Data and Methodology
The option price data is provided by Optionmetrics and covers the period from the first expiration after the introduction of QQQ options on March 19, 1999 to May 31, 2009. The QQQ, NDX, Treasury bill and VIX data is provided by Datastream4, while mutual fund data is provided by Morningstar. Business cycle announcement data are provided by the National Bureau of Economic Research.
4 NDX, VIX and Initial Unemployment Claims data is collected from March 1998 to ensure sufficient lag time for signal generation.
In order to assess the performance of active and passive collar strategies, we construct indices which represent the return streams generated by such strategies. The passive strategies follow a fixed set of option selection rules defining the initial moneyness and time to expiration 6
of the calls and puts, regardless of market conditions. In contrast, the active5 collar strategies base their option selection rules on a combination of three simple market/economic based signals (momentum, volatility, and a macroeconomic factor) and thus adjust to various market conditions.
5 It should be noted that while we use the term active to represent these strategies, they are not truly actively managed. They still follow an established set of selection rules, but the rules include a dynamic element conditioned on economic variables.
6 This is the Friday prior to the first expiration Saturday following the introduction of QQQ options.
Passive Collar Strategy: We generate a daily time series of returns for each of the passive strategies beginning on March 19, 19996. At the close on this day a 1-month call is written and a 1, 3 or 6-month put is purchased. Depending on the particular passive implementation, the initial moneyness of the calls and puts are set at either 5%, 4%, 3%, 2%, 1% OTM or ATM. At the close on the Friday prior to the following expiration, we take one of two actions: If 1-month puts are used, the puts and calls are settled at intrinsic value and we roll into new 1-month puts and calls with the specified moneyness. If a 3- or 6-month put is used, the calls are settled at intrinsic value and new 1-month calls with the specified moneyness are rolled into, while the longer term put is held for another month. When the new 1-month calls are written, the net proceeds from the sale of the calls and the expiration of the previous calls are fully invested in the strategy and the position is rebalanced to ensure a 1:1:1 ratio of the underlying, puts and calls. Once the 3- or 6-month put expires, it is settled at intrinsic value and we once again roll into new puts and calls with the specified moneyness and time to expiration. In order to include the impact of transaction costs, the puts are purchased at the ask price and the calls are written at the bid price when each new put or call position is established. Each trading day in between roll dates, the options are priced at the mid-point between the bid and ask prices. In this manner, daily returns 7
are calculated for each passive strategy implementation. The following example illustrates this process:
Passive 2% OTM 1-Month Call 6-Month Put Implementation
Date Exdate Quantity Wealth Roll In 3/ 19/ 1999 Purchase QQQ 1.000 @ 102.44 $ 108.69 $ (Initial) Purchase a 6-month 2% OTM put expiring on: 9/17/1999 (Strike price = 100) 1.000 @ 9.50 $ (at ask) Sell a 1-month 2% OTM call expiring on: 4/16/1999 (Strike price = 104) 1.000 @ (3.25) $ (at bid) Roll Out 4/ 16/ 1999 QQQ value 1.000 @ 103.94 $ 112.37 $ Keep the put (now 5-month 4% OTM) expiring on: 9/17/1999 (Strike price = 100) 1.000 @ 8.44 $ (mid of bid/ask) Payout value of previous call at expiration: 4/16/1999 (Strike price = 104) 1.000 @ - $ (intrinsic value) Roll In 4/ 16/ 1999 Purchase QQQ 1.037 @ 103.94 $ 112.37 $ Keep the put (now 5-month 4% OTM) expiring on: 9/17/1999 (Strike price = 100) 1.037 @ 8.44 $ (mid of bid/ask) Sell a 1-month 2% OTM call expiring on: 5/22/1999 (Strike price = 106) 1.037 @ (4.00) $ (at bid) Repeat until put expires Roll Out 9/ 17/ 1999 QQQ value 1.045 @ 126.63 $ 123.31 $ Payout value of the put at expiration: 9/17/1999 (Strike price = 100) 1.045 @ - $ (intrinsic value) Payout value of the call at expiration: 9/17/1999 (Strike price = 118) 1.045 @ (8.63) $ (intrinsic value) Roll In 9/ 17/ 1999 Purchase QQQ 0.924 @ 126.63 $ 123.31 $ Purchase a 6-month 2% OTM put expiring on: 3/18/2000 (Strike price = 124) 0.924 @ 10.25 $ (at ask) Sell a 1-month 2% OTM call expiring on: 10/16/1999 (Strike price = 129) 0.924 @ (3.38) $ (at bid)
Active Strategy Market Signals
For the active implementations, a series of three market signals determine the choice of initial call and put moneyness, as well as the ratio of the number of calls written to the number of puts and QQQ shares purchased, while the time to expiration is fixed at one month for the calls and 6 months for the puts.
