Ways to Develop an Optimal Trading Strategy Finance Essay

The main objective of the dissertation is to develop an optimal trading strategy also considering the execution cost of each trading step, using stochastic dynamic programming. More explicitly, the following problem is proposed and solved: Given a fixed block of shares to be executed within a fixed finite number of periods, and given price dynamics that capture price impact, i.e., the execution price of an individual trade as a function of the share traded and other state variables, find the optimal sequence of trades (as a function of the state variables) that will minimize the expected cost of executing within periods. There has been a tremendous interest and consequent growth in terms of equity trading, partly due to the advent of a large number of mutual and pension funds. In these situations, the impact of trading costs has been assuming increasing importance. Trading costs or execution costs are costs that are associated with the execution of investment strategies which include commissions, bid/ask spreads, opportunity costs of waiting, and price impact from trading. There has been studies where although the performance certain funds have been showed to perform very well compared to market, but actual performance was significantly different (Perold (1988)). The difference arose due to the inclusion of the execution costs. This shortfall is surprisingly large and underscores the importance of execution-cost control, particularly for institutional investors whose trades often comprise a large fraction of the average daily volume of many stocks. So the problem of developing an optimal trading strategy, considering the execution costs also comes in to perspective. There are various methods to do this. Here dynamic programming is used to derive an optimal trading strategy considering the execution costs. The use of dynamic programming though not new in financial economics, is novel here by the fact that the trading steps is takes time and the present steps affects the price and thus the costs in the future. Dynamic Programming Dynamic programming is a method for solving complex problems by breaking them down into simpler sub problems. It is applicable to problems exhibiting the properties of overlapping sub problems which are only slightly smaller and optimal substructure. When applicable, the method takes far less time than the usual methods. The key idea behind dynamic programming is quite simple. In general, to solve a given problem, we need to solve different parts of the problem (sub problems), then combine the solutions of the sub problems to reach an overall solution. Often, many of these sub problems are really the same. The dynamic programming approach seeks to solve each sub problem only once, thus reducing the number of computations. This is especially useful when the number of repeating sub problems is exponentially large. Top-down dynamic programming simply means storing the results of certain calculations, which are later used again since the completed calculation is a sub-problem of a larger calculation. Bottom-up dynamic programming involves formulating a complex calculation as a recursive series of simpler calculations. History The term dynamic programming was originally used in the 1940s by Richard Bellman to describe the process of solving problems where one needs to find the best decisions one after another. By 1953, he refined this to the modern meaning, referring specifically to nesting smaller decision problems inside larger decisions, and the field was thereafter recognized by the IEEE as a systems analysis and engineering topic. Bellman’s contribution is remembered in the name of the Bellman equation, a central result of dynamic programming which restates an optimization problem in recursive form. The word dynamic was chosen by Bellman to capture the time-varying aspect of the problems, and also because it sounded impressive. The word programming referred to the use of the method to find an optimal program, in the sense of a military schedule for training or logistics. This usage is the same as that in the phrases linear programming and mathematical programming, a synonym for optimization. Overview Dynamic programming is both a mathematical optimization method and a computer programming method. In both contexts it refers to simplifying a complicated problem by breaking it down into simpler subproblems in a recursive manner. While some decision problems cannot be taken apart this way, decisions that span several points in time do often break apart recursively; Bellman called this the “Principle of Optimality”. Likewise, in computer science, a problem which can be broken down recursively is said to have optimal substructure. If subproblems can be nested recursively inside larger problems, so that dynamic programming methods are applicable, then there is a relation between the value of the larger problem and the values of the subproblems.