A Confidence Interval Triggering Method for Stock Trading Via Feedback Control

A Confidence Interval Triggering Method for Stock Trading Via Feedback Control S. Iwarere and B. Ross Barmish Abstract This paper builds upon the robust control paradigm for stock trading established in []. To this end, the contribution of the current work is an algorithm for triggering a trade. Whereas previous work considered the management of a trade, this paper concentrates on entry into the trade. That is, based on historical prices, we generate, three possible signals: long, short or no trade. These signals are derived using an Ito process model based on geometric Brownian motion. The parameters of this model, the Ito process drift µ and the volatility σ, are estimated and adaptively updated as each new piece of price data arrives. The confidence interval for µ determines when a trade is triggered. If a trade is triggered, then the amount invested in stock is obtained using the saturation-reset linear feedback controller described in []. The performance of this trading method is studied in both idealized markets and realworld markets. I. INTRODUCTION This paper is part of a relatively new branch of technical analysis which involves the application of control theoretic concepts to stock and option trading. The key idea in this literature is to formulate the trading law as a feedback control on the price sequence. Subsequently, buy and sell signals are generated over time and the trader s holdings correspondingly change; e.g., see [] for the author s approach and the earlier work in [2]-[5]. In the control theory literature to date, the triggering mechanism for entering or exiting a trade has not been emphasized. This issue is the main focal point of this paper. We begin by first considering the triggering issue in a so-called idealized market. To this end, an underlying Ito process with unknown drift parameter µ and volatility parameter σ is assumed for the discrete-time stock price process S(k). Then, given n price measurements, we proceed to create an estimate µˆ of µ and use the corresponding confidence interval to decide whether to enter a trade or walk away. Then, if the lower confidence level L satisfies L, a long trade is triggered. On the other hand, if the upper confidence level U satisfies U, this dictates going short. Finally, for the case when L < < U, the trader is deemed to have insufficient confidence about the direction of drift. Hence, in this case, no trigger results. When the confidence levels are such that stock should S. Iwarere, Graduate Student, ECE Department, University of Wisconsin, Madison, WI 5376, e-mail: iwarere@wisc.edu B. Ross Barmish, Professor ECE Department, University of Wisconsin, Madison, WI 5376, e-mail: barmish@engr.wisc.edu be traded, we use the linear feedback saturation-reset controller of [] to modulate the amount invested I(k). Subsequently, we study the performance of the triggering plus feedback combination in two types of markets. The first of these markets, per description above, is called the Ito Market. In this context, we carry out a large number of Monte Carlo simulations to see if our confidence interval method can successfully latch on to the correct sign for µ. In such cases, the trading scheme is seen to be successful in a statistically justifiable sense. The second type of market which we consider is a real-world market. That is, using historical time series for a number of well-known stocks, we study the efficacy of our method in a back-testing context. Whether it be an Ito market or a real-world market, our method requires adaptively updating the trading signal. At the close on each day, the n-day estimation window for µ is updated by dropping the oldest price point and adding the newest one. Subsequently, this leads to an update of the triggering signal as the trade proceeds. II. THE ITO MARKET AND ASSOCIATED CONFIDENCE INTERVALS In this section, we describe the so-called Ito Market, an idealization of a real market. This market will serve as a vehicle for an initial test of the efficacy of the triggering mechanism which we are proposing for trading. Recognizing that real-world stock prices can have price variations which are quite different from those in this market, we view success in the Ito Market as necessary but far from sufficient. Our trading philosophy, consistent with much classical literature, is that we first derive theoretical results in an idealized market as a pre-filter to aid in a decision on whether extensive back-testing on real data is worthwhile. For example, in the celebrated Black-Scholes model [7], idealizing assumptions such as continuous trading and no brokerage costs characterize the market under consideration. In summary, before expending time and effort to conduct extensive back-testing in a real market, our point of view is that success should be demonstrated in a variety of idealized markets. The Ito Market is one particular idealization; see [] for an example of another idealized market. In the Ito Market, the trajectory of stock prices is generated via geometric Brownian motion. Indeed, with Δt representing the time interval between potential trades measured in years, the one-step propagation of the stock

price S(k) is governed by S(k + ) = ( + µδt + σe(k) Δt)S(k) where µ is the so-called annualized drift or expected annualized return, E(k) is a normally distributed random variable with zero mean and unit variance and σ represents the annualized volatility of the stock. Our point of view is that the two parameters, µ and σ, are unknown to the trader. At time k, we use n days of price data S(k), S(k ),..., S(k n + 2), S(k n + ) to obtain an estimate µˆ(k) for µ. Then with desired confidence level α as given, we estimate a confidence interval [L(k), U(k)] for µˆ(k). Student s t-distribution with n degrees of freedom; see [] for more details. Remarks: Notice that the long trade trigger condition L(k) and the short trade trigger condition U(k) are mutually exclusive. Intuitively, we expect a long trigger to occur when a stock s price trajectory is drifting upward corresponding to µ > while a short trigger should occur when µ <. A false trigger occurs when a stock is trending upward and the short trigger condition is satisfied or when a stock is trending downward and the long trigger condition is satisfied. If neither a short nor long signal occurs, then no trade is exercised on the stock. When the lower confidence limit satisfies L(k), a long trade is triggered. Similarly, when the upper confidence limit satisfies the condition U(k), this indicates that the stock should be shorted; i.e., in this case, the trader borrows shares from the broker which are sold immediately in the market. These shares must be returned to the broker at a later date and, putting aside margin and brokerage fees, the key idea is that the trader will profit if the stock price goes down. A. Drift Estimate µˆ(k) and its Confidence Interval To obtain µˆ(k), L(k) and U(k), we first note that for k, with the one-period return is given by. S(k + ) S(k) ρ(k) =. S(k) Then, substituting the stock price dynamics above, we obtain ρ(k) = µδt + σe(k) Δt which is a normally distributed random variable with mean µδt and standard deviation σ Δt. Noting that these random variables are independent and identically distributed, at time k n, having the n most recent price observations in hand, we form the estimate nt µˆ(k) = ρ(k i) III. THE FEEDBACK CONTROL DYNAMICS For the sake of a self-contained exposition, we now briefly describe the saturation-reset feedback trading law used. This trading law is a minor modification of the one used in []. Whereas previous work has only one trading trigger at k =, in the current paper, we need to account for the possibility that we may see many triggers which either initialize a trade or shut down a trade. Going Long: Indeed, suppose at time k = k, the trader sees an account value V (k ) and a long signal L(k ) is encountered. Then, we initialize a long trade with initial investment I(k ) = I = γ V (k ) where γ is a controller parameter; e.g., γ =.5 requires half of the trader s current account to go into the long stock position. Over the period when L(k + j) remains non-negative, we also constrain the controller to satisfy the saturation condition I(k + j) I max = γ max V (k ); j where γ max γ indicates the maximum allowable nδt trade. For example, suppose V (k ) = $,, i= γ = /2 and γ max =.5, then we obtain I = $5 and its associated variance estimate and I max = $5,. In the sequel, however, we keep nt γ max to reduce the need for margin. σˆ2(k) = (ρ(k i) µˆ(k)) 2 n i= Now, at the entry point, we begin with trading gain g(k ). Now, to obtain the lower and upper daily confidence limits L(k) and U(k), for a given tolerable risk level α, we use estimates of µˆ(k) and σˆ(k). The lower confidence limit Then, as the stock price evolves from S(k ) to S(k + j), assuming no margin or commissions, we see the resulting trading gain updated as z α.,n σˆ(k) g(k +j +) = g(k +j)+ρ(k +j)i(k 2 +j); j =,, 2... L(k) = µˆ(k) n Similarly, the account value begins at V (k ) and becomes and upper confidence limit V (k + j + ) = V (k + j) + ρ(k + j)i(k + j). α. z 2,n σˆ(k) U(k) = µˆ(k) + n Now, for operation in the non-saturation regime, we update the amount invested via the linear feedback where z α,n is the value at which α = 2F n z α, 2 2,n and F n (y) is the cumulative normal distribution of the I(k + j + ) = I(k + j) + KΔg(k + j)