Active Collar Adjustment Strategy: Three different sets of active market signals are used for the strategy implementations, differing by their time horizon; short, medium and long-term. The three signals are based on momentum, volatility and a compound macroeconomic indicator 8
(unemployment claims and business cycle), respectively. In order to ensure that the strategies are investable, all signals use contemporaneously lagged data7.
7 The signals are designed so that they are based only on data which existed prior to the date on which the signal would have been generated in practice. For example, a signal for the March 19, 1999 option roll-in date would only use data which existed on March 18, 1999 or earlier.
8 The use of the NDX rather than the QQQ provides us with historical data beyond the introduction of the QQQ. In this way, we can generate signals from the beginning of the QQQ data series.
9 Additional evidence of the existence of momentum and potential explanations for its existence can be found in Jegadeesh and Titman  and Schneeweis, Kazemi and Spurgin .
10 In this paper they do not take short positions. They use the signals as in/out position indicators.
Momentum Signal: The momentum signal is a simple moving average cross-over (SMACO) of the NASDAQ-100 index (NDX)8. A SMACO compares a short-term moving average (SMA) and a long-term moving average (LMA) to determine whether an upward or downward trend exists. The rule is defined by the number of days covered by each of the moving averages. For example, a 5/150 SMACO rule compares a 5 day SMA with a 150 day LMA. If the SMA is greater (less) than the LMA, then an upward (downward) trend indicated, suggesting a buy (sell) signal. Our choice of signals is based on Szakmary, Davidson, Schwarz  and Lento 9, which both consider 1/50, 1/150, 5/150, 1/200 and 2/200 SMACO rules on the NDX. Szakmary et al apply NASDAQ index SMACOs as buy/sell signals for individual stocks for the period from 1973 to 199110. They find some significant excess returns, although their significance does not survive transactions costs. Similarly, Lento finds some significant forecasting abilities in the same SMACO rules on the NASDAQ at a 10-day lag over the period of 1995 to 2008. Following their methodology, we use 1/50, 5/150, and 1/200 SMACO rules on the NDX. This provides us with a short, medium and long-term momentum signal. Each roll date, we calculate the SMA and LMA for each of the three momentum rules and use them to generate the momentum signals. All else equal, if the calculation results in a buy signal, the 9
collar would widen (increasing upside participation with a corresponding reduction in downside protection). In contrast, all else equal, the collar would be tightened in response to sell signal (increasing downside protection while reducing upside participation).
