A Generalized Framework for Simultaneous Long-Short Feedback Trading
AA Generalized Framework for Simultaneous Long-Short FeedbackTrading
Joseph D. O’Brien, Mark E. Burke, and Kevin BurkeMACSI, Department of Mathematics and Statistics, University of Limerick, Ireland
Abstract
We present a generalization of the Simultaneous Long-Short (SLS) trading strategy de-scribed in recent control literature wherein we allow for different parameters across the shortand long sides of the controller; we refer to this new strategy as Generalized SLS (GSLS).Furthermore, we investigate the conditions under which positive gain can be assured withinthe GSLS setup for both deterministic stock price evolution and geometric Brownian motion.In contrast to existing literature in this area (which places little emphasis on the practicalapplication of SLS strategies), we suggest optimization procedures for selecting the controlparameters based on historical data, and we extensively test these procedures across a largenumber of real stock price trajectories (495 in total). We find that the implementation ofsuch optimization procedures greatly improves the performance compared with fixing controlparameters, and, indeed, the GSLS strategy outperforms the simpler SLS strategy in general.
1. Introduction
The use of feedback in a control-theoretic scenario has been well studied within a variety ofdifferent fields – its use dates back at least two millennia where the flow of water was regulatedto improve the accuracy of water clocks. The application of feedback models became widespreadduring the Industrial Revolution and since then their use has become ubiquitous in engineeredsystems [2]. In recent years, the methodology has been applied in the setting of equity tradingwherein a closed-loop feedback system is used to modulate the investment level, I ( t ), in responseto changes in the equity price, p ( t ). As this basic system reacts to changes in price, rather thanmaking predictions about future price movements, the resulting strategy is often described as“model free”.The initial framework was originally developed from a purely long investment perspective [3],but was later extended so that one takes both long and short positions in the equity – the so-called“simultaneous long-short” (SLS) strategy [4]. The key feature of the SLS strategy is that the re-sulting gain function, g ( t ), is guaranteed to be positive under (potentially restrictive) assumptionsof deterministic p ( t ) (albeit this is not assumed to be known), continuous trading, perfect liquid-ity, and the absence of transaction fees (for more details see [4]). Further developments in this1 a r X i v : . [ q -f i n . T R ] S e p rea included the consideration of interest rates and collateral requirements [5, 6, 15, 7]. Thiswork culminated in [8] which laid the foundations for many future research directions such asusing a controller with delay [14], different price process models [10, 9], time varying price evo-lution parameters [20] and also the use of a proportional-integral (PI) controller rather than theproportional controller which was used in the original SLS strategy [16].While various extensions to the SLS framework have been proposed to date, the basic under-lying model structure has remained essentially unchanged in the sense that the initial investmentand feedback parameters are the same for both the long and short investments. However, a realtrader may wish to tune these components of the controller in different ways. We also note thatin the current literature, although much theoretical work has been done (mainly on investigat-ing the positive gain property under different scenarios), testing such SLS strategies in practicalsituations has been much more limited. Specifically, testing has been typically carried out on avery small number of stock price series (one or two), and, hence, there is very little sense of thegeneral performance of these control-based strategies. Furthermore, there has also been a lackof guidance on how one should select the feedback parameter in practice, where, apparently, thischoice has been quite arbitrary in applications shown to date. However, with the level of gainbeing quite modest in some applications (as was mentioned in [20, 16]), one wonders whether ornot greater gains can be made through a more objective selection process, e.g., by optimizingsome criterion.The aim of the current article is to tackle the issues raised in the previous paragraph by:a) generalizing SLS to allow parameters which differ across the short and long side – the resultingstrategy we call ”Generalized SLS” (GSLS), thereby permitting greater flexibility beyond theclassical SLS approach,b) proposing an optimization procedure for selecting control parameters based on historical data,providing an objective process for their selection, andc) extensively analyzing the performance of GSLS (and SLS) with our proposed optimizer on alarge number (495) of real stock price trajectories to determine the practical usefulness of thecontrol-based trading concept.The rest of this article is set out as follows. In Section 2 we introduce the classical SLS strategy of[8], after which, in Section 3, we extend to our proposed GSLS strategy, deriving a number of newanalytical results. Section 4 provides a suggestion on how one might objectively choose controlparameters within GSLS (and, hence, SLS and other varieties thereof) which is then tested onhistorical data in Section 5. Finally, we conclude with some discussion in Section 6.
