Abstract. Keywords: Equity Options, Synthetic Stock, Investment

Transcription

1 Abstract This paper examines the historical time-series performance of seventeen trading strategies involving options on the S&P 500 Index. Each option strategy is examined over different maturities and moneyness, incorporating transaction costs and margin requirements. An initial analysis was constructed by assuming a theoretical option starting in 1970 and allowing for a continuum of trading comparing historical performance via returns and Sharpe Ratios as compared to the S&P 500 as a benchmark. A second analysis generated portfolios for a typical investor, using the past 34 years and 10 years to examine rates of return given certain trading restrictions. The analysis revealed significant profitability in investing in certain option strategies, in particular, an ATM Synthetic Stock. Keywords: Equity Options, Synthetic Stock, Investment

2 Equity Options Strategies for the Individual Investor Robert C. Hamernik Honors Committee: Dr. David Peterson Dr. James S. Doran Dr. R. Mark Isaac

3 The members of the Committee approve the thesis of Robert C. Hamernik defended on December 5 th, Dr David Peterson Professor Directing Thesis Dr Mark Isaac Outside Committee Member Dr James Doran Committee Member 2

4 Abstract This paper examines the historical time-series performance of seventeen trading strategies involving options on the S&P 500 Index. Each option strategy is examined over different maturities and moneyness, incorporating transaction costs and margin requirements. An initial analysis was constructed by assuming a theoretical option starting in 1970 and allowing for a continuum of trading comparing historical performance via returns and Sharpe Ratios as compared to the S&P 500 as a benchmark. A second analysis generated portfolios for a typical investor, using the past 34 years and 10 years to examine rates of return given certain trading restrictions. The analysis revealed significant profitability in investing in certain option strategies, in particular, an ATM Synthetic Stock. Keywords: Equity Options, Synthetic Stock, Investment 3

5 I. Introduction Most current research focusing on options can be segmented into three subfields; the first involves the theoretical and empirical estimation of various option-pricing models, 1 the second details the role option play in hedging risk exposure 2, while the third attempts to price exotic option constructs. 3 To this point, very little attention has been dedicated to the effect options have from an individual investor standpoint. Explicitly, what effect holding or writing options have on portfolio returns? The purpose of this paper is to examine, from an historical perspective, the return and risk to holding various option strategies from a representative investor standpoint. Our representative investor is considered distinct from an institutional investor, since the individual has limited net worth, and faces the burden of higher transaction costs given bid-ask spreads, taxes, and overall relative trade size. Consequently, the results are designed to provide investment options and strategies across various levels of risk aversion while still maintaining a diversified portfolio. Hopefully, the results will provide new insight into investment options that can actually be utilized in today s market. The underlying asset to which the portfolio will be compared to is the Standard and Poor s 500 Index (S&P 500), which is a capitalization-weighted index of 500 Blue-Chip stocks. The S&P 500 is typically used as the benchmark for the overall performance of the market. From a theoretical standpoint, investing directly into the index would eliminate all nonsystematic risk. In general, most managers who are active in the market accept beating the market as a measure of positive abnormal returns. As such, utilizing options on the index to 1 For example, Bakshi, Cao, and Chen (1997), Scott (1997), Heston (1993), Benzoni (2001), Driessen and Maenhout (2003), Rubinstein (1976), and Santa-Clara and Shu Yan (2004) 2 For example, Buraschi and Jackwerth (2001), and Jackwerth (2000) 3 For example, Merton, Scholes, and Gladstein (1978) and Merton, Scholes, and Gladstein (1982) 4

6 examine various option strategies, will allow the comparison relative to this benchmark for a wide array of investor risk preferences. This research further focuses on three types on individual investors: high-risk aversion, medium-risk aversion, and low-risk aversion, where the medium risk averse investor would accept the return and risk associated with the market. Each strategy is compared and generated with the idea of classifying the strategy within a risk aversion class. It does not examine the reasons an investor falls into each category, but the results should provide useful alternatives for each of the three types of investors. When discussing risk, it should be clear that this research is not trying to define risk nor is it trying to discover riskless investments. The S&P 500 Index s level of risk will be the baseline for the medium risk-averse investor. The highly risk averse investor will have a low risk tolerance based primarily on lower standard deviation of returns. Similarly, the low risk averse investor will have a higher risk tolerance, which allows for high volatility in returns. Rates of return and Sharpe ratios will be used in evaluating the strategies but only after classify the strategy to an investor type based on the volatility of that strategy. The options market today is a highly liquid, low-transaction-cost, and penetrable market. An individual investor, today, can easily trade small quantities of contracts through a broker or an individual on-line brokerage account. The options market today is nothing like it was twenty or even thirty years or ago. Thirty years ago the options market barely existed and was primarily an institutional investment vehicle; the options market had just become standardized, allowing an individual investor to invest in index options but at extremely high costs and without out the fluidity of today s market, as noted in Williams and Hoffman (2001). In today s market, option prices instantly change in value as prices fluctuate in underlying assets, according to market 5

7 maker s valuation estimates. The ease of entry and exit is as fluid as trading exchange listed stocks. Gains and losses can be easily magnified by the leverage options provide, and through the research, a solution to maximize gains and realize the potential risk of losses will be highlighted for each investor. In particular, seventeen separate strategies are examined over multiple time-horizons and holding periods. The selection of these strategies is inline with the most popular option strategies as per the Chicago Board of Option Exchange and similar to those examined by Santa-Clara and Saretto (2005). While Santa-Clara and Saretto (2005) used actual option prices from 1982 through 2002, their study excluded several key strategies, and examines the returns over a shorter-time horizon. While some of the conclusions are consistent, the perspective within this paper is from a pure investment standpoint, demonstrating the rates of returns to portfolios of options through time. In addition, this research will further their analysis by including other major financial events, such as the October 1987 crash, which is important from a long-term investment perspective. The results from the analysis will demonstrate that investors who engage in monthly atthe-money synthetic stocks will have the overall highest returns over the past 34 years. However, this does not result in the highest Sharpe ratio, as a 1-year at-the-money equity collar surpasses the synthetic stock by over 6 times. This reveals the nature of risk and return over the various strategies, and how certain strategies are best suited for different levels of risk tolerance. II. Data Seventeen strategies were analyzed under multiple criteria over several periods. The S&P 500 was used as the base index and as the index leveraged in the experiments. The periods 6

8 span from January 1970 until December The seventeen strategies evaluated are as described in Table 1. Although options did not exist much before the mid 1970 and they were not very liquid until recently, the data gathered shows that over long periods of times and varying market condition how each strategy would perform and how these conditions affected the strategies. Prices for the option strategies were generated using Black-Scholes (1972), with the S&P 500 as the underlying asset. The risk-free comes from the Federal Reserve, using nominal yields on the 1-year Treasury, as shown in Figure 8. The maturity of the option is set at 1-month and 1-year. The volatility input used came for a GARCH (1,1) analysis on the returns of the S&P 500 described below, µ t 1 σ 2 t ε = σ υ t = µ + ε = α + βµ t t t 2 t 1 + γσ 2 t 1 (1) (2) (3) where µ t is the return from t-1 to t,σ t is the conditional volatility, and υ Ν(0,1). From this estimate, daily time-series of volatility was constructed an the input to the Black-Scholes model. Each of the seventeen strategies was evaluated over three levels of moneyness; Out-ofthe money (OTM), at-the money (ATM), and in-the money (ITM). The strategies are all based on leveraging and investing in the S&P 500 Index. For example, the Equity Collar investment is a covered and protected position; one hundred shares of the index were purchased, one 5% OTM Call contract was written, and one 5% OTM Put contract was purchased for the ATM Equity Collar position. This enables the investor to collar the upward movement of his position by selling the 5% OTM Call and collar the downward potential of his position by buying the 5% 7

9 OTM put. For an ITM Equity Collar position, the collar would appreciate 5% up the S&P 500 Index. Then 100 shares of the Index would be purchased at market, one 10% OTM Call contract would be written, and one ATM Put position would be purchased. Table 1 displays the construction of each strategy. All index purchases are made at current market value. Each call and put s strike price is made at exactly the value required by the percent in, at, or out of the money. For example, if the index is valued at $ then the strike price for an ATM Call would be $ This is included since the availability of options on the index were not standardized in 1970 so true option values would have been unattainable or non-existent. The implications of implied volatility versus historical volatility were a concern in the setup of the experiment. Since the implied volatility can only be inferred through exploration of actual option prices and the period examined includes time from before the options were standardized, historical volatility proved to be a more viable, realistic, and attainable value for use in the Black-Scholes formula. Therefore, with historic option prices were generated using the Black-Scholes formula 2004 back to 1970 and typical tick prices have been ignored and the bid/ask spreads have been ignored. The bid/ask tick prices are usually at intervals of every five cents have been removed for a more exact price generated from the formula. For example, if the S&P 500 index in 1970 had a value of $50 and a call option was bought out of the money, then the option premium paid may be $.01; quite different from the typical minimum of $.05 tick intervals. The bid/ask spread on such a small option premium would also be impossible to determine since the market s liquidity has changed drastically over 35 years and some of the option s premiums are so small. 8

10 Figure 1 displays the annual GARCH (1,1) volatility of the S&P 500. The average annual volatility is figured daily and the chart displays the changes in the average volatility over the 34-year period starting January 1971 through December Table 2 describes the average market rates of return without brokerage fees or dividends for the S&P 500 Index and the average historical volatility that corresponds with each of the periods. This research s intent is to find investment opportunities that can outperform the S&P 500 based on risk and rates of return. Therefore, a control group portfolio was created by investing the monthly investment amount straight into the S&P 500 Index every month to establish a baseline performance for all portfolios. The rates of return are determined by the value in the account at the end of the period compared to the total amount invested during the entire period. The monthly investment rate for each strategy is $1000 in 2004 dollars. To adjust for inflation and the overall change in buying power over time, the Consumer Price Index (CPI) was used to index the value of each monthly investment. Table 3 displays the conversion ratios used to determine the accurate monthly investment rate. Each strategy is evaluated over three levels of maturity for theoretical analysis and two intervals of maturity for the practical application. The three theoretical maturities are 30 days, 90 days, and 1 year. Each of the theoretical maturity is held as closely as possible to the number of days it specifies. A theoretical option bought on May 1 st, 1987 is exercised 21 trading days later. Each one-year option is exercised as close to 365 days later as possible. While the theoretical evaluation seems to be worthless by today s established expiration dates and trading practices, this system was established to demonstrate the average returns and standard deviations from those returns no matter the initial investment date. The theoretical data is also used to 9

11 determine Sharpe ratios and average leverage of each strategy. The practical application of the data is used to display the actual results if the given prices and investments had been made. Each option price is determined by use of the Black-Scholes formula. Since the options for the theoretical data were not exercised prior to expiration and based solely from closing prices, the effect of dividends has been mitigated and the Black-Scholes formula for European options was used. The following formula is the Black-Scholes used in all the option calculations: Call Put d d 1 2 = S N = S N = ln = d 1 r t ( d1) K e N( d2 ) r t ( d ) + K e N( d ) 1 2 ( S / K ) + ( r +.5σ ) σ t σ t t 2 (4) (5) (6) (7) Where S = Index Price, t = time to expiration in years, K = Strike price Although investing directly in the index is impossible, the Electronic Traded Funds (ETFs) do allow an individual investor today to trade in the stocks that comprise an index. For the S&P 500 index, the SDPR (ticker SPY) is the ETF. This research did not utilize the ETF since they are a recent addition to the markets; but the theory behind the ETF was used in this research since the research requires and investor to invest directly into the index. A transfer of application from this research would be to invest in the ETF and options in the ETF, or the investor could invest in the appropriate proportion of the ETF to equate to the weight of the index. For example, the SPDR is a tenth of the value of the S&P 500; therefore, an investor could invest in 40 shares of the ETF for each share of the index he would be required to own. 10

