Is There Money to Be Made Investing in Options? a Historical Perspective

Is There Money to be Made Investing in Options? A Historical Perspective

James S. Doran Department of Finance Florida State University

and

Robert Hamernick Florida State University

Revised: January 4, 2006

JEL : G11, G12, G13 Keywords : Portfolio Returns, Option Strategies, Option Pricing Sharpe Ratios, S&P 500

Acknowledgments: The authors acknowledge the helpful comments and suggestions of James Ang, Gary Benesh, Jim Carson, Yingmei Cheng, Jeﬀ Clark, John Inci, Bong-Soo Lee, Ehud Ronn, Dave Peterson, and Participants at the FSU Seminiar.

Communications Author: James S. Doran

Address: Department of Finance College of Business Florida State University Tallahassee, FL. 32306 Tel.: (850) 644-7868 (Oﬃce) FAX: (850) 644-4225 (Oﬃce) E-mail: [email protected] Abstract

This paper examines the historical time-series performance of twelve 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. The analysis was conducted by assuming monthly trading starting in 1970 and 1995, comparing historical performance via returns and Sharpe ratios as compared to the buy and hold returns on the S&P 500 as a benchmark. The analysis generates returns for a “typical” investor, where the option investment is a portion of the total portfolio. The analysis revealed that investing in options as an enhancement to a buy and hold portfolio can result in returns and Sharpe ratios that exceed the S&P 500. However, the performance is conditional on that the investments made in options are small as compared to the portfolio as a whole.

i 1 Introduction

Since the introduction of option contracts as a liquid financial asset, market participants have constructed these financial instruments to utilize the non-linear payoff feature tailored to many investors needs. Additionally, it has allowed investors the ability to hedge downside exposure as well as lock in potential profits. From a research standpoint, since the seminal work of Black and Scholes (1973), there has been extensive research done on the theoretical and empirical properties of option prices, leading to a diverse and rich literature.1 From a practical standpoint, the use of options as a investment for the individual investor has received little attention. Harvey and Whaley (1992) and Figlewski (1989, 1994) examined the arbitrage properties option prices and concluded that the transaction costs faced by investors limit any potential arbitrage opportunity found through mispricing. However, these works ignore the role options may have to enhance a portfolio as a complementary investment. Explicitly, what effect does holding or writing options have in addition to a long portfolio returns? The purpose of this paper is to examine, from an historical perspective, the return and risk to holding a variety of different option strategies from a representative investor standpoint. The focus of this paper will be on an individual investor who 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, margin requirements, 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 options as an investment vehicle that that can be utilized in today’s market. While the possible combinations of option contracts is boundless, this paper will focus on the most common, or popular, strategies that can be implemented, for a variety of holding

1Major theoretical extension have included the incorporation of stochastic volatility models such as Hull and White (1987) and Heston (1993); the inclusion of jumps, Bates (1996), and double jump models of the type in Duﬃe, Pan, and Singleton (2000). Empirical Work such as Bakshi, Cao, and Chen (1997), have examined the ﬁt of these models to option pricing data, as well as examine the hedging implications from such prices.

1 periods and moneyness. In particular, twelve options strategies on the S&P 500 are examined over a thirty four- and ten-year holding period, and compared to the index as a benchmark. These strategies range from speculative, i.e naked positions, to conservative, writing covered calls or protective puts. To analyze the benefit of investing in options, the representative investor portfolio was then constructed by investing in the given option contract on a monthly basis, where the payoff to the position was then re-invested into the underlying asset. In setting up the portfolio in this fashion, it allowed for direct comparison of the risk-return trade-off of investing in the given strategy versus a buy and hold position in the market using a dollar cost averaging scheme. The twelve option strategies are designed to cover a wide range of investment periods, taking advantage of periods of high and low volatility, bear and bull markets, as well as covering varying degrees of risk aversion. The results will demonstrate the returns to the various strategies, highlighting the benefit and cost to each. In particular, it will reveal that investment in a constant strategy underperforms the market in almost every case, except for strategies that involve sell put contracts. This highlights the finding of Coval and Shumway (2000), Bakshi and Kapadia (2003), and Doran (2005), who demonstrate that puts are expensive due to investor aversion to volatility and/or jump risk. However, the length of option contract plays an important role, as the use of long-term contracts can be used to leverage other portions of the portfolio to the investor’s benefit. While the work here is similar in scope to that of Santa-Clara and Saretto (2005), some of the conclusion and the message are distinct. In particular, some of the results presented provide alternative evidence to the Santa-Clara and Saretto (2005) findings, suggesting that even in the presence of transaction costs and margin calls, investing in option strategies can be profitable. In addition, the construction of the investor portfolio is unique, which combines both buying options and holding the underlying, which places the central focus of this paper on the returns to the portfolio, and not just the option position. The results from the analysis demonstrate three conclusions for option investors. First, consistent with Bakshi and Kapadia (2003), Coval and Shumway (2000), and Doran (2005), selling options is profitable, especially covered calls, but only in the short-term. Second,

2 certain strategies are best implemented in an attempt to market time, such as combination strategies, due to the high cost of investing. Lastly, and contrary to prior evidence, the Sharpe ratio from a long synthetic stock position outperforms the market even accounting for transaction costs and margin. However, this is conditioned on the option position being a small portion of the portfolio, as large negative realizations can eliminate signiﬁcant portfolio wealth. This suggests that through the use of leverage, investors can take advantage of either mis-pricing and/or diﬀering levels of risk-aversion through the options market, as long as the option position is a small portion of the overall portfolio. The rest of the article is organized as follows. Section II will address the portfolio formation. Section III will describe the data and methodology. Section IV details the results. Section V will conclude.

