Pairs Trading: An Optimal Selling Rule with Constraints

The focus of this paper is on identifying the most effective selling strategy for pairs trading of stocks. In pairs trading, a long position is held in one stock while a short position is held in another. The goal is to determine the optimal time to sell the long position and repurchase the short position in order to close the pairs position. The paper presents an optimal pairs-trading selling rule with trading constraints. In particular, the underlying stock prices evolve according to a two dimensional geometric Brownian motion and the trading permission process is given in terms of a two-state {trading allowed, trading not allowed} Markov chain. It is shown that the optimal policy can be determined by a threshold curve which is obtained by solving the associated HJB equations (quasi-variational inequalities). A closed form solution is obtained. A verification theorem is provided. Numerical experiments are also reported to demonstrate the optimal policies and value functions

In-depth optimization of stock market data mining technologies

University of Technology, Sydney. Faculty of Information Technology.Stock trading is a number of stocks to be exchanged from one trader to another trader. It consists of a trader selling a number of stocks at a price and a volume, and another trader buying the same stocks at the same price and the same volume. Most traders want to buy a stock at a low price and to sell the stock at a high price in order to make a profit. However, it is difficult to know whether the current trading (buy/sell) price is low or high. Some researchers have presented technical trading rules which are mathematical formulas with many parameters to solve this problem, such as moving average rules, filter rules, support and resistance, channel break-out rules, and so on. All these rules are based on historical data to generate the best parameters and use the same parameters in future trading to make a profit. When the parameters of a trading rule are set properly, the trading rule can help the traders to make a profit (buy/sell at a low/high price). Experiments have shown that technical trading rules are profitable. However, there are still some disadvantages and limitations to the technical trading mies in real stock market trading. First, the technical trading rules do not integrate domain knowledge (expert experiences and domain constraints, etc). For example, some trading rules pattern maybe only generate three signals during one year trading to get the most profit. However, the pattern is unreasonable and it is unprofitable in future trading, because the pattern is only a mathematical maximum, but it is impracticable in stock trading. Second, the output of a parameter for a trading rule is only one single value. Sometimes, it may be a noise so the trading rule is inapplicable in future trading. Third, present algorithms to calculate parameters of trading rules are inefficient. Most trades are performed through internet such that they can buy and sell stocks in online and a trade is completed in a second. Real markets are dynamic such that trading rules have to be updated all the time depending on changing situations (new data come in, new parameters will be recomputed). Current enumerate algorithms waste too much time to get new parameters. However, a one-second short delay in real stock trading will lose the best trading chance. Fourth, when we evaluate the performance of a stock, we need not only to consider its performance (profit and return), but also to compare it to other stocks performance. At present, trading rules do not compare to the other stocks performance when they are selected to generate a signal, so the selected stocks or rules may be not the best ones. Fifth, in stock markets, there are many stocks and many trading rules. The problem is how to match and rank stocks and rules to combine a profitable and applicable pair. However, trading rules do not solve this problem. Lastly, trading rule techniques do not consider the sizes of investments. However, in real market trading, different investments will result in a different performance of a pair. We propose in-depth data mining methodologies based on technical trading rules to overcome these disadvantages and limitations mentioned above. In this thesis, we present the solutions to combat the existing six problems. To address the first problem, we designed a domain knowledge database to store domain knowledge (expert experience and domain constraints). During the computing procedure, we integrated domain knowledge and constraints. We observed the output more reasonable as we considered domain knowledge. To address the second problem, we optimized a sub-domain output instead of a single value, in the sub-domain all combinations of parameter can get a near-best result. Moreover, in the sub-domain, some experienced traders can also set or micro-tune parameters by themselves and a better performance is guaranteed. To address the third problem, we adopted genetic algorithms and robust genetic algorithms to improve the efficiency. Genetic algorithms and robust genetic algorithms can get a near-optimal result in an endurable execution time, and the result is also near to the best one. To address the fourth problem, we applied fuzzy sets and multiple fitness functions to evaluate stocks. Because many factors influence the performance of a stock, it is necessary to create a multiple fitness function for genetic algorithms and robust genetic algorithms. To address the fifth problem, we built a stock-rule performance table to rank stock-rule pairs and find the best matching pairs. The stock-rule pair results showed that the ranked performance is better than that of randomly matched pairs. Finally, to address the sixth problem, we drew a graph of the relationship between investments and number of stock-rule pairs to search maximal points, and to decide the number of pairs for different sizes of investments. In summary, the purpose of this thesis is to identify optimal methodologies in stock market trading, to make more profit with less risk for investors. The experimental results showed that the methodologies are more profitable and predictable

Optimal Pairs Trading Rules

This thesis derives an optimal trading rule for a pair of historically correlated stocks. When one stock\u27s price increases and the other one\u27s decreases, a trade of the pair is triggered. The idea is to short the winner and to long the loser with the hope that the prices of the two assets will converge again. In this thesis the spread of the two stocks is governed by a mean-reverting model. The objective is to trade the pair in such a way as to maximize an overall return. The same slippage cost is imposed on every trade. Furthermore, a local-time process to the spread is introduced in order to avoid infinitely large gains. We use the associated Hamilton-Jacobi-Bellman equations to characterize the value functions which are solved by using the smooth-fit method. It is shown that the solution of the optimal pairs trading problem can be obtained by solving a set of nonlinear equations. Additionally, a set of sufficient conditions is provided in form of a verification theorem. The thesis concludes with a numerical example

Bollinger Bands Thirty Years Later

The goal of this study is to explain and examine the statistical underpinnings of the Bollinger Band methodology. We start off by elucidating the rolling regression time series model and deriving its explicit relationship to Bollinger Bands. Next we illustrate the use of Bollinger Bands in pairs trading and prove the existence of a specific return duration relationship in Bollinger Band pairs trading.Then by viewing the Bollinger Band moving average as an approximation to the random walk plus noise (RWPN) time series model, we develop a pairs trading variant that we call "Fixed Forecast Maximum Duration' Bands" (FFMDPT). Lastly, we conduct pairs trading simulations using SAP and Nikkei index data in order to compare the performance of the variant with Bollinger Bands