12 With today s index values and options prices, investing directly into the index or buying 40x round lots of the ETF are impractical. Buying several call options directly in the index is also out side the range of the $1000 monthly investment. The research ignored the need to trade options in contractual sizes of a hundred. The reason to ignore this market standardization is to see the maximization of leverage and to allow for several of the strategies would not be feasible for an investor limited to investing $1000. Although the purpose of the paper is to discover successful strategies for the individual investor, ignoring several of these strategies only because of the investor not being able to purchase round lots would have left to many holes in the analysis. Some of the more typically defensive and risk reducing strategies, such as the Equity Collar, Protective Put, and any of the Delta Hedges would have been excluded. Finally, the margin requirements for selling the Index or writing options were determined using the Chicago Board of Exchange s (CBOE) minimum requirements formula. Margin = Leverage * (Premium Proceeds +.2* Aggregate Contract Value Amount OTM) (8) Many brokerages require higher margin requirements, but the market minimum was used to create consistency in the research. The margin required for the construction of the strategy is a cost above the $1000 monthly investment and is assumed that the investor already possess the margin to prevent receiving a margin call and liquidating the investment assets prior to expiration. An additional goal of each account is to be self-sustaining concerning margin requirements. Since an investor is required to deposit money into the account and that money could significantly influence the portfolio investment, the influence of the additional start-up capital was not figured into the results of the rates of return for the 153 strategies in Tables 8 through

13 III. Empirical Results The first phase of the experiment established the strategies as if they were traded daily, for a full 30 days, 90 days, and one year, and exercised if profitable. So every business-trading day between Jan 1970 and Dec 2004, a position was created in each strategy. For example, if buying a Call, the option would be established 8,581 times between 1970 and Each trading day one call option would be purchased. So the 30-day positions are held for 21 trading days, 90-day positions are held for 63 trading days, and one-year positions are held for 252 trading days. At expiration, the option is exercised if profitable, and the strategy s profitability, standard deviation of returns, and Sharpe ratio are established. All seventeen strategies were evaluated over the three periods of 30 days, 90 days, and one year and at the three levels on moneyness. Thus, the experiment determined the profitability of 153 different positional strategies. For each of the 153 positions, the rates of return, the standard deviation of returns and the leverage was calculated. The rates of return were determined using the cost of the position s establishment as the base compared to the total profit or loss. The standard deviation was determined using all 8,581 executions; no outliers or exclusion of any data points was made. Both the rates of return and the standard deviations were then annualized. The leverage was determined using the base price of an ATM call for that specific time period. Therefore, the leverage of a 30-day ATM Synthetic Stock position is compared to a 30-day ATM Call. The ATM Synthetic Stock position has a % average leverage ratio, meaning one could buy times the number of Synthetic Stock positions on average as one Call position. For complete details of the 153 position, reference Table 4. 12

14 Table 4 displays the 17 strategies at each level on moneyness in the rows and the different periods in the main columns. Each main column is then divided into the rates of return (RR), the standard deviation (SD), and leverage. The Sharpe ratios were figured by dividing the rates of return by the standard deviations. Each strategy was then ranked by the standard deviation and the Sharpe ratio. The standard deviation of the S&P 500 was used as a baseline for determining each investor category. A medium risk averse investor s strategies ranged from the S&P 500 s standard deviation to half its value. The highly risk averse investor s strategies were all strategies with half the standard deviation of the S&P 500 s standard deviation or less. The low risk averse investor s strategies were all the strategies with standard deviations greater than the S&P 500 s. Table 5 displays the results of the standard deviation analysis; all 153 strategies and the S&P 500 are displayed in sequential order and color-coded by investor type. Additionally, Table 6 examines the positive respective Sharpe ratios for each strategy. A unique relationship emerges from the comparison of the Sharpe ratios and the standard deviations. Higher Sharpe ratios are almost directly related to the lower standard deviations. The rates of return seem to have minor significance in chance the sequence of the ordering of the 98 strategies that have a positive Sharpe ratio. The second phase of the experiment was to trade one position daily on each of the 153 strategies, to determine the effect of the average leverage of each strategy. Table 4 displays the average leverage from each position. Each day, money was invested in each strategy in the same amount as the cost of the corresponding period s ATM Call value. Phases 1 and 2 of the experiment yielded some interesting but not overly surprising results. Not surprising was the high standard deviations of most of the strategies. Most of the strategies utilize leverage over buying the index, so the deviation of the returns were magnified. 13

15 Some strategies showed very low standard deviations but many of them were the covered positions in which shares of the index were also invested, thus dramatically lowering leverage. Additionally, the standard deviations of the short-term positions, e.g. 30-day, were significantly higher than the standard deviations of the long-term securities. Some of the results of the rates of return were very surprising. The volatility of the market over the last 35 years has led to Buying the Strangle and Straddle to be quite positive despite the natural intuition of the investment to be unprofitable or marginally so. Most surprising was the seemingly more secure investments had some very low Sharpe ratios; for example, the 30-Day OTM Call Delta Hedge and 12.5% of all highly risk averse and medium risk averse strategies had negative Sharpe ratios. IV. Strategy Test The final phase of the experiment utilized the true trading days for the strategies experiment. Each position is created on the first trading day of the month and exercised according to its period. The 30-day investment is exercised the third Friday following the first trading day of the month. The one-year investment is exercised the third Friday of the next January. Additionally, the one-year investment must have a minimum of 7 months to exercise. Therefore, all July through December strategies are not exercised until the second January. For example, an August 1981 one-year Call would expire the third Friday in January This does restrict the 30-day strategy somewhat significantly form a full 21 trading days to as little as 11, and the one-year strategy from the 252 trading days to as little as 137, but reflects the intent of the experiment by separating strategy profitability over the long and short maturities. Overall, the theory behind choosing these restrictive periods is an investor s money is deposited into their brokerage account directly from their payroll on the first day of the month. 14

16 Therefore, investing in the remaining of the month s time for the 30-day strategy appeared the most logical short-term investment choice. The one-year strategy system was based off the fact Long-term Equity Anticipation Securities or LEAPS an investor can invest in expire on the third Fridays in January. Some indexes have different LEAPS expirations from the standard January but the majority of the ETFs have January expirations. Since an investor cannot directly invest in the index, he is limited to investing in ETFs based on the index for the covered positions. Table 7 displays the results from the experiment and tracking the growth in the accounts over time. All 17 strategies were evaluated despite the outcome of their standard deviation and Sharpe ratio analysis under the Empirical Results. All the strategies were evaluated at the three levels of moneyness and over the two levels of maturity. Then each position was compared on a 34-year run and a 10-year run of a baseline portfolio constructed of investing in the S&P 500 Index. Table 2 displays the S&P 500 average for each period and Table 7 displays the three most profitable strategies in each period. Of note, the buying Put was extremely successful in the one-year investments over the last 10 years because of the recession and massive market decline. The leverage shows a direct refection as to the profitability of the position. The more leverage the more profitable buying a put was. Because of the recession from 2000 until 2002, the results of the strategy could be considered unrepresentative of the general market. The 34- year run of the one-year investments shows a drastically different picture; however, the 30-day strategy shows the buying OTM puts as a highly profitable investment over the 10-year and 34- year runs. Tables 8 through 11 display the profitability of each strategy over the two time runs (10- year and 34-year). The OTM put is a highly profitable strategy; however, the percentage of the 15

17 time the investment is profitable would pale and hinder even most of the truly low risk averse investors. Table 8 shows the OTM Put as profitable only 3.2% of the times over the 34-year run. In the one-year strategies, the OTM put is profitable around 20% of the time due to the long period and depth of the market decline from 2000 until Despite the large market decline in the latter part of the experiment and due to the length of the periods the experiment was conducted over, a significant number of the strategies proved to be profitable above the S&P 500. In fact, almost a full third of the strategies investigated provided returns higher than that of the S&P 500. Tables 8 through 11 display the percent times each strategy was profitable, the number times over the period it was profitable, the rates of return, and how much that rates of return beat the corresponding period s S&P 500 return. Overall, based on standard deviations and rates of return, the following strategies were chosen as the overall best investments for each investor type: Highly Risk Averse 1 Year ATM, ITM, or OTM Equity Collar 30 Day ITM Protective Put 1 Year ATM Covered Call Medium Risk Averse 30 Day ATM or OTM Equity Collar 30 Day ATM Covered Call 30 Day ATM Protective Put Low Risk Averse 30 Day OTM Selling Straddle 30 Day OTM, ATM, or ITM Call Bull Spread 16

18 1 Year OTM, ATM, or ITM Buying Strangle 1 Year ATM Synthetic Stock The highly risk averse strategies all have standard deviations half of the S&P 500, but some of the rates of return were similar to the S&P 500. Many strategies had rates of return less than the S&P 500, as one would expect. Nevertheless, in the case of each of the three best strategies for the highly risk averse investor, the use of options as an investment strategy provided minimal difference in returns but significant lower volatility within the portfolio. This is most evident within the 10-year investment runs. Figures 2 through 7 displays the growth of the portfolio compared to the growth in the S&P 500 portfolio. In Figures 2, 4, and 6, the strategy is traded daily under the empirical experiment to see if any significant deviations from the returns of the S&P 500. Figures 3, 5, and 7 display the real portfolio growths during the 34- year and 10-year time runs. The lower volatility shows mostly in the 10-year time runs. The longer the time runs the more it appears to match the S&P 500. This is a tremendous benefit showing that the strategy does not deviate significantly in the long-term rates of return but the lower standard deviation proves a lower volatility. This clearly explains the significantly higher Sharpe ratios despite similar returns. As shown in the 10-year time runs, the more level growth rate of the strategy compared to the S&P 500 growth shows that each of the three risk averse strategies have significantly less volatility than the S&P 500 and, therefore, the market as a whole. The ATM, ITM and OTM one-year Equity Collar rates of return edge out the S&P 500 in 34-year run. The ATM one-year Equity Collar also outperforms the S&P 500 over the 10-year run. The ITM 30-day Protective Put and the ATM one-year Covered Call are barely outperformed by the S&P 500 over both the 34-year and 10-year runs. The highly risk averse 17

19 investor is going to seek the best possible rates of return for his risk tolerance and these three strategies provide very equitable solutions. The medium risk averse strategies all have standard deviations less than or equal to the S&P 500. The theory behind the choice of the limits of the standard deviation are the medium risk averse investor is willing to accept the average market risk represented by the standard deviation of the S&P 500. Therefore, a medium risk averse investor would seek lower standard deviations of at least the S&P 500 but not as low as the highly risk averse investor seeking protection of wealth rather then possible creation of it. In Figures 9 through 14, the medium risk averse investor s best three strategies mirror the S&P 500 performance almost exactly. These performances ignore margin, transaction costs, and taxes on the profitable option strategies. Because of the results of the experiment using the real data as in Figures 10, 12, and 14, the risk neutral investor would likely benefit more from investing directly into the Index or ETF, thus minimizing transaction costs, taxes, and margin requirements. The low risk averse investor is willing to accept greater risks for potentially greater returns. The low risk averse strategies are those with the higher standard deviations. The low risk averse investments for this experiment were the strategies with standard deviations higher than the S&P 500 s. Although some low risk averse strategies may have had negative Sharpe ratios, the Tables 8 through 11 display some investments with incredible rates of returns; thus the significant difference between the strategies recommended for the low risk averse investor and those strategies with very high rates of return. The 30-day OTM Selling straddle has the investor selling a Call 5% OTM and selling a Put 5% ITM. With the general upward trend in the market over time, the risk seeking investor 18