where K > is the controller gain and. Δg(k + j) = g(k + j + ) g(k + j) Saturation and Reset: Now, suppose at some point the formula above dictates I(k +j ) > I max. Then, we simply reduce amount invested to I max. Another possibility is that we are already in the saturation regime with I(k + j) = I max and the stock price declines; i.e., S(k + j +) < S(k + j). In this case, we use the update above for I(k +j +) which implies that the investment level sinks below I max. This is the so-called reset action of the controller. In summary, we can combine the two possibilities above to obtain the single update formula I(k + j + ) = min{i(k + j) + KΔg(k + j), I max } with the added restriction that we do not allow I(k +j) < in a long trade. saturation constants are given by γ = /2 and γ max =. These parameters were selected small enough so that margin considerations would not come into play. Next, we see from Figure that U() <. 4 2 98 96 94 92 Ito Stock Price Going Short: The updating formulae for short trading are identical to those above provided the following is understood: Instead of K >, we now use K <. Furthermore, when the short is triggered via U(k ), in the formulae above, we begin with I <. 9 2 3 4 5 6 7 8 9 2 days: 2 Fig.. Ito Market Stock Price with µ =.2 and σ =.2 IV. TWO ILLUSTRATIVE EXAMPLES In this section, we provide two examples illustrating the trading method under consideration. In the first example, the idealized Ito Market is considered with n = and we illustrate how the controller effectively identifies the correct sign of µ with adequate confidence; a winning trade results for the sample path considered. The second example involves real-world trading. To this end, we consider a one hundred day trading period for Goldman Sachs (GS) beginning on October 6, 28. A. The Ito Market We illustrate the trading algorithm by considering a stock in an Ito Market. The simulation begins with initial stock price S() = and the stochastic variations are driven via our geometric Brownian motion model with underlying drift parameter µ =.2 and volatility σ =.2. Upon arrival at Day, for the sample path considered, stock price S() = 94, and the estimated process parameters were µˆ() =.32; σˆ() =.96 and associated confidence interval given by L() =.63; U() =.. The estimates above provide the initial conditions for trading. Indeed, we assume initial account value V () = $,. Furthermore, to describe the closed loop system, given that daily closing prices are assumed, the incremental time interval in the model is taken to be Δt = /252 representing a trading year with 252 days. Finally, for illustrative purposes, we take feedback gain as K = and the trade entry and Hence, we begin the trade with a short on Day with a plan to trade for one hundred additional days. Recalling that the underlying drift is µ =.2, we note that the controller is initially acting incorrectly. That is, our trading algorithm is dictating that we go short which is inconsistent with the positive drift. From the plot of the confidence limits L(k) and U(k) in Figure 2, we see that the controller switches from short to long around Day 43..3.2...2.3 Fig. 2. L(k) and U(k) 2 3 4 5 6 7 8 9 2 day Confidence Limits L(k) and U (k) for Ito Market Stock The remaining plots summarizing this simulation are given in Figures 3 and 4. Two points to note is that the investment never entered into the saturation regime and that the account value terminates with V (2) =, 24.32 which is quite L U

significant given that the account reaches a low of 973 around Day 78 and trading only occurs for one hundred days. 5 4 3 2 I(k) value V () = $, and allowed trading to occur for seventy-five additional days with feedback gain K = 6 and trade entry and saturation constants given by γ = /2 and γ max = ; i.e., we initialize trading with I = $5 and saturate with I max = $, In Figure 5, a plot is given for the stock price over the trading period. Interestingly, except for two days near the end of the trading period, the feedback control requires the stock to be shorted. 2 GS Stock Price 2 3 4 9 5 2 3 4 5 6 7 8 9 2 days: 2 Fig. 3. Amount Invested I(k) for Ito Market Stock 8 7 x Account Value 6.3.2. Fig. 5. 8 9 2 3 4 5 [/6/28 2/3/29] GS Stock Price: Green/Red Denotes Short/Long This is consistent with the plots of the confidence interval limits L(k) and U(k) given over this same trading period in Figure 6..99.5 L(k) and U(k).98 2 3 4 5 6 7 8 9 2 days: 2 Fig. 4. Account Value V (k) for Ito Market Stock.5.5 B. Trading Goldman Sachs We now illustrate the trading algorithm by considering stock price variations for Goldman Sachs (GS) for the seventyfive day period beginning October 5, 28. Note that the seventy-five preceding days are used for estimation of the drift µ; i.e., n = 75. At day zero, we begin with S() = 47.44. Indeed, we carried out the trading simulation with initial stock price S(75) =.9 per Figure 5, with stochastic variations now being driven by the real market. Upon arrival at Day 75, the start of trading, we obtained estimated process parameters µˆ(75) =.45; σˆ(75) =.96 with associated confidence interval given by L(75) =.86; U(75) =.23. Noting that U(75) <, the stock is initially shorted. We ran the trading algorithm beginning with initial account 2 2.5 3 3.5 8 9 2 3 4 5 Fig. 6. day Confidence Limits L(k) and U(k) for GS In Figure 7, we plot the amount invested during trading. To be noted is the approximate five week period beginning around Day 2 when the trade is saturated; i.e., I(k) = I max = $,. Finally, in Figure 8, a plot of the account value is given. V. MORE EXTENSIVE ITO MARKET SIMULATION To motivate the work described in this section, we note that the data for the Ito Market trading example in the previous section was obtained from just one sample path; i.e., only L U

6 4 2 2 4 6 8 2 x I(k) 8 9 2 3 4 5 [/6/28 2/3/29] Fig. 7. Amount Invested I(k) for GS Account Value level; i.e, α =.9. For each sample path, per theory described in this paper, the amount invested in the stock is dynamically updated each day with corresponding daily updates daily in the estimates µˆ(k) and σˆ(k) as well. In turn, daily updates are obtained for the confidence limits L(k) and U(k) and the triggering rule often leads to switches between long and short positions over the course of a trade. In the simulations, it is often the case that the no-trade condition L(k) < < U(k) occurs. We point this out because this implies that the rate of return on trading is actually higher than one might infer by simply looking at the histograms of the final account value to follow. To illustrate, if one obtains a $, trading gain on the initial $, in the account, if this was achieved being in the market only 5% of the time, the rate of return is arguably much higher than ten percent. E[V(end)]=886.946 [σ=783.5645].6 9 93.342% Positive Return [9875 Trials: 4934 Short, 494 Long].5 8 7.4 6.3 5 4.2 3. 2 8 9 2 3 4 5 [/6/28 2/3/29] Fig. 8. Account Value V (k) for GS.9..2.3.4.5.6.7 V(end) [end=] [min=97.8852, max=653.39] Fig. 9. Histogram of Account Value for Low Volatility Case E[V(end)]=736.35 [σ=374.9388] 5 x one realization of the underlying geometric Brownian motion was used. To demonstrate that our positive trading result was not simply a matter of good luck, we now report on a rather extensive set of three Monte Carlo simulations, each involving, trials. To seed each simulation, we randomly select a drift µ and a volatility σ. We differentiate among the three simulations by referring to the low, medium and high volatility cases. In all three simulations, µ is selected from a normal distribution with zero mean and standard deviation.5 while the selection of σ depends on the degree of volatility being considered. In the low volatility case, σ is chosen using a uniform distribution over [,.]. In the medium volatility case, we use uniform distribution over [,.5] and in the high volatility case, we use uniform distribution over [,.9]. For each (µ, σ) sample, we generated a stock price trajectory for 2 days using Δt = 252 in the Ito process model. The first days were used to obtain an initial estimate of µˆ() and its confidence limits L() and U(). For the next days, trading occurred using the saturation-reset controller with gain K =, saturation parameters γ = /2, γ max = and 9% confidence 5 Fig.. 72.2972% Positive Return [9555 Trials: 4852 Short, 473 Long].5.5 2 2.5 3 V(end) [end=] [min=6987.7696, max=26632.3284] x Histogram of Account Value for Medium Volatility Case Tables I and II provide quantitative results describing Figures 9, and. Some observations that are evident from these tables are as follows: Positive returns (on the average) were realized in each scenario. The returns ranged from 5.85% to 8.86%. In all scenarios, the stock was traded over 94% of the time with approximately half the trades being short

2 E[V(end)]=584.562 [σ=255.73] 8 6 6.6927% Positive Return [9429 Trials: 4675 Short, 4754 Long] using 75 days, we also trade for 75 days. In all trading experiments, we took the initial amount invested to be I = 5 corresponding to γ = /2 and saturation value I max = corresponding to γ max =. In our simulations, we used feedback gains K = 2, 4, 6, 8. 4 2 Fig...5.5 2 2.5 3 V(end) [end=] [min=573.392, max=25889.9558] Histogram of Account Value for High Volatility Case and half the trades being long. The higher the volatility of the stock, the lower the percentage of positive returns. The lowest volatility stock had 93% positive returns while the most volatile stock had approximately 62% positive returns. The higher the volatility of the stock, the higher the percentage of trials where the stock was not traded. The lowest volatility stock resulted in no trading.25% of the time while the highest volatility stock resulted in no trading 5.7% of the time. Overall the results were encouraging. Even in the high volatility case, positive returns resulted for nearly nearly 6% of the simulations with returns of nearly 6% on the average. TABLE I HISTOGRAM DATA FROM FIGURES 9,, E [V ] σ (V ()) V min V max Return (%) 886.95 783.56 97.89 653.3 8.87 736.35 374.94 6987.77 26632.33 7.36 584.56 255.7 573.4 25889.96 5.85 TABLE II HISTOGRAM DATA FROM FIGURES 9,, No. Trades Short Long Positive Trades (%) 9875 4934 494 93.3 9555 4852 473 72.3 9429 4675 4754 6.69 VI. ADDITIONAL SIMULATIONS FOR REAL-WORLD STOCKS In this section, we supplement the Goldman Sachs example with summarizing data for a number of other stocks which we traded in simulations according to our theory. These simulations were carried out using training window sizes n = 5, 75,. In each case, the number of trading days was also n. For example, when we estimate µ x Thirteen stock price trajectories from a variety of sectors in the U.S. market were obtained from finance.yahoo.com to simulate real-world stock trading beginning with the closing price on July, 28 and ending with the closing price on April 6, 29. The trading results were mixed; see Table III. We see returns ranging from 25.43% on the negative end to 39.68% on the positive end. The trading results varied both among different stocks and within the same stock. Some stocks produced positive results using one sliding trading window (for example, days) and negative results for a different trading window (for example, 5 days). This phenomena is reflected in Table III with the GE (General Electric) and GS (Goldman Sachs) stocks. This also occurs with the S&P 5. In this regard, the following point should be noted: While we are demonstrating the mechanics of the trading method, we are not claiming that our method guarantees exceptional returns. This issue is addressed in more detail in the next section. As indicated above the variability of performance as a function of trading parameters is readily demonstrated by one of our simulations in Table III for Goldman Sachs. In contrast to our earlier simulation with n = 75, the use of n =, with all other trading parameters the same, leads to inferior performance. This n = scenario for GS and the associated volatility and confidence limits are shown in Figures 2-5. For the type of price variations in GS over the time interval of interest, the smaller trading window of 75 days is more effective than the day window. It allows the controller to react quickly to the wild swings in the trajectory of the stock. As a result, by capitalizing on volatility, the smaller trading window results in a large positive return of 38.96% while the longer trading window results in a large negative return of 25.43%. TABLE III YAHOO RESULTS DATA FOR REAL-WORLD STOCKS Symbol K n V (2n) Return (%) CVS 6 5 643.26 6.43 DOW 4 349.5 34. DUK 8 9929. -.7 GE 4 75 967.8 9.68 GE 4 8783.7-2.7 GOOG 8 5 958.8-4.9 GS 6 75 3968.4 39.68 GS 6 7457.25-25.43 S&P5 4 5 39.3 3.9 S&P5 8 75 9328.33-6.72 SWY 6 9366.97-6.33 UPS 8 75 528.4 5.28

GS Stock Price I(k) 2 6 4 9 2 8 7 2 6 4 2 3 4 5 6 7 8 9 2 [/2/28 4/6/29] Fig. 2. GS Stock Price: Green/Red Denotes Short/Long Account Value 6 2.5 2 3 4 5 6 7 8 9 2 [/2/28 4/6/29] Fig. 4. Amount Invested I(k) for GS L(k) and U(k) 95 2.5 9.5 85.5 8.5 75 2 3 4 5 6 7 8 9 2 [/2/28 4/6/29] Fig. 3. Account Value V (k) for GS 2 2.5 Fig. 5. 2 3 4 5 6 7 8 9 2 day Confidence Limits L(k) and U(k) for GS L U VII. CONCLUSIONS AND FUTURE WORK The consistently positive trading returns in the Ito Market provide impetus for continuation of this direction of research. When simulations were carried out using realworld data, results were rather mixed. In some cases, the strategy vastly out-performs classical benchmarks such as buy and hold; in other cases, the performance fell short. The reader is best served by taking the point of view that numerical results in this paper are solely for demonstration of the mechanics of the trading algorithm. In this regard, a rigorous evaluation of performance vis-a-vis standard benchmarks is relegated to future research. Furthermore, it is important to note that the quality of trading results depends critically on the strategy parameters: the chosen confidence level α for triggering, the number of training days n for data acquisition, the feedback gain K and the prescribed initial and saturation investment levels I and I max respectively. This strongly suggests that trading results can be dramatically improved by carrying out an in-sample pre-optimization on the training data. That is, the trader can take the time series for price and carry out an optimization with respect to the strategy parameters over the last n days. To illustrate the pre-optimization idea above, suppose that n = and trading begins at k =. Then, using the preceding training points S( ), S( 2),..., S( 99), S( ), one can carry out a large number of simulations which amount to fictitious trading on the training data before the real trading begins. For each such simulation, say one begins with V ( ) = $,. Then the objective is to maximize the final value of the account. V = V () = V (α, n, K, I, I max ) to obtain optimal starting values for the strategy. Subsequently, each day, as the new price arrives and the trading gain g(k) is calculated, the strategy parameters can be adapted by discarding the oldest price point and adding the newest one and re-running the optimization. REFERENCES [] Barmish, B., On Trading of Equities: A Robust Control Paradigm, Proceedings of the IFAC World Congress, Seoul, Korea, pp. 62 626, 28. [2] Meindl, P., and J. Primbs, Dynamic Hedging with Stochastic Volatility Using Receding Horizon Control,, Proceedings of Financial Engineering Applications, MIT, Nov. 8-, 24.

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