High-dimensional statistical arbitrage with factor models and stochastic control
HHigh-dimensional statistical arbitrage with factor models andstochastic control
Jorge Guijarro-Ordonez ∗ October 2019
Abstract
The present paper provides a study of high-dimensional statistical arbitrage that com-bines factor models with the tools from stochastic control, obtaining closed-form opti-mal strategies which are both interpretable and computationally implementable in a high-dimensional setting. Our setup is based on a general statistically-constructed factor modelwith mean-reverting residuals, in which we show how to construct analytically market-neutral portfolios and we analyze the problem of investing optimally in continuous time andfinite horizon under exponential and mean-variance utilities. We also extend our model toincorporate constraints on the investor’s portfolio like dollar-neutrality and market frictionsin the form of temporary quadratic transaction costs, provide extensive Monte Carlo sim-ulations of the previous strategies with 100 assets, and describe further possible extensionsof our work.
Keywords: statistical arbitrage, factor models, algorithmic trading, Ornstein-Uhlenbeck pro-cess, mean reversion, stochastic control.
Word count: ∗ Department of Mathematics, Stanford University. Email address: [email protected]. a r X i v : . [ q -f i n . M F ] D ec Introduction
Modeling of pairs trading based on stochastic control has been an active research topic inmathematical finance for the last few years. After the papers by Jurek and Yang (2007) andMudchanatongsuk et al. (2008), an increasing number of models have been proposed in thisframework (see, for example, Chiu and Wong (2011), Tourin and Yan (2013), and Liu andTimmerman (2013)), in which generally they assume that some statistically-designed relationbetween the prices of two assets is a mean-reverting stochastic process and find a dynamicoptimal allocation in continuous time in some version of the classical Merton framework. Morerecently, a number of papers have also studied the optimal entry and exit points when trading acouple of cointegrated assets, such as Leung and Li (2015), Lei and Xu (2015), Ngo and Pham(2016), and Kitapbayev and Leung (2018).In the high-dimensional case, however, relatively little model-based research has been conducted.Cartea and Jaimungal (2016) and Lintilhac and Tourin (2016) investigate a multidimensionalgeneralization of the model in Tourin and Yan (2013) and apply stochastic control to solve aMerton-like problem in continuous time on a collection of cointegrated assets, with exponentialutility and finite horizon. In a different direction which is not exactly statistical arbitrage,Cartea et al. (2018) address an optimal execution problem with transaction costs on a basket ofmultiple cointegrated assets, which they also solve with control techniques. Finally, without us-ing stochastic control, the paper by Avellaneda and Lee (2010) carries out a data-based study ofstatistical arbitrage in the US equity market by proposing a factor model with mean-revertingresiduals and a threshold-based bang-bang strategy. This model is further analyzed and ex-tended by Papanicolaou and Yeo (2017), who discuss risk control and develop an optimizationmethod to allocate the investments given the trading signals.The previous papers in this high-dimensional framework thus either apply stochastic control toa mean-reverting process they already have or use a factor model to construct this process andthen choose the trading signals based on residuals, but none of them considers the combinationof these two powerful techniques. The present paper aims to fill this gap by providing a study ofstatistical arbitrage in a high-dimensional setting that combines factor models and the tools fromstochastic control, extending the previous studies and obtaining closed-form optimal strategieswhich are interpretable and easy to implement computationally.More precisely, in our framework an investor observes the returns of a high-dimensional collec-tion of risky assets and, similar to Avellaneda and Lee (2010) and Papanicolaou and Yeo (2017),uses historical data to statistically construct a factor model such that the cumulative residualsare assumed to be mean-reverting and following an Ornstein-Uhlenbeck process. However, un-like the previous literature, these residuals may be correlated and interdependent and, basedon their behavior, the investor must decide how to optimally allocate her wealth in the riskyassets and a riskless security so that the expected utility of her terminal wealth is maximizedand she is market-neutral. There are three main results in this paper:First, for a big class of statistically-constructed factor models that includes PCA we show howthe investor may theoretically construct market-neutral portfolios without solving any opti-mization problem (unlike the approach followed in Papanicolaou and Yeo (2017) or Boyd etal. (2017), for example) provided that the factor model holds, and we show how this makesthe optimal allocation problem analytically tractable and guarantees market-neutrality by con-struction. These portfolios are explicitly computable and depend quadratically on the factormodel loadings and, to the best of our knowledge, using this construction to connect factormodels and stochastic control theory in statistical arbitrage is new.2econd, using these explicit market-neutral portfolios as control variables, we show how theinvestor should trade optimally in continuous time to maximize either an exponential utility ora Markowitz-inspired mean-variance objective, obtaining explicit analytic forms of the optimalstrategies in both cases in this high-dimensional setting. The structure of these optimal strate-gies is related to the classical solution of the Merton problem and is affine in the deviation ofthe residuals from their statistical mean, thus giving a precise estimate of how much we shouldbuy when the assets are underpriced and how much we should sell when they are overpriced, asin classical pairs trading. The coefficients are given by the solution of matrix Riccati differentialequations and depend quadratically on the factor model loadings, and the strategies in boththe exponential and the mean-variance case are surprisingly similar except for a non-myopiccorrection term that does not appear in the classical framework under a geometric Brownianmotion. This arises from the fact that in our case the drift of the underlying Ornstein-Uhlenbeckprocess is stochastic.The structure and the techniques to find these affine strategies are thus similar in spirit tothose in the affine process literature in finance (see Duffie et al. (2003) for a broad survey),to the more recent affine control literature in algorithmic trading (see, for example, Carteaet al. (2015) and the references therein), and to the literature on extensions of the Mertonproblem incorporating an Ornstein-Uhlenbeck process (see, for example, Benth and Karlsen(2005), Liang et al. (2011), Fouque et al. (2015) and Moutari et al (2017), which deal with asingle risky asset in the context of the Schwarz model or in geometric Brownian motion withstochastic drift or volatility; and Brendle (2006) and Bismuth et al. (2019), which consider themultiasset case again in the setting of geometric Brownian motion with stochastic drift). Whilethe techniques that we use to find the optimal strategies are therefore classical, the frameworkis new because the mean-reverting behavior of the underlying stochastic process arises from theresiduals of a factor model and in the context of statistical arbitrage, and we consider the generalcase of an arbitrary number of assets with a market-neutrality restriction. Moreover, the explicitsolutions allow us to understand the dependence of the optimal strategies on specific elements ofa statistical arbitrage strategy (such as the factor model, its loadings matrix and its connectionwith market-neutrality, and the mean-reversion speed of the residuals and their correlationstructure), and to compare arbitrageurs with exponential and mean-variance utilities.Third and finally, we extend the previous results in two directions by discussing how to incorpo-rate into the model soft constraints frequently imposed by arbitrageurs (such as dollar-neutrality,limitations on the money spent on each asset, leverage restrictions, etc.) and also market fric-tions in the form of quadratic transaction costs, inspired by the papers of Garleanu and Pedersen(2013, 2016) and also by the more general quadratic transaction cost and linear price impactliterature in portfolio theory (see, for example, Moreau et al. (2017) and Muhle-Karbe et al.(2017) for some new research directions and Obizhaeva and Wang (2013), Rogers and Singh(2010), Almgren and Chriss (2001) and Bertsimas and Lo (1998) for some classical papers). Inboth extensions, we again find explicit analytic strategies which are easily interpretable, andwhich quantitatively correspond to quadratic corrections in the structure of the original optimalstrategies (when adding soft constraints like dollar neutrality) or to “tracking” averages of thefuture original optimal portfolios (when adding quadratic transactions costs). Moreover, in bothcases these new strategies depend quadratically on the loadings of the factor model. Again, thenovelty of the results comes from the study of these questions (dollar neutrality, transactioncosts, etc.) in a new context in which they are crucial (statistical arbitrage with an arbitrarynumber of assets and a market-neutrality restriction, in particular using control techniques anda factor model), and this framework and the strategies that we find are new to the best of ourknowledge. 3o conclude the paper with a more empirical analysis, we also perform extensive numericalsimulations with a high-dimensional number of assets (100). This gives further insights aboutthe behavior of the previous strategies that are not obvious when looking at the correspondingequations, and allows us to understand the sensitivity of the model parameters and the depen-dence on the underlying factor model. This high-dimensional numerical study is also new withrespect to the existing literature, and the main conclusions are that (1) the exponential-utilitystrategies are more profitable than the mean-variance strategies (and they also take more ex-treme positions), (2) after some initial up and downs the sample paths of the different wealthprocesses progressively stabilize due to the asymptotic properties of the Ornstein-Uhlenbeckprocess, (3) increasing the risk-control parameters (like the parameter controlling dollar neu-trality) consistently produces a concentration of the distribution of the terminal wealth aroundsmaller values, and (4) imposing market neutrality when the loadings of the factor model getbigger leads to more aggressive strategies whose terminal wealth has a higher variance.The remainder of the article is organized as follows. In section 2 we introduce our model,construct the market-neutral portfolios that make the problem analytically tractable, and for-mulate the control problems. Next, section 3 presents the basic results under the exponentialand the mean-variance frameworks, whereas section 4 extends these results by considering theaddition of soft constraints and of quadratic transaction costs. Section 5 then allows us tounderstand in greater depth the behavior of the previous strategies and models by performingsome Monte Carlo simulations, and section 6 presents the main conclusions and proposes futurenew directions of research. Finally, an appendix contains all the proofs.
In the remainder of this paper we will consider the following general framework. We will assumethat an investor observes the returns of a large number N of risky assets and, like in classicalportfolio theory based on stochastic control, she must decide how to dynamically allocate herwealth by investing in them or in a riskless asset with constant interest rate r so that theexpected utility of her wealth at a finite terminal time T is maximized. However, unlike theclassical framework and the existing literature, to do so she will execute a statistical arbitragestrategy based on a factor model, in which instead of trading depending on the state of theoriginal returns she will trade depending on the behavior of the residuals, which will be thetrading signals. For example, in the case of two assets, this is equivalent to classical pairstrading, in which the investor may perform a simple linear regression on the returns of twohistorically correlated securities and, depending on how far the oscillation of the residual isfrom its historical average, she decides if there is a mispricing and opens and closes long andshort positions in the original assets in a market-neutral way. In this paper, we will study thegeneralization of this to the high-dimensional case of an arbitrary number of assets, in which wesubstitute the simple linear regression by a statistical factor model and we study the optimalallocations under the framework of stochastic control, assuming a mean-reverting stochasticmodel for the behavior of the residuals.More precisely, we make the following three general assumptions on how the investor will gen-erate these residuals and what dynamics they will have:(1) Assumption 1:
The investor has computed a factor model for the returns of the risky4ssets, which will hold during the the investment (finite) horizon and is given by dR t = Λ dF t + dX t , (2.1)where R t is the cumulative asset returns process, Λ is the (constant-in-time) loadingsmatrix, F t is the cumulative factors process, and X t is the cumulative residuals process .(2) Assumption 2:
This factor model has been computed statistically by using some versionof PCA , so the rows of Λ are the largest eigenvectors of some square matrix and thediscrete-time version of dF t (i.e., the daily, hourly, etc. factors returns) is then computedby linearly regressing the discrete-time version of dR t (i.e., the daily, hourly, etc. assetsreturns) on some rescaling of Λ, so dF t = ˜Λ dR t (2.2)for some rescaling ˜Λ of Λ. (In fact, the only thing we need about this assumption isthat (2.2) holds for some matrix ˜Λ, which allows for a bigger class of factor models thanclassical PCA).(3) Assumption 3:
The process X t given by the cumulative residuals is mean-reverting. Inparticular, for analytic tractability we assume that it is a matrix N -dimensional Ornstein-Uhlenbeck process satisfying the following stochastic differential equation with knownparameters dX t = A ( µ − X t ) dt + σdB t , where A is a constant N -dimensional square matrix whose eigenvalues have positive realparts so that there is mean-reversion, µ is a constant N -dimensional vector, B t is avector of m independent Brownian motions in the usual complete filtered probabilityspace (Ω , F , P , ( F t ) ≤ t ≤ T ), and σ is a constant N × m matrix such that the instantaneouscovariance matrix σσ (cid:48) is invertible.The previous framework thus combines high-dimensional statistical arbitrage, factor modelsand stochastic control in a way which is new to the best of our knowledge, and it extendsseveral models in the existing literature. For example, statistical arbitrage models based ona more particular case of Assumptions 1, 2, 3 (in which the residuals are assumed to be in-dependent one-dimensional Ornstein-Uhlenbeck processes, so A and σ are diagonal) and inwhich no stochastic control methods are applied have been studied empirically in the US eq-uity market by Avellaneda and Lee (2010) and Papanicolaou and Yeo (2017). In a differentdirection, if we consider the particular case of removing the factor model by making Λ = 0,we have the situation in which the returns themselves are globally mean-reverting following amatrix Ornstein-Uhlenbeck process, which has also been studied empirically and analyticallyusing stochastic control techniques in the context of optimal execution in Cartea, Gan, andJaimungal (2018), and in the context of statistical arbitrage in Cartea and Jaimungal (2016) inthe particular case in which A has rank one. Here we have decided to write the factor model in a somewhat unusual differential, continuous-time formin terms of the cumulative residuals and returns because of notational simplicity for this section of the paper.In practice, however, the factor model will be estimated in discrete time, by replacing the differentials by thecorresponding discrete increments (so, for instance, dR t should be replaced by the daily, hourly, etc. asset returns, dF t would be just the corresponding daily, hourly, etc. factors returns, and so forth). In any case we will onlyuse this notation and framework in this section of the paper, and the reader may look at Avellaneda and Lee(2010) for essentially the same continuous/discrete time framework and some estimation techniques. Note alsothat the constant loadings assumption is realistic in short time horizons. See Letteau and Pelger (2018) and Pelger and Xiong (2018) for some new versions of high-dimensional PCAthat might be particularly interesting for this problem. .2 Making the model tractable for stochastic control and imposing market-neutrality Unlike the classical literature on portfolio choice based on stochastic control, choosing as controlvariables the amount of capital that the agent invests in each of the N risky assets of the previousframework might make the optimal allocation problem intractable. Indeed, since we only haveinformation about the dynamics of the residuals (and not directly about the returns like in theclassical framework), these residuals are not independent, and the factors themselves dependon the returns, the classical approach would lead to complicated interdependencies. Moreover,since the investor is executing a statistical arbitrage strategy, we would need to incorporateadditional market neutrality constraints so that the returns of the strategy do not depend onthe model factors, but just on the idiosyncratic component of the model given by the residuals.This would complicate the problem further, and might require numerical optimization methodsas done in Papanicolaou and Yeo (2017) and Boyd et al. (2017).In this paper, on the contrary, we deal with both problems simultaneously and we solve themanalytically by following a new approach. This is based on the following proposition, whichshows that, by using the N risky assets at our disposal, it is actually possible to constructanalytically N market-neutral portfolios whose returns only depend on one coordinate of X ,which greatly simplifies the complexity of the problem and makes it analytically tractable: Proposition 2.1.
Under the previous assumptions, it is possible to construct explicitly N market-neutral portfolios such that investing any real number π it of dollars in the i -th oneat time t yields an instantaneous return of π it dX ti (and hence a combined return of π t · dX t ).Moreover, the total amount of capital invested at time t by doing so is π t · p for an explicitconstant-in-time vector p ∈ R N , which depends quadratically on the factor model loadings.Proof. The mathematical construction of the market-neutral portfolios under the given assump-tions is surprinsingly straightforward and involves just a linear projection. Indeed, (2.1) impliesthat dR ti = (cid:88) j Λ ij dF tj + dX ti , whereas (2.2) yields dF tj = (cid:88) k ˜Λ jk dR tk . Combining the two previous equations we find that, for c ik := (cid:80) j ˜Λ jk Λ ij , dR ti = (cid:88) k (cid:88) j ˜Λ jk Λ ij dR tk + dX ti = (cid:88) k c ik dR tk + dX ti . Thus, if at time t we hold the (explicitly constructible) constant-in-time portfolio given by˜ p i := ( − c i , − c i , . . . , − c i,i − , − c ii , − c i,i +1 , . . . , − c iN )(i.e., we invest − c i dollars in the first asset, − c i dollars in the second one, and so on), weautomatically obtain an instantaneous return of dX ti , which is market neutral and depends onlyon the i th coordinate of the process X t . Further, from the above equations it is also obviousthat for any real number π it , π it ˜ p i will also be market-neutral and yielding a return of π it dX ti ,and the same applies to (cid:80) i π it ˜ p i , which will have a return of (cid:80) i π it dX ti = π t · dX t .6inally, regarding the last part of the statement just observe that the total amount of capitalinvested in the strategy π t = ( π it ) ≤ i ≤ N at time t is simply (cid:88) i ( π it ˜ p i ) · = (cid:88) i π it (˜ p i · ) = π t · p where p := (˜ p i · ) ≤ i ≤ N , which concludes our proof. Remark 2.1.
Note in particular that, if Λ or ˜Λ are sparse matrices, then most of the c ik inthe above construction will be 0, so the investor will be investing in a few number of assetsin each market-neutral portfolio and this could significantly reduce his transaction costs whilerebalancing his positions. In particular, Pelger and Xiong (2018) discuss a way of obtaining thiskind of sparse factor model.The key consequence of the above proposition is that, if we choose as control variables theamount of capital π t that we wish to invest in these N market-neutral portfolios (instead ofdirectly in the original assets) at time t , the dynamics of the problem get remarkably simpler,they only depend separately on the coordinates of X , and we have market-neutrality by con-struction. This solves simultaneously the two problems we discussed before and allows us toconnect stochastic control and the factor model in a simple way, and it is therefore the approachwhich we will adopt in the remainder of this paper.Note also that, under these new control variables, all the information about the factor modeland in particular about its loadings matrix is now encoded in the vector p , which will playan important role in the remaining sections. Moreover, some statements about the strategiesmust be rewritten in terms of it within this new framework. For instance, in the new setting astrategy ( π t ) ≤ t ≤ T is dollar-neutral at t if p · π t = 0, since as we mentioned before p · π t is thetotal capital spent at time t . Under the previous framework, now we formulate rigorously the control problems we will studyin the paper. We suppose that the investor executes the following trading strategy: at eachtime t ∈ [0 , T ], she invests π t dollars in the risky market-neutral portfolios we constructed inProposition 2.1, and she invests her remaining wealth (or borrow money if the remaining wealthis negative) in the risk-free asset with constant interest rate r , so that the resulting strategy isself-financing. Thus, assuming for the moment no market frictions or other constraints (whichwill be both considered in section 4), the evolution of her wealth is given by the equation dW t = π t · dX t + ( W t − π t · p ) rdt (2.3)and she aims to choose π t to maximize the expected utility of her terminal wealth (that is, u ( W T ) for a given utility function u ).Supposing further that she trades continuously in time, this means that mathematically shemust solve the high-dimensional non-linear stochastic optimization problem given by H ( t, x, w ) = sup π ∈A [ t,T ] E t,x,w [ u ( W T )] (2.4)subject to dW t = (cid:0) π (cid:48) t A ( µ − X t ) + ( W t − π (cid:48) t p ) r (cid:1) dt + π (cid:48) t σdB t X t = A ( µ − X t ) dt + σdB t , where the admissible set A [ t,T ] is the set of all the F s -predictable and adapted processes( π s ) s ∈ [ t,T ] in R N with the minimal technical restrictions that E [ (cid:82) Tt || π s || ds ] < ∞ (so Ito’sformula may be applied and doubling strategies are excluded) and the above SDEs have anunique (strong) solution, and (cid:48) indicates transposition.Finally, the associated dynamic programming equation of the problem is non-linear and ( N +2)-dimensional, and is given by0 = ∂ t H + ( µ − x ) (cid:48) A (cid:48) ∇ x H + 12 Tr( σσ (cid:48) ∇ xx H ) +sup π (cid:18)(cid:0) π (cid:48) A ( µ − x ) + ( w − π (cid:48) p ) r (cid:1) ∂ w H + 12 π (cid:48) σσ (cid:48) π∂ ww H + π (cid:48) σσ (cid:48) ∇ xw H (cid:19) (2.5)with terminal condition H ( T, x, w ) = u ( w ).The problem is therefore formally related to the classical Merton framework, but instead ofa geometric Brownian motion there is a multidimensional Ornstein-Uhlenbeck process whichmakes it impossible to combine the dynamics of W and X into a single equation and to getrid of the N -dimensional state variable x . Moreover, unlike the previous studies on extensionsof the Merton problem with an Ornstein-Uhlenbeck process discussed in section 1, in (2.4) and(2.5) the mean-reverting behavior of the underlying stochastic process arises in the context ofstatistical arbitrage and from the residuals of a factor model (which is encoded in the vector p of the equations above and which will play an important role in the following sections), andwe consider the general case of an arbitrary number of assets with a market neutral restriction.Furthermore, the model will be extended in section 4 to incorporate other important featuresof statistical arbitrage strategies, like dollar neutrality restrictions and transaction costs, andwe will analyze the impact of the factor model on these extensions.Quite surprisingly, the previous problems admit interpretable closed-form solutions – which iscomputationally useful in this high-dimensional setting, and which allows us to understand theinfluence of the model parameters and especially of the factor model – in the cases in which theutility is exponential or of a Markowitz-inspired mean-variance type (but not for other usualchoices of utility functions, like the HARA family). This is what we will show in the followingtwo sections, first for the simple setup of (2.4) and (2.5) in section 3, and then extending themodel in section 4 to incorporate soft constraints on the investor’s portfolio and quadratictransaction costs. In this section we therefore present the closed-form, optimal strategies for the problem given by(2.4) and (2.5) in the cases in which the utility is exponential or of a mean-variance type, dis-cussing the former in the first subsection and the latter in the second one. While the techniquesthat we use are classical, in both cases the explicit solutions allow us to gain insight on the newframework of statistical arbitrage with a factor model and will be the basis for the extensionsof section 4. 8 .1 The exponential utility case
In this first setting, the complete, explicit description of the optimal strategy is given by thefollowing main theorem (see Cartea and Jaimungal (2016) and Lintilhac and Tourin (2016) forrelated results with an exponential utility):
Theorem 3.1.
Under an exponential utility (so u ( w ) = − e − γw for some γ > ) and thetechnical condition described in our verification theorem (Proposition 3.2 below), the optimalportfolio to have at time t according to (2.4) is explicitly computable and given by π ∗ t = ( σσ (cid:48) ) − A ( µ − X t ) − prγe r ( T − t ) + A (cid:48) ( σσ (cid:48) ) − γe r ( T − t ) (cid:18) ( A ( µ − X t ) − pr )( T − t ) − Apr ( T − t ) (cid:19) . The result follows from the following two propositions, whose proof is given in Appendix A.1using classical stochastic control techniques:
Proposition 3.1 (Solving the PDE) . The value function H associated to (2.4) and (2.5) when u ( w ) = − e − γw is explicitly computable and admits the probabilistic representation H ( t, x, w ) = − exp( − γ ( we r ( T − t ) + h ( t, x ))) where h ( t, x ) = E ∗ t,x (cid:20)(cid:90) Tt γ ( A ( µ − Y s ) − pr ) (cid:48) ( σσ (cid:48) ) − ( A ( µ − Y s ) − pr ) ds (cid:21) and dY t = rpdt + σdB ∗ t for a new Brownian motion B ∗ under a new probability law P ∗ . Theassociated optimal control in feedback form is then π ∗ = − ( σσ (cid:48) ) − D H∂ ww H (3.1) where D H = ( A ( µ − x ) − pr ) ∂ w H + σσ (cid:48) ∇ xw H . Proposition 3.2 (Verification) . The strategy given in Theorem 3.1. is indeed optimal if ≤ s ≤ T || Λ ( s ) || < and
32 max ≤ s ≤ T || Λ ( s ) || < , where Λ ( s ) and Λ ( s ) are the diagonal matrices containing, respectively, the eigenvalues of Ω / ( C + C (cid:48) )Ω / ( s ) and Ω / C C (cid:48) Ω / ( s ) , for C ( s ) = A (cid:48) ( σσ (cid:48) ) − A ( I N + A ( T − s )) , C ( s ) = A (cid:48) ( σσ (cid:48) ) − ( I N + A ( T − s )) σ Ω( s ) = (cid:90) s e − A ( s − u ) σσ (cid:48) e − A (cid:48) ( s − u ) du. Besides being a closed-form strategy which is easily implementable in our high-dimensionalsetting, the above optimal portfolio is also interpretable. Indeed, the first term of the optimalpolicy is Merton-like in that it represents the drift of the underlying process (which here isstochastic unlike in the classical geometric Brownian motion) minus the adjusted risk-free rate(which here depends on the loadings of the factor model via p ), and divided by a measureof the volatility (which is given by σσ (cid:48) , the instantaneous quadratic covariation of X ) and theArrow-Pratt coefficient of absolute risk-aversion of the value function with respect to the wealth w (i.e., − ∂ ww H/∂ w H ), which is the product γe r ( T − t ) , where γ is the risk aversion parameter ofour utility function and the factor e r ( T − t ) measures the gains from interest between t and T .9n the other hand, the second summand is a non-myopic correction term which again dependslinearly on the drift of X , and whose effect vanishes when we approach the terminal time T .Moreover, note that, while the first term does not depend explicitly on the terminal time T ,this correction term does, reflecting the fact that, since there are non-zero interest rates andmoreover the behavior of the residuals is oscillating, the investor must keep in mind the finalhorizon to decide if she bets on the mean-reversion cycle before that time. Finally, observe that,quite naturally, in both terms as the risk-aversion parameter γ , the instantaneous volatility σσ (cid:48) ,or the interest rate r increase, the optimal portfolio vector π ∗ t gets closer to 0, implying thatthe investor will simply invest most of her wealth in the riskless asset.The above strategy also seems intuitive within our particular framework of statistical arbitragewith a factor model, and sheds further light on the problem. Indeed, note that the currentstate of the residual process X t only appears in the strategy through the terms in A ( µ − X t ),which essentially tells us to invest more in the risky assets the further their residuals are fromtheir historical mean µ and in a way proportional to the historical mean reversion speed givenby A , like in classical pairs trading. Moreover, all the remaining terms depend jointly on thefactor model and the interest rate through the term pr , which reflects the cost of the leverageassociated to imposing market-neutrality through the loadings of the factor model. In particular,note that, the bigger the loadings of the factor model are (and hence the bigger p is), the morewe will need to invest to achieve market neutrality (again like in pairs trading with a big beta)and the bigger our leverage will be, and this will affect the optimal strategy depending on theinterest rate r .Finally, regarding the technical optimality conditions, intuitively they arise from the fact that H ( t, X t , W ∗ t ), the value function evaluated at the wealth process W ∗ t corresponding to theoptimal strategy, may blow up because of the exponential function coming from the exponentialutility. In particular, since W ∗ t ends up being an Ito process depending quadratically on X t and X t is Gaussian, the term exp( − γW ∗ t e r ( T − t ) ) is related to the moment generating functionof a chi-squared distribution, which blows up far away from 0. Thus, these technical conditionsare just ensuring that the corresponding functions are integrable. Interestingly, this does notdepend on the risk-aversion parameter γ , the interest rate r , or the factor model used (capturedby p ), but just on the parameters of X and the terminal time T . In the second, Markowitz-inspired mean-variance framework, the investor tries to maximize herexpected terminal wealth but at the same time she continuously penalizes at each instant the in-stantaneous variance (i.e., the volatility) of her wealth process according to a volatility-aversionfunction γ ( t ). The optimal strategy in this case is again available in closed form and inter-pretable and, quite remarkably, for an appropriate choice of this volatility-aversion function, weobtain exactly the same optimal policy as in the exponential case but without the correctionterm. This is shown in the following theorem, whose proof is given in Appendix A.2. usingclassical control techniques: Theorem 3.2. If γ ( t ) is continuous and positive on [0 , T ] , the problem in (2.4) with the fol-lowing mean-variance objective function H ( t, x, w ) = sup π ∈A t,T E t,x,w (cid:20) W T − (cid:90) Tt γ ( s )2 ddτ Var s ( W τ ) | τ = s ds (cid:21) as explicit optimal portfolio at t given by π ∗ t = ( γ ( t ) σσ (cid:48) ) − ( A ( µ − X t ) − pr ) e r ( T − t ) . In particular, for γ ( t ) = γe r ( T − t ) , the above optimal policy is the same as the first term of thecorresponding one in Theorem 3.1. This unexpected connection between the mean-variance and the exponential utility cases mayin fact be explained heuristically. Indeed, supposing that r = 0 for the sake of simplicity andconsidering a second order approximation of the exponential we get − exp( − γW T ) ≈ − γW T − γ W T / , and for maximization purposes when conditioned on t this behaves essentiallyas E t,x,w [ γW T − γ Var t ( W T ) / γ we obtain exactly the previous objective function,showing moreover that the correction term of Theorem 3.1 that does not appear in this case isheuristically associated to the moments of order higher than 2 of the exponential utility.Regarding the interpretation of the mean-variance strategy within our context of statisticalarbitrage and its connection with the exponential-utility arbitrageur, there are two importantremarks.First, as we mentioned, the optimal strategy here is the same as the myopic part of the expo-nential case modulo the value of γ ( t ). In particular, this means that, unlike the exponentialarbitrageur, the mean-variance arbitrageur will not take into account the expected number ofmean-reversion cycles until the terminal time T . Moreover, for a non-zero interest rate and aconstant volatility aversion γ ( t ), the mean-variance arbitrageur is bolder than the correspondingexponential investor with the same γ , since she will invest significantly more money (quantita-tively, by a factor of e r ( T − t ) ) in going long or short, taking more aggresive positions the higherthe interest rate is and the sooner it is with respect to the terminal date.Second, the optimal strategy has two components like in section 3.1: one term in A ( µ − X t )which measures how far the residuals are from their historical mean and how fast they willmean-revert (like in classical pairs trading), and a second term in pr linked to the factor model,which measures the cost of the leverage associated to imposing market neutrality. In particular,note that, the bigger the loadings of the factor model are (and hence the bigger p is), the moreaggressive the positions will be and the more leverage the investor will have if r (cid:54) = 0. In this final theoretical section of the paper, we complete the picture described in the previoustwo sections by considering two important and new extensions within the context of statisticalarbitrage with a factor model. In the first subsection, we show how to incorporate in theabove strategies soft constraints frequently imposed by arbitrageurs with the example of dollar-neutrality, whereas in the second one we introduce market frictions in the form of quadratictransaction costs. In both cases, we obtain again closed-form analytic solutions which areinterpretable, convenient from a computational perspective in our high-dimensional setting, andwhich shed further light on the influence of the factor model and its connection with marketneutrality. 11 .1 Incorporating soft constraints on the admissible portfolios
While imposing restrictions on the portfolios by introducing hard constraints directly on theadmissible set A t,T leads in general to problems that must be solved numerically (and hencepotentially unfeasible in a high-dimensional setting), it is still possible to impose many additionalsoft constraints in the two frameworks of section 3 without increasing significantly the difficultyof the problems, by just adding a carefully chosen penalty term to the corresponding objectivefunction.As an illustration of this, we give in the next corollary the corresponding optimal strategies whena dollar-neutrality restriction is softly enforced. To do so, recall that, within the framework ofsection 2 that imposed market-neutrality within the factor model, a strategy π t is dollar neutralif p · π t = 0, which means that the total amount of capital invested at time t is 0. Hence, wecan softly enforce dollar neutrality by replacing the wealth process of Theorems 3.1 and 3.2 bythe penalized wealth process defined by d ˜ W t := dW t − α ( t )( π t · p ) / dt for a certain generaltime-dependent penalty function α ( t ). This penalizes non dollar-neutrality (i.e., π t · p (cid:54) = 0) ateach time and is quadratic to simplify the optimization process.The proof follows exactly the same lines as in the previous two cases and is obtained from themby small modifications, so we will omit it for the sake of brevity. Corollary 4.1.
Suppose that dollar neutrality is softly enforced by replacing the wealth processof Theorems 3.1 and 3.2 by the penalized wealth process defined by d ˜ W t := dW t − α ( t )( π t · p ) / dt .Then(1) The problem with mean-variance utility has optimal portfolio at t given by π ∗ t = ( γ ( t ) σσ (cid:48) + α ( t ) pp (cid:48) ) − ( A ( µ − X t ) − pr ) e r ( T − t ) . (2) The problem with exponential utility has optimal portfolio at t given by π ∗ t = ( γe r ( T − t ) σσ (cid:48) + α ( t ) pp (cid:48) ) − (cid:0) A ( µ − X t ) − pr − γσσ (cid:48) ( b ( t ) + c ( t ) X t ) (cid:1) . where c ( t ) is an N × N symmetric matrix and b ( t ) is an N -dimensional vector, vanishingwhen t → T , and with coordinates depending on A, σ, rp, γ and α ( t ) . In particular, c ( t ) isgiven by the solution of the matrix Riccati ODE ∂ t c − A (cid:48) c − cA − γcσσ (cid:48) c + e r ( T − t ) ( A + γσσ (cid:48) c ) (cid:48) M ( t )( A + γσσ (cid:48) c ) and b ( t ) is the solution of the linear system of ODEs ∂ t b − A (cid:48) b + cAµ − e r ( T − t ) ( A + γσσ (cid:48) c ) M ( t )( Aµ − pr − γσσ (cid:48) b ) − γcσσ (cid:48) b, both with terminal conditions b ( T ) = c ( T ) = 0 and where M ( t ) = ( γσσ (cid:48) e r ( T − t ) + α ( t ) pp (cid:48) ) − . The resulting optimal policies have therefore the same structure as the two previous strategiesof section 3, but now the additional term α ( t ) pp (cid:48) has been introduced in the inverse to enforcethe dollar-neutrality condition. This again depends on the factor model via p and is related tohow extreme the capital positions will be because of the market-neutrality restriction, whichdepends directly on the loadings matrix and hence on p . Note in particular that, the bigger theloadings of the factor model are, the bigger α ( t ) pp (cid:48) will be and hence the bigger the impact ofthe dollar neutrality restriction will be. 12 .2 Incorporating quadratic transaction costs In this subsection, we finally extend our model to incorporate market frictions in the formof transaction costs, which play a crucial role when executing statistical arbitrage strategies.We consider in particular quadratic transaction costs, which are in general a measure of priceimpact or illiquidity and which make the model anaytically tractable.To do so, rather than looking directly at the amount of capital π t invested in the risky assetsat time t as the control variables, we consider instead the trading intensity I t at which theseinvestements will be made at time t , which is therefore given by dπ t = I t dt . We can now adaptto our setting the model for temporary transaction costs introduced in Garleanu and Pedersen(2016), who posit (providing a market microstructural justification and referring to empiricalresearch) that these transaction costs at time t may be represented quadratically as I (cid:48) t CI t fora certain symmetric positive-definite matrix C , which essentially comes from the assumptionthat the price impact of the investor’s actions is linear on its trading intensity I t .Under this framework, we can then rewrite the performance criteria of Theorem 3.2 (for thesake of brevity, we will just deal with the mean-variance case) by incorporating the adverseeffect caused by these transaction costs on the investor’s wealth as a running penalty, obtainingthe stochastic optimization problem given by H ( t, x, w, π ) = sup I ∈A ∗ E t,x,w,π (cid:20) W T − (cid:90) Tt γ ( s )2 ddτ Var s ( W τ ) | τ = s ds − (cid:90) Tt I (cid:48) s CI s ds (cid:21) (4.1)in which as we mentioned the new control variable is I ; t, x, w, π are now state variables;and A ∗ is the set of all F -adapted predictable processes I t such that the corresponding SDEshave a unique (strong) solution for any initial data and both I t and the resulting π t given by dπ t = I t dt are again in L (Ω × [0 , T ]). Thus, the investor aims to maximize her terminal wealth,but penalizing at each instant both for the risk of her strategy (measured by the volatily ofher wealth process) and for the price impact caused by her actions (reflected in the quadratictransaction costs).In this new setting, it is again possible to find explicitly the optimal strategy that the investorshould follow, which is described in detail in the next theorem: Theorem 4.1. If γ ( t ) ≥ (i.e., non-negative volatility aversion) and is continuous, the optimalstrategy in the above problem “tracks” a moving aim portfolio Aim( t, X t ) with a tracking speed of Rate( t ) , according to the following ODE describing the evolution of the optimal trading intensity I t = dπ t /dt I t = Aim( t, X t ) + Rate( t ) π t , where Rate( t ) is a N × N negative-definite matrix tending to 0 when t → T , and Aim( t, X t ) admits the probabilistic representation Aim( t, x ) = (cid:90) Tt f ( s ) E t,x [Frictionless( s )] ds where Frictionless( s ) is the optimal portfolio at time s in the frictionless case of section 3.2.and f ( s ) is a certain averaging function given in Proposition 4.3 below. The assumption that C is symmetric is without loss of generality, since if the transaction costs are givenby I (cid:48) t ˜ CI t for a non-symmetric ˜ C , then one can see that by considering the symmetric part of ˜ C (given by C := ( ˜ C + ˜ C (cid:48) ) /
2) we have that I (cid:48) t ˜ CI t = I (cid:48) t CI t . and given by the solution of a matrix Riccati ODE specified in the Porposition 4.2 below. urthermore, the optimal portfolio is then π ∗ s = π t + (cid:90) st : exp (cid:18)(cid:90) su Rate( v ) dv (cid:19) : Aim( u, X u ) du, where the notation : exp( (cid:82) tu · ds ) : represents the time-ordered exponential . Remark 4.1.
If in particular the investor has constant volatility aversion (so γ ( t ) = γ ), thematrix Riccati ODE is explicitly solvable andRate( t ) = C − / D tanh( D ( t − T )) C / for D := ( γC − / σσ (cid:48) C − / ) / . Moreover, if the transaction costs are proportional to thevolatility (i.e., C = λσσ (cid:48) for λ >
0, see Garleanu and Pedersen (2013, 2016) for a marketmicrostructural justification) then this rate is indeed a scalar given by (cid:113) γλ tanh (cid:16)(cid:113) γλ ( t − T ) (cid:17) and : exp (cid:0)(cid:82) su Rate( v ) dv (cid:1) := cosh (cid:16)(cid:113) γλ ( s − T ) (cid:17) / cosh (cid:16)(cid:113) γλ ( u − T ) (cid:17) .The result follows from the following sequence of three propositions, which are proved in Ap-pendix A.3: Proposition 4.1 (Conjectured solution) . The solution of the HJB equation associated to theproblem is H ( t, x, w, π ) = e r ( T − t ) w + π (cid:48) a ( t ) π + π (cid:48) b ( t, x )+ d ( t, x ) if there exist a N × N symmetricmatrix a ( t ) , a N -dimensional vector b ( t, x ) and a scalar function d ( t, x ) satisfying(1) The matrix Riccati ODE ∂ t a − γ ( t ) σσ (cid:48) + aC − a = 0 (4.2) with terminal condition a ( T ) = 0 .(2) The vector-valued and the scalar linear parabolic PDEs ( ∂ t + L X ) b + e r ( T − t ) ( A ( µ − x ) − rp ) + a (cid:48) C − b = 0 (4.3)( ∂ t + L X ) d + 12 b (cid:48) C − b = 0 (4.4) with terminal conditions b ( T, x ) = d ( T, x ) = 0 and where L X is the infinitesimal generatorof X , acting coordinatewise.The hypothesized optimal trading intensity at ( t, x, w, π ) is then I ∗ = C − ( a ( t ) π + b ( t, x )) . Proposition 4.2 (Existence of solutions) . .(1) If γ ( t ) ≥ (non-negative volatility-aversion) and is continuous, then the Riccati equation(4.2) has a symmetric, bounded and negative definite solution on all [0 , T ] . In particular,for γ ( t ) = γ , the solution is a ( t ) = C / D tanh( D ( t − T )) C / for D := ( γC − / σσ (cid:48) C − / ) / . Recall that the time-ordered exponential of a time-dependent matrix A ( s ) is defined as : exp( (cid:82) tu A ( s ) ds ) :=lim ||P||↓ (cid:81) n P i =1 exp( A ( t i )∆ t i ), where P := { u = t , t , . . . , t n = t } is a partition of [ u, t ], ∆ t i := t i − t i − , and theproduct is ordered increasingly in time. If A ( s ) is a scalar, then obviously : exp( (cid:82) tu A ( s ) ds ) := exp( (cid:82) tu A ( s ) ds )
2) Moreover, under this condition the parabolic PDEs (4.3) and (4.4) have an unique solutionsatisfying a polynomial growth condition in x , and this solution admits the probabilisticrepresentation b ( t, x ) = E t,x (cid:20)(cid:90) Tt : exp (cid:18)(cid:90) st a (cid:48) ( u ) C − du (cid:19) : e r ( T − s ) ( A ( µ − X s ) − rp ) ds (cid:21) (4.5) d ( t, x ) = 12 E t,x (cid:20)(cid:90) Tt b ( t, X s ) (cid:48) C − b ( t, X s ) ds (cid:21) (4.6) Furthermore, b has linear growth in x u and d has quadratic growth in x , both uniformlyin t . Proposition 4.3 (Verification) . Under the assumptions of the previous proposition, the tradingintensity given in Theorem 4.1 is indeed optimal with the choices
Aim( t, x ) = C − b ( t, x ) , Rate( t ) = C − a ( t ) , f ( s ) = C − : exp (cid:18)(cid:90) st Rate( u ) (cid:48) du (cid:19) : γ ( s ) σσ (cid:48) . The interpretation of the above strategy (which is again explicit and hence easily implementablein practice) is again intuitive and complementary to the infinite-horizon model of Garleanu andPedersen: in the presence of temporary quadratic transactions costs, the investor trades with acertain decreasing rate Rate( t ) towards an aim portfolio Aim( t, X t ) depending on the time andthe mean-reversion state of the signals X t . This aim portfolio is given by a weighted average ofthe future optimal strategies in the frictionless case, reflecting the fact that now trading is notfree and thus to enter a trade the investor must weight the future outcomes derived from thestrategy. Moreover, as shown in the above remark, the trading rate is bounded by 1 because ofthe properties of tanh, depends on t unlike the infinite-horizon case, and is naturally decreasingin the transaction cost parameter λ (or in general in C ) and increasing in the volatility aversionparameter γ .Finally, regarding the influence of the factor model and the imposition of market neutralityin this new setting, note that Rate( t ) is insensitive to it (since it only depends on the riskaversion parameter γ , the volatility of the residual process σσ (cid:48) , and the transaction cost matrix C ) and in Aim( t, x ) it only appears through the term E t,x [Frictionless( s )] and hence only whenconsidering the future optimal strategies in the frictionless case, which has been described insection 3. Likewise, the residual process X s only affects the strategy through this same termand hence, as seen in section 3 when studying these frictionless cases, only through the distancebetween this residual and its historical mean, like in classical pairs trading. We finally conclude the paper by providing some high-dimensional numerical simulations thatgive new insights about the behavior of the previously discussed strategies and their sensitivityto the different parameters, emphasizing in a separate simulation the role of the factor modeland its connection with market-neutrality. To this end, we first simulate a large number ofpaths of X in a high-dimensional setting (in particular, we choose N = 100) by using exactMonte Carlo sampling along a discrete time grid, and we then execute the previous strategies forsome parametric choices of X and some values of p to compute sample paths of π t and W t and15istograms of the terminal Profit & Loss (P&L). We have therefore opted to defer systematic(out-of-sample) experiments with real data to a separate paper, since examining carefully thedelicate issues of asset selection, rebalance frequency, construction of the factor models andobtention of X , high-dimensional parameter estimation and updating, risk control, etc. thatthe problem requires would be impossible to consider here without prohibitively extending thelength of the paper.During all this section, we therefore fix the following parameters for our model: N = 100 , µ = 0 , X = µ,A is diagonal with entries drawn i.i.d from a normal distribution of mean 0.5 and standarddeviation 0 .
1, and the coefficients of σ are drawn i.i.d from a uniform distribution in [ − . , . , . p = for the first simulations, and we will also perturb it later to study differentfactor model regimes and the impact of imposing market neutrality. We also fix a finite horizonof T = 20 and a temporal grid 0 = t < t < . . . t L = T obtained by discretizing [0 , T ] withconstant ∆ t = T /L = 20 /
400 = 0 . X are correlated and mean-revert with similar speeds (given by the eigenvalues of A ) to anequilibrium of 0, describing an average number of approximately 10 oscillation cycles of upsand downs before the terminal time (given by the product of T and the average mean-reversionspeed). The choice of p = arises when the asset returns themselves are mean-reverting andmay be modeled directly by X so we can take Λ = 0 in our factor model, while the perturbationsof p will imply departing from this assumption to factor models with heavier loadings, in whichimposing market neutrality leads to more leveraged positions. As an illustration, the readermay look at sample paths of the first four coordinates of such a process X in Figure 1 below.Figure 1: Sample paths of the first four coordinates of X in [0 , T ]We then sample M = 1 ,
000 paths of X on this grid exactly with standard Monte Carlo tech-niques by using the fact that, since X t +∆ t = e − A ∆ t X t + ( I − e − A ∆ t ) µ + (cid:90) t +∆ tt e − A (∆ t + t − s ) σdB s , t +∆ t | X t ∼ N ( µ ( X t , ∆ t ) , Σ(∆ t )), where µ ( X t , ∆ t ) = e − A ∆ t X t + ( I − e − A ∆ t ) µ, Σ(∆ t ) = (cid:90) ∆ t e − A (∆ t − s ) σσ (cid:48) e − A (cid:48) (∆ t − s ) ds, and execute the following strategies at the corresponding times t l ’s, with W = π = 0 and π t constant between consecutive times:(1) The exponential utility strategy of Theorem 3.1 with γ ( t ) = 1 , , , r = 0 , γ ( t ) = 1 , , , α ( t ) = 0 , ,
50 and r = 2%.(3) The mean-variance utility strategy with quadratic transaction costs of Theorem 4.1 with γ ( t ) = 1 , , , r = 2%, and C = λσσ (cid:48) for λ = 0 . , . , p , which encapsulates allthe factor model information and which we perturb to simulate the effect of going away fromthe case where the returns themselves are mean-reverting (which corresponds to the previouscase p = ) and of having progressively more leveraged (and more extreme) market-neutralportfolios:(4) The three strategies above with γ ( t ) = 1 , α ( t ) = 0 and r = 2% (and λ = 1 for the thirdstrategy) for p = + (cid:15) a and a = 1 , , ,
8, where (cid:15) a is a N -dimensional vector whosecomponents are drawn i.i.d. from a uniform distribution in [ − a, a ].We finally present the simulated path of a sample wealth process ( W t ) t ∈ [0 ,T ] , the simulatedpath of the first coordinate of a sample allocation process ( π t ) t ∈ [0 ,T ] , and the histogram for theterminal wealth W T for each of the above cases in the following four subsections, along with afinal analysis: We have just simulated some simple cases of the previously discussed strategies for space limitation reasons,but of course it would also be possible to include further constraints (like a leverage restriction, for instance) ortime-varying hyperparameters with no additional effort, by just using the previously derived formulae. It wouldbe interesting as well to execute the strategies with some perturbations of the real parameters to simulate possiblemicrostructural noise and imperfect estimation. .1 Simulations of the exponential-utility strategy (a) Sample paths of W t when r = 0 for γ = 1 , , , W t when r = 0 .
02 for γ = 1 , , , π t when r = 0 for γ = 1 , , , π t when r = 0 .
02 for γ = 1 , , , W T when r = 0 for γ =1 , , , W T when r = 0 .
02 for γ = 1 , , , Figure 2: Results for the exponential utility18 .2 Simulations of the mean-variance strategy with dollar neutrality (a) Sample paths of W t for γ =1 , , , α = 0 (b) Sample paths of W t for γ =1 , , , α = 20 (c) Sample paths of W t for γ =1 , , , α = 50(d) Sample paths of π t for γ =1 , , , α = 0 (e) Sample paths of π t for γ =1 , , , α = 20 (f) Sample paths of π t for γ =1 , , , α = 50(g) Histograms of W T for γ =1 , , , α = 0 (h) Histograms of W T for γ =1 , , , α = 20 (i) Histograms of W T for γ =1 , , , α = 50 Figure 3: Results for the mean-variance utility when r = 0 .
02 with different dollar-neutralityrestrictions 19 .3 Simulations of the mean-variance strategy with quadratic transactioncosts (a) Sample paths of W t for γ =1 , , , λ = 0 . W t for γ =0 , , , λ = 0 . W t for γ =1 , , , λ = 1(d) Sample paths of π t for γ =1 , , , λ = 0 . π t for γ =1 , , , λ = 0 . π t for γ =1 , , , λ = 1(g) Histograms of W T for γ =1 , , , λ = 0 . W T for γ =1 , , , λ = 0 . W T for γ =1 , , , λ = 1 Figure 4: Results for the mean-variance utility when r = 0 .
02 with different quadratic transac-tion costs C = λσσ (cid:48) .4 Simulations for different factor model and market-neutrality regimes (a) Sample paths of W t for theexponential utility (b) Sample paths of W t for themean-variance utility (c) Sample paths of W t forthe mean-variance utility withtransaction costs(d) Sample paths of π t for theexponential utility (e) Sample paths of π t for themean-variance utility (f) Sample paths of π t forthe mean-variance utility withtransaction costs(g) Histograms of W T for the ex-ponential utility (h) Histograms of W T for themean-variance utility (i) Histograms of W T for themean-variance utility withtransaction costs Figure 5: Results for the three strategies above with different p ’s, when r = 0 . , α = 0 , γ =1 , λ = 1 We now present our main conclusions after observing the previous plots, analyzing the behaviorof the histograms of the final wealth W T , the sample paths of the wealth process ( W t ) t ∈ [0 ,T ] , andthe sample paths of the positions ( π t ) t ∈ [0 ,T ] , with a final subsubsection discussing the effect ofimposing market-neutrality under different p ’s.21 .5.1 Histograms of the final wealth Looking first at the above histograms (Figures 3-4 (g)-(i), and 2 (e)-(f)), we see that, forour parametric choice and our setting in which X is effectively a multidimensional Ornstein-Uhlenbeck process with known parameters,(1) The most profitable strategy is the one derived from the exponential utility (Figure 2, (e)and (f)) with the lowest risk-aversion parameter γ , even in the most adverse scenarios ofthe histogram and both with and without zero interest rates. Moreover, even for biggervalues of γ this strategy significantly performs better under any regime of r and α thanthe mean-variance strategy (Figure 3, (g)-(i)).(2) We observe the following outcomes when changing one of the parameters for each of thethree strategies (Figures 3-4 (g)-(i), and 2 (e)-(f)): • Increasing the value of the risk-aversion parameter γ produces a concentration of thedensity of W T around smaller values, i.e., the expected wealth decreases and so doesthe dispersion around it. • Increasing the dollar-neutrality penalty α has this same negative effect, but makeslittle difference unless the increments in α are considerable. • Increasing the value of the interest-rate r has an overall positive effect, which is morepronounced in the mean-variance case (since as we mentioned at the end of section2.2 the investor is then bolder than the exponential agent). • Increasing the transaction cost parameter λ decreases the expected terminal wealth,but it also skews its distribution producing a considerable right-tail (whereas all theother distributions are essentially symmetric).These outcomes have a natural interpretation: since the model is perfectly specified andthe parameters are known, the derived strategies will always produce benefits by con-struction, and they will be bigger the fewer additional constraints we impose (such asrisk-aversion, dollar-neutrality, and transaction costs) and the more we can take advan-tage of previous success (by increasing r ). This situation, however, might not apply underparameter misspecification, where the additional constrains would help the investor miti-gate the model risk. W t Examining next the sample paths for the particular simulation which is plotted (Figures 3,4(a)-(c), and 2 (a)-(b)), we can observe exactly the same patterns as discussed in the previousparagraph when modifying the parameters γ, α , r and γ . There are, however, two new andinteresting remarks:(1) In the three strategies, after an initial period of ups and downs and similarity between thedifferent strategies, there is a tendency towards stabilization because of the asymptoticproperties of the Ornstein-Uhlenbeck process, and of differentiation depending on theparametric choices.(2) This phenomenon is especially pronounced with the exponential utility and with biggervalues of r (Figure 2 (a)-(b)) since it takes more aggressive positions, reflecting the factthat sometimes the agent will invest more money than what she will make at that moment22and sometimes even having temporary negative wealth and borrowing aggressively) tocontinue executing the strategy. π t Considering now the plots of the sample paths of the positions π t (Figures 3,4 (d)-(f), and 2(c)-(d)), we similarly notice that(1) The positions become more extreme when decreasing γ , α and λ (i.e., the risk-aversionparameter, the non-dollar-neutrality penalty and the transaction cost parameter) andwhen increasing r (the interest rate). The greatest overall impact is produced by γ and λ and then r , especially in the mean-variance case for the same reasons as before.(2) The exponential utility strategy takes much more extreme positions than the mean-variance strategies, which in this idealized setting of perfect estimation partially explainswhy the exponential agent obtains a greater wealth at the terminal time.(3) The cycles in the positions π t match the oscillations of X t depicted in Figure 1, asdescribed theoretically in the corresponding equations. Finally, looking separately at the effect of imposing market neutrality under various factormodel regimes depending on p (which, as we mentioned, depends quadratically on the factormodel loadings), we observe the following (Figure 5):(1) As the parameter p gets bigger, the market neutral portfolios of section 2 become moreextreme and the adopted positions π t also become more aggressive, especially in theexponential utility case (Figure 5, (d)-(f)).(2) Since the strategy is more aggressive but we have perfect calibration and estimation,with bigger p the mean-variance and especially the exponential strategy become moreprofitable. However, the wealth process also has more ups and downs (Figure 6, (a)-(c)),the standard deviation of the terminal wealth increases considerably (Figure 6, (g)-(h)),especially in the mean-variance case, and with the biggest p there are also heavy losseswhen transaction costs are incorporated (Figure 6, (c),(i)). The strategies are thereforeriskier, but a relatively large value of p is needed to appreciate its effect.(3) Lastly, note that the influence of p on the strategies also depends most of the time onthe value of r , since they normally appear combined as a factor of rp in the equationsdescribing the strategies. In particular, when r = 0 there is no theoretical effect associatedto p (apart from possible model risk and high leverage in a real-world setting) unless thedollar-neutrality parameter α ( t ) (cid:54) = 0. In this paper we have aimed to provide a systematic study of high-dimensional statistical arbi-trage combining both stochastic control and factor models. To this end, we have first proposeda general framework based on a statistically-constructed factor model, and then shown how23o obtain analytically explicit market-neutral portfolios and rephrase our problem in terms ofthem to make it tractable and get market neutrality by construction. Using this insight, wehave then been able to study the question of optimizing the expected utility of the investor’sterminal wealth in continuous time under both an exponential and a mean-variance objective.In both cases, we have obtained explicit closed-form solutions ready for numerical implemen-tation, analyzed the corresponding strategies from the perspective of statistical arbitrage andthe underlying factor model, and discussed extensions involving the addition of soft constraintson the admissible portfolios (like dollar-neutrality) and the presence of temporary quadratictransaction costs. Finally, we have run some high-dimensional Monte Carlo simulations to ex-plore the behavior of the previous strategies, and analyzed their qualitative aspects and theirsensitivity to the relevant parameters and the underlying factor model.There are four natural extensions to our work, on which we are conducting research at themoment and which we intend to publish in separate papers. First, one could investigate amore realistic version of the problem in which, rather than in continuous time, the investormay only trade more realistically at an increasing sequence of optimally chosen stopping times,generalizing in multiple directions the literature initiated by the influential work of Leung and Li(2015) and developing robust and efficient numerical methods. Second, it would be interestingto study more realistic modelizations of market frictions, illiquidity, and transaction costs, or todevelop a model considering issues of parameter misspecification. Third, on a more empiricalside and as we mentioned at the start of the section 5, one should consider in this settingthe problems of construction of the factor models, high-dimensional parameter estimation, andrisk control, along with (out-of-sample) experiments with real market data under the strategiesdeveloped in this paper. Fourth and finally, one could study a more data-driven version of theproblem, where the fixed stochastic model is replaced by new tools from reinforcement learning.
The author would like to thank George Papanicolaou for suggesting the topic of the previousresearch and for many insightful discussions about the problem and the presentation of theresults, and the editor and an anonymous reviewer for their very helpful suggestions to improvethe quality of the paper. 24
Appendix. Proofs
A.1 Proof of Theorem 3.1
Proof of Proposition 3.1:
The dynamic programming principle suggests that the value function H should satisfy thedynamic programming equation (2.5) with terminal condition H ( T, x, w ) = − e − γw , and theoptimal control may then be found in feedback form by looking at the first order condition ofthe term inside the supremum since the corresponding function is quadratic and concave in π (if ∂ ww H <
