Data Collection

Chart 1 – Screenshot of RTOptions program

            Chart 1 is a sample screen of the RTOptions program that was written to gather the data used in this study. The program connects to a data provider over the internet to retrieve real-time bid/ask prices for a given stock and its options. It then runs the data through 5 potential arbitrage formulas and displays color coded results. A red background means an attempted arbitrage trade would result in a guaranteed loss, yellow indicates positive return but less than the risk-free rate and green means a return above the risk-free rate.

            The put-call parity function is as follows:

where

: stock price

: put price

: call price

: strike price

: short term risk-free interest rate

: time until expiration

 

            The “Parity Return” column is the result given by applying the Put-Call Parity strategy where the trader would buy the stock, sell a covered call and buy a put at the same strike as the call and the same expiration date. Ideally, the trader would short a treasury bond and use the money received to finance the purchase of the stock and put. The “Short Parity” column (also called a Reversal) also uses the Put-Call Parity equation but with the trader’s actions reversed. The trader short sells the stock, sells a covered put and buys a matching call. He then takes the money received from the short sale to buy a treasury bond.

            The “Call Arb” column shows the return of a call spread. To create a spread the trader sells a call with a high price and buys another call with a low price relative to each other. This typically results in a profit potential as shown in chart 2. With correct pricing there will always be an area on the chart that results in a loss but if prices are out of step then it could be possible to sell the spread without any risk potential.

 

Figure 1: Left image shows normal price differentials with a loss area (shown in red). Right image shows a mispriced spread without any loss area.

 

            The Value at Risk (VaR) of spreads is calculated as follows:

Where

K1: strike price of option 1

K2: strike price of option 2

C1: sale price of option 1

Cs: purchase price of option 2

 

 

            Setting VaR to 0 reveals that if  then an arbitrage opportunity exists and the trader would see a profit graph like the right side of figure 1. That is, the difference between the strikes is less than the difference between the prices of the 2 options. In order for this to occur, option 1 must be selling at a large premium and option 2 must be selling at a large discount.

            The “Put Arb” column is very similar to the “Call Arb” column. The same strategy is employed but with puts instead of calls. The “Double” column shows the result of doing both call spreads and put spreads at the same time. It is the sum of the “Put Arb” and “Call Arb” columns.

            The “Odds” and “WRet” columns are not true arbitrage. These columns are used for what could be wishfully called “Almost Arbitrage” or jokingly, “Arbitrage WITH risk”. These columns use a formula that predicts the odds that an option will expire in-the-money to search for options that can be sold at prices higher than the price they should be trading at given the risk level they represent. The goal is to sell out-of-the-money options and wait for them to expire worthless while limiting risk.

Figure 2: Selling put spreads and call spreads at out of the money levels.

            Figure 2 illustrates how selling put and call spreads at out of the money strike prices creates a wide zone of profitability. If the stock is within the green area when the options reach maturity then the option seller walks away with the premium they received. If there is a large decrease or increase in the stock price before maturity then the option seller will suffer a loss. Both the maximum profit and maximum loss are limited. The RTOptions program searches for positions that give the highest and widest green area while minimizing the red areas.

            The program uses a formula that is commonly used in decision trees:

Where

: Odds of success

: Reward

: Odds of failure

: Loss

 

            If the product of the odds of success multiplied by the reward is greater than the product of the odds of failure multiplied by the loss then the activity in question is deemed to be a bet worth making. In fact, the bigger the success side of the equation is compared to the loss side the better. This concept can be used when comparing various potential trades to each other.

            The odds of success and failure are calculated using the famous Black-Scholes equation:

Where

C: price of a call

S: current price of the stock

N(d1): Odds the option will expire out of the money

K: strike price of the call

r: short term risk-free interest rate

T: time remaining until expiration

N(d2): Odds the option will expire in the money

: volatility of the stock

            Since the object of this study was to sell options that are out of the money and have them stay out of the money, N(d1) is the odds of success and N(d2) is the odds of failure. The profit of a spread is calculated as the difference in the premiums (). The value at risk is calculated from the VaR formula. The reward and potential loss amounts were converted into percentages as shown:

 ;

            The RTOptions program looks for trades where R%* N(d2) is greater than L%* N(d1). These values are shown in the program under the WRet columns. A green background highlights the fact that the position is statistically worth taking. That is, it is selling for a higher price than the price it should be at given the risk level it represents.

Unmanaged Results – Almost Arbitrage

If one was to blindly take the positive predictions displayed by the RTOptions program and let them run until expiration without concern for the changing conditions of the market then they would have averaged a return of 5.02% of the VaR during the one month period ending March 21, 2008. Remember that the program simply applies the math to find spreads that are relatively overpriced for the risk they represent. The program occasionally finds high risk positions that are so overpriced that they are displayed in the output despite the fact that they are likely to fail.

Chart 2 – Comparing Black-Scholes N(d2)[red line] to actual data.

            Chart 2 shows how well the N(d2) formula preformed compared to the data collected from the stock market. In cases where the formula predicted an 80% or higher chance that the option spread would not be in the money at expiration, the formula preformed at or better than expectations. In fact, the actual results show that only 4 out of the 95 positions that were in this range failed (became in the money). Above the 80% prediction rate, there is an observed success rate of 95.7%.

