A General Framework for Pairs Trading with a Control-Theoretic Point of View
aa r X i v : . [ q -f i n . S T ] A ug A General Framework for Pairs Tradingwith a Control-Theoretic Point of View
Atul Deshpande and B. Ross Barmish Abstract — Pairs trading is a market-neutral strategy that ex-ploits historical correlation between stocks to achieve statisticalarbitrage. Existing pairs-trading algorithms in the literature re-quire rather restrictive assumptions on the underlying stochas-tic stock-price processes and the so-called spread function . Incontrast to existing literature, we consider an algorithm forpairs trading which requires less restrictive assumptions thanheretofore considered. Since our point of view is control-theoretic in nature, the analysis and results are straightforwardto follow by a non-expert in finance. To this end, we describea general pairs-trading algorithm which allows the user todefine a rather arbitrary spread function which is used in afeedback context to modify the investment levels dynamicallyover time. When this function, in combination with the priceprocess, satisfies a certain mean-reversion condition, we deemthe stocks to be a tradeable pair. For such a case, we provethat our control-inspired trading algorithm results in positiveexpected growth in account value. Finally, we describe testsof our algorithm on historical trading data by fitting stockprice pairs to a popular spread function used in literature.Simulation results from these tests demonstrate robust growthwhile avoiding huge drawdowns.
I. I
NTRODUCTION
A pairs-trading algorithm is a market-neutral strategy whichexploits historical correlation between stocks to achievestatistical arbitrage. Such algorithms usually involve takingcomplementary positions in the two constituent stocks of thepair; i.e., long one stock and short the other. This occurswhen the stock prices, which are otherwise historically re-lated, temporarily diverge from their proven behavior. Undersuch conditions, the trader bets that the prices will move ina manner so as to return to their historical relationship. Ex-amples of correlated/paired stocks involve Exchange TradedFunds (ETFs), certain currency pairs or stocks of companiesin the same industry such as Home Depot and Lowes orWalMart and Target.Literature such as [1]–[5] deal with the more practicaldetails of pairs trading and include considerations of theperformance of pairs-trading methods, including the impactof transaction costs on profitability. A co-integration model between the logarithm of two stock prices is suggested in [6],and the results are used to determine the magnitude ofdeviation of the spread from its equilibrium, which in turntriggers appropriate long/short positions on the pair. Atul Deshpande is a graduate student working towards his doctoral disser-tation in the Department of Electrical and Computer Engineering, Universityof Wisconsin, Madison, WI 53706. [email protected] B. Ross Barmish is a faculty member in the Department of Electricaland Computer Engineering, University of Wisconsin, Madison, WI 53706. [email protected]
In [7] and [8], the logarithmic relationship between stockprices is modeled as an Ornstein-Uhlenbeck (O-U) process.The same model is used for the spread function of stockprices in the continuous-time setting in [9]. However, thespread in this case is a function of stock returns and notthe stock prices themselves. Whereas the literature discussedso far deals with just one spread for a pair of stocks,reference [10] deals with multiple spreads, each involvingbaskets of underlying securities. Each of the aforementionedspreads is assumed to be modelled on an independent O-Uprocess, and the paper proposes an optimal distribution ofinvestment among the spreads.More recent papers such as [11] and the
Cointelation modelin [13] build up on the co-integration model of the spreadfunction to design a stochastic control approach for pairstrading. To summarize, a vast preponderance of the theorydeveloped to date requires rather specific assumptions on theunderlying stock price processes and the spread function.Unlike the existing literature discussed above, the workdescribed in this paper applies to not just one specific modelof reversion for the spread as in [8]–[10] and [13]. Moreover,we do not limit the spread to be a particular function ofthe underlying stock prices as in [6], [8], [10] and [11]or make any assumptions on the underlying stock prices asin [11]. We propose a general framework which works foran arbitrary spread function of the underlying stock prices,provided it satisfies the minimal requirements of mean-reversion as represented by an expectation condition givenin the following section. Popular models like the Ornstein-Uhlenbeck process satisfy our conditions.
