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 [1]. 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 [1]. The performance of this trading method is studied in both idealized markets and real-world 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 [1] 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 σis 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 ≥ 0, a long trade is triggered. On the other hand, if the upper confidence level U satisfies U ≤0, this dictates going short. Finally, for the case when L <0< 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 53706, e-mail: [email protected]
B. Ross Barmish, Professor ECE Department, University of Wisconsin, Madison, WI 53706, e-mail: [email protected]
be traded, we use the linear feedback saturation-reset controller of [1] 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 [1] 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
Proceedings of American Control Conference,
Baltimore, Maryland 2010
price S(k)is governed by
√
S(k+ 1)=(1 +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 − 1),...,S(k − n + 2), S(k − n + 1)
to obtain an estimate µˆ(k) for µ. Then with desired confidence level 1−α as given, we estimate a confidence interval [L(k), U(k)]for µˆ(k).
When the lower confidence limit satisfies L(k) ≥ 0, a long trade is triggered. Similarly, when the upper confidence limit satisfies the condition U(k)≤0, 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≥1, with the one-period return is given by
. S(k+1)−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 ran dom variables are independent and identically distributed, at time k≥n, having the nmost recent price observations in hand, we form the estimate
n t 1 µˆ(k) = ρ(k−i) nΔt i=1
and its associated variance estimate
n t 1 σˆ2(k) = (ρ(k−i)−µˆ(k))2 n−1 i=1
Now, to obtain the lower and upper daily confidence lim its L(k)and U(k), for a given tolerable risk level 1−α, we use estimates of µˆ(k)and σˆ(k). The lower confidence limit
Student’s t-distribution with n−1 degrees of freedom; see [10] for more details.
Remarks: Notice that the long trade trigger condition L(k) ≥ 0 and the short trade trigger condition U(k) ≤ 0
are mutually exclusive. Intuitively, we expect a long trigger to occur when a stock’s price trajectory is drifting upward corresponding to µ > 0 while a short trigger should occur when µ <0. 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.
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 [1]. Whereas previous work has only one trading trigger at k = 0, 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∗)≥ 0
is encountered. Then, we initialize a long trade with initial investment
I(k∗) =I0=γ0V(k∗)
where
0≤γ0≤1
is a controller parameter; e.g., γ0= 0.5 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)≤Imax=γmaxV(k∗); j≥0
where γmax ≥ γ0 indicates the maximum allowable
trade. For example, suppose V(k∗) = $10,000, γ0 = 1/2 and γmax = 1.5, then we obtain I0 = $5000
and Imax = $15,000. In the sequel, however, we keep
γmax≤1 to reduce the need for margin.
Now, at the entry point, we begin with trading gain g(k∗). 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
,n−1σˆ(k) g(k∗+j+1)=g(k∗+j)+ρ(k∗+j)I(k∗+j); j= 0,1,2... √ z . L(k) =µˆ(k)− α 2 α 2 n
Similarly, the account value begins at V(k∗)and becomes and upper confidence limit
V(k∗+j+ 1)=V(k∗+j) +ρ(k∗+j)I(k∗+j).
,n−1σˆ(k) √
U(k) =. µˆ(k) +z
n Now, for operation in the non-saturation regime, we update the amount invested via the linear feedback
where z ,n−1is the value at which α= 2Fn−1 −zα 2,n−1 , α
and Fn−1(y) is the cumulative normal distribution of the I(k∗+j+ 1)=I(k∗+j) +KΔg(k∗+j) 2
saturation constants are given by γ0 1/2 and γmax 1.
These parameters were selected small enough so that margin considerations would not come into play. Next, we see from Figure 1 that U(100)<0. 110 120 130 140 150 160 170 180 190 200 90 92 94 96 98 100 102 104 days: 101 − 200 Ito Stock Price
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 143.
110 120 130 140 150 160 170 180 190 200 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 day L(k) and U(k) L U
where K >0 is the controller gain and .
Δg(k∗+j) =g(k∗+j+1)−g(k∗+j)
Saturation and Reset: Now, suppose at some point the for mula above dictates I(k∗+j∗)> Imax. Then, we simply re
duce amount invested to Imax. Another possibility is that we
are already in the saturation regime with I(k∗+j) =Imax
and the stock price declines; i.e., S(k∗+j+ 1)< S(k∗+j). In this case, we use the update above for I(k∗+j+ 1)which implies that the investment level sinks below Imax. 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+1)=min{I(k∗+j) +KΔg(k∗+j), Imax}
with the added restriction that we do not allow I(k∗+j)<0
in a long trade.