2. Simultaneous Long-Short (SLS) Strategy
As is customary, we assume that the strategy is applied in the setting of an “idealized market”where the primary assumptions are: (i) continuous-time trading is possible, (ii) the equity is2erfectly liquid so that shares can be purchased with no gap between the bid and ask prices, and(iii) there are no transaction fees or interest rates. See [4] for further details.We now consider the key components of the SLS strategy. Let p ( t ) represent the price of theequity at time t ≥
0. Furthermore, let I ( t ) be the level of investment at time t , and g ( t ) be thegain function, i.e., the cumulative trading gain over the period [0 , t ] where g ( t ) < g (0) = 0. In the simpler long strategy of [3](i.e., no short component), the amount invested is given by I ( t ) = I + Kg ( t ) , where K ≥ I > dg = dpp I = dpp ( I + Kg )whose solution yields g ( t ) = I K (cid:8) q ( t ) K − (cid:9) , where q ( t ) = p ( t ) /p (0) > I ( t ) = I q ( t ) K .The SLS strategy extends the above (but follows along the same lines) by introducing simul-taneous long and short investments, I L ( t ) and I S ( t ) respectively, and their associated cumulativegain functions, g L ( t ) and g S ( t ). The strategy is defined by I L ( t ) = I + Kg L ( t ) ,I S ( t ) = − I − Kg S ( t ) , where g L ( t ) = I K (cid:8) q ( t ) K − (cid:9) ,g S ( t ) = I K (cid:8) q ( t ) − K − (cid:9) . Note that the overall investment is I ( t ) = I L ( t ) + I S ( t ) = K { g L ( t ) − g S ( t ) } and, in particular,that I (0) = 0, i.e., the simultaneous long and short positions (which are opposite and equal when t = 0 with magnitude I ) lead to an initial net zero investment position. As shown in [4], underthe idealized market, the cumulative gain at time t for this SLS strategy is3 ( t ) = g L ( t ) + g S ( t ) = I K (cid:8) q ( t ) K + q ( t ) − K − (cid:9) , (1)and, hence, g ( t ) > q ( t ) (cid:54) = 1, i.e., this strategy always makes a profit except when p ( t ) = p (0).
3. Generalized SLS Strategy
As seen in Section 2, the basic SLS strategy is composed of the simpler long strategy with theaddition of a short component. In particular, the short component mirrors the long componentin the sense that it shares the same initial investment, I , and feedback parameter, K . However,we now view this as an interesting special case of a more general framework in which the longand short components have distinct parameters as follows: I L ( t ) = I + Kg L ( t ) ,I S ( t ) = − αI − βKg S ( t ) , where α, β >
0, and g L ( t ) = I K (cid:8) q ( t ) K − (cid:9) ,g S ( t ) = αI βK (cid:110) q ( t ) − βK − (cid:111) . We refer to this as the Generalized SLS (GSLS) strategy, within which α = β = 1 corresponds toSLS, and whose overall gain function is g ( t ) = I K (cid:20) q ( t ) K − αβ (cid:110) q ( t ) − βK − (cid:111)(cid:21) (2)Notwithstanding the guaranteed positive gain property of the important SLS special case, inpractice, this specific strategy cannot uniformly yield the optimal gain. We will discuss this inthe following sections. β = 1 Before investigating the GSLS strategy, it is instructive to first consider the special case where β = 1, i.e., the case where the long and short initial investments differ, but the feedback parameteris the same in both components. In this specific case note that the gain is given by g ( t ) = I K (cid:2) q ( t ) K − α (cid:8) q ( t ) − K − (cid:9)(cid:3) , (3)where we see that g ( t ) = 0 when q ( t ) = 1 and when q ( t ) = α /K . Furthermore, viewed as afunction of q ( t ), the gain function has a single turning point (a global minimum of − I ( α / − /K ≤
0) at q ( t ) = α / (2 K ) which lies between the two roots. Note that in the classic SLS case,where α = 1, these three points coincide to become a single root and global minimum of zero at q ( t ) = 1, yielding the associated positive gain property.Assume that α <
