Mean Reverting Asset Trading Research Topic Presentation CSCI-5551 Grant Meyers

Table of Contents Introduction + Associated Information Problem Definition Possible Solution 1 Problems with Solution 1 Possible Solution 2 / Research Topic Specific Questions to be Answered

1. Introduction Asset Definition + Properties of a Mean Reverting Asset

Asset Definition 1. A resource with economic value that an individual, corporation or country owns or controls with the expectation that it will provide future benefit. 2. A balance sheet item representing what a firm owns. This presentation will cover only – stocks which represent an ownership interest in a business. Asset Definitions take from: http://www.investopedia.com/terms/a/asset.asp Due to the vast array of financial instruments that would qualify under the term ‘asset’ I limited this paper to stocks. The paper that I am basing this on does have a chapter associated to using options though.

Properties of a Mean Reverting Asset Needs some level of volatility in price. Needs to vacillate around a center value; rising / falling around a dependable ‘Mean’ value. Needs some level of volatility in price. – obviously if there is no low to buy at and no high to sell at, you can’t make any profit. Preferably a seasonal or otherwise dependable cycle up and down. – Good candidates for this are things like oil company stocks or retailers, as they have ‘good’ and bad seasons. Another is the summertime lul in stock market activity and ‘the earnings’ season when lots of companies are giving good or bad news to their stock holders. High liquidity, being able to buy and sell at optimum prices. – You can’t pick a ‘thinly’ traded stock and expect to dependably hit an expected price. Another problem is ‘spread’ namely, that what people are ‘selling’ for and what people are ‘bidding’ / actually purchasing for. Minimal chance of ‘insider trading’ or other ‘exceptional’ events. – Things like the BP oil spill or the collapse of Enron are good examples of ‘exceptional’ events that cost a lot of people a lot of money.

Properties of a Mean Reverting Asset Required for a good Mean Reverting Asset: Preferably a seasonal or otherwise dependable cycle up and down. High liquidity, being able to buy and sell at optimum prices. Minimal chance of ‘insider trading’ or other ‘exceptional’ events. Needs some level of volatility in price. – obviously if there is no low to buy at and no high to sell at, you can’t make any profit. Preferably a seasonal or otherwise dependable cycle up and down. – Good candidates for this are things like oil company stocks or retailers, as they have ‘good’ and bad seasons. Another is the summertime lul in stock market activity and ‘the earnings’ season when lots of companies are giving good or bad news to their stock holders. High liquidity, being able to buy and sell at optimum prices. – You can’t pick a ‘thinly’ traded stock and expect to dependably hit an expected price. Another problem is ‘spread’ namely, that what people are ‘selling’ for and what people are ‘bidding’ / actually purchasing for. Minimal chance of ‘insider trading’ or other ‘exceptional’ events. – Things like the BP oil spill or the collapse of Enron are good examples of ‘exceptional’ events that cost a lot of people a lot of money.

Examples of a Mean Reverting Asset Chevron over last 5 years

Examples of a Mean Reverting Asset Disney this year

Table of Contents Problem Definition Introduction + Associated Information Problem Definition Possible Solution 1 Problems with Solution 1 Possible Solution 2 / Research Topic Specific Questions to be Answered

Problem Definition What does ‘Mean Reverting Asset Trading’ encompass?

Core Questions Can you make money from the ‘Stock Market’ by trading? Which companies do you choose? What are the costs?

Problem Components 1 - Timing Can you make money from the ‘Stock Market’ by trading? Maximize profit from buying low + selling high. When do you buy? (A) $10,000 of Netflix (NFLX) bought on 16 Dec 2014 @ $45.21 / share = 221 shares (B) $10,000 of Netflix (NFLX) bought on 6 Aug 2015 @ $126.45 / share = 79 shares When do you sell? (A) 221 shares sold on 6 Aug 2015 @ $126.45 / share = $27,945.45 (+$17,945.45) (B) 79 shares sold on 22 Oct 2015 @ $97.32 / share = $7,688.28 (-$2,311.72)

Problem Components 2 – Options Which companies do you choose? There are 1,868 stocks listed on New York Stock Exchange. There are 3,300 stocks listed on the Nasdaq. There are 1,299 stocks listed on Euronext. This doesn’t even include the Asia markets, nor options, nor mini-options, nor futures, nor private securities.

Problem Components 3 - Costs Transaction Cost Online Self Directed Trade - $8.90 Broker Assisted Trade - $30.99 Opportunity Cost $10,000 of Amazon (AMZN) bought on 24 Oct 2014 sold today is worth $20,367.02 $10,000 of Apollo Education Group (APOL) on 22 Dec 2014 sold today is worth $2,151.8 Emotional Loss Aversion - Humans fear loss much more than possible winnings Price of Amazon pulled on 24 Oct 2015, this was before their latest earnings figures were released; it has sense gone up quite a bit.

Variables + Unknowns Maximize Gain, Minimize Loss Timing the Buy Timing the Sell Minimizing costs There is no obvious solution, no method always works. Hindsight may be perfect, but predicting the future with precision is literally impossible.

Table of Contents Possible Solution 1 Introduction + Associated Information Problem Definition Possible Solution 1 Problems with Solution 1 Possible Solution 2 / Research Topic Specific Questions to be Answered

Possible Solution 1 Based on analytic solution to asset price prediction algorithm.

ⅆ𝑿 𝒕 = 𝜶 𝜷−𝑿 𝒕 ⅆ𝒕+ 𝝈ⅆ𝑾 𝒕 where 𝑿 𝟎 =𝒙 Possible Solution 1 Using the equation: ⅆ𝑿 𝒕 = 𝜶 𝜷−𝑿 𝒕 ⅆ𝒕+ 𝝈ⅆ𝑾 𝒕 where 𝑿 𝟎 =𝒙 𝛼>0 is the rate of reversion to the mean. 𝛽 is the equilibrium level / mean value. 𝑋 𝑡 is the stock price’s model. 𝜎>0 is the volatility of the asset. 𝜎ⅆ𝑊(𝑡) is the stochastic term causing ups and downs around the mean. Based on a ‘Standard Brownian’ motion. 𝛼 𝛽−𝑋 𝑡 ⅆ𝑡 operates much like Hooke’s law, in that when the price is higher than the ‘equilibrium’ level this term will pull it back down to the mean at the rate of 𝛼. If price is lower than the ‘equilibrium’ level this term will push it back up to the mean at the rate of 𝛼. Stochastic – “The term stochastic occurs in a wide variety of professional or academic fields to describe events or systems that are unpredictable due to the influence of a random variable.“ per Wikipedia. Effectively that is to say that the indeterminate part of the equation is ‘accounted’ for in the stochastic term.

Possible Solution 1 – Analytical Solution Using the equation: ⅆ𝑿 𝒕 = 𝜶 𝜷−𝑿 𝒕 ⅆ𝒕+ 𝝈ⅆ𝑾 𝒕 where 𝑿 𝟎 =𝒙 A solution based on 𝜶 and 𝜷 is possible. In practice, these values are not stable and not that easy to find for a given asset.

Possible Solution 1 – Analytical Solution Buy at x1 and sell at x2 Taken from the Zhang and Zhang paper referenced in the references section. 𝜂(𝑡) is an equation all to itself based on the reward function, and a over time. K is percentage of slippage (loss between execution price and desired price). a the rate of reversion to the mean. b is the equilibrium level / mean value. All this really says is that the optimum buy and sell points are related.

