The CBOE S&P 500 PutWrite Index (PUT)

Transcription

1 The CBOE S&P 500 PutWrite Index (PUT) Introduction Interest in option-based investment strategies continues to grow as portfolio managers become increasingly sensitive to the excess returns such types of strategies have generated. The Chicago Board Options Exchange (CBOE ) has been instrumental in supporting this investment style by its introduction of the CBOE S&P 500 BuyWrite Index (BXM SM ), the first option-based benchmark. The BXM tracks the performance of a hypothetical S&P 500 covered call strategy. The CBOE has now developed a new option-based benchmark for the equally popular option strategy of selling puts rather than calls. The new index, called the CBOE S&P 500 PutWrite Index (PUT) tracks the performance of a hypothetical investment strategy (PUT strategy) that overlays short S&P 500 puts over a money market account. The number of puts is set to collateralize the exposure to S&P 500 downturns. This design provides higher leverage than the BXM strategy, and it can also capture the potentially rich premia of S&P 500 put options documented in several academic studies. These studies have found that short option strategies, and especially short put strategies, appear to generate high risk-adjusted returns. Reasons cited for the excess returns are the negative risk-premium garnered by volatility, and, in the case of puts, the high demand for portfolio protection. For more information on the PUT Index, please visit the website or send an to institutional@cboe.com. Month End Index Values, June 1988 to May PutWrite BXM SPTR 3 Month T-Bill /1/1988 6/1/1989 6/1/1990 6/1/1991 6/1/1992 6/1/1993 6/1/1994 6/1/1995 6/1/1996 6/1/1997 6/1/1998 6/1/1999 6/1/2000 6/1/2001 6/1/2002 6/1/2003 6/1/2004 6/1/2005 6/1/2006

2 Overview of PUT Index The PUT strategy invests cash at one- and three-month Treasury Bill rates and sells a sequence of one-month at-the-money S&P 500 puts (SPX SM ). The short put position is collateralized in the sense that the Treasury bills can finance the maximum possible loss from final settlement of the SPX puts. The PUT portfolio is rebalanced on the third Friday of the month when the puts expire and a new batch of puts is sold. This procedure is referred to as the roll. On every third roll, the total cash in the PUT portfolio is reinvested at the three-month Treasury bill rate. The rebalanced portfolio is long three-month Treasury bills and short one-month SPX puts. On other roll dates, the cash obtained from selling new SPX puts is invested at the one-month Treasury bill rate, and the cash required to settle expiring in-the-money puts is financed first by one-month Treasury bills and second by three-month Treasury bills - if necessary. On such roll dates, the rebalanced portfolio is typically long one and threemonth Treasury bills and short one-month SPX puts. PUT and BXM Rates of Return at Expiration PUT and BXM Sold At-The-Money PUT and BXM OptionsSold Out-Of-The-Money 1+ R PUT (Case 1) 1+ R PUT 1 + R BXM 1 + R BXM 1+ R PUT (Case 2) 1 + R s 1 + R s R s = S&P 500 Index rate of return R PUT = PUT Index rate of return R BXM = BXM Index rate of return

3 The PUT strategy trades off an exposure to S&P 500 risk for a premium over the risk-free rate. If markets were perfectly efficient and prices continuous, the gist of the strategy would be to offer a more leveraged version of the BXM 1 with greater risk and expected return. In reality, strike price are discrete, and the PUT and BXM options are usually sold out-of-the-money. As shown in the chart above, the PUT remains more leveraged than the BXM but the BXM is now variable over a wider range of S&P 500 returns. As a result, the greater leverage of the PUT does not necessarily translate to greater riskiness as measured by a greater standard deviation of its return. In addition, markets are fraught with frictions and pressure points that can affect prices. Several academic studies have puzzled over the apparent richness of S&P 500 put options. Consistent with this, the implied volatility of the PUT put option has often be greater than the implied volatility of the BXM call option. Last, asset managers should find the PUT strategy a convenient method to utilize disposable cash to enhance returns. 45% Roll Date Implied Volatilities 40% 35% PUT put BXM Call 30% 25% 20% 15% 10% 5% 0% 6/1/1988 6/1/1990 6/1/1992 6/1/1994 6/1/1996 6/1/1998 6/1/2000 6/1/2002 6/1/2004 6/1/ See Appendix A for a formal analysis of the PUT return between roll dates and a comparison of the PUT and BXM.

4 Historical Performance of the CBOE S&P 500 PutWrite Index, June 1988 May 2007 From June 1988 to May 2007, the PUT had an annualized monthly return of 12.65% compared to 12.30% for the S&P 500 Total Return Index (SPTR), 11.86% for the BXM and 4.67% for three-month Treasury bills. The PUT had a smaller standard deviation than the BXM and SPTR % 60.00% 50.00% PutWrite BXM SPTR 40.00% 30.00% 20.00% 10.00% 0.00% % -10% -7.50% -5% -2.50% 0% 2.50% 5% 7.50% 10% 15% Based on commonly used measures of risk and return, the risk-adjusted performance of the PUT also compares favorably with the performance of SPTR and BXM. Its Sharpe and modified Sharpe ratios are greater. The PUT also does well by the Stutzer 2 Index, an alternative measure of risk-adjusted returns that take into account leptokurtic returns. 2 The Stutzer index is an alternative to the Sharpe ratio that takes into account the asymmetry of option returns. The Stutzer measure equal to the square root of twice the Stutzer index is on the same scale as the Sharpe ratio. When returns are normally distributed, the Stutzer measure is equal to the Sharpe ratio.

5 Statistics to 5/31/07 PutWrite BXM SPXTR 3M T- Bill Arithmetic Mean Monthly Return 1.03% 0.97% 1.05% 0.38% Annualized Monthly Std. Dev. 8.27% 9.16% 13.75% Annualized Geometric Mean Monthly Return 12.65% 11.86% 12.30% 4.67% Monthly Skew Excess Kurtosis Sharpe Ratio :(ER - Tbill) / Std.Dev Modified Sharpe: (ER - TBill) / Semi Dev Stutzer Measure % 10% Mean Monthly PUT and BXM Returns for Different SPTR Returns June 1988 to May 2007 PUT BXM SPTR 5% 0% -5% -10% -15% -20% Below % Btw -12.5% and -10% Btw -10% and -7.5% Btw-7.5% and -5% Btw -5% and -2.5% Btw -2.5% and 0 Btw 0 and 2.5% SPTR Monthly Return Btw 2.5% and 5% Btw 5% and 7.5% Btw 7.5% and 10% Btw 10% and 12.5% As expected, PUT monthly returns tend to (a) increase with the return on the S&P 500, (b) be greater than the returns of the BXM and SPTR when SPTR returns are negative or small, and (c) be smaller when SPTR returns are larger. More specifically, the PUT tends to perform better when the monthly return of SPTR is at or below 2.5%. The solid relative performance of the PUT is explained by the fact that this occurred 67% of the time between June 1988 and May 2007.

6 Construction of PUT Portfolio The PUT tracks the value of an initial investment of $100 in a portfolio that passively follows the PUT strategy. The portfolio is managed and calculated as follows: On June 1, 1988, the inception date, $100 is invested at the three-month Treasury bill rate 3. This investment is carried until June 17, 1988 which is the first date when SPX at-the-money puts are sold. We call this the first roll date. The cash proceeds from the sale are invested at the one-month Treasury bill rate. If the puts expire in the money at the second roll date, July 15, 1988, the final settlement loss is financed by the Treasury bills, and a new batch of puts is sold. The revenue from the sale of the puts is invested again at the one-month Treasury bill rate. On the third roll date, August 19, 1988, both the one-and three-month Treasury investments are liquidated and the cash is used to finance possible losses from the expiring puts. New puts are sold and the total net cash balance is now reinvested at the three-month Treasury bill rate. This cycle is repeated from thereon. Final Settlement Price of Expiring Put Options At expiration, the put options are settled to a Special Opening Quotation (SOQ, ticker SET ) of the S&P The SOQ is a special calculation of the S&P 500 Index compiled from the opening prices of S&P 500 stocks. The SOQ is calculated when all S&P 500 stocks have opened for trading; this typically happens before 11:00 a.m. ET 5. The aggregate settlement value of the expiring puts is equal to the number of puts times the maximum of 0 and the difference between the strike price of the puts and the SOQ (N*max[0,K-S OQ)]). Selection of the At-the-Money Strike Price 3 The intra-day cash from selling puts at the open is deemed to be invested at the close of the roll date. Similarly, settlement losses are deemed to be financed at the close. 4 If the third Friday is an exchange holiday, the put option will be settled against the SOQ on the previous business day and the new put option will be selected on that day as well. 5 If one or more stocks in the S&P 500 Index do not open on the day the SOQ is calculated, the final settlement price for SPX options is determined in accordance with the Rules and By-Laws of the Options Clearing Corporation.

7 The strike price of the new options is the strike price of the listed CBOE SPX put option that is closest to but not greater than the last value of the S&P 500 Index reported before 11:00 a.m. ET. For example, if the last S&P 500 Index value reported before 11:00 a.m. ET is and the closest listed SPX put option strike price below is 1430 then 1430 strike SPX put options are sold. Sale Price of Put Options 6 The new put options are deemed sold at a price equal to the volume-weighted average of the traded prices ( VWAP ) of put options with that strike during the half-hour period beginning at 11:30 a.m. ET. CBOE calculates the VWAP in a two-step process: first, CBOE excludes trades between 11:30 a.m. and 12:00 p.m. ET that are identified as having been executed as part of a spread, and then CBOE calculates the weighted average of all remaining transaction prices at that strike between 11:30 a.m. and 12:00 p.m. ET, with weights equal to the fraction of total non-spread volume transacted at each price during this period. The source of the transaction prices used in the calculation of the VWAP is CBOE s Market Data Retrieval ( MDR ) System 7. If no transactions occur at the new put strike between 11:30 a.m. and 12:00 p.m. ET, the new put options are deemed sold at the last bid price reported before 12:00 p.m. ET. Number of Puts Sold Unlike an investor in the BXM strategy who sells a single call at every roll, the PUT investor sells a different number of puts at every roll. The number of puts is chosen to ensure that the maximum final settlement loss can be financed by Treasury bills. Hence in the worst case scenario, when the S&P 500 falls to zero, the value of the PUT portfolio is zero. The number of puts sold increases with Treasury bill rates and the price of the put. It decreases with the strike price. Index Calculation CBOE calculates the PUT once per day at the close of trading. On any given date, the index represents the mark-to-market value of the initial $100 invested in the PUT strategy. At the close of every business date, the value of the PUT is equal to the value of the Treasury bill account less the mark-to-market value of the puts: 6 A slightly different roll procedure is used to calculate the historical series of the CBOE S&P 500 PutWrite Index. This is to take into account the changes in the timing of the expiration of S&P 500 options, and to mimic the changes made in the calculation of the BXM series over time. Up to November 20, 1992, the roll is deemed to take place at the close of the 3rd Friday, the strike price of the new put is determined at 4:00 p.m. EST. and the new puts are deemed sold at the last bid price before 4:00 p.m. EST. After this date, the index is rolled at 11:00 a.m. ET instead. And starting on March 17, 2006, the new puts were sold at the VWAP. 7 Time & Sales information from CBOE s MDR System is disseminated through the Options Price Reporting Authority (OPRA) and is publicly available through most price quote vendors.

8 PUT t = M t N last P t where M t is the Treasury bill balance at the close of date t, N last is the number of put options sold at the last roll date, and P t is the arithmetic average of the last bid and ask prices of the put option reported before 4:00 p.m. ET on date t. On all but roll dates, the Treasury bill balance is obtained by compounding the one and three-month Treasury balances at the previous business close at their respective daily rates. i i i M t = ( 1+ rt M t 1 1) where i = 1 and 3 for one and three-month Treasury bills, and r i t-1 is the corresponding Treasury bill rate from the previous to the current close. The Treasury bill rates between two roll dates are obtained by compounding the daily rates. On every third roll date, the Treasury bills are deemed to mature, the cash is used to pay for final settlement of the puts if they expire in-the-money, and new puts are sold. The net cash balance available for reinvestment is: i i t = ( 1+ rt 1 ) M t 1 N last Max[0, K old SOQt ] i M + N new P vwap where K old is the strike price of the put options sold at the previous roll date, SOQ t is the final settlement price on roll date t, N new is the number of new puts sold and P vwap is the volume-weighted average price at which the new options are sold. This balance is reinvested at the three-month Treasury bill rate. Hence in the month following a third roll date, the one-month Treasury balance is zero. The number of new puts sold on any roll date t is set such that the Treasury balance at the next roll date covers the maximum put settlement loss: N new = M K /(1 + R ) P ) Third roll dates t /( new 3 vwap N new [ vwap + R1 = M (1 + R ) + M (1 + R )]/ K P /(1 )] Other roll dates where K new is the strike price at which the new puts are sold, and R 1 and R 3 are the oneand three-month Treasury bill rates to the next roll date, Example:

9 Number Treasury balances of puts Strike price SOQ Settlement Loss 1 Month 3 Months 11/20/ Pre- 11/21/03settlement Put bid Effective Threemonth Tbill rate to next roll Post settlement November 21, 2003 was a third roll date. Daily compounding of the one-month and three months Treasury balances outstanding at the close of November 20, 2003 (daily compounding rates and respectively) yielded one-and three-month settlement balances of $22.08 and $ Since the SOQ was 1038, the 1040 put expired in the money with a settlement loss of $ =.644* ( ). The number of new puts sold was N = M / [K/(1+R) P ] = / (1030/ ) = Equivalently, N*K = (M+ N*P )* (1+R) =.6612*1030. Suppose that the S&P 500 had decreased to 0 at the next roll date (December 19, 2003.) The settlement loss on the puts would have been N*K=.6612 * By construction, this would have been exactly covered by the Treasury investment. The calculation on other roll dates is similar to that on third roll dates but the cash from sale of the puts is invested at the one-month Treasury bill rate. Bibliography Bakshi, Gurdip and Nikunj Kapadia (2003) Delta-hedged gains and the negative market volatility risk premium, Review of Financial Studies 16, Bollen, Nicholas and Robert Whaley, (2004) Does Net Buying Pressure Affect the Shape of Implied Volatility Functions?" Journal of Finance April 2004, Vol.59 No.2 Bondarenko, Oleg (2003) Why are put options so expensive? Working paper. Carr, Peter and Liuren Wu (2006) A Tale of two indices, Journal of Derivatives, Spring 2006 Broadie, Mark, Mikhail Cherno and Michael Johannes (2007) Understanding Index Option Returns, draft

