CBOE S&P 500 Implied Correlation Index

Transcription

1 CBOE S&P 0 Impled Correlaton Opton prces reflect the rsk of a stock or stock ndex. The level of rsk conveyed by opton prces s often referred to as mpled volatlty. The mpled volatlty of a sngle-stock opton smply reflects the market s expectaton of the future volatlty of that stock s prce returns. Smlarly, the mpled volatlty of an ndex opton reflects the market s expectaton of the future volatlty of that ndex s prce returns. However, ndex volatlty s drven by a combnaton of two factors: the ndvdual volatltes of ndex components and the correlaton of ndex component prce returns. Intutvely, one would expect that the mpled volatlty of an ndex opton would rse wth a correspondng change n the mpled volatltes of optons on the ndex components. Yet, there are tmes when ndex opton mpled volatlty moves and there s no correspondng shft n mpled volatltes of optons on those components. Ths outcome s due to the market s changng vews on correlaton. The relatonshp between the mpled volatltes of optons on an ndex and the mpled volatltes of a weghted portfolo of optons on the components of that ndex, therefore, becomes a measure of the market s expectaton of the future correlaton of the ndex components - the mpled correlaton of the ndex. The sgnfcance of mpled correlaton s that t reflects changes n the relatve premum between ndex optons and sngle-stock optons, provdng tradng sgnals for a strategy known as volatlty dsperson (correlaton) tradng. Commonly, a long volatlty dsperson trade s characterzed by sellng at-the-money ndex opton straddles and purchasng at-the-money straddles n optons on ndex components. One nterpretaton of ths strategy s that when mpled correlaton s hgh, ndex opton premums are rch relatve to sngle-stock optons. Therefore, t may be proftable to sell the rch ndex optons and buy the relatvely nexpensve equty optons. Begnnng n July 009, CBOE wll begn dssemnatng daly values for the CBOE S&P 0 Impled Correlaton. The CBOE wll dssemnate two ndexes ted to two dfferent maturtes January 00 ( ICJ ) and January 0 ( JCJ ). Both ICJ and JCJ are measures of the expected average correlaton of prce returns of S&P 0 components, mpled through SPX opton prces and prces of sngle-stock optons on the largest components of the SPX. Each day, CBOE wll publsh the ndex values four tmes per mnute, and provde on ts webste the market value weghts of each of the top stocks n the S&P 0. Hstorcal nformaton datng back to 007 wll also be avalable. Dervaton of the CBOE S&P 0 Impled Correlaton An ndex measures the value of a dversfed holdng of assets. In the case of a stock ndex such as the S&P 0, the assets are 0 ndvdual stocks that are among the largest and most actvely traded n the world. Generally, the varance of such an ndex s gven by: CBOE Propretary Informaton Copyrght (c) 009, Chcago Board Optons Exchange, Incorporated. All rghts reserved.

2 Where: w + w w ρ () >, Volatlty of th, th ndex components w, w Weght of th, th ndex components ρ Par-wse correlaton of ndex components For the purposes of calculatng the CBOE S&P 0 Impled Correlaton the weght of an ndex component s determned as follows: where: w S PS P () P Prce of the th ndex component S Float-adusted shares outstandng of the th ndex component The s desgned to reflect the market-captalzaton weghted average correlaton of the components, ρ Average. As such, equaton () can be smplfed and solved for ρ Average : ρ Average > w w w Sample Calculaton (3) The CBOE S&P 0 Impled Correlaton measures expected average correlaton of the S&P 0 usng SPX opton mpled volatltes and a weghted portfolo of the mpled volatltes of optons on stocks n an SPX trackng basket, a subset of the S&P 0 comprsed of the largest components as measured by market captalzaton. The optons used to calculate ICJ (January 00) are SPX optons exprng n December 009 and ndvdual stock LEAPS exprng n January 00. Lkewse, JCJ (January 0) uses SPX optons exprng n December 00 and LEAPS exprng n January 0. ICJ wll be calculated through ovember 009 opton expraton; JCJ wll be calculated through ovember 00 opton expraton. On the busness day mmedately followng ovember expraton, CBOE wll ntroduce a new maturty of the CBOE S&P 0 Correlaton. For example, on Monday, ovember 3, 009, CBOE wll begn calculatng KCJ (January 0), usng SPX optons exprng n December 0 and stock LEAPS exprng n January 0. The followng sample calculaton of ICJ s based on prces at the close of tradng on May 9, 009, and assumes an nterest rate of %, the yeld-to-maturty of the U.S. Treasury CBOE Propretary Informaton Copyrght (c) 009, Chcago Board Optons Exchange, Incorporated. All rghts reserved.