20 can generate a steady income that may yield very significantly returns over time as the Figure 16 displays. Figure 15 displays the Empirical Results that an investor would lose his investment constantly over the 34-year period if the investment were made for the full 30 days, every day. In reality, if the 30-day investment is made on the first trading day of the month and exercised if profitable on the third Friday, then the investor can beat the returns of the S&P 500. The 30-day OTM Selling Straddle has the highest Sharpe ratio of the low risk averse strategies, but a relatively high standard deviation even for the riskier strategies and it only marginally outperforms the S&P 500. The 30-day Call Bull Spread results display differences depending on the 34 or 10-year runs. As Figures 17 and 18 display, the OTM Call Bull Spread is more profitable in the long run but over the 10-year time run the ATM Call Bull Spread is significantly more profitable than the OTM Bull Call Spread and the S&P 500 Index. In fact, the 30-day ATM Call Bull Spread is the fourth most profitable ATM position for the 30-day investments. The risk seeking investor takes advantage of the general upward trend in the market yet reduces his average cost of the ATM Call bought by selling the OTM Call position. The investor sacrifices maximum upside potential in the market place for a higher percent chance of the strategy being in the money. Buying the Strangle position, no matter the moneyness, proves to be a viable position for the low risk averse investor. As Figures 19 and 20 display, the Strangle provides the investor with an ability to capitalize on the markets volatility, but as Table 5 displays the Buying Strangle position has a relatively low standard deviation compared to many of the low risk averse strategies. The Strangle investment has a 5% spread from the midpoint, but the spread is easily compensated by the volatility over the one-year investment period. The results are not very surprising since the S&P 500 average return is almost 10% annually according to Table 2. The 19

21 ability for the position to move 5% in a single year into profitability would almost be a forgone conclusion. The premium cost of creating the position can be significant, and those premiums become the hurdles that the market must surpass for the position to offer larger returns than the S&P 500. The final low risk averse position of note is the ATM Synthetic Stock. The Synthetic Stock position offers almost the same returns as buying the index directly at a fraction of the cost. The longer the maturity of the option the further the price spread between the Call and Put. The further the spread between the Call and the Put the less the leverage and the lower the gains from the position. The historical trend of the market is generally increasing at the rate of 10% per year. Then buying a synthetic position at a fraction of the cost should yield significantly higher standard deviations and returns. The 30-day ATM Synthetic Stock drastically outperforms the market over the 34 and 10-year time runs. As Figures 21 and 22 display, the One-year ATM position shows a phenomenal rates of return and the effects of compounding those returns over time. For the low risk averse investor willing to accept significant risks but willing to invest in the market for the long run, the ATM synthetic stock position offers a very viable investment solution to outperforming the S&P 500 Index. V. Margin, Taxes, and Transaction Costs. Most of the strategies within the experiment required margin in one form or another. Many of the portfolios used the investments within the index as the margin needed to establish the position. Some of the strategies did require substantial amounts of margin, especially when the positions were first established. Tables 12 and 13 display the amount of margin required at the beginning of the experiment, the number of months before the account could self-sustain the 20

22 margin requirement, and the number of times the account would receive a margin call or need additional funds as the strategy built leverage. Table 13 also contains the maximum amount of margin that would have been required during the experiment. The margin rate was calculated using the CBOE initial margin requirements. The formula for computing the margin was: Margin = #options * (Premium+.2*underlying value-otm) Individual brokerages may require more or less margin. For example, ETrade.com requires a 40% margin of the underlying value instead of the 20% like Interactivebrokers.com and the CBOE. Some of the positions did require a substantial amount of initial margin that maybe outside the realm of the individual investor s typical capabilities. But since each investor may or may not start with any cash in his account before he begins his strategy, drawing an arbitrary value to his starting cash would limit the leverage and the true effectiveness of the strategies through time. It would have given the strategies that do not require margin, such as the Call, more leverage in the beginning years than strategies like the Synthetic Stock. The Synthetic Stock would then have been limited from its full potential and biased the results and rates of return for the beginning years. Therefore, margin requirements were not included into the rates or return calculations and were not a limiting factor for the investor. The purpose of Tables 12 and 13 are to display the requirements and the amount of time before the account could sustain itself for each strategy. Thus allowing the investor to understand the gains he could expect from his monthly investment and to know the amount of initial funding he may be required to front to begin trading the strategy. 21

23 Table 14 shows an example of the impacts margin may have had on the different strategies. The Synthetic Stock was one of the highest leveraged positions that required margin, and it had a significant rate of return over the S&P 500. The first column in Table 14 displays the rates of return without margin. The second column displays the rates of return for the strategy assuming the maximum margin requirement was placed into the account at the beginning of the investment period in cash. The margin placed into the account as cash did not receive any interest and was not invested in the S&P 500 like the monthly funds or the profits from investing in the strategy. This way the cash for the margin would not further benefit or hinder the investment strategy for the investor. The table displays that for the 30-day investment, the margin barely reduced the rates of return; however, margin reduced the rates of return significantly for the one-year strategy. The one-year investment required vastly higher quantities of margin because the strategy establishes 12 continuous months of investments and as high as 18 months of investments at any given point. Transaction costs were another difficult consideration during the experiment. Since options were not standardized during the earlier years of the experiment, transaction costs would have been astronomical and a barrier to entering the market. Transaction costs of the 1980s compared to today s costs seem almost as expensive. Today, an investor can buy option contracts for as little as $.75 from his online brokerage account compared to a flat fee of $50 plus a per contract cost for most broker-assisted transactions. Since the internet has reduced cost so dramatically over the last several years it would be unrealistic to charge the same rates as in 1975 as is Therefore, the costs to establish the position and liquidate the position every month were set at a flat $100 per month after the CPI adjustments were made. This allows for higher transaction costs during the early years when barriers were significantly higher and more costly 22

24 and lower transaction costs during the last few years since the cost-reducing emergence of online brokering. Taxes are a final aspect that an individual investor would have to consider. Despite recent beneficial changes to the tax law for capital gains, the laws concerning the profits of trading options may not be considered long-term capital gains. This paper is not a discussion on how to minimize the tax effects on a portfolio, but the effects of taxes on a portfolio are quite real. Therefore, a flat rate of 28% was used when the strategy was deemed profitable. An individual investor should seek advice from his Personal Accountant before investing in any strategy to help understand his potential tax liabilities. However, since the tax laws have been very dynamic over the last 35 years, the flat tax seemed the best strategy to display how all the cost could shape the profitability of the strategies. While applying the transaction costs is not an exact replication the effect of the transaction costs, and the tax rate is only a hypothetical tax rate, the margin and costs of investing in each strategy had a significant impact in diluting the rates of return. Table 14 displays the overall effects of margin, brokerage fees, and taxes. The effects from transaction costs taxes are much more significant to the 30-day strategy than the one-year strategy. The effects of margin had a much more significant impact on the one-year strategy than the transaction costs and taxes did. Figures 23 and 24 display the account performance considering the effects of brokerage fees and taxes. So despite the seemingly extreme profitability of the ATM Synthetic Stock strategy, due consideration must be paid to the effects of margin, taxes, and fees in establishing the strategies. VI. Conclusions 23

25 The overall theory that investing in options is a risky choice has been demonstrated to be false. Some of the strategies evaluated do demonstrate ways for an investor to significantly reduce risk while not sacrificing the returns he may be seeking in the long run. Options have an allure for the low risk averse investor and the speculator; potentially huge profits for very little investment. The effects of speculating and taking risks in the long run can easily be disastrous if the investment strategy is not fully investigated. An investor is not speculating in the market if he truly understands his risks/reward potential, knows his percentile chances of profitability, and has a disciplined investment approach. A disciplined investor can take significant risks to lead towards the significant payoff that should accompany those risks. The strategies discussed in Section IV show that for each investor s risk tolerance there is an appropriate strategy. Surprisingly, the medium risk averse investor s best strategy is the buy and hold strategy within the index. Buy the index and hold it. This simple strategy has proven to be tough to outperform throughout the whole experiment. For the low risk averse investor, potentially huge gains are possible but the losses that accompany those risks can be very intimidating. For many of the strategies the additional risks in investing in the options are not out weighted by the potential profits. The low risk averse strategies that are profitable have dramatic downside potential. The most significant strategy investigated based on astounding returns and ability for the individual investor to establish the position has to be the ATM Synthetic Stock. A young investor has the ability to absorb large short-term losses to receive the eventually huge gains. The risks of the ATM Synthetic Stock have shown to be very rewarding in the long run; therefore, the individual investor that has the time to benefit from long upward trends in the market that can be maximized from the leverage afforded in the Synthetic Stock. 24

26 The risks associated with the one-year versus the 30-day strategy do not seem to be worth the risk and margin requirements for the potential rewards. Therefore, the best strategy for the individual long-term investor is the 30-day ATM Synthetic Stock. 25

27 Bibliography Bakshi, G., C. Cao, and Z. Chen, 1997, Empirical Performance of Alternate Option Pricing Models, Journal of Finance 52, Bakshi, G., C. Cao, and Z. Chen, 2000, Pricing and Hedging Long-term Options, Journal of Econometrics 94, Bates, D., 2000, Post 87 Crash Fears in S&P 500 Futures Options. Journal of Econometrics 94, Benzoni, Luca, 2001, Pricing Options Under Stochastic Volatility: An Empirical Investigation, Working paper. Buraschi, Andrea, and Jens Jackwerth, 2001, The Price of a Smile: Hedging and Spanning in Options Markets, Review of Financial Studies 14, Chicago Board Options Exchange (CBOE), Product Specifications, Downloaded 7 November Chicago Board Options Exchange, Understanding Index Options, , Downloaded 26 May Christensen, Bent J., and Nagpurnanand R. Prabhala, 1998, The Relation Between Implied and Realized Volatility, Journal of Financial Economics 50, Coval, Joshua D., and Tyles Shumway, 2001, Expected Option Returns, Journal of Finance 56, Driessen, Joost, and Pascal Maenhout, 2003a, A Portfolio Perspective on Option Pricing Anomalies, Working paper. Fisher, Donald E. and Jordan, Ronald J., 1995, Security Analysis and Portfolio Management, New Jersey, Prentice Hall. 26

28 George, Thomas J., and Francis A. Longstaff, 1993, Bid-ask Spreads and Trading Activity in the S&P 100 Index Option Market, Journal of Financial and Quantitative Analysis 28, Google Definitions, S & P 500, Downloaded 5 October Hardouvelis, Gikas A., 1990, Margin Requirements, Volatility, and the Transitory Component of Stock Prices, American Economic Review 80, Heston, S., and S. Nandi, 2000, A closed-form GARCH Option Valuation Model, Review of Financial Studies 13, Hull, John C, 2002, Fundamentals of Futures and Options Markets, New Jersey, Prentice Hall. Hull, John C, 2003, Options, Futures. & Other Derivatives, New Jersey, Prentice Hall. Jackwerth, Jens Carsten, 2000, Recovering Risk Aversion from Option Prices and Realized Returns, Review of Financial Studies 13, Jones, Christopher S., 2004, A Nonlinear Factor Analysis of S&P 500 Index Options Returns, Working paper. Merton, Robert C., Myron S. Scholes, and Mathew L. Gladstein, 1978, The Returns and Risk of Alternative Call Option Portfolio Investment Strategies, Journal of Business 51, Merton, Robert C., Myron S. Scholes, and Mathew L. Gladstein, 1982, The Returns and Risk of Alternative Put Option Portfolio Investment Strategies, Journal of Business 55, Rubinstein, Mark, 1976, The Valuation of Uncertain Income Streams and the Pricing of Options, Bell Journal of Economics 7,