1, so that we invest in the short component to a lesser extent than thelong component. In this case, g ( t ) > q ( t ) (cid:54)∈ [ α /K , α = 1. This is,theoretically, risk free in the sense that g ( t ) > q ( t ) (cid:54) = 1. On the other hand, inselecting α < q ( t ) >
1. Indeed, it is easy to show that g α< ( t ) > g α =1 ( t ) when the price rises ( q ( t ) >
1) so that the reward goes up by choosing α < g ( t ) < t . In this case, the investment is fixed at I ( t ) = I and g ( t ) = I { q ( t ) − } . In a similar manner to GSLS with α <
1, we are anticipating a pricerise. However, we are guaranteed to make a loss if q ( t ) <
1, whereas, with GSLS, we can stillprofit if q ( t ) < α /K . In other words, we are reducing the risk compared with simply goinglong. Furthermore, although with GSLS the risk is reduced relative to going long, the gain is notuniformly lower (for all control parameter values) even when q ( t ) > . . . . . . − . . . qg α = 0 . , β = 1 , K = 1 α = 0 . , β = 1 K = 3 α = β = 1 , K = 1 α = β = 1 , K = 3“going long” Figure 1: Plots of the gain for various values of q ( t ) when the trading strategy is GSLS ( α = 0 . β = 1) (blue), SLS (red), and simply going long (green). Furthermore, for GSLS and SLS both K = 1 (solid) and K = 3 (dashed) cases are shown.The above discussion relates to the case where α <
1. When α > g ( t ) > q ( t ) (cid:54)∈ [1 , α /K ], and, moreover, we can show that g α> ( t ) > g α =1 ( t ) when the price falls( q ( t ) < α = 1 is sub-optimal in the sense that, if onepossesses knowledge on the likely direction of the stock price, a greater return can be made bychoosing α (cid:54) = 1. In practical situations, a trader is likely to require some input into his/her tradingstrategy such that GSLS may be more attractive than SLS. Furthermore, GSLS may be seen as5ying somewhere between SLS and standard trading in that trader knowledge can be incorporatedas in standard trading, but where the risk is reduced as in SLS. Interestingly, since α /K → K → ∞ , the condition for positive gain when α (cid:54) = 1 is simply that q ( t ) (cid:54) = 1 when K is large. Thissuggests that a large enough K value can overcome a poor α choice; indeed, GSLS with large K essentially behaves as SLS. In the previous section we discussed GSLS under the restriction that β = 1. We now considerthe more general scenario where β is not necessarily equal to 1. In particular, we investigate theconditions for positive gain within GSLS. Theorem 1.
In the idealized market, the gain made by the GSLS strategy is positive ∀ K, β provided that (1 − α ) ln { q ( t ) } ≥ . Proof : First we write the gain function, (2), as g ( t ) = I q ( t ) K − K + αI q ( t ) − βK − βK . (4)Now, consider the inequality q x ≥ x ln q (5)for q > x ∈ R , which is due to the fact that the function m ( x ) = q x − − x ln q is convexwith global minimum of zero at x = 0.Then, from (5), setting x = K yields q K − K ≥ ln q, (6)while setting x = − βK yields q − βK − βK ≥ − ln q. (7)Hence, combining (4), (6), and (7), we get g ( t ) ≥ I (1 − α ) ln { q ( t ) } and, since I >
0, we require that (1 − α ) ln { q ( t ) } ≥ g ( t ) ≥ (cid:3) The above result generalizes the positive gain result of SLS which we recover by setting α = β = 1. Furthermore, note, in fact, that the result does not depend on the value of β (providedthat it is finite) so that positive gain is assured once α = 1 even if the feedback parameters onthe short and long side differ. The requirement that (1 − α ) ln { q ( t ) } ≥ q ( t ) >
1, we should set α <
1, and if we expect q ( t ) <
1, we should set α >
1. It is also worth noting that, for large K , the behaviour of g ( t ) is as follows: g ( t ) ∼ (cid:40) I q ( t ) K ln { q ( t ) } , q ( t ) > − αI q ( t ) − βK ln { q ( t ) } , q ( t ) < K can yield positivegain irrespective of α and β . These findings mirror those of Section 3.1 which were for GSLS with β = 1. Indeed GSLS with β = 1 is qualitatively similar to the general case. Theorem 2.
