Is There Money to be Made Investing in Options? AHistorical Perspective

James S. DoranDepartment of FinanceFlorida State University

and

Robert HamernickFlorida State University

Revised: January 4, 2006

JEL : G11, G12, G13Keywords : Portfolio Returns, Option Strategies, Option Pricing

Sharpe Ratios, S&P 500

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

CommunicationsAuthor: James S. Doran

Address: Department of FinanceCollege of BusinessFlorida State UniversityTallahassee, FL. 32306

Tel.: (850) 644-7868 (Office)FAX: (850) 644-4225 (Office)

E-mail: jsdoran@cob.fsu.edu

Abstract

This paper examines the historical time-series performance of twelve trading strategiesinvolving options on the S&P 500 Index. Each option strategy is examined over differentmaturities and moneyness, incorporating transaction costs and margin requirements. Theanalysis was conducted by assuming monthly trading starting in 1970 and 1995, comparinghistorical performance via returns and Sharpe ratios as compared to the buy and hold returnson 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 thatinvesting in options as an enhancement to a buy and hold portfolio can result in returns andSharpe ratios that exceed the S&P 500. However, the performance is conditional on thatthe investments made in options are small as compared to the portfolio as a whole.

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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 suchas Hull and White (1987) and Heston (1993); the inclusion of jumps, Bates (1996), and double jumpmodels of the type in Duffie, Pan, and Singleton (2000). Empirical Work such as Bakshi, Cao, andChen (1997), have examined the fit of these models to option pricing data, as well as examine thehedging implications from such prices.

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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,

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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 combina-

tions. 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.

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Figure 1: Investment Timeline

t=1 Monthly Investment: α1

Buy & Hold: Portfolio Value: X0S1 + (α1+R0)/S1 =X1 shares + R1 =(X0000+X1111)S1111+ R1111 Option Position: Portfolio Value: (Y0P1+R0)/S1 = Z1 shares + (α1+R0)/C1 = Y1 options + R1 =Z1111S1111+Y1111+ R1111

Buy & Hold: Portfolio Value: (X0+X1)S2 + (α2+R1)/S2 =X2 shares + R2 =(X0000+X1111+X2222)S2222+ R2222 Option Position: Portfolio Value: Z1S2 + (Y1P2+R1)/S2 = Z2 shares + (α2+R1)/C2 = Y2 options + R2 =(Z1111+Z2222)S2222+Y2222+ R2222

Invest α0 in either C0 or S0 Initial Portfolio Position 1. Buy & Hold: α0/S0 =X0 shares + R0 in risk-free rate 2. Option Position: α0/C0 =Y0 options + R0 in risk-free rate

t=2 Monthly Investment: α

t=0 Monthly Investment: α0

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.

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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 install-

ments, 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.

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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

V cBS,ψ = StN(d1)− e−rt ∗KjN(d1 − σiv

√(ψ)) (1)

V pBS,ψ = −StN(−d1) + e−rt ∗KjN(−(d1 − σiv

√(ψ))) (2)

with (3)

d1 =ln(Ft/X) +

σ2iv

2ψ

σiv√ψ

(4)

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 priceversus the actual option price. This methodology was implemented for two reasons. First, sinceit was necessary to generate a simulated option price for earlier periods, we wished to remainconsistent for later periods. Second, the simulated option values generated very similar results toactual prices, but added the flexibility of allowing for changes is underlying volatility to demonstratethe effect small miscalculations have on the portfolio returns.

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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 histor-

ical 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 volatil-

ity 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 + εt (5)

σrv,t = α0 + αiσpiv,t + εt (6)

where σrv,t is a 22-day realized volatility,6 σciv,t is the implied volatility from a call option,

and σpiv,t is the implied volatility from a put option.7 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

ˆσiv,t =σrv,t − α0

αi(7)

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 variable7Results and available upon request.8See Bakshi, Cao, and Chen (1997) for evidence.

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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 condi-

tions. 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 thenadding in average differences in the volatility skew. The results yielded little difference in overallportfolio returns.

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price, V iBS,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,

such that a 1-year, 5% OTM put that costs $44.20 at V iBS,t, will cost $46.20 at the bid. Each

bid and ask price was calculated as such,

V ibid,t = V i

BS,t(1 +BA(i, ψ)

2) (8)

V iask,t = V i

BS,t(1−BA(i, ψ)

2) (9)

for each moneyness and maturity combination adjusting the price to the nearest $0.05 incre-

ment. 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 sce-

nario, 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

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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 sig-

nificantly 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 op-

tion 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 constraintson options, and the resulting impact on returns through increased leverage.

12The ability to sell naked positions is dependent on the trading platform used. Some brokeragesrequire that the short position is completely covered with cash, others allow leverage up to andexceeding 100% of portfolio value.

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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,

Ri =

(ΩiT − Ci

Ci

)360

days(10)

where ΩiT is the portfolio value at the end of the holding period, Ci 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 re-

sults 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 Associatesexamined the returns to BXM in the paper “Passive Options-based Investments Strategies: TheCase 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. Theresults are comparable to the 10-year ATM Covered Call strategy with transaction costs.

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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.

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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, theportfolio 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, becauseof 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 volatilityfrom Black-Scholes. For specific references read Jorion (1995), Canina and Figlewski (1993), andHentschel (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 mea-

surement 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 volatil-

ity 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 portfo-

lio 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 posi-

tion 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 areat 5% levels, rounded to the closest $0.05 increment. The ATM is constructed in a similar fashionusing the current level of the S&P 500. This is done for both monthly and yearly options. Thesingle 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 optionsas listed to form the positions.

Strategy ITM ATM OTM

Call C C CCovered Call -C -C -C

Naked Call -C -C -CPut P P P

Protective Put -P -P -PNaked Put -P -P -P

Bear Spread P -PBull Spread C -C

Butterfly C -2C CStraddle C PStrangle C -P

Synthetic Stock C -PSynthetic Stock C -PSynthetic 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 YearStrategy 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. Thetable 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 hasa strike price that 5% less/more than the current level of the index. The returns to the monthlyoptions are calculated using an option that is bought at the beginning of the month, and expireson the third Friday of the month. The yearly options are calculated using an option that is boughtat the beginning of the month, held for a year, and expires on the third Friday of that option’smonth.

34 Years 10 YearsMonth Year Month Year

Option Type Exerc. Profit Exerc. Profit Exerc. Profit Exerc. Profit

CATM 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%PATM 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 fordegrees of moneyness, where an ITM/OTM option has a strike price that 5% less/more than thecurrent level of the index. The returns to the monthly options are calculated using an option thatis bought at the beginning of the month, and expires on the third Friday of the month. The yearlyoptions 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 notransaction costs (Ri

BS,t), incorporating bid-ask spreads (RiBA,t), and incorporating both bid-ask

spreads, commissions, and taxes (RiBAC,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 YearStrategy Ri

BS,t RiBA,t Ri

BSC,t RiBS,t Ri

BA,t RiBSC,t

CATM 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%PATM 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%

NCATM 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%NPATM 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 YearStrategy Ri

BS,t RiBA,t Ri

BSC,t RiBS,t Ri

BA,t RiBSC,t

CATM 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%PATM 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%

NCATM 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%NPATM 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 hasa strike price that 5% less/more than the current level of the index. The returns to the monthlyoptions are calculated using an option that is bought at the beginning of the month, and expireson the third Friday of the month. The yearly options are calculated using an option that is boughtat the beginning of the month, held for a year, and expires on the third Friday of that option’smonth. The returns are calculated assuming no transaction costs (Ri

BS,t), incorporating bid-askspreads (Ri

BA,t), and incorporating both bid-ask spreads, commissions, and taxes (RiBAC,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 YearStrategy Ri

BS,t RiBA,t Ri

BSC,t RiBS,t Ri

BA,t RiBSC,t

CCATM 17.0% 15.0% 11.9% N/A N/A N/ACCITM 10.6% 7.7% 5.1% N/A N/A N/ACCOTM 15.3% 14.9% 13.0% N/A N/A N/APPATM 6.8% 6.5% 5.6% 4.8% 4.8% 3.8%PPITM 8.5% 7.6% 6.5% 5.9% 5.8% 4.8%PPOTM 6.4% 5.9% 5.1% 3.6% 3.3% 2.6%

