Statistical Arbitrage vs. Long-Investing: Case of Options

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1 Statistical Arbitrage vs. Long-Investing: Case of Options Website: Statistical Arbitrage vs. Long-Investing: Case of Options Page 1

2 Statistical Arbitrage vs. Long Investing: Case of Options Overview: This paper discusses some possible ways of constructing a pairs-trading strategy for investing in stocks and options (the main idea is applicable to any asset class). A statistical arbitrage framework lends itself well to the equity market neutral style of portfolio construction. Some difficulties are apparent in the way with which practitioners determine the size, speed, direction and timing of a future reversal in the spread between two assets to some historical trend. We suggest ways to overcome such problems while still keeping as close to a profitable outcome as possible and to adequate risk controls. Keywords: Statistical Arbitrage, Pairs-trading, Strategy, Options, Stocks, Insurance, Hedging, Market Neutral, Trend, Reversal, Spread, Time Series Analysis. Word Count: 5320 (16 pages) Statistical Arbitrage vs. Long-Investing: Case of Options Page 2

3 O ye who believe! When ye deal with each other in transactions involving future obligations in a fixed period of time, reduce them to writing; let a scribe write down faithfully as between the parties; let not the scribe refuse to write; as Allah has taught him, so let him write... Surah Al-Baqarah, verse 282 Why are Options so risky? Is there a better way to invest using options and to make them more suitable for risk management purposes? Options (and to a large extent Futures) are one of the greatest financial contracts ever conceived by man. They are non-interest bearing contracts that can be used for hedging and insurance purposes. These crucial economic activities have a bearing on real economic decisions such as whether or not to own a stock and ownership claim to a company. And most importantly of all these instruments allow different groups of agents to transact business across time and in an environment where the future value of assets is uncertain. Money managers when they take a long position in a stock will also often hedge by buying insurance. For example, the purchase of a put option on that stock gives them the right but not the obligation to sell the stock at a specified price at some specified date in the future. However the option has its downside as well: it will reduce the profits made on the long position in the stock as hedging is expensive should you turn out to be right and the option expires worthless. If for instance your stock went up when you had feared before that it may go down then the insurance purchased through the put option has now cost you the premium price you had to pay to buy it. It is the feature that the stock price can end up being anywhere that gives an option a claim against a future payment stream based on an underlying security s price movement a value but that also introduces an element of risk that the option payoff turns out to be very different than was initially perceived. Some predictive model or technique is needed to direct the investor to those trading decisions that are the more reasonable ones to make. This paper shall describe one approach inspired by Statistical Arbitrage and also by a desire to make finance and investing safer for the general public. Using Options to Hedge Risk: A Primer We begin by introducing the main elements and motivation behind options trading. The presentation is kept simple and we avoid bringing in the complicated expressions needed to derive the actual Black-Scholes options pricing formula or its variants. The reader may refer to Option Trading: Pricing and volatility strategies and techniques by Euan Sinclair for a more detailed description of these and other related issues in the options literature. Statistical Arbitrage vs. Long-Investing: Case of Options Page 3

4 Example Let s assume the last closing price (which we ll refer to simply as the current stock price) on June 28 for a particular company Shire Pharmaceuticals is 610. Where will this stock price be in the future? Let s take a look at what options traders think about the future prospects for this stock. The London International Financial Futures & Options Exchange (LIFFE) is a nice options exchange that provides data for exactly this purpose. Example reproduced from p.189 of Using the Financial Pages by Romesh Vaitilingam CASE OF THE PUT OPTION Notice that with the stock price currently at 610 you get more value with the 650 put option compared to the 600 put option for any given month of expiry. This is because with the stock currently at 610, if this situation remained at expiry then you could sell shares of Shire Pharm at a higher price of 650. Sell >>>>>>> Buy Profit! But nothing is for free. What are the risks here? **The stock price may rise too much too fast** The put option contract that we have seen above gives the holder the right to sell the stock if they wish, not at the new lower stock price but instead at the higher price of 650. From this perspective you don t want the current stock price (S) to end up higher than the 650 Put Strike price (X) at the time of expiry in March. In fact you want to see the stock fall so that the Put contract becomes more valuable. For S < X Stock is lower than Strike Put Value at expiration: (X S) For S > X Stock is higher than Strike Put Value at expiration: 0 So the Put value will be higher the lower the Stock price is relative to the Strike price. If the stock price rises before expiration then the option has lost part of its remaining value. Statistical Arbitrage vs. Long-Investing: Case of Options Page 4

5 Upon expiry, if the stock price is higher than the strike price the option is not in the money (i.e. is not valuable) and it expires worthless. It is the exact reverse situation for the Call Option. Below are the correct strategies to follow with trading in options if you knew what the path would be for the future movement in the stock. TRADING OPTIONS UNDER DIFFERENT SCENARIOS (For Puts & Calls) 1 Future Stock Price Fall is believed If Stock Falls >>>>>>>> Buy Put Option If Stock Falls >>>>>>>> Sell Call Option (right to sell at higher price) (right to buy at lower price) Future Stock Price Rise is believed If Stock Rises >>>>>>>> Buy Call Option If Stock Rises >>>>>>>> Sell Put Option (right to buy at lower price) (right to sell at higher price) **You must pay a premium for the Option** Because of the advantages to the buyer of the option contract where the strike price at which you may buy or sell can be different to the stock price existing at the expiration date the seller of this option will want to charge you some money in the form of a premium. That amount then becomes the price of the option. These are the cell figures in the table. An option is not free. You pay for the privilege. You expect to find higher premiums charged on put option contracts with strikes that are higher than the current stock price for a given time expiration. And you expect to find higher premiums charged on call option contracts with strikes that are below the current stock price for a given time to expiration (the Call example is omitted here). Although the strike levels for the option and indeed all terms for the option contract are fixed and specified ahead of time, the stock price can move at any time. The beauty of the Black-Scholes formula is that it provides a method to continuously price the option (which is fixed) given the stock price (which is not). **Time acts against you** You have a limited amount of time before expiration to experience the desired movement in the stock price. Time value decays and becomes the enemy of the option buyer but a friend for the option seller. As mentioned before, upon expiry if the option is not in the money (i.e. is not valuable) then it expires worthless. You expect the option price whether a put or a call to increase when the time to expiry increases. This is because there is greater chance of the stock to move relative to the strike price level specified in the contract. 1 Knowing when exactly the stock will rise or fall and by how much and how fast is of course an interesting question to think about. One solution is to regress the underlying stock A with another stock B and then to visually inspect when the spread between the two has drifted too far from the mean line of zero. By specifying the various ways in which the spread can evolve so as to revert back to the zero mean line this will then say something about how stock A and stock B will likely behave going forward. The connection to the option trading strategy then follows logically from the forecasted movement of the stock. Statistical Arbitrage vs. Long-Investing: Case of Options Page 5

6 Clearly then we find there are significant hurdles to using options in risk management within an overall investment strategy. We may turn out to be disappointed with the way the option contract performs from the time of purchase to the time of sale or expiry. The question is whether there can be a way ahead of time to offset some of the disadvantages of owning and trading in options? OPTIONS STRATEGY WITH STOCK PRICE FORECASTING Motivation We are typically concerned in our ability to get the forecast correct with regard to the future price path of the stock. The logic being that the option price will then respond almost one-for-one with the underlying movement of the asset around which the option is based. This setup is justified according to the standard Black-Scholes framework of pricing options because the only unknown that needs to be inputted is the (implied) volatility of the stock itself. All other parameters are known with certainty and are specified at any time: stock price, strike price, time to maturity, interest rate etc. Knowledge of where the stock is useful for calculating how the option is priced. Traders use their assessment of the future stock price to determine fair value for the option at any given moment. And either through a pricing formula or through the general process of exchange and public information discovery the market arrives at a fair price for the option. Stock move >>>>>>> Option move The traditional way to think about value investing is to try and figure out whether an asset is undervalued or overvalued, which then indicates if it will likely go up or down over a period of time, assuming that the price converges to the security s value. Is there some way, some technique to forecast the future price of a stock and then use that information to form an options trading strategy without having to bother about whether stocks are fundamentally valued or whether the option is fairly priced? In order to answer this and many other questions let us consider an experiment that lists all the major possible future payoffs from a long-short Stat-Arb investment strategy in two stocks A and B. This will prove instructive in understanding where exactly a statistical arbitrage market neutral strategy can work as well as the instances where it cannot. PAIRS-TRADING OUTCOMES WITH STOCKS Let s assume that you identify a mispricing in the market and that, while you want to try to exploit this opportunity, you also want to be exactly market neutral in your portfolio on both the long and the short sides. We need to choose the proportion of shares to invest in some Stock A and the proportion of shares to invest in some Stock B. We start our analysis by assuming an amount of $2000 to invest across the long and short side of the position. Statistical Arbitrage vs. Long-Investing: Case of Options Page 6

7 Then we determine the future payoffs of our holding position in the long/short portfolio by assuming different scenarios with regard to the evolution of both stocks. Share prices Stock A: $20 per share Stock B: $10 per share At time t let s open the trade: << PAIRS TRADE >> Long 1000 shares of Stock A = - $2,000 Short 200 shares of Stock B = + $2,000 Net long / short = $0 This is a dollar neutral position. Scenarios for the Future Payoff (Stocks A & B) Scenario 1: Both Stocks rise by 50% Scenario 2: Both Stocks fall by 50% Scenario 3: Stock A rises by 50% and Stock B rises by 40% Scenario 4: Stock A falls by 40% and Stock B falls by 50% Scenario 5: Stock A rises by 40% and Stock B rises by 50% Scenario 6: Stock A falls by 50% and Stock B falls by 40% Scenario 7: Stock A falls by 50% and Stock B rises by 50% Scenario 8: Stock A rises by 50% and Stock B falls by 50% Statistical Arbitrage vs. Long-Investing: Case of Options Page 7

8 Let s work out the dollar amounts made or lost for each of these eventualities. Scenario 1 [Zero profit] A: Net position = $3,000 - $2,000 = $1,000 B: Net position = $2,000 - $3,000 = - $1,000 Profit >>> 0 Scenario 2 [Zero profit] A: Net position = $1,000 - $2,000 = - $1,000 B: Net position = $2,000 - $1,000 = $1,000 Profit >>> 0 Scenario 3 [Positive return] A: Net position = $3,000 - $2,000 = $1,000 B: Net position = $2,000 - $2,800 = - $800 Profit >>> 200 Scenario 4 [Positive return] A: Net position = $1,200 - $2,000 = - $800 B: Net position = $2,000 - $1,000 = $1,000 Profit >>> 200 Scenario 5 [Negative return] A: Net position = $2,800 - $2,000 = $800 B: Net position = $2,000 - $3,000 = - $1,000 Profit >>> Scenario 6 [Negative return] A: Net position = $1,000 - $2,000 = - $1,000 B: Net position = $2,000 - $1,200 = $800 Profit >>> Statistical Arbitrage vs. Long-Investing: Case of Options Page 8

9 Scenario 7 [The worst outcome] A: Net position = $1,000 - $2,000 = - $1,000 B: Net position = $2,000 - $3,000 = - $1,000 Profit >>> - $2,000 Scenario 8 [The best outcome] A: Net position = $3,000 - $2,000 = $1,000 B: Net position = $2,000 - $1,000 = $1,000 Profit >>> $2,000 Scenarios 1 and 2 are zero profit. Both stocks A and B move in the same direction by exactly the same percentage, thereby resulting in no change to the long/short position. Scenario 3 is positive profit because, while both stocks move in the same direction, the stock that you were long in went up by proportionately more than the stock that you were short in. Scenario 4 is positive profit because, while both stocks move in the same direction, the stock that you were long in went down by proportionately less than the stock that you were short in. Scenario 5 is negative profit because, while both stocks move in the same direction, the stock that you were long in went up by proportionately less than the stock that you were short in. Scenario 6 is negative profit because, while both stocks move in the same direction, the stock that you were long in went down by proportionately more than the stock that you were short in. Scenario 7 is the worst outcome because you were long the stock that went down and short the stock that went up. Scenario 8 is the best outcome because you were long the stock that went up and short the stock that went down. Statistical Arbitrage vs. Long-Investing: Case of Options Page 9

10 Pairs-Trading Strategies From Stocks to Options In the above example we saw how one may construct a pairs-trading strategy between two stocks, going long Stock A and going short some other Stock B. Now we consider the case of how to trade speculatively in options associated with stocks. For simplicity we assume a pairs trading strategy for only one option on a single stock. This translates into 4 possible trades or 2 sets of long-short positions: one long-short set for the initial opening position and one long-short set for the closing position. 2 PAIRS-TRADING USING OPTIONS The challenge now is to construct a paired option trade in a dynamic space-time going long and short in the same option but over different months. Consider again the case of Shire Pharmaceuticals. Assume you are only interested in near- or at-the-money options i.e. options with strikes that are close to the stock price. Focusing attention on these options is likely to isolate the most liquid options on offer which has the added bonus of reducing order slippage i.e. the likelihood of an order submitted to an options exchange not being quickly filled or filled at all. Suppose you think that: Mar Call is overvalued Dec Call is undervalued >>> Stock will (eventually) fall from now till March >>> Stock will (eventually) rise from now till December Notice how this traces out the structure of the paired option trade. Somehow you decided that the stock will initially rise and then fall. This analysis has led you to take a long and short trading position (and therefore an investment strategy) in the option for that stock. Say on September 15 th Open the trade at time t : (i) Sell the 44.5 (Mar) Call (ii) Buy the 32.0 (Dec) Call both at the 650 strike price. Close trade at time t+m for the December Call and at t+n for the March Call with m<=n : (i) Buy the??.? (Mar) Call (ii) Sell the??.? (Dec) Call again both at the 650 strike price. [?.?? denotes the price is to be set in the future] 2 Notice that a maximum of 8 possible options trades can be specified when considering a paired strategy on two stocks A and B. The ratio of possible options trades to the quantity of underlying stocks is therefore 4:1. Statistical Arbitrage vs. Long-Investing: Case of Options Page 10

11 A range of strategies may be followed. If the call position that you sold is out of the money it could be allowed to run and expire worthless without needing to buy it back. Similarly if the call position that you bought and which is in the money is allowed to run to expiry then you could choose to accept delivery of shares in lieu of a cash settlement. Finally note that in the interim if you are concerned that the stock may move adversely then you may like to take decisive action to close out the trade early and book any profit or loss on the position. **Notice the beauty of the equity market neutral strategy: we use the income from the sale of one asset as the proceeds for the purchase of another asset** **As options offer the investor an alternative universe of security specifications to choose from, we can make a play today based on how the future might turn out tomorrow** When will this strategy be profitable? The stock needs to rise from now till December and fall from now till March. Specifically, since we are considering options for the Shire Pharm stock above: (i) >>> If Dec Call at time t+m > 32.0 (ii) >>> If Mar Call at time t+n < 44.5 = Profit at t+m on one leg = Profit at t+n on the other leg For you to make money on this paired call trade, two things therefore need to happen. You profit on the long option trade The price of your purchased Dec Call option (which you subsequently sell) needs to rise within the time frame in question i.e. before option expiry. You want the stock price to rise before December expiry so that the price at which you sell that option is hopefully higher than when you bought it. You profit on the short option trade The price of your sold Mar Call option (which you subsequently buy back) needs to fall within the time frame in question i.e. before option expiry. You want the stock price to fall before March expiry so that the price at which you buy back that option is hopefully lower than when you sold it. Money in the bank Do we go neutral with options too? As a seller of the first Call option you obtain the 44.5 premium. As a buyer of the second Call option you pay the 32.0 premium. You now have ( ) = 12.5 net cash (at time t). Statistical Arbitrage vs. Long-Investing: Case of Options Page 11