3 3 securty maturng closest to January 00 opton expraton. As descrbed below, the stocks comprsng the trackng basket would have been selected on Aprl 30, 009. Step : The -stock SPX trackng basket. On the last busness day of each month, the components of the S&P 0 are ranked by market captalzaton (closng prce tmes float-adusted shares ), wth the top stocks comprsng the SPX trackng basket for the followng month. Addtonally, stocks wth a market captalzaton rank of 5 through 55 are kept n a replacement pool n the event that one or more of components s acqured or otherwse removed from the S&P 0. The table n Appendx A lsts the SPX trackng basket n effect for May 009 based on the closng prces of S&P 0 components on Aprl 30, 009. Step : Select the optons to be used n the mpled correlaton calculaton; determne the mpled volatlty for SPX optons, ndex, and the mpled volatltes for optons on the stocks comprsng the SPX trackng basket,. For each stock n the SPX trackng basket, the put opton wth a strke prce ust below and the call opton wth a strke prce ust above the current stock prce are selected. The opton prce s deemed to be the average of the bd / ask quote on CBOE. ext, for each opton, an mpled volatlty s calculated usng the Barone-Ades Whaley opton valuaton model. The mpled volatltes of each put/call par are then weghted through a lnear nterpolaton to arrve at a sngle at-the-money mpled volatlty for each stock. For example, on May 9, 009, the closng prce of Apple, Inc. (AAPL) was As shown n the followng table, the sngle mpled volatlty used for AAPL was determned by nterpolatng the mpled volatltes for the 35 Put and 40 Call. Stock Expraton Strke Put/Call Md Stock prce Impled Vol Weght AAPL Impled Vol 35 P AAPL January C The nterpolaton weghts shown n the above table were determned n the followng manner: w p X X c c S X p w c p w where: X c strke prce of call opton ust above the current stock prce X p strke prce of put opton ust below the current stock prce S current stock prce Float-adusted are nvestable shares as defned by Standard & Poor s; that s, as shares that are freely avalable to be bought by the nvestng publc. S&P calculates the float-adusted shares for each component of the S&P 0 components by multplyng a company s shares outstandng by t nvestable weght factor, or IWF. CBOE Propretary Informaton Copyrght (c) 009, Chcago Board Optons Exchange, Incorporated. All rghts reserved.

4 4 AAPL [ w p And AAPL 35P ] + [ w c AAPL 40C ] The SPX optons used to calculate ICJ are the put opton wth a strke prce ust below and the call opton wth a strke prce ust above the forward SPX level to the optons expraton date. The forward ndex level s determned usng at-the-money SPX optons prces, where the strke prce wth the smallest absolute dfference between the call and put prces s consdered to be the at-the-money strke prce. Each opton prce s deemed to be the average of ts bd / ask quote. Usng the Black opton valuaton model for stock ndex futures, the mpled volatltes for the two SPX optons are calculated. Followng the same method descrbed for the stock optons, a sngle at-the-money mpled volatlty for SPX s calculated. Contnung wth the example, the SPX at-the-money strke prce was 95, and the forward ndex level for ICJ, F ICJ, was 909.8: F ICJ r* t X + e * ( C P) 95+ e ( *03 / 365) * ( ) where: X at-the-money strke prce r yeld to maturty of applcable U.S. Treasury securty t tme to opton expraton C md-pont of the bd/ask of the at-the-money strke call opton P md-pont of the bd/ask of the at-the-money strke put opton The same strke-weghted nterpolaton shown above usng equty optons s used to determne the at-the-money mpled volatlty for SPX optons. Based on the calculated forward prce of 909.8, the mpled volatltes of the 900 put and the 95 call were nterpolated as shown n the followng table to obtan an mpled volatlty for the SPX of 8.7. Expraton date Exercse prce Put/Call Md Forward Level Impled Volatlty Weght 9-Dec P Dec C SPX Impled Volatlty 8.7 Step 3: Calculate the captalzaton weght, w, of each component n the -stock basket. The weght, w, assgned to the mpled volatlty of each component s the float-adusted market captalzaton of that component dvded by the total float-adusted market captalzaton of the -stock basket: w P S PS CBOE Propretary Informaton Copyrght (c) 009, Chcago Board Optons Exchange, Incorporated. All rghts reserved.