Panel B: 10 year horizon

Month YearStrategy Ri

BS,t RiBA,t Ri

BSC,t RiBS,t Ri

BA,t RiBSC,t

CCATM 10.6% 9.8% 8.6% N/A N/A N/ACCITM 8.4% 6.8% 5.1% N/A N/A N/ACCOTM 10.8% 10.6% 10.0% N/A N/A N/APPATM 8.4% 8.3% 7.7% 2.5% 2.2% 1.7%PPITM 7.2% 6.9% 6.3% 1.7% 1.0% 0.5%PPOTM 5.3% 5.2% 4.8% 2.8% 2.6% 2.0%

24

Table 6: Return to Combinations Strategies

Table 6 reports the returns over 34 and 10 years to 5 combination strategies; straddles, strangles,butterfly, bull spread, and a bear-spread. Each strategy centered around the current level of theindex, such that the straddle buys an ATM put and call, while the strangle buys an OTM put andcall. In each case the ITM/OTM option has a strike price that 5% less/more than the current levelof the index. The returns to the monthly options are calculated using an option that is boughtat the beginning of the month, and expires on the third Friday of the month. The yearly optionsare 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 notransaction costs (Ri

BS,t), incorporating bid-ask spreads (RiBA,t), and incorporating both bid-ask

spreads, commissions, and taxes (RiBAC,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 YearStrategy Ri

BS,t RiBA,t Ri

BSC,t RiBS,t Ri

BA,t RiBSC,t

Straddle 3.0% 2.8% 2.2% 5.5% 5.2% 4.5%Strangle 0.3% 0.3% 0.1% 5.3% 4.9% 4.4%

Butterfly 11.3% 9.2% 7.5% 7.4% 0.4% -0.1%Bull-Spread 9.2% 9.1% 8.2% 8.4% 6.3% 5.9%Bear-Spread 8.7% 8.5% 7.8% 22.2% 14.0% 12.6%

Panel B: 10 year horizon

Month YearStrategy Ri

BS,t RiBA,t Ri

BSC,t RiBS,t Ri

BA,t RiBSC,t

Straddle 1.5% 1.4% 1.0% 6.4% 5.8% 5.1%Strangle -1.6% -1.6% -1.6% 7.4% 7.2% 6.5%

Butterfly 7.6% 5.5% 5.1% -0.2% -0.9% -0.7%Bull-Spread 7.1% 6.8% 6.9% 7.2% 5.4% 5.4%Bear-Spread 4.4% 3.8% 4.2% 7.2% 4.2% 4.0%

25

Table 7: Return to Synthetic Stock

Table 7 reports the returns over 34 and 10 years to a synthetic stock strategy (SYS), where theposition is created by going long a call and short a put. Three moneyness combinations areexamined; ATM the calls and puts, ATM call with an OTM put, and OTM call with an OTMput. In each case the ITM/OTM option has a strike price that 5% less/more than the current levelof the index. The returns to the monthly options are calculated using an option that is boughtat the beginning of the month, and expires on the third Friday of the month. The yearly optionsare 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 notransaction costs (Ri

BS,t), incorporating bid-ask spreads (RiBA,t), and incorporating both bid-ask

spreads, commissions, and taxes (RiBAC,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 YearStrategy Ri

BS,t RiBA,t Ri

BSC,t RiBS,t Ri

BA,t RiBSC,t

SYS 12.9% 11.3% 9.0% 11.3% 9.8% 9.3%SYS (CATM , POTM ) 10.4% 10.0% 8.4% 10.5% 9.3% 8.5%SYS (COTM , POTM ) 11.4% 10.1% 8.6% 20.9% 14.7% 14.0%

Panel B: 10 year horizon

Month YearStrategy Ri

BS,t RiBA,t Ri

BSC,t RiBS,t Ri

BA,t RiBSC,t

SYS 7.4% 6.2% 5.3% 18.5% 17.4% 15.3%SYS (CATM , POTM ) 7.7% 7.1% 6.2% 15.1% 14.1% 14.6%SYS (COTM , POTM ) 6.5% 6.2% 5.6% 20.6% 20.8% 19.6%

26

Table 8: Sharpe Ratios

Table 8 reports the returns, risk, and the Sharpe ratio calculated from the cash flows for all strategiesfor both the monthly and yearly time horizons. The table ranks each strategy by Sharpe ratio forboth the month and year options. The ratios were calculated using the 10-year holding period fromJanuary 1995 through December 2004

