A Generalization of the Robust Positive Expectation Theorem for Stock Trading via Feedback Control
AA Generalization of the Robust Positive ExpectationTheorem for Stock Trading via Feedback Control
Atul Deshpande and B. Ross Barmish Abstract — The starting point of this paper is the so-called
Robust Positive Expectation (RPE) Theorem, a result whichappears in literature in the context of Simultaneous Long-Shortstock trading. This theorem states that using a combination oftwo specially-constructed linear feedback trading controllers,one long and one short, the expected value of the resultinggain-loss function is guaranteed to be robustly positive withrespect to a large class of stochastic processes for the stockprice. The main result of this paper is a generalization of thistheorem. Whereas previous work applies to a single stock, inthis paper, we consider a pair of stocks. To this end, we maketwo assumptions on their expected returns. The first assumptioninvolves price correlation between the two stocks and the secondinvolves a bounded non-zero momentum condition. With knownuncertainty bounds on the parameters associated with theseassumptions, our new version of the RPE Theorem providesnecessary and sufficient conditions on the positive feedbackparameter K of the controller under which robust positiveexpectation is assured. We also demonstrate that our resultgeneralizes the one existing for the single-stock case. Finally, itis noted that our results also can be interpreted in the contextof pairs trading. I. I
NTRODUCTION
The primary motivation for this paper is the so-called
RobustPositive Expectation
Theorem for Simultaneous Long-Short(SLS) trading of a single stock; see [1] and [2]. This resultis a stochastic version of an arbitrage theorem originallyintroduced for continuously differentiable stock prices in [3].It tells us that a combination of two controllers, one forthe long trade and one for the short trade, provides aguarantee that the expected value of the gain-loss functionis robustly positive with respect to a family of underlyingstock prices which are Geometric Brownian Motions (GBM)with unknown drift µ and unknown volatility σ . Whereasrobust portfolio balancing strategies have been presented inpapers such as [4], the earliest contribution we find on robustpositive expectation can be found in papers such as [5] andother related work by the same authors, such as [6]. Incontrast to the above, we focus here on the linear feedbackcontrol framework which is covered in papers such as [1]–[3]and [7]–[12].The body of literature motivating this paper includes anumber of flavors for the underlying stock prices and thecontrol structure. For example, in reference [9], robustnessresults are given for stock prices generated by Merton’s jump 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] diffusion model and references [10]–[12] address variants ofthe SLS controller for the discrete-time case. To concludethis brief survey, we note that most of the literature citedabove falls within the robust control paradigm formulatedin [13]. Less closely related to this line of research arereferences [14]–[20], which, unlike the papers on robustcontrol, are based on rather specific stock-price models. Forexample, in [14], stock prices are modeled as GBM processescoupled by a finite-state Markov chain, and in [15], tradingsignals are modeled as Ito processes based on GBM models.On the other hand, in [16]–[20], either the asset being tradedor a relationship between multiple assets, is modeled as amean-reverting Ornstein-Uhlenbeck process.Whereas the SLS literature focuses on trading shares of asingle stock, in this paper, we consider scenarios involvingsimultaneously trading two stocks. One simple method toextend the single-stock theory to two stocks would be toimplement separate SLS controllers for each stock. Thatis, a robustly positive expected (RPE) gain for each stockindividually implies that the pairs trade has RPE too. Inthis paper, we study a different approach