2 Data

2.1 Portfolio Formation

To analyze the effect incorporating options has on portfolio returns, twelve distinct strategies were implemented for two holding periods using different moneyness and maturity combinations. The portfolio is designed to be directly comparable to an investor that just buys and holds the underlying index. As such, if an option is profitable at expiration, the investor will take the proceeds and re-invest in the underlying asset. Of course, the alternative is to take the gain from the options and subsequently buy more options, but this is unrealistic since there is almost certain probability that at some point the options will expire worthless, and the investor will lose all her money. Consequently, by setting up the portfolio in this fashion, analysis can be made to the marginal effect of holding options in a portfolio, and demonstrate the impact through various stages of the business cycle Explicitly, the investor will make monthly installments in one of the twelve strategies throughout the holding period. This is designed to mimic the typical investor who makes monthly contributions to a 401K plan or IRA. To make the investment process transparent, we present the following simple two-period illustration below.

3 Figure 1: Investment Timeline

t=2 t=0 t=1 Monthly Investment: Monthly Investment: α0 Monthly Investment: α1 α

Buy & Hold: Invest α0 in either C0 or S0 Portfolio Value: (X +X )S + {(α +R )/S =X shares} + R 0 1 2 2 1 2 2 2 =(X0+X1+X2)S2+ R2 Initial Portfolio Position Option Position: 1. Buy & Hold: α0/S0 =X0 shares + R0 in risk-free rate Portfolio Value: Z1S2 + {(Y1P2+R1)/S2 = Z2 shares } 2. Option Position: α0/C0 =Y0 options + R0 in risk-free rate + (α2+R1)/C2 = Y2 options + R2 =(Z1+Z2)S2+Y2+ R2

Buy & Hold: Portfolio Value: X0S1 + {(α1+R0)/S1 =X1 shares} + R1 =(X0+X1)S1+ R1 Option Position: Portfolio Value: (Y0P1+R0)/S1 = Z1 shares + (α1+R0)/C1 = Y1 options + R1 =Z1S1+Y1+ R1

Each month t the option investor will invest αt dollars in Ct, an option strategy, yielding

Yt contracts. Since whole contracts of the option must be purchased, the reminder of the cash will be invested at the risk-free rate Rt. Assuming monthly expirations of the option contracts, the investor will then take the proﬁts from each option, Pt+1, plus any money earned from the risk-free rate, and re-invest the proceeds in the underlying asset at a price

St+1, buying Zt+1 shares. These shares will then be held in the portfolio to the terminal date and will only be sold to cover potential margin calls against the option positions. At the same point αt+1 dollars will purchase new option contracts at Ct+1. This process is repeated until the end of the holding period.

4 The alternative strategy is to buy and hold the underlying asset each month at a price

St and buying Xt shares. The difference between these two strategies is that by initially purchasing option contracts, the cash earned, positively or negatively, in the next period will results in different values of Xt and Zt. As the investor continues with monthly installments, the impact between the two shares will be reflected in the final portfolio value. By applying this very general framework, the impact of option investing can be assessed. As to application, the representative investor will have $1000 a month to invest in her chosen strategy, with each investment made at the beginning of every month. The initial amount was set for an investor who could afford these monthly installments from an annual median income ranging from $70,000 to $130,000. The amount will then be adjusted through time by the CPI to account for inflation.2

2.2 Option Strategies

The twelve strategies that are examined are listed in table 1, and were chosen according to the “basic strategies” outlined in accordance with the Chicago Board of Options Exchange (CBOE). While there are many more combinations of options, especially across time and moneyness, by examining these twelve combinations, the basic intuition for understanding return and risk can be inferred for more complex constructs. The strategies can be further segmented, where the first six strategies are single option strategies, while the other six are combination strategies.3 Consequently, some strategies will cost more, resulting in fewer contracts bought, and the reduced chance of profitability due to less leverage. Since some strategies require the shorting of option positions, the potential leverage increases, as does the overall risk to the portfolio. The risk of an option position varies greatly across the strategies. For example, taking a negative naked position in put(call) options can result in large losses given the potentially large negative(positive) movements in the index. However, writing covered calls or protective puts provides a natural hedge if there is a long position in the underlying asset. The difference

2The initial monthly installment for the investor in 1970, using CPI adjusted dollars, was $228. In 1995, the installment value was $807/month. 3The synthetic stock position is analyzed at diﬀerent levels of moneyness.

5 in these two strategies is just in the long position in the underlying, but the consequence for the portfolio returns are large. In addition, the timing and moneyness of the option can have large ramiﬁcations for the position. As such, while a covered call seems an appropriate hedge, the question to the maturity of the option contract remains.