Going Short: The updating formulae for short trading are identical to those above provided the following is understood: Instead of K > 0, we now use K < 0. Furthermore, when the short is triggered via U(k∗)≤0, in the formulae above, we begin with I0<0.
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 = 100 and we illustrate how the controller effectively identifies the correct sign of µwith 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 16, 2008.
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(0) = 100 and the stochastic variations are driven via our geometric Brownian motion model with underlying drift parameter µ = 0.2 and volatility σ = 0.2. Upon arrival at Day 100, for the sample path considered, stock price S(100) = 94, and the estimated process parameters were
µˆ(100)=−0.132; σˆ(100)=0.196
and associated confidence interval given by
L(100)=−0.163; U(100)=−0.01.
The estimates above provide the initial conditions for trading. Indeed, we assume initial account value V(100)=$10,000. Furthermore, to describe the closed loop system, given that daily closing prices are assumed, the incremental time inter val in the model is taken to be Δt= 1/252 representing a trading year with 252 days. Finally, for illustrative purposes, we take feedback gain as K = 1 and the trade entry and
= =
Fig. 1. Ito Market Stock Price with µ=.2and σ=.2
Hence, we begin the trade with a short on Day 100 with a plan to trade for one hundred additional days. Recalling that the underlying drift is µ= 0.2, we note that the controller is initially acting “incorrectly.” That is, our trading algorithm
Fig. 2. 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(200) = 10,240.32 which is quite
significant given that the account reaches a low of 9730
around Day 178 and trading only occurs for one hundred days. 110 120 130 140 150 160 170 180 190 200 −5000 −4000 −3000 −2000 −1000 0 1000 2000 3000 4000 5000 I(k) days: 101 − 200
γmax 1; i.e., we initialize trading with I0 $5000 and
saturate with Imax=$10,000
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.
80 90 100 110 120 130 140 150 60 70 80 90 100 110 120 [10/16/2008−2/3/2009] GS Stock Price
Fig. 5. 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. 80 90 100 110 120 130 140 150 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 day L(k) and U(k) L U
value V(100)= $10,000 and allowed trading to occur for seventy-five additional days with feedback gain K= 6and trade entry and saturation constants given by γ0= 1/2 and
= =
Fig. 3. Amount Invested I(k)for Ito Market Stock
110 120 130 140 150 160 170 180 190 200 0.98 0.99 1 1.01 1.02 1.03 x 10 Account Value days: 101 − 200
Fig. 4. Account Value V(k)for Ito Market Stock
B. Trading Goldman Sachs
We now illustrate the trading algorithm by considering stock price variations for Goldman Sachs (GS) for the seventy-five day period beginning October 15, 2008. 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(0) = 147.44. Indeed, we carried out the trading simulation with initial stock price S(75) = 111.91 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)=−1.045; σˆ(75)=0.96
with associated confidence interval given by
L(75)=−0.86; U(75)=−1.23.
Noting that U(75) < 0, the stock is initially shorted. We ran the trading algorithm beginning with initial account
Fig. 6. 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 102 when the trade is saturated; i.e., I(k) =
Imax=$10,000. 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
80 90 100 110 120 130 140 150 −12000 −10000 −8000 −6000 −4000 −2000 0 2000 4000 6000 I(k) [10/16/2008−2/3/2009]
Fig. 7. Amount Invested I(k)for GS
80 90 100 110 120 130 140 150 1 1.1 1.2 1.3 1.4 1.5 1.6 x 10 Account Value [10/16/2008−2/3/2009]
Fig. 8. Account Value V(k)for GS
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 10,000 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 0.5 while the selection of σ depends on the degree of volatility being considered. In the low volatility case, σis chosen using a uniform distribution over [0,0.1]. In the medium volatility case, we use uniform distribution over [0,0.5] and in the high volatility case, we use uniform distribution over [0,0.9].
For each (µ,σ) sample, we generated a stock price
1
trajectory for 200 days using Δt = 252 in the Ito process model. The first 100 days were used to obtain an initial estimate of µˆ(100) and its confidence limits L(100) and U(100). For the next 100 days, trading occurred using the saturation-reset controller with gain K = 1, saturation parameters γ0 = 1/2 , γmax = 1 and 90% confidence
level; i.e, 1−α= 0.9. 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)<0< 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 $1,000 trading gain on the initial $10,000 in the account, if this was achieved being in the market only 50% of the time, the rate of return is arguably much higher than ten percent.