for trading a pair,where one arm of a controller goes long on one of thestocks and the other arm goes short on the other stock.This new control structure is motivated by the desire toexploit correlated price behavior between two stocks ratherthan treating them separately. To this end, we make certainassumptions on the stock dynamics, namely the satisfactionof directional correlation and bounded momentum condi-tions. Letting g ( N ) denote the cumulative gain or loss upto stage N , we describe a generalized SLS controller withfeedback parameter K , which is constructed using the knownuncertainty bounds. Our main result for the two-stock caseprovides necessary and sufficient conditions on K underwhich robust satisfaction of the condition E [ g ( N )] > isguaranteed with respect to parameter variations associatedwith the conditions above. We also show how these resultsgeneralize the RPE Theorem for the single-stock scenario.Given that our formulation is aimed at two stocks withcorrelated price dynamics, this paper provides a new per-spective on “pairs-trading” literature. Unlike this literature,however, we do not include assumptions of price reversion,either through reliance on models such as those of Ornstein-Uhlenbeck as seen in [17]–[20] or more general models forthe spread function as in [21] and [22]. Existing Result Being Generalized:
The take-off point forthis paper is the Robust Positive Expectation Theorem for anSLS controller used to trade a single stock. Indeed, assuminga stock with prices represented by a discrete-time stochastic a r X i v : . [ q -f i n . S T ] M a r ig. 1. The SLS Controller price process S ( k ) over k = 0 , , . . . N , let ρ ( k ) denote thereturn in the k -th period; i.e., ρ ( k ) . = S ( k + 1) − S ( k ) S ( k ) , are taken to be independent, with an unknown con-stant mean µ . = E [ ρ ( k )] for k = 0 , , , . . . , N − .Given the setup above, the Simultaneous Long-Short (SLS)controller, depicted in Figure 1, determines the net invest-ment level I ( k ) in the stock at stage k . This is accom-plished by summing the outputs of two linear time-invariantcontrollers. The first uses an initially positive I ( k ) for thelong trade and the second uses an initially negative I ( k ) for the short trade. To elaborate, a long position I ( k ) > represents the trader holding the appropriate number ofshares of the stock and making profit as S ( k ) increases.On the other hand, a short position I ( k ) < leads to aprofit when there is a decrease in the stock price. We take I ( k ) . = I + Kg ( k ) ; I ( k ) . = − I − Kg ( k ) with initial investment I > , feedback parameter K > and g ( k ) , g ( k ) being the cumulative gain-loss functionsof the two controllers, with initial values g (0) = g (0) = 0 .Subsequently, the trader’s net investment level in thestock I ( k ) is obtained as I ( k ) = I ( k ) + I ( k ) = K ( g ( k ) − g ( k )) . The robust positive expectation result from which we take offtells us: Except for the degenerate break-even case obtainedwith µ = 0 , the cumulative gain-loss function g ( k ) = g ( k ) + g ( k ) is robustly positive in expectation. That is, without knowl-edge of µ , the condition E [ g ( N )] > is satisfied. Further-more, as seen in existing work such as [10], the expectedgain-loss function is explicitly given by E [ g ( N )] = I K (cid:104) (1 + Kµ ) N + (1 − Kµ ) N − (cid:105) , with the positivity of the above expression guaran-teed for all non-zero µ by virtue of the basic factthat (1 + x ) N + (1 − x ) N > for all x (cid:54) = 0 and N ≥ .Since this result is the starting point for our current work,for the sake of a self-contained exposition, we provide anelementary derivation of the formula for E [ g ( N )] above inthe appendix. II. T WO -S TOCK S ETUP AND M ARKET A SSUMPTIONS
In this section, we consider two stocks instead of one andnow describe the assumptions which are in force. Theseassumptions are not only on the price processes for the twostocks, but also on the market within which we operate.
