Trading by estimating the forward distribution using quantization and volatility information

Transcription

1 Tradng by estmatng the forward dstrbuton usng quantzaton and volatlty nformaton Attla Ceffer, Janos Levendovszky Abstract In ths paper, novel algorthms are developed for electronc tradng on fnancal tme seres by usng quantzaton and volatlty nformaton for achevng Hgh Frequency Tradng (HFT). The proposed methods are estmaton based and tradng actons are carred out after estmatng the Forward Condtonal Probablty Dstrbuton (FCPD) on the quantzed return values. Ths s motvated by the fact, that the forward dstrbuton can gve a more relable predcton on the future values of the correspondng tme seres than smply usng mean-square-error predcton. From the past samples t s easy to learn the Condtonal Expected Value (CEV) f the possble past return values are small, whch needs quantzaton. From the CEV one needs to obtan the FCPD, whch can be acheved by ntroducng a specal encodng scheme on the observed prces. For estmatng FCPD, a FeedForward Neural Network (FFNN) wll be used, whch can provde a good estmaton f the return levels are quantzed properly and a specal encodng scheme s appled. Based on ths estmaton, a tradng sgnal s launched f the probablty of prce change becomes sgnfcant. Ths s measured by a quadratc crteron. Due to the encodng scheme and quantzaton, the complexty of learnng and estmaton have been reduced, whch paves the way towards HFT. The effcency of tradng s further ncreased by usng volatlty nformaton also estmated from the FCPD. The performance analyss of the new method has proven to be proftable accordng to the dfferent performance measure (acheved proft, average proft, maxmum drawdown etc.) on all hstorcal tme seres taken from FOREX. However, postve gan has usually been materalzed on md-prces. In order to beat the secondary effects, such as bd-ask spread and transacton costs, we needed to further optmze the method by optmzng meta-parameters wth Smulated Annealng. Ths postve effect was demonstrated n the case of EUR/USD exchange rates. Keywords: algorthmc tradng, neural networks, condtonal probablty dstrbuton, quantzaton JEL Classfcaton: C45 1. Introducton The selecton of portfolos whch are optmal n terms of rsk-adjusted returns has been an ntensve area of research n the recent decades [1]. Furthermore, the man focus of portfolo optmzaton tends to move towards the applcaton of Hgh Frequency Tradng (HFT) when a huge amount of fnancal data s taken nto account wthn a very short tme nterval and tradng wth the optmzed portfolo s also to be performed at hgh frequency wthn these ntervals []. HFT presents a challenge to both algorthmc and archtectural development, because of the need for developng algorthms runnng fast on specfc archtectures (e.g. GPGPU, FPGA chpsets) where speed s the most mportant attrbute. On the other hand, proftable portfolo optmzaton and tradng needs the evaluaton of rather complex goal functons wth dfferent constrants whch sometmes cast the problem n the NP hard doman [3] [4] [5]. As a result, the computatonal paradgms emergng from the feld of neural computng, whch support fast parallel mplementaton, are often used n the feld algorthmc tradng [6] [7] [8].