The GSLS gain function is increasing in K and β . Proof : In this case we write the gain function, (2), as g ( t ) = I q ( t ) K L − K L + αI q ( t ) − K S − K S . (8)where K L and K S are the feedback parameters on the long and short sides respectively, and theincremental gain equation is given by dg = ∂g∂K L dK L + ∂g∂K S dK S , (9)where ∂g∂K L = I q K L ( K L ln q −
1) + 1 K L ,∂g∂K S = αI q − K S ( − K S ln q −
1) + 1 K S . By replacing x with − x in (5) and multiplying by q x , we can establish another inequality, q x ( x ln q −
1) + 1 ≥ . (10)Using (10) with x = K L and x = − K S , respectively, gives ∂g/∂K L ≥ ∂g/∂K S ≥ dK L and dK S are positive, so that g ( t )is increasing in K L and K S , and, hence, in K and β (since K L = K and K S = βK ). (cid:3) While the above result suggests that a trader can simply increase the feedback parametersto increase profits, this may not be feasible in practice. Firstly, large feedback parameters willcause the controllers (and associated investments) to vary wildly in response to (potentially small)changes in gains which introduces a large degree of variability into the system. Moreover, one ofeither the long or short investments will become large, and, of course, all traders will have limitedresources; for the same reason, one cannot simply increase I arbitrarily.The basic results of this section can be seen in Figure 2 for some specific parameter values.7 − − − − Kg α = 0 . , q = 0 . α = 0 . , q = 2 α = 2 , q = 0 . α = 2 , q = 2 Figure 2: GSLS gain viewed as a function of K with β = 1, I = 1. The dots are given at K = log q ( t ) α which is a root of (3). The cases where this root lies in the negative K region satisfythe requirement that (1 − α ) ln { q ( t ) } ≥
0. In these cases, positive gain is assured for all K valuessince negative K values are infeasible. We now show that the GSLS strategy is robust in the case where the price evolution of the equityis determined by Geometric Brownian Motion (GBM) such that dpp = µ dt + σ dW where W represents a Weiner process, µ is the drift and σ is the volatility [13],[19]. Our prooffollows along similar lines to that of [8].As in [8], the gain made by the long controller is g L ( t ) = I K (cid:110) q ( t ) K e σ ( K − K ) t − (cid:111) and, similarly, the gain made by the short controller is g S ( t ) = αI βK (cid:110) q ( t ) − βK e σ ( βK − ( βK ) ) t − (cid:111) and, hence, the total gain is g ( t ) = I K (cid:104) q ( t ) K e σ ( K − K ) t − αβ (cid:110) q ( t ) − βK e σ ( βK − ( βK ) ) t − (cid:111)(cid:21) . The expected gain can then be derived by noting that the k th moment of a log-normally dis-tributed random variable X with log( X ) ∼ N (cid:16)(cid:0) µ − σ (cid:1) t, σ t (cid:17) is given by E (cid:104) X k (cid:105) = e (cid:0) kµ − σ (cid:1) t + k σ t . E { g ( t ) } = I K (cid:26) e Kµt − αβ (cid:16) e − βKµt − (cid:17)(cid:27) . (11)As the expected gain function under GBM, (11), is of the same form as the gain function underdeterministic price evolution, (2), with q ( t ) = e µt . Theorem 1 follows immediately and, hence,positive expected gain follows when (1 − α ) µ ≥
0. Similarly, Theorem 2 applies so that theexpected gain increases in K and β .Note that the variance of the gains may also be calculated and it is given byVar { g ( t ) } = E (cid:8) g ( t ) (cid:9) − [ E { g ( t ) } ] = I K (cid:26) e Kµt (cid:16) e σ K t − (cid:17) + αβ e − βKµt (cid:16) e βK σ t − (cid:17) (cid:18) αβ e − βKµt + 2 e Kµt − βK σ t (cid:19)(cid:27) . (12)We also note that these results extend to the time-varying price dynamics case described in [20].
