2015（其他）Dynamic Portfolio Optimisation with Intermediate Costs A Least-Squares Monte-Carlo Simulation Approach

Electronic copy available at: /abstract=2696968 Dynamic Portfolio Optimisation with Intermediate Costs:A Least-Squares Monte-Carlo Simulation ApproachRongju ZhangMonash UniversitySchool of Mathematical SNicolas LangrenThe Commonweath Scientific and Industrial Research OrganisationReal Options and Financial Risknicolas.langrenecsiro.auYu TianNational Australian BankPricing and Risk Analysis, Retirement Solutions, NAB W.auZili ZhuThe Commonwealth Scientific and Industrial Research OrganisationReal Options and Financial Riskzili.zhucsiro.auFima KlebanerMonash UniversitySchool of Mathematical SKais HamzaMonash UniversitySchool of Mathematical S1 Electronic copy available at: /abstract=2696968 AbstractWe propose a dynamic portfolio optimisation strategy that takes liquidity cost into account. Ourliquidity cost model is built upon the so-called Marginal Supply-Demand Curve which describes theasset price as a function of the trading volume. We extend the least-squares Monte Carlo algorithmto a stochastic control problem with switching costs and endogenous state variables. This approachis simulation-based, with great flexibility in the choice of investor objective, asset dynamics, portfolioconstraints, intermediate consumption and incorporation of different sources of costs, making it easyto implement in practice. We study a portfolio investing in four major sectors in the U.S. market.We benchmark our dynamic strategy against several alternative portfolio strategies in terms of theout-of-sampledistributionofterminalwealthandtherealmarketperformance. Overall, ourdynamicstrategy outperforms other standard asset allocation strategies.Keywords: portfolio selection; liquidity cost; marginal supply-demand curve; least-squares Monte Carlo;stochastic dynamic programming; stochastic control.21 IntroductionDynamic portfolio optimisation is commonly used by financial institutions, such as investment banks,pension funds, hedge funds and other asset management companies. The portfolio optimisation pro-cedure basically deals with the problem of finding an asset allocation strategy that maximises a givenperformance criterion, with possible risk constraints.Following on the single-period mean-variance framework proposed by Markowitz (1952), widely acceptedas the starting point of modern portfolio theory, many improvements have been proposed. Mathemat-ically, the most natural framework to deal with dynamic portfolio optimisation is stochastic control;see Samuelson (1969) or Musumeci and Musumeci (1999) for example. The main difficulty is then toestimate the numerical solution of the associated Hamilton-Jacobi-Bellman (HJB) equations.The main aim of this paper is to incorporate liquidity cost into the multi-period portfolio optimisationproblem. Indeed, managing a portfolio without considering liquidity impact may substantially misvaluetheportfolioandmisleadinvestors. Unfortunately, theresearchonportfoliooptimisationassociatedwithliquidity impact is constrained by the lack of formalisation of the limit order book. Almgren, Thum,Hauptmann, and Li (2005) decompose the market impact into two components: a permanent componentthat reflects the movement of the market due to the order imbalance and a temporary component thatreflects the price concession needed to fill a demanding amount of market orders within a short timeinterval. We interpret this temporary component as the liquidity cost.A popular approach to take liquidity cost into account is to assume it proportional to the total portfolioturnover. A more general framework has been developed in Acerbi and Scandolo (2008), where theauthors formalise the valuation of a liquidity-adjusted portfolio by defining a so-called Marginal Supply-Demand Curve (MSDC), which provides the market price of an asset as a function of the trading volume.The loss in portfolio value generated by this liquidity adjustment is then defined as liquidity cost. Tian,Rood, and Oosterlee (2013) calibrated the MSDC to the European equity market, obtaining a square-root relation for large- or medium-cap equities and a square relation for small-cap equities. Thisframework forms the basis of our liquidity cost modelling.To solve our dynamic portfolio optimisation problem with liquidity cost, we resort to numerical methodsfor stochastic control problems. There are three popular