12 Thus each long/short trade is self-generating in terms of income (disregarding transaction costs and other fees). The sell position provides the capital for the long position. Notice that the difference between the long and short side of each paired trade is kept in reserve and can possibly be reinvested into more pair trades. Because options are derivatives i.e. they derive their value from other securities and have no intrinsic value other than that of a promise between two parties to a transaction, it makes no sense to be neutral also in the options trading position. An option s value stems only from the fact that somebody else is willing to pay for its claim and nothing else. You are rather trying to arbitrage on the mistaken value attached to all options when they are assumed to be fairly priced according to a popular calculation method such as the Black- Scholes formula. The source of the mistaken value cannot be contained in options: the source of volatility must be real and this automatically excludes instruments which cease to exist after a certain point in time. The source of volatility is rather contained in the stock. With stocks there is an underlying value and a source of volatility, so therefore the question does arise on how to achieve neutrality when it comes to equity investing. How can up be down and down still be up? Recall the example of the stocks A and B above in the market neutral portfolio. Note that we require only a RELATIVE PRICE CHANGE. We do not require that the security rises or falls in precisely the way we would like to observe, though that would be best. It is this flexibility combined with our predictive skill that is the true hidden value of the Stat-Arb technique when it comes to a pairs-trading strategy. What if the stock goes up in both periods? Both option contracts for December and March can go up and you can still make money, provided the stock in which you are effectively long in (Dec) goes up by proportionately more than the stock in which you are effectively short in (Mar) rises. What if the stock goes down in both periods? Again you can still make money provided the stock in which you are effectively long in (Dec) goes down by proportionately less than the stock in which you are effectively short in (Mar) falls. But to achieve either of these outcomes you still need to be correct in judging the overall strength of direction, size, speed and timing of the stock (or pair of stocks if modelling more than one) as we move forward in time. Statistical Arbitrage vs. Long-Investing: Case of Options Page 12

13 Long-Short Trading Decisions: When to Open & Close / One versus Two legs With both the initial short-long trade and the subsequent long-short trade now cancelling out, all risk is removed and any profit or loss gets booked. Notice that for both the Dec Call and the Mar Call we could have assumed that the longshort trades both at the open and at the close are made simultaneously. However this need not necessarily be the case as we showed above with the staggered nature of the long and short positions for the Dec and Mar Calls. These decisions are left to the executor of the strategy. In fact if the forecasting at the beginning turns out to be true one does not have to construct exactly market neutral long/short trades at all times. Depending on the strength of the forecast you could go with a long-only leg or a short-only leg with the trade until the end, provided that the capital exists for you to do so. However you still require statistical arbitrage techniques to highlight these opportunities, so the choice between going strictly long, short and neutral is a rather technical and arguably moot point. This point regarding the non-neutral nature of investment is returned to in the conclusion. The need for good econometric forecasting You are in need of a good time series forecasting model to suggest when a change in the relative stock prices (the spread) is most likely as you move through time. That way you will know when it is favourable to open a long/short trade and when to subsequently close out the same trade, which in turn helps to pinpoint which of the many candidates of adjacent option months might land you with the best pair trades to invest in. Why Pairs-trading Strategies often fail Suppose we observe that the distance or spread between prices of the two securities becomes pronounced. The spread, rather than going up and down and crossing the zero line often, instead drifts largely in one direction, say upward, for a long period of time. This highlights a possible miss-pricing between the two securities and thus a high likelihood that the spread will eventually reverse direction to eliminate the sizeable discrepancy. When a large deviation in the price spread between two securities from some long-term trend occurs a trader can interpret this as a signal that the situation is unlikely to continue. You then would buy the undervalued asset and sell the overvalued one. When the spread reverts back again the position is unwound and any profit or loss gets booked. The risk however is that, rather than see the spread revert to the mean, it will continue to grow. Many Stat-Arb hedge funds have experienced losses because their positions did not mean revert as anticipated within the time frame they were expecting. Losses from pairstrading strategies that failed to time the reversal, speed, size and direction of the spread were made worse by a desire to see an eventual profit in the end. Liquidating into a tough market resulted in very poor performance. We see an example of how this can be below. Statistical Arbitrage vs. Long-Investing: Case of Options Page 13

