Building Option Price Index

Transcription

1 Building Opion Price Index Chris S. Xie Polyechnic Insiue New York Universiy (NYU), New York Phone: Augus 8, 2008

2 Absrac In his paper, I use real daa in building call and pu opion price indices for 30 US blue-chip socks boh individually and collecively, by applying divisor-adjusing mehodologies (mimic mehods used for DJIA Index) in order o achieve coninuiy, comparabiliy and consisency of he indices. The resul shows ha i is feasible. As horizonal measures, hese indices provide beer iming gauges o follow opion price movemens for he underlying asses, while implied volailiies could provide verical measures o idenify which opions are under- or over-valued. Wih opion price indices and implied volailiies, invesors could make beer and informed invesmen decisions. Keywords: Opion Price Average Index, Divisor Adjusing Mehodology, Implied Volailiy, Annualized Volailiy, Benchmark Opion Price Index JEF Classifier: C02, G00

3 Building Opion Price Index INTRODUCTION Currenly here is no index for call and pu opion prices movemens over he ime horizon. I do believe ha invesors could make more-informed and beer decisions if here are opion prices indices available such as Opions Price Average Indices (OPAