Technical Description of S&P 500 Buy-Write Monthly Index Composition

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1 Technical Descripion of S&P 500 Buy-Wrie Monhly Index Composiion The S&P 500 Buy-Wrie Monhly (BWM) index is a oal reurn index based on wriing he nearby a-he-money S&P 500 call opion agains he S&P 500 index porfolio each monh on he day he previous nearby conrac expires. The hisorical reurn series begins June 1, 1988, he firs day ha Sandard and Poors began reporing he daily cash dividends for he S&P 500 index porfolio, and currenly exends hrough December 31, 001. The purpose of his noe is o describe he mechanics underlying he index compuaion and o show is performance resuls. 1. S&P 500 index reurn compuaion. The S&P 500 index reurn series makes he assumpion ha any daily cash dividends paid on he index are immediaely invesed in more shares of he index porfolio. 1 The daily reurn of he S&P 500 index porfolio is herefore compued as R S, S S + D S 1 =, 1 where S is he repored S&P 500 index level a he close of day, and D is he cash dividend on paid on day. The numeraor conains he income over he day, which comes in he form of price appreciaion, S S 1, and dividend income, D. The denominaor is he invesmen oulay, ha is, he level of he index as of he previous day s close, S 1.. Buy-wrie monhly (BWM) index reurn compuaion. The reurn of he S&P 500 buy-wrie monhly (BWM) is he reurn on a porfolio ha consiss of a long posiion in he S&P 500 index and a shor posiion in a S&P 500 call opion. The daily reurn of he BWM index is defined as 1 Sandard & Poor s makes he same assumpion in is compuaion of he oal annualized reurn for he S&P 500 index. 1

2 R BWM, S + D S 1 ( C C 1) =, S C 1 1 where C is he repored call price a he close of day and all oher noaion is as previous defined. The numeraor in his expression conains he price appreciaion and dividend income of he index less he price appreciaion of he call, C C 1. The income on he BWM index exceeds he S&P 500 index when he call depreciaes in price over he day, and vice versa. The invesmen cos in he BWM is he level of he S&P 500 index level a he close on he previous day less he closing price of he call, ha is, he proceeds from he sale of he index call are used o offse he purchase of he index porfolio in forming he combined porfolio. 3. Prices used in compued daily reurns. The daily prices/dividends used in he reurn compuaions are aken from he following sources. Firs, he S&P 500 closing index levels and cash dividends were aken from monhly issues of Sandard & Poor s S&P 500 Index Focus Monhly Review. The Review is published by Sandard & Poor s, 55 Waer Sree, New York. Second, he daily S&P 500 index opion prices were drawn from he CBOE s MDR daa file. The call opion price is based on he midpoin of he las pair of bid/ask quoes appearing before or a 3PM (CST) each day, ha is, C 3PM, bid price + ask price 3PM 3PM =. On expiraion day, he call opion price is based on he index selemen price, ha is, C = max(0, S X), sele, sele, where S sele, is he selemen price of he call, and X is he exercise price. Where he exercise price exceeds he selemen index level, he call expires worhless. The hisorical reurns of he BWM index sar in June 1988 and currenly exend hrough December 001. During he period, he mos acive S&P 500 index opions swiched from he close-expiry o he open-expiry selemen opions. In consrucing he series, he close-expiry

3 opions were used hrough he Ocober 199 conrac. Saring wih he November 199 conrac, he open-expiry opions were used. This means ha afer he Ocober 199 expiraion, he expiring call opion posiion is closed a he opening selemen on expiraion day, and new calls are wrien a 10AM. Thus, he buy-wrie reurn on an expiraion day is compued as: R = (1 + R ) (1 + R ) 1, BWM, ON, ID, where R ON, is he overnigh reurn of he buy-wrie sraegy based on he expiring opion, and R ID, is he inraday buy-wrie reurn based on he newly wrien call. The overnigh reurn is compued as R ON, S10AM, + D Sclose, 1 ( Csele, Cclose, 1) =, S C close, 1 close, 1 where S 10AM, is he repored level of he S&P 500 index a 10AM on expiraion day, and C sele, is he selemen price of he expiring opion. The selemen price is based on he special opening S&P 500 index level compued on expiraion days and used for he selemen of S&P 500 index opions and fuures. Noe ha he daily cash dividend, D, is assumed o be paid overnigh. The inraday reurn is defined as R ID, Sclose, S10AM, ( Cclose, C10AM, ) =, S C 10AM, 10AM, where he call prices are for he newly wrien opion. The exercise price of he call is he jus ou-of-he-money opion based on he repored 10AM S&P 500 index level. Finally, each ime a new call opion is wrien, i is assumed o be wrien a a price ha is half way beween he bid price and he bid/ask midpoin. For he expiraion days before and including he Ocober 199 conrac, he sales price used is 3