3 Data

To establish the portfolios, and the subsequent returns, option prices were calculated from historical price on the S&P 500. Using the S&P 500 as the underlying asset was an obvious choice as it is the standard benchmark to which most portfolios are compared to. Since the analysis will be done assuming the options will be held until maturity, the options prices will be calculated using Black-Scholes, as shown below,4

c −rt q VBS,ψ = StN(d1) − e ∗ KjN(d1 − σiv (ψ)) (1)

p −rt q VBS,ψ = −StN(−d1) + e ∗ KjN(−(d1 − σiv (ψ))) (2) with (3) 2 σiv ln(Ft/X) + 2 ψ d1 = √ (4) σiv ψ To derive the option prices required several inputs as data on options is not available for some of the holding period, and options were not standardized until 1973. The risk-free rate was the 1-year nominal rate, which came from the Federal Reserve. The maturity of the options was calculated to be consistent with expiration schedule set by the exchange, stated in a years. As such, the time to maturity, ψ, was calculated for each option as the diﬀerence in days from the start of the month to the third Friday of the month for the one-month option. For the one-year option, the maturity is the diﬀerence in days from the start of a given month to the third Friday of the same month in the following year. The strike prices,

4A distinct disadvantage of this methodology is relying on the derivation of the option price versus the actual option price. This methodology was implemented for two reasons. First, since it was necessary to generate a simulated option price for earlier periods, we wished to remain consistent for later periods. Second, the simulated option values generated very similar results to actual prices, but added the ﬂexibility of allowing for changes is underlying volatility to demonstrate the eﬀect small miscalculations have on the portfolio returns.

6 K, are set to be as close to at-the-money (ATM), 5% out-of-the money (OTM), and 5% in-the money (ITM), in one dollar increments. For example, if the underlying price was $45.60, then the ATM strike would be $46. While small errors in these parameters have minimal impact on the potential option price, incorrect inference on the implied volatility can result in large mis-pricingings.5 The historical implied volatility was constructed using the relationship between implied and realized volatility. Doran and Ronn (2005), Christensen and Prabhala (1998), and others have noted the strong predictive link between implied and realized volatility, noting that implied volatility is the most eﬃcient, yet biased estimate of realized volatility. Using this information, and information about current day volatility skews for both puts and calls, will allow for an historical estimate of implied volatility, and the volatility skew. This was accomplished by running a regression of the following,

σrv,t = α0 + αiσciv,t + t (5)

σrv,t = α0 + αiσpiv,t + t (6)

6 where σrv,t is a 22-day realized volatility, σciv,t is the implied volatility from a call option, 7 and σpiv,t is the implied volatility from a put option. This was done for puts and calls, one year and one month maturities, and using ATM, 5% OTM, and 5% ITM implied volatilities. The data for the implied volatility came from OptionMetrics, using a sample of options on the S&P 500 from 1996-2002. Using the coeﬃcients estimates, and constructing realized volatility back through 1970, estimated implied volatility was derived as

σrv,t − α0 σiv,tˆ = (7) αi for all twelve implied volatility series. While Black-Scholes is quite accurate for ATM options, it is inaccurate for ITM and OTM options.8 In deriving the volatility in this fashion reduces

5See Hentschel (2002). 6Refer to Doran and Ronn for deﬁnition of this variable 7Results and available upon request. 8See Bakshi, Cao, and Chen (1997) for evidence.

7 the problems associated with model mis-specification in Black-Scholes by incorporating the negative skews typically observed for index options. In addition, in separating the puts and the calls, controls for the findings by Bates (1996) that OTM puts tend to be expensive relative to all other options. This is also consistent with the negative volatility risk premium shown by Bakshi and Kapadia (2003), Coval and Shumway (2000), and Doran and Ronn (2005).9 Controlling for maturity was also necessary, as the skew is more muted for longer maturity options due to reduced jump concerns. By ignoring this stylized fact, OTM long- term puts would be overly expensive, and would have distorted the final portfolio returns. The estimated values were then compared against the actual values for the 1996-2002 period to test for consistency. The findings suggested less than a 3% pricing error on average. For robustness, Measurement error in volatility will be addressed later. Solving for the option price required an additional price control for the tick size. Since option prices are quoted in $0.05 increments, the Black-Scholes price was rounded up to the next $0.05 increment in the case of a long position, and down in the case of a short position. This was done in part to control for potential estimation error and bias the returns of the option portfolio downward. To control for low values of the index and/or OTM positions, a long position in the option could not fall below $0.10 while the equivalent short position had a floor of $0.05. With these controls in place, option values for calls and puts were generated at the start of every month, for both the month and year, ATM, ITM, and OTM options, such that the twelve strategies could be implemented and compared to the S&P 500.

4 Estimation Results

4.1 Implementation

The implementation of the twelve strategies was done under three separate trading conditions. The ﬁrst assumed that all options were bought at the price given by the Black-Scholes

9An alternative methodology was implemented regressing VIX on realized volatility, and then adding in average diﬀerences in the volatility skew. The results yielded little diﬀerence in overall portfolio returns.