Stock Price Dynamics:
We consider stocks S and S withstochastically varying prices S ( k ) and S ( k ) respectivelyfor k = 0 , , , . . . , N and N > . The returns on the stocks,given by ρ i ( k ) . = S i ( k + 1) − S i ( k ) S i ( k ) for i = 1 , are respectively assumed to be in-dependent for k = 0 , , , . . . , N − , with con-stant means µ . = E [ ρ ( k )] , µ . = E [ ρ ( k )] . The relationshipbetween these returns are assumed to satisfy the follow-ing conditions:
Directionally Correlated Returns Assumption:
We as-sume that there exists a constant β (cid:54) = 0 suchthat µ = βµ with β = (1 + ε ) β , β (cid:54) = 0 known to the trader and ε uncertain, withknown bounds ≤ ε ≤ ε max . Note that the above impliesthat sign β = sign β for all admissible ε , that is, there isuncertainty in the magnitude of β , but not its sign. Bounded Non-Zero Momentum Assumption:
It is as-sumed that there are positive constants µ min and µ max known to the trader, such that µ min ≤ | µ | ≤ µ max . Remarks:
The bounds on µ above in combination with theassumption of directionally correlated returns lead to boundson µ given by < | β | µ min ≤ | µ | ≤ (1 + ε max ) | β | µ max . For the special case when the two stocks are one and thesame, β ≡ with ε max = 0 , and this formulation reducesto a restricted version of the single-stock problem describedin literature. Idealized Market Assumptions:
The trading is assumed tobe carried out under idealized market conditions. That is,there are no transaction costs such as brokerage commission,fees or taxes for buying or selling shares. For such a market,it is also assumed that there is perfect liquidity; i.e., there isno gap between the bid and ask prices, and the trader can buyor sell any number, including fractions , of shares as desired atthe currently traded price. That is, the trader is a price taker,investing small enough amounts so as not to affect the pricesof the stocks. These assumptions are similar to those madein finance literature in the context of “frictionless markets”going as far back as [23]. everage, Margin and Interest:
In practice, brokers usu-ally impose a limit on the investment levels based on theaccount value V ( k ) . For example, a trader may be boundby the constraint | I ( k ) | + | I ( k ) | ≤ γV ( k ) , where γ ≥ denotes the so-called leverage which is extended. In thetheory to follow, it is assumed that leverage is never alimiting factor. That is, sufficient resources are available tocover any desired investment levels in the respective stocks.Accordingly, issues involving margin interest are not in play.Finally, it is noted that there is no mention of ordinaryinterest on idle cash in the trading account. The explanationfor this is that the results in this paper focus entirely onthe gains and losses g ( k ) and g ( k ) which are attributableto trading. III. T HE T WO -S TOCK C ONTROLLER
Beginning with stocks S and S , the two-stock generalizedSLS controller which we now describe has the same structureas the one in Figure 1. However, for this more general case,we have both stock prices as inputs to the controller, allowfor different initial investments I , , I , instead of I andhave K , K instead of K . The linear feedback controllersinvesting I ( k ) in S and I ( k ) in S are given by I ( k ) . = I , + K g ( k ) ; I ( k ) . = − I , − K g ( k ) , with parameters I ,i and K i chosen by the trader as explainedbelow and with g ( k ) and g ( k ) being the cumulativegain-loss functions of the investments with initial valuesof g (0) = g (0) = 0 . Choice of Parameters:
We first select initial investmentparameter I > and feedback parameter K > . Then, thetwo controllers, defined in terms of these two parameters,have initial investment levels I , . = I ; I , . = I β and feedback parameters K . = K ; K . = Kβ . Remarks:
We observe that the choices of I ,i and K i above compensate against the differing momenta of thetwo stocks. When β > , notice that the initial investmentssatisfy I (0) > and I (0) < . However, the signs of oneor both these quantities may change at a later stage k . Thus,despite being initially long on S and short on S , our stockpositions at later stage k can be different. A similar statementcan be made for β < . Starting Point for the Analysis:
A simple adaptation of thesingle-stock formula in Section I leads us to E [ g ( N )] = I , K (cid:104) (1 + K µ ) N − (cid:105) + I , K (cid:104) (1 − K µ ) N − (cid:105) = I K (cid:104) (1 + Kµ ) N + (1 − Kµ (1 + ε )) N − (cid:105) . = G N ( K, µ , ε ) for the two-stock case. Note that with ε max = 0 , theformula above reduces to the one for the single-stock case.The notation G N ( K, µ , ε ) above making the dependenceon K , µ and ε explicit will be useful in the sequel forpresentation and proof of the results.IV. M AIN R ESULTS
In the theorem to follow, we characterize the set of K leading to the satisfaction of the robust positive expectationof g ( N ) with respect to µ and ε within their respectivebounding sets. We also provide a corollary which leads tothe recovery of the existing single-stock result when β = 1 and ε max → . All the proofs for the results in this sectionare furnished in Section V. Robust Positive Expectation Theorem:
Suppose twostocks S and S have directionally correlated re-turns and satisfy the bounded non-zero momentum con-dition, with associated uncertainty bounds ≤ ε ≤ ε max and < µ min ≤ | µ | ≤ µ max . Then, for N odd, the two-stock generalized SLS controller with K > guaranteesrobust satisfaction of the condition G N ( K, µ , ε ) > forall admissible µ and ε , if and only if G N ( K, µ min , ε max ) > G N ( K, µ max , ε max ) > . For N even, robust satisfaction is guaranteed ifand only if either K > N − /µ min , or whenboth K ≤ / ( µ min (1 + ε max )) and G N ( K, µ min , ε max ) > . Remarks:
To accurately estimate the set of K whichguarantees satisfaction of the robust positive expectationconditions above, as demonstrated in Section VI, we cansimply conduct a parameter sweep over a suitably large rangewith K > . Unlike existing results for the single-stockcase, one possible outcome is that the set of K satisfyingthe theorem requirements is empty. This can occur when theuncertainty bounds are “too large.” The corollary below isapropos to the special case when both stocks are one and thesame; i.e. β = 1 and we consider ε max → to recover theexisting result for the single-stock case is presented below. Corollary:
Given any
K > , for ε max suitably small,robust satisfaction of the condition E [ g ( N )] > for alladmissible µ and ε is guaranteed. V. P
ROOF OF THE T HEOREMS
This section can be skipped by the reader seeking toavoid technicalities. Recalling that G N ( K, µ , ε ) repre-sents E [ g ( N )] for a fixed K , µ and ε , the starting point forour analysis is the fact that the robust positive expectationproperty holds if and only if G N ( K, µ , ε ) > for all admissible pairs ( µ , ε ) . We first present some nota-tion, a preliminary definition and a few lemmas which willbe instrumental to the proofs to follow. Indeed, for fixed θ ,we define the polynomial G θ ( ε ) . = (1 + θ ) N + (1 − θ (1 + ε )) N − or ε ≥ . Note the similarity between theexpressions for G θ ( ε ) and G N ( K, µ , ε ) . Indeed,when θ = Kµ , G θ ( ε ) > if and only if G N ( K, µ , ε ) > . Definition (Critical Uncertainty Bound):
For a fixed θ , the critical uncertainty bound is defined as ε c ( θ ) . = inf { ε > G θ ( ε ) ≤ } . Remarks:
Given K and µ , the quantity ε c ( Kµ ) tells usthe smallest ε for which the expected gain G N ( K, µ , ε ) isnon-positive. Using the convention that the infimum over anempty set is + ∞ , if θ < , since G θ ( ε ) > for all ε > , weobtain ε c ( θ ) = + ∞ . Furthermore, when θ = 0 , G θ ( ε ) = 0 for all ε > , thus ε c (0) = 0 . Finally, for θ > , noticethat the continuity of G θ ( ε ) in combination with the factthat G θ (0) > ensures that ε c ( θ ) > . The lemmas tofollow more fully characterize the function ε c ( θ ) for thenon-trivial case when θ > . Notational Convention:
In the proof to follow, there arenumerous occasions where root operations are required. Toavoid ambiguities attributable to non-unique or complexroots, the following notational conventions are in force:If
X > is real and N is a positive integer, then X /N istaken to be the unique positive N -th root of X . For X < and N odd, we take X /N = −| X | /N which is obtainedusing the definition for the positive variable case. We provideno definition when X < for N even since this caseis never encountered in the sequel. There are also caseswhen we consider expressions of the form X /N − m for aninteger m . In this case, this quantity is defined as Y /N ,where Y = X − mN , and evaluated in a manner consistentwith the convention above. Lemma 1 (Critical Uncertainty) : Given θ > , it fol-lows that ε c ( θ ) = + ∞ N even , θ > N − − ( − (1+ θ ) N ) N θ − otherwise . Proof:
For N even and θ > /N − , since G θ ( ε ) ≥ (1 + θ ) N − for all ε > , it follows that the set of ε for which G θ ( ε ) ≤ is empty. Hence, ε c ( θ ) = + ∞ . For all other θ > for N odd or even, ε c ( θ ) must be the smallest finite ε > solvingthe equation G θ ( ε ) = 0 . This is easily found to be ε c ( θ ) = 1 − (cid:16) − (1 + θ ) N (cid:17) N θ − . Definition:
To facilitate the proof of the following lem-mas, we define the function f ( θ ) on the set of posi-tive θ (cid:54) = 2 /N − as f ( θ ) . = (cid:16) − (1 + θ ) N (cid:17) N − θ (1 + θ ) N − (cid:16) − (1 + θ ) N (cid:17) N − . Furthermore, in the sequel, we also use its derivativefor θ (cid:54) = 2 /N − . This is calculated to be f (cid:48) ( θ ) = 2 ( N − θ (1 + θ ) N − (cid:104) − (1 + θ ) N (cid:105) N − . Lemma 2 (Monotonicity) : For < θ < /N − , thefunction ε c ( θ ) is monotonically increasing with derivative ε (cid:48) c ( θ ) = f ( θ ) /θ . Proof:
In the interval < θ < /N − , a straightforwardcalculation leads to ε (cid:48) c ( θ ) as given above. To complete theproof, it suffices to show that f ( θ ) > in the intervalof interest. Since f (0) = 0 , and f (cid:48) ( θ ) > in theinterval, by inspection, it follows that f ( θ ) > for all θ in the interval. Lemma 3 (Maximality) : For N odd and θ > /N − , thefunction ε c ( θ ) has a unique stationary point where it attainsits maximum value, with its derivative ε (cid:48) c ( θ ) = f ( θ ) /θ . Proof:
For N odd and θ > /N − , we show that ε c ( θ ) initially increases, thereafter achieves a maximum and de-creases as θ continues to increase. To this end, it suffices toshow that ε (cid:48) c ( θ ) is initially positive, later crosses zero andthereafter stays negative. For θ > /N − , straightforwardcalculation leads to ε (cid:48) c ( θ ) as given above. Indeed, it nowsuffices to show that its numerator f ( θ ) behaves in themanner described above. Indeed, for θ > /N − , f ( θ ) tendsto infinity as θ approaches /N − from above. However,it decreases monotonically thereafter as f (cid:48) ( θ ) is negativefor all θ > /N − . Finally, the fact that f ( θ ) eventuallybecomes negative is immediate since lim θ → + ∞ f ( θ ) = − . Hence, ε (cid:48) c ( θ ) , while initially positive for θ > /N − , de-creases to cross zero and turns negative, which in turn impliesthat ε c ( θ ) increases to its maximum and decreases thereafteras θ increases to infinity. This completes the proof. Proof of Robust Positive Expectation Theorem:
To provenecessity, we assume G N ( K, µ , ε ) > for all admissible µ and ε , and consider two cases. For the case when N is odd,the claimed necessary condition G N ( K, µ min , ε max ) > G N ( K, µ max , ε max ) > follows trivially from the fact that ( µ min , ε max ) and ( µ max , ε max ) are both admissible pairs. Forthe case when N is even, the second necessarycondition G N ( K, µ min , ε max ) > is immediate usingthe same argument as for N odd. To complete the proofof necessity, we assume K ≤ (2 /N − /µ min and mustshow that K ≤ µ min (1 + ε max ) . Indeed, proceeding by contradiction, if
K > µ min (1 + ε max ) , t is straightforward to verify that G N (cid:18) K, µ min , Kµ min − (cid:19) = I K (cid:104) (1 + Kµ min ) N − (cid:105) ≤ , which contradicts the assumed positivity of G N ( K, µ , ε ) .To establish sufficiency, we assume feedback gain K > satisfying the conditions in the theorem and must showthat G N ( K, µ , ε ) > for all admissible µ and ε . Tothis end, we choose an arbitrary admissible pair ( µ , ε ) , anddivide the analysis into three cases: Case 1 ( µ < ): In this case, whether N is even or odd, thecondition G N ( K, µ , ε ) > is trivially satisfied by virtue ofthe fact that G N ( K, µ , ε ) ≥ I K (cid:104) (1 − K | µ | ) N + (1 + K | µ | ) N − (cid:105) > with the last inequality following from the single-stockresult; see Section I and the Appendix. Case 2 ( µ > , N odd): Assuming satisfaction ofthe theorem requirements G N ( K, µ min , ε max ) > and G N ( K, µ max , ε max ) > , and noting that ∂G N /∂ε < ,for ε ≤ ε max , we obtain G N ( K, µ min , ε ) > and G N ( K, µ max , ε ) > . Using this fact in conjunctionwith the definition of the critical uncertainty bound ε c ( θ ) ,we obtain ε max < min { ε c ( Kµ min ) , ε c ( Kµ max ) } . Invoking Lemma , we see that ε c ( θ ) is increasingwhen < θ < /N − and from Lemma , ε c ( θ ) mono-tonically decreases after achieving a unique maximum atsome θ > /N − . Thus, irrespective of the position of thismaximal point, considering θ = Kµ min and θ = Kµ max ,we have ε c ( Kµ ) ≥ min { ε c ( Kµ min ) , ε c ( Kµ max ) } for all µ min ≤ µ ≤ µ max . Thus, for the arbitrarilychosen µ , it follows that ε max < ε c ( Kµ ) , which impliesthat G N ( K, µ , ε ) > . Case 3 ( µ > , N even): The first subcase which weconsider is when
K > (2 /N − µ min holds. To show that G N ( K, µ , ε ) > , we first note that theabove strict inequality and the positivity of µ and N beingeven implies that G N ( K, µ , ε ) ≥ I K (cid:104) (1 + Kµ ) N − (cid:105) > . For the second subcase with K ≤ /N − µ min , K ≤ µ min (1 + ε max ) and G N ( K, µ min , ε max ) > , it follows that − (1 + Kµ min ) N ≥
0; 1 − Kµ min (1 + ε ) ≥ and − (1 + Kµ min ) N < (1 − Kµ min (1 + ε )) N . Hence, (cid:104) − (1 + Kµ min ) N (cid:105) N < − Kµ min (1 + ε max ) , which, upon rearrangement and use of Lemma 1 leads to ε max < − (cid:16) − (1 + Kµ min ) N (cid:17) N Kµ min − ε c ( Kµ min ) . Now, in the sub-subcase where µ > (2 /N − /K , we have G N ( K, µ , ε ) ≥ I K (cid:104) (1 + Kµ ) N − (cid:105) > . In the other sub-subcase, µ min ≤ µ ≤ (2 /N − /K , fromLemma 2, we know that ε c ( Kµ min ) ≤ ε c ( Kµ ) . There-fore, ε max < ε c ( Kµ ) , implying that for the pair ( µ , ε ) ,we have G N ( K, µ , ε ) > . Proof of Corollary:
Given
K > , it suffices to showthat for ε max sufficiently small, the requirements ofthe RPE theorem are satisfied. For N odd, this fol-lows since G N ( K, µ min , ε max ) and G N ( K, µ max , ε max ) are continuous functions of ε max , G N ( K, µ min , > and G N ( K, µ max , > . That is, there exists a δ o > such that for ε max < δ o , G N ( K, µ min , ε max ) > G N ( K, µ max , ε max ) > . For N even and ε max suitably small, the condition /N − ≤