[5] In the optimization literature this relationship is called the Bellman equation. Dynamic programming in mathematical optimization In terms of mathematical optimization, dynamic programming usually refers to simplifying a decision by breaking it down into a sequence of decision steps over time. This is done by defining a sequence of value functions V1 , V2 , … Vn , with an argument y representing the state of the system at times i from 1 to n. The definition of Vn(y) is the value obtained in state y at the last time n. The values Vi at earlier times i=n-1,n-2,…,2,1 can be found by working backwards, using a recursive relationship called the Bellman equation. For i=2,…n, Vi -1 at any state y is calculated from Vi by maximizing a simple function (usually the sum) of the gain from decision i-1 and the function Vi at the new state of the system if this decision is made. Since Vi has already been calculated for the needed states, the above operation yields Vi -1 for those states. Finally, V1 at the initial state of the system is the value of the optimal solution. The optimal values of the decision variables can be recovered, one by one, by tracking back the calculations already performed. Bellman`s Principle of Optimality The principle that an optimal sequence of decisions in a multistage decision process problem has the property that whatever the initial state and decisions are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decisions. Literature review The tremendous growth in equity trading over the past 20 years, fuelled largely by the burgeoning assets of institutional investors such as mutual and pension funds has created a renewed interest in the measurement and management of trading costs. Such costs often called “execution costs” because they are associated with the execution of investment strategies include commissions, bid/ask spreads, opportunity costs of waiting, and price impact from trading (see Loeb, 1983 and Wagner, 1993 for further discussion), and they can have a substantial impact on investment performance. For example, Perold (1988) observes that a hypothetical or paper portfolio constructed according to the Value Line rankings outperforms the market by almost 20% per year during the period from 1965 to 1986, whereas the actual portfolio the Value Line Fund outperformed the market by only 2.5% per year, the difference arising from execution costs. This implementation shortfall is surprisingly large and underscores the importance of execution-cost control, particularly for institutional investors whose trades often comprise a large fraction of the average daily volume of many stocks. There has also been considerable interest from the regulatory perspective in defining “best execution”, especially in the wake of recent concerns about NASDAQ trading practices, the impact of tick size on trading costs, and the economic consequences of market fragmentation. Indeed, Macey and O`Hara (1996) observe that “. . . while the obligation to give customers the benefits of best-execution of orders is one of the most well-established principles of securities law, and despite the fact that the concept of best execution is continually referred to in cases, treatises, law review articles, exchange rules, and administrative proceedings, no clear definition of best execution exists”. Bertsimas and Lo (1998) tries to provide one clear definition of best execution, based on the minimization of the expected cost of execution using stochastic dynamic programming. While dynamic optimization is certainly not new to financial economics, the use of dynamic programming in defining best execution is novel. In particular, this approach explicitly recognizes the fact that trading takes time, and that the very act of trading affects not only current prices but also price dynamics which, in turn, affects future trading costs. Therefore, defining and controlling execution costs are fundamentally dynamic problems, not static ones, a fact recognized implicitly by Perold (1988) and most professional portfolio managers, and developed explicitly here. Indeed, recent studies by Chan and Lakonishok (1995) and Keim and Madhavan (1995a,b,c) show that because the typical institutional investor’s trades are so large, they are almost always broken up into smaller trades executed over the course of several days. Chan and Lakonishok call such sequences “packages”, and using a sample of 1.2 million transactions of 37 large investment management firms during the period from July 1986 to December 1988, they show that only 20% of the market value of these packages is completed within a day and that over 53% are spread over four trading days or more. For this reason, best execution cannot be defined as a single number or in the context of a single trade, it is a strategy that unfolds over the course of several days and which ought to adapt to changing market conditions. Dynamic optimization provides a compelling economic rationale for trading in packages: properly parcelled packages minimize the expected costs of execution over a fixed time horizon. In particular, we propose and solve the following problem in this paper: given a fixed block SM of shares to be executed within a fixed finite number of periods, and given price dynamics that capture price impact, i.e., the execution price of an individual trade as a function of the shares traded and other “state” variables, find the optimal sequence of trades (as a function of the state variables) that will minimize the expected cost of executing within periods. Using stochastic dynamic programming, we obtain explicit closed-form expressions for these optimal trading strategies, which we call best-execution strategies, for various specifications of price dynamics. We show that best execution strategies can sometimes be expressed as a linear combination of a naive (and not uncommon) strategy breaking up shares evenly into a package of trades each of size and a correction factor that adjusts each trade size up or down according to additional information, e.g., stock-specific private information, opportunity costs, changing market conditions, etc. In the absence of such information, we derive conditions under which the naïve strategy is optimal: an arithmetic random walk for prices with linear price. We also show by construction that apart from these rather restrictive and empirically implausible conditions, the naive strategy is not optimal in general. We also obtain the expected cost of best execution-the optimal-value function which is given recursively by the Bellman equation as a by-product of the optimization process, which may serve as a useful benchmark for pricing principal-bid and negotiated-block transactions. The