8 i price, VBS,t, where i = c is the call price given by eq. 1 and i = p is the put price given by eq. 2. The second accounted for the bid-ask spread, BA(i, ψ), by adjusting the Black-Scholes price by a given percentage depending on option type and maturity length, ψ. As shown in Table 2, the eﬀect of the bid-ask spread can increase the price of the option signiﬁcantly,

i such that a 1-year, 5% OTM put that costs $44.20 at VBS,t, will cost $46.20 at the bid. Each bid and ask price was calculated as such,

BA(i, ψ) V i = V i (1 + ) (8) bid,t BS,t 2 BA(i, ψ) V i = V i (1 − ) (9) ask,t BS,t 2 for each moneyness and maturity combination adjusting the price to the nearest $0.05 increment. Incorporating the spread in the fashion can highlight the quantitative impact bid-ask quotes have on portfolio returns. However, this effect must be considered a worse-case scenario, as most transactions occur within the spread. Finally, commissions and taxes were incorporated with the bid-ask spread costs to reflect all potential transaction costs. The cost of the option contract was given by a fairly expensive commission schedule of $20 plus $0.02 of the dollar trade amount. Buying the underlying incurred a fee of $20 a purchase for hundred shares or less, and $80 a purchase for over a hundred shares. This value will be subtracted from cash left over that would have initially gone in the risk-free rate. If there is not enough cash to cover the commissions, one less share or option contract will be purchased, as the portfolio is constrained to buy whole units. An additional assumption made is that the underlying index can actually be purchased. This was simply done for tractability, and does not have a significant impact on the results. Since it is possible to buy options on exchange traded funds (ETF) such as SPDR’s, the results here are quite transferable to actual application. For the 34-year holding period, the investor is assumed to just be starting out, with no holdings or cash. This was done for two specific reasons. First, this will result in a highly levered position earlier in the portfolio, and second, will highlight the impact of the options through time. This distinction will become apparent in examining the long synthetic stock position, through the effect of margin calls and over-weighting the portfolio in options. For

9 the 10-year holding position, the investor will have a starting position of $50,000. Since the holding period is shorter, the returns to the portfolio would have stronger time-dependence if there were no initial cash balance. By having a larger cash balance, the purchase of the option contracts represent less than 2% of the overall portfolio. As such, the returns through time will be not be as dependent on when they are bought, and more a function of the type of contract.10 When a short position in the option was taken, the margin was set such that only 10% of the balance of the portfolio plus whatever the monthly cash installment was could be used to cover the short sale. While the 10% value was set arbitrarily, and could have been increased or decreased to change the overall risk of the portfolio,11 The value was set so the investor could take a moderately risky position. Since the selling of options, especially naked, is considered a highly risky strategy, using 10% margin is a conservative estimate.12 Table 3 reports the percentage of option exercised and profitable for the given strategies. The results reveal that on average, short-term options are less profitable than long-term options, and calls are more profitable than puts. When comparing across maturities, OTM call options become 40-45% more profitable when buying longer-term options. This is quite interesting since the OTM profitability, as compared to the ATM and ITM options, is significantly less that than when bought over shorter maturities. Combination strategies reveal that a butterfly spread, used to take advantage of low volatility is highly profitable over short periods, while straddles and strangles, which are used in anticipation of large price swings, are best over longer maturities. As to the spreads, since there is positive price drift, bull-spreads are more profitable than bear-spreads, especially over the long-term. While this reveals information in regards to the differences in exercise and profitability across option types, it will be interesting to assess how these differences translate to actual portfolio

10Clearly returns are a function of how the markets are performing, but with no initial position, the returns can be highly distorted. 11This is an interesting topic itself. It would be interest to examine the eﬀect of margin constraints on options, and the resulting impact on returns through increased leverage. 12The ability to sell naked positions is dependent on the trading platform used. Some brokerages require that the short position is completely covered with cash, others allow leverage up to and exceeding 100% of portfolio value.

10 returns.

4.2 Portfolio Returns

The initial results for the single option positions are reported in table 4 for both the 34 year and 10 year time horizon. The results are shown for the three pricing scenarios, for both the monthly and yearly options, and show the annual returns, Ri, to the given option portfolio, Ωi, accounting for the initial investment and monthly cash installments calculated as such,

Ωi − Ci ! 360 Ri = T (10) Ci days i i where ΩT is the portfolio value at the end of the holding period, C is the total dollar investment over the life of the portfolio, and days is the number of days from the start of the investment to the end. There are several key insights worth noting. First, the effect of transaction costs has a 1% impact on portfolio returns for the long strategies, and a large effect for naked short positions, about 2-5%. Second, short-term naked positions are highly profitable, consistent with the findings of Coval and Shumway (2000) and others. Third, most long strategies, once accounting for transaction costs, do not outperform the S&P 500, similar to the results of Harvey and Whaley (1992), Figlewski (1989) and Santa-Clara and Sarreto (2005). The exceptions are the long-term calls in the 10-year holding period, with the OTM calls performing best. The profitability of hedging strategies shown in table 5 is, by comparison, quite obvious. Since the covered call and protective strategy must protect the whole portfolio, this strategy is distinct from naked position which only sells the amount covered by 10% of the portfolio plus the monthly cash investment. As has been a recent trend, executing covered calls has outperformed the market.13 The profitability of using monthly options, over the 10 year

13This particular strategy is now monitored by the CBOE’s BXM index. Ibbotson Associates examined the returns to BXM in the paper “Passive Options-based Investments Strategies: The Case of the CBOE S&P 500 Buy Write Index.” Since the start of measurement period, 06/01/1988, the BXM index has outperformed the S&P 500 by 1.7% based upon annualized daily returns. The results are comparable to the 10-year ATM Covered Call strategy with transaction costs.

11 holding period, is highlighted in figure 2. As can be seen, the results show that the final portfolio value of the covered call exceeds that of the S&P 500 by over $50,000, a difference of over 20%. By comparison, the protective put value doesn’t exceed the S&P, but has little to no risk. This result is directly related to the notion that the covered call limits upside gains, for positive returns, while the protective put eliminates downside risk. However, these strategy must be executed using only short-term options, and only using ATM or OTM options.14The long-term option results suggest that using options to hedge portfolio risk are limited at best. This is because the hedge is either too expensive, or restricts too much upside potential. The third set of portfolio returns examine combination strategies, and is reported in table 6. Only one strategy outperforms the S&P, the long-term bear-spread over 34 years, and is a function of the 1973 and 1987 crash. Over the 10-year time horizon, the returns for this particular strategy are 8.6% lower, and significantly underperform the market. Only the yearly strangle and bull-spread approached the S&P returns, after all transaction costs, over the 10 year horizon. These findings suggest that implementing these types of strategies over long periods will underperform not only because they are expensive, but also because these strategies are designed to take advantage of certain market conditions, such as periods of low volatility with the butterfly spread. As with many investment criteria, implementing a combination strategy needs superior market timing. Finally, table 7 reports the returns to a synthetic stock position. The returns are quite revealing, and contrary to many prior findings. Long-term synthetic positions can exceed market returns by over 5%, since it takes advantage of the short position in the expensive put, and the long leveraged position in the call. This return is substantial, highlighted by synthetic position in the OTM call and OTM put, especially over the 10 year holding period. However, these results must be taken with caution. Within the 34-year period, the synthetic position actually fell to a negative portfolio balance. This arises since the option position initially is a large portion of the portfolio. From February 1973 through December 1974, the market fell from 114.93 to 66.91, a percentage decrease of 41%. With many long-term naked

14Executing a long-term covered call would be inconsistent with the portfolio formation.

12 positions in the put, the portfolio suffered huge losses and fell to a negative balance. This resulted in the case when no position could be taken since the monthly installments had to cover the losses. Only on February 1977, after paying down several years of debt could the investment strategy be resumed. Once the portfolio was positive, it took 20 years for the synthetic position to achieve the level of S&P 500, reaching an equivalent portfolio value around June 1997. This is highlighted in figure 3. This is very different than the scenario that played out in 2000-2003. With a significant portion of the portfolio in the underlying, there was little concern of losing the entire portfolio. Consequently, the strategy could always be implemented, and was highly profitable in the recovery after 2003 as shown in figure 4. The results for this position should not be considered surprising, but what is surprising is that the position has not been utilized more. The synthetic stock is just a levered market position, just like holding a high beta portfolio. However, unlike a high beta portfolio, selling OTM puts generates high returns in part due to the “crashaphobia”. This appears to be a key determinant in the high returns for the position.

4.3 Sharpe Ratios

To make a full assessment of the portfolios, it is necessary to examine the risk-adjusted performance. The Sharpe ratio was calculated in this case by taking the returns in the portfolio, minus the de-annualized 1-year risk-free rate, over the standard deviation of the returns. Since the calculations were done on the portfolio, and not the options, the Sharpe ratio can be utilized since the portfolio returns are close to normal.15 The Sharpe ratios are reported in table 8, and the returns include all transaction costs. All calculations were done using the 10-year time horizon. The results indicated that most strategies underperform the S&P 500, which is similar to prior ﬁndings. What is diﬀerent is that there are certain strategies whose risk-adjusted performance exceeds that of the S&P 500. In particular, selling short-term options, calls or puts, tend to outperform the market through increased levels of returns. Protective strategies

15A problem in using the Sharpe ratio for options is the non-linear payoﬀ of an option contract, where the payoﬀ in any given period is some positive value or zero. In using portfolio return, the portfolio either goes up or down as function of the option contract and the underlying index.

13 such as covered calls or protective puts also outperform, but in the protective put case, it is due to the reduction in risk. The long-term strategies reveal that the synthetic position also has a higher risk-adjusted performance than the market, which suggest that there is still high risk-aversion to market crashes with long-term puts.16 It is interesting to observe that the long call position is also profitable. This result is a function of leverage and that the percentage of options profitable is higher than the short-term counterpart. Why is this the case? Is it mis-pricing due to measurement error in the volatility or do the results actually reflect true market conditions. One potential explanation is due to the portfolio formation, resulting in increased shares in the underlying asset when the options are in the money. As the number of shares bought increases, the long call strategy will exceed the buy and hold position, and thus out-pace the market for a higher return with almost equivalent risk. It is here where the marginal impact of owning options can be observed, and how the use of options enhances returns.