11 + ε max is easily seen to be satisfied. Now, arguing as in the case of N odd, for ε max suitably small, we again obtain G N ( K, µ min , ε max ) > . Thus, there exists δ e > such that for ε max < δ e , thesufficient conditions for N even are satisfied.VI. I LLUSTRATIVE E XAMPLE
This section demonstrates the construction and use of thetwo-stock controller for a toy example with GeometricBrownian Motion (GBM) as the underlying price process.For the first stock, the discrete-time GBM which we use fordaily updates is described by S ( k + 1) − S ( k ) S ( k ) = µ + σ w ( k ) where µ is the drift, σ the volatility and the w ( k ) areindependent standard normal random variables. To illustratethe application of the Robust Positive Expectation Theo-rem, we begin with uncertainty bounds µ min = 0 . and µ max = 0 . . Assuming each time step above rep-resents a daily return, these bounds correspond to varia-tions of and respectively on an annualized basis.In addition, the two stocks are assumed to be direction-ally correlated with a nominal β = 1 and uncertaintybound ε max = 0 . ; i.e., ≤ β ≤ . . Therefore, the discrete-time GBM for the second stock is described by uncertaindrift µ = (1 + ε ) µ and volatility σ as S ( k + 1) = (1 + (1 + ε ) µ + σ w ( k )) S ( k ) ith the w ( k ) being independent random variables, eachhaving standard normal distribution. It is important to notehere that the trader does not know the direction of theunderlying stock-price movement. That is, the sign of µ is unknown; only the bounds on | µ | are available. In theanalysis to follow, we take N = 125 , which represents, giventhe daily update equations, about six months of trading. Controller Design:
Beginning with initial investment I = 10 , in dollars, we seek to find a suitable K > satisfying the requirements of the RPE Theorem. Since N isodd, we work with the inequalities G N ( K, µ min , ε max ) > G N ( K, µ max , ε max ) > . For the given uncertainty bounds, we obtain the conditions (1 + 0 . K ) + (1 − . K ) > . K ) + (1 − . K ) > as being necessary and sufficient for robust positive expecta-tion. Conducting a parameter sweep with K > , the inequal-ities above are satisfied if and only if . < K < . Some Practical Considerations:
In this subsection, we con-tinue the analysis of the example above by introducing somepractical considerations into a simulation not covered by thetheorem. Recalling the discussion of leverage in Section II,we now take γ = 2 , use K = 25 and assume an initialaccount value V (0) = 10 , in dollars. Then, to remain incompliance with the leverage constraint, any time the con-troller in Section III encounters | I ( k ) | + | I ( k ) | > γV ( k ) ,the two investments are scaled back using the formula I i ( k ) = I i ( k ) | I ( k ) | + | I ( k ) | γV ( k ) ; i = 1 , . Controller Performance Over a Sample Path:
For oursimulations, we use daily volatilities of σ = σ = . ,and admissible GBM drift parameter µ = − . ,with ε = 0 . . This corresponds to β = 1 . between the twostocks and a drift of µ = 0 . . First, we illustrate thecontroller performance for a single sample path for each ofthe two stock prices; see Figure 2 where we observe pricedeclines of approximately . for S and for S over the trading period. Figure 3 shows the performanceof the controller. We see an overall return of onthe initial $10 , during the trading period. It is alsonoteworthy that most of the gains come in the trading periodbetween stages 70 and 100. This is the period when thelargest downward stock price movement occurs. Aggregate Statistics over Many Sample Paths:
Now,instead of the single sample path analysis, we considerthe performance against the entire family of GBM pro-cesses under consideration. We now calculate the re-turns X = V ( N ) − V (0) /V (0) using one million samplepaths with µ and ε chosen using the uniform distributionover their respective admissible ranges. Figure 4 shows theempirically estimated probability density function of X . Thecontroller yields an average return of about . and amedian return of about . with a probability of profit Stage $0.65$0.7$0.75$0.8$0.85$0.9$0.95$1$1.05 S t o ck P r i c e Stock 1Stock 2
Fig. 2. Simulated Prices Along One Sample Path
Stage $9000$10000$11000$12000$13000$14000$15000$16000$17000$18000 A cc oun t V a l ue V ( k ) Fig. 3. Controller Gain-Loss Function for Scenario in Fig. 2 -1 0 1 2 3 4 5
Returns x f X ( x ) Fig. 4. Probability Distribution of Returns of . . Interestingly, the statistics indicate positive expectedvalue for g ( N ) even with the added leverage constraints.Most notably, even among the unprofitable scenarios, weobserve that the controller limits the losses. For exam-ple, 99.99% of all the unprofitable sample paths show losseslimited to less than 10% of the initial account value.II. C ONCLUSION
The main result in this paper is a new version of the RobustPositive Expectation (RPE) Theorem for the case of tradingtwo directionally correlated stocks with bounded non-zeromomenta. Given the uncertainty bounds µ min , µ max and ε max , the theorem provided necessary and sufficientconditions on K under which robustly positive expectedtrading gain E [ g ( N )] is guaranteed. If the conditions ofthe theorem result in no positive K satisfying the RPEcondition, we deem the pair as not tradable. This reflectsthe fact that the uncertainty bounds are too large to enablerobustness guarantees.By way of future research, a logical step would be to back-test our new two-stock controller using historical data and tocompare the performance to that of traditional pairs-tradingalgorithms. It is also worth noting that the investmentlevels I ( k ) and I ( k ) of the two arms of our new controllerevolve independently. A potential research direction involvesthe development of new controllers with cross-couplingin their investment levels; each controller depends on theperformance of the other. Another interesting direction ofresearch would be to generalize the theory presented hereto a basket of more than two directionally-correlated stocks.A PPENDIX
Here, we obtain the expression for E [ g ( N )] for anSLS controller operating on a single stock. Giventhe price process S ( k ) over k = 0 , , , , . . . , N ,having independent returns ρ ( k ) with constantmean µ = E [ ρ ( k )] , beginning with the SLScontroller I ( k ) . = I + Kg ( k ) , I ( k ) . = − I − Kg ( k ) described in Section I with the update equations g i ( k + 1) = g i ( k ) + I i ( k ) ρ ( k ) for i = 1 , and g (0) = g (0) = 0 , substituting for I i ( k ) , g ( k + 1) = (1 + Kρ ( k )) g ( k ) + I ρ ( k ) ; g ( k + 1) = (1 − Kρ ( k )) g ( k ) − I ρ ( k ) . Taking the expectation in both the equations while notingthat ρ ( k ) and g i ( k ) are independent, we obtain E [ g ( k + 1)] = (1 + Kµ ) E [ g ( k )] + I µ ; E [ g ( k + 1)] = (1 − Kµ ) E [ g ( k )] − I µ. Since each equation above has a simple scalar state-spaceform x ( k + 1) = ax ( k ) + bu ( k ) with zero initial conditionand constant input u ( k ) = I µ , a straightforward calculationleads to E [ g ( N )] = I K (cid:16) (1 + Kµ ) N − (cid:17) ; E [ g ( N )] = I K (cid:16) (1 − Kµ ) N − (cid:17) . Now, summing the two solutions above, we obtain E [ g ( N )] = I K (cid:104) (1 + Kµ ) N + (1 − Kµ ) N − (cid:105) . R EFERENCES[1] B. R. Barmish and J. A. Primbs, “On Arbitrage Possibilities ViaLinear Feedback in an Idealized Brownian Motion Stock Market,”
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