classes of numerical methods for HJB equationsin the literature: analytic solutions, partial differential equations (PDEs) and Monte Carlo methods.Analytic solutions are the easiest but are few and far between and limited to very simple models; seeBellman (2010), Merton (1971), Brennan, Schwartz, and Lagnado (1997) or Cvitanic, Polimenis, andZapatero (2008). The PDE approach is more versatile and can encompass more features (proportionaltransaction costs for example). However, the asset dynamics tractable by PDE are still limited. More-over, this approach cannot deal with a high-dimensional portfolio, which is a major limitation for theasset allocation problem. Finally, the Monte Carlo method is the most versatile. It can easily deal withany asset dynamics and is suitable for high-dimensional problems.TheuseofMonteCarlosimulationstosolveoptimalstoppingproblemswaspopularisedbyLongstaffandSchwartz (2001) on the American option pricing problem. Their method, known as least-squares MonteCarlo (LSMC), was later extended to more general stochastic control problems. The implementationof LSMC on the portfolio selection problem was pioneered by Brandt, Goyal, Santa-Clara, and Stroud(2005), where the authors determine a semi-closed form of the optimal policy by solving the first order3condition of the Taylor series expansion of the investors future value function. However, there are twolimitations of this approach: first it cannot handle the problems where the control variable depends onthe endogenous state variables such as portfolio wealth; second it cannot deal with the intermediatetransaction cost of rebalancing a portfolio. Few years later, a multi-period asset allocation problemwith intermediate transaction costs using LSMC was studied in Bao, Zhu, Langren, and Lee (2014).The authors approximate the transaction cost as proportional to the portfolio turnover and wrap thetransaction cost into an additive time-separable utility function. However, an additive time-separableutility function cannot deal with large intermediate costs such as the liquidity cost of trading a largevolume of assets, i.e., the approximation error explodes due to the non-linearity of the utility function.In this paper, we extend Bao et al. (2014)s approach to deal with a non-linear liquidity cost. The trickto avoid using the additive time-separable utility function is to conditionalise the objective function onthe post-decision state variables. Both exogenous and endogenous state variables will be considered. Todeal with the endogenous state variable, we rely on the control randomisation technique provided byKharroubi, Langren, and Pham (2014).To sum up, our two main contributions are the following:We propose a dynamic portfolio optimisation model that can deal with liquidity risk. It is basedon the literature in Acerbi and Scandolo (2008).We generalize the LSMC algorithm to problems with switching costs and endogenous variables. Itis based on the control randomisation technique in Kharroubi et al. (2014).In particular, our numerical experiments illustrate the importance of incorporating liquidity cost intomulti-period asset allocation.The outline of the paper is as follows. Section 2 describes our portfolio optimisation problem, withtransaction costs and liquidity costs taken into account. Section 3 describes in detail our extendedLSMC algorithm. After that, Section 4 provides numerical applications, with several comparisons of ourdynamic portfolio strategy to alternative investment strategies. Finally, Section 5 concludes the paper.2 The Portfolio Optimisation ProblemThis section describes the portfolio selection problem faced by the investor. Subsection 2.1 provides amathematical description of the objective function. We assume that the investor tries to maximise theexpected utility of final wealth under the constraint of transaction cost and liquidity cost. Subsection2.2 describes the dynamics of the portfolio value with these two costs. Subsection 2.3 outlines morespecifically how we model liquidity costs within this portfolio optimisation problem. Finally, Subsection2.4 introduces the form of the transaction cost.2.1 Objective functionWe consider a multi-period portfolio optimisation problem over a finite time horizon T. Suppose thereareN possibleassets. LetSt = Sit i=1;:;N denotetheassetpricesattimet20;T. Weassumediscreteportfolio rebalancing times and denoteT =f1;:;T 1gas the set of possible rebalancing