10 Feldman, Barry E. Roy, Dhruv Roy (2005), Passive Options-Based Investment Strategies: The Case of the CBOE S&P 500 BuyWrite Index. The Journal of Investing. (Summer 2005). Santa-Clara, Pedro and Alessio Saretto (2004), Option Strategies: Good Deals and Margin Calls, Working paper. Robert Whaley and Schneeweis, Thomas, and Richard Spurgin. (2001) "The Benefits of Index Option-Based Strategies for Institutional Portfolios" The Journal of Alternative Investments, Spring 2001, pp Whaley, Robert. (2002) "Risk and Return of the CBOE BuyWrite Monthly Index" The Journal of Derivatives (Winter 2002) pp

11 APPENDIX A Rates of Return of the PUT and BXM between Roll Dates The effect of put premia on the rate of return of the PUT is more transparent if we examine its rate of return between roll dates. For convenience of exposition, assume it is a third roll date. The PUT portfolio is therefore long three-month Treasury bills and short SPX puts. Define the richness of the put as π = P / K and its moneyness as m where K = (1-m) S, and S is the value of the S&P 500 at the time the puts are sold. The expression (m+r s ) - is equal to zero if the put expires out-of-the-money and to m+r s if it expires in the money. At the next roll date, the rate of return of the PUT is equal to 1+ R PUT ( m+ R ) = s ( 1 )(1 ) π π m + R 1 + R Selling puts leverages the Treasury bill rate by a factor that increases with the richness of the put. The tradeoff is a leveraged exposure to movements of the S&P 500 below the strike price. This leverage increases with the richness of the put as does the volatility of the rate of return. To clarify the difference between the PUT and the BXM, assume first that both the call and put are sold exactly at-the-money, and that put-call parity holds. Define R s - =0 if R s > 0 and R s - = R s if R s < = 0. Then: R 1+ R = ( 1 ) s PUT and π π + R 1 + R 1 + R BXM = + π Rd π s R 1 d 1 1 R The exposure of the PUT to negative S&P 500 rates of return is more leveraged than the exposure of the BXM, and when the S&P 500 rate of return is positive, the PUT has a greater rate of return than the BXM. When the call and put are sold out-of-the money, the rate of return of the PUT is as shown above and that of the BXM is 1+ R BXM 1+ d + m' ( Rs m') = + 1 π ' 1 π '

12 where d is the dividend yield of the S&P 500 Index, and m and π represent the moneyness and richness of the BXM call, defined analogously to those of the PUT put. The PUT remains more leveraged than the BXM but its standard deviation can be smaller or greater depending on the levels of moneyness of the two indexes.

13 The CBOE S&P 500 PutWrite Index (PUT SM ) is designed to represent a proposed hypothetical short put strategy. Like many passive indexes, the PUT Index does not take into account significant factors such as transaction costs and taxes and, because of factors such as these, many or most investors should be expected to underperform passive indexes. In the construction of the hypothetical PUT index, the SPX puts are assumed to be written at a certain price on the third Friday of the month. However, there is no guarantee that all investors will be able to sell at this price, and investors attempting to replicate the PUT Index should discuss with their brokers possible timing and liquidity issues. Transaction costs for a put writing strategy such as the PUT could be significantly higher than transaction costs for a passive strategy of investing in Treasury bills. Past performance does not guarantee future results. Standard & Poor's, S&P, and S&P 500 are registered trademarks of The McGraw-Hill Companies, Inc. and are licensed for use by the Chicago Board Options Exchange, Incorporated (CBOE). CBOE, not S&P, calculates and disseminates the PUT Index. CBOE and Chicago Board Options Exchange are registered trademarks of the CBOE, and SPX SM, BXM SM and PUT SM are servicemarks of the CBOE. The methodology of the CBOE S&P 500 PutWrite Index is owned by CBOE and may be covered by one or more patents or pending patent applications. Options involve risk and are not suitable for all investors. Prior to buying or selling an option, a person must receive a copy of Characteristics and Risks of Standardized Options (ODD). Copies of the ODD are available from your broker, by calling OPTIONS, or from The Options Clearing Corporation, One North Wacker Drive, Suite 500, Chicago, Illinois Supporting documentation for claims, comparisons, recommendations, statistics or other technical data is available by calling OPTIONS, sending an to help@cboe.com, or by visiting