Transaction-Cost-Conscious Pairs Trading via Approximate Dynamic Programming

Transcription

1 Transacion-Cos-Conscious Pairs Trading via Approximae Dynamic Programming Undergraduae Honors Thesis Deparmen of Compuer Science Sanford Universiy Xiang Yan Advisor: Benjamin Van Roy June 1, 2006

2 Absrac In his paper, we develop an algorihm ha opimizes logarihmic uiliy in pairs rading. We assume price processes for wo asses, wih ransacion cos linear wih respec o he rae of change in porfolio weighs. We hen solve he opimizaion problem via a linear programming approach o approximae dynamic programming. Our simulaion resuls shoha when asse price volailiy and ransacion cos are sufficienly high, our ADP sraegy offers significan benefis over he chosen baseline sraegy. Our baseline sraegy is an opimized version of a pairs rading heurisic sudied in he lieraure [3]. 1 Inroducion Pairs rading is a saisical arbirage echnique wih which Wall Sree raders garnered wild success in he 1980s. The pairs rading scheme is quie simple: idenify wo socks ha have hisorically moved ogeher. When heir price gap widens, shor he winner and buy he loser, in he hope ha he gap will evenually narroo he hisorical mean. When his even occurs, he arbirageur profis. In [3], Gaev e al. examined daily sock daa from 1962 o 1997 and idenified a few sock pairs wih minimum disance in hisorical normalized price space. They hen applied sraighforward self-financing pairs rading rules o hem over six-monh rading periods. Their resul was an average annualized excess reurn of abou 12 percen. In his paper, we assume an evoluion process of wo highly correlaed asse prices. Furhermore, we assume ha he ransacion cos during rading is linear wih respec o he rae of change in porfolio weighs. To maximize logarihmic uiliy of our wealh, we develop a sraegy by applying approximae dynamic programming, and in paricular, he linear programming approach [6]. We also develop a baseline sraegy by opimizing a heurisic proposed by [3]. In our simulaion, we compare he risk-free equivalen raes of reurn yielded by our ADP sraegy and he baseline sraegy. The ADP algorihm ha we develop exends he linear programming approach in ha i reas coninuous processes, governed by sochasic differenial equaions. Wih he excepion of [4] and [5], prior work on he linear programming approach has focused on discree sae spaces. We work wih coninuous models because hey faciliae compuaion of greedy policies and reamen of consrains in he linear program. In a coninuous model, even when he acion space is coninuous and high-dimensional, given a sae and an approximae value funcion, a greedy acion can be 1

3 compued by solving a racable convex program. We acknowledge ha his paper does no assess he applicabiliy of he assumed pairs rading model. Insead, our conribuion lies in he applicaion of approximae dynamic programming o an insance of sochasic opimizaion problems wih coninous ime and coninuous sae space. 2 Problem Formulaion Consider wo asses, asse 0 and asse 1, wih highly correlaed price processes S 0 and S 1. One migh consider he following sochasic differenial equaion ha models he mean-reversion propery of heir log-prices x 0 and x 1 : dx 0 = γ e δ( u) (dx 1 u dx 0 u) d + µ d + σ db 0 (1) u=0 and dx 1 = γ e δ( u) (dx 0 u dx 1 u) d + µ d + σ db 1, (2) u=0 where µ and σ are he drif and diffusion coefficiens, respecively, and B 0 and B 1 are Brownian moions. Noe ha he firs erm of each equaion characerizes he dependence of fuure log prices on he differences of pas log prices. To sudy he evoluion of log prices, we define ransformed sae variable y as, y = u=0 e δ( u) (dx 1 u dx 0 u). (3) Hence, dx i = ( 1) i γy d + µ d + σ db i, for i = 0, 1. (4) Noe ha given y, disribuion of x is known. So in our model, y compleely characerizes he marke condiion. Consider an invesor in he wo asses. A any given ime, his wealh has hree componens: he amoun invesed in asse 0, he amoun invesed in asse 1, and cash. We use η = (η 0 η 1 ) o denoe his porfolio weighs a ime, where η i denoes he fracion of invesed in asse i. 2