2 In ths paper the, tradng s done based on the estmated Forward Dstrbuton (FD) based on the artcle [9]. Snce FD takes ts values on the possble asset prces (or returns), the number of probabltes to be estmated explodes exponentally wth respect to the to the length of the memory. As a result, for the sake of accurate estmatons, we need a very large tranng set, whch prevents HFT due to the low speed of learnng. In order to speed up data collecton and learnng (usng a small number of samples), we need to quantze the asset prces. Instead of quantzng the prce tself, we quantze the change of the prces (returns), whch vares n a smaller nterval. In the paper, we use the Lloyd-ax algorthm for quantzaton to attan a good tradng performance. Snce the FCPD are calculated only on the quantzaton levels, the performance of the method hghly depends on the qualty of these levels. In ths way, the error n the estmaton of FCPD can be greatly reduced by usng a proper quantzaton algorthm. In order to beat the bd-ask spread, we have ntroduced further adjustments n the algorthm: tradng only n hgh volatlty perods; optmzng the meta-parameters by Smulated Annealng. Wth these adjustments, we could secure postve proft n the presence of bd-ask spread on EUR/USD currency exchange rates. The materal summarzed above s organzed as follows: n Secton, the theoretcal background of tradng by FFNN s outlned; n Secton 3, encodng schemes are ntroduced to obtan FCPD; n Secton 4, we apply optmal quantzaton for better FCPD estmaton; n Secton 5, we present the further adjustments n the tradng algorthm; n Secton 6, the computatonal model of the tradng algorthm s mapped out; n Secton 7, we valdate the methodology on generated and real hstorcal fnancal tme seres.. Theoretcal background tradng wth FFNN Let us assume that there s a vector valued random asset prce process (e.g. the values of s ( t) s ( t),..., s ( t). The return seres of s(t) s currency cross rates), whch s denoted by defned as s () t rt ( ) 1. s ( t1) 1 A portfolo s expressed by a portfolo vector ( t) w ( t),..., w ( t) w yelds a lnear 1 combnaton of asset returns (beng the portfolo return) as a tme seres N T xt : wr ( t) wr ( t). The possble returns are taken from a dscrete set 1 r ( t), x( t) Q q,..., q. Let 1 P P x( t 1) q x( t) q,..., x t L 1 q, 1,..., j denote the condtonal probabltes havng L past observatons at hand. Wth these condtonal probabltes, one can devse a Bayesan tradng strategy, as follows:

3 Or more precsely, f P x ( t 1) 0 then buy at t and sell at t 1; f P x ( t 1) 0 then sell at t and buy back at t 1. f P P then buy at t and sell at t 1; : : f P P then sell at t and buy at t 1. : : (.1) One can also see that the larger the value P P the more relable decson we can : : make on tradng. As a result, the rsk of the tradng can be fne-tuned by choosng a proper 0 for whch tradng can only take place f P P. : : In order to perform tradng as detaled above, one needs the estmaton of the condtonal probabltes P P x( t 1) q x( t) q,..., x t L 1 q, 1,...,. The estmaton of j P, 1,..., can be gven by a FFNN, based on observng the hstorcal part of the tme seres of a gven portfolo x(t). Based on these observatons, one can construct a tranng set contanng some samples followed by the observed forward return x(k+1) gven as follows: ( K ) x k, x( k 1), k 1,..., K where x k x( k),..., x( k L 1). Let us then construct an FFNN based predctor x t Net Θx, where x ( ),..., ( 1) we obtan ( 1), t k 1 t x t x t L. After learnng K ( K ) 1 Θ : mn ( 1), opt x k Net Θ x k (.) Θ K and the FFNN wll provde the optmal non-lnear predcton Net, t E x( t 1) t Θ x x, because 1 lm ( 1), ( 1), K K K x k Net Θ x k E x t Net Θ x t and k1 t t t mn E x( t 1) Net Θ, x Net Θ, x E x( t 1) x (for further detals see [10] [11] Θ [1]). 3. Codng scheme to obtan FCPD To perform the tradng algorthm elaborated by formulas (.1), one needs the condtonal probabltes. In order to obtan them, let us encode the possble values of the return of the portfolo nto an orthonormal vector set:

4 ( l) ( l) 1 f l ql r : r l 0 otherwse and rewrte the tranng set accordng to the encodng mechansm: ( K ) () l r x, where xk,, 1,..., k 1 k k K Then by mnmzng the error functon 1 K K r r k1 f 1 q. r k1 Net Θ, xk E r Net Θ, x, k1 ( one wll obtan Net K ) opt, E Θ x r x, where due to the encodng, component l of the condtonal expected value wll yeld the correspondng condtonal probablty as r x ( ) ( ) ( ) ( l) r x r x r x ( 1) x E r p p p p x t q P l l l l l 1 1 Havng the condtonal dstrbuton at hand, one can then mplement the tradng strategy dscussed above, n the form of f P P then buy at t and sell at t 1; : : f P P then sell at t and buy at t 1. : : Unfortunately, the method outlned above, requres hgh complexty neural network as the y Net Θx, s dm( y ) whch s the number of possble dmenson of the output returns. The archtecture s shown on Fgure 1. l (3.1) Fgure 1: The neural archtecture to estmate FCPD