29 Sahr, Robert, 2005, Inflation Conversion Factors for Dollars 1665 to Estimated 2015, Conversion Factors in 2004 Dollars, using CPI-U-X1 for 1950 to estimated Santa-Clara, Pedro, and Shu Yan, 2004, Jump and Volatility Risk and Risk Premia: A New Model and Lessons from the S&P 500 Options, UCLA working paper. Santa-Clara, Pedro and Saretto, Alessio, 2005, Option Strategies: Good Deals and Margin Calls*, UCLA working paper. Thomsett, Michael C, 2001, Getting Started in Options, New York, Wiley & Sons, Inc. U.S. Federal Reserve, 2005, Selected Interest Rates: U.S. Treasuries Constant Maturities Nominal 1-Year, Downloaded 26 May Williams, Michael S. and Hoffman, Amy, 2001, Fundamentals of the Options Market, New York. McGraw-Hill. 28

30 Table 3 CPI Data Table The table below represents the conversion factors for the monthly investment in each option strategy. Dr. Sahr bases the conversion factors on his Consumer Price Index calculations, which adjust for changes in the formula over time. The base year is 2004 and the base dollar amount is $1000. To figure an individual year s monthly investment, multiply the $1000 by the conversion factor for that year. Year Conversion Factor (c) 2005 Robert C. Sahr, Political Science Department, Oregon State University, Corvallis, OR

31 Table 2 Description of Data Table The following table represents the rate of return and the GARCH (1,1) average volatility and the average rate of return for the S&P 500 Index over two time frames. The first period starts Jan 1970 and ends Dec 2004, and thus spans 34 years of time. The second period starts Jan 1994 and ends Dec 2004, and thus spans 10 years of time. 34 Year 10 Year Rate of Return S&P % % 1 Volatility Average % % 1 Based on Rate of Return from Annual Monthly Investment Plan 30

32 Table 6 Sharpe Ratios The following table depicts the positive Sharpe ratios for the experiment. The Sharpe ratio was determined by dividing the average rate of return by the standard deviation. Green is the highly risk averse investor, which was determined by the standard deviation of returns being half of the S&P 500 Index. The medium risk averse investor is color coded tan and has a standard deviation less than or equal to the standard deviation of the S&P 500 Index. Red is the low risk averse investor that is willing to accept higher standard deviations of returns for potentially higher rates of return. Strategy SHARPE RATIO 1 Year Equity Collar ATM Year Equity Collar ITM Day Protective Put ITM Year Equity Collar OTM Day Equity Collar ITM Day Covered Call ITM Day Call Delta Hedge ITM Day Covered Call ITM Day Call Delta Hedge ITM Day Equity Collar ATM Year Call Delta Hedge ITM Year Covered Call ITM Day Equity Collar ITM Day Equity Collar OTM Day Protective Put ITM Day Covered Call ATM Day Equity Collar OTM Year Covered Call ATM Day Equity Collar ATM Day Covered Call ATM Year Call Delta Hedge ATM Day Call Delta Hedge ATM Year Protective Put ITM Day Call Delta Hedge ATM Year Covered Call OTM Day Protective Put ATM Day Covered Call OTM Year Protective Put ATM Day Protective Put ATM Day Covered Call OTM Year Protective Put OTM Year Call Delta Hedge OTM Day Protective Put OTM Day Protective Put OTM Day Call Delta Hedge OTM 0.57 S & P Day Selling Straddle OTM Day Call Bull Spread ITM Day Call Bull Spread ITM Year Call Bull Spread IOM Day Selling Straddle OTM Year Buying Straddle ITM Day Put OTM Year Call ATM Year Buying Straddle ATM Day Buying Strangle ATM Day Call Delta Hedge OTM Year Call Bull Spread ATM Year Buying Strangle ATM Day Call Bull Spread ATM Year Buying Strangle ITM Day Put Bear Spread OTM Day Buying Straddle ITM Day Call Bull Spread ATM Year Call ITM Day Buying Strangle ITM Day Long Synthetic Stock ATM Day Buying Straddle ITM Year Call OTM Year Call Bull Spread ITM Day Call ITM Day Buying Strangle ITM Day Buying Strangle ATM Day Long Synthetic Stock ATM Day Call ITM Day Call Bull Spread OTM Year Buying Strangle OTM Day Put Bull Spread ITM Day Call ATM Year Buying Straddle OTM Year Long Synthetic Stock ITM Day Long Synthetic Stock ITM Year Long Synthetic Stock ATM Day Long Synthetic Stock ITM Day Call ATM Day Call OTM Day Buying Straddle ATM Day Call Bull Spread OTM Day Call OTM Day Put OTM Day Naked Put ITM Year Put OTM Day Buying Straddle ATM Day Put Bull Spread ATM Day Selling Strangle OTM Day Put Bull Spread ITM Year Put Bear Spread OTM Year Put ATM Day Put Bull Spread ATM Day Put Bear Spread OTM Year Put Bull Spread ITM Day Naked Put ITM Year Put ITM Day Put ATM Day Put ATM Year Put Bear Spread ATM Day Selling Strangle OTM Year Long Synthetic Stock OTM

33 Table 5 Low Risk Aversion Chart The following chart displays the categories of investments color coded to the risk aversion of the investor. Red is the low-risk averse investor that is willing to accept higher standard deviations of returns for potentially higher rates of return. Strategy Annualized ROI Annualized St Deviation SHARPE RATIO 1 Year Long Synthetic Stock OTM % % Day Long Synthetic Stock ATM % % Day Put OTM % % Day Naked Put OTM % % (0.29) 30 Day Put Bull Spread OTM % % (0.26) 30 Day Put Bear Spread OTM % % Day Call OTM % % Day Call Bear Spread OTM % % (0.16) 30 Day Call Bull Spread OTM % % Day Buying Strangle ATM % % Day Selling Strangle ATM % % (0.28) 90 Day Long Synthetic Stock ATM % % Day Put OTM % % Day Naked Put OTM % % (0.15) 90 Day Put Bull Spread OTM % % (0.09) 90 Day Put Bear Spread OTM 69.24% % Day Put ATM 22.65% % Day Naked Put ATM % % (0.03) 30 Day Selling Strangle OTM 10.15% % Day Buying Strangle OTM % % (0.02) 1 Year Put OTM 88.05% % Year Naked Put OTM % % (0.14) 90 Day Long Synthetic Stock OTM % % (0.17) 30 Day Put Bull Spread ATM 66.49% % Day Put Bear Spread ATM % % (0.13) 30 Day Buying Strangle ITM % % Day Selling Strangle ITM % % (0.23) 30 Day Call ATM 89.82% % Day Call OTM 72.71% % Day Call Bull Spread ATM % % Day Call Bear Spread ATM % % (0.27) 90 Day Call Bear Spread OTM % % (0.20) 90 Day Put ATM 9.50% % Day Naked Put ATM -9.50% % (0.02) 90 Day Call Bull Spread OTM 81.95% % Year Long Synthetic Stock ATM 69.05% % Day Buying Strangle ATM 81.11% % Day Selling Strangle ATM % % (0.20) 1 Year Put Bear Spread OTM 43.58% % Year Put Bull Spread OTM % % (0.12) 1 Year Put ATM 37.03% % Year Naked Put ATM % % (0.11) 90 Day Selling Strangle OTM 42.35% % Day Buying Strangle OTM % % (0.13) 30 Day Long Synthetic Stock OTM % % (0.22) 90 Day Put Bull Spread ATM 32.37% % Day Put Bear Spread ATM % % (0.11) 30 Day Buying Straddle ATM 47.95% % Day Selling Straddle ATM % % (0.16) 30 Day Naked Put ITM 40.40% % Day Put ITM % % (0.14) 30 Day Long Synthetic Stock ITM 46.77% % Day Call ATM 43.42% % Day Selling Straddle OTM % % Day Buying Straddle OTM % % (0.47) 90 Day Buying Strangle ITM 52.23% % Day Selling Strangle ITM % % (0.20) 90 Day Naked Put ITM 15.74% % Day Put ITM % % (0.07) 90 Day Long Synthetic Stock ITM 40.92% % Day Call ITM 48.28% % Year Put ITM 14.43% % Year Naked Put ITM % % (0.06) 30 Day Buying Straddle ITM 54.15% % Day Selling Straddle ITM % % (0.26) 90 Day Call Bull Spread ATM 49.10% % Day Call Bear Spread ATM % % (0.24) 30 Day Put Bull Spread ITM 37.45% % Day Put Bear Spread ITM % % (0.19) 1 Year Put Bear Spread ATM 4.07% % Year Put Bull Spread ATM -4.07% % (0.02) 90 Day Buying Straddle ATM 22.68% % Day Selling Straddle ATM % % (0.13) 1 Year Call OTM 37.89% % Day Call ITM 33.54% % Day Selling Straddle OTM 48.87% % Day Buying Straddle OTM % % (0.29) 90 Day Put Bull Spread ITM 20.24% % Day Put Bear Spread ITM % % (0.12) 30 Day Call Bear Spread ITM % % (0.33) 30 Day Call Bull Spread ITM 53.63% % Year Long Synthetic Stock ITM 27.69% % Year Buying Strangle OTM 28.06% % Year Selling Strangle OTM % % (0.19) 90 Day Buying Straddle ITM 32.30% % Day Selling Straddle ITM % % (0.22) 1 Year Buying Strangle ATM 39.47% % Year Selling Strangle ATM % % (0.27) 1 Year Put Bull Spread ITM 9.27% % Year Put Bear Spread ITM -9.27% % (0.07) 1 Year Call ATM 37.81% % Year Buying Strangle ITM 30.97% % Year Selling Strangle ITM % % (0.26) 90 Day Call Bull Spread ITM 37.89% % Day Call Bear Spread ITM % % (0.33) 1 Year Call Bull Spread IOM 35.50% % Year Call Bear Spread OTM % % (0.31) 1 Year Call ITM 23.89% % Year Buying Straddle ATM 26.92% 93.34% Year Selling Straddle ATM % 93.34% (0.29) 1 Year Buying Straddle OTM 16.95% 93.00% Year Selling Straddle OTM % 93.00% (0.18) 1 Year Call Bull Spread ATM 23.59% 84.96% Year Call Bear Spread ATM % 84.96% (0.28) 1 Year Buying Straddle ITM 24.13% 82.97% Year Selling Straddle ITM % 82.97% (0.29) 1 Year Call Bull Spread ITM 13.88% 64.88% Year Call Bear Spread ITM % 64.88% (0.21) 30 Day Put Delta Hedge OTM % 57.94% (0.43) 30 Day Call Delta Hedge OTM 7.79% 27.81% Year Put Delta Hedge OTM % 27.81% (0.45) 90 Day Put Delta Hedge OTM % 25.37% (0.44) 1 Year Put Delta Hedge ATM % 18.05% (0.56) 32