0, i.e., if there is risk aversion). The first order condition gives that0 = σσ (cid:48) ∂ ww Hπ + ( A ( µ − x ) − pr ) ∂ w H + σσ (cid:48) ∇ xw H, and solving for π we find the control given in the proposition’s statement. Putting it back into(2.5) we get the following non-linear and ( N + 2)-dimensional PDE0 = ∂ t H + ( A ( µ − x )) (cid:48) ∇ x H + 12 Tr( σσ (cid:48) ∇ xx H ) + wr∂ w H − D H (cid:48) ( σσ (cid:48) ) − D H ∂ ww H . (A.1)Now, looking at the terminal condition, we guess that the solution of this PDE will be of theform H ( t, x, w ) = − exp( − γ ( we r ( T − t ) + h ( t, x ))) for some function h ( t, x ) to be determined andsuch that h ( T, x ) = 0. Some easy computations then show that ∂ t H = − γH ( − rwe r ( T − t ) + ∂ t h ) ∂ w H = − γe r ( T − t ) H ∂ ww H = γ e r ( T − t ) H ∇ xw H = γ e r ( T − t ) H ∇ x h ∇ x H = − γH ∇ x h ∇ xx H = − γH ( ∇ xx h − γ ∇ x h ∇ x h (cid:48) ) D H = − γe r ( T − t ) H ( A ( µ − x ) − pr − γσσ (cid:48) ∇ x h ) . Plugging all this into (A.1), dividing everything by − γH , and doing some simple algebra toexpand the last term yields0 = − rwe r ( T − t ) + ∂ t h + ( A ( µ − x )) (cid:48) ∇ x h + 12 Tr( σσ (cid:48) ( ∇ xx h − γ ∇ x h ∇ x h (cid:48) )) + wre r ( T − t ) +12 γ ( A ( µ − x ) − pr ) (cid:48) ( σσ (cid:48) ) − ( A ( µ − x ) − pr ) + γ ∇ x h (cid:48) σσ (cid:48) ∇ x h − ( A ( µ − x ) − pr ) (cid:48) ∇ x h and we can see that almost miraculously the non-linear terms in h , the terms in w , and thethird and part of the last term of the PDE get cancelled and the equation gets dramaticallysimplified, obtaining0 = ∂ t h + 12 Tr( σσ (cid:48) ∇ xx h ) + rp (cid:48) ∇ x h + 12 γ ( A ( µ − x ) − pr ) (cid:48) ( σσ (cid:48) ) − ( A ( µ − x ) − pr ) . This is now a parabolic linear PDE in h and we can find explicitly its solution by using theFeynman-Kac formula. Indeed, if we consider the stochastic process given by dY t = rpdt + σdB ∗ t (A.2)we can rewrite the above equation in terms of the infinitesimal generator L ∗ of Y as0 = ( ∂ t + L ∗ ) h + 12 γ ( A ( µ − x ) − pr ) (cid:48) ( σσ (cid:48) ) − ( A ( µ − x ) − pr )25nd then we can express its solution via the following conditional expectation, which is theprobabilistic representation given in the proposition’s statement: h ( t, x ) = E ∗ t,x (cid:20)(cid:90) Tt γ ( A ( µ − Y s ) − pr ) (cid:48) ( σσ (cid:48) ) − ( A ( µ − Y s ) − pr ) ds (cid:21) = 12 γ ( Aµ − pr ) (cid:48) ( σσ (cid:48) ) − ( Aµ − pr )( T − t ) − γ ( Aµ − pr ) (cid:48) ( σσ (cid:48) ) − A E ∗ t,x (cid:20)(cid:90) Tt Y s ds (cid:21) + 12 γ E ∗ t,x (cid:20)(cid:90) Tt Y (cid:48) s A (cid:48) ( σσ (cid:48) ) − AY s ds (cid:21) . Finally, to find h explicitly, notice that we can easily solve the SDE (A.2), obtaining, for s ≥ t , Y s = x + rp ( s − t ) + σ ( B ∗ s − B ∗ t ) . and this allows us to compute the two expectations in our expression for h above. Indeed,Fubini’s theorem and elementary facts about the Brownian motion immediately yield E ∗ t,x (cid:20)(cid:90) Tt Y s ds (cid:21) = (cid:90) Tt E ∗ t,y [ Y s ] ds = x ( T − t ) + rp ( T − t ) E ∗ [( B ∗ s − B ∗ t ) (cid:48) σ (cid:48) A (cid:48) ( σσ (cid:48) ) − Aσ ( B ∗ s − B ∗ t )] = ( s − t )Tr( σ (cid:48) A (cid:48) ( σσ (cid:48) ) − Aσ ) , we similarly find out that E ∗ t,x (cid:20)(cid:90) Tt Y (cid:48) s A (cid:48) ( σσ (cid:48) ) − AY s ds (cid:21) = (cid:90) Tt ( x + rp ( s − t )) (cid:48) A (cid:48) ( σσ (cid:48) ) − A ( y + rp ( s − t ))+( s − t )Tr( σ (cid:48) A (cid:48) ( σσ (cid:48) ) − Aσ ) ds = x (cid:48) A (cid:48) ( σσ (cid:48) ) − Ax ( T − t )+ (cid:0) x (cid:48) A (cid:48) ( σσ (cid:48) ) − Arp + Tr( σ (cid:48) A (cid:48) ( σσ (cid:48) ) − Aσ ) (cid:1) ( T − t ) r p (cid:48) A (cid:48) ( σσ (cid:48) ) − Ap ( T − t ) , which gives us the complete explicit solution of the DPE, and hence the explicit form of theoptimal strategy π ∗ by using equation (3.1). Proof of Proposition 3.2:
Since in the previous proof we have found explicitly the classical smooth solution H of thedynamic programming equation, we just have to check that π ∗ ∈ A [0 ,T ] and that the usualregularity conditions hold for the classical proof to apply. More precisely, this means that thelocal martingale dH −L πt,x,w Hdt is a supermartingale for any admissible π and a true martingalefor π ∗ , where L πt,x,w is the infinitesimal generator of the controlled process ( X, W π ), or somesufficient condition for this like the one we stated in Proposition 3.2 in terms of the modelparameters, which is what we will show here.As for the first issue, it is easy to see that π ∗ ∈ A [0 ,T ] . Indeed, it is obviously F t -adaptedand predictable (in fact, it has continuous paths) and, using the trivial inequalities || x + y || ≤ || x || + 2 || y || and || Ax || ≤ || A |||| x || and the fact that X t is a Gaussian process, it is easy tosee that (cid:82) T E [ || π ∗ s || ] ds < ∞ . Moreover, applying Ito’s formula to the process e − rt W t yields d ( e − rt W t ) = − re rt W t dt + e − rt dW t = π t · e − rt dX t − π t · e − rt prdt and, therefore, W t = w + e rt (cid:18)(cid:90) t π s · e − rs dX s − (cid:90) t π s · e − rs prds (cid:19) (A.3)26or any t ≥ π . Thus, the SDE for W has a unique strong solution W ∗ for the particular case π = π ∗ for any initial data, given by the above integral (note that thestochastic integral is well defined, since dX s = A ( µ − X s ) ds + σdB s , π ∗ and X are continuous,and again (cid:82) T e − rs E [ || π (cid:48)∗ s σ || ] ds < ∞ ).As for the regularity conditions, we simply adapt the proof of Theorem 2.1. of Lintilhac andTourin (2016) for a related model, which guarantee the uniform P -integrability of the familyof random variables ( H ( τ, X τ , W ∗ τ )) τ ∈ [0 ,T ] where τ is a F -stopping time, and which we simplyadapt to the parameters of the present model obtaining the sufficient conditions stated inProposition 3.2.The key observation to adapt their proof is that in our case we also have that the hypothesizedvalue function is of the form H ( t, x, w ) = − exp( − γwe r ( T − t ) − x (cid:48) A ( t ) x − A ( t ) x − A ( t )) forsome explicit smooth functions A i ( t ) that we computed in the proof of Proposition 3.1, and our X is also a matrix Ornstein-Uhlenbeck process under P with SDE dX t = A ( µ − X t ) dt + σdB t ,and γW ∗ τ e r ( T − τ ) = γwe r ( T − τ ) + γ (cid:90) τ π ∗ s · e r ( T − s ) ( A ( µ − X s ) − pr ) ds + γ (cid:90) τ π ∗ s · e r ( T − s ) σdB s as we showed in (A.3). Thus, using the Cauchy-Schwarz inequality as in their proof, the partcorresponding to − X (cid:48) τ A ( τ ) X τ − A ( τ ) X τ − A ( τ ) in the above expression for H ( τ, X τ , W ∗ τ )may be bounded exactly as they do. As for the part corresponding to − γW ∗ τ e r ( T − τ ) , we canagain repeat their exact reasoning, but noting that the quadratic term in X s in the first integralabove is now X (cid:48) s C ( s ) X s for the matrix C ( s ) that we defined before, and likewise the termin X s in the second integral is X (cid:48) s C ( s ), which following their proof gives respectively the twoexplicit sufficient conditions that we stated in Proposition 3.2. A.2 Proof of Theorem 3.2
The proof of this follows the same lines as the previous one and is actually much simpler, so wejust indicate the relevant changes. The HJB equation is now0 = ∂ t H + ( µ − x ) (cid:48) A (cid:48) ∇ x H + 12 Tr( σσ (cid:48) ∇ xx H ) +sup π (cid:18)(cid:0) π (cid:48) A ( µ − x ) + ( w − π (cid:48) p ) r (cid:1) ∂ w H + 12 π (cid:48) σσ (cid:48) π∂ ww H + π (cid:48) σσ (cid:48) ∇ xw H − γ ( t )2 π (cid:48) σσ (cid:48) π (cid:19) with terminal condition H ( T, x, w ) = w. Guessing that the value function will now be of the form H ( t, x, w ) = we r ( T − t ) + a ( t ) + b ( t ) (cid:48) x + x (cid:48) c ( t ) x for a scalar a ( t ), an N -dimensional vector b ( t ), and a symmetric N × N matrix c ( t ),and plugging this into the above equation, we obtain the hypothesized optimal control given inthe statement of the theorem and the above PDE gets reduced to the following system of threefirst-order linear matrix ODEs0 = ∂ t c − A (cid:48) c − cA + e r ( T − t ) A (cid:48) ( γ ( t ) σσ (cid:48) ) − A ∂ t b − A (cid:48) b + cAµ − e r ( T − t ) A (cid:48) ( γ ( t ) σσ (cid:48) ) − ( Aµ − pr )0 = ∂ t a + 12 (cid:0) µ (cid:48) A (cid:48) b + b (cid:48) Aµ (cid:1) + 12 Tr( σσ (cid:48) c ) + e r ( T − t ) Aµ − pr ) (cid:48) ( γ ( t ) σσ (cid:48) ) − ( Aµ − pr )27ith terminal conditions a ( T ) = b ( T ) = c ( T ) = 0.This system has an explicit bounded solution in [0 , T ], since the classical solution of the generalfirst-order linear matrix ODE ∂ t y + uy + v ( t ) = 0 with y ( T ) = 0 is given by y ( t ) = (cid:90) Tt exp (( s − t ) u ) v ( s ) ds, if v ( s ) is continuous on [0 , T ], in which case it is automatically bounded on [0 , T ] as well; andsimilarly the classical solution of ∂ t y + uy + yu (cid:48) + v ( t ) = 0 with y ( T ) = 0 for a symmetric v isgiven by y ( t ) = (cid:90) Tt exp (( s − t ) u ) v ( s ) exp (cid:0) ( s − t ) u (cid:48) (cid:1) ds. Thus, the HJB equation has an explicit classical solution which has quadratic growth in thestate variables uniformly in t . A classical verification result (cf. for example Theorem 4.3 ofGuyon & Labord`ere (2013)) then yields that our hypothesized optimal control is indeed optimalprovided that it is admissible, which may be checked exactly as in the proof of Theorem 3.1. A.3 Proof of Theorem 4.1
Proof of Proposition 4.1:
The corresponding dynamic programming equation is in this case0 = ( ∂ t + L X ) H + (cid:0) π (cid:48) A ( µ − x ) + ( w − π (cid:48) p ) r (cid:1) ∂ w H + 12 π (cid:48) σσ (cid:48) π∂ ww H ++ π (cid:48) σσ (cid:48) ∇ xw H − γ ( t )2 π (cid:48) σσ (cid:48) π + sup I (cid:18) I (cid:48) ∇ π H − I (cid:48) CI (cid:19) with terminal condition H ( T, x, w, π ) = w and where the supremum is obviously attained at I ∗ = C − ∇ π H .Substituting this back in the above equation and plugging the stated ansatz we obtain that0 = 12 π (cid:48) ∂ t aπ + π (cid:48) ( ∂ t + L X ) b + ( ∂ t + L X ) d + π (cid:48) ( A ( µ − x ) − pr ) e r ( T − t ) − γ ( t )2 π (cid:48) σσ (cid:48) π + 12 ( aπ + b ) (cid:48) C − ( aπ + b ) . Matching the coefficients for π (cid:48) ( · ) π , π (cid:48) ( · ), and the constant yields the above differential equa-tions.Before we prove the next proposition, we state here the following result for comparison andexistence of solutions of matrix Riccati ODEs (cf. Theorem 2.2.2 in Kratz (2011)), which wewill use in our proof. Theorem A.1.