typical broker/dealer engaging in such transactions will not willingly hold large positions for long, and will seek to trade out of these positions as quickly and as cost-effectively as possible, i.e., he will seek best-execution strategies for his holdings. Of course, risk aversion, adverse selection, and inventory and opportunity costs may change the objective function to be minimized, in which case our benchmark may only be a lower bound on the fair market price of a block transaction. Nevertheless, even in these cases the problem of best execution is still a dynamic optimization problem and our approach is still applicable (although closed form expressions for best-execution strategies may not be available). Moreover, one can show that the basic approach described in the coming sections can be extended in several important ways: (1) Specifying more general price-impact functions and deriving numerical solutions (Section 4) (2) Trading a portfolio of stocks simultaneously (3) Imposing constraints such as no-sales or, in the portfolio case, a maximum amount of money invested. These results comprise a systematic and quantitative approach to defining and controlling execution costs, measuring the liquidity of large-block transactions, and rationalizing within an economic paradigm the kind of informal trading practices that characterize many institutional equity investors. The model Consider an investor seeking to acquire a large block of S shares of some stock over a fixed time interval [0, 1]. Since it is well-known that the short term demand curves for even the most actively traded equities are not perfectly elastic, a market order at date 0 for the entire block is clearly not an optimal trading strategy.5 A more effective strategy would be to break into smaller purchases distributed throughout the interval [0, 1], but how should such purchases be parcelled out? The answer depends, of course, on the degree to which a purchase affects the market price, i.e., the “price impact” and the dynamics of future market prices. Given a particular price-impact function and a specification for the price dynamics, e.g., a random walk, a dynamic optimal trading strategy that minimizes the expected total cost of acquiring SM in [0, 1] may be obtained by stochastic dynamic programming. Defining best execution To illustrate this approach, suppose that at time 0 the investor begins his program to acquire shares, and this program must be completed by time. With little loss in generality, let time be measured in discrete intervals of unit length. Since the length of a “period” is arbitrary, it can be set to accommodate even the finest trading-decision interval that is of practical relevance. For example, if the decision to acquire is made at the start of the day and the acquisition must be completed by the day`s end, setting 13 yields 30 minute intervals from the 9:30 am market open to the 4:00 pm market close. If the acquisition is part of an end-of-quarter portfolio rebalancing, the trading horizon may be extended to three or four days, in which case increases proportionally. Although all of our results are qualitatively independent of both the time horizon and the number of trading periods (with the exception of numerical examples, of course), for concreteness the length of each period should be regarded as some fraction of a single day and related parameters should be calibrated accordingly. Denote by St be the number of shares acquired in period t at price Pt, where t=1, 2 … T. Then the investor`s objective of minimizing execution costs may be expressed as: subject to the constraint: One may also wish to impose a no-sales constraint, i.e., St ≥ 0 (after all, it is difficult to justify selling stocks as part of a buy-program), but for expositional convenience these constraints may be ignored for now. To complete the statement of the problem, we must specify the “law of motion” for Pt. This includes two distinct components: the dynamics of Pt in the absence of our trade (the trades of others may be causing prices to fluctuate), and the impact that our trade of St shares has on the execution price Pt. For simplicity, suppose that the former component is given by an arithmetic random walk, and the latter component is simply a linear function of trade size so that a purchase of St shares may be executed at the prevailing price Pt-1 plus an impact premium of θSt, θ>0. Then the law of motion for Pt may be expressed as: where et is assumed to be a zero-mean independently and identically distributed (IID) random shock, i.e., white noise. One can observe that the two components, price impact and price dynamics can be separated. A nonlinear price impact function can easily be incorporated into the random walk specification, and non-random-walk dynamics can be combined with a linear price impact function. However, these two components interact in important ways. For example, the above equation implies that price impact has a “permanent” effect on the price level because of the random-walk specification of the price dynamics. It is this interaction between price impact and price dynamics that makes execution-cost control a dynamic optimization problem. This interaction also explains the difficulties in developing a clear economic definition of best execution: such a definition requires the specification of price dynamics a well as price impact, and these vary from one stock to another, and may well vary over time. Despite the fact that the equation has some implausible empirical implications independent price increments, positive probability of negative prices, percentage price impact that decreases with price, permanent price impact, etc. It provides a concrete illustration of the more general and considerably more complex analysis. In later sections one can see that the equation is precisely the dynamics