwok Cheung for carryng prelmnary computatons that helped me form some of the deas analysed n ths paper. The usual dsclamer apples. 2
Abstract We mplement some recent Monte Carlo estmators for opton prcng and assess ther performance n fnte samples. We fnd that the accuracy of these estmators s remarkable, even when more exotc fnancal dervatves are consdered. Fnally, we mplement the Glasserman and Yu (2004b) methodology to prce Asan Bermudan optons and basket optons. Key Words: Amercan optons, Monte Carlo method JEL Classfcaton: G10, G13 3
1. Introducton Monte Carlo methods to prce Amercan style optons seem to be now an actve research area. The reason s manly due to ts sutablty to prce path dependent optons and to solve hgh dmensonal problems. It s now standard to mplement Monte Carlo methods usng regresson methods to prce dervatves wth Amercan features. For example, Longstaff and Schwartz (2001) suggest usng Least Squares approxmaton to approxmate the opton prce on the contnuaton regon and Monte Carlo methods to compute the opton value (LS). Proof of asymptotc convergence of the opton prce estmator s derved under varous assumptons and therefore more work s needed n ths case. Recently Clement et al (2002) undertake a theoretcal analyss of the LS estmator, and show that the opton prce converges, n the lmt, to the true opton prce. But the theoretcal proof n Clement et al (2002) has some lmtatons n that t s based on a sequental rather than ont lmt. Glasserman et al (2004a) consder the lmtatons n Clement et al (2002) and prove convergence of the LS estmator as the number of paths and the number of polynomals functons ncrease together. Further assumpton of martngales polynomals s requred here. Glasserman et al (2004b) (GY) mplement a weghted Monte Carlo Estmator (WME) to prce Amercan dervatves and show that ther estmator produces less dsperse estmates of the opton prce. However, no fnte-sample proof of convergence of the proposed estmator s provded n that study. Furthermore, proof of Theorem 1s based on a two perod framework. Applcatons of Monte Carlo estmators to prce fnancal dervatves generally requre usng varance reducton technques. One common feature of some of the studes cted above and other recent emprcal ones s that they all have consdered antthetc varates. As we shall see, partcularly when prcng Amercan style dervatves usng one method rather than another makes the dfference when determnng the early exercse value. In ths paper we analyse the fnte sample approxmaton of the LS (2001) and GY (2004b) by extendng emprcal studes such as Stentoft (2004). As shown n Glasserman and Yu (2004a) the choce of the bass functon used n the regresson s very mportant snce (unform) convergence of the opton prce to the true prce can only be guaranteed f polynomals span the true optmum. To address ths ssue, we consder dfferent bass functons and suggest possble optmal polynomals. We also dscuss ways to mplement varance reducton technques n ths context and study the contrbuton of these methods to varance reducton and bas. Fnally, our study s the frst emprcal study on the WME as n Glasserman et al (2004b) and t also extends that methodology to prce 4
optons on a maxmum of n assets and Bermudan-Asan optons. We show that, even when more dffcult payoffs are consdered, the WME estmator produces reasonably accurate prces. 2. The Least Squares Monte Carlo Methods We consder a probablty space ( Ω, Α, Ρ) and ts dscrete fltraton ( F ) = 0,..., n, wth n beng an nteger. Defne wth X,... 0, X 1 X n a d R valued Markov chan representng the state varable recordng all the relevant nformaton on the prce of a certan underlyng asset. Assume that V (x), d x R, s the value of an opton f exercsed at tme under the state x. Usng a dynamc programmng framework the value of the opton s gven by: V ( x) = supτ Γ E[ Θτ ( Xτ ) X = x] (1) wth V ( x) = Θ ( x) (2) n n V ( x) = max{ Θ ( x), E[( V+ 1( X + 1) X = x]} (3) To determne the opton value V 0 one has to () approxmate the condtonal expectatons n (3) n some ways, and () obtan a numercal (Monte Carlo) evaluaton of the latter. If the opton payoff s a square ntegrable functon, then V (.) wll be a functon spannng the Hlbert space and we can approxmate the condtonal expectatons n (3) by the orthogonal proecton on the space generated by a fnte number of bass functons k = 0,1,..., K, such that φ, = 1,..., n and k V ( x) = φ ( x) (5) n n V ( x) = max{ φ ( x), E[( V + 1( X + 1) X x]} (6) = 5
Usng a smple regresson approach: V K + 1 ( X + 1) c, k, k ( X ) + ε + 1 k = 0 φ (7) Therefore, we have transformed the dynamc programmng scheme n (6) nto a smple regresson requrng the estmaton of K + 1 coeffcents (7). At ths pont we need to evaluate the condtonal expectaton numercally. Ths can be done by smulatng paths of the Markov process X, wth = 1,..., M, and calculatng, at each stoppng tme τ, recursvely, the payoff Ρ * τ = φ( τ, X ). Remark. In Equaton 7 we have ncluded the error termε. As ponted out n Grasserman and Yu (2004b), ths s necessary for Equaton 7 to hold at each. Assumpton 1. If E( ε + 1 / X ) = 0 and E[ φ ( X ) φ ( X ) ] s non-sngular, then V V for all ' ^ = 0,1,..., n, where V^ s the estmated opton prce. Proof of convergence n LS (2001) apples to the smplest possble case of only one exercse tme and one state varable. Clement et al (2002) extend that proof to a mult perod framework under the assumpton that K s fxed. Ths would mply that the regresson used s correct, therefore no sample bas s consdered. GY (2004a) generalse the proof n Clement et al (2002) and show that the opton prce obtaned by regresson methods converges to the true prce as ( K, M ). However, martngales bass are consdered n ths case. All the theoretcal results mentoned above are very mportant, partcularly from a theoretcal pont of vew. However, for practcal applcatons of these methodologes we are more concerned wth ther performance n fnte sample. In Equaton (7) we have approxmated the condtonal expectaton by usng current bass functons (that s φ X ) ). However one would expect the opton prce at tme + 1to be more ( closely correlated wth the bass functon φ X ) rather than φ X ). GY (2004b) develop a + 1 ( + 1 method based on Monte Carlo smulatons where the condtonal expectaton s approxmated by ( 6
φ ( X 1) rather than φ X ). They show that ther Monte Carlo scheme has a regresson + 1 + representaton gven by: ( ^ K + 1 X + 1) = kφ+ 1, k ( X + 1) + k = 0 V ^ ( ϖ ε (8) + 1 However an mportant assumpton s necessary n ths case: Assumpton 2. (Martngale property of bass functon) E( φ + 1( X + 1) X ) = φ ( X ), for all. GY (2004b) call ths method regresson later, snce t nvolves usng functons φ X ). On the + 1 ( + 1 other hand, they call the LS (2001) method regresson now snce t uses functons φ X ). Although Theorem 1 n GY (2004b) provdes a ustfcaton for usng regresson later as opposed to regresson now, proof of that theorem s based on a sngle perod framework. Furthermore, GY (2004b) nether provde an emprcal applcaton of ther proposed estmator nor suggest ways of obtanng martngale bass. ( 3. A Smple Example To motvate ths study, n ths secton, we provde a smple example where we estmate early exercses values for Amercan put optons by crude Monte Carlo methods and usng varance reducton technques. Table 1 below shows the results. Monte Carlo EU-BS Early Exercse Value Bnomal Early Exercse Dfference 5.265 4.84 0.425 5.265 0.425 0.00% 6.234 5.96 0.274 6.244 0.284 1.00% 7.374 7.14 0.234 7.383 0.243 0.90% Antthetc Varates 5.261 4.84 0.421 5.265 0.425 0.40% 6.24 5.96 0.28 6.244 0.284 0.40% 7.384 7.14 0.244 7.383 0.243-0.10% Control Varates 5.264 4.84 0.424 5.265 0.425 0.10% 6.246 5.96 0.286 6.244 0.284-0.20% 7.387 7.14 0.247 7.383 0.243-0.40% Table 1. Monte Carlo refers to crude Monte Carlo method. EU-BS s the prce of an equvalent 7
European opton obtaned by Black & Scholes formula. Bnomal s the prce of the opton obtaned by bnomal methods. Early exercse refers to the estmates of the early exercse value. We consder three n-the-money put optons wth strke $45, ntal prce $40, maturty seven months and rsk free rate of nterest 4.88% and volatltes 20%, 30% and 40% respectvely. The last column shows the dfference, n percentages, between estmates of the early exercse value by crude Monte Carlo, Monte Carlo mplemented by varance reducton technques, and Bnomal methods. As we can see usng varance reducton technques reduces the bas by an order of 80% on average. Ths s lkely to have a substantal mpact on the estmate of the put opton prce. 4. Valung Amercan Put Optons Table 1 above shows that t s mportant to mplement Monte Carlo methods wth varance reducton technques snce, n ths way, we can reduce the bas n the estmaton of the early exercse value and acheve a more accurate prce of the opton. Therefore varance reducton technques reduce the probablty of generatng sub-optmal exercse decsons. In ths secton we frst apply the LS (2001) and GY (2004b) methods to prce Amercan style put optons and thereafter mplement the same methodologes by usng dfferent bass functons and dfferent varance reducton technques. As we ponted out above, we can only expect convergence of the estmated opton prce to the true prce f polynomals used n the regresson are optmal polynomals. We start wth a smple applcaton where we do not use varance reducton technques. Prces reported are averages of 50 trals. We report standard errors and root mean square errors as a measure of the bas n the estmaton of the condtonal expectaton n (6). As a benchmark, we consder the Bnomal method wth 20,000 tme steps. Table 2 below shows the emprcal results. To mplement the GY (2004b) estmator we specfy the followng martngale bass under geometrc Brownan moton and exponental polynomal: k 2 / 2 k ( 0 φ X ) = ( X ) exp ( kr + k( k 1) σ )( t t ) (9) On the other hand we could not fnd a vald martngale specfcaton for polynomals when Laguerre bass were used. Fnally, followng GY (2004a), Hermte polynomals ( H k ) defne martngales as: k k / 2 φ ( X ) = t H (10) k 8
Longstaff-Schwartz (2001), Glasserman-Yu (2004b) Methods Exponental Laguerre BIN. 2 3 4 2 3 4 GY 0.2/0.0833 4.9968 4.997 4.997 0 0 0 5 SE [0.00122] [0.00114] [0.00095] 0 0 0 RMSE [0.00331] [0.00299] [0.00328] 0 0 0 LS 02/0.0833 4.9968 4.9967 4.997 4.995 4.996 4.996 5 SE [0.00021] [0.00114] [0.0011] [0.000703] [0.000407] [0.000217] RMSE [0.00325] [0.00326] [0.00285] [0.00535] [0.00368] [0.00376] GY 0.2/0.3333 5.0927 5.0857 5.082 0 0 0 5.087 SE [0.00749] [0.0055] [0.00678] 0 0 0 RMSE [0.09267] [0.1023] [0.1009] 0 0 0 LS 0.2/0.3333 5.0922 5.0856 5.0881 5.077 5.087 5.0852 5.087 SE [0.04687] [0.00752] [0.0082] [0.00684] [0.00647] [0.00668] RMSE [0.0922] [0.1005] [0.10272] [0.00098] [0.00448] [0.00184] GY 0.2/0.5833 5.2523 5.2614 5.2651 0 0 0 5.265 SE [0.00968] [0.0066] [0.0124] 0 0 0 RMSE [0.2839] [0.2935] [0.29498] 0 0 0 LS 0.2/0.5833 5.2489 5.2635 5.2641 5.2528 5.265 5.265 5.265 SE [0.00566] [0.0114] [0.008195] [0.00677] [0.009518] [0.00838] RMSE [0.2865] [0.2926] [0.2871] [0.01221] [0.001313] [0.000188] GY 0.3/0.0833 5.0597 5.0591 5.0611 0 0 0 5.06 SE [0.00591] [0.005903] [0.00711] 0 0 0 RMSE [0.05975] [0.06372] [0.0685] 0 0 0 LS 0.3/0.0833 5.0595 5.0591 5.0581 5.054 5.061 5.061 5.06 SE [0.00633] [0.00541] [0.0056] [0.005489] [0.006804] [0.00679] RMSE [0.05951] 0.06372] [0.06371] [0.000651] [0.004389] [0.000094] GY 0.3/0.3333 5.6941 5.7042 5.7086 0 0 0 5.706 SE [0.01056] [0.0098] [0.0131] 0 0 0 RMSE [0.7172] [0.7300] [0.73087] 0 0 0 Table 2. Note that GY and LS are respectvely the methodologes proposed by Longstaff and Swartz (2001) and Glasserman and Yu (2004b). SEs are standard errors and RMSEs are root mean square errors. Exponental and Laguerre are the polynomals used n ths applcaton. 2-4 refer to the number of bass used. BIN s the prce of the opton gven by a Bnomal method. The zeros refer to cases when we were not able to obtan martngales bass for a specfc polynomal and therefore we could not mplement the GY(2004b) method. The frst column of Table 2 shows the methodologes used (.e. Glasserman and Yu, 2004b and Longstaff and Schwartz, 2001). In the second column we report the volatltes used to prce the opton and the tme to expry. 1 The strke of the opton s assumed to be $45 and the ntal stock prce $40. Therefore we only consder n the money optons. The rsk free rate of nterest s assumed to be 4.88% per year. Ffty tme steps are consdered n combnaton wth 100,000 Monte Carlo replcatons. We consder two dfferent polynomal bass, namely exponental and Laguerre. The 9
numbers of bass used are 2, 3 and 4 bases. Followng Brode and Kaya (2004), the RMSE s defned as 2 ( bas 2 + varance ) 1/ 2. Results n Table 2 favour Laguerre polynomals n qute few cases. Standard errors are n general small. The RMSE confrms what has been found n other studes, that s, the convergence mpled by these estmators s not unform. In fact, by ncreasng the number of bass one does not necessarly reduces the bas. Table 2 contnued LS 0.3/0.3333 5.6941 5.6991 5.7034 5.689 5.699 5.706 5.706 SE [0.00918] [0.0115] [0.01309] [0.01321] [0.01385] [0.00123] RMSE [0.7185] [0.7279] [0.73380] [0.01735] [0.006130] [0.00467] GY 0.3/0.5833 6.2232 6.2379 6.2455 0 0 0 6.244 SE [0.0143] [0.01952] [0.01487] 0 0 0 RMSE [1.268] [1.2838] [1.2875] 0 0 0 LS 0.3/0.5833 6.2221 6.2314 6.2439 6.227 6.2427 6.234 6.244 SE [0.0068] [0.01547] [0.01326] [0.01182] [0.0133] [0.01428] RMSE [1.2741] [1.2866] [1.2818] [0.01678] [0.00134] [0.00592] GY 0.4/0.0833 5.2775 5.2864 5.2881 0 0 0 5.286 SE [0.01027] [0.00855] [0.00763] 0 0 0 RMSE [0.28247] [0.29144] [0.2952] 0 0 0 LS 0.4/0.0833 5.2758 5.2859 5.2832 5.2749 5.2889 5.2908 5.286 SE [0.00622] [0.00874] [0.01032] [0.00089] [0.00936] [0.01071] RMSE [0.28319] [0.29066] [0.2958] [0.0111] [0.00287] [0.00479] GY 0.4/0.3333 6.4911 6.5097 6.5131 0 0 0 6.51 SE [0.01522] [0.01657] [0.0123] 0 0 0 RMSE [1.521] [1.537] [1.5386] 0 0 0 LS 0.4/0.3333 6.4988 6.5006 6.5121 6.4954 6.511 6.514 6.51 SE [0.00732] [0.0141] [0.01898] [0.01522] [0.01479] [0.01719] RMSE [1.522] [1.532] [1.5412] [0.01459] [0.00104] [0.00370] GY 0.4/0.5833 7.3631 7.3781 7.382 0 0 0 7.383 SE [0.01906] [0.02537] [0.02382] 0 0 0 RMSE [1.4086] [1.4287] [1.4321] 0 0 0 LS 0.4/0.5833 7.3701 7.3760 7.3824 7.3621 7.376 7.374 7.383 SE [0.007797] [0.02166] [0.01782] [0.02478] [0.001341] [0.01291] RMSE [1.4107] [1.421] [1.4301] [0.02094] [0.5611] [0.00949] Followng other studes such as LS (2001) and Stentoft (2004), n Table 3 below we have mplemented these methodologes usng standard antthetc varates. We prce the same opton (.e. wth the same parameters) as the one consdered n Table 2. Although, as we mentoned above, antthetc varates have already been consdered n other emprcal studes usng the LS (2001) method, they have never been used to mplement the estmator proposed n GY (2004b). Therefore as far as we know the present study s the frst emprcal study to mplement the GY (2004b) estmator to prce fnancal dervatves. 1 For example, 0.2/0.0833 should be read as 20% volatlty and 1 month to expry. 2 Refer to Brode and Kaya (2004) for further detals. 10
Longstaff-Schwartz (2001), Glasserman and Yu (2004b) Methods Exponental Laguerre BIN. 2 3 4 2 3 4 GY 0.2/0.0833 4.996 4.997 4.997 0 0 0 5 SE [0.000173] [0.000259] [0.0002537] 0 0 0 RMSE [0.003621] [0.003447] [0.003454] 0 0 0 LS 02/0.0833 4.9966 4.996 4.997 4.994 4.996 4.996 5 SE [0.000351] [0.0002892] [0.000279] [0.000658] [0.00039] [0.000266] RMSE [0.003424] [0.003559] [0.003459] [0.00591] [0.00367] [0.003565] GY 0.2/0.3333 5.079 5.085 5.084 0 0 0 5.087 SE [0.006513] [0.00667] [0.005471] 0 0 0 RMSE [0.008158] [0.00207] [0.00304] 0 0 0 LS 0.2/0.3333 5.079 5.086 5.0865 5.082 5.084 5.084 5.087 SE [0.006342] [0.005806] [0.00441] [0.00650] [0.00448] [0.00529] RMSE [0.008438] [0.001535] [0.000493] [0.005289] [0.00297] [0.00281] GY 0.2/0.5833 5.251 5.261 5.262 0 0 0 5.265 SE [0.006765] [0.008092] [0.00567] 0 0 0 RMSE [0.01367] [0.003912] [0.00305] 0 0 0 LS 0.2/0.5833 5.254 5.259 5.2634 5.253 5.262 5.261 5.265 SE [0.005042] [0.006498] [0.005579] [0.00839] [0.00571] [0.00604] RMSE [0.01113] [0.00608] [0.00158] [0.01271] [0.00296] [0.004253] GY 0.3/0.0833 5.054 5.059 5.061 0 0 0 5.06 SE [0.006252] [0.004472] [0.00404] 0 0 0 RMSE [0.00605] [0.001031] [0.000162] 0 0 0 Table 3: Antthetc varates. GY and LS are respectvely the methodologes proposed by Longstaff and Schwartz (2001) and Glasserman and Yu (2004b). SEs are standard errors and RMSEs are the root mean square errors. Exponental and Laguerre are the polynomals used n ths applcaton. 2-4 refer to the number of bass functons used. BIN s the prce of the opton gven by the Bnomal method. The zeros refer to cases when we were not able to obtan martngales bass for a specfc polynomal and therefore we could not mplement the GY (2004b) method. Both the methodologes produce an accurate prce of the opton. Very small standard errors sgnal that estmates are accurate and not dsperse. In general estmates of the opton prce gven by the LS (2001) method seem to be less dsperse than others. Ths result may not fully support Theorem 1 n Glasserman and Yu (2004b). Our result may mply that, once a mult perod framework s consdered, Theorem 1 n GY (2004b) no longer holds 3. In fact, evdence of unform convergence s much stronger when the LS (2001) method, n conuncton wth Laguerre bass, s used than the GY (2004b) method. In fact, n ths case, n general, the bas decreases as we ncrease the number of bass 4. 3 Ths result mght be due to the specfc martngales bases used n ths study. We shall look at ths ssue n more detals n a separate study and suggest ways of desgnng martngales bass wth bounded varance. 4 Of course, we do not clam here that (unform) convergence of the estmated opton prce to the true prce s due to the varance reducton methodology employed. In fact, t may well be due to the polynomal chosen (.e. Laguerre) n ths emprcal example. 11
Table 3 contnued LS 0.3/0.0833 5.055 5.059 5.068 5.054 5.059 5.061 5.06 SE [0.00581] [0.002595] [0.00379] [0.00579] [0.00439] [0.00432] RMSE [0.00521] [0.00122] [0.000279] [0.21134] [0.000929] [0.000554] GY 0.3/0.3333 5.691 5.702 5.707 0 0 0 5.706 SE [0.01595] [0.007062] [0.00737] 0 0 0 RMSE [0.01595] [0.00425] [0.000681] 0 0 0 LS 0.3/0.3333 5.693 5.704 5.704 5.690 5.702 5.705 5.706 SE [0.007459] [0.006206] [0.00634] [0.00823] [0.00742] [0.00562] RMSE [0.01342] [0.001563] [0.001874] [0.4254] [0.004271] [0.001261] GY 0.3/0.5833 6.228 6.24 6.239 0 0 0 6.244 SE [0.00572] [0.009582] [0.00613] 0 0 0 RMSE [0.01583] [0.00506] [0.005132] 0 0 0 LS 0.3/0.5833 6.229 6.235 6.238 6.224 6.239 6.240 6.244 SE [0.00829] [0.00758] [0.00779] [0.00999] [0.00803] [0.00647] RMSE [0.01479] [0.00873] [0.006014] [0.02019] [0.00471] [0.003669] GY 0.4/0.0833 5.275 5.285 5.289 0 0 0 5.286 SE [0.008957] [0.005144] [0.00535] 0 0 0 RMSE [0.01085] [0.0006409] [0.002555] 0 0 0 LS 0.4/0.0833 5.274 5.284 5.286 5.274 5.284 5.287 5.286 SE [0.00565] [0.005249] [0.00546] [0.00499] [0.00643] [0.00629] RMSE [0.01239] [0.001781] [0.0001743] [0.012403] [0.00205] [0.001198] GY 0.4/0.3333 6.494 6.509 6.509 0 0 0 6.51 SE [0.008389] [0.009061] [0.00754] 0 0 0 RMSE [0.01627] [0.000939] [0.0002757] 0 0 0 LS 0.4/0.3333 6.496 6.507 6.508 6.4923 6.506 6.51 6.51 SE [0.007467] [0.005948] [0.00846] [0.00734] [0.000437] [0.00709] RMSE [0.01419] [0.003064] [0.001931] [0.017695] [0.004269] [0.000199] GY 0.4/0.5833 7.366 7.371 7.378 0 0 0 7.383 SE [0.00901] [0.000234] [0.00915] 0 0 0 RMSE [0.0167] [0.0025] [0.00469] 0 0 0 LS 0.4/0.5833 7.361 6.379 7.384 7.366 7.376 7.384 7.383 SE [0.009083] [0.000342] [0.00953] [0.00836] [0.000897] [0.00720] RMSE [0.02167] [0.00291] [0.000509] [0.01692] [0.006972] [0.000523] To measure the mpact of antthetc varates on the estmates of the opton prces n Table 3, we have calculated the varance reducton (VR) factor, as the rato of the estmate of naïve varance and the estmate of antthetc varate varance, for a reasonable sample of the optons n Table 3. We have consdered optons wth volatltes 20-40% and tme to expry 1 and 4 months. When exponental bass s used the VR factor ranges from 0.68 to 3.34 for GY (2004b) method and 0.96 to 6.62 for LS (2001) method. On the other hand when Laguerre bass s used the VR factor ranges 12
from 0.10 to 7.47 5. The bggest gan from usng antthetc varates methods seems to come from mplementng the LS (2001) method by usng Laguerre bass. 4.1 Regresson Methods and Moment Matchng One mportant ssue when prcng dervatves by smulaton s that we can confdently prce a dervatves securty f, n the frst place, we can correctly smulate the dynamcs of the underlyng asset. The methodology we present below accomplshes ths task. d We follow Boyle et al (1997) and consder the R valued Markov chan sequence X 0, X 1,... X n and assume that we know the expectaton E( X ) = exp( rt) X0. The sample mean process of the sequence above can be wrtten as: X = 1 M M = 1 X (11) In fnte sample we know that E ( X ) X. However we can adust the smulated paths such that the followng equalty holds for all : ~ X ( t) = X ( t) + E[ X ( t)] X ( t) (12) ~ X where ( t) s the new smulated path after the transformaton. Consequently, we have that E[ X ~ ( t)] = E( X ( t)] holds and the mean of the smulated sample path matches the populaton mean exactly. Apart from matchng the frst moment of the process, we can also match hgher order moments such as varance for example. In ths case one re-wrtes the process n (12) n the followng form: ~ σ X ( t) [ X ( t) X ( t)] X = + s X E[ X ( t)] (13) 5 We have consdered a sample of 30 optons. Results are not reported to save space. We have also dropped n some cases some extreme values (.e. very hgh or low VR factors) that mght have generated outlers. 13
where σ X and s X are, respectvely, the populaton and the sample varance. One mportant drawback of the process n (12) s that sample paths are correlated and therefore t s unlkely that the ntal and the smulated processes wll have the same dstrbuton. The correlaton also makes estmates of standard errors meanngless. To overcome these drawbacks, n the emprcal applcaton, we have mplemented the addtve process n (12) to the standard Brownan moton process B (t), n the followng way: ~ s B t B t B t B ( ) = [ ( ) ( )]/ (13) t To preserve ndependence between sample paths, we have rescaled the ncrements of the process B (t) as follows: ~ Z ( t k ) = = 1 t t 1 Z Z s where Z are standard normal varables, and s 2 n 1 2 = ( Z Z ). n 1 = 1 4.2 Emprcal Results In ths Secton we show an applcaton of moment matchng when prcng one of the optons consdered n Table 2 and Table 3. We consder a put opton wth seven months to expry, volatlty 40%, ntal stock prce $40. The rate of nterest s 4.88%p.a. We set the number of steps equal to 50 n all the experments we conduct. We compute standard errors and root mean squares errors for sample sze of 16, 70, 300, 1000 based on 2000 smulatons. Values are reported n log term. 14
0-0.5 GYA LSA LSMM GYMM -1 RMSE -1.5-2 -2.5 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 Sample Sze Fgure 1 Standard errors versus sample sze n prcng an Amercan Put opton wth strke $45 and ntal stock prce $40. In Fgure 1 we have compared standard errors versus sample sze for the GY (2004b) and LS (2001) methods mplemented wth antthetc varates (A) and moment matchng (MM). Antthetc varates outperform moment matchng n ths case. Interestngly standard errors for GY (2004b) and LS (2004) methods are narrower when moment matchng s used. In Fgure 2 we compare the root mean squares error for LS (2001) and GY (2004b) methods when mplemented wth the same varance reducton methods as n Fgure 1. 1 0.5 0 GYMM LSMM GYA LSA -0.5 RMSE -1-1.5-2 -2.5-3 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 Sample Sze Fgure 2 Root mean squares error versus sample sze n prcng an Amercan Put opton wth strke $45 and ntal stock prce $40. 15
It s nterestng that the root mean squares errors for the LS (2001) and GY (2004b) methods are almost ndstngushable when mplemented wth the same varance reducton method. The lowest root mean square error s obtaned when the Longstaff and Schwartz method s mplemented wth antthetc varates 6. In the next secton we shall present an alternatve approach based on control varates to mplement methodologes such as the LS (2001) and GY (2004b). 4.3 Regresson Methods and Control Varates The method of control varates s one of the most popular varance reducton technques and has many analoges wth moment matchng. Applcatons of ths method n fnance for prcng, (Rubnsten, et al, 1985), or model calbraton (Glasserman and Yu, 2003) are very common. In ths secton we mplement the methodologes presented n Secton 2 by usng control varates. Suppose that, gven a stoppng tme τ Γ( t, T ), and the state varable X, we want to estmate the prce of an opton that, as n (1), can be found by solvng the followng condtonal expectaton: V ( x) supτ E[ Θτ ( Xτ ) X x] (14) = Γ = for the set of all possble stoppng tmesτ. If we consder functons φ (x) and mpose that: k Assumpton 3. 2 For = 1,..., n 1 φk (x) s n L ( φ ( )) ; Pr( V ( x) = V ) = 0; Π denotes the orthogonal proecton 2 Ω X from L ( ) onto the vector space generated by φ ( x ), φ ( x ),..., φ ( )}, ^ { 1 2 k x It follows that, as ( K, M ), the sample estmator of the opton: ^ 1 = M =1 Π V Vτ = V (15) M 6 Note we do not clam that ths s an unversal result. In fact moment matchng methods can also be mplemented n other dfferent ways. For example one could use moments for the stock prce paths for the adustment n (13) nstead of ~ usng B. 16