4 To inroduce he noion of rading sraegy, we define sae a ime as z = (, η, y ) S = R + R 2 R. Sae is defined his way o capure all he informaion needed o make an invesmen decision a ime. We now define rading sraegy as ψ : S R 2, which maps each sae o raes of change in porfolio weighs. Given his definiion, we someimes also refer o ψ as rading rae or policy. Fuhermore, we someimes wrie ψ as shorhand for ψ(z ), and in he appropriae conex, abuse noaion and wrie ψ as a vecor in R 2. Assuming ha ransacion cos is linear wih respec o he rae of change in porfolio weighs and ha ineres rae in he money marke is zero, he wealh process evolves as follows: d = η ds S c ψ 1 d, (5) where c 0 is he linear scaling facor of ransacion cos. Our objecive is o solve he following infinie-horizon, ime-discouned opimizaion problem: [ ] max E e α u( ) d, ψ =0 where uiliy funcion u( ) = log( ), which models he risk-averse psychology of he invesor. 3 Dynamic Programming Approach 3.1 Seup To apply ADP o our opimizaion problem, we would like o shoha he wealh process is a diffusion. To do so, we firs demonsrae ha he marke condiion process y is a diffusion. Take differenial of boh sides of (3) o obain dy = dx 1 dx 0 δy d. (6) Wrie ou (4) as, dx 0 = γy d + µ d + σ db 0 (7) 3

5 and dx 1 = γy d + µ d + σ db 1. (8) Subrac (7) from (8): dx 1 dx 0 = 2γy d + σ(db 1 db 0 ) = 2γy d + 1 σ db, (9) 1 where db = db0. From (6) and (9): db 1 dy = (2γ + δ)y d + 1 σ db 1 = µ y (y ) d + σ y db. (10) Hence, y is a diffusion wih drif and diffusion coefficiens µ y (y ) = (2γ + δ)y and σ y = ( σ σ), respecively. We now shoha he price process S is also a diffusion. From (4), we have dx = = dx0 dx 1 γy + µ d + σ db γy + µ = µ x (y ) d + σ x db, (11) where µ x (y ) = γy + µ γy + µ 4

6 and σ x = σ 0. 0 σ Since x i = log(s i ), for i = 0, 1, by Iô s Formula and (11), we can hen express S as he following diffusion: ds S = µ s (y ) d + σ s d, (12) where µ s (y ) = µ x (y ) diag(σ xσ x ) 1 and σ s = σ x = σ 0. 0 σ Based on (5) and (12), we have he wealh process as he following diffusion: d = η µ s (y ) d + η σ s db c ψ 1 d = µ w (z, ψ ) d + σ w (z ) db, where µ w (z, ψ ) = η µ s (y ) c ψ 1 and σ w (z ) = η σ s. Before applying dynamic programming, we derive he dynamics of he porfolio weighs process η. We use ξ R 2 o denoe he dollar amoun invesed in he wo asses. Tha is, ξ = η. (13) 5

7 I can be shown ha dξ = ξ ds S + ψ d. (14) Noe ha (14) illusraes wo facors ha influence invesmen allocaion in dollars, namely change in asse prices and change in porfolio weighs. By Iô s Formula, i follows ha = d( ξ(i) ) dη (i) = 1 dξ (i) = ψ (i) = ψ (i) ξ(i) w 2 d + η (i) ds (i) d + ξ(i) w 3 (d ) 2 1 w 2 S (i) ψ (i) d d d + η (i) ds (i) S (i) η (i) ds (i) S (i) η (i) ds (i) S (i) η (i) d η (i) d dξ (i) d + η (i) ( d + η (i) ( d ) 2 η (i) ) 2 η (i) ds (i) S (i) ds (i) S (i) d d. Noe ha ( dw ds (i) S (i) ) 2 = σ w (z )σ w (z ) d, and d = σ s (i ) σw (z ) d, where σ (i ) s is he i-h row of σ s. Hence, dη = µ η (z, ψ ) d + σ η (z ) db, (15) where µ (i) η (z, ψ ) = ψ (i) σ (i) η (z ) = η (i) ( + η (i) (σ (i ) µ (i) s s σ w (z )). ) (y ) µ w (z, ψ ) + σ w (z )σw (z ) σ s (i ) σw (z ), 6