5 Hgh complexty FFNN contans a large number of free parameters whch, n turn, requres a large learnng set to tran. Ths wll prevent fast executon of the strategy and, as a result, hnders HFT. Thus, the present effort n ths secton s focused on decreasng the number of outputs, whch wll also reduce the number of free parameters to optmzed. In order to acheve ths, we quantze the tme seres. 4. Optmal quantzaton for mprovng the estmaton of FCPD The returns of a fnancal tme seres can be approxmated by Gaussan random varables. However, equdstant quantzaton s only optmal for samples followng unform dstrbuton. In order to overcome ths shortcomng, we have quantzed the expected value of the squared quantzaton error (.e. the squared dfference between orgnal and quantzed sgnals) can be reduced by applyng non-equdstant quantzaton, due to the fact that quantzng the components whch occur wth smaller probablty wth larger error than the components whch occur wth hgher probablty, the overall error can be made smaller [13] [14]. In ths way, one can obtan a more accurate estmaton of the FCPD, whch may yeld better tradng decsons. To determne the optmal quantzaton levels, we used the Lloyd-ax algorthm. The Lloyd-ax algorthm: 1. use an ntal set of representatve levels: q 1,,...,. assgn each sample xt () n tranng set 3. calculate new representatve levels: ( K ) to closest representatve ( C x K ) : Q( x),3,..., 1 q x 1,,..., C 4. repeat. and 3. untl no further dstorton reducton (or applyng a stoppng crteron). Our smulatons have proven that by runnng the Lloyd ax algorthm, the quantzaton error drops to 10 tmes lower than usng equdstant quantzaton due to mnmzng the objectve 1 C1 x q p( x) dx. 1 C functon x B 5. Further adjustments n the tradng algorthm The results have shown, that proftable tradng s feasble usng neural networks by predctng the forward condtonal dstrbuton on md prces. However, to acheve profts n the presence of the bd ask spread and transacton costs, further refnements are needed Tradng n hgh volatlty perods Based on the predcted standard devaton we can mprove the tradng effcency. Each tme the neural network gves an entry sgnal (ether long or short ), we calculate the standard devaton from the forward condtonal dstrbuton as t P q Pq 1 1. If the standard devaton reaches a gven threshold (hgh volatlty), we enter nto the trade, otherwse (low volatlty), we stay away from the market. q :

6 f P P and t > then buy at t and sell at t 1; : : f P P and t > then sell at t and buy at t 1. : : (5.1) 5.. Optmzng the meta-parameters We ntroduced three methods to mprove the performance: 1. applyng Lloyd-ax quantzaton algorthm to obtan accurate condtonal probablty estmaton (see Secton 4.);. applyng a flter whch let us enterng the trade only f the probablty of upsde (or downsde) movement reaches a gven threshold ε (see formula 3.1); 3. applyng a flter whch let us enterng the trade only f the standard devaton of the predcted condtonal forward dstrbuton reaches a gven threshold η (see Secton 5.1.). In order to get the best results we should not only test these methods separately, but the parameters of the model must be optmzed together leadng to meta-model optmzaton performed by Smulated Annealng [15]. We optmze the followng parameters:, the number of quantzaton levels; L, the memory of the process (number of nputs of FFNN); ε, threshold for probabltes; η, threshold for standard devaton; N, the number of neurons n the hdden layer;, the length of tranng data expressed n hours; T tran T retran, the retranng tme wndow expressed n hours. These parameters are optmzed by Adaptve Smulated Annealng algorthm, whch s avalable e.g. n ATLAB. 6. Computatonal approach We wrote a comprehensve optmzaton framework to optmze the metaparameters on a parallel manner. The model s shown on Fgure. Fgure : Optmzaton procedure