The estmator n (15) therefore converges almost surely toward the prce of the opton gven by (14) 7. Defne the path estmator of the opton usng control varates as follows: ^ ^ Π Z = Π V + λ [ ΠΥ E ( Υ)] (16) where λ s a prevsble process n F wth E F ( λ ) < and compute the condtonal expectaton. Υ s a random varable for whch we can The sample estmator n (15) can be wrtten as: Π ~ V = 1 m m = 1 ( Zτ ) (17) ^ = Π V + λ [ ΠY E ( Y )] (18) lm λ E [( ΠY E ( Y ))] = 0 (19) Therefore the followng result follows E V ) = V ( ~ (20) From (16) t follows that Var ( λ )], partcularly we have Var( Z ) Var( ) f [ Z ~ ~ ^ V * Cov V, Y ] λ = (21) Var[ Y ] [ ^ Therefore effcency can be ganed by mnmsng λ n (16). To acheve ths goal, we can use a smple Least Squares approach, that, we already use to compute estmates of the condtonal 7 See Clement et al (2002) amongst others. 17
expectaton n (14). The estmaton of λ, n ths case, wll ntroduce some bas, however ths wll vansh as the number of replcatons ncreases. As ponted out n Boyle, Broade and Glasserman (1997), the estmator of λ need not be very precse to acheve a reducton of varance n the case of usng only one control. It becomes nstead mportant when multple controls are ntroduced. In the emprcal applcaton n ths paper we have fxed λ = 1 for all. 4.4 Emprcal Results In ths Secton we mplement the LS (2001) and GY (2004b) methodologes by usng control varates. To mplement the LS (2001) and GY (2004b) by control varates, we use the approach descrbed above. Emprcal results are reported n Table 4. Longstaff-Schwartz (2001), Glasserman and Yu (2004a) Methods Exponental Laguerre 2 3 4 2 3 4 BIN. GY 0.2/0.0833 4.999 4.996 4.997 0 0 0 5 SE 0.00689 0.00771 0.0077 0 0 0 RMSE 0.00083 0.00395 0.003052 0 0 0 LS 02/0.0833 4.996 4.995 4.995 4.996 4.9948 4.999 5 SE 0.00594 0.00871 0.00646 0.000519 0.0069 0.00682 RMSE 0.00409 0.0048 0.005019 0.000373 0.000418 0.000101 GY 0.2/0.3333 5.077 5.086 5.085 0 0 0 5.087 SE 0.01185 0.009746 0.0107 0 0 0 RMSE 0.01015 0.000829 0.00194 0 0 0 LS 0.2/0.3333 5.078 5.083 5.091 5.08 5.085 5.084 5.087 SE 0.00999 0.00777 0.0071 0.00522 0.00835 0.01024 RMSE 0.00908 0.003678 0.00331 0.00659 0.002344 0.00274 GY 0.2/0.5833 5.251 5.262 5.26 0 0 0 5.265 SE 0.0094 0.01269 0.0146 0 0 0 RMSE 0.01514 0.002854 0.00531 0 0 0 LS 0.2/0.5833 5.253 5.261 5.263 5.251 5.266 5.264 5.265 SE 0.0132 0.00957 0.0096 0.00968 0.010665 0.01412 RMSE 0.01179 0.00373 0.00218 0.00149 0.0007153 0.000145 GY 0.3/0.0833 5.056 5.056 5.059 0 0 0 5.06 SE 0.00819 0.00809 0.00811 0 0 0 RMSE 0.003608 0.003745 0.0006428 0 0 0 LS 0.3/0.0833 5.057 5.057 5.06 5.055 5.0595 5.063 5.06 SE 0.01011 0.00706 0.009211 0.0059 0.00786 0.00887 RMSE 0.002734 0.00269 0.0004334 0.00504 0.000462 0.0003185 Table 4. Control varates. SEs are standard errors and RMSEs are the root mean square errors. Exponental and Laguerre are the polynomals used n ths applcaton. 2-4 refers to the number of bass functons used. BIN s the prce of the opton gven by a Bnomal method. The zeros refer to cases when we were not able to obtan martngales bases for a specfc polynomal and therefore we were not able to obtan martngales bass and could not mplement the GY (2004b) method. 18
We have mplemented the method of control varates by samplng the prce of a smlar European * opton at each possble stoppng tme and settng the value of λ equal to 1. Table 4 contnued GY 0.3/0.3333 5.691 5.705 5.702 0 0 0 5.706 SE 0.00701 0.00886 0.01252 0 0 0 RMSE 0.01536 0.000733 0.004363 0 0 0 LS 0.3/0.3333 5.688 5.707 5.706 5.692 5.702 5.704 5.706 SE 0.0117 0.00888 0.01082 0.00756 0.00952 0.00884 RMSE 0.018356 0.000463 0.0004135 0.0144 0.00422 0.00171 GY 0.3/0.5833 6.238 6.236 6.243 0 0 0 6.244 SE 0.00981 0.0146 0.00959 0 0 0 RMSE 0.01636 0.00768 0.000871 0 0 0 LS 0.3/0.5833 6.229 6.241 6.242 6.226 6.239 6.246 6.244 SE 0.01249 0.01081 0.01385 0.001369 0.0161 0.0156 RMSE 0.01538 0.002699 0.001753 0.01792 0.000532 0.000152 GY 0.4/0.0833 5.278 5.282 5.289 0 0 0 5.286 SE 0.00765 0.00704 0.008454 0 0 0 RMSE 0.007675 0.00418 0.002976 0 0 0 LS 0.4/0.0833 5.275 5.285 5.289 5.275 5.282 5.285 5.286 SE 0.01115 0.00978 0.00535 0.00875 0.00642 0.00705 RMSE 0.010527 0.001442 0.002635 0.0115 0.003856 0.00102 GY 04/0.3333 6.491 6.507 6.513 0 0 0 6.51 SE 0.01421 0.015 0.01444 0 0 0 RMSE 0.0201 0.00264 0.003587 0 0 0 LS 0.4/0.3333 6.491 6.504 6.51 6.496 6.513 6.51 6.51 SE 0.01076 0.0131 0.01243 0.0128 0.01025 0.0105 RMSE 0.01554 0.005928 0.002473 0.0144 0.0003199 0.000489 GY 0.4/0.5833 7.364 7.374 7.38 0 0 0 7.383 SE 0.01583 0.0111 0.0123 0 0 0 RMSE 0.01915 0.008729 0.003454 0 0 0 LS 0.4/0.5833 7.363 7.381 7.382 7.365 7.377 7.383 7.383 SE 0.01011 0.01657 0.0145 0.01286 0.013356 0.0162 RMSE 0.02024 0.002854 0.001329 0.01776 0.005812 0.0004751 Results n Table 4 show that control varates estmator produces a very accurate prce regardless the functon used n the regresson. Three bass are suffcent to acheve a low RMSE. Even n ths applcaton, t s not always the case that the GY (2004b) method produces the smallest standard errors. Ths may further support what we ponted out n Secton 3 8. The LS (2001) method mplemented wth Laguerre bases seems to produce the lowest RMSE and the strongest evdence of unform convergence. Fnally, the VR factor, n ths case, ranges between 0.45 and 5. 8 The assumpton of fnte varance on the bass functons (see Assumpton C1 n Glasserman and Yu, 2004b) may also be another reason why varaton of the estmates of the opton n ths case s not as stable as Theorem 1 would suggest.. 19