Month YearStrategy Return Risk Sharpe Strategy Return Risk Sharpe

PPITM 9.46% 3.57% 1.475 SY S(CO, PO) 16.92% 21.65% 0.587PPATM 9.80% 6.09% 0.920 SY S(CA, PO) 14.48% 17.77% 0.578NCOTM 11.62% 15.39% 0.482 COTM 13.15% 16.75% 0.534CCOTM 11.54% 15.40% 0.476 SYS 14.73% 19.77% 0.532NPOTM 11.49% 18.00% 0.405 CATM 11.51% 16.43% 0.445CCATM 10.02% 16.27% 0.358 CITM 10.54% 16.22% 0.391NCATM 10.12% 16.76% 0.353 S&P 500 9.66% 15.72% 0.347NPATM 12.45% 23.65% 0.349 Strangle 9.51% 16.25% 0.327

S&P 500 9.66% 15.72% 0.347 NPOTM 9.19% 16.97% 0.294Bull-Spread 9.87% 16.96% 0.334 NPATM 8.70% 17.62% 0.255

PPOTM 7.65% 11.50% 0.300 Straddle 8.22% 16.08% 0.250CITM 9.18% 16.82% 0.296 Bull-Spread 7.90% 15.76% 0.235

SY S(CA, PO) 9.40% 19.60% 0.265 NPITM 7.87% 19.29% 0.190CATM 8.67% 17.37% 0.257 PATM 6.00% 15.10% 0.119

Butterfly 8.08% 15.54% 0.250 PITM 5.84% 15.21% 0.108PITM 7.75% 14.34% 0.248 Bear-Spread 5.53% 15.48% 0.086

SY S(CO, PO) 8.55% 18.36% 0.237 POTM 5.15% 15.01% 0.064SYS 9.08% 25.35% 0.193 NCITM 5.58% 22.26% 0.062

NPITM 11.28% 36.90% 0.192 NCOTM 4.86% 19.77% 0.033Bear-Spread 6.59% 14.11% 0.169 NCATM 4.87% 20.71% 0.032

PATM 5.60% 13.60% 0.103 PPOTM 2.94% 6.02% (0.210)NCITM 6.54% 34.10% 0.068 Butterfly 0.44% 15.69% (0.240)CCITM 6.23% 31.49% 0.064 PPATM 1.99% 6.14% (0.360)

Straddle 3.88% 16.71% -0.019 PPITM -1.81% 7.77% (0.774)POTM 2.10% 14.26% -0.147 CCITM n/a n/a n/aCOTM 0.17% 16.31% -0.247 CCOTM n/a n/a n/a

Strangle -0.77% 15.93% -0.312 CCATM n/a n/a n/a

27

Table 9: Volatility Sensativity

Table 9 reports the returns to two long-synthetic stock positions and two short-term covered callgiven three levels of volatility. The initial input for implied volatility was adjusted by 5% in eitherdirection for the long maturity strategy, and 3% for the short maturity strategy. This was done inparticular to show the sensitivity of the portfolio returns to measurement error in implied volatility.

Panel A: Long-term Synthetic Stock

Strategy +5% 0% -5%

SYS 14.7% 15.3% 18.7%SY S(CO, PO) 18.4% 19.6% 22.6%

Panel B: Short-term Covered Call

Strategy +3% 0% -3%

CCATM 13.5% 8.6% 4.9%CCOTM 12.5% 10.0% 8.2%

28

Table 10: Sub-Period Estimation

Table 10 reports the returns to various option strategies over the two sub-periods; the Bull marketof January 1996 through April 2000, and the Bear market of April 2000 through December 2004.The period of January 2004 through December 2004 was included since long-term options werebought and allowed to expire.

Panel A: Month

Strategy Bull Market Bear MarketRFR 5.4% 3.6%

Straddle 11.97% -10.08%Strangle 5.97% -12.15%

Butterfly 18.73% -2.96%Bull-Spread 23.53% -3.26%Bear-Spread 16.19% -0.35%

S&P 500 17.63% -0.96%

Panel B: Year

Strategy Bull Market Bear MarketRFR 5.4% 3.6%

Straddle 20.80% -7.78%Strangle 23.51% -7.97%

Butterfly 5.78% -9.14%Bull-Spread 19.63% -4.24%Bear-Spread 5.74% 3.75%

S&P 500 17.63% -0.96%

29

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2004

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32