The following example illustrates the process for the momentum signal calculation:
Momentum Signal Calculation for the 3/19/1999 Roll Date
1 Day SMA 5 Day SMA 50 Day SMA 150 Day SMA 200 Day SMA 2102.77 2061.98 1998.76 1629.73 1554.89 Short Term Momentum Signal Calculation: 1 Day SMA = 2102.77 > 50 Day SMA = 1998.76 Medium Term Momentum Signal Calculation: 5 Day SMA = 2061.98 > 150 Day SMA = 1629.73 Long Term Momentum Signal Calculation: 1 Day SMA = 2102.77 > 200 Day SMA = 1554.89 LONG NDX Momentum Signal MEDIUM NDX Momentum Signal SHORT NDX Momentum Signal +1 +1 +1 Note: All moving averages using data up to the prior day's close (e.g. 3/18/1999) Since the 1 day SMA is greater than the 50 day SMA, the NDX is trending upwards. This is a bullish signal, so the momentum signal = +1. Holding the macroeconomic signal constant, this would widen the collar (move the put 1% further OTM and the call 1% further OTM). Since the 5 day SMA is greater than the 50 day SMA, the NDX is trending upwards. This is a bullish signal, so the momentum signal = +1. Holding the macroeconomic signal constant, this would widen the collar (move the put 1% further OTM and the call 1% further OTM). Since the 1 day SMA is greater than the 200 day SMA, the NDX is trending upwards. This is a bullish signal, so the momentum signal = +1. Holding the macroeconomic signal constant, this would widen the collar (move the put 1% further OTM and the call 1% further OTM).
Volatility Signal: The volatility signal is based on Renicker and Mallick . Renicker and Mallick create an "enhanced" S&P 500 buy-write strategy and back test it over the period from 1997 to September 2005.11 They find excess returns to a strategy which writes 0.75 calls to each long index position when the markets short-term anxiety level is high (as indicated by a situation
11 Note that since the Renicker and Mallick study reported results based on the period used in this study, the use of this variable is not independent from the period used to analze its impact on the collar strategy. 10
in which the 1-month ATM S&P 500 implied volatility is more than 1 standard deviation above its current 250-day moving average level), and writes 1.25 calls per index position when the anxiety level is low (when the 1-month implied volatility is more than 1 standard deviation below the 250-day average level)12. Their goal in varying the quantity of written calls is to have a longer exposure to the market in times of high anxiety and shorter exposure in times of complacency. We make two minor modifications to their strategy. First, we use the daily VIX close as an indicator of implied volatility levels. Second, we consider a short, medium and long-term time frame to generate the 3 corresponding signals. In order to match the time frames of our momentum signals our short, medium and long-term volatility signals use 50, 150 and 250-day windows respectively. In keeping with the methodology of Renicker and Mallick, on roll dates we sell 0.75 (1.25) calls per index position when the previous day’s VIX close is more than 1 standard deviation above (below) its current moving average level, otherwise we sell 1 call per index position as illustrated by the following formula:
12 When the 1-month implied volatility level is within the 1 standard deviation bounds, they follow a standard 1:1 ratio buy-write.
# of Calls Written per Long Put and Long QQQ Position = 1 + (0.25 * Volatility Signal),
where the volatility signal is -1, 0 or +1. 11
Volatility Signal Calculation for the 3/19/1999 Roll Date
Spot VIX VIX 250-Day Standard Deviation VIX 250-Day Moving Average VIX 150-Day Standard Deviation VIX 150-Day Moving Average VIX 50-Day Standard Deviation VIX 50-Day Moving Average 24.3 6.5 27.0 5.9 30.4 2.5 27.8 Short Term Volatility 1-Standard Deviation Range Calculation: 1-Standard Deviation Range = VIX 50-Day Moving Average +/- VIX 50-Day Standard Deviation = 27.8 - 2.5 to 27.8 + 2.5 = 25.3 to 30.3 Medium Term Volatility 1-Standard Deviation Range Calculation: 1-Standard Deviation Range = VIX 150-Day Moving Average +/- VIX 150-Day Standard Deviation = 30.4 - 5.9 to 30.4 + 5.9 = 24.6 to 36.3 Long Term Volatility 1-Standard Deviation Range Calculation: 1-Standard Deviation Range = VIX 250-Day Moving Average +/- VIX 250-Day Standard Deviation = 27.0 - 6.5 to 27.0 + 6.5 = 20.5 to 33.5 VIX 50-Day 1 Std. Dev. Range VIX 150-Day 1 Std. Dev. Range VIX 250-Day 1 Std. Dev. Range 25.3 to 30.3 24.6 to 36.3 20.5 to 33.5 Short Term Momentum Signal Calculation: Medium Term Momentum Signal Calculation: Long Term Momentum Signal Calculation: LONG Volatility Signal MEDIUM Volatility Signal SHORT Volatility Signal 0 +1 +1 Note: Spot VIX level and all calculations use data up to the prior day's close (e.g. 3/18/1999) Spot VIX = 24.3 < Lower Bound of the 50-Day 