34 Table 5 (Con t) Medium and High Risk Aversion Chart The following chart displays the categories of investments color coded to the risk aversion of the investor. Green is the high-risk averse investor, which was determined by the standard deviation of returns being half of the S&P 500 Index. The medium-risk averse investor is color coded tan and has a standard deviation less than or equal to the standard deviation of the S&P 500 Index. Strategy Annualized ROI Annualized St Deviation SHARPE RATIO S & P % 16.53% Day Call Delta Hedge OTM 8.34% 14.72% Day Covered Call OTM 9.70% 13.72% Day Protective Put OTM 8.95% 13.50% Year Put Delta Hedge ITM -8.60% 13.46% (0.64) 1 Year Protective Put OTM 9.04% 13.24% Day Put Delta Hedge ATM -6.88% 13.17% (0.52) 90 Day Protective Put OTM 8.49% 12.66% Day Put Delta Hedge ATM -6.62% 12.10% (0.55) 1 Year Protective Put ATM 8.86% 11.93% Day Covered Call OTM 8.93% 11.60% Day Equity Collar ATM 9.84% 11.19% Year Protective Put ITM 8.47% 10.49% Day Protective Put ATM 7.57% 10.25% Year Call Delta Hedge OTM 6.83% 10.18% Day Call Delta Hedge ATM 7.82% 9.86% Year Covered Call OTM 7.73% 9.86% Day Protective Put ATM 7.45% 9.59% Day Equity Collar OTM 8.50% 9.52% Day Call Delta Hedge ATM 7.83% 9.48% Day Equity Collar OTM 9.49% 9.44% Day Covered Call ATM 8.16% 9.36% Day Covered Call ATM 8.10% 8.78% Year Covered Call ATM 7.30% 8.26% Day Equity Collar ITM 8.40% 8.13% Day Put Delta Hedge ITM -5.85% 7.89% (0.74) 1 Year Call Delta Hedge ATM 6.70% 7.80% Day Equity Collar ATM 8.61% 7.70% Day Protective Put ITM 6.76% 7.13% Year Equity Collar OTM 8.90% 6.86% Year Covered Call ITM 7.10% 6.81% Year Call Delta Hedge ITM 6.74% 6.31% Day Equity Collar ITM 8.15% 6.30% Day Call Delta Hedge ITM 6.92% 6.19% Day Covered Call ITM 7.13% 6.15% Day Covered Call ITM 6.60% 5.14% Day Call Delta Hedge ITM 6.50% 5.09% Year Equity Collar ATM 7.76% 5.06% Day Put Delta Hedge ITM -5.11% 4.90% (1.04) 1 Year Equity Collar ITM 7.30% 4.89% Day Protective Put ITM 5.93% 4.54%

35 Table 12 Margin Requirements The following table represents each of the strategies that will require margin, the initial margin requirements for the position, the number of months before the value of the account is self-maintaining and a margin minimum is no longer required, and the number of margin calls over the period that the strategy had. The period of investment was from Jan 1971 to Dec 2004 for the 34-year data, and the period of investment as from Jan 1995 to Dec 2004 for the 10-year data. The investment is established on the first trading day of the month and held to exercise. Therefore, the 30-day position is established on the first trading day of the month and then exercise the third Friday of that same month. 34 Year 10 Year Months Before Selfsustaining Months Before Selfsustaining 30 DAY Initial Amount Margin Calls Initial Amount Margin Calls Sell Naked OTM Put Infinite Never Constant Infinite Never Constant Sell Naked ATM Put $3, $16, Sell Naked ITM Put $ $1, OTM Synthetic Stock $ $1, ATM Synthetic Stock $24, $37, ITM Synthetic Stock $ $1, OTM Covered Call $ $ ATM Covered Call $ $ ITM Covered Call $ $ OTM Bull Call Spread $29, $317, Never Constant ATM Bull Call Spread $2, $7, ITM Bull Call Spread $ $3, OTM Bear Call Spread $58, Never Constant $634, Never Constant ATM Bear Call Spread $ $11, ITM Bear Call Spread $ $1, OTM Bull Put Spread Infinite Never Constant Infinite Never Constant ATM Bull Put Spread $3, $2, ITM Bull Put Spread $ $2, OTM Bear Put Spread $53, Infinite Never Constant ATM Bear Put Spread $3, $17, ITM Bear Put Spread $ $3, OTM Straddle (Sell) $ $3, ATM Straddle (Sell) $3, $13, ITM Straddle (Sell) $ $2, OTM Strangle (Sell) $4, $22, ATM Strangle (Sell) $75, Never Constant $1,006, Never Constant ITM Strangle (Sell) $3, $14, OTM Equity Collar $ $ ATM Equity Collar $ $ ITM Equity Collar $ $ OTM Call Delta Hedge $ $4, ATM Call Delta Hedge $ $ ITM Call Delta Hedge $ $ OTM Put Delta Hedge $5, $114, ATM Put Delta Hedge ($130.77) 0 0 ($369.60) 0 0 ITM Put Delta Hedge ($281.67) 0 0 ($1,013.82)

36 Table 13 Margin Requirements The following table represents each of the strategies that will require margin, the initial margin requirements for the position, the number of months before the value of the account is self-maintaining and a margin minimum is no longer required, and the number of margin calls over the period that the strategy had. The period of investment was from Jan 1971 to Jan 2004 for the 34-year data, and the period of investment as from Jan 1994 to Jan 2004 for the 10-year data. The investment is established on the first trading day of the month and held to exercise. Therefore, the one-year position is established on the first trading day of the month and then exercised on the third Friday of January the following year. The minimum holding time for any one-year strategy is 7 months and the maximum is less than 18 months. Therefore, a position established on the first trading day in August of 1987 is not exercised until January Year Investment 34 Year 10 Year Months Before Selfsustaining Margin Initial Calls Amount MAX Margin Months Before Selfsustaining 1 YEAR Initial Amount MAX Margin Margin Calls Sell Naked OTM Put $ $635, Never Constant $5, $317, Sell Naked ATM Put $ $220, Never Constant $3, $138, Sell Naked ITM Put $ $107, $1, $49, OTM Synthetic Stock Infinite Infinite ATM Synthetic Stock $ $11, $2, $51, ITM Synthetic Stock $ $11, $ $16, OTM Covered Call Covered 0 0 Covered 0 0 ATM Covered Call Covered 0 0 Covered 0 0 ITM Covered Call Covered 0 0 Covered 0 0 OTM Bull Call Spread $1, $28, $3, $57, ATM Bull Call Spread $1, $37, $4, $74, ITM Bull Call Spread $1, $51, $5, $88, Never Constant OTM Bear Call Spread Infinite Infinite ATM Bear Call Spread $1, $36, $4, $85, Never Constant ITM Bear Call Spread $1, $33, $4, $73, Never Constant OTM Bull Put Spread Infinite Infinite ATM Bull Put Spread $2, $325, Never Constant $9, $275, Never Constant ITM Bull Put Spread $1, $197, Never Constant $6, $160, Never Constant OTM Bear Put Spread $1, $62, $5, $247, Never Constant ATM Bear Put Spread $1, $97, $6, $191, Never Constant ITM Bear Put Spread $0.00 $55, $6, $135, Never Constant OTM Straddle (Sell) $ $12, $1, $43, ATM Straddle (Sell) $ $6, $1, $34, ITM Straddle (Sell) $2, $449, $19, $1,052, Never Constant OTM Strangle (Sell) $ $37, $3, $105, Never Constant ATM Strangle (Sell) $ $20, $3, $86, Never Constant ITM Strangle (Sell) $ $11, $3, $86, OTM Equity Collar $21.38 $ $60.92 $1, ATM Equity Collar $38.76 $ $ $2, ITM Equity Collar $58.95 $ $ $3, OTM Call Delta Hedge $61.88 $ ATM Call Delta Hedge $72.78 $ ITM Call Delta Hedge $69.15 $ OTM Put Delta Hedge $ $37, $1, $288, Never Constant ATM Put Delta Hedge $62.69 $4, $ $10, ITM Put Delta Hedge ($52.87) ($52.87) 0 0 ($257.60)

37 Table 4 OTM Ratios on Option Strategies (Theoretical) The following table displays each of the 17 option strategies Out of the Money (OTM). For each strategy, the 30-day, 90-day, and one-year rates of return (RR), Standard Deviation (SD), and Leverage are calculated. Leverage is based on the price of buying an ATM call for each time period; option pricing is based on Black-Scholes computations. Therefore, if buying 30-day OTM Calls, a person could buy % more OTM Calls for the same price as one 30-day ATM Call. Each option is held for the exact time regardless of day of the week and then exercised. Strategy 30 Day 90 Day 1 Year Leverage Leverage RR 1 SD 1 2 RR 1 SD 1 3 RR SD Leverage 4 S & P % 16.53% 100% Out Of The Money (OTM) Buying Straddle % % 37.86% % % 59.14% 16.95% 93.00% 84.27% Buying Strangle % % % % % % 28.06% % % Call % % % 72.71% % % 37.89% % % Call Bear Spread % % % % % % % % % Call Bull Spread % % % 81.95% % % 35.50% % % Call Delta Hedge 7.79% 27.81% 16.16% 8.34% 14.72% 12.20% 6.83% 10.18% 19.27% Covered Call 9.70% 13.72% 2.04% 8.93% 11.60% 4.00% 7.73% 9.86% 10.73% Equity Collar 8.50% 9.52% 2.00% 9.49% 9.44% 3.86% 8.90% 6.86% 10.10% Long Synthetic Stock % % 46.24% % % % % % % Naked Put % % % % % % % % % Protective Put 8.95% 13.50% 2.02% 8.49% 12.66% 3.88% 9.04% 13.24% 9.68% Put % % % % % % 88.05% % % Put Bear Spread % % % 69.24% % % 43.58% % % Put Bull Spread % % % % % % % % % Put Delta Hedge % 57.94% 38.44% % 25.37% 29.52% % 27.81% 75.99% Selling Straddle % % 37.86% 48.87% % 59.14% % 93.00% 84.27% Selling Strangle 10.15% % % 42.35% % % % % % Represent more than 100% rate of return Represents less than 0% or a negative return Represents more than 1000% leverage over the price of an ATM Call Represents less than 100% leverage or the position costs more than the price of an ATM Call 1 Annualized 2 Based on 30 Day ATM Call Price 3 Based on 90 Day ATM Call Price 4 Based on 1 Year ATM Call Price 36

38 Table 4 (Con t) ATM Ratios on Option Strategies (Theoretical) The following table displays each of the 17 option strategies At the Money (ATM). For each strategy, the 30-day, 90-day, and one-year rates of return (RR), Standard Deviation (SD), and Leverage are calculated. Leverage is based on the price of buying an ATM call for each time period; option pricing is based on Black-Scholes computations. Therefore, if buying 30-day OTM Calls, a person could buy % more OTM Calls for the same price as one 30-day ATM Call. Each option is held for the exact time regardless of day of the week and then exercised. Strategy 30 Day 90 Day 1 Year Leverage Leverage RR 1 SD 1 2 RR 1 SD 1 3 RR SD Leverage 4 S & P % 16.53% 100% At The Money (ATM) Buying Straddle 47.95% % 58.91% 22.68% % 64.95% 26.92% 93.34% 76.68% Buying Strangle % % % 81.11% % % 39.47% % % Call 89.82% % % 43.42% % % 37.81% % % Call Bear Spread % % % % % % % 84.96% % Call Bull Spread % % % 49.10% % % 23.59% 84.96% % Call Delta Hedge 7.82% 9.86% 3.76% 7.83% 9.48% 6.94% 6.70% 7.80% 16.38% Covered Call 8.16% 9.36% 2.07% 8.10% 8.78% 4.09% 7.30% 8.26% 11.06% Equity Collar 9.84% 11.19% 2.03% 8.61% 7.70% 3.96% 7.76% 5.06% 10.47% Long Synthetic Stock % % % % % % 69.05% % % Naked Put % % % -9.50% % % % % % Protective Put 7.45% 9.59% 1.99% 7.57% 10.25% 3.82% 8.86% 11.93% 9.55% Put 22.65% % % 9.50% % % 37.03% % % Put Bear Spread % % % % % % 4.07% % % Put Bull Spread 66.49% % % 32.37% % % -4.07% % % Put Delta Hedge -6.62% 12.10% 4.84% -6.88% 13.17% 10.91% % 18.05% 43.87% Selling Straddle % % 58.91% % % 64.95% % 93.34% 76.68% Selling Strangle % % % % % % % % % Represent more than 100% rate of return Represents less than 0% or a negative return Represents more than 1000% leverage over the price of an ATM Call Represents less than 100% leverage or the position costs more than the price of an ATM Call 1 Annualized 2 Based on 30 Day ATM Call Price 3 Based on 90 Day ATM Call Price 4 Based on 1 Year ATM Call Price 37