8 3.2 Dynamic Programming In he previous subsecion, We have esablished as a diffusion whose drif and diffusion coefficiens are funcions of sae z and policy ψ. We now proceed o solve he opimizaion problem wih dynamic programming echniques. Recall ha we desire o [ ] max E z,ψ e α u( ) d. ψ =0 For each sraegy ψ, we define value funcion J ψ (z) = E z,ψ [ =0 ] e α u( ) d, where he subscrips of he expecaion operaor mean ha he iniial sae of he process is z and sraegy ψ is employed. Furher, we define he opimal value funcion, J (z) = sup J ψ (z). ψ For each sraegy ψ, we now define he coninuous-ime dynamic programming operaor H ψ : H ψ J(z) = D ψ J(z) αj(z) + u(w), z, where infiniesimal generaor D is defined as D ψ J(z) = J w wµ w + ( ) J µ y + y J 2 w 2 w2 σ w σw r ( ) J µ η ( 2 J y 2 σ yσy + 2 J w y σ yσ w w + 2 J w σ ησ w w + r ) r ( 2 J 2 σ ηση ( 2 ) J y σ ησy. ) Then, he opimal value funcion J is soluion o he Hamilon-Jacobi-Bellman Equaion: max ψ R 2(H ψj)(z) = 0, z. (16) 7

9 Correspondingly, we define he opimal rading sraegy ψ (z) : S R 2 as a funcion ha saisfies 3.3 Logarihmic Uiliy ψ (z) argmax H ψ J (z). (17) ψ R 2 Wih logarihmic uiliy, he opimal value funcion decomposes in an ineresing way: J (z) = log(w) α + V (η, y), for some funcion V. So we resric our ques for J wihin he following se of funcions, { J log = J J(w, η, y) = log(w) α } + V (η, y) for some V V. Here, V is a se of funcions saisfying some echnical regulariy condiions. We will no describe he echnical regulariy condiions here because hey are complicaed and no relevan o his work. Noe ha for all J J log, we have w J w = 1 α, w2 2 J w 2 = 1 α, 2 J w = 0, 2 J w y = 0. We can hen simplify he HJB quaion in (16) o 0 = max ψ R 2 { µ wα + µ y V y + µ η V 1 2α σ wσw ( ) ( ) r 2 V σ y 2 y σy r 2 V σ 2 η ση ( ) } +r 2 V y σ ησy αv. (18) Noe ha ψ as defined in (17) sill saisfies (18). Observing ha ψ appears only in µ w and µ η in (18), we simplify (17) o ψ arg max ψ R 2 {( ( V ) ) η 1 c ψ 1 + α ( ) V ψ}. (19) The funcion being maximized here is piecewise linear wih a kink a ψ = 0. Consequenly, 8