4.4 Volatility Control

As noted earlier, a potential source of error is that the implied volatility could be measured inaccurately, especially for the OTM options. To control for this potential mis-measurement, the implied volatility that was used as input in the Black-Scholes calculation was adjusted in both a positive and negative direction.17 This is demonstrated on two particular strategies, the long-synthetic stock and the short-term covered call. These two strategies were selected to demonstrate the impact the potential measurement error in implied volatility would have on long versus short strategy, and a combination versus a single option strategy. The bounds selected were ±5% for the long-term options, and ±3% for the short-term options. The size of the bounds were selected to over-estimated the potential distortion, and thus bias the results. Changing volatility by 3%-5% results in tremendous price swings

16Typical ﬁndings suggest that OTM puts are expensive, especially short-term options, because of aversion to market crashes. Long-term results are a function of volatility risk. 17There has been extensive research done on the measurement error in inferring implied volatility from Black-Scholes. For speciﬁc references read Jorion (1995), Canina and Figlewski (1993), and Hentschel (2002).

14 in the value of the option, but has, by comparison, minimal impact on the return to the portfolio. Accounting for this possible measurement error highlights two important features of the setup. First, the results are dependent on the constant investment in the strategy, which can overcome large possible mis-pricing. Second, while the use of estimated prices is a drawback, in creating pricing bounds demonstrates the returns in high and low period of volatility. The result for the analysis are shown in table 9. As can be seen, the returns for the synthetic positions are not driven by potential measurement error in volatility. While the results do differ for the different levels of volatility, the changes are small by comparison as compared to the actual changes in the option prices. Regardless of the volatility, the strategy still outperforms the market. The effect on the covered call is greater, since writing the call is a one-sided contract. Lowering the volatility reduces the value of the option, and the overall return to the portfolio. For the ATM strategy, this does reduce the returns below the market, but the results are not consistent with the findings of prior studies. This suggests the volatility bounds are wider than appears realistic, and thus the current results appear feasible.

4.5 Returns through time

It is worth examining the returns to the combination strategies in certain market conditions, since these strategies are designed to take advantage of specific market scenarios. Two sub- periods are examined; the bull run-up from 1996 through April 2000 and the market decline from April 2000 through December 2004. For each sub-period, the portfolio is assumed to start investing in the given strategy at the given point in time with a minimum starting balance of $50,000 to make the returns comparable. The returns shown in table 10 demonstrate the effect of good and bad market conditions on the given strategies for both the monthly and yearly option maturities. In particular, the difference between the bull- and bear-spread is quite striking. In the bull run-up starting in 1996, the bull-spread outperforms the S&P 500 by 5.9%(2.0%), while the bear-spread underperforms by 1.44%(11.89%) for the monthly(yearly) options. After April 2000, the findings are reversed, with the bear-spread outperforming the S&P by 0.51%(2.79%), and

15 the bull-spread underperforming by 2.30%(3.28%). However, while this is not surprising, what is interesting is that choosing the wrong strategy for the given time-period will always result in worse losses than the gains when choosing the correct strategy. This suggests that the combination strategies are too expensive a consistent long-term investment versus just buying and holding the S&P 500.

5 Conclusion

The findings here present alternative conclusions to prior literature. While the evidence presented here suggests that there are profitable scenarios in investing in options, the results do not contradict the results of Figlewski (1989, 1994), Harvey and Whaley (1992), and Santa-Clara and Sarreto (2005). The focus of this work is on the marginal impact of options to a traditional buy and hold portfolio, and not on the return-risk relationship of options alone. The work here highlights the benefit of investing in options. While investing in most option contracts are expensive, and underperforms the market, it is not the case for all contracts. This appears to be a function of two specific aspects. First, and consistent with prior findings, selling put contracts is beneficial due to volatility risk aversion. Second, the use of leverage will enhance portfolio returns without bearing excessive downside exposure conditional on that a small percentage of the overall portfolio is invested in options. The ramifications for the findings suggest that investors can use a portion of their portfolio to enhance returns, as long as it is a small portion. Over-investing can have large negative ramifications, and the consequences can be severe. Take the case of the long synthetic position in 34-year scenario. In this case the percentage of option used was large. This resulted in a negative portfolio position when the market crashed in 1973-1974. By comparison, the 10-year portfolio was never in danger, since the overall investment was small by comparison. The results here can lead to some interesting research in portfolio formation. Since there is opportunity to take advantage of the option markets, it seems worthwhile to investigate what the optimal portfolio composition should be to take advantage of this potential opportunity.

16 Since various trading houses have diﬀerent margin requirements, the results would be highly dependent on what leverage the investor was allowed to take, and whether those restrictions limit any potential gains.

17 References

[1] Bakshi, G., C. Cao, and Z. Chen, 1997. “Empirical Performance of Alternative Option Pricing Models,” Journal of Finance, 52, 2003-2049.