5 5 On May 9, 009, the total captalzaton of the -stock basket was $4.5 trllon. The weght of Exxon Mobl Corp (XOM), the largest component n the -stock basket, was 8.3% ($343 bllon / $4.5 trllon), compared to 4.3% n the S&P 0. It s mportant to note that the ndvdual stock weghts used n the mpled correlaton calculaton are determned relatve to the captalzaton of the SPX trackng basket, and not the SPX tself. Step 4: Calculate the mpled correlaton, ρ Average Once all of the mpled volatltes are determned and component weghts are assgned, the actual calculaton of the CBOE S&P 0 Impled Correlaton Indcator s relatvely straghtforward. However, n order to llustrate further, the two terms expanded below: w and ww, are + + w & w w49 49 w w + +, ww {[ ww + ww3 3 + K + ww ww ]+ +[ w w K + w w + w w ] [ w ] w } The table shown n Appendx A ncludes the mpled volatltes and component weghts for the SPX components used to calculate the mpled correlaton ndex on May 9, 009. In addton, the mpled volatlty for SPX,, was 8.7. Based on the nputs shown n Appendx A and the values calculated above the Impled Correlaton for May 9, 009 was calculated as follows: ρ Average > w w w where: w (0.087 ) ( ) (0.008 ) ( ) CBOE Propretary Informaton Copyrght (c) 009, Chcago Board Optons Exchange, Incorporated. All rghts reserved.

6 6 > ww *(0.087*0.0385*30.99* *0.0067*58.8*45.7) ( ) ρ Average ICJ 00* CBOE Propretary Informaton Copyrght (c) 009, Chcago Board Optons Exchange, Incorporated. All rghts reserved.

7 Hstory of the CBOE S&P 0 Impled Correlaton CBOE has constructed a hstory of the CBOE S&P 0 Impled Correlaton from January 3, 007 through May 9, 009. As shown n the chart below, the Impled Correlaton fluctuated substantally over the last 9 months. The hghest closng ndex level of occurred on ovember 0, 008 for KCJ wth a maturty of January 009. On ths day, the CBOE Volatlty (VIX) reached ts record hgh close of 80.86, as the S&P 0 closed down 6.7% to Impled Correlaton (ICJ, JCJ, KCJ) VIX Jan-07 3-Apr-07 3-Jul-07 3-Oct-07 3-Jan-08 3-Apr-08 3-Jul-08 3-Oct-08 3-Jan-09 3-Apr-09 JCJ January 008 KCJ January 009 ICJ January 00 JCJ January 0 VIX Smlar to the VIX, mpled correlaton exhbts a tendency to ncrease when the S&P 0 decreases. Whle the nverse relatonshp to the SPX s smlar t s not as strong for the mpled correlaton ndexes. Ths relatonshp suggests that the benefts of dversfcaton offered by nvestng n broad-based equty ndexes are lmted. The followng chart llustrates the nverse relatonshp between the mpled correlaton ndex (ICJ 00) and the SPX from ovember 008 through May 9, 009. CBOE Propretary Informaton Copyrght (c) 009, Chcago Board Optons Exchange, Incorporated. All rghts reserved.