7 The FFNN model parameter dentfcaton and the evaluaton of predcton by calculatng the condtonal probabltes s made on multple portfolos, whch helps us select an optmal portfolo. Then, the tradng sgnal for takng the approprate tradng actons s made on the bass of the dentfed portfolo and the cost functon. Our computatonal framework s shown by Fgure and detaled as follows: T select a sngle asset portfolo vector quantze the tme seres; w w,..., 1 w N and the correspondng tme seres; ft a Feed Forward Neural Network to the tme seres of the selected portfolo by usng the Levenberg-arquardt learnng algorthm; evaluate the objectve functon by predctng the future condtonal probablty dstrbuton accordng to the dentfed model gven the portfolo; contnue the optmzaton process untl the optmal parameters are obtaned. The objectve functon we use s maxmzng the average proft per trade; form a tradng sgnal based on the prce behavour of the optmal portfolo to decde on whch tradng acton s to be launched; Fnally, one can carry out a performance analyss by testng and evaluatng varous numercal ndcators for the sake of comparng the proftablty of the dfferent methods. Snce Lloyd-ax quantzaton algorthm works on an teratve manner, t sgnfcantly ncreases the computng tme. However, we have shown, that usng ths optmzed quantzaton, better proft can be acheved on each tme seres we tested on. 6. Performance analyss An extensve back-testng framework has been created to handle tradng actons on varous nput data and provde numercal results for the sake of comparng dfferent methods on dfferent tme seres (ether FOREX or artfcally generated data). At frst, we nvestgate the estmaton performance of the proposed model on generated data. The results showed, that FFNN can successfully predct the forward dstrbuton, furthermore n some case t s more accurate, than the standard hstogram method (smply calculatng the relatve frequences). Table 11 n the Appendx shows the correspondng results. For a detaled comparatve analyss, the followng performance measures were calculated for each experments on the correspondng tme seres: Proft ganed - the money realzed by the agent; axmum drawdown - the maxmum loss from a peak to a trough of the balance; of trades the number of trades the agent made on the on the evaluaton perod; Wnnng rate - rato of the total number of wnnng trades to the number of all trades; trade duraton the average holdng perod of an asset or portfolo n seconds. proft per trade (n ponts, whch s the smallest possble prce change) - the money realzed by the agent dvded by the number of trades. In ths secton, we show the numercal results obtaned on the followng foregn exchange data sets: EUR/USD;

8 GBP/USD and NZD/USD tck data on the day of We only tested on a sngle asset tme seres, however the procedures can be used for multasset portfolos. In each case the length of memory of neural network was 3 and we used the last 30 mnutes of tck data observatons from the past to ft FFNNs, whle we retraned the network after 3 mnutes of tck data. The number of quantzaton levels was =5 n each smulaton. For the sake of comparson, we used several FFNNs, each wth dfferent number of neurons n the hdden layer. The results are shown on the followng tables Numercal results obtaned n the case of usng equdstant quantzaton For the sake of comparson, we ran the tradng algorthm n the case of usng equdstant quantzaton. Hdden layer sze Hdden layer sze Hdden layer sze axmum Drawdown of trades Wnnng rate trade duraton proft n ponts Proft % 0.49 % 50.0 % % 7.45 % % % 0.54 % % % 7.0 % % % 8.0 % % % 7.3 % % % 7.76 % % % 6.4 % % Table 1: Tradng on the EUR/USD axmum Drawdown of trades Wnnng rate trade duraton proft n ponts Proft % 1.11 % 0.00 % % 6.37 % % % 5.37 % % % 4.64 % % % 6.4 % % % 4.76 % % % 5.76 % % % 5.97 % % Table : Tradng on the GBP/USD axmum Drawdown of trades Wnnng rate trade duraton proft n ponts Proft % 1.5 % % % 3.03 % % % 3.9 % %