5 Valung Amercan Bermuda Asan Optons We consder the prevous methodologes when prcng more complex optons such as Amercan Asan optons and optons wrtten on a maxmum of n assets. It s wth ths type of optons that the LS (2001) and GY (2004b) methodologes become nterestng. As n Longstaff and Schwartz (2001), we consder prcng an Amercan Asan opton havng also an ntal lockout perod. In order to use the optons prces reported n Longstaff and Schwartz (2001) as benchmark, we consder an Amercan call opton that after an ntal lock out perod of three months can be exercsed at any tme up to maturty T. We assume T = 2 years. The average s the (contnuous) arthmetc average of the underlyng stock prce calculated over the lock out perod. We mplement the LS (2001) and GY (2004b) methodologes by usng control varates method. The choce of the control n ths case falls, obvously, on the prce of an equvalent geometrc opton. Therefore, we use the methodology descrbed above and choose the prce of a geometrc average opton as a control. As n Longstaff and Schwartz (2001) the strke prce s $100, the rsk free rate of nterest 6% and volatlty 20%. We use dfferent scenaros for the stock prces (S) and assume 200 steps for both stock prce and average. The results are reported n Table 5: Amercan Bermudan Asan Optons (LS 2001 Method) Expon. Lagu. Expon. Lagu. Expon. Lagu. S m = 30,000 M = 50,000 m = 75,000 80 0.9211 0.9218 0.937 0.945 0.9422 0.950 90 3.080 3.106 3.210 3.312 3.320 3.312 100 7.492 7.522 7.679 7.845 7.843 7.873 110 13.23 13.89 14.188 14.234 14.355 14.501 120 20.09 21.2 22.081 22.111 22.189 22.311 Table 5. S s the stock prce, m the number of smulatons, whle Expon. and Lagu. are respectvely exponental and Laguerre bass functons. As n Longstaff and Schwartz (2001), we use the frst eght Laguerre bass 9 and 50,000 replcatons. In our applcaton, we have also used exponental bass. Usng fnte dfference methods to prce these optons LS (2001) report opton prces equal to $0.949 ($80), $3.267 ($90), $7.889 ($100), $14.538 ($110) and $22.423 ($120) 10. In general, our results support those reported n Tables 3 of Longstaff and Schwartz (2001). That s the LS (2001) method produces a very accurate prce of the opton. If we calculate the early exercse value n ths case and compare t wth what reported n LS (2001) for the same optons but usng antthetc varates, we have that, for Laguerre bases and, 9 That s, frst two Laguerre bases on the stock prce and average plus ther cross products ncludng an ntercept. 10 Number n the brackets are ntal stock prces and the ntal average value for the stock prce s assumed to be 90. 20
m = 75,000, dfferences n the early exercse values n LS (2001) ranges between 0.007 and 0.050, whle n the present study the range s between 0.001 and 0.042. Ths s n lne wth what we ponted out at the begnnng. The choce and the correct mplementaton of varance reducton technques s mportant when prcng opton wth Amercan features snce t reduces the probablty of generatng sub-optmal strateges. In Table 6, we extend the Glasserman and Yu (2004b) method to prce Amercan Asan optons. We use Hermte bass ( φ KH ) to satsfy Assumpton 2 as follows, f kφ KH, wth The method seems to underestmate the opton prce. k / 2 f k = t. Amercan Bermudan Asan Optons (GY, 2004b Method) Hermte S m = 30,000 M = 50,000 M = 75,000 80 0.925 0.936 0.940 90 3.188 3.310 3.166 100 7.521 7.544 7.563 110 13.83 14.223 14.321 120 20.11 21.645 22.022 Table 6. S s the stock prce, m the number of smulatons. However, n general, more work s necessary to mplement ths method snce the choce of a martngale bass mght be fundamental. On the other hand t seems that ths fundamental problem has become more, to use Chrs Rogers`s words, an art than a scence. As ponted out above we shall address ths mportant ssue n a separate study. 6 Valung Amercan Basket Optons Fnally, we consder an addtonal hgh dmensonal problem. We consder an Amercan call opton wrtten on a maxmum of fve rsky assets payng a proportonal dvdend. We assume that each asset return s ndependent from the other. Once agan, we use the same parameter specfcatons as n Longstaff and Schwartz (2001) and Broade and Glaserman (1997) such that we can use prces reported n these papers as benchmark. We mplement the methodologes by usng antthetc varates. Broade and Glasserman (1997) use stochastc mesh to solve ths type of problems and report confdence nterval for the opton prces. We consder three dfferent optons wth ntal stock prces of 90,100, and 110 respectvely. The assets pay a 10% proportonal dvdend, the strke prce of the opton s 100, the rsk free rate of nterest s 10% and volatlty s 20%. Confdence ntervals 21