[2] Bakshi, G., and N. Kapadia, N, 2003, “Delta-hedged Gains and the Negative Volatility Risk Premium. Review of Financial Studies, 16, 527-566.

[3] Bates, D., 1996, “Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options,” Review of Financial Studies, 9, 69-107.

[4] Bates, D., 2000, “Post ’87 Crash Fears in S&P 500 Futures Options,” Journal of Econometrics, 94, 181-238

[5] Black, F., and M. Scholes, 1973, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, 81, 637-654.

[6] Canina, L., and S. Figlewski (1993), “The Information Content of Implied Volatility,” Review of Financial Studies, 6, 659-681.

[7] Christensen, B., and N. Prabhala, 1998, “The Relationship between Implied and Realized Volatility,” Journal of Financial Economics, 50, 125-150.

[8] Coval, J., and T.Shumway, 2000, Expected Option Returns. Journal of Fi- nance

[9] Doran, J., 2005, “The inﬂuence of tracking error on volatility risk premium estimation.” Florida State University Working Paper.

[10] Doran, J., and E. Ronn, 2005, “On the Market Price of Volatility Risk,” Florida State University Working Paper.

[11] Duffie, D., J.Pan, and K. Singleton, 2000, “Transform Analysis and Asset Pricing for Affine Jump Diffusions”, Econometrica, 68(6), 1343-1376.

18 [12] Figlewski, S., 1994, “ How to Lose Money in Derivatives.” The Journal of Finance Vol. 2 No. 2.

[13] Figlewski, S., 1989, “Options Arbitrage in Imperfect Markets.” Journal of Finance, Vol 44, No. 5.

[14] Harvey, C., and R. Whaley., 1992, “Market Volatility Prediction and the Eﬃ- ciency of the S&P 100 Index Option Market,” Journal of Financial Economics 31, (1992): 43-73

[15] Heston, S., 1993, “A Closed-Form Solution of Options with Stochastic Volatil- ity with Applications to Bond and Currency Options,” Review of Financial Studies, 6, 327-343.

[16] Hentschel, L., 2002. “Errors in Implied Volatility Estimation”. Journal of Fi- nancial and Quantitative Analysis

[17] Hull, J., White, A., 1987, “The Pricing of Options on Assets with Stochastic Volatilities,” Journal of Finance, 42, 281-300.

[18] Jorion, P., 1995. “Predicting Volatility in the Foreign Exchange Markets,” Journal of Finance, 6, 327-343.

[19] Pan, J., 2000, “The Jump-Risk Premia Implicit in Options: Evidence from an Integrated Time-Series Study”, Journal of Financial Economics, 63(1), 3-50.

[20] Rubinstein, M., 1994, “Implied Binomial Trees,” Journal of Finance, 49, 771- 818.

[21] Santa-Clara, P., and A. Saretto, 2005, “Option Strategies: Good Deals and Margin Calls,” UCLA Working Paper

19 Table 1: Option Construction

Table 1 demonstrates how the option strategies were constructed. A long position in the call(put) is denoted with A C(P ); a short position is denoted as −C(−P ). The ITM and OTM options are at 5% levels, rounded to the closest $0.05 increment. The ATM is constructed in a similar fashion using the current level of the S&P 500. This is done for both monthly and yearly options. The single option strategies (the ﬁrst six in the table) are analyzed at all three levels of moneyness. The combination strategies (the next ﬁve, including the three synthetic positions) use the options as listed to form the positions.

Strategy ITM ATM OTM

Call C C C Covered Call -C -C -C Naked Call -C -C -C Put P P P Protective Put -P -P -P Naked Put -P -P -P

Bear Spread P -P Bull Spread C -C Butterﬂy C -2C C Straddle CP Strangle C -P

Synthetic Stock C -P Synthetic Stock C -P Synthetic Stock C -P

20 Table 2: Bid-Ask Spreads

Table 2 shows the average percentage bid-ask spread for both calls and puts for short- and long- term options. The average was calculated over the period from 1996-2002 for all options ATM, OTM (5%), and ITM (5%) on the S&P 500.

Month Year Strategy OTM ATM ITM OTM ATM ITM

PUT 7.27% 4.69% 3.97% 9.10% 6.88% 5.29% CALL 9.72% 4.60% 3.62% 7.17% 4.84% 3.24%

21 Table 3: Percentage of Options Exercised and Proﬁtable

Table 3 reports the percentage of options exercised (option finishes in the money) and profitability (option finishes in the money plus the option premium) for the various options strategies. The table shows the percentages for monthly and yearly options for the 34 year and 10 year horizons. For each case the strategy is examined for degrees of moneyness, where an ITM/OTM option has a strike price that 5% less/more than the current level of the index. The returns to the monthly options are calculated using an option that is bought at the beginning of the month, and expires on the third Friday of the month. The yearly options are calculated using an option that is bought at the beginning of the month, held for a year, and expires on the third Friday of that option’s month.