            However, below the 80% certainty rate the performance of N(d2) drops. The 75% to 80% level only experienced a 67% success rate. This is compensated by the fact that the 70-75 level did better than expected with an 83% success rate. It seems clear that below the 80% level, N(d2) comes close to the correct value within a widening range.

Chart 3 – Adding linear regressions of out of the money success [yellow line] and profitability success [green line].

            Chart 3 shows how closely the N(d2) line coincides with linear regressions of the actual data. The yellow line represents the actual level of option contracts that remained out of the money. This line crosses the N(d2) line at approximately the 80% certainty level. The success rate was higher than expected above 80% and was worse than expected at levels below 80% with the difference growing as the certainty level drops.

            The green line represents price change success. Because the RTOptions program searches for overvalued options to sell, some of the contracts resulted in a positive return even though they failed to remain out of the money. The net result is that profits from these positions were higher than projected. The green money line is above the red N(d2) line across the entire spectrum of certainty levels. The entire data set earned more money than expected when taken as a whole.

Chart 4 – The average return for each certainty level.

            Chart 4 represents the average, high and low returns for each certainty level. As the reader can see there was at least one failure in all data ranges except the 90%-95% level. There was 1 failure at 80-85, 2 at 85-90 and 1 at the 95-100 level. These failures resulted in an average loss of 81.9% of their VaR. Obviously this hurts the average return for those 3 levels but it is observed that the average persisted to be positive.

            The blue bars represent the average return at each certainty level. The 80, 85, 90, 95+ ranges had average returns of 8.5%, 3.9%, 6.8% and 1.9% respectively. The average return was negative in 4 levels below the 80% level. The white bars show the maximum projected return of a position in each of the certainty levels. It is observed that as certainty drops (risk increases) the potential return rises, resulting in a peak at the 45% certainty of making a 94% monthly return. A risk hungry trader could attempt to take advantage of this peak by trading positions with a 50% certainty level. Indeed, the 50% level offered the highest average return of 18.2%.

            Finally the purple bars represent the maximum lost for each certainty level. Of the 169 spreads monitored for this study, 29 lost money. Of those 29, 17 lost 100% of their VaR resulting in an average loss of 79.5%. It is important to note that there were failures even above the 80% certainty level.

Managed Results (Losses) – Almost Arbitrage

            This section will attempt to determine if active monitoring of the portfolio would have changed the final results. Minimum certainty levels will be established for both entering the trade and maintaining the position. That is, only trades at a given initial level and higher will be entered into and they will be exited if they drop below the maintenance level.

            This strategy doesn’t improve results. The data shows that most of the positions that would have been exited early (at a loss) recovered before their expiration date. It would have been more advantageous to simply hold the losing positions and hope for the best. If the trader closed a position on the day it produced a loss then he would have made that loss permanent. If he continued to hold the position then time value would have continued to work in his favor causing the position to become profitable.

            The optimal return was 18.1% which resulted from entering trades in the 50-55 certainty level and holding them to expiration. The second best return (17.6%) resulted from positions starting in the 70-75 level and exiting trades if the fell below 45% certainty. There was only 1 position at that level that dropped below 45% certainty and it did not recover by expiration.

            The best odds of recovery for positions that could have resulted in a loss was 94% for positions initiated at the 80-85 certainty level. The average recovery rate for trades initiated above the 50% level was 80.7%. The average for trades started below the 50% level was only 36.6%. Given the high odds of recovery for trades above the 50% level the situation is clearly in the trader’s favor if he simply holds onto positions regardless of their day to day fluctuations.

Managed Results (Gains) – Almost Arbitrage

           This section investigates if it is possible to close out winning positions early rather than waiting for expiration. Traders are faced with the reality of limited funds. This forces them to pick and choose which trades they want to commit resources to. Unlike the RTOptions program, traders can not trade every single signal. Each position creates a potential loss. Those potential losses add up to eat away at the trader’s margin requirements. It makes sense to remove those liabilities (close the position) if it has already produced the targeted profit before expiration.

            The data show that 134 out of the 169 spread positions in the data could have been closed early with their targeted return. Among those 134 positions the average time to maximum profit was 9.2 trading days. The average return for these positions was 15.3% of their VaR.

Conclusion

            The Black-Scholes N(d2) equation seems to be a reliable indicator of the odds that an option will be in the money at maturity. When compared to a linear regression of the profits returned, N(d2) seems to be a bit conservative. N(d2) underestimated the odds of earning a profit. Selling out of the money spreads where the odds of success times the reward was greater than the odds of failure times the loss produced profitable results in 133 out of 169 cases during the 30 days ending March 21, 2008.

            The most predictable trades were those initiated at or above the 80% certainty level of success. Trades in this range sometimes returned a rate of 18% to 22% of their value at risk in a single month. When positions go against the trader it would be wise to hold the losing positions rather then closing them due to the observation that 80.7% of losses are temporary. Losing positions are very likely to recover before maturity. In the case of winning positions, it would be wise to close them once they can no longer produce a profit. These positions still represent a liability on the portfolio and if they can no longer earn a return they should be closed. It was observed that the average time to reach this point was 9.2 trading days.