necessary to render the naive strategy of dividing into trades each of size , the optimal one. The investor`s problem is now well-posed: find the sequence of trades {St} that minimizes the expected execution costs , subject to the constraint , and given a linear price-impact function incorporated into the law of motion for Pt. This is a classical optimal control problem which can be solved by stochastic dynamic programming, and we define the best-execution strategy as its solution. The Bellman equation The basic ingredients for any dynamic programming problem are the state of the environment at time t, the control variable, the randomness, the cost function, and the law of motion. In our context, the state at time t=1.2 … T consists of the price Pt-1 realized at the previous period, and Wt , the number of shares that remain to be purchased. The state variables summarize all the information the investor requires in each period t to make his decision regarding the control. The control variable at time t is the number of shares St purchased. The randomness is characterized by the random variable et. an additional state equation which measures the remaining number of shares to be traded is given as: where the boundary condition WT+1=0 is equivalent to the constraint that must be executed by period T. The dynamic programming algorithm is based on the observation that a solution or “optimal control” {S*1 , S*2, …, S*T} must also be optimal for the remaining program at every intermediate date t. That is, for every t, for every t, 0<t<T. The sequence {S*t , S*t+1, …, S*T} must still be optimal for the remaining program . This important property is summarized by the Bellman equation, which relates the optimal value of the objective function in period t to its optimal value in period t+1: By starting at the end (time T) and applying the Bellman equation and the law of motion for Pt, and Wt recursively, the optimal control can be derived as functions of the state variables that characterize the information that the investor must have to make his decision in each period. In particular, the optimal-value function VT(.), as a function of the two state variables PT-1 and WT, is given by Since this is the last period and WT+1 must be set to 0, there is no choice but to execute the entire remaining order WT, hence the optimal trade size S*T is simply WT. Substituting the law of motion into PTWT yields VT as a function of PT-1 and WT. In the next-to-last period T-1, the Bellman equation is less trivial: This may be cast as an explicit function of ST-1 which can be minimized by taking its derivative with respect to ST-1 and solving for its zero. This yields Continuing in this fashion, the optimal trades S*T-k and the optimal-value function VT-k(PT-k-1,WT-k) may be obtained recursively as: until we reach the beginning of the program and find: The best execution strategy Substituting the initial conditional W1= , then yields the optimal trade size S*1 as an explicit function of , and the expected best-execution cost V1 as an explicit function of , P0, and the price-impact parameter θ: By forward substitution we find that S*1 =S*2 =2 … =S*T = The best-execution strategy is simply to divide the total order into T equal “waves” and trade them at regular intervals. This remarkably simple trading strategy comes from the fact that the price impact θSt does not depend on either the prevailing price Pt-1 or the size of the unexecuted order Wt, hence the price-impact function is the same in each period and independent from one period to the next. But since each period`s execution cost PtSt is a convex (quadratic) function of St, the sum of these single-period execution costs will be minimized at the point where the marginal execution costs are equated across all periods. There is no advantage to shifting trades to one period or another – they all offer the same trade-offs to the objective function – hence the trade sizes should be set equal across all periods. Note that in this case the optimal controls MS*t N are all non-negative hence the non-negativity constraints could have been imposed trivially. The optimal-value function at time 1, V1 (P0, W1), gives the expected cost of the best-execution strategy and we see that this cost is the sum of two terms: the no-impact cost P0 and the cumulative price impact θ2(1+(1/T))/2. Observe that while the impact term is a decreasing function of T – having more time to acquire can never increase the expected cost Ð the cumulative price impact does not vanish as T increases without bound. This seems counterintuitive since one might expect price impact to become negligible if there is no time limit on completing the purchase. However, observe that the law of motion for Pt implies that the price impact θSt of an individual trade has a permanent effect on Pt, hence even infinitesimally small trades will have an impact on next period`s price, and the limiting sum of all these infinitesimal trades multiplied by infinitesimally increased prices is finite and non-zero: θ2/2. These results underscore the importance of the law of motion`s specification in determining the total expected cost of executing SM. Of course, equation of Pt empirically implausible for a number of reasons. However, it serves a useful purpose in demonstrating the basic approach to best execution, as well as in rationalizing the rather common practice of parcelling a large trade into smaller pieces of equal size and submitting them at regular intervals over some fixed time span. This naive strategy is indeed optimal if the price-impact function and price dynamics of In the next section we present a closed-form solution for the best-execution strategy under a more complex price-impact function, one which depends both on the trade size and a serially-correlated state variable that proxies for information such