34 Years 10 Years Month Year Month Year Option Type Exerc. Profit Exerc. Profit Exerc. Profit Exerc. Profit

CAT M 59.8% 36.0% 72.5% 51.0% 61.7% 41.7% 71.3% 62.0% CITM 96.3% 53.4% 79.0% 54.0% 95.8% 60.0% 73.1% 63.9% COTM 7.6% 6.4% 64.1% 43.9% 7.5% 5.0% 68.5% 58.3% PAT M 40.2% 27.0% 27.5% 21.7% 38.3% 25.8% 28.7% 25.9% PITM 92.4% 42.6% 35.9% 28.0% 92.5% 37.5% 31.5% 26.9% POTM 3.7% 3.2% 21.0% 16.4% 4.2% 3.3% 26.9% 22.2% Straddle 100.0% 33.6% 100.0% 44.9% 100.0% 31.7% 100.0% 60.2% Strangle 11.3% 7.6% 85.1% 42.7% 11.7% 6.7% 95.4% 60.2% Butterﬂy 90.7% 58.6% 15.6% 13.6% 90.8% 56.7% 4.6% 4.6% Bull-Spread 59.8% 38.7% 72.1% 66.2% 61.7% 44.2% 71.3% 69.4% Bear-Spread 40.2% 29.4% 27.9% 25.3% 38.3% 29.2% 28.7% 27.8%

22 Table 4: Return to Traditional Option Portfolios

Table 4 reports the returns over 10 and 34 years to four simple options strategies; long a call (C), long a put (P), short a call (NC), and short a put (NP). For each case the strategy is examined for degrees of moneyness, where an ITM/OTM option has a strike price that 5% less/more than the current level of the index. The returns to the monthly options are calculated using an option that is bought at the beginning of the month, and expires on the third Friday of the month. The yearly options are calculated using an option that is bought at the beginning of the month, held for a year, and expires on the third Friday of that option’s month. The returns are calculated assuming no i i transaction costs (RBS,t), incorporating bid-ask spreads (RBA,t), and incorporating both bid-ask i spreads, commissions, and taxes (RBAC,t). For comparison, the S&P 500 had an annualized return of 9.42% over the 34 year period and a 6.9% return over the 10 year period.

Panel A: 34 year horizon

Month Year i i i i i i Strategy RBS,t RBA,t RBSC,t RBS,t RBA,t RBSC,t CAT M 8.8% 8.5% 7.6% 8.7% 8.5% 7.7% CITM 9.9% 9.6% 8.3% 8.7% 8.3% 7.4% COTM 5.1% 5.1% 4.6% 9.0% 8.6% 7.7% PAT M 7.4% 7.0% 6.2% 10.3% 9.7% 8.9% PITM 9.4% 9.1% 7.8% 11.9% 11.4% 10.2% POTM 2.2% 0.8% 0.7% 7.2% 6.8% 6.0% NCAT M 17.2% 15.1% 12.6% 8.4% 7.7% 6.4% NCITM 10.5% 7.4% 5.3% 9.2% 8.6% 7.0% NCOTM 15.4% 14.9% 13.5% 8.1% 7.5% 6.4% NPAT M 23.6% 22.0% 21.0% 11.5% 11.2% 10.0% NPITM 20.9% 16.9% 15.2% 10.6% 10.1% 8.9% NPOTM 18.2% 17.7% 17.1% 13.4% 12.9% 11.5% Panel B: 10 year horizon

Month Year i i i i i i Strategy RBS,t RBA,t RBSC,t RBS,t RBA,t RBSC,t CAT M 6.4% 5.9% 5.5% 10.9% 10.4% 9.4% CITM 7.1% 6.7% 6.4% 8.7% 8.7% 8.2% COTM -1.0% -1.0% -1.1% 12.8% 13.1% 11.6% PAT M 3.8% 3.4% 3.3% 4.8% 4.5% 4.7% PITM 5.7% 5.4% 5.1% 5.1% 4.6% 4.5% POTM 0.6% 0.4% 0.4% 4.5% 4.4% 3.7% NCAT M 10.5% 9.7% 8.8% 3.7% 3.3% 2.7% NCITM 8.2% 6.4% 5.2% 4.1% 4.0% 3.3% NCOTM 11.0% 10.7% 10.2% 3.9% 3.6% 2.9% NPAT M 12.4% 11.6% 10.6% 5.5% 5.0% 5.3% NPITM 11.7% 9.6% 8.3% 4.6% 4.6% 4.1% NPOTM 10.6% 10.4% 9.7% 6.4% 6.7% 6.0%

23 Table 5: Return to Covered Calls and Protective Puts

Table 5 reports the returns over 34 and 10 years to covered calls (CC) and protective put (PP). For each case the strategy is examined for degrees of moneyness, where an ITM/OTM option has a strike price that 5% less/more than the current level of the index. The returns to the monthly options are calculated using an option that is bought at the beginning of the month, and expires on the third Friday of the month. The yearly options are calculated using an option that is bought at the beginning of the month, held for a year, and expires on the third Friday of that option’s i month. The returns are calculated assuming no transaction costs (RBS,t), incorporating bid-ask i i spreads (RBA,t), and incorporating both bid-ask spreads, commissions, and taxes (RBAC,t). For comparison, the S&P 500 had an annualized return of 9.42% over the 34 year period and a 6.9% return over the 10 year period.

Panel A: 34 year horizon