reported n Brode and Glasserman (1997) are [16.602, 16.710] when the ntal asset value s 90; [26.101, 26.211] wth ntal asset value of 100, and fnally [36.719, 36.842] when the ntal value s 110. The opton prces n Longstaff and Schwartz (2001) are respectvely, 16.657, 26.182, and 36.812 and they all fall wthn the Broade and Glasserman `s confdence nterval above. Amercan Basket Opton (LS 2001 Method) Expon. Hermte Expon. Hermte Expon. Hermte S m = 30,000 M = 50,000 m = 75,000 90 16.6895 16.677 16.6555 16.6171 16.6632 16.642 100 26.1758 26.1744 26.1708 26.1033 26.0804 26.12 110 36.7697 36.7642 36.7826 36.7482 36.8214 36.748 Table 7. S s the stock prce, m the number of smulatons, whle Expon. and Hermte are respectvely exponental and Hermte bass functons. Amercan Basket Optons (GY, 2004b Method) Hermte S m =30. m =50. m = 75. 90 16.5935 16.623 16.4759 100 26.0789 26.181 25.6802 110 36.286 36.71 36.1032 Table 8. S s the stock prce, m the number of smulatons. We note that regardless of the number of replcatons or bass functons used, we acheve, n all cases, a prce that les wthn the above nterval. We have also extended the GY (2004b) method to prce basket optons (see Table 8). As n the prevous applcaton, we have used Hermte polynomals to satsfy Assumpton 1 n GY (2004b). We note that opton prces estmates fall wthn the Broade and Glasserman `s confdence nterval when 50,000 paths are consdered. Therefore the martngale bass used n ths case seems to be approprate. 22
8. Conclusons From an academc and even a practtoner`s pont of vew, prcng Amercan optons stll remans an nterestng research area, partcularly when Monte Carlo methods s employed. Ths s manly due to the flexblty of ths method to accommodate hgh dmensonal features. Recently, Longstaff and Schwartz (2001) and Glasserman and Yu (2004b) propose two opton prcng estmators based on Monte Carlo smulatons. The general obectve of ths paper s to undertake an emprcal analyss to nvestgate the fnte sample approxmatons of these estmators. Apart from the above specfed obectve, we also () estmate the bas nduced by these estmators, () suggest an optmal polynomal functon, () extend these methodologes by mplementng varous varance reducton technques. Fnally, ths s the frst emprcal study on the estmator proposed n Glasserman and Yu (2004b) and t extends that method to solve hgh dmensonal problems. One general result n the lterature on prcng Amercan optons by Monte Carlo methods (regresson methods) s that Monte Carlo methods generate sub-optmal polces when used to estmate early exercses values and consequently they generate estmated prces that are below the true prce (see for example LS, 2001, and Clement et al, 2002, for a dscusson). Rogers (2002) formulate the problem n Equaton (3) as the dual and show that one can use a martngale approach to reduce the probablty of choosng sub-optmal polces when determnng the early exercse value. However, ths approach requres desgnng an optmal martngale and there s no clear cut rule yet on how to do that. In ths study we pont out that varance reducton technques, f correctly mplemented, can help us to reduce the probablty of generatng sub-optmal polces. Overall, we fnd that opton prces estmates by LS (2001) and GY (2004b) methodologes are accurate regardless the type of opton consdered. A large part of the sample bas can be elmnated wth an acceptable number of replcatons (.e. 100,000). However, n general, the LS (2001) estmator performs the best. Wth ths estmator we found Laguerre polynomals and control varates to out-perform the others 11. Therefore, n practcal applcatons, we recommend usng Laguerre polynomals. In general, a number of bass equal to three, 100,000 replcaton and control varate seem to be the rght combnaton to acheve a substantal level of accuracy. 11 We have also consdered Hermte polynomals but the results were not satsfactory and therefore were not reported n ths study. However, results are avalable upon request. 23
References Boyle, P., P., M., Broade and P., Glasserman, 1997, Monte Carlo Methods for Securty Prcng, Journal of Economc, Dynamcs and Control, 21, 1267-1321. Broade, M. and P., Glasserman, 1997, A stochastc Mesh Method for Prcng Hgh Dmensonal Amercan Optons, Workng Papers, Columba Unversty. Broade, M., and O., Kaya, 2004, Exact Smulaton of Stochastc Volatlty and other Affne Jump Dffuson Processes, Columba Unversty, Graduate School of Busness. Clement, E., D., Lamberton and P., Protter, 2002, An Analyss of a Least Squares Regresson Method for Amercan Opton Prcng, Fnance and Stochastc, 6, 449-471. Cox, J., C., S.A.Ross and M.,Rubnsten, 1979, Opton Prcng:A Smplfed Approach, Journal of Fnancal economcs, 7, 229-263. Egloff, D., 2003, Monte Carlo Algorthms for Optmal Stoppng and Statstcal Learnng, Workng Paper, Zurch Kantonalbank, Zurch Swtzerland. Glasserman, P., and B., Yu, 2004a, Number of Paths Versus Number of Bass Functons n Amercan Opton Prcng, The Annals of Appled Probablty, Vol.14. Glasserman, P., and B., Yu, 2004b, Smulaton for Amercan Optons:Regresson Now or Regresson Later?, n H., Nederreter edtor, Monte Carlo and Quas Monte Carlo Methods, 2002. Longstaff, F., A., and E., S., Schwartz, 2001, Valung Amercan Optons by Smulaton: A Smple Least-Squares Approach, The Revew of Fnancal Studes, Vol. 14, No.1. Rogers, L., C., 2002, Monte Carlo Valuaton of Amercan Optons, Mathematcal Fnance, 12, 271-286. Rubnsten, R., Y., and R., Marcus, 1985, Effcency of Multvarate Control Varates n Monte Carlo Smulatons, Operatons Research, 33. Stentoft, L., 2004, Assessng the Least Squares Monte Carlo Approach to Amercan Opton Valuaton, Revew of Dervatves research, 7(3). 24