10 wih some parameer seings, he funcion is unbounded from above, resuling in no soluion o he opimizaion problem. In such cases, he opimal rading rae ψ involves jumps. Tha is, insananeous shifs in porfolio weighs. To avoid his effec, we inroduce a regulaion erm εψ ψ for some ε > 0 o he HJB equaion in (18) o arrive a max F ε ψ R 2 ψv = 0, (20) where operaor F ε ψ is defined as Fψ ε V = µ w α + V µ y r ( 2 V 2 σ ησ η y + V µ η 1 ) + r 2α σ wσw + 1 ( 2 2 r V ( 2 V y σ ησ y ) y 2 σ yσy ) αv εψ ψ. Accordingly, he opimal rading sraegy ψ is now approximaed by ψ ε, which is defined as ψε argmax Fψ ε V ε, ψ R 2 where V ε is soluion o (20). Tha is, ψ ε arg max ψ R 2 {( ( V ε ) ) η 1 c ψ 1 + α ( ) V ε ψ εψ ψ}. (21) 4 Approximae Dynamic Programming To obain V ε, we can solve he following opimizaion problem wih a coninuum of variables and a coninuum of consrains: min V (y, η) ρ(d(y, η)) subjec o max F ε ψ R 2 ψ V 0 V V, (22) where he funcion V is he variable being opimized and ρ is a given posiive measure. As his opimizaion problem is unsolvable analyically, we consider he approximae linear 9

11 programming approach proposed by [6, 1]. This approach fis V ε wih a weighed combinaion of seleced basis funcions φ 1,, φ K by solving a variaion of (22). Specifically, consider he opimizaion problem: min subjec o (y,η) (Φr)(y, η) ρ(d(y, η)) max F ε ψ R 2 ψ (Φr) 0. (23) Here, Φ(y, η) = (Φ 1 (y, η),, Φ K (y, η)) and Φr = Φ(y, η)r, where r R K is he variable being opimized. Noe ha we drop he consrain V V because normally, seleced basis funcions ψ 1,, ψ K V. We now subsiue non-linear consrains in his problem wih heir linear equivalences o aain he following linear program: min (y,η) (Φr)(y, η) ρ(d(y, η)) subjec o F ε ψ (Φr) 0, ψ R2. (24) We hope o use he soluion o his problem, r ADP, o derive a policy ψ ADP, which will, in urn, approximae he opimal rading policy. Tha is, ψ ψ ε ψ ADP (25) and ψ ADP argmax ψ R 2 {( ( Φr ADP ) ) η 1 c ψ 1 + α ( ) Φr ADP ψ εψ ψ}. Unforunaely, our new linear program is sill inracable as is objecive enails an inegral and i has a coninuum of consrains. To address hese difficulies, we approximae he soluion o he linear program by applying a sae sampling echnique proposed by [2]. Specifically, we sample idenical independenly disribued saes (y (1), η (1) ),, (y (Q), η (Q) ) and pose he following linear program: min Q (Φr)(y (i), η (i) ) i=1 subjec o (F ε ψ Φr)(y(i), η (i) ) 0, i = 1,, Q, ψ. 10

12 Noing ha he newly formulaed problem sill has infiniely many consrains due o a coninuum of ψ, we perform he following ieraive heurisic o achieve an approximaion o r ADP : 1. Pick an iniial guess r Based on he approximaion Φr k, compue he greedy acion ψ k (i) for each (y (i), η (i) ). Tha is, ψ k (i) argmax ψ R 2 {( ( Φrk ) ) η 1 c ψ 1 + α ( ) Φrk ψ εψ ψ}. 3. Solve Linear Program: Q min (Φr k )(y (i), η (i) ) i=1 subjec o (F ψ j (i)φr k )(y (i), η (i) ) 0, (26) i = 1,, Q, j = 0,, k. o obain r k Reurn o Sep 2 unil r k appears o have nearly converged. Upon compleion of his algorihm, we achieve r ADP, an approximaion o r ADP. From r ADP, we can derive is corresponding opimal policy ψ ADP : ψ ADP { ( ( argmax Φ r ADP + ψ R ( 2 Φ r ADP ) ψ εψ ψ}. ) ) η 1 α c ψ 1 (27) ψ ADP approximaes ψ ADP, which in urn approximaes ψ ε and ψ, as shown in (25). 5 Experimens 5.1 Baseline Sraegy We considered wo sraegies o serve as he baseline, he performance of which will be compared wih ha of our ADP sraegy. The firs candidae is a smoohed version of he opimal sraegy 11