4. Selection of Parameters
While results such as those in the previous section provide insight into the behaviour of feedback-based strategies (especially the conditions leading to positive gain), they fall short of yieldinga practical implementation in the sense of suggesting values of the control parameters in realapplications. Therefore, in this section, we focus on possible criteria that one may optimize inorder to select the control parameters within GSLS (and, hence, also SLS).As a first step we will make the assumption that the price evolution process can be modelledby GBM since this assumption yields explicit solutions for both E { g ( t ) } and Var { g ( t ) } , and oursuggested criteria will be based on these quantities; of course other price evolution models couldbe used. Now, let g ∗ t be some prespecified target gain for time-point t , and define the trading“bias” as bias { g ( t ) } = E { g ( t ) } − g ∗ t (13)which is the expected difference between the realized gain, g ( t ), and the target gain, g ∗ t . Underthe GBM assumption, this quantity will depend on the parameter µ as well as the control pa-rameters, α , β , and K . Therefore, as a first step, the GBM parameters can be estimated usingstandard inference procedures [1], and, following this, the control parameters can be selected byminimizing, [bias { g ( t ) } ] , i.e., these are the control parameters which minimize the differencebetween E { g ( t ) } and g ∗ t .The suggestion above does not take account of the volatility of the stock price, as evidenced bythe fact that the objective function does not depend on σ . Thus, an alternative criterion wouldinvolve both the so-called bias and the variability in gains. We therefore propose the trading“mean squared error” (MSE) asMSE { g ( t ) } = E { g ( t ) − g ∗ t } = [bias { g ( t ) } ] + Var { g ( t ) } (14)9 Gain D en s i t y s (a) Bias Optimizer −0.4 −0.2 0.0 0.2 0.4 Gain D en s i t y s (b) MSE Optimizer Figure 3: Density of gains over 1000 simulated GBM stock prices over one year where controlparameters were selected on the basis of (a) bias and (b) MSE. The target gain here is g ∗ t = 0 . σ (via the variance term) in addition to µ , and can be minimized with re-spect to the control parameters. Such control parameters might result in E { g ( t ) } < g ∗ t , but wherethe lower variation in gains justifies the choice, i.e., control parameters selected via MSE { g ( t ) } will, in any given run, tend be closer to the target gain due to the lower variability in gains.
5. Testing GSLS with Optimized Control Parameters
Before testing our proposed methods on real data, we first carry out a simulation study. Wesimulated stock prices according to GBM with drift µ = 0 .
1, and a range of different volatilityvalues, σ = { . , . , . } . Since I simply plays the role of scaling up and down the gain, wefix I = 1, and optimize both (13) and (14) with respect to the control parameters wherein theknown values of the GBM parameters are inserted prior to this optimization. Furthermore, weassume that, after one year of trading (over 252 days), the trader is aiming for a 15% return, i.e, g ∗ = 0 .
15, so that he/she beats the market drift ( µ = 0 .
1) by 5 percentage points.Although the two objective functions can be optimized using standard optimization algo-rithms, since there are only three parameters (
K, α, β ), we performed a discrete grid searchover the parameter space. In particular, we used 10 equally spaced values for each parameterwith K ∈ [0 , α ∈ [0 , β ∈ [0 ,
5] yielding 1000 parameter combinations. For each ofthe 3 simulation scenarios, we found the optimal control parameters according to both bias andMSE minimization, and then applied the resulting strategy over 252 trading days for each of 1000simulated GBM trajectories.Figure 3 shows the results based on optimizing both bias and MSE in terms of a density plotof the realized gains over each of the 1000 simulation replicates. We can see from Figure ?? that10ptimizing the bias does not appear to perform well as the volatility increases, with most of themass being on negative gain for the larger volatility value, σ = 0 .