39 Table 4 (Con t) ITM Ratios on Option Strategies (Theoretical) The following table displays each of the 17 option strategies In the Money (ITM). For each strategy, the 30-day, 90-day, and one-year rates of return (RR), Standard Deviation (SD), and Leverage are calculated. Leverage is based on the price of buying an ATM call for each time period; option pricing is based on Black-Scholes computations. Therefore, if buying 30-day OTM Calls, a person could buy % more OTM Calls for the same price as one 30-day ATM Call. Each option is held for the exact time regardless of day of the week and then exercised. Strategy 30 Day 90 Day 1 Year Leverage Leverage RR 1 SD 1 2 RR 1 SD 1 3 RR SD Leverage 4 S & P % 16.53% 100% In The Money (ITM) Buying Straddle 54.15% % 32.87% 32.30% % 46.73% 24.13% 82.97% 64.63% Buying Strangle % % 98.76% 52.23% % 95.13% 30.97% % 91.24% Call 48.28% % 34.54% 33.54% % 51.87% 23.89% % 73.62% Call Bear Spread % % 55.71% % % % % 64.88% % Call Bull Spread 53.63% % 55.71% 37.89% % % 13.88% 64.88% % Call Delta Hedge 6.50% 5.09% 2.45% 6.92% 6.19% 5.29% 6.74% 6.31% 14.79% Covered Call 6.60% 5.14% 2.15% 7.13% 6.15% 4.24% 7.10% 6.81% 11.49% Equity Collar 8.40% 8.13% 2.07% 8.15% 6.30% 4.07% 7.30% 4.89% 10.90% Long Synthetic Stock 46.77% % 36.68% 40.92% % 59.67% 27.69% % 89.80% Naked Put 40.40% % % 15.74% % 80.05% % % % Protective Put 5.93% 4.54% 1.93% 6.76% 7.13% 3.72% 8.47% 10.49% 9.39% Put % % % % % 80.05% 14.43% % % Put Bear Spread % % 61.47% % % % -9.27% % % Put Bull Spread 37.45% % 61.47% 20.24% % % 9.27% % % Put Delta Hedge -5.11% 4.90% 2.66% -5.85% 7.89% 6.70% -8.60% 13.46% 29.97% Selling Straddle % % 98.76% % % 46.73% % 82.97% 64.63% Selling Strangle % % 98.76% % % 95.13% % % 91.24% Represent more than 100% rate of return Represents less than 0% or a negative return Represents more than 1000% leverage over the price of an ATM Call Represents less than 100% leverage or the position costs more than the price of an ATM Call 1 Annualized 2 Based on 30 Day ATM Call Price 3 Based on 90 Day ATM Call Price 4 Based on 1 Year ATM Call Price 38

40 Table 8 Profitability Table The following table represents the percent of times each of the 17 options strategies OTM, ATM, and ITM are profitable. The investment is established on the first trading day of the month and held to exercise. Therefore, the 30-Day position is established on the first trading day of the month and then exercise the third Friday of that same month. The investment period is from Jan 1971 to Dec Upon exercise, the value of position is compared to the value of the position to establish. If the value is greater at the end of the period then it is considered profitable. For example, a $1000 premium was received from selling ATM strangles. If the value of the positions at the time of exercise is still positive then the strategy was profitable. 30 Day Investment over 34 Years OTM ATM ITM % Times Beat Times Beat % Times Beat Strategy Profitable Profitable RR S&P Strategy % Profitable Profitable RR S&P Strategy Profitable Profitable RR S&P Put 3.19% % 34.8% Synthetic Stock 56.13% % 57.5% Strangle (Buy) 40.69% % 1.8% Bear Put Spread 3.19% % 32.0% Strangle (Buy) 9.80% % 12.2% Bull Put Spread 40.93% % 0.4% Call 6.62% % 7.8% Call 39.95% % 1.7% Naked Put 56.86% % 0.4% Bull Call Spread 6.62% % 7.5% Bull Call Spread 41.42% % 1.6% Synthetic Stock 56.37% % 0.4% Strangle (Buy) 30.64% % 1.1% Straddle (Buy) 42.65% % 1.2% Straddle (Buy) 53.92% % 0.4% Synthetic Stock 56.13% % 0.4% Put 30.88% % 1.0% Call 55.39% % 0.4% Straddle (Sell) 57.84% % 0.3% Bear Put Spread 31.37% % 0.5% Bull Call Spread 68.63% % 0.1% Covered Call 60.29% % 0.0% Equity Collar 60.05% % 0.0% Equity Collar 72.79% % 0.0% Protective Put 58.58% % 0.0% Protective Put 44.36% % 0.0% Covered Call 97.30% % 0.0% Equity Collar 44.61% % 0.0% Covered Call 72.79% % 0.0% Protective Put 81.13% % 0.0% Straddle (Buy) 42.16% % 0.0% Bull Put Spread 68.63% % 0.0% Bear Call Spread 31.37% % 0.0% Call Delta Hedge 34.07% % 0.0% Put Delta Hedge 0.00% 0 9.4% 0.0% Straddle (Sell) 46.08% % 0.0% Put Delta Hedge 24.75% % 0.0% Call Delta Hedge 0.00% 0 9.3% 0.0% Put 43.14% % 0.0% Strangle (Sell) 69.36% % 0.0% Naked Put 69.12% % 0.0% Bear Put Spread 59.07% % 0.0% Bear Call Spread 93.38% % 0.0% Straddle (Sell) 57.35% % 0.0% Put Delta Hedge 0.00% 0 8.6% 0.0% Naked Put 96.81% * 0.0% Bear Call Spread 58.58% % 0.0% Call Delta Hedge 0.00% 0 8.6% 0.0% Bull Put Spread 96.81% * 0.0% Strangle (Sell) 90.20% % 0.0% Strangle (Sell) 59.31% % 0.0% Represents that the strategy is profitable more than 70% of the time. Represents that the strategy is profitable over the S&P 500 by more than 30% Represents that the strategy is profitable over the S&P 500 by more than 10% but less than 30% Represents that the strategy is profitable over the S&P 500 but by less than 10% * Unknown percent return since value is incredibly negative. 39

41 Table 9 Profitability Table The following table represents the percent of times each of the 17 options strategies OTM, ATM, and ITM are profitable. The investment is established on the first trading day of the month and held to exercise. Therefore, the 30-Day position is established on the first trading day of the month and then exercise the third Friday of that same month. The investment period is from Jan 1995 to Dec Upon exercise, the value of position is compared to the value of the position to establish. If the value is greater at the end of the period then it is considered profitable. For example, a $1000 premium was received from selling ATM strangles. If the value of the positions at the time of exercise is still positive then the strategy was profitable. 30 Day Investment over 10 Years OTM ATM ITM % Times Beat % Times Beat % Times Beat Strategy Profitable Profitable RR S&P Strategy Profitable Profitable RR S&P Strategy Profitable Profitable RR S&P Put 3.33% % 9.5% Synthetic Stock 61.67% % 39.4% Bear Call Spread 30.83% % 3.1% Bear Call Spread 95.00% % 7.2% Bull Call Spread 46.67% % 3.6% Strangle (Buy) 47.50% % 2.9% Bear Put Spread 3.33% 4 9.2% 6.6% Bull Put Spread 69.17% % 2.8% Naked Put 61.67% % 1.3% Strangle (Sell) 70.00% % 2.5% Call 45.83% % 2.5% Bull Put Spread 46.67% % 1.0% Straddle (Sell) 63.33% % 1.5% Naked Put 70.00% % 2.4% Straddle (Buy) 60.00% % 0.9% Synthetic Stock 61.67% % 1.0% Strangle (Sell) 91.67% % 2.2% Call 60.83% % 0.9% Covered Call 62.50% % 0.0% Straddle (Buy) 42.50% % 0.5% Synthetic Stock 61.67% % 0.8% Protective Put 61.67% % 0.0% Equity Collar 61.67% % 0.0% Bull Call Spread 69.17% % 0.7% Equity Collar 46.67% % 0.0% Covered Call 73.33% % 0.0% Equity Collar 73.33% % 0.0% Call Delta Hedge 36.67% % 0.0% Protective Put 46.67% % 0.0% Covered Call 96.67% % 0.0% Put Delta Hedge 24.75% % 0.0% Straddle (Sell) 57.50% % 0.0% Protective Put 51.67% % 0.0% Straddle (Buy) 36.67% % 0.0% Put Delta Hedge 0.00% 0 2.0% 0.0% Straddle (Sell) 40.00% % 0.0% Strangle (Buy) 30.00% % 0.0% Call Delta Hedge 0.00% 0 1.9% 0.0% Bear Put Spread 53.33% % 0.0% Bull Put Spread 96.67% % 0.0% Strangle (Buy) 8.33% % 0.0% Put 38.33% % 0.0% Bull Call Spread 5.00% 6-4.5% 0.0% Put 30.00% % 0.0% Call Delta Hedge 0.00% 0 1.2% 0.0% Call 5.00% 6-4.8% 0.0% Bear Put Spread 30.83% % 0.0% Put Delta Hedge 0.00% 0 1.2% 0.0% Naked Put 96.67% % 0.0% Bear Call Spread 53.33% % 0.0% Strangle (Sell) 52.50% % 0.0% Represents that the strategy is profitable more than 70% of the time. Represents that the strategy is profitable over the S&P 500 by more than 30% Represents that the strategy is profitable over the S&P 500 by more than 10% but less than 30% Represents that the strategy is profitable over the S&P 500 but by less than 10% * Unknown percent return since value is incredibly negative. 40