1-Standard Deviation Range = 25.3 Spot VIX = 24.3 < Lower Bound of the 150-Day 1-Standard Deviation Range = 24.6 Spot VIX = 24.3 is Between the Lower Bound of the 250-Day 1- Standard Deviation Range = 20.5 and the the Upper Bound of the 250-Day 1-Standard Deviation Range = 33.5 Spot VIX is below the lower bound of the 1-standard deviation range around the 50-day moving average of VIX. This indicates a low level of anxiety, suggesting that we should sell more calls. This a bearish signal, so the volatility signal = +1. This signal would result in selling 1.25 calls for each long put and long QQQ position. Spot VIX is below the lower bound of the 1-standard deviation range around the 150-day moving average of VIX. This indicates a low level of anxiety, suggesting that we should sell more calls. This a bearish signal, so the volatility signal = +1. This signal would result in selling 1.25 calls for each long put and long QQQ position. Spot VIX is between the lower bound and the upper bound of the 1-standard deviation range around the 250-day moving average of VIX. This indicates a medium level of anxiety, suggesting that we should sell the standard number of calls. This a neutral signal, so the volatility signal = 0. This signal would result in selling 1 call for each long put and long QQQ position.
It is worth noting that the volatility signal only affects the call writing portion of the strategy, puts are always purchased at a 1:1 ratio with the index13.
13 While we could apply these signals to both the put and call positions, we chose to apply them only to the call writing to be consistent with Renicker and Mallick. 12
Macroeconomic Signal: The final variable used in the active collar adjustment strategy signal process is based on the trend of initial unemployment claims and the state of the economy with respect to the business cycle. Boyd, Hu and Jagannathan  consider the impact of unemployment rate surprise on the stock market in the period from 1973 to 2000. They find that in expansionary periods, stocks typically rise on bad unemployment news, while the opposite relationship holds in contractionary periods14. This is consistent with Veronesi  which suggests that bad news in expansionary periods and good news in contractionary periods are typically correlated with an increase in uncertainty and an increase in the equity risk premium (corresponding to an increase in expected returns and reduction in current prices). We use these findings to construct a signal based on initial unemployment claims. The announcements from the NBER’s Business Cycle Dating Committee are used to identify the state of the business cycle. It is worth noting that NBER does not define a recession as two consecutive quarters of negative GDP growth. They define it as follows: "A recession is a significant decline in economic activity spread across the economy, lasting more than a few months, normally visible in real GDP, real income, employment, industrial production and wholesale-retail sales"15. These announcements are generally considered the authority on the current state of the business cycle. Since there is often a significant delay in announcement dates, we base the signals on announcement dates to avoid hindsight biases. For example, the December 2007 peak was
14 These results are somewhat counter-intuitive in the case of expansionary economies. One might expect rising unemployment to negatively affect stock prices regardless of the business cycle, but the literature cited above suggests that rising unemployment in expansionary economies causes expected future interest rates to decline, increasing the value of equities, while rising unemployment in contractions indicates slower future earnings growth rates, reducing the value of equities.
15 See http://www.nber.org/cycles.html 13
announced about one year later on December 1, 2008. Our signal would be based on an expansionary economy until December 1, 2008. Since initial unemployment claims are released on a weekly basis, we include a one-week lag in the calculations to ensure the investability of our strategy. In order to closely match the macroeconomic signal methodology with those of the momentum and volatility signals, we base our short, medium and long-term macroeconomic signals on 1/10, 1/30 and 1/40 week SMACOs for weekly initial unemployment claims. Since rising unemployment claims in an expansionary economy is a bullish stock market price and volatility signal, if the SMA is greater than the LMA, we shift the collar towards the ATM put and OTM call (increasing both strike prices) thereby increasing the portfolio’s exposure to upside moves as well as increasing its vega16. In contractionary periods, rising unemployment claims would cause us to shift the strike prices in the opposite direction.