8 SPX Level ICJ and VIX Level ov-07 9-Dec-07 9-Jan-08 9-Feb-08 9-Mar-08 9-Apr-08 9-May-08 9-Jun-08 9-Jul-08 9-Aug-08 9-Sep-08 9-Oct-08 9-ov-08 9-Dec-08 9-Jan-09 9-Feb-09 9-Mar-09 9-Apr-09 9-May-09 SPX ICJ VIX Optons nvolve rsk and are not sutable for all nvestors. Pror to buyng or sellng an opton, a person must receve a copy of Characterstcs and Rsks of Standardzed Optons (ODD). Copes of the ODD are avalable from your broker, by callng OPTIOS, or from The Optons Clearng Corporaton, One orth Wacker Drve, Sute 0, Chcago, Illnos Supportng documentaton for clams, comparsons, recommendatons, statstcs or other techncal data s avalable by callng -888-OPTIOS, contactng CBOE at or by vstng Past performance does not guarantee future results. CBOE, Chcago Board Optons Exchange, CBOE Volatlty and VIX are regstered trademarks of Chcago Board Optons Exchange, Incorporated (CBOE). Standard & Poor's, S&P, and S&P 0 are regstered trademarks of The McGraw-Hll Companes, Inc. and are lcensed for use by the CBOE. CBOE Propretary Informaton Copyrght (c) 009, Chcago Board Optons Exchange, Incorporated. All rghts reserved.

9 Appendx A SPX Stocks used to calculate the CBOE S&P 0 Impled Correlaton May 9, 009 Tcker Symbol Company ame Share Prce Float- Adusted Shares Outstandng (mllons) Market Captalzaton ($ mllons) Impled Volatlty Basket Weght AAPL Apple Inc. $ $ 0, % ABT Abbott Laboratores $ 45.06, $ 69, % AMG Amgen Inc $ 49.94, $ 5, % BAC Bank of Amerca Corp $.7 6, $ 7, % BMY Brstol-Myers Squbb $ 9.9,979.9 $ 39, % CMCSA Comcast Corp A $ 3.77, $ 39, % COP ConocoPhllps $ 45.84,480.4 $ 67, % CSCO Csco Systems Inc $ 8. 5, $ 07, % CVS CVS Caremark Corp. $ 9.80, $ 43, % CVX Chevron Corp $ 66.67, $ 33, % DIS Walt Dsney Co $ 4., $ 44, % GE General Electrc Co $ , $ 4, % GILD Glead Scences Inc $ $ 39, % GOOG Google Inc $ $ 99, % GS Goldman Sachs Group Inc $ $ 7, % HD Home Depot Inc $ 3.6, $ 39, % HPQ Hewlett-Packard Co $ 34.35, $ 8, % IBM Intl Busness Machnes Corp $ 06.8, $ 4, % ITC Intel Corp $ 5.7 5, $ 87, % JJ Johnson & Johnson $ 55.6, $ 5, % JPM JP Morgan Chase & Co $ , $ 38, % KFT Kraft Foods Inc A $ 6., $ 38, % KO Coca-Cola Co $ 49.6, $ 97, % LLY Llly, El & Co $ 34.57,0.34 $ 34, % LOW Lowe's Cos Inc $ 9.0, $ 7, % MCD McDonald's Corp $ 58.99,3.6 $ 65, % MDT Medtronc Inc $ 34.35,8.5 $ 38, % MMM 3M Co $ $ 39, % MO Altra Group Inc $ 7.09, $ 35, % MO Monsanto Co. $ $ 45, % MRK Merck & Co Inc $ 7.58,07.7 $ 58, % MSFT Mcrosoft Corp $ , $ 59, % ORCL Oracle Corp $ , $ 75, % OXY Occdental Petroleum $ $ 54, % PEP PepsCo Inc $ 5.05,556.9 $ 8, % PFE Pfzer Inc $ 5.9 6, $ 0, % PG Procter & Gamble $ 5.94, $ 5, % PM Phlp Morrs Internatonal $ 4.64,003.9 $ 85, % QCOM QUALCOMM Inc $ 43.59,649.4 $ 7, % SGP Scherng-Plough Corp $ 4.40,66.4 $ 39, % SLB Schlumberger Ltd $ 57.3, $ 68, % T AT&T Inc $ , $ 46, % UPS Unted Parcel Servce Inc B $ $, % USB US Bancorp $ 9.0, $ 33, % UTX Unted Technologes Corp $ $ 49, % VZ Verzon Communcatons Inc $ 9.6, $ 83,5 3..0% WFC Wells Fargo & Co $ 5. 4, $ 08, % WMT Wal-Mart Stores $ 49.74, $, % WYE Wyeth $ 44.86,33.46 $ 59, % XOM Exxon Mobl Corp $ , $ 34, % CBOE Propretary Informaton Copyrght (c) 009, Chcago Board Optons Exchange, Incorporated. All rghts reserved.