Project 3. Trade stocks to make as much money as possible.

Transcription

1 Project. Trade stocks to make as much money as possible. CEE 2L. Uncertainty, Design, and Optimization Department of Civil & Environmental Engineering Duke University Henri P. Gavin and Steve Lattanzio, Duke CEE 7 Due Wednesday, April 8, 25 Think big, think positive, never show any sign of weakness. Always go for the throat. Buy low, sell high. Fear? That s the other guy s problem. Nothing you have ever experienced will prepare you for the absolute carnage you are about to witness. Super Bowl, World Series - they don t know what pressure is. In this building, it s either kill or be killed. You make no friends in the pits and you take no prisoners. One minute you re up half a million in soybeans and the next, boom, your kids don t go to college and they ve repossessed your Bentley. Are you with me? Dan Aykroyd as Louis Winthorpe III, to Eddie Murphy as Billy Ray Valentine, in Trading Places, 98. Greg Smith, Why I am Leaving Goldman Sachs, New York Times, March 4, 2. Randall Munroe, Aug. 25

2 Introduction In this project, your group will design and optimize a methodology for selling and buying shares of stock. Naturally, your objective is to make as much money as possible! Groups will use 2 days of actual stock closing prices to design, test, and optimize an automatic stock trading method. Once groups are satisfied with the performance of their method, the method will be applied to the subsequent 2 days of stock closing prices. Ten points of the final grade will be apportioned according to how well your automatic stock trading algorithm works on the subsequent 2 days of stock closing prices. The closing price of stock s on day d is denoted p d,s. Day-to-day changes in stock prices can be highly volatile; the change from yesterday to today can have very low correlation with the change from today to tomorrow. For this reason, it can be helpful to estimate trends over a recent period of time from a smoothed record of recent stock prices. One simple way to smooth the series of stock prices is to make a weighted average of the most recent N closing prices of a particular stock. The day-today changes of the smoothed stock prices, and the rates of those changes, will have better day-to-day correlations, and might therefore serve as a better prediction for decision-making purposes. 2

3 2 Financial Data Analysis... quantifying quality The decision to buy and sell stock will be based on a stock statistic which we can call quality Q. Stocks with higher quality are better candidates for purchase. Stocks with lower quality are better candidates for sale. A measure of quality of stock s on day d (Q d,s ) could be a weighted sum of the fractional change in a stock price, the rate of the fractional change of a stock price, and the volatility of a stock price over a recent period of time. where Q d,s = q p d,s + q 2 p d,s + q v d,s, () q, q 2, q are weighting coefficients with values to be found via optimization. p d,s is a weighted average of the price of stock s on day d, N p d,s = w i p (d i),s (2) i= where the weights w i emphasize more recent data more significantly, and the sum of the weights is. [ w i = exp 5i ] ( ) N Ni=. () exp[ 5i/N] Examples of exponential weights for N = 5, N =, and N = 2 are shown in Figure. The effect of the number of trading days involved in the weighted average of stock prices, N, on the smoothed price, the fractional change of smoothed price, and the rate of fractional change of smoothed price is shown in Figure 2. p d,s is the fractional change of a weighted average of the price of stock s on day d, p d,s = ( p d,s p (d 2),s ) / ( p d,s + p (d 2),s ) (4) p d,s is the rate of the fractional change of the weighted average of stock s on day d, p d,s = ( p d,s p (d 2),s ) / 2 (5) v d,s is the volatility of stock s on day d defined here as the standard deviation of the stock price over the previous N days, as a fraction of the weighted average stock price, p d,s, v d,s = N p (d i),s 2 N p (d j),s. (6) N N p d,s i= v d,s is like a running coefficient of variation over the last N days of data. j=

4 weight, w i days in the past Figure. Exponential weights from equation () for N = 5, N =, and N = 2. fractional change rate of fractional change volatility smooth price Figure 2. Effect of the number of trading days N involved in the weighted average of stock prices on the smoothed price, the fractional change of smoothed price, and the rate of fractional change of smoothed price, for N = 5 (green), N = (red), and N = 2 (black). 4

5 7 closing price per share trading days Figure. 2 days of stock closing prices for nineteen stocks day price per share stock number day 2 / day price per share day 2 price per share stock number stock number Figure 4. Initial prices, final price, and ratio of final-to-initial price 5

6 Stock Trading Decisions and Rules If Q d,s < S d on day d then stock s is potentially sellable. If Q d,s > B d on day d then stock s is potentially buyable. The sell- and buy-thresholds S d and B d are variables that can change over time. The sell threshold S d shall not be greater than the buy threshold B d, so a stock can not be both potentially buyable and potentially sellable. The set of stocks to buy and sell are governed by the following rules: All shares of stock s are sold on day d if: shares of stock s are owned (no shorting); Q d,s < S d (Stock s is potentially sellable); (optionally) Q d,. > B d, for another stock, not stock s, Shares of one stock may be sold if a different stock is potentially buyable. Shares of stock s are bought on day d if: Q d,s > B d, for any stock not sold on day d. Each time M s shares of stock s are sold, the cash in hand, C, to purchase stocks is increased by the current value of the shares. A transaction cost of two dollars is paid for every stock sold or bought. C d+ = C d + s sold p s,d M s 2. s sold 2. s bought (7) A fraction of the cash in hand ( < f c < ) is then invested in equal parts in all of the stocks to be purchased. Each day the buy threshold B d and the sell threshold S d are updated to the maximum and minimum quality metric of all stocks owned on that day. Doing so attempts to ensure that future decision will only improve the investments. But this updating of the buy and sell thresholds does not guarantee that the value of the investments will grow with time. 6