16 Since vega is highest for ATM options, moving the short call further OTM and moving the long put towards the ATM will increase the vega of both option positions. 14
These calculations can be illustrated with the following example:
Macroeconomic Signal Calculation for the 3/19/1999 Roll Date
NBER Announcements Date Indication 12/22/1992 Trough 11/26/2001 Peak (3/19/1999 is during an expansionary economy) So on 3/19/1999, a downward trend in unemployment is a bearish signal. 1 Week SMA 40 Week SMA 30 Week SMA 10 Week SMA LONG Unemployment Trend MEDIUM Unemployment Trend SHORT Unemployment Trend 308.0 317.1 311.8 311.4 Down Down Down Short Term Macroeconomic Signal Calculation: 1 Week SMA = 308.0 < 10 Week SMA = 311.4 Medium Term Macroeconomic Signal Calculation: 1 Week SMA = 308.0 < 30 Week SMA = 311.8 Long Term Macroeconomic Signal Calculation: 1 Week SMA = 308.0 < 40 Week SMA = 317.1 LONG Macroeconomic Signal MEDIUM Macroeconomic Signal SHORT Macroeconomic Signal -1 -1 -1 Note: All moving averages using data up to the prior week's close (e.g. 3/12/1999) Since the 1 week SMA is less than the 10 week SMA, unemployment is falling. In an expansionary economy this is a bearish signal, so the signal = -1. Holding the momentum signal constant, this would shift the collar up (move the put 1% less OTM and the call 1% further OTM). Since the 1 week SMA is less than the 30 week SMA, unemployment is falling. In an expansionary economy this is a bearish signal, so the signal = -1. Holding the momentum signal constant, this would shift the collar up (move the put 1% less OTM and the call 1% further OTM). Since the 1 week SMA is less than the 40 week SMA, unemployment is falling. In an expansionary economy this is a bearish signal, so the signal = -1. Holding the momentum signal constant, this would shift the collar up (move the put 1% less OTM and the call 1% further OTM).
Trading Rules: We combine the momentum, volatility and macroeconomic signals for each time frame to generate our short, medium and long-term active strategies. Due to the excessive transactions costs that would be associated with daily rolling of option positions, changes in the signals are not incorporated into the strategies on any days except the roll dates17. On each roll
17 In the case of strategies where the put and call expirations are not coincident, such as the 1-month call/3-month put strategies, the put moneyness will only be reset when it is rolled (in this example, once every 3 months), while the call moneyness is reset at each call roll (every month, since we only consider strategies with 1-month calls). 15
date, the initial moneyness of the puts and calls is determined based on the momentum and macroeconomic signals and the ratio of written calls is determined by the volatility signal. Our rules are constructed in such a manner to ensure that the target initial percentage moneyness of the options will be an integer which falls between ATM and 5% OTM. The signals adjust the initial moneyness of the puts and calls from a level near the center of the range at 3% OTM and 2% OTM, respectively18. From this central point, the momentum signal will serve to widen or tighten the collar by increasing or decreasing the amount OTM, respectively. The macroeconomic signal will shift the collar up by increasing the amount OTM of the calls and decreasing the amount OTM of the puts, or shift the collar down by moving the strikes in the opposite direction. The net effect can be illustrated by the following formulas for the call strikes:
18 Puts tend to cost more than calls for a given level of moneyness, so we start the puts further OTM to allow the option component of the strategy to be close to zero cost.
Call % OTM = 2 + (Momentum signal + Macroeconomic signal),
and for puts:
Put % OTM = 3 + (Momentum signal - Macroeconomic signal),
where the momentum signal and the macroeconomic signal are +1/-1 binary signals.
The following example provides an illustration of the trading signal calculation: 16