Let L ( t ) , L ( t ) , L ( t ) , N ( t ) , N ( t ) ∈ R d × d be piecewise continuous on R . More-over, suppose L ( t ) , L ( t ) , N ( t ) , N ( t ) and S , S ∈ R d × d are symmetric. Let T > and S ≥ S , L ≥ L ≥ , N ≥ N , on [0 , T ] . Assume that the terminal value problem ∂ t H + H L H + M H + H M + N = 0 , H ( T ) = S , as a (symmetric) solution H on [0 , T ] . Then the terminal value problem ∂ t H + H L H + M H + H M + N = 0 , H ( T ) = S , has a (symmetric) solution H on [0 , T ] and H ( t ) ≥ H ( t ) for all t ∈ [0 , T ] . We are now in a position to give the following:
Proof of Proposition 4.2: (1) The first statement follows directly from the comparison Theorem A.1 stated before, sincethe matrix Riccati ODE ∂ t a + aC − a = 0with terminal condition a ( T ) = 0 has the obvious symmetric solution a ( t ) = 0 definedon all [0 , T ]. Thus, (4.2) has a symmetric classical solution a ( t ) defined on all [0 , T ] with a ≤
0, which is bounded because [0 , T ] is compact and a is differentiable hence continuous.As for the particular solution when γ ( t ) = γ , simply note that pre- and post-multiplying(4.2) by C − / and defining ˜ a := C − / aC − / gives the new Riccati ∂ t ˜ a − γC − / σσ (cid:48) C − / + ˜ a = 0 , whose solution is ˜ a ( t ) = D tanh( D ( t − T )).(2) The existence of solutions with polynomial growth and their probabilistic representationin the above form follow from a vector-valued version of the Feynman-Kac theorem (seeAppendix A.3 of Cartea et al. (2018) for a proof of how to adapt the one-dimensional case)provided that the appropriate regularity conditions hold. Using, for example, Condition2 of Appendix E in Duffie (2010), it is sufficient that all the functions of ( t, x ) A ( µ − x ), σ , a ( t ) (cid:48) C − , e r ( T − t ) ( A ( µ − x ) − rp ) (and b ( t, x ) (cid:48) C − b ( t, x ) for the existence of d ) areuniformly Lipschitz in x , they and their first and second derivatives in x are continuouswith polynomial growth in x uniformly in t , and a ( t ) ≤
0. All of these properties arestraightforward to check in this case because all the corresponding functions are givenexplicitly and are simple, and the required properties for a follow from (1).The fact that b has linear growth in x uniformly in t is then a consequence of the proba-bilistic representation (4.5). Indeed, Fubini’s theorem implies that b ( t, x ) = (cid:90) Tt : exp (cid:18)(cid:90) st a (cid:48) ( u ) C − du (cid:19) : e r ( T − s ) ( A E t,x [ µ − X s ] − rp ) ds whereas the fact that X t +∆ t = e − A ∆ t X t + ( I − e − A ∆ t ) µ + (cid:90) t +∆ tt e − A (∆ t + t − s ) σdB s yields E t,x [ µ − X s ] = e − A ( s − t ) ( µ − x ) . Combining the two pieces and using the boundedness of a and the compactness of [0 , T ]gives the desired uniform bound in t .The quadratic growth of d in x uniformly in t is then obvious looking at its probabilisticrepresentation and using the linear growth of b .29 roof of Proposition 4.3: Combining the two previous propositions, we have already found an explicit classical solutionof the associated HJB equation with quadratic growth in the state variables uniformly in t ,so using again Theorem 4.3 in Guyon and Labord`ere (2013), we just have to verify that thecandidate intensity given in Proposition 4.1 is admissible.For this, first of all note that the corresponding SDEs controlled by the above intensity have anunique (strong) solution for any initial data. Indeed, given I ∗ and the definition of I as dπ = Idt ,we can solve explicitly the corresponding first-order linear matrix ODE for π ∗ yielding, for s ≥ t , π ∗ s = π t + (cid:90) st : exp (cid:18)(cid:90) su Rate ( v ) dv (cid:19) : Aim ( u, X u ) du, and this π ∗ in turn defines W ∗ like in the proof of Theorem 3.1.Finally, from the above construction it is obvious that both π ∗ t and I ∗ t are F t -adapted andpredictable (in fact, they have continuous paths), and the property that π ∗ is in L ([0 , T ] × Ω)(i.e., that (cid:82) T E [ || π ∗ s || ] ds < ∞ ) stems from the observation that Rate ( u ) is deterministic andbounded (because of Proposition 4.3.1), Aim ( t, x ) has linear growth in x uniformly in t (byProposition 4.3.2), and X is a Gaussian process (so it is in L ([0 , T ] × Ω)). I ∗ t is likewise in L ([0 , T ] × Ω) since, as we saw in Proposition A.2, I ∗ t = C − ( a ( t ) π ∗ t + b ( t, X t )) andwe can therefore use the triangular inequality, the just shown fact that π ∗ t is in L ([0 , T ] × Ω),and the same arguments as above that a ( t ) is bounded (because of Proposition 4.3.1), that b ( t, x ) has linear growth in x uniformly in t (by Proposition 4.3.2), and that X is a Gaussianprocess, to conclude. 30 eferences [1] Almgren, R. and Chriss, N. (2001). Optimal execution of portfolio transactions. Journal ofRisk , 3, 5-40.[2] Avellaneda, M., and Lee, J. H. (2010). Statistical arbitrage in the US equities market.
Quantitative Finance , 10(7), 761-782.[3] Benth, F. E and Karlsen, K. H. (2005). A note on Merton’s portfolio selection problem forthe Schwartz mean-reversion model.
Stochastic analysis and applications , 23(4), 687-704.[4] Bertsimas, D. and Lo, A. (1998). Optimal control of execution costs.
Journal of FinancialMarkets , 1, 1-50.[5] Bismuth, A., Gu´eant, O. and Pu, J. (2019). Portfolio choice, portfolio liquidation, andportfolio transition under drift uncertainty.
Mathematics and Financial Economics , 13(4),661-719.[6] Boyd, S., Busseti, E., Diamond, S., Kahn, R., Koh, K., Nystrup, P., and Speth, J. (2017),Multiperiod trading via convex optimization.
Foundations and Trends in Optimization,
Stochastic processesand their applications,
Algorithmic and high frequency trading.
Cambridge University Press, Cambridge.[9] Cartea, ´A. and Jaimungal, S. (2016). Algorithmic trading of co-integrated assets.
Interna-tional Journal of Theoretical and Applied Finance , 19(06), 165038.[10] Cartea, ´A., Gan, L., and Jaimungal, S. (2018). Trading cointegrated assets with priceimpact. To appear in
Mathematical Finance .[11] Chiu, M. C. and Wong, H. Y. (2011). Mean-variance portfolio selection of cointegratedassets.
J. Econ. Dyn. Control , 35, 1369-1385.[12] DeMiguel, V., Mei, X. and Nogales, F. J. (2016). Multiperiod portfolio optimization withmultiple risky assets and general transaction costs.
Journal of Banking and Finance , 69,p. 108-120.[13] Duffie, D. (2010). Dynamic asset pricing theory. Princeton University Press, Princeton, 3rdedition.[14] Duffie, D., Filipovic, D., and Schachermayer, W. (2003). Affine processes and applicationsin finance.
The Annals of Applied Probability , 13(3), 984-1053.[15] Fouque, J.-P., Papanicolaou, A., and Sircar, R. (2015). Filtering and portfolio optimiza-tion with stochastic unobserved drift in asset returns.
Communications in MathematicalSciences , 13(4), 935-953.[16] Garleanu, N. and Pedersen, L. H. (2013). Dynamic trading with predictable returns andtransaction costs.
J. Finance , 68(6), 23092340.[17] Garleanu, N. and Pedersen, L. H. (2016). Dynamic portfolio choice with frictions.
J. Econ.Theory , 164, 487-516. 3118] Guo, X., Lai, T. L., Shek, H., and Wong, S (2016). Quantitative trading: algorithms,analytics, data, models, optimization. Chapman and Hall/CRC, Boca Raton.[19] Guyon, J., and Labord`ere, P. H. (2013). Nonlinear option pricing. Chapman and Hall/CRC,Boca Raton.[20] Jurek, J. W. and Yang, H. (2007) Dynamic portfolio selection in arbitrage. Working Paper,Harvard University.[21] Kitapbayev, Y. and Leung, T. (2018). Mean reversion trading with sequential deadlinesand transaction costs.
International Journal of Theoretical and Applied Finance , 21(1),1850004.[22] Kratz, D.-M. P. (2011). Optimal liquidation in dark pools in discrete and continuous time.Ph. D. thesis, Humboldt-Universitat zu Berlin.[23] Lei, Y. and Xu, J. (2015), Costly arbitrage through pairs trading.
J. Econ. Dyn. Control ,56, 119.[24] Leung, T., and Li, X. (2015). Optimal mean reversion trading with transaction costs andstop-loss exit.
Int. J. Theor. Appl. Finance , 18(3), 1-31.[25] Lettau, M, and Pelger, M. (2018). Estimating latent asset-pricing factors, Working Paper,Stanford University.[26] Liang, Z.,Yuen, K. C., and Guo, J. (2011). Optimal proportional reinsurance and invest-ment in a stock market with OrnsteinUhlenbeck process.
Insurance: Mathematics andEconomics , 49(2), 207215.[27] Lintilhac, P., and Tourin, A. (2016). Model-based pairs trading in the bitcoin markets.
Quantitative Finance , 17(5), 703-716.[28] Liu, J. and Timmermann, A. (2013) Optimal convergence trade strategies.
Review of Fi-nancial Studies , 26(4), 10481086.[29] Ludwig, S. (2012). Optimal portfolio allocation of commodity related assets using a con-trolled forward-backward stochastic algorithm. Ph. D. thesis, Rubrecht-Karls Universit¨atHeidelberg.[30] Merton, R. (1971). Optimum consumption and portfolio rules in a continuous time model.
Journal of Economic Theory , 3, 373-413.[31] Moreau, L., Muhle-Karbe, J., and Soner, H.M. (2017). Trading with small price impact.
Mathematical Finance , 27(2), 350-400.[32] Moutari, S., Sandjo, A. N., and Colin, F. (2017). An Explicit Solution for a PortfolioSelection Problem with Stochastic Volatility.
Journal of Mathematical Finance , 7, 199-218.[33] Mudchanatongsuk, S., Primbs, J. A., and Wong, W. (2008). Optimal pairs trading: Astochastic control approach. , 1035-1039.[34] Muhle-Karbe, J., Liu, R., Weber, M. (2017). Rebalancing with linear and quadratic costs.
SIAM Journal on Control and Optimization,
J. Math. Anal. Appl. , 441(1), 403425.3236] Obizhaeva, A. and Wang, J. (2013). Optimal trading strategy and supply/demand dynam-ics.
Journal of Financial Markets , 16 (1), 1-32.[37] Papanicolaou, G. and Yeo, J. (2017). Risk control of mean-reversion time in statisticalarbitrage.
Risk and Decision Analysis,
6, 263-290.[38] Pelger, M. and Xiong, R. (2018). Interpretable proximate factors for large dimensions,Working Paper, Stanford University.[39] Rogers, L. C. G. and Singh, S. (2010). The cost of illiquidity and its effects on hedging.
Mathematical Finance , 20(4), 597-615.[40] Tourin, A. and Yan, R. (2013). Dynamic pairs trading using the stochastic control approach.