On Sochasic and Wors-case Models for Invesing Elad Hazan IBM Almaden Research Cener 650 Harry Rd, San Jose, CA 9520 ehazan@cs.princeon.edu Sayen Kale Yahoo! Research 430 Grea America Parkway, Sana Clara, CA 95054 skale@yahoo-inc.com Absrac In pracice, mos invesing is done assuming a probabilisic model of sock price reurns known as he Geomeric Brownian Moion (GBM. While ofen an accepable approximaion, he GBM model is no always valid empirically. This moivaes a wors-case approach o invesing, called universal porfolio managemen, where he objecive is o maximize wealh relaive o he wealh earned by he bes fixed porfolio in hindsigh. In his paper we ie he wo approaches, and design an invesmen sraegy which is universal in he wors-case, and ye capable of exploiing he mosly valid GBM model. Our mehod is based on new and improved regre bounds for online convex opimizaion wih exp-concave loss funcions. Inroducion Average-case Invesing: Much of mahemaical finance heory is devoed o he modeling of sock prices and devising invesmen sraegies ha maximize wealh gain, minimize risk while doing so, and so on. Typically, his is done by esimaing he parameers in a probabilisic model of sock prices. Invesmen sraegies are hus geared o such average case models (in he formal compuer science sense, and are naurally suscepible o drasic deviaions from he model, as winessed in he recen sock marke crash. Even so, empirically he Geomeric Brownian Moion (GBM ([Osb59, Bac00] has enjoyed grea predicive success and every year rillions of dollars are raded assuming his model. Black and Scholes [BS73] used his same model in heir Nobel prize winning work on pricing opions on socks. Wors-case Invesing: The fragiliy of average-case models in he face of rare bu dramaic deviaions led Cover [Cov9] o ake a wors-case approach o invesing in socks. The performance of an online invesmen algorihm for arbirary sequences of sock price reurns is measured wih respec o he bes CRP (consan rebalanced porfolio, see [Cov9] in hindsigh. A universal porfolio selecion algorihm is one ha obains sublinear (in he number of rading periods T regre, which is he difference in he logarihms of he final wealhs obained by he wo. Cover [Cov9] gave he firs universal porfolio selecion algorihm wih regre bounded by O(log T. There has been much follow-up work afer Cover s seminal work, such as [HSSW96, MF92, KV03, BK97, HKKA06], which focused on eiher obaining alernae universal algorihms or improving he efficiency of Cover s algorihm. However, he bes regre bound is sill O(log T. This dependence of he regre on he number of rading periods is no enirely saisfacory for wo main reasons. Firs, a priori i is no clear why he online algorihm should have high regre (growing wih he number of ieraions in an unchanging environmen. As an exreme example, consider a seing wih wo socks where one has an upward drif of % daily, whereas he second sock remains a he same price. One would expec o figure ou his paern quickly and focus on he
firs sock, hus aaining a consan fracion of he wealh of he bes CRP in he long run, i.e. consan regre, unlike he wors-case bound of O(log T. The second problem arises from rading frequency. Suppose we need o inves over a fixed period of ime, say a year. Trading more frequenly poenially leads o higher wealh gain, by capializing on shor erm sock movemens. However, increasing rading frequency increases T, and hus one may expec more regre. The problem is acually even worse: since we measure regre as a difference of logarihms of he final wealhs, a regre bound of O(log T implies a poly(t facor raio beween he final wealhs. In realiy, however, experimens [AHKS06] show ha some known online algorihms acually improve wih increasing rading frequency. Bridging Wors-case and Average-case Invesing: Boh hese issues are resolved if one can show ha he regre of a good online algorihm depends on oal variaion in he sequence of sock reurns, raher han purely on he number of ieraions. If he sock reurn sequence has low variaion, we expec our algorihm o be able o perform beer. If we rade more frequenly, hen he per ieraion variaion should go down correspondingly, so he oal variaion says he same. We analyze a porfolio selecion algorihm and prove ha is regre is bounded by O(log Q, where Q (formally defined in Secion.2 is he sum of squared deviaions of he reurns from heir mean. Since Q T (afer appropriae normalizaion, we improve over previous regre bounds and reain he wors-case robusness. Furhermore, in an average-case model such as GBM, he variaion can be ied very nicely o he volailiy parameer, which explains he experimenal observaion he regre doesn increase wih increasing rading frequency. Our algorihm is efficien, and is implemenaion requires consan ime per ieraion (independen of he number of game ieraions.. New Techniques and Comparison o Relaed Work Cesa-Bianchi, Mansour and Solz [CBMS07] iniiaed work on relaing wors case regre o he variaion in he daa for he relaed learning problem of predicion from exper advice, and conjecured ha he opimal regre bounds should depend on he observed variaion of he cos sequence. Recenly, his conjecured was proved and regre bounds of Õ( Q were obained in he full informaion and bandi linear opimizaion seings [HK08, HK09], where Q is he variaion in he cos sequence. In his paper we give an exponenial improvemen in regre, viz. O(log Q, for he case of online exp-concave opimizaion, which includes porfolio selecion as a special case. Anoher approach o connecing wors-case o average-case invesing was aken by Jamshidian [Jam92] and Cross and Barron [CB03]. They considered a model of coninuous rading, where here are T rading inervals, and in each he online invesor chooses a fixed porfolio which is rebalanced k imes wih k. They prove familiar regre bounds of O(log T (independen of k in his model w.r.. he bes fixed porfolio which is rebalanced T k imes. In his model our algorihm aains he igher regre bounds of O(log Q, alhough our algorihm has more flexibiliy. Furhermore heir algorihms, being exensions of Cover s algorihm, may require exponenial ime in general. Our bounds of O(log Q regre require compleely differen echniques compared o he Õ( Q regre bounds of [HK08, HK09]. These previous bounds are based on firs-order gradien descen mehods which are oo weak o obain O(log Q regre. Insead we have o use he second-order Newon sep ideas based on [HKKA06] (in paricular, he Hessian of he cos funcions. The second-order echniques of [HKKA06] are, however, no sensiive enough o obain O(log Q bounds. This is because progress was measured in erms of he disance beween successive porfolios in he usual Euclidean norm, which is insensiive o variaion in he cos sequence. In his paper, we inroduce a differen analysis echnique, based on analyzing he disance beween successive predicions using norms ha keep changing from ieraion o ieraion and are acually sensiive o he variaion. A key echnical sep in he analysis is a lemma (Lemma 6 which bounds he sum of differences of successive Cesaro means of a sequence of vecors by he logarihm of is variaion. This lemma, Cross and Barron give an efficien implemenaion for some ineresing special cases, under assumpions on he variaion in reurns and bounds on he magniude of he reurns, and assuming k. A ruly efficien implemenaion of heir algorihm can probably be obained using he echniques of Kalai and Vempala. 2
which may be useful in oher conexs when variaion bounds on he regre are desired, is proved using he Kahn-Karush-Tucker condiions, and also improves he regre bounds in previous papers..2 The model and saemen of resuls Porfolio managemen. In he universal porfolio managemen model [Cov9], an online invesor ieraively disribues her wealh over n asses before observing he change in asse price. In each ieraion =, 2,... he invesor commis o an n-dimensional disribuion of her wealh, x n = { i x i =, x 0}. She hen observes a price relaives vecor r R n +, where r (i is he raio beween he closing price of he i h asse on rading period and he opening price. In he h rading period, he wealh of he invesor changes by a facor of (r x. The overall change in wealh is hus (r x. Since in a ypical marke wealh grows a an exponenial rae, we measure performance by he exponenial growh rae, which is log (r x = log(r x. A consan rebalanced porfolio (CRP is an invesmen sraegy which rebalances he wealh in every ieraion o keep a fixed disribuion. Thus, for a CRP x n, he change in wealh is (r x. The regre of he invesor is defined o be he difference beween he exponenial growh rae of her invesmen sraegy and ha of he bes CRP sraegy in hindsigh, i.e. Regre := max log(r x log(r x x n Noe ha he regre doesn change if we scale all he reurns in any paricular period by he same amoun. So we assume w.l.o.g. ha in all periods, max i r (i =. We assume ha here is known parameer r > 0, such ha for all periods, min,i r (i r. We call r he marke variabiliy parameer. This is he only resricion we pu on he sock price reurns; hey could be chosen adversarially as long as hey respec he marke variabiliy bound. Online convex opimizaion. In he online convex opimizaion problem [Zin03], which generalizes universal porfolio managemen, he decision space is a closed, bounded, convex se K R n, and we are sequenially given a series of convex cos 2 funcions f : K R for =, 2,.... The algorihm ieraively produces a poin x K in every round, wihou knowledge of f (bu using he pas sequence of cos funcions, and incurs he cos f (x. The regre a ime T is defined o be Regre := T = f (x min x K T f (x. Usually, we will le denoe T =. In his paper, we resric our aenion o convex cos funcions which can be wrien as f (x = g(v x for some univariae convex funcion g and a parameer vecor v R n (for example, in he porfolio managemen problem, K = n, f (x = log(r x, g = log, and v = r. Thus, he cos funcions are paramerized by he vecors v, v 2,..., v T. Our bounds will be expressed as a funcion of he quadraic variabiliy of he parameer vecors v, v 2,..., v T, defined as Q(v,..., v T := min µ = T v µ 2. This expression is minimized a µ = T T = v, and hus he quadraic variaion is jus T imes he sample variance of he sequence of vecors {v,..., v }. Noe however ha he sequence can be generaed adversarially raher han by some sochasic process. We shall refer o his as simply Q if he vecors are clear from he conex. Main heorem. In he seup of he online convex opimizaion problem above, we have he following algorihmic resul: Theorem. Le he cos funcions be of he form f (x = g(v x. Assume ha here are parameers R, D, a, b > 0 such ha he following condiions hold: 2 Noe he difference from he porfolio selecion problem: here we have convex cos funcions, raher han concave payoff funcions. The porfolio selecion problem is obained by using log as he cos funcion. = 3
. for all, v R, 2. for all x K, we have x D, 3. for all x K, and for all, eiher g (v x [0, a] or g (v x [ a, 0], and 4. for all x K, and for all, g (v x b. Then here is an algorihm ha guaranees he following regre bound: Regre = O((a 2 n/b log( + bq + br 2 + ard log(2 + Q/R 2 + D 2. Now we apply Theorem o he porfolio selecion problem. Firs, we esimae he relevan parameers. We have r n since all r (i, hus R = n. For any x n, x, so D =. g (v x = (v, and hus x g (v x [ r, 0], so a = r. Finally, g (v x = (v x, so 2 b =. Applying Theorem we ge he following corollary: Corollary 2. For he porfolio selecion problem over n asses, here is an algorihm ha aains he following regre bound: ( n Regre = O r 2 log(q + n. 2 Bounding he Regre by he Observed Variaion in Reurns 2. Preliminaries All marices are assumed be real symmeric marices in R n n, where n is he number of socks. We use he noaion A B o say ha A B is posiive semidefinie. We require he noion of a norm of a vecor x induced by a posiive definie marix M, defined as x M = x Mx. The following simple generalizaion of he Cauchy-Schwarz inequaliy is used in he analysis: x, y R n : x y x M y M. We denoe by A he deerminan of a marix A, and by A B = Tr(AB = ij A ijb ij. As we are concerned wih logarihmic regre bounds, poenial funcions which behave like harmonic series come ino play. A generalizaion of harmonic series o high dimensions is he vecor-harmonic series, which is a series of quadraic forms ha can be expressed as (here A 0 is a posiive definie marix, and v, v 2,... are vecors in R n : v (A + v v v, v2 (A + v v + v 2 v2 v 2,..., v (A + τ= v τ vτ v,... The following lemma is from [HKKA06]: Lemma 3. For a vecor harmonic series given by an iniial marix A and vecors v, v 2,..., v T, we have T v (A + [ τ= v τ vτ A + ] T τ= v log v τ vτ. A = The reader can noe ha in one dimension, if all vecors v = and A =, hen he series above reduces exacly o he regular harmonic series whose sum is bounded, of course, by log(t +. 2.2 Algorihm and analysis We analyze he following algorihm and prove ha i aains logarihmic regre wih respec o he observed variaion (raher han number of ieraions. The algorihm follows he generic algorihmic scheme of Follow-The-Regularized-Leader (FTRL wih squared Euclidean regularizaion. Algorihm Exp-Concave-FTL. In ieraion, use he poin x defined as: ( x arg min f τ (x + x n 2 x 2 τ= Noe he mahemaical program which he algorihm solves is convex, and can be solved in ime polynomial in he dimension and number of ieraions. The running ime, however, for solving his ( 4
convex program can be quie high. In he full version of he paper, for he specific problem of porfolio selecion, where f (x = log(r x, we give a faser implemenaion whose per ieraion running ime is independen of he number of ieraions, using he more sophisicaed online Newon mehod of [HKKA06]. In paricular, we have he following resul: Theorem 4. For he porfolio selecion problem, here is an algorihm ha runs in O(n 3 ime per ieraion whose regre is bounded by ( n Regre = O r 3 log(q + n. In his paper, we reain he simpler algorihm and analysis for an easier exposiion. We now proceed o prove he Theorem. Proof. [Theorem ] Firs, we noe ha he algorihm is running a Follow-he-leader procedure on he cos funcions f 0, f, f 2,... where f 0 (x = 2 x 2 is a ficiious period 0 cos funcion. In oher words, in each ieraion, i chooses he poin ha would have minimized he oal cos under all he observed funcions so far (and, addiionally, a ficiious iniial cos funcion f 0. This poin is referred o as he leader in ha round. The firs sep in analyzing such an algorihm is o use a sabiliy lemma from [KV05], which bounds he regre of any Follow-he-leader algorihm by he difference in coss (under f of he curren predicion x and he nex one x +, plus an addiional error erm which comes from he regularizaion. Thus, we have Regre f (x f (x + + 2 ( x 2 x 0 2 f (x (x x + + 2 D2 = g (v x [v (x x + ] + 2 D2 (2 The second inequaliy is because f is convex. The las equaliy follows because f (x = g (x v v. Now, we need a handle on x x +. For his, define F = τ=0 f τ, and noe ha x minimizes F over K. Consider he difference in he gradiens of F + evaluaed a x + and x : F + (x + F + (x = τ=0 f τ (x + f τ (x = τ= [g τ (v τ x + g τ (v τ x ]v τ + (x + x = τ= [ g τ (v τ ζ τ (x + x ]v τ + (x + x (3 = τ= g τ (v τ ζ τ v τ v τ (x + x + (x + x. (4 Equaion 3 follows by applying he Taylor expansion of he (muli-variae funcion g τ (v τ x a poin x, for some poin ζτ on he line segmen joining x and x +. The equaion (4 follows from he observaion ha g τ (v τ x = g τ (v τ xv τ. Define A = τ= g (v ζ τ v v + I, where I is he ideniy marix, and x = x + x. Then equaion (4 can be re-wrien as: F + (x + F (x g (v x v = A x. (5 Now, since x minimizes he convex funcion F over he convex se K, a sandard inequaliy of convex opimizaion (see [BV04] saes ha for any poin y K, we have F (x (y x 0. Thus, for y = x +, we ge ha F (x (x + x 0. Similarly, we ge ha F + (x + (x x + 0. Puing hese wo inequaliies ogeher, we ge ha Thus, using he expression for A x from (5 we have ( F + (x + F (x x 0. (6 x 2 A = A x x = ( F + (x + F (x g (v x v x g (v x [v (x x + ] (from (6 (7 5
Assume ha g (v x [ a, 0] for all x K and all. The oher case is handled similarly. Inequaliy (7 implies ha g (v x and v (x x + have he same sign. Thus, we can upper bound g (v x [v (x x + ] a(v x. (8 Define ṽ = v µ, µ = + τ= v τ. Then, we have v x = ṽ x + T =2 x (µ µ x µ + x T + µ T, (9 where ṽ = v µ, µ = + Then we bound T =2 x (µ µ x µ + x T + µ T τ= v. Now, define ρ = ρ(v,..., v T = T = µ + µ. T =2 x µ µ + x µ + x T + µ T Dρ + 2DR. (0 We will bound ρ momenarily. For now, we urn o bounding he firs erm of (9 using he Cauchy- Schwarz generalizaion as follows: ṽ x By he usual Cauchy-Schwarz inequaliy, ṽ A x A ṽ 2 A x 2 A from (7 and (8. We conclude, using (9, (0 and (, ha a(v x a ṽ 2 ṽ A x A. ( A ṽ 2 A a(v x a(v x + adρ + 2aDR. This implies (using he AM-GM inequaliy applied o he firs erm on he RHS ha a(v x a 2 ṽ 2 + 2aDρ + 4aDR. A Plugging his ino he regre bound (2 we obain, via (8, Regre a 2 ṽ 2 A + 2aDρ + 4aDR + 2 D2. The proof is compleed by he following wo lemmas (Lemmas 5 and 6 which bound he RHS. The firs erm is a vecor harmonic series, and he second erm can be bounded by a (regular harmonic series. Lemma 5. ṽ 2 A 3n b log [ + bq + br 2]. Proof. We have A = τ= g (v ζτ v v +I. Since g (v ζτ b, we have A I +b v v. Using he fac ha ṽ = v µ and µ = + τ v τ, we ge ha ( ( ṽ τ ṽτ = + v s vs + s + [v r vs + v s vr ]. τ= s= τ=s (τ+ 2 s= r<s τ=s (τ+ 2 Now, τ=s (τ+ + 2 s x dx = 2 s +. Since (v r + v s (v r + v s 0, we ge ha v r vr + v s vs [v r vs + v s vr ], and hence we have ṽ τ ṽτ τ= s= ( + s vs v s + s= r<s + [v rv r + v s v s ] s= ( 2 + s vs v s 3 v s vs. Le à = 3 I +b ṽṽ. Noe ha he inequaliy above shows ha 3à A. Thus, using Lemma 3, we ge ṽ 2 = ṽ A A ṽ 3 b [ bṽ ] à [ [ ] bṽ ] 3 b log à T. (2 Ã0 To bound he laer quaniy noe ha Ã0 = I =, and ha ÃT = I + b ṽṽ ( + b ṽ 2 2 n = ( + b Q n where Q = ṽ 2 = ṽ µ 2. Lemma 7 (proved in he full version of he paper, we show ha Q Q + R 2. This implies ha ÃT ( + bq + br 2 n and he proof is compleed by subsiuing his bound ino (2. s= 6
Lemma 6. ρ(v,..., v T 2R[log(2 + Q/R 2 + ]. Proof. Define, for τ 0, he vecor u τ = v τ µ T +. Noe ha by convenion, we have v 0 = 0. We have T τ=0 u τ 2 = µ T + 2 + T τ= v τ µ T + 2 = R 2 + Q. Furhermore, µ + µ = + +2 τ=0 v τ + τ=0 v τ = + τ=0 u τ +2 + τ=0 u τ (+ 2 τ=0 u τ + + u + Summing up over all ieraions, µ + µ ( (+ 2 τ=0 u τ + + u + 2 u 2R[log(2+Q/R 2 +]. The las inequaliy follows from Lemma 8 (proved in he full version below by seing x = u /R, for. Lemma 7. Q Q + R 2. Lemma 8. Suppose ha 0 x and x2 Q. Then T = x / log( + Q +. 3 Implicaions in he Geomeric Brownian Moion Model We begin wih a brief descripion of he model. The model assumes ha socks can be raded coninuously, and ha a any ime, he fracional change in he sock price wihin an infiniesimal ime inerval is normally disribued, wih mean and variance proporional o he lengh of he inerval. The randomness is due o many infiniesimal rades ha jar he price, much like paricles in a physical medium are jarred abou by oher paricles, leading o he classical Brownian moion. Formally, he model is parameerized by wo quaniies, he drif µ, which is he long erm rend of he sock prices, and volailiy σ, which characerizes deviaions from he long erm rend. The parameer σ is ypically specified as annualized volailiy, i.e. he sandard deviaion of he sock s logarihmic reurns in one year. Thus, a rading inerval of [0, ] specifies year. The model posulaes ha he sock price a ime, S, follows a geomeric Brownian moion wih drif µ and volailiy σ: ds = µs d + σs dw, where W is a coninuous-ime sochasic process known as he Wiener process or simply Brownian moion. The Wiener process is characerized by hree facs:. W 0 = 0, 2. W is almos surely coninuous, and 3. for any wo disjoin ime inervals [s, ] and [s 2, 2 ], he random variables W W s and W 2 W s2 are independen zero mean Gaussian random variables wih variance s and 2 s 2 respecively. Using Iō s lemma (see, for example, [KS04], i can be shown ha he sock price a ime is given by S = S 0 exp((µ σ 2 /2 + σw. (3 Now, we consider a siuaion where we have n socks in he GBM model. Le µ = (µ, µ 2,..., µ n be he vecor of drifs, and σ = (σ, σ 2,..., σ n be he vecor of (annualized volailiies. Suppose we rade for one year. We now sudy he effec of rading frequency on he quadraic variaion of he sock price reurns. For his, assume ha he year-long rading inerval is sub-divided ino T equally sized inervals of lengh /T, and we rade a he end of each such inerval. Le r = (r (, r (2,..., r (n be he vecor of sock reurns in he h rading period. We assume ha T is large enough, which is aken o mean ha i is larger han µ(i, σ(i, ( µ(i σ(i 2 for any i. 7