9 % 4.98 % % % 4.58 % 55 7 % %.90 % % % 3.3 % % % 3.40 % % Table 3: Tradng on the NZD/USD One must note, that the wnnng rato (on md prces) was very hgh (75 % or more), whch means the FFNN can predct the future prces. To calculate proftablty of the tradng algorthm n real crcumstances, one must takng nto account the spread. We measured the average spread of the 3 assets on the testng perod: EUR/USD 5.49 ponts; GBP/USD 9.55 ponts; NZD/USD 1.9 ponts. Unfortunately, the prce movement s not enough to cover the bd-ask spread (the average spread s hgher than the average proft per trade). It can be also seen that by ncreasng the neurons n the hdden layer, the average trade duraton s decreasng and the number of trades s ncreasng. 6.. Numercal results obtaned n the case of usng Lloyd-ax quantzaton Next, we appled Lloyd-ax quantzaton on the return seres. The results are shown below. Hdden layer sze Hdden layer sze axmum Drawdown of trades Wnnng rate trade duraton proft n ponts Proft % 0.00 % % %.67 % 50.0 % % 0.04 % % % 0.7 % % % 3.47 % % % 4.34 % % % 5.04 % % % 3.15 % % Table 4: Tradng on the EUR/USD axmum Drawdown of trades Wnnng rate trade duraton proft n ponts Proft % 0.00 % % % 0.14 % % % 9.07 % % % 3.55 % % % 4.31 % % % 7.83 % % % 5.05 % % % 7.16 % % Table 5: Tradng on the GBP/USD

10 Hdden layer sze axmum Drawdown of trades Wnnng rate trade duraton proft n ponts Proft % 0.00 % % % 1.7 % % %.18 % % % 1.75 % % % 1.96 % % % 1.60 % % % 3.08 % % % 3.0 % % 0.93 Table 6: Tradng on the NZD/USD One can see, that the proft generated by Lloyd-ax optmzer tradng algorthm on each tme seres s ncreased compared to the profts acheved by equdstant quantzaton. On the other hand, the maxmal drawdown s decreased, whch s also favourable. Unfortunately, the average proft per trade dd not ncreased, whch s essental for beatng the bd-ask spread and transacton costs. 7. Further optmzng the tradng performance The next numercal results are obtaned by mprovng the tradng wth the methods lsted n Secton 5. Frst, we examned the effect of the flter specfed n Secton 5.1. on EUR/USD return seres. We set η from 0.5E-6 to 4E-6 to dscover the best parameter settng. The results are shown below. nmal standard devaton - η (E-6) axmum Drawdown of trades Wnnng rate trade duraton proft n ponts Proft %.08 % % %.1 % % % 1.69 % % % 0.91 % % % 0.91 % % % 0.41 % % % 0. % % % 0. % % % 0. % % % 0.13 % % Table 7: Tradng on the EUR/USD One can see, that the predcted standard devaton has a hgh mpact on the tradng results. To ensure proftablty, one must beat the bd-ask spread and transacton cost e.g. the average profts on the trades must be greater, than the average spread. By settng the flter to.5e-6 on EUR/USD, we can secure almost 3 ponts per trade.

11 7.1. Optmzng the meta-parameters To acheve more proft, we optmzed the meta-parameters of the model. The optmzaton were performed on a nd generaton Intel Core 7 machne, usng 4 parallel threads at a tme. Durng hours, the optmzer run the strategy for approxmately 100 dfferent states. The followng tables show the 10 best results for the three FOREX tme seres. Case Proft ax drawdown proft n ponts of trades Ht rato % 1.33 % % 5.31 % 1.34 % % % 1.34 % % % 0. % % % 0.18 % % % 1.77 % % % 0.31 % % % 1.50 % % % 0.0 % % % 1.65 % % Table 8: Tradng on the EUR/USD Case Proft ax drawdown proft n ponts of trades Ht rato % 0.4 % % 0.84 % 0.18 % % % 0.33 % % % 0.36 % % % 0.05 % % 6.1 % 0.35 % % % 1.79 % % % 0.78 % % % 0.35 % % % 0.37 % % Table 9: Tradng on the GBP/USD Case Proft ax drawdown proft n ponts of trades Ht rato % 4.98 % % 1.64 % 0.16 % % % 1.96 % % % 1.76 % % % 1.10 % % %.36 % % % 1.83 % % % 0.76 % %