times. Let4 t = it i=1;:;N be the portfolio weights in each assets at time t2T. In particular, PNi=1 it = 1. LetCt RN be the set of admissible portfolio strategies at each rebalancing time t2T. The definitionof these sets can include constraints defined by the investor (weight limits in each individual asset forexample). Finally, let Wt denote the portfolio value, or wealth, at time t20;T. We suppose that theinvestor tries to maximise the expected utility of final wealth EU (WT) over all the possible strategiesf t2Ctgt2T, for a given utility function U : R!R.To sum up, we are facing a multivariate stochastic control problem. The state variables of the problemare:Exogenous state variables. The N asset prices are exogenous stochastic risk factors.Endogenous state variables. As rebalancing a portfolio will generate transaction cost andliquidity cost, the portfolio weight and portfolio wealth will be endogenous risk factors.LetF =fFtgt20;T be the filtration generated by all the state variables. The objective function readsVt = supf s2Csg;s2t;T)EU (WT)jFt; t2T; (2.1)where Vt and t areFt-adapted. In particular, Vt = Vt (St;Wt ; t ) and t = t (St;Wt ; t ). Themainunknownsoftheproblemarethereforethewholerebalancingfunctions t : (s;w;a)! t (s;w;a)2Ct for every t2T.2.2 Portfolio value dynamicsFor every rebalancing time t 2T, let qt = qit i=1;:;N describes the number of units held in eachasset. In particular, Wt = qt:St = PNi=1qitSit where : denotes the dot product between two vectors.We assume that these quantities are constant between two rebalancing dates. Let TCt and LCt denoterespectively the transaction costs and liquidity costs at time t2T when rebalancing the portfolio fromqt 1 to qt. More generally, TCt and LCt depend on both exogenous state variables and endogenousstate variables.Given an initial portfolio value W0, the wealth of the investor evolves as follows:W1 = W0 + q0:(S1 S0)W1 = W1 TC1 LC1W2 = W1 + q1:(S2 S1).Wt = Wt TCt LCt (2.2)W(t+1) = Wt + qt:(St+1 St).WT = WT 1 + qT 1:(ST ST 1)WT = WT 5There is no transaction at time T1.There are two possible descriptions of the portfolio positions:1. Absolute positions using the quantity (number of units) in each asset qt, as done in this subsection.2. Relative positions using the proportions of wealth in each asset t, as done in the previous sub-section.One can easily switch from one description to the other, as qitSit = itWt for each asset i = 1;:;N andrebalancing time t2T. Both descriptions will be needed, as transaction costs and liquidity costs willdepend on q, while portfolio strategies are described using .2.3 Liquidity costsUnlike transaction costs, liquidity costs are rarely considered within dynamic portfolio selection prob-lems. In this paper, we adopt Acerbi and Scandolo (2008)s notion of liquidity-adjusted portfolio valuewhich is characterised by the MSDC. The fundamental idea of the MSDC concept is that the real priceof an asset depends on the trading volume. We will use the MSDC as a representation of the limit orderbook so as to evaluate the liquidity cost of reallocating the portfolio at each rebalancing time.Definition 2.1. The Marginal Supply-Demand Curve of an asset at a given time is a map m : Rnf0g!R that provides a description of the limit order book. If q is the trading volume in the asset, then whenq 0 (bid orders), m(q) represents the market price of the asset after selling q units. Conversely, whenq 0.Thefirstconditionensuresano-arbitrageassumption. Denotemit(q) astheMSDCoftheith assetattimet. Fromthesecondcondition, thebestbidpriceandthebestaskpricearegivenbySiB;t := limq!0+ mit(q)and SiA;t := limq!0 mit(q). Using these two prices, we can further define the rate of bid-ask spread itas SiA;t := SiB;t(1 + it), which is a useful liquidity indicator. See an example of the MSDC in Figure 2.1.As convention, we define qt := qt 1 qt = ( qit)i=0;:;N as the vector of trading volume at timet2T, then this quantity coincides with the definition of a portfolio in Acerbi and Scandolo (2008).Thus, the corresponding MSDC in our context is mit( qit). We now define the liquidity cost generatedby the transaction qt using this MSDC:Definition2.2. The liquidation Mark-to-Market (MtM) value at timetof the transaction qt is definedasL( qt) :=NXi=1Z qit0mit(u)du:1For every t2T, Wt Wt, and Wt = Wt if there is no rebalancing, which is the optimal thing to do at time Twith the objective function (2.1).6Figure 2.1: This figure shows an