Table of Contents Problem Definition Problems with Solution 1 Introduction + Associated Information Problem Definition Possible Solution 1 Problems with Solution 1 Possible Solution 2 / Research Topic Specific Questions to be Answered

Problems with Solution 1

Problems with Solution 1 Requires model for underlying asset to set calculation constants and determine the rate of reversion to the mean, and the equilibrium level / mean value. Allows adjustments via main 2 parameters only. Nearly impenetrable math…

Table of Contents Possible Solution 2 / Research Topic Introduction + Associated Information Problem Definition Possible Solution 1 Problems with Solution 1 Possible Solution 2 / Research Topic Specific Questions to be Answered

Possible Solution 2 / Research Topic Stochastic Approximation Methods and Applications in Financial Optimization Problems - Chapter 2: Mean-Reverting Asset Trading

Mean Reverting Asset Prediction Equation This equation can be used to ‘Predic t ′ a stocks price: ⅆ𝑿 𝒕 = 𝜶 𝜷−𝑿 𝒕 ⅆ𝒕+ 𝝈ⅆ𝑾 𝒕 Instead of using the analytical solution, you can use simulation with actual values and ‘estimate’ a solution.

Components 1 Stochastic Approximation Noise Sources Used to recursively estimate some quantities based on noise corrupted observations. Originally introduced in 1950s. Noise Sources Imperfect sampling period. Multiple trades executing ‘simultaneously’. Sampling technique. Midpoint between bid / sell, or last trade price

Components 2 Model Recursive Stochastic Approximation Equation 𝜃 𝑛+1 = 𝜃 𝑛 + 𝜀 𝑛 𝑋 𝑛 𝜃 𝑛 is an exact parameter of the system. 𝑋 𝑛 is a ‘random’ independent variable of the system and ‘corrupted’ by noise. 𝜀 𝑛 is the step size.

Mean Reverting Asset Prediction Equation - Estimation Testable Estimator: 𝜃 𝑛+1 = 𝜃 𝑛 + 𝜀 𝑛 𝐷 Φ ( 𝜃 𝑛 , 𝜉 𝑛 ) 𝜀 𝑛 is the step size (seconds, minutes, …) 𝐷 Φ ( 𝜃 𝑛 , 𝜉 𝑛 ) is the Gradient of vectors of correlated and uncorrelated noise 𝜃 𝑛 is known data values 𝐷 Φ

Advantages Over Solution 1 No model for the underlying asset. Less rigid, less dependent on human ‘intuition’. Easily updated for new data & ‘paradigm’ shifts in whole sectors. Data for stocks is easily available & in an easily processed format. Model is the a and b values of the prediction equation.

Advantages Over Solution 1, continued Multiple asset data time-resolutions allow for variable scaled action speeds. If broker takes, on average, 10 seconds to execute a trade, having a regression based on faster time would not necessarily work well. Using 24 hour scale data, may allow for a more macroscopic view of the asset’s movement.

Table of Contents Specific Questions to be Answered Introduction + Associated Information Problem Definition Possible Solution 1 Problems with Solution 1 Possible Solution 2 / Research Topic Specific Questions to be Answered

Specific Questions to be Answered 1 Data Sample Related Does the algorithm work when there is a macroscopic change in the overall market? Does changing the training & applying time windows affect the return? How much? Do longer windows fair better or shorter ones? Are there any dependable seasonal fluctuations? Does the asset ‘class’ affect the effectiveness of the algorithm?

Specific Questions to be Answered 2 Performance Related How fast can the Xeon server crunch the numbers? How fast can the Hydra server crunch the numbers? Is there a better way to format the data than the default JSON format? Given the use of common mathematical operations, could they be switched out to a format that uses matrix multiplication?

References Human Loss Aversion - http://www.sciencemag.org/content/313/5787/684 Asset Definition - http://www.investopedia.com/terms/a/asset.asp NYSE Listing Size: https://en.wikipedia.org/wiki/New_York_Stock_Exchange NASDAQ Listing Size: https://en.wikipedia.org/wiki/NASDAQ EuroNext Listing Size: https://en.wikipedia.org/wiki/Euronext Average Online Trading Cost: http://www.valuepenguin.com/average-cost- online-brokerage-trading Zhang and Zhang Reference: Hanqin Zhang, Qing Zhang, Trading a mean- reverting asset: Buy low and sell high, Automatica, Volume 44, Issue 6, June 2008, Pages 1511-1518, ISSN 0005-1098, http://0- dx.doi.org.skyline.ucdenver.edu/10.1016/j.automatica.2007.11.003.