42 Table 10 Profitability Table The following table represents the percent of times each of the 17 options strategies OTM, ATM, and ITM are profitable. The investment is established on the first trading day of the month and held to exercise. Therefore, the one-year position is established on the first trading day of the month and then exercised on the third Friday of January the following year. The minimum holding time for any one-year strategy is 7 months and the maximum is less than 18 months. Therefore, a position established on the first trading day in August of 1987 is not exercised until January The investment period is from Jan 1971 to Jan Upon exercise, the value of position is compared to the value of the position to establish. If the value is greater at the end of the period then it is considered profitable. For example, a $1000 premium was received from selling ATM strangles. If the value of the positions at the time of exercise is still positive then the strategy was profitable. 1 Year Investment over 34 Years OTM ATM ITM % Times Beat % Times Beat % Times Beat Strategy Profitable Profitable RR S&P Strategy Profitable Profitable RR S&P Strategy Profitable Profitable RR S&P Bear Put Spread 21.54% % 12.5% Synthetic Stock 59.49% % 6.5% Strangle (Buy) 54.62% % 1.4% Strangle (Buy) 49.23% % 4.0% Strangle (Buy) 54.36% % 2.9% Straddle (Buy) 56.92% % 1.3% Straddle (Buy) 51.79% % 2.2% Bear Put Spread 24.62% % 2.4% Protective Put 83.08% % 0.1% Bull Call Spread 58.72% % 1.3% Straddle (Buy) 58.97% % 2.0% Bear Call Spread 24.62% % 0.1% Put 20.51% % 0.8% Equity Collar 78.97% % 1.8% Equity Collar 85.64% % 0.1% Equity Collar 93.59% % 0.2% Bull Call Spread 66.41% % 0.8% Covered Call 87.95% % 0.0% Protective Put 69.74% % 0.1% Protective Put 66.92% % 0.1% Bull Call Spread 75.38% % 0.0% Covered Call 80.00% % 0.0% Covered Call 82.56% % 0.0% Bear Put Spread 33.85% % 0.0% Bear Call Spread 41.28% % 0.0% Bear Call Spread 33.59% % 0.0% Straddle (Sell) 43.08% % 0.0% Straddle (Sell) 48.21% % 0.0% Straddle (Sell) 41.03% % 0.0% Strangle (Sell) 45.38% % 0.0% Call Delta Hedge 0.00% 0 7.4% 0.0% Strangle (Sell) 45.64% % 0.0% Synthetic Stock 60.51% % 0.0% Put Delta Hedge 0.00% 0 6.7% 0.0% Put 23.59% % 0.0% Put Delta Hedge 0.00% 0 6.6% 0.0% Strangle (Sell) 50.77% % 0.0% Call Delta Hedge 0.00% 0 7.0% 0.0% Call Delta Hedge 0.00% 0 6.5% 0.0% Call 47.69% % 0.0% Put Delta Hedge 0.00% 0 6.8% 0.0% Call 56.15% % 0.0% Naked Put 79.49% %* 0.0% Call 53.59% % 0.0% Put 27.18% % 0.0% Synthetic Stock 66.41% %* 0.0% Naked Put 76.41% % 0.0% Naked Put 72.82% % 0.0% Bull Put Spread 78.46% %* 0.0% Bull Put Spread 75.38% %* 0.0% Bull Put Spread 66.15% % 0.0% Represents that the strategy is profitable more than 70% of the time. Represents that the strategy is profitable over the S&P 500 by more than 30% Represents that the strategy is profitable over the S&P 500 by more than 10% but less than 30% Represents that the strategy is profitable over the S&P 500 but by less than 10% * Unknown percent return since value is incredibly negative. 41

43 Table 11 Profitability Table The following table represents the percent of times each of the 17 options strategies OTM, ATM, and ITM are profitable. The investment is established on the first trading day of the month and held to exercise. Therefore, the one-year position is established on the first trading day of the month and then exercised on the third Friday of January the following year. The minimum holding time for any one-year strategy is 7 months and the maximum is less than 18 months. Therefore, a position established on the first trading day in August of 1987 is not exercised until January The investment period is from Jan 1994 to Jan Upon exercise, the value of position is compared to the value of the position to establish. If the value is greater at the end of the period then it is considered profitable. For example, a $1000 premium was received from selling ATM strangles. If the value of the positions at the time of exercise is still positive then the strategy was profitable. 1 Year Investment over 10 Years OTM ATM ITM % Times Beat % Times Beat % Times Beat Strategy Profitable Profitable RR S&P Strategy Profitable Profitable RR S&P Strategy Profitable Profitable RR S&P Put 20.00% % 89.7% Put 24.17% % 58.8% Put 30.00% % 44.9% Call 57.50% % 12.3% Synthetic Stock 60.83% % 31.4% Strangle (Buy) 60.00% % 6.9% Strangle (Buy) 63.33% % 10.2% Strangle (Buy) 65.00% % 10.2% Call 60.00% % 4.7% Naked Put 80.00% % 5.1% Call 60.00% % 7.4% Straddle (Buy) 61.67% % 4.2% Straddle (Buy) 65.83% % 5.0% Straddle (Buy) 67.50% % 4.9% Synthetic Stock 60.83% % 4.1% Bull Call Spread 59.17% % 2.3% Naked Put 75.83% % 4.6% Naked Put 70.00% % 3.1% Protective Put 65.00% % 0.0% Equity Collar 82.50% % 0.3% Bull Call Spread 73.33% % 0.0% Equity Collar 84.17% % 0.0% Bull Call Spread 63.33% % 0.0% Protective Put 69.17% % 0.0% Covered Call 77.50% % 0.0% Protective Put 64.17% % 0.0% Bear Call Spread 26.67% % 0.0% Put Delta Hedge 0.00% 0 2.6% 0.0% Bear Call Spread 36.67% % 0.0% Equity Collar 82.50% % 0.0% Bear Call Spread 40.83% % 0.0% Covered Call 80.83% % 0.0% Covered Call 85.83% % 0.0% Call Delta Hedge 0.00% 0 0.1% 0.0% Put Delta Hedge 0.00% 0 2.1% 0.0% Put Delta Hedge 0.00% 0 1.4% 0.0% Bear Put Spread 22.50% % 0.0% Call Delta Hedge 0.00% 0-0.1% 0.0% Bear Put Spread 36.67% % 0.0% Straddle (Sell) 34.17% % 0.0% Bear Put Spread 26.67% % 0.0% Call Delta Hedge 0.00% 0-0.4% 0.0% Bull Put Spread 77.50% % 0.0% Straddle (Sell) 32.50% % 0.0% Straddle (Sell) 38.33% % 0.0% Strangle (Sell) 36.67% % 0.0% Bull Put Spread 73.33% % 0.0% Strangle (Sell) 40.00% % 0.0% Synthetic Stock 61.67% 74-10%* 0.0% Strangle (Sell) 35.00% % 0.0% Bull Put Spread 63.33% % 0.0% Represents that the strategy is profitable more than 70% of the time. Represents that the strategy is profitable over the S&P 500 by more than 30% Represents that the strategy is profitable over the S&P 500 by more than 10% but less than 30% Represents that the strategy is profitable over the S&P 500 but by less than 10% * Unknown percent return since value is incredibly negative. 42

44 Table 7 Real ROR The following table represents the top three strategies over each of the investment periods. The ranking were determined by a combination of Real rates of return against the S&P 500. The first investment and period is the 30-day investment for 34 years. The investment is established on the first trading day of the month and held to exercise. Therefore, the 30-day position is established on the first trading day of the month and then exercise the third Friday of that same month. The investment period is from Jan 1971 to Dec The second investment and period is the 30- day investment for 10 years. The investment period is from Jan 1995 to Dec The third investment and period is the one-year investments for 34 years. The one-year position is established on the first trading day of the month and then exercised on the third Friday of January the following year. The minimum holding time for any one-year strategy is 7 months and the maximum is less than 18 months. Therefore, a position established on the first trading day in August of 1987 is not exercised until January The investment period is from Jan 1971 to Jan The final investment and period is the one-year investments over 10 years. The investment period is from Jan 1994 to Jan Investment Rank Strategy Real ROR 30 Day - 34 Year 1 ATM Long Synthetic Stock 67.55% 2 OTM Put 44.86% 3 OTM Bear Put Spread 42.01% 30 Day - 10 Year 1 ATM Long Synthetic Stock 41.99% 2 OTM Put 12.09% 3 OTM Bear Call Spread 9.80% 1 Year - 34 Year 1 OTM Bear Put Spread 22.89% 2 ATM Synthetic Stock 16.88% 3 OTM Strangle (Buy) 14.36% 1 Year - 10 Year 1 OTM Put 95.31% 2 ATM Put 64.49% 3 ITM Put 50.55% 43

45 Table 14 Actual Return for ATM Synthetic Stock The following table represents the rates of return (ROR) for the 30-day and one-year ATM Synthetic Stock strategy traded over two time intervals. The 30-Day strategy is established on the first trading day of the month and held to exercise. Therefore, the 30-day position is established on the first trading day of the month and then exercise the third Friday of that same month. The one-year strategy is established on the first trading day of the month and then exercised on the third Friday of January the following year. The minimum holding time for any one-year strategy is 7 months and the maximum is less than 18 months. Therefore, a position established on the first trading day in August of 1987 is not exercised until January The 34-year time period is from Jan 1971 to Jan 2004, and the 10-year time period is from Jan 1994 to Jan The first column labeled Investments is the ROR for the strategy with no consideration for Margin requirements, taxes, or brokerage fees. The second column labeled With Margin is the ROR for the strategy with consideration for market margin requirements but no consideration for taxes or brokerage costs. The final column labeled With Margin, Broker Fees, and Taxes is the ROR for the strategy considering Market Margin requirements, 28% tax rate, and $50 opening and $50 closing transaction costs. 34 Year 10 Year 34 Year 10 Year 34 Year 10 Year Investment With Margin With Margin, Broker Fees, and Taxes 30 DAY 67.55% 41.99% 61.26% 28.55% 30.01% 24.69% 1 YEAR 16.88% 37.04% 7.89% 16.79% 7.68% 14.39% 44

46 Table 1 Option Strategies The following table describes how the option strategies were constructed. The theoretical data options covered three different maturities; 30-day, 90-day and one-year. The options were examined at different levels of moneyness; OTM, ATM, and ITM. The amount OTM and ITM was 5% of the S&P 500 Index value. The practical application of the strategies was examined at the same levels of moneyness but only over 30-day and one-year maturities. The strategies are described as a function of price. Price = Time( % above or below index value, buying or selling). For example buying a OTM put is p(-5%, b). If the S&P 500 Index needs to be bought or sold for the strategy it is represented by (I) for buying or (-I) for selling. ITM ATM OTM Buying Straddle c(5%, b) p(-5%, b) c(0%, b) p(0%, b) c(-5%, b) p(5%, b) Buying Strangle c(0%,b) p(-10%, b) c(-5%,b) p(-5%, b) c(-10%,b) p(0%, b) Call c(5%, b) c(0%, b) c(-5%, b) Call Bear Spread c(5%, s) c(0%, b) c(0%, s) c(-5%, b) c(-5%, s) c(-10%, b) Call Bull Spread c(5%, b) c(0%, s) c(0%, b) c(-5%, s) c(-5%, b) c(-10%, s) Call Delta Hedge c(5%, s) I c(0%, s) I c(-5%, s) I Covered Call c(5%, s) I c(0%, s) I c(-5%, s) I Equity Collar c(-10%,s) I p(0%, b) c(-5%,s) I p(-5%, b) c(0%,s) I p(-10%, b) Long Synthetic Stock c(5%, b) p(-5%, s) c(0%, b) p(0%, s) c(-5%, b) p(5%, s) Naked Put p(5%, s) p(0%, s) p(-5%, s) Protective Put I p(5%, b) I p(0%, b) I p(-5%, b) Put p(5%, b) p(0%, b) p(-5%, b) Put Bear Spread p(0%, s) p(-5%, b) p(-5%, s) p(0%, b) p(-10%, s) p(-5%, b) Put Bull Spread p(0%, b) p(5%, s) p(-5%, b) p(0%, s) p(-10%, b) p(-5%, s) Put Delta Hedge -I p(5%, s) -I p(0%, s) -I p(-5%, s) Selling Straddle c(5%, s) p(-5%, s) c(0%, s) p(0%, s) c(-5%, s) p(5%, s) Selling Strangle c(0%,s) p(-10%, s) c(-5%,s) p(-5%, s) c(-10%,s) p(0%, s) 45

47 Figure 2 Highly Risk Averse 1 st Strategy 1-Year Investment ATM, ITM, and OTM Equity Collar (Traded Daily) The following figure represents the account growth for the One-Year ATM, ITM, and OTM Equity Collar strategies versus the S&P 500 over 34 years traded daily. The One-Year ATM Equity Collar strategy is established each trading day of the month and then exercised exactly one year later. The 34-year time period is from Jan 1971 to Jan $1,200, $1,000, $800, $600, $400, $200, $ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/2003 S & P Year ATM Equity Collar 1 Year ITM Equity Collar 1 Year OTM Equity Collar 46