as proprietary research or market conditions. With information, the best-execution strategy differs in important ways from the naive strategy S*t =. In particular, the best-execution strategy becomes a nontrivial function of the information variable and can sometimes exhibit counterintuitive trading patterns. Linear price impact with information Suppose that the price-impact function is linear in St as in Eq. (2.3), but now let Xt denote a serially-correlated state variable which also affects the execution price Pt linearly, hence where et and ηt are independent white noise processes with mean 0 and variances σ2e and σ2η , respectively. The presence of Xt in the law of motion for Pt captures the potential impact of changing market conditions or of private information about the security. For example, Xt might be the return on the BSE index, a common component in the prices of most equities. Broad market movements affect all securities to some degree, and c measures the sensitivity of this particular security to such market movements. Alternatively, Xt might represent some private information about the security, and γ the importance of that information for Pt. In particular, Xt may denote the output of an “alpha model” which incorporates stock-specific analysis to yield an excess return not yet impounded into market prices. In either case, the impact of Xt on the execution price, and the time series properties of Xt have important implications for the best-execution strategy. Having specified the linear price-impact function with information, the best-execution strategy and optimal-value function can be obtained by dynamic programming as before, and is given by For k=0,1,2…T-1 where And Since it is assumed θ>0, ak is positive, and ck and dk are negative for all k>0. The sign of bk can vary, but is positive for all k’0 if h, c, and o are all positive. In contrast to the case of a linear price-impact function with no information, the best-execution strategy varies over time as a linear function of the remaining shares WT-k and the information variable XT-k. In particular, the first term of the equation is simply the naive strategy of dividing the remaining shares WT-k at time T-k evenly over the remaining k+1 periods. The second term of the equation is an adjustment that arises from the presence of serially correlated information XT-k. Observe that this term vanishes if ρ=0. When ρ>0 this implies that XT-k is unpredictable, and while XT-k still has an impact on the current execution price, observing XT-k tells us nothing about expected future execution prices hence it can no longer affect the best-execution strategy. If ρ>0 and it is assumed, without loss of generality, that γ>0, then δx,k in the equation is also positive, implying that positive realizations of XT-k increases the number of shares purchased at T-k, ceteris paribus. This may seem counterintuitive at first because a positive XT-k necessarily increases the execution price PT-k by γXT-k, so why trade more? The answer may be found in the fact that XT-k is positively serially correlated, hence XT-k>0 implies that future realizations are likely to be positive which, in turn, implies additional expected increases in future execution prices. Therefore, although a positive XT-k makes it more costly to purchase shares in period T-k, this additional cost is more than offset by the sequence of expected future price increases that arise from positively serially-correlated information. Alternatively, if ρ<0 so that XT-k exhibits reversals, the equation shows that a positive realization of XT-k decreases the number of shares purchased, ceteris paribus: it is more expensive to trade in period T-k and XT-k is likely to reverse next period making it less expensive to trade then, hence it is optimal to trade less now. The impact of an increase in XT-k on expected best-execution costs may be measured explicitly by the derivative of the optimal-value function VT-k with respect to XT-k: Suppose γ and ρ are positive so that bk is positive. Since ck is always negative, the impact of an increase in XT-k on the expected best-execution cost depends on whether bkWT outweighs 2ckXT-k. For empirically plausible parameter values, the first term will generally dominate the second, hence increases in XT-k will typically increase the expected best-execution cost, a sensible implication given that an increase in XT-k increases current and all future expected prices. It is also not surprising that VT-k is an increasing function of WT-k for empirically plausible parameter values – the larger is the unexecuted portion of the initial block, the higher the expected best-execution cost. In the next section, a numerical example is provided to illustrate the behaviour of the best-execution strategy under several simulated scenarios. A numerical example Tables 1 provides illustrative numerical examples of the best-execution strategies under the linear price-impact function with information for three simulated realizations of the information variable Xt and pricing shocks et. The goal is to minimize the expected execution costs of a 100,000-share purchase over T=20 periods for a stock currently trading at P=$50, given the following parameter values: θ=5X10-5 γ=5.0 ρ=0.50 σ2ε=(0.125)2 σ2η=0.001 To develop some intuition for these parameters, observe that the no-impact cost of acquiring is 100,000XP0=$5M, and the expected full-impact cost is 100,000XE[P0 +θ + γX1 + ε1] = 100,000X$55 = $5.5 million since E[Xt]=0, hence θ is calibrated to yield an impact of $500,000 on a 100,000-share block purchase. It also follows that Hence the standard deviation of the information component is approximately 18 cents (per period). Finally, the standard deviation of et is calibrated to be 12.5 cents or one “tick” (per period)

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