12 % 1.80 % % % 1.58 % % Table 10: Tradng on the NZD/USD Note that n each of these cases we could secure postve proft on the md prces. The largest ncrease of average proft was attaned on the EUR/USD rate, whch may, n ths case, wll pave the way towards proftable tradng on bd-ask prce seres also. In the Appendx, the parameters of the ten cases are gven. Whle, on the other two assets, the strategy does not have enough predcton power to yeld proftable tradng. Conclusons As the numercal results demonstrated, the new methods were able to yeld proft on the mdprces of the underlyng rates (EUR/USD, GBP/USD, NZD/USD). Ths proft was even achevng 1.85 % n the best case. However, when the bd-ask spread was also taken nto account n the numercal analyss, then postve proft tended to reman only n the case of EUR/USD. As a result, further adjustments needed to beat the bd-ask spread. These further adjustments ncluded: applyng optmal quantzaton by Lloyd-ax algorthm; tradng only n hgh volatlty perods; optmzng the meta-parameters by Smulated Annealng. Wth these adjustments, we could secure postve proft n the presence of bd-ask spread. In the case of EUR/USD, the proft was more than 5.3 % for the tested one-day perod. Nevertheless, these adjustments needed heurstc consderatons (such as tryng out emprcally the sze of memory and dentfyng hgh volatlty perods). There s also a slowdown of the executon of algorthm due to runnng Smulated Annealng.

13 References [1] K. Anagnostopoulos and G. amans, The mean-varance cardnalty constraned portfolo optmzaton problem: An expermental evaluaton of fve multobjectve evolutonary algorthms, Expert Systems wth Applcatons, pp , 011. [] E. P. Chan, Quanttatve Tradng, Wley, 008. [3] A. D'Aspremont, Identfyng small mean-revertng portfolos, Quanttatve Fnance, pp , 011. [4] I. R. Spos and J. Levendovszky, Optmzng sparse mean revertng portfolos, Algorthmc Fnance, pp , 013. [5] N. Fogaras and J. Levendovszky, Sparse, mean revertng portfolo selecton usng smulated annealng, Algorthmc Fnance, pp , 013. [6] I. Kaastra and. Boyd, Desgnng a Neural Network for Forecastng Fnancal and Economc Tme Seres, Neurocomputng, vol. 10, no. 3, pp , [7] E. W. Saad, D. V. Prokhorov and D. C. Wunsch, Comparatve study of stock trend predcton usng tme delay, recurrent and probablstc neural networks, IEEE Transactons on Neural Networks, vol. 9, no. 6, pp , [8] J. Levendovszky és F. Ka, Predcton based hgh frequency tradng on fnancal tme seres, Perodca Polytechnca, %1. kötet1, pp. 9-34, 01. [9] J. Levendovszky, I. Reguly, A. Olah and A. Ceffer, "Low complexty algorthmc tradng by Feedforward Neural Networks," 016. [10] K. Hornk,. Stnchcombe and H. Whte, ultlayer Feedforward Networks are Unversal Approxmators, Neural Networks, VOL., pp , [11] S. Haykn, Neural Network Theory: a Comprehensve Foundaton, Prentce Hall, [1] K. Funahash, On the Approxmate Realzaton of Contnuous appngs by Neural Networks, Neural Networks, VOL., pp , [13] S. Lloyd, Least squares quantzaton n PC, IEEE Transactons on Informaton Theory, vol. 8, no., pp , 198. [14] J. ax, Quantzng for mnmum dstorton, IRE Transactons on Informaton Theory, vol. 6, no. 1, pp. 7-1, [15] L. Ingber, Adaptve smulated annealng (ASA): Lessons learned, Control and Cybernetcs, %1. kötet5, %1. szám1, pp , 1996.

14 Appendx In the tables below, we gve the results n detals. Table 11 shows the performance of predctng the FCPD by FFNN and standard hstogram method. SE L=1 L= L=3 N= N= N= N= N= N= N= N= Hstogram Table 11: Valdatng the predcton model The next table shows the metaparameters of the ten cases of EUR/USD. Case ε L N η Ttran Tretran E E E E E E E E E E Table 1: The metaparameters of the ten cases on EUR/USD