A. Control-Theoretic Point of View
This paper falls within the recently emerging body of workon the application of control-theoretic concepts to stocktrading; see [12] and its bibliography for an overview of therelevant literature. At the same time, we note that little hasbeen said about the application of control theory to pairstrading. Our setup for the problem at hand is shown inFigure 1. The stock price pair ( p ( k ) , p ( k )) is processedby the controller to create a spread function S ( p ( k )) onwhich trading is based. The controller determines the numberof shares ( n ( k ) , n ( k )) to be held in the respective stocksduring each trading period.One desirable property of our controller is to ensure positiveexpected change in account value at each step. In thefollowing section, we state our assumptions regarding themarket and the stocks which form a tradeable pair . Given ig. 1. Pairs Trading as a Stochastic Feedback Control Problem these assumptions, we provide a trading algorithm and provethat it results in positive expected growth in account value.We further test this algorithm on historical data by employinga spread function which is frequently used in literature, andsee robust gains and low drawdowns in simulations.II. G ENERAL F RAMEWORK FOR P AIRS T RADING
In this section, we state our assumptions of the market andthe stock price processes. We further define a mean-revertingspread , and provide a trading algorithm which guaranteespositive expected growth in account value at each step whenthe investor holds positions in the stocks.
A. Idealized Market Assumption
The trader is assumed to be working under the followingidealized market conditions. These requirements are nearlyidentical to those used in finance literature in the context of“frictionless markets” dating back to [14] and used in manypapers thereafter.
Zero Transaction Costs : The trader does not incur anytransaction costs, such as brokerage commission, transactionfees or taxes for buying or selling shares.
Price Taker : The trader is a price taker who is smallenough so as not to affect the prices of the stocks.
Perfect Liquidity Conditions : There is no gap between thebid/ask prices, and the trader can buy or sell any number,including fractions , of shares at the currently traded price.
Prices, Bounded Returns and Density Functions:
Thestocks under consideration have strictly positive prices p ( k ) and p ( k ) . In addition, the price vector p ( k ) is assumedto have a continuous probability density function which isunknown to the trader. For the stochastic stock-price process p ( k ) . = (cid:20) p ( k ) p ( k ) (cid:21) with p i ( k ) > for i = 1 , , the return on the i -th stockduring the k -th period is given by X i ( k ) . = p i ( k + 1) − p i ( k ) p i ( k ) for i = 1 , . Finally, we assume bounded returns.That is, there is a constant < γ < suchthat | X i ( k ) | ≤ γ for i = 1 , . B. Mean-Reverting Spread Function Assumption
To motivate the definition to follow, we imagine a pair ofstocks represented by the price process p ( k ) and view it asa tradeable pair if we can define a function S ( p ) such thatthe dynamics of the price processes cause S ( p ( k )) to be mean-reverting along sample paths. That is, the change inthe spread function ∆ S ( k ) . = S ( p ( k + 1)) − S ( p ( k )) is expected to decrease the absolute value of the spreadfunction. For example, when S ( p ( k )) is positive along asample path, we expect the price dynamics of its constituentstocks to force | S ( p ( k )) | to reduce and move towardszero, with a “pull” proportional to its distance from zero.In the formal definition to follow, this is manifested asthe assumption that there exists some η > such thatwhen S ( p ( k )) > , E [∆ S ( k ) | S ( p ( k ))] ≤ − η | S ( p ( k )) | .Similarly, when S ( p ( k )) < , we expect symmetricalbehaviour, that is, E [∆ S ( k ) | S ( p ( k ))] ≥ η | S ( p ( k )) | .In the above expression, the constant η is representative ofthe degree of reversion of the spread function. The expected“pull” is in a direction intended to reduce the spread, andthe magnitude of the pull gets higher as the spread valueincreasingly deviates from zero. Definition:
A given twice continuously differentiable func-tion S : (0 , ∞ ) × (0 , ∞ ) → R with no stationary points issaid to be mean-reverting with respect to the price process p ( k ) if there exists a constant η > such that the conditionalexpectation condition E [ sign ( S ( p ( k )))∆ S ( k ) | S ( p ( k ))] ≤ − η | S ( p ( k )) | is satisfied. C. The Trading Algorithm
With the above assumptions and definitions in place, wenow describe an algorithm for trading the pair of stocks.To this end, let V ( k ) be the value of the trading accountat stage k , with initial value V (0) and take ∇ S ( p ( k )) to be the gradient vector of the spread function withrespect to price p evaluated at p ( k ) . Define |∇ S ( p ( k )) | as the vector of the absolute value of the elements of thisvector. Let L > be the allowed leveraging factor, that is,the trader is allowed to invest up to LV ( k ) in absolute value. Trading Threshold:
We first describe the set of all possiblevalues that p ( k + 1) can attain under the bounded returnsassumption described in the previous section. Noting that | p i ( k + 1) − p i ( k ) | ≤ γp i ( k ) . and observing that p ( k + 1) is contained in the set B γ ( p ( k )) . = { p : | p i − p i ( k ) | ≤ γp i ( k ) for all i } , we choose the trading threshold, a function of p ( k ) , to be τ ( k ) . = 12 η max p ∈B γ ( p ( k )) (cid:12)(cid:12)(cid:12) ( p − p ( k )) T ∇ S ( p )( p − p ( k )) (cid:12)(cid:12)(cid:12) where ∇ S ( p ) is the Hessian of the spread function. hreshold-Based Trading Algorithm: Let the vector n ( k ) represent the number of shares held by the trader in thestocks at the k -th stage, with n i ( k ) being the number ofshares held in the i -th stock. The following rule specifiesthe trader’s holdings in the stocks: Under favorable tradingconditions, characterized by | S ( k ) | > τ ( k ) , we take n ( k ) = − λ ( k ) sign ( S ( p ( k ))) ∇ S ( p ( k )) where λ ( k ) . = LV ( k ) |∇ S ( p ( k )) | T p ( k ) is a positive constant. Otherwise, we take n ( k ) = 0 . Recalling that since S has no stationary points, λ ( k ) isalways well defined. Note that the idealized markets allowus to trade fractional quantities of shares, a negative n i ( k ) indicates shorting that many shares of the i -th stock,whereas a positive value indicates that the trader must buy as many shares. On the other hand, n ( k ) = 0 implies thatthe trader chooses not to hold any positions in the stocks.The formulae above guarantee that when | S ( p ( k )) | > τ ( k ) ,the trader is fully invested up to the limit allowed by thebroker, as total invested amount | n ( k ) | T p ( k ) = LV ( k ) . Forthe special case when L = 1 , the trader is said to be selffinanced. At stage k + 1 , the account value V ( k + 1) isredistributed in the stocks according to the trading algorithmdescribed above, but with spread S ( p ( k + 1)) as the inputvariable. Remark:
The change in account value during the k -thinterval is evaluated as ∆ V ( k ) = V ( k + 1) − V ( k )= n T ( k )∆ p ( k ) where ∆ p ( k ) . = p ( k + 1) − p ( k ) is the change in the pricevector during the k -th period. In the following section,we show that the trading algorithm described above yieldspositive expected growth in account value.III. M AIN R ESULT
According to our trading algorithm, for all k suchthat S ( p ( k )) ≤ τ ( k ) , the trader does not hold any positionsin the stocks, and hence ∆ V ( k ) = 0 . The followingtheorem, the main result of the paper, tells us that whenconditions are favorable for trading, the expected change inaccount value must be positive.
Theorem (Positive Expected Growth) : Let ( p ( k ) , p ( k )) be a stock-price pair with bounded returns | X i ( k ) | ≤ γ for i = 1 , and associated spread function S ( p ) whichis mean reverting with respect to sample paths p ( k ) . Thenthe trading strategy with threshold τ ( k ) guarantees that theexpected change in the account value, ∆ V ( k ) , is positive forall k for which trading occurs. That is, for all k such that P ( | S ( p ( k )) | > τ ( k )) > , it follows that E (cid:2) ∆ V ( k ) (cid:12)(cid:12) | S ( p ( k )) | > τ ( k ) (cid:3) > . A. Proof of the Postive Expected Growth Theorem
We first state and prove a preliminary lemma which willlater be used in the proof of the theorem:
Lemma (Bounded Approximation Error) : Along the samplepaths p ( k ) , the difference between change in the spreadfunction during the k -th period and its linear approximationhas bound (cid:12)(cid:12) ∆ S ( k ) − [ ∇ S ( p ( k ))] T ∆ p ( k ) (cid:12)(cid:12) ≤ ητ ( k ) . Proof:
We consider the first-order Taylor series of the spreadfunction for a given price change vector ∆ p from the pricepoint p ( k ) ; i.e., S ( p ( k ) + ∆ p ) = S ( p ( k )) + [ ∇ S ( p ( k ))] T ∆ p + R (∆ p ) where the error term R (∆ p ) is the first-order La-grange remainder. In accordance with the Taylor-Lagrangeformula [15], there exists a price point p ∗ and con-stant < h ∗ < , such that with p ∗ = p ( k ) + h ∗ ∆ p, it follows that R (∆ p ) = S ( p ( k ) + ∆ p ) − S ( p ( k )) − [ ∇ S ( p ( k ))] T ∆ p = 12 ∆ p T ∇ S ( p ∗ )∆ p. Recalling that the change in the price vector ∆ p ( k ) , althoughnot known a priori , is bounded, the range of possible valuesof p ( k + 1) is limited to the previously defined set B γ ( p ( k )) .The error term for this unknown ∆ p ( k ) is thus bounded by | R (∆ p ( k )) | ≤
12 max p ∈B γ ( p ( k )) (cid:12)(cid:12)(cid:12) ( p − p ( k )) T ∇ S ( p )( p − p ( k )) (cid:12)(cid:12)(cid:12) = ητ ( k ) . Recalling the formula for ∆ S ( k ) and the above boundon R (∆ p ( k )) , we obtain (cid:12)(cid:12) ∆ S ( k ) − [ ∇ S ( p ( k ))] T ∆ p ( k ) (cid:12)(cid:12) ≤ ητ ( k ) . Proof of the Theorem:
Recalling that the change in accountvalue ∆ V ( k ) = n T ( k )∆ p ( k ) and substituting for n ( k ) from the definition in the previoussection, when | S ( p ( k )) | > τ ( k ) , ∆ V ( k ) = − sign ( S ( p ( k ))) λ ( k ) (cid:0) [ ∇ S ( p ( k ))] T ∆ p ( k ) (cid:1) . Since we are only interested in proving that the sign of theexpected change in account value is positive, and λ ( k ) > is a constant for a given k , without loss of generality, weassume λ ( k ) = 1 . Thus, ∆ V ( k ) = − sign ( S ( p ( k ))) (cid:0) [ ∇ S ( p ( k ))] T ∆ p ( k ) (cid:1) . From the
Bounded Approximation Error lemma, we identifythe bounds ∆ S ( k ) − ητ ( k ) ≤ [ ∇ S ( p ( k ))] T ∆ p ( k ) ≤ ∆ S ( k ) + ητ ( k ) . sing these inequalities, we obtain sign ( S ( p ( k )))[ ∇ S ( p ( k ))] T ∆ p ( k ) ≤ sign ( S ( p ( k )))∆ S ( k )+ ητ ( k ) . Negating and taking expectation on both sides conditionedon S ( p ( k )) leads to a lower bound for the expected changein account value conditioned on S ( p ( k )) , namely E (cid:2) ∆ V ( k ) (cid:12)(cid:12) S ( p ( k )) (cid:3) = − E (cid:2) sign ( S ( p ( k ))) (cid:0) [ ∇ S ( p ( k ))] T ∆ p ( k ) (cid:1)(cid:12)(cid:12) S ( p ( k )) (cid:3) ≥ − E [ sign ( S ( p ( k )))∆ S ( k ) | S ( p ( k ))] − ητ ( k ) . Now invoking the mean-reversion assumption on S ( p ( k )) ,we obtain E (cid:2) ∆ V ( k ) (cid:12)(cid:12) S ( p ( k )) (cid:3) ≥ − E [ sign ( S ( p ( k )))∆ S ( k ) | S ( p ( k ))] − ητ ( k ) > η ( | S ( p ( k )) | − τ ( k )) . Let f S ( p ( k )) ( s ) be the probability density functionon S ( p ( k )) , perhaps discontinuous, induced by p ( k ) .Since P ( | S ( p ( k )) | > τ ( k )) > , the set I τ ( k ) . = { s : | s | > τ ( k ) and f S ( p ( k )) ( s ) > } is non-empty with non-zero length. Hence, notingthat E (cid:2) ∆ V ( k ) (cid:12)(cid:12) S ( p ( k )) (cid:3) > for all S ( p ( k )) ∈ I τ ( k ) and using the Law of Total Expectation , we obtain E (cid:2) ∆ V ( k ) (cid:12)(cid:12) | S ( p ( k )) | > τ ( k ) (cid:3) = Z s ∈ I τ ( k ) E (cid:2) ∆ V ( k ) (cid:12)(cid:12) S ( p ( k )) (cid:3) f S ( p ( k )) ( s ) ds > . IV. S
IMULATIONS AND R ESULTS
In this section, we first describe our general simulationsetup. Then we use a candidate pair of securities and spreadfunction, and simulate our trading algorithm using historicaldata. Finally, we present and discuss the results and comparethe performance of our algorithm to buy-and-hold strategieson the constituent securities of the pair. All the simulationsto follow use the leveraging factor L = 1 ; that is, we assumea self-financed account. Additionally, the algorithm ensuresthat the trader is fully leveraged whenever he takes positionsin the stocks. A. Simulation Setup