13 in he absence of ransacion cos. I urns ou ha in he case of log uiliy, his sraegy is myopic. The second candidae is a refined version of a heurisic proposed by [3]. This candidae sraegy akes long-shor posiions upon observing significan gaps in asse prices, and liquidaes a he nex crossing of he prices. We evenually seleced he second candidae because he firs one performed poorly in simulaion. Below, we discuss our chosen baseline and pospone a discussion of he unseleced candidae ill he appendix. To moivae our baseline sraegy, a descripion of he heurisic used by Gaev e al. is in order [3]. The auhors prepared a pool of socks wih eigheen monhs of price daa available. For each pair of socks, hey compued he sum of squared differences in normalized prices over he firs welve monhs. They hen seleced a few pairs wih he smalles such sums. To simulae rading on hese pairs by using he price daa in he remaining six monhs, he auhors applied he following sraegy: when price difference of a sock pair exceeds wice he hisorical sandard deviaion, hey would long he winner and shor he loser wih an equal amoun of money in magniude. They would hen unwind heir posiions when he wo prices converge or he six-monh rading period erminaes. In he former case, hey would re-ener rade should deviaion in sock prices occurs again. Our baseline sraegy is a refined version of he aforemenioned heurisic. Specifically, i rades as follows: we sar ou wih no posiions. When he measure of pas asse price differences y exceeds some hreshold y baseline, we ener rades. The fracion of he curren wealh ha we inves in he winner and he loser asses are η baseline and η baseline, respecively. A he nex crossing of he asse prices, we liquidae boh posiions. Given a problem insance, y baseline and η baseline are chosen o opimize reurns via a discree, simulaion-based search over η baseline [0, η] and y baseline in some inerval ha conains reasonable values of y. Here, η is an upper bound ha we impose on porfolio weighs during simulaion, as will be discussed in he following subsecion. 5.2 Simulaions We conduced experimens wih an insance of he model presened in he previous secion. Guided by real marke and rading processes, we considered ime o be in unis of a year and seleced model parameers α = 0.2, δ = 0.5, and ε =

14 We seled on he following 9 basis funcions, which seem o yield favorable resuls. Noe ha φ 1 hrough φ 7 are Chebyshev polynomials of he firs kind, up o he second degree. They are chosen o achieve numerical sabiliy. φ 1 (z) = 1, φ 2 (z) = η 1, φ 3 (z) = η 2, φ 4 (z) = y, φ 5 (z) = 2η1 2 1, φ 6 (z) = 2η2 2 1, φ 7 (z) = 2y 2 1, φ 8 (z) = η 1 y, φ 9 (z) = η 2 y. To solve for basis funcion weighs, we sampled saes in he following manner o consruc he linear program: we sampled η 1 and η 2 according o a Gaussian disribuion wih mean 0 and sandard deviaion 40/3. We hen le each sample ha fell ouside inerval [ 20, 20] ake on 20 or 20, whichever was closer. The disribuion and inerval were chosen o conain he range of porfolios ha migh be held by a reasonable sraegy. To sample y, we simulaed he process described in (10) for an exponenially disribued amoun of ime wih mean 1/α and we recorded he las sample. For boh our approximae dynamic programming policy and he baseline policy, we generaed sample pahs according o (1) and (2), wih each discree ime sep equal o 1 day (i.e., 1/365 of a year). Similar o sae sampling, we also bounded our porfolio weighs wihin he range [ 20, 20] during simulaion. In addiion, we imposed a range of [ 500, 500] on rading raes. This inerval was chosen o conain he range of reasonable policies. We compued he average uiliy u over all sample pahs and hen compued he equivalen rae of reurn by solving for r in u = =0 e α u(e r ) d, 13