2. However, this is perhaps to beexpected since this approach does not take account of the volatility. In contrast, minimizing MSE(see Figure ?? ) yields more consistent results in terms of the target gain even as the volatilityparameter increases (albeit the variability of the gain naturally increases with the volatility).This suggests that parameter selection using MSE as an objective criterion may be preferablethan using bias. As mentioned, in the existing literature, the degree to which feedback-based trading strategieshave been tested on real data has been somewhat limited, i.e., the number of stock price series hasbeen very small, and the choice of control parameters has been apparently quite arbitrary. Thus,more extensive testing is required if real-world traders are to be convinced that the adoption ofsuch strategies can be fruitful in general.With the above goal in mind, we consider the daily closing prices for 495 members of theStandard and Poor’s 500 (S&P 500) index over the course of two-year period, January 2016 -December 2017 (of the current 500 members, it was not possible to obtain complete data for thetime period in question in 5 cases which were, therefore, omitted). The justification for choosingthis data is based on the fact that the members of the S&P 500 are the largest companies traded inthe United States and eligibility is based on a number of factors including market capitalization ,liquidity , and the company being domiciled [12]. Thus, these members have highly liquid stocksso that feedback-based trading strategies could plausibly be implemented in a real-world scenario,and, moreover, the members of this index are intended to provide a reasonably good representationof the stock market as a whole. Figure 4 shows the median adjusted closing price calculated overall 495 members on each day over the two-year period, along with 2.5%, and 97.5% quantiles; itis clear that the majority of the stock prices have increased over time.For each of the stock price series, we take the year 2016 to be the training period for which theGBM parameters are estimated using maximum likelihood [1], and then, based on the estimatedGBM parameters, we select the GSLS control parameters via the bias and MSE optimizationapproaches described in Section 4 with a target gain of g ∗ = 0 .
15 for each stock. FollowingSection 5.1, when optimizing GSLS control parameters, we use a grid search over 1000 parametercombinations resulting from 10 equally spaced values for each parameter, K ∈ [0 , α ∈ [0 , β ∈ [0 , I = 1 since this simply scales the gains. Market capitalization of $5.3bn or more. Annual dollar value traded to float-adjusted market capitalization should be greater than 1, and the stockshould trade a minimum of 250,000 shares in each of the six months leading up to the evaluation date. The company must be a U.S company as defined by the US Index Committee.
100 200 300 400 500
Number of Days C l o s i ng P r i ce Figure 4: The 2.5% (bottom, dashed), 50% (middle, solid), and 97.5% (top, dashed) quantiles ofclosing prices for 495 members of the S&P 500 over the period January 2016 - December 2017.Before applying any optimized strategies, we first test the classical SLS strategy in a similarway to previous literature, i.e., a value for the control parameter is simply selected and tested;we use K = { , , , , } . The advantage here, however, is that we are testing over a much largersample of price series than that of previous literature. So that the results are comparable withthe case where we carry out optimization, we test these five strategies in the year 2017 only (2016is not used to provide insight since we are simply fixing K from the offset). Figure 5 displays theaverage gain on each trading day (average was computed over the 495 stock series). We can seethat the average gain increases with the value of K , however the trajectory of the gain is quiteerratic over time; most of the growth appears as a sharp jump occurring in the last c.50 days. . . . . Days G a i n K = 1K = 2K = 3K = 4K = 5
Figure 5: Average gain (computed over all stock series) on each trading day in 2017 from the SLSstrategy with various K values.We then applied each of the 1000 GSLS parameter combinations to all stocks in the year 2017(again without using 2016 to optimize), and computed the average gain over all stocks at the endof the year; the results are visualized in Figure 6 along with the five values from the five SLS12trategies mentioned in the previous paragraph. Interestingly, there are a large number of GSLSstrategies which lead to a loss. These mainly correspond to cases where α > −0.50−0.250.000.25 0 1 2 3 4 5 K g Alpha
Beta
Gain for different K, b and a Figure 6: Average gain (computed over all stocks) after 252 days of trading in 2017 for each of1000 GSLS strategies. The colour of the points represent the β value, while their size representsthe α value. Also shown are five SLS strategies (the dark coloured points).The tests described above (both SLS and GSLS) mimic those of previous literature, i.e., wehave simply set the control parameters at the start of 2017 without drawing insight from historicaldata. In other words, the parameters could be thought of as essentially randomly selected, and arenot tuned on a per-stock basis. It is unlikely that a real-world trader would adopt such a strategyfrom which he/she is quite detached. We, therefore, implement our optimization approach foreach stock based on the 2016 data (as described above), and apply the optimized strategies to the2017 data. Figure 7 shows the average gain using optimized strategies (both bias and MSE) overeach trading day, where we find that the MSE approach performs better than the bias approach,and gets quite close to the target, g ∗ = 0 .