example of the MSDC, where the horizontal axis represents the transac-tionvolumeoftheassetandtheverticalaxisrepresentsthemarketpricecorrespondingtothetransactionvolume.Definition 2.3. The uppermost MtM value at time t of the transaction qt is defined asUp( qt) :=Xqit0SiB;t qit:Definition 2.4. The liquidity cost (liquidation cost) of the transaction qt is defined asLC( qt) := Up( qt) L( qt):The liquidation MtM value can be viewed as the total market value being liquidated when reallocatingthe assets. The uppermost MtM value is the best market value of the trading quantity given that theliquidation has not taken place. The difference between these two values is the liquidity cost. We referto Acerbi and Scandolo (2008) for more details.As shown by these definitions, the evaluation of the liquidity cost depends on the MSDC. In real markets,the MSDC comes at anytime in the form of a non-increasing piecewise-constant function of the tradingvolume. Many researchers have calibrated parametric models to the limit order books, for example a3=5 relation between the temporary market impact and the trading volume for equities listed on NYSEand NASDAQ in Almgren et al. (2005) and a square-root relation between the price change and tradingvolume for S&P 500 equities in Cont, Kukanov, and Stoikov (2014). In Tian et al. (2013), the authorscalibrated the MSDC to the European equity market, interpreting a square-root relation for large- ormedium-cap equities and a square relation for a small-cap equities.In our paper, we choose to approximate the liquidity cost of rebalancing the portfolio using the square-root MSDC shape. This meansmit(q) =80; (2.3)7where the parameter k2R+ is called the liquidity risk factor. Using equation (2.3) and Definition 2.4,the liquidity cost is given byLC( qt) =Xqit0SB;t qit + i( qit) ;wherei( qit) = 2k2kqqite kpqit e kpqit 1:(2.4)2.4 Transaction costsTransaction costs (or commission fees) of buying or selling assets can take various shapes: fixed costs,proportional costs, a combination of both or a more general form. In this paper, we consider transactioncosts proportional to the total trading value. Due to the bid-ask spread, transaction costs are differentwhen selling assets and when buying assets. Denote the proportional rate as . Then, the transactioncost at time t2T is given byTC( qt) =NXi=1SiA;tj qitja49f qit0g; (2.5)where, as in the previous subsection, qt = ( qit)i=0;:;N is the vector of trading volume at time t2T,and SiB;t and SiA;t are the best bid price and the best ask price of the ith asset at time t2T.3 Least-Squares Monte-Carlo SimulationThis section describes the numerical algorithm we use to solve the portfolio selection problem (2.1).Namely, we adopt the so-called least-squares Monte Carlo simulation approach. Simply speaking, theidea is to use dynamic programming with conditional expectation approximated by simple least-squaresregression across simulated paths of the underlying risk factors.This approach was used in Longstaff and Schwartz (2001) to value American option, and later extendedto more general optimal switching problems such as gas storage valuation Boogert and de Jong (2008) orinvestment in power plants Ad, Campi, Langren, and Pham (2014). On the portfolio section problem,Brandt et al. (2005) and Bao et al. (2014) used similar extension of the least-squares Monte Carloalgorithm. These two papers mainly differ on how the optimal portfolio weights are computed in thebackward loop: Brandt et al. (2005) determine semi-closed form portfolio weights by maximising aTaylor series expansion of the investors future value function, while Bao et al. (2014) perform a morestraightforward maximisation over a fixed grid of discrete portfolio weights.This paper adopts the approach of Bao et al. (2014), extended to liquidity costs, with a greater emphasison how the endogenous wealth is handled.8This Section describes in detail how the least-squares Monte Carlo algorithm works on the portfolioselection problem introduced in Section 2. The algorithm is made of two main parts:1. A forward loop during which the risk factors are simulated (Subsection 3.2)2. A backward dynamic programming loop during which the conditional future utility is estimatedby least-squares regression and then used to estimate the optimal policy (Subsection ?).However, before getting into the description of these two time loops, we need some preliminary remarkson how to switch from absolute