To test our algorithm on historical data, we first selectcandidate securities for pairs trading and a candidate spreadfunction. Then, via use of historical data, we fit the securityprices to the candidate spread function, and check for statis-tical satisfaction of the mean-reversion property.Once trading begins, using the data withheld, we use a stag-gered sliding window method to estimate model parameterson the fly. This departure from strict application of the theoryis done because the relationship between the stock pricesis not necessarily stationary in practice. In this framework, we use training windows of length N , followed by tradingwindows of length m . At the end of the training window,current model parameters specific to the spread function areestimated. Then, this model is used to calculate the spreadfunction and the threshold during the trading window whichimmediately follows. In our simulations, we use N = 40 and m = 5 .During the training window, we also calculate the returnsusing the prices of the securities. The maximum absolutevalue of these returns leads to our estimate ˆ γ ( k ) , namely, ˆ γ ( k ) . = max i =1 , k − N ≤ j
The pair of securities chosen for testing were the exchange-traded funds Direxion Daily FTSE China Bull 3X ETF(YINN) and the Direxion Daily FTSE China Bear 3X ETF(YANG). These are related to the same market, namelyChina, albeit with different outlooks. Also, since both theETFs are 3X leveraged in the markets, they are more volatile,leading to more frequent trading opportunities. Figure 2shows the daily closing prices of these two securities forthe period from July 1, 2011 to December 31, 2015. Notingthe price corrections made by the fund management inYANG around trading periods and respectively, wecorrespondingly adjust these prices before using them foranalysis.First, we select the co-integration model used in prior liter-ature for the spread function; namely S ( p ) = log ( p ) − βlog ( p ) − µ.
200 400 600 800 1000 1200
Trading period $0$50$100$150 p ( k ) YINNYANG
Fig. 2. Daily Closing Prices of YINN and YANG
Trading period -0.100.10.20.30.40.50.60.70.80.9 ˆ η ( k ) Fig. 3. ˆ η ( k ) Estimated from the Spread Function
Once trading has commenced, we fit the price data to ourchosen model using a regression on S above to obtain theestimates ˆ β and ˆ µ during the training window.Finally, using this model, we compute the spread functionretrospectively over the training window, and also use itduring the trading window. We use the constructed spreadfunction over the training window to estimate ˆ η as describedbefore. Figure 3 shows the estimated ˆ η ( k ) versus tradingperiod. We note that a near-zero or negative ˆ η ( k ) is inter-preted as unfavorable conditions for pairs trading. That is, therequirement | S ( p ( k )) | > τ ( k ) becomes nearly impossible tosatisfy.We now use our knowledge of ˆ β to compute an estimate ofthe Hessian ∇ S ( p ( k )) using the formula ∇ ˆ S ( p ( k )) . = " − ˆ β ( k ) p ( k ) p ( k ) . The running estimate ˆ η ( k ) and ∇ ˆ S ( p ( k )) are used tocompute ˆ τ ( k ) . For simplicity of computation, in the calcu-lations to follow, we approximate the Hessian as a constant Trading period ˆ τ ( k ) Fig. 4. ˆ τ ( k ) Calculated Using ˆ η , ˆ γ ( k ) and ∇ S ( p ) over B γ ( p ( k )) and work with the estimate ˆ τ ( k ) = ( ˆ γ η (cid:12)(cid:12)(cid:12) p ( k ) T ∇ ˆ S ( p ( k )) p ( k ) (cid:12)(cid:12)(cid:12) if ˆ η > ∞ if ˆ η ≤ Figure 4 shows the trend for ˆ τ ( k ) over time. The y-axis isbroken to better represent the variation in the lower valuesof ˆ τ ( k ) while simultaneously capturing the occasional highvalue. Note that the plot of ˆ τ ( k ) is discontinuous in k ,and the breaks indicate times when ˆ τ ( k ) = ∞ ; this occurswhen ˆ η ≤ . The values of the computed spread functionand the ˆ τ ( k ) are compared to determine whether conditionsare favorable for trading, and if they are, the share holdingsare determined in accordance with the trading rule presentedin the previous section. Results:
To evaluate the performance of our trading algo-rithm, we consider three separate scenarios. The first two ofthese correspond to a straightforward buy-and-hold begin-ning with $10,000 worth of YINN securities and $10,000worth of YANG securities respectively. The third scenariocorresponds to using our threshold-based algorithm to tradethe two securities with a starting account value of $10,000.Figure 5 shows the performance of the three scenarios duringthe period under consideration. As seen in the figure, a traderinvested solely in YANG initially sees a 78% profit, buteventually loses nearly 88% of the account value. On theother hand, a trader invested solely in YINN loses 41% ofthe account value after seeing a peak profit of 126%. Bydesign, these securities are bullish and bearish respectivelyon the same index, and in an ideal world, one would expectthe losses in one portfolio to be offset by profits in the other.But as seen from Figure 5, both these scenarios eventuallyturn out to be loss-making. This can be explained by thefact that the ETFs often fail to accurately track their targetindices, and the operation of leveraged ETFs comes withadditional risks and overheads, as explained in [16].The portfolio which trades using our algorithm shows 60%
200 400 600 800 1000 1200$0$5000$10000$15000$20000$25000
Trading period A cc o un t V a l u e PortfolioYINNYANG
Fig. 5. Pairs Trading Compared to Performance of YINN and YANG profits over the same period. It is also noteworthy thatdespite the high volatility in the securities, the pairs-tradingstrategy results in minimal drawdowns. The coincidence ofa majority of the gains shown by the portfolio with theperiods when ˆ η ( k ) achieves high values in Figure 3 pointsto the potential for future work involving the efficacy of η as an indicator of fit quality between a spread function andthe price data: see Section 5 for further discussion. Also,during the worst period for the YINN portfolio (900-1000),the pairs never trade as a result of a high ˆ τ ( k ) . This suggestsa possible explanation as to why we avoid the disastrousdrawdowns which wiped out the gains in the buy-and-holdtrading scenarios used for comparison.V. C ONCLUSION
In this paper, we presented an algorithm for trading a pairof securities under rather weak hypotheses on the the priceprocess and the spread function being used. Under the as-sumptions of bounded returns and mean-reversion describedin Section 2, we described a threshold-based trading schemewhich guarantees positive expected growth in the accountvalue. To illustrate how the trading algorithm works inpractice, we provided simulation results involving a pairof exchange-traded funds. Our results show robust growthcompared to an alternative buy-and-hold strategy which onemight use on the constituent securities.The first point to note is that the theory presented in thispaper includes three assumptions which were made solely forthe purpose of simplifying the exposition. First, we assumedthat only two stocks are involved in the spread. In fact, ifwe consider a spread which is comprised of more than twostocks, the analysis of the account value is nearly identicalto that given here. This type of more general portfolio-likeproblem will be pursued in our future work. The secondassumption we made is that each price p i ( k ) is a randomvariable with a continuous probability density function. Infact, the proof of the main theorem can easily be extended tohandle the case when only a probability measure is available. Finally, we assumed that the stocks have bounded returns.However, even when these assumptions are dropped, webelieve it should be possible to analyze the case when thereturns are bounded with an appropriately high probability,and obtain similar results.By way of future research, further study of the estimatedmean-reversion parameter ˆ η ( k ) seems promising. Given apair of securities, by observing this variable using trainingdata, it would be of interest to study the extent to which ˆ η ( k ) is a predictor as to the “promise” of a pairs trade. A secondtopic for future study is that of trading frequency. Given thatour simulations were carried out using daily closing prices,it would be of interest to see how our algorithm performswhen prices arrive more frequently. Studies of this natureshould be possible to carry out using available tick data.More generally, there may be a number if important opti-mization problems associated with the issues raised aboveand our approach to pairs-trading problems. From a practicalperspective, it would be of interest to include a numberof considerations such as margin, risk-free securities andtransaction costs in future analyses.R EFERENCES[1] M. Whistler,
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