15. It is interesting to note that, in both cases, the gainincreases reasonably steadily over time (in contrast to those of Figure 5).To get a sense of the variability in gains, Figure 8 displays the 2.5% and 97.5% quantilesfor the gains over all the stocks. Unsurprisingly, the level of variability associated with thebias-optimized strategies is higher than that of the MSE-optimized strategies. Furthermore,note that the median gain is essentially zero for the bias-optimized strategies, whereas, for theMSE-optimized strategies, it is very close to the average gain observed in Figure 7. Clearly,from the quantiles shown, the distribution of gains is somewhat skewed for the bias-optimizedstrategies with some ”lucky” cases having done very well over the period in question. However,the performance of the MSE approach is more consistent which is in line with the findings ofSection 5.1.In the optimization procedures discussed above, the target gain was fixed at 15% for all stocks.13
50 100 150 200 250 . . . . Days G a i n MSEBiasTarget Gain
Figure 7: Average gain (computed over all stocks) on each trading day in 2016 for optimizedstrategies. Both the bias-optimized (red) and MSE-optimized (black) strategies were based onthe data from 2016. A horizontal green reference line highlights the target gain, g ∗ = 0 . − . . . . Days G a i n MSEBiasTarget Gain
Figure 8: Median gain (computed over all stocks) for both the bias-optimized (red, solid) andMSE-optimized (black, solid) strategies, along with 2.5% and 97.5% quantiles (dashed). A hori-zontal green reference line highlights the target gain, g ∗ = 0 . i = 1 , . . . , g ∗ t,i = | ˆ µ i | + C where ˆ µ i is the estimatedGBM drift parameter and C is some constant which describes the amount by which we wish tobeat the inherent drift in the price evolution. The reason we take the absolute value of the driftis that profit can also be made in the case of negative drift via the short component. Figure 9displays the average gain over time with C = 0 .
05. While the average gain in the MSE-optimizedcase does not change much from that of Figure 7, the average gain for the bias-optimized caseis almost doubled – and is much closer to that of the MSE case. This finding is perhaps to be14
50 100 150 200 250 . . . . Days G a i n MSEBiasAverage Target Gain
Figure 9: Average gain (computed over all stocks) on each trading day in 2016 for optimizedstrategies with g ∗ = | µ | + 0 .
05. Both the bias-optimized (red) and MSE-optimized (black) strate-gies were based on the data from 2016. Although the target gain changes here on a stock-by-stockbasis, we show a horizontal green reference line at g = 0 .
15 for comparison with earlier results.expected since the bias optimization is solely chasing a target, whereas the MSE approach alsotakes account of variability.Table 1: Summary of gains over all stock at the end of the year 2017 using different strategies
Strategy
GSLS SLS
Parameters MSE MSE Bias Bias K = 1 K = 2 K = 5 MSE MSE Bias BiasTarget Fixed Varied Fixed Varied — — — Fixed Varied Fixed VariedSummary 1st Quartile 0.004 0.006 -0.158 -0.154 -0.029 -0.057 -0.131 -0.008 -0.016 -0.083 -0.114of Median 0.155 0.160 -0.006 0.026 < .
001 -0.002 -0.020 < . < .
001 -0.006 -0.017Gains Average 0.161 0.164 0.077 0.142 0.026 0.053 0.138 0.015 0.022 0.108 0.1243rd Quartile 0.293 0.300 0.185 0.264 0.060 0.115 0.237 0.023 0.039 0.128 0.217Inter-Quartile Range 0.289 0.294 0.343 0.418 0.089 0.172 0.368 0.031 0.055 0.211 0.331 “MSE” and “Bias” indicate that parameters were selected by optimizing these criteria for each stock based on2016 data where “Fixed” means that g ∗ = 0 .
15 and “Varied” means g ∗ = | µ | + 0 .
05. Inter-Quartile Range = 1stQuartile − Table 1 contains a summary of the gains for the various strategies considered so far, alongwith optimized versions of SLS (i.e., α = β = 1 and optimization is done with respect to K only).We can see that GSLS with MSE optimisation is the strategy with the highest mean and mediangain. Furthermore, it is the only strategy where the mean and median gains are numerically verysimilar (e.g., SLS with K = 5 has an average gain of 0.138 but a median gain of -0.02); in fact, themedian gains for all other strategies are either negative or negligibly low. Also GSLS with MSEoptimisation is the only strategy with a first quartile gain which is non-negative (although it isclose to zero). In terms of variability of gains (via the inter-quartile range), the bias optimizerhas higher variability than the MSE optimizer which we would expect. It is noteworthy that SLSwith MSE optimization has the lowest level of variability, but also the lowest level of gain.15 . Discussion The GSLS strategy expands classical SLS so that the parameters of the long and short controllerscan differ (i.e., the initial investment and feedback parameters). By allowing different initial longand short investments such that the overall net initial investment is nonzero, I (0) (cid:54) = 0, the GSLSstrategy can be thought of as a paradigm lying between SLS and more standard trading strategies(e.g., simply going long or short). This provides the opportunity to include some insight into thetrading strategy while lowering risk relative to the standard trading strategies.The robustness of GSLS with differing long and short initial investments ( α (cid:54) = 1) is weakenedrelative to SLS in the sense that theoretical (expected) gain is no longer guaranteed to be positive;interestingly, GSLS with differing feedback parameters ( β (cid:54) = 1) maintains robustness providedthat α = 1. However, should the trader possess some knowledge of the price evolution, and select α (cid:54) = 1, a greater level of gain can be achieved than that of SLS. Indeed, the primary driver of anyrealized gain is, as one would expect, the price evolution of the stock itself – which is why onemight wish to make estimates about its likely evolution at the expense of some robustness.Throughout the current paper trader “knowledge” entered in the form of a GBM assumptionof price evolution which is common in financial literature [11, 18]. Of course, this is a simplifyingassumption, and, in practice, more general models could be used which could include other formsof market knowledge exogenous to the historical price series itself. The secondary reason forusing the GBM model is that it admitted a closed form solution for the expected gain and thevariance of the gain. In more complex models, these quantities could be obtained by simulation.The model assumption could potentially be avoided altogether by using some model-free estimateof E { g ( t ) } such as the last realized gain value achieved in a historical sequence { g , g , . . . , g n } .Another alternative might be a weighted average over these realized gains (cid:80) nj =1 w j g j where theweights, w j , grow with j so that more weight is placed on more recent observations. In any case,in our practical application, the methods appeared to perform well despite the GBM assumption.Selection of control parameters which are “optimal” in some sense has not previously beenconsidered in the literature to our knowledge. The lack of such selection procedures presents amajor hurdle in the wider adoption of feedback-based trading strategies. To this end, we haveproposed two possibilities which we call bias and MSE optimization. The bias optimizationfocusses on the expected gain, whereas the MSE approach additionally takes account of thevariation of gains. Under the GBM assumption, the expected gain is connected to the drift ofthe process, while the variation incorporates the process volatility. While our suggestion involvesspecifying a target gain (for which a reasonable value can be informed by the estimated GBMmodel), we might, alternatively, have simply selected the best performing strategy over the testingperiod. This is equivalent to minimizing bias with some arbitrarily large target gain. Note alsothat our MSE objective function, whilst being a very natural quantity in itself, places equalweight on the bias and variance of the gain; more generally, a tuning parameter might controlthe trade-off between these two quantities. Another approach still (akin to the so-called “efficientfrontier” in portfolio optimization [17]) would be based on maximizing the expected gain subjectto a fixed level of variance, or minimizing the variance subject to a fixed expected gain. We couldalso imagine an optimization procedure which updates throughout time such that the controlparameters are altered dynamically (albeit the initial investments are obviously fixed from the16tart). Thus, while we have provided the some initial suggestions for selecting control parameters,which perform well in our applications, there are a variety of avenues for extending these; this isa focus of our future work.Overall, despite being a relatively new area of research, feedback-based trading strategiesclearly have interesting theoretical properties as well as promising performance in practice. Throughthe extension of classical SLS to GSLS, the investigation of parameter selection procedures, andthe extensive analysis of performance in a large sample of real stock price data, this paper con-tributes to a greater understanding of the potential of such strategies. Moreover, it moves thetheory towards the goal of the wider adoption of feedback-based trading strategies by real-worldtraders. Acknowledgment
The authors would like to thank James P. Gleeson (MACSI, University of Limerick) for his helpfulcomments on an earlier draft of this paper. This work was funded by Science Foundation IrelandGrant 16/IA/4470 (J.D.O’B.).
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