48 Figure 3 1-Year Investment ATM, ITM, and OTM Equity Collar (Real Data) The following figures represent the account growth for the monthly investor of the One- Year ATM, ITM and OTM Equity Collar strategies versus monthly investor of the S&P 500 over 34 years and 10 years. The One-Year ATM, ITM, and OTM Equity Collar strategies are established on the first trading day of the month and then exercised on the third Friday of January the following year. The minimum holding time for any one-year strategy is 7 months and the maximum is less than 18 months. Therefore, a position established on the first trading day in August of 1987 is not exercised until January The 34-year time period is from Jan 1971 to Jan 2004, and the 10-year time period is from Jan 1994 to Jan Years 10 Years 47

49 Figure 6 3 rd Strategy 1-Year Investment ATM Covered Call (Traded Daily) The following figure represents the account growth for the One-Year ATM Covered Call strategy versus the S&P 500 over 34 years traded daily. The One-Year ATM Covered Call strategy is established each trading day of the month and then exercised exactly one year later. The 34-year time period is from Jan 1971 to Jan $1,200, $1,000, $800, $600, $400, $200, $ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/2003 S & P Year ATM Covered Call 48

50 Figure 7 1-Year Investment ATM Covered Call (Real Data) The following figures represent the account growth for the monthly investor of the One-Year ATM Covered Call strategy versus monthly investor of the S&P 500 over 34 years and 10 years. The One-Year ATM Covered Call strategy is established on the first trading day of the month and then exercised on the third Friday of January the following year. The minimum holding time for any one-year strategy is 7 months and the maximum is less than 18 months. Therefore, a position established on the first trading day in August of 1987 is not exercised until January The 34-year time period is from Jan 1971 to Jan 2004, and the 10-year time period is from Jan 1994 to Jan Years 10 Years 49

51 Figure 4 2 rd Strategy 30-Day ITM Protective Put (Traded Daily) The following figure represents the account growth for the 30-Day ITM Protective Put strategy versus the S&P 500 over 34 years traded daily. The 30-Day ITM Protective Put strategy is established each trading day of the month and then exercised exactly 30 days later. The 34-year time period is from Jan 1971 to Jan $250, $200, $150, $100, $50, $ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/2003 S & P Day ITM Protective Put 50

52 Figure 5 30-Day ITM Protective Put (Real Data) The following figures represent the account growth for the monthly investor of the 30-Day ITM Protective Put strategy versus monthly investor of the S&P 500 over 34 years and 10 years. The 30-Day ITM Protective Put strategy is established on the first trading day of the month and then exercised on the third Friday of the same month. The 34-year time period is from Jan 1971 to Dec 2004, and the 10-year time period is from Jan 1995 to Dec Years 10 Years 51

53 Figure 9 Medium Risk Averse 1 st Strategy 30-Day Investment ATM and OTM Equity Collar (Traded Daily) The following figure represents the account growth for the 30-Day ATM and OTM Equity Collar strategies versus the S&P 500 over 34 years traded daily. The 30-Day ATM and OTM Equity Collar strategies are established each trading day of the month and then exercised exactly 30 days later. The 34-year time period is from Jan 1971 to Dec $250, $200, $150, $100, $50, $ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/2003 S & P Day ATM Equity Collar 30 Day OTM Equity Collar 52

54 Figure Day Investment ATM and OTM Equity Collar (Real Data) The following figures represent the account growth for the monthly investor of the 30-Day ATM and OTM Equity Collar strategies versus monthly investor of the S&P 500 over 34 years and 10 years. The 30-Day ATM and OTM Equity Collar strategies are established on the first trading day of the month and then exercised on the third Friday of the same month. The 34-year time period is from Jan 1971 to Dec 2004, and the 10-year time period is from Jan 1995 to Dec Years 10 Years 53

55 Figure 11 2 nd Strategy 30-Day Investment ATM Covered Call (Traded Daily) The following figure represents the account growth for the 30-Day ATM Covered Call strategy versus the S&P 500 over 34 years traded daily. The 30-Day ATM Covered Call strategy is established each trading day of the month and then exercised exactly 30 days later. The 34-year time period is from Jan 1971 to Dec $250, $200, $150, $100, $50, $ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/2003 S & P Day ATM Covered Call 54

56 Figure Day Investment ATM Covered Call (Real Data) The following figures represent the account growth for the monthly investor of the 30-Day ATM Covered Call strategy versus monthly investor of the S&P 500 over 34 years and 10 years. The 30-Day ATM Covered Call strategy is established on the first trading day of the month and then exercised on the third Friday of the same month. The 34-year time period is from Jan 1971 to Dec 2004, and the 10-year time period is from Jan 1995 to Dec Years 10 Years 55

57 Figure 13 3 rd Strategy 30-Day ATM Protective Put (Traded Daily) The following figure represents the account growth for the 30-Day ATM Protective Put strategy versus the S&P 500 over 34 years traded daily. The 30-Day ATM Protective Put strategy is established each trading day of the month and then exercised exactly 30 days later. The 34-year time period is from Jan 1971 to Dec $250, $200, $150, $100, $50, $ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/2003 S & P Day ATM Protective Put 56

58 Figure Day Investment ATM Protective Put (Real Data) The following figures represent the account growth for the monthly investor of the 30-Day ATM Protective Put strategy versus monthly investor of the S&P 500 over 34 years and 10 years. The 30-Day ATM Protective Put strategy is established on the first trading day of the month and then exercised on the third Friday of the same month. The 34-year time period is from Jan 1971 to Dec 2004, and the 10-year time period is from Jan 1995 to Dec Years 10 Years 57

59 Figure 15 Low Risk Averse 1 st Strategy 30-Day Investment OTM Selling Straddle (Traded Daily) The following figure represents the account growth for the 30-Day OTM Selling Straddle strategy versus the S&P 500 over 34 years traded daily. The 30-Day OTM Selling Straddle strategy is established each trading day of the month and then exercised exactly 30 days later. The 34-year time period is from Jan 1971 to Dec $250, $200, $150, $100, $50, $0.00 -$50, /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/2004 S & P Day OTM Selling Straddle 58

60 Figure Day Investment OTM Selling Straddle (Real Data) The following figures represent the account growth for the monthly investor of the 30-Day OTM Selling Straddle strategy versus monthly investor of the S&P 500 over 34 years and 10 years. The 30-Day OTM Selling Straddle strategy is established on the first trading day of the month and then exercised on the third Friday of the same month. The 34-year time period is from Jan 1971 to Dec 2004, and the 10-year time period is from Jan 1995 to Dec Years 10 Years 59

61 Figure 17 2nd Strategy 30-Day Investment OTM, ATM, and ITM Call Bull Spread (Traded Daily) The following figure represents the account growth for the 30-Day OTM, ATM, and ITM Call Bull Spread strategies versus the S&P 500 over 34 years traded daily. The 30-Day OTM, ATM, and ITM Call Bull Spread strategies are established each trading day of the month and then exercised exactly 30 days later. The 34-year time period is from Jan 1971 to Dec $300, $250, $200, $150, $100, $50, $ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/ /29/2004 S & P Day OTM Call Bull Spread 30 Day ATM Call Bull Spread 30 Day ITM Call Bull Spread 60

62 Figure Day Investment OTM, ATM, and ITM Call Bull Spread (Real Data) The following figures represent the account growth for the monthly investor of the 30-Day OTM, ATM, and ITM Call Bull Spread strategies versus monthly investor of the S&P 500 over 34 years and 10 years. The 30-Day OTM, ATM, and ITM Call Bull Spread strategies are established on the first trading day of the month and then exercised on the third Friday of the same month. The 34-year time period is from Jan 1971 to Dec 2004, and the 10-year time period is from Jan 1995 to Dec Years 10 Years 61

63 Figure 21 4 rd Strategy 1-Year Investment ATM Synthetic Stock (Traded Daily) The following figure represents the account growth for the One-Year ATM Synthetic Stock strategy versus the S&P 500 over 34 years traded daily. The One-Year ATM Synthetic Stock strategy is established each trading day of the month and then exercised exactly one year later. The 34-year time period is from Jan 1971 to Jan $1,400, $1,200, $1,000, $800, $600, $400, $200, $0.00 -$200, /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/2003 S & P Year ATM Synthetic Stock 62

64 Figure 22 1-Year ATM Synthetic Stock (Real Data) The following figures represent the account growth for the monthly investor of the One-Year ATM Synthetic Stock strategy versus monthly investor of the S&P 500 over 34 years and 10 years. The One-Year ATM Synthetic Stock strategy is established on the first trading day of the month and then exercised on the third Friday of January the following year. The minimum holding time for any one-year strategy is 7 months and the maximum is less than 18 months. Therefore, a position established on the first trading day in August of 1987 is not exercised until January The 34-year time period is from Jan 1971 to Jan 2004, and the 10-year time period is from Jan 1994 to Jan Years 10 Years 63

65 Figure 19 3 rd Strategy 1-Year Investment OTM, ATM, and ITM Buying Strangle (Traded Daily) The following figure represents the account growth for the One-Year OTM, ATM, and ITM Buying Strangle strategies versus the S&P 500 over 34 years traded daily. The One-Year OTM, ATM, and Buying Strangle strategies are established each trading day of the month and then exercised exactly one year later. The 34-year time period is from Jan 1971 to Jan $1,400, $1,200, $1,000, $800, $600, $400, $200, $ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/ /30/2003 S & P Year ATM Buying Strangle 1 Year ITM Buying Strangle 1 Year OTM Buying Strangle 64

66 Figure 20 1-Year Investment OTM, ATM, and ITM Buying Strangle (Real Data) The following figures represent the account growth for the monthly investor of the One-Year OTM, ATM, and ITM Buying Strangle strategies versus monthly investor of the S&P 500 over 34 years and 10 years. The One-Year OTM, ATM, and ITM Buying Strangle strategies are established on the first trading day of the month and then exercised on the third Friday of January the following year. The minimum holding time for any one-year strategy is 7 months and the maximum is less than 18 months. Therefore, a position established on the first trading day in August of 1987 is not exercised until January The 34-year time period is from Jan 1971 to Jan 2004, and the 10-year time period is from Jan 1994 to Jan Years 10 Years 65

67 Figure Day ATM Synthetic Stock w/ Costs vs. S & P 500 The following figures represent the account growth for the monthly investor of the 30-Day ATM Synthetic Stock strategy including brokerage costs versus monthly investor of the S&P 500 over 34 years and 10 years. The 30-Day ATM Synthetic Stock strategy is established on the first trading day of the month and then exercised on the third Friday of that same month. Brokerage costs are figured at $100 per month. The 34-year time period is from Jan 1971 to Jan 2004, and the 10-year time period is from Jan 1994 to Jan Year 10 Years 66

68 Figure 24 1-Year ATM Synthetic Stock w/ Costs vs. S & P 500 The following figures represent the account growth for the monthly investor of the One-Year ATM Synthetic Stock strategy including brokerage costs versus monthly investor of the S&P 500 over 34 years and 10 years. The One-Year ATM Synthetic Stock strategy is established on the first trading day of the month and then exercised on the third Friday of January the following year. The minimum holding time for any one-year strategy is 7 months and the maximum is less than 18 months. Therefore, a position established on the first trading day in August of 1987 is not exercised until January Brokerage costs are figured at $100 per month. The 34-year time period is from Jan 1971 to Jan 2004, and the 10- year time period is from Jan 1994 to Jan Year 10 Years 67