14 Example of a long/short pair that doesn t revert to the mean as expected Let s turn our attention away from stocks and options for a moment and look now at the relationship between the US 10-year Treasury Yield and the US 2-year Treasury Yield. These data are available at daily frequency from January 2001 to January Both time series are plotted below on the same graph and beneath is the spread graph (one minus the other). But notice that the deviation of the spread from its historical level the gap between the two series is observed to last for years before returning to trend once again. It would seem very hard to profitably trade this as a long-short pair UST 10yr UST 2yr SPREAD Statistical Arbitrage vs. Long-Investing: Case of Options Page 14

15 Conclusion Do I Invest Long-Short, only Long or only Short with the Pairs-Trading Strategy? This paper has looked at formulating a statistical arbitrage strategy with an application to pairs-trading in stocks and options. The discussion of market neutral investing can be applied more generally to any financial asset. The main motivation for opening the pairs-trade is that you spot a deviation in the spread which you suspect will correct at some point later, leading to a rise in the asset you are long in and a fall in the asset you are short in. This will lock in a profit if the two assets do indeed adjust to eliminate the discrepancy in the spread. However there is a need to judge correctly the direction, speed and crucially the timing of the future change. When exactly will the spread mean revert? Of course, assuming that you were initially prepared to hold the position for years, and that the prices offered for the long and short trades were favourable given the timeframe in question, then this wait may not be a problem: in fact it may prove beneficial if the eventual correction is rather large and the initial prices reasonable. However such information regarding the future behaviour of the spread is not known with certainty today. If only it was! The moral of this story is no matter how good your statistical analysis or how long your dataset is, finding a spread that mean-reverts in the way you anticipate is very difficult. One suggestion might be to disregard the spread altogether as a main signal to open and close the trade, or at least to supplement it with other metrics to help identify trading opportunities that are more consistently spaced out. But we think there is a third way. An alternative approach to market neutral trading (which assumes you have money left over from previous pairs-trading positions that are yet to be concluded) is to consider taking the risk to invest purely long or purely short in a single asset or combinations of different assets. This would be the case in the graph of the US 10-year and 2-year Treasury where a deviation is observed for long periods of time but we possess some idea of the future path of the asset which is known beforehand. In that case it makes more sense to go only in one direction that is, to either go long or short rather than going both long and short and thereby suffer a loss on at least one leg and possibly both legs of a position. References: Douglas, F. E and G. Vainberg (2007). Option Pricing Models & Volatility Using Excel VBA. John Wiley & Sons Inc, New Jersey. Schap, K. (2005). The Complete Guide to Spread Trading. McGraw-Hill. Sinclair, E. (2010). Option Trading: Pricing and Volatility Strategies and Techniques, John Wiley & Sons Inc, New Jersey. Thomsett, M. C. (2010). Options Trading For the Conservative Investor, 2 nd edition. FT Press, New Jersey, Pearson Education Ltd. Vaitilingam, R. (2006). The Financial Times Guide to Using the Financial Pages, 5 th edition. FT Prentice Hall, Pearson Education Ltd Statistical Arbitrage vs. Long-Investing: Case of Options Page 15

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