Then using he facs of he Wiener process saed above, we can prove he following lemma, which shows ha he expeced quadraic variaion, and is variance, is he essenially he same regardless of rading frequency. The proof is a sraighforward calculaion and deferred o he full version of his paper. Lemma 9. In he seup of rading n socks in he GBM model over one year wih T rading periods, here is a vecor v such ha [ T E = r v 2] σ 2 ( + O( T and [ T VAR = r v 2] 6 σ 2 ( + O( T, regardless of how he socks are correlaed. Applying his bound in our algorihm, we obain he following regre bound from Corollary 2. Theorem 0. In he seup of Lemma 9, for any δ > 0, wih probabiliy a leas 2e δ, we have Regre O(n(log( σ 2 + n + δ. Theorem 0 shows ha one expecs o achieve consan regre independen of he rading frequency, as long as he oal rading period is fixed. This resul is only useful if increasing rading frequency improves he performance of he bes consan rebalanced porfolio. Indeed, his has been observed empirically (see e.g. [AHKS06], and more empirical evidence is given in he full version of his paper.. To obain a heoreical jusificaion for increasing rading frequency, we consider an example where we have wo socks ha follow independen Black-Scholes models wih he same drifs, bu differen volailiies σ, σ 2. The same drif assumpion is necessary because in he long run, he bes CRP is he one ha pus all is wealh on he sock wih he greaer drif. We normalize he drifs o be equal o 0, his doesn change he performance in any qualiaive manner. Since he drif is 0, he expeced reurn of eiher sock in any rading period is ; and since he reurns in each period are independen, he expeced final change in wealh, which is he produc of he reurns, is also. Thus, in expecaion, any CRP (indeed, any porfolio selecion sraegy has overall reurn. We herefore urn o a differen crierion for selecing a CRP. The risk of an invesmen sraegy is measured by he variance of is payoff; hus, if differen invesmen sraegies have he same expeced payoff, hen he one o choose is he one wih minimum variance. We herefore choose he CRP wih he leas variance. We prove he following lemma in he full version of he paper: Lemma. In he seup where we rade wo socks wih zero drif and volailiies σ, σ 2, he variance of he minimum variance CRP decreases as he rading frequency increases. Thus, increasing he rading frequency decreases he variance of he minimum variance CRP, which implies ha i ges less risky o rade more frequenly; in oher words, he more frequenly we rade, he more likely he payoff will be close o he expeced value. On he oher hand, as we show in Theorem 0, he regre does no change even if we rade more ofen; hus, one expecs o see improving performance of our algorihm as he rading frequency increases. 4 Conclusions and Fuure Work We have presened an efficien algorihm for regre minimizaion wih exp-concave loss funcions whose regre sricly improves upon he sae of he ar. For he problem of porfolio selecion, he regre is bounded in erms of he observed variaion in sock reurns raher han he number of ieraions. Recenly, DeMarzo, Kremer and Mansour [DKM06] presened a novel game-heoreic framework for opion pricing. Their mehod prices opions using low regre algorihms, and i is possible ha our analysis can be applied o opions pricing via heir mehod (alhough ha would require a much igher opimizaion of he consans involved. Increasing rading frequency in pracice means increasing ransacion coss. We have assumed no ransacion coss in his paper. I would be very ineresing o exend our porfolio selecion algorihm o ake ino accoun ransacion coss as in he work of Blum and Kalai [BK97]. 8
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