7 4 Example As an example, consider the optimization of a stock trading method that is calibrated for two-hundred days of closing prices for nineteen stocks, as shown in Figure. Figure 4 shows the day prices, the day 2 prices and the ratio of the day 2 prices to the day prices for each of the nineteen stocks. Applying the methodology shown above with the following parameter values: N =, q =, q 2 = 2, q = 2, f c =.9, B =. and S =. gives the results shown in Figures 5, 6, and value investments cash shares owned Figure 5. (a) Value of investments and value of cash on hand. (b) Number of shares owned over time. The change of the value of the investments and the cash on hand is shown in the top part of Figure 5. Initially the cash on hand is $.. The value of cash and investments after 2 trading days is $.4. The change in the shares owned is shown in the bottom part of Figure 5. A number of metrics for a select set of stocks is shown in Figure 6. The evolution of the buy threshold B d and the sell threshold S d is shown in Figure 7. no bonus for me this year 7

8 change of % change % change volatility quality price Figure 6. Price, fractional change, rate of fractional change, volatility, and quality of stocks,,, and 6.4 buy threshold sell threshold.2 threshold values Figure 7. buy and sell thresholds, B d and S d. 8

9 5 Project Assignment Study and adjust the method described in this document to make as much money as possible with the same data as used in the example. Download the data file of stock prices, stock prices.dat, and an example stock trading analysis program, exchange analysis.m. If the stock price data file opens into a new browser window, you can right click in the window to save the data file. function [ cost, constraint ] = exchange_analysis ( param, Plots ) 2 % c o s t = t r a d e s t o c k s ( param ) % a n a l y z e a p o l i c y f o r buying and s e l l i n g s t o c k s 4 % 5 % INPUT VARIABLES 6 % p () = N.... p o i n t s in a v e r a g i n g 7 % p (2) = q ().... q u a l i t y s l o p e c o e f f 8 % p () = q (2).... q u a l i t y c u r v a t u r e c o e f f 9 % p (4) = q ().... q u a l i t y v o l a t i l i t y c o e f f % p (5) = f c.... f r a c t i o n o f cash to i n v e s t ( must be between and ) % p (6) = B.... buy t h r e s h o l d % p (7) = S.... s e l l t h r e s h o l d % 4 % OUTPUT VARIABLES 5 % c o s t.... t h e n e g a t i v e o f t h e v a l u e o f cash and i n v e s t m e n t s 6 % a f t e r 2 days o f t r a d i n g. The stock trading methodology that your group develops will be parameterized by at least the following design variables: N, q, q 2, q, f c, B, and S, where B is the buy threshold on day and Ŝ is the sell threshold on day. Tasks. Read and study the provided stock exchange program exchange analysis.m. 2. To help study the method of weighted averages used in this project, reproduce Figure 2 of this document for stock number 5 and describe the effect of N on p, p and p in a few sentences.. (Optional) You may change the stock trading algorithm by modifying exchange analysis.m in five places: The definitions of quality Q (lines 9-96); The decision to sell stocks and hold cash (lines 8-); The distribution of purchased shares (lines -5); The method used to update the buy and sell thresholds, B d and S d (lines 48-57); and The smallest allowable difference between the thresholds, B d and S d (lines 59-6); 4. Using the code as provided (or as modified by your group in the task, above), optimize the values of N, q, q 2, q, f c, B, and S and/or other parameters your group may have introduced in order to maximize value. 9

10 (a) Write a Matlab script exchange opt.m that will use the Matlab function exchange analysis.m along with optimization routines, such as ORSopt.m and/or NMAopt.m, to optimize the design parameters. The first two lines of code in exchange opt.m must load in the stock price data file as follows: global stock_prices 2 stock_prices = load( stock_prices. dat ); % l o a d s t o c k p r i c e data (b) Try different initial guesses for the design parameters, list your initial guesses, the resulting optimal values, and the resulting earnings. Are your optimal values and optimal earnings very sensitive to the initial guesses? (c) Run a sensitivity analysis. Run separate analyses with each of your parameters perturbed by +5% individually. That is, for analysis i the i-th parameter value is.5 times its optimal value, and all the other parameters are at their optimal value. What are your earnings with each of the perturbed parameter sets? Did your earnings increase or decrease significantly? What pararameters have the greatest effect on your earnings? (d) If an optimal solution shows q <, q 2 >, and q >, what inference can be drawn regarding investment policy? 5. After completing questions through 4: (a) Specify your recommended set of parameters: N, q, q 2, q, f c, B, and S. and any other parameters you may have introduced. (b) Produce versions of Figure 5 and Figure 7 associated with your recommended set of parameters. You can do this by simply running my cost = exchange analysis( my params, ) (c) If your group changed the methodology in any way in question, describe your changes in words and show the lines of Matlab your group wrote to replace the corresponding sections of exchange analysis.m. (d) What is the greatest amount of money your group was able to make? I will apply your group s method to the set of closing stock prices for the 2 trading days subsequent to the 2 days that you used to optimize your system. I will distribute versions of Figure 5 for each group s results. Have fun and good luck!

11 6 Report contents Each group of two or three people should prepare a report including: Your written responses to questions 2,, 4, and 5. Plots from questions 2 and 5. Print-outs of the Matlab code you wrote. If your group changed exchange analysis.m send your revised version to me in an attachment. A table of your final optimized design parameter values: N q q 2 q f c B S day -2 value day 2-4 value. (Leave the last column blank.) 7 Grading points: The final value of cash and investments at the end of the first two-hundred days of stock prices, in comparison with other group projects. points: The final value of cash and investments at the end of the subsequent twohundred days of stock prices, in comparison with other group projects. 2 points: Quality of the report: (typed, pt or pt,.5 space, inch margins, page numbers, figure numbers, figure captions, etc.) points: Completeness and correctness of descriptions and discussions points: Answers to questions 2-5. (Question is optional.)