4 C bid price3pm + ask price3pm bid price3pm + =. For he expiraions afer he Ocober 199 conrac, he sales price is C bid price10am + ask price10am bid price10am + =. 4. Monhly performance resuls. To compue monhly reurns, he daily reurns of he various indexes are linked geomerically. In his way, hey represen acual reurns over he monh. These are repored in Table 1. Also repored is he rae of reurn of a Eurodollar ime deposi whose number of days o mauriy maches he number of days in he monh. A he boom of Table 1 are he mean monhly reurns. The average monhly rae of reurn for he shor-erm money marke insrumens as 0.483%. The S&P 500 index porfolio produced a reurn of 1.187% a monh wih a sandard deviaion of 4.103%. The BWM produced a reurn of 1.151% a monh wih a sandard deviaion of.67%. In oher words, BWM produced a monhly reurn abou equal o he S&P 500 index, bu a less han 65% of he risk level (i.e.,.67%/4.103%). The The BWM index reurn can be levered up o mach he risk level of he S&P 500 index. M performance meric says ha he BWM index earned.3% a monh more han he S&P 500 index porfolio a a level of risk equal o he S&P 500 index porfolio. In heory, he M performance measure should be equal o zero. The fac ha i exceeds zero indicaes ha S&P 500 index calls were overpriced during he period June 1988 hrough December 001. Indeed, he average implied volailiy of he call opions wrien in he BWM, 15.79%, exceeds he average realized volailiy over he opion s life, 14.1%. The average (median) difference beween he ATM call implied volailiy and realized volailiy is abou 167 (34) basis poins. The risk-free reurn.00604% is he 30-day Eurodollar ime deposi rae as of he close on 6/1/1988 applied over a holding period of 9 days (i.e., he number of days beween 6/1/1988 and 6/30/1988). 4

5 To show ha he high levels of implied volailiy for S&P 500 index opions are responsible for generaing he abnormal reurns of he BWM index, he index is reconsruced using heoreical opion values raher han observed opion prices. The assumed volailiy parameer used in he Black-Scholes formula o generae opion values is he realized volailiy over he life of he opion. Table 1 includes he resuls. Under he heading Alernaive buy-wrie sraegies, he row labeled Theoreical values shows ha he his buy-wrie sraegy is 0.060%, very near zero. M performance measure of 5. Reurn disribuions Figures 1 and conain he disribuions of sandardized reurns for he S&P 500 index and he BWM index over he period June 1988 hrough December 001. Also shown is he normal disribuion. The figures show ha he reurns of he S&P 500 index and he BWM index are symmeric bu lepokuric. The chances of a zero reurn or a slighly posiive reurn are much higher han is suggesed by he normal disribuion. On he oher hand, he chances of a large posiive or a large negaive reurn for he S&P 500 and he BWN indexes are slighly higher han is suggesed under he normal. Figures 3 and 4 compare he rae of reurn disribuions for he S&P 500 and BWM indexes on a daily and on a monhly basis. As he figures show, boh sraegies produce symmeric disribuions, however, he BWM index disribuion has lower dispersion. 6. Toal reurn indexes Figure 5 shows he performance of he BWM index vis-à-vis he S&P 500 index over he period June 1988 hrough December 001. From he ouse, he BWM racked he S&P 500 index closely. Saring in 199, he BWM begins o rise faser han he S&P 500. By mid- 1995, he level of he S&P 500 oal reurn index surpasses he BWM. Beginning in 1997, he S&P 500 index charges upward in a fas bu volaile fashion. The buy-wrie sraegy lags behind, as should be expeced. When he marke reverses in mid-000, he BWM again moves ahead of he S&P

6 TABLE 1: Summary saisics for he monhly reurns of monery marke marke deposis, he S&P 500 index porfolio, and he CBOE s BWM index during he period June 1988 hrough December 001. No. of Monhly reurns Cum. Annualized Sharpe monhs Average SDev. reurn reurns raio M-squared Risk-free: % 0.15% S&P 500: % 4.103% % CBOE's BWM: %.67% % % Alernaive buy-wrie sraegies: Theoreical values: %.65% % % Midpoin prices: %.671% % % Bid prices: %.673% % % 6

7 FIGURE 1: Disribuion of sandardized daily raes of reurn for S&P 500 index and CBOE s BWM index for he period June 1, 1988 hrough December 31, 001. Probabiliy S&P 500 BWM Normal Sandardized daily rae of reurn 7

8 FIGURE : Disribuion of sandardized monhly raes of reurn for S&P 500 index and CBOE s BWM index for he period June 1, 1988 hrough December 31, 001. Probabiliy S&P 500 BWM Normal Sandardized monhly rae of reurn 8

9 FIGURE 3: Disribuion of daily raes of reurn for S&P 500 index and CBOE s BWM index during he period June 1, 1988 hrough December 31, 001. Frequency 1,00 1, Daily rae of reurn 9

10 FIGURE 4: Disribuion of monhly raes of reurn for S&P 500 index and CBOE s BWM index during he period June 1, 1988 hrough December 31, 001. Frequency Monhly rae of reurn 10

11 FIGURE 5: Monh-end oal reurn indexes for S&P 500 and CBOE s BWM index for he period June 1, 1988 hrough December 31, 001. Level,500,000 S&P 500 BWM 1,500 1, Monh 11

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