0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 x 10 0 100 200 300 400 500 600 700 800 900 1000
V(end) [end=100] [min=9070.8852, max=16530.3091]
93.1342% Positive Return [9875 Trials: 4934 Short, 4941 Long]
E[V(end)]=10886.946 [σ=783.5645]
Fig. 9. Histogram of Account Value for Low Volatility Case
0.5 1 1.5 2 2.5 3 x 10 0 500 1000 1500
V(end) [end=100] [min=6987.7696, max=26632.3284]
72.2972% Positive Return [9555 Trials: 4852 Short, 4703 Long]
E[V(end)]=10736.3501 [σ=1374.9388]
Fig. 10. Histogram of Account Value for Medium Volatility Case
Tables I and II provide quantitative results describing Figures 9, 10 and 11. 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
0.5 1 1.5 2 2.5 3 x 10 0 200 400 600 800 1000 1200
V(end) [end=100] [min=5703.1392, max=25889.9558]
61.6927% Positive Return [9429 Trials: 4675 Short, 4754 Long]
E[V(end)]=10584.562 [σ=2055.7131]
Fig. 11. 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 1.25%
of the time while the highest volatility stock resulted in no trading 5.71% of the time.
Overall the results were encouraging. Even in the high volatility case, positive returns resulted for nearly nearly 60%
of the simulations with returns of nearly 6%on the average.
TABLE I
HISTOGRAM DATA FROM FIGURES 9, 10, 11
E[V100] σ(V(100)) Vmin Vmax Return (%)
10886.95 783.56 9070.89 16530.31 8.87 10736.35 1374.94 6987.77 26632.33 7.36 10584.56 2055.71 5703.14 25889.96 5.85
TABLE II
HISTOGRAM DATA FROM FIGURES 9, 10, 11
No. Trades Short Long Positive Trades (%)
9875 4934 4941 93.13 9555 4852 4703 72.30 9429 4675 4754 61.69
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 =50,75,100. In each case, the number of trading days was also n. For example, when we estimate µ
using 75 days, we also trade for 75 days. In all trading experiments, we took the initial amount invested to be I0 = 5000 corresponding to γ0 = 1/2 and saturation
value Imax = 10000 corresponding to γmax = 1. In our
simulations, we used feedback gains K= 2,4,6,8.
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 1, 2008 and ending with the closing price on April 16, 2009. 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, 100days) and negative results for a different trading window (for example, 50
days). This phenomena is reflected in Table III with the GE (General Electric) and GS (Goldman Sachs) stocks. This also occurs with the S&P500. 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=100, with all other trading parameters the same, leads to inferior performance. This n=100scenario for GS and the associated volatility and confidence limits are shown in Figures 12-15.
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 100 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 50 10643.26 6.43 DOW 4 100 13409.51 34.10 DUK 8 100 9929.10 -.71 GE 4 75 10967.80 9.68 GE 4 100 8783.07 -12.17 GOOG 8 50 9581.18 -4.19 GS 6 75 13968.40 39.68 GS 6 100 7457.25 -25.43 S&P500 4 50 11390.31 13.90 S&P500 8 75 9328.33 -6.72 SWY 6 100 9366.97 -6.33 UPS 8 75 10528.41 5.28
110 120 130 140 150 160 170 180 190 200 60 70 80 90 100 110 120 [11/20/2008−4/16/2009] GS Stock Price 110 120 130 140 150 160 170 180 190 200 −6000 −4000 −2000 0 2000 4000 6000 I(k) [11/20/2008−4/16/2009]
Fig. 12. GS Stock Price: Green/Red Denotes Short/Long
110 120 130 140 150 160 170 180 190 200 7500 8000 8500 9000 9500 10000 Account Value [11/20/2008−4/16/2009]
Fig. 13. Account Value V(k)for GS
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 real-world 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 1−αfor triggering, the number of training days n for data acquisition, the feedback gain K and the prescribed initial and saturation investment levels I0
and Imax 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.
Fig. 14. Amount Invested I(k)for GS
110 120 130 140 150 160 170 180 190 200 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 day L(k) and U(k) L U
Fig. 15. Confidence Limits L(k)and U(k)for GS
To illustrate the pre-optimization idea above, suppose that n = 100 and trading begins at k = 0. Then, using the 100 preceding training points S(−1), S(−2),...,S(−99), S(−100), 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(−100) = $10,000. Then the objective is to maximize the final value of the account
.
V0=V(0)=V0(α,n,K,I0, Imax)
to obtain optimal starting values for the strategy. Subse quently, 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.
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