15 where u(w) = log(w) is he uiliy funcion. Inuiively, r represens he rae of reurn required for a risk-free porfolio o deliver he same level of uiliy u. 5.3 Resuls Figures 1, 2, and 3 explore he dependence of reurns on he ransacion cos scaling c, given differen σ. The figures sugges ha when volailiy is sufficienly high, he ADP sraegy offers significan benefis over he baseline sraegy. Even when volailiy is low, he ADP sraegy ouperforms he baseline sraegy in reurns as well, if ransacion cos is high. In addiion, as volailiy increases, he ADP sraegy seems o yield higher reurns. 6 Exensions and Open Issues Our experimenal resuls have added confidence o our linear programming approach o opimizing gains in he pairs rading problem ha we defined. More heoreic work, however, is needed o prove he reasonableness of our approximaions in his coninuous-ime, coninuous-sae-space problem. The sudy of pairs rading presened in his paper represens only a saring poin o explore he problem fully. Consrucing a model wih more sophisicaed ransacion cos and modifying our algorihm accordingly o opimize for ha model will be a nex sep in making our approach pracical. 14

16 7 Risk-free raes of reurn Baseline 2 ADP Transacion cos scaling (c) Figure 1: Equivalen risk-free raes of reurn versus ransacion cos scaling. Parameer values: µ = 0, σ = 0.2, γ = 52, ψ = 500, ψ = 500, η = 20, η = 20, α = 0.2, δ = 0.5, ε = Risk-Free raes of reurn ADP 2 Baseline Transacion cos scaling (c) Figure 2: Equivalen risk-free raes of reurn versus ransacion cos scaling. Parameer values: µ = 0, σ = 0.3, γ = 52, ψ = 500, ψ = 500, η = 20, η = 20, α = 0.2, δ = 0.5, ε =

17 Risk-free raes of reurn Baseline Transacion cos scaling (c) ADP Figure 3: Equivalen risk-free raes of reurn versus ransacion cos scaling. Parameer values: µ = 0, σ = 0.4, γ = 52, ψ = 500, ψ = 500, η = 20, η = 20, α = 0.2, δ = 0.5, ε = Acknowledgmens The auhor would like o hank Jiarui Han and Benjamin Van Roy for heir significan conribuion o all phases of his research projec. References [1] D. P. de Farias and B. Van Roy, The Linear Programming Approach o Approximae Dynamic Programming. Operaions Research, Vol. 51, No. 6, 2003, pp [2] D. P. de Farias and B. Van Roy, On Consrain Sampling in he Linear Programming Approach o Approximae Dynamic Programming. Mahemaics of Operaions Research, Vol. 29, No. 3, 2004, pp [3] E. Gaev, W. N. Goezmann and K. G. Rouwenhors, Pairs Trading: Performance of a Relaive Value Arbirage Rule. Yale School of Managemen working paper [4] J. Han, Dynamic Porfolio Managemen: An Approximae Linear Programming Approach. Docoral Disseraion, Sanford Universiy,

18 [5] M. Hauskrech and B. Kveon, Linear Program Approximaions for Facored Coninuous-Sae Markov Decision Processes. In Advances in Neural Informaion Processing Sysems 16, 2004, pp [6] P. Schweizer and A. Seidmann, Generalized Polynomial Approximaions in Markovian Decision Processes. Journal of Mahemaical Analysis and Applicaions, Vol. 110, 1985, pp

19 Appendix: Myopic Sraegy The following is a differen formulaion of policy in he same pairs rading problem wihou ransacion cos. This formulaion assumes ha an invesor can insananeously reshuffle his porfolio weighs upon observing marke condiions. Specifically, porfolio weighs, raher han rading rae, is regarded as policy. Our original hope is o use a smoohed version of he opimal sraegy derived in his formulaion as our baseline. As we discuss, however, he poor performance of such a sraegy led us o abandon his hough. A formulaion of he wealh process is in order. In he absence of ransacion cos, evolves according o d = η(z ) ds S d (28) Subsiuing (12) ino (28), we express as he following diffusion: d = µ w (z, η ) d + σ w (z, η ) db, where µ w (z, η ) = η(z ) µ s (z ) and σ w (z, η ) = η(z ) σ s. We aim o [ ] max E e α u( ) d, η R 2 =0 where uiliy funcion u(w) = log(w). Noe ha our policy here is porfolio weigh vecor η, no longer rading rae vecor ψ. For each sraegy η, we define a value funcion J η (z) = E z,η [ 0 ] e α u( ) d, where he subscrips of he expecaion operaor mean ha he iniial sae of he process is z and 18

20 sraegy η is employed. Furher, we define he opimal value funcion as J (z) = sup J η (z). η From dynamic programming, we knoha J saisfies he Hamilon-Jocabi-Bellman Equaion: max η R 2 H ηj(z) = 0, z, (29) where dynamic operaor H η is defined as H η J(z) = D η J(z) αj(z) + u(w), z, and infiniesimal operaor D η is D η J(z) = J w wµ w + ( ) J µ y + 1 y 2 2 J w 2 w2 σ w σw + 1 ( 2 ) 2 r J y 2 σ yσy + 2 J w y σ yσw w = J ( ) J w wµ s η + µ y J y 2 w 2 w2 η σ s σs η r ( 2 J y 2 σ yσ y ) + 2 J w y σ yσ s ηw. As explained in secion 3.3, given our goal o opimize logarihmic uiliy, J decomposes according o J (z) = log(w) α + V (η, y), (30) for some V. I follows hen w J w = 1 α, w2 2 J w 2 = 1 α, 2 J w = 0, 2 J w y = 0. (31) Subsiuing (30) and (31) ino he HJB Equaion in (29) yields 0 = max η R 2 { ( ) 1 V α µ s η + µ y 1 y 2α η σ s σs η + 1 ( 2 ) } 2 r V y 2 σ yσy. 19

21 Noe ha he funcion being maximized here is quadraic in η. Hence, he opimal rading sraegy η has a closed-form soluion, η (z) = ( ) 1 σ s σs µ s (z). Noe ha he above expression of η does no involve V. Tha means η is a myopic sraegy. We denoe his sraegy wih η myopic in he following discussion. We define ψ myopic as he rae of change in porfolio weighs under policy η myopic. In order o reduce ransacion cos, we consider rading rae of he form aψ myopic, for some consan a [0, 1]. Given a problem insance, his consan is seleced o opimize ransacion coss via a discree search over he uni inerval and simulaion-based evaluaion of alernaives. We refer o he resuling sraegy as he smoohed myopic sraegy because i smoohs he rading aciviy ha would resul from he myopic sraegy. In our experimen, we esed our ADP sraegy and he smoohed myopic sraegy on sample pahs of asse price evoluion, and compared he performances of hese wo sraegies. We noiced ha on some sample pahs, he smoohed myopic sraegy delivers significanly higher reurns han he ADP sraegy. On oher sample pahs, however, he smoohed myopic sraegy quickly dragged wealh o below zero, rendering he value reurned by he log uiliy funcion undefined. This numerical difficuly precluded us from compuing he average uiliy of he smoohed myopic sraegy. We can, however, infer from his experimenal resul ha he smoohed myopic sraegy yields poor overall performance when ransacion cos is non-zero. We conjecure ha such performance is largely due o an underlying assumpion of he formulaion for he myopic policy: namely, he invesor can insananeously change porfolio weighs upon observing marke condiions. This assumpion does no hold in our formulaion for he ADP sraegy and hence is no adhered o in simulaion. I follows, hen, ha he smoohed myopic sraegy compued for he curren ime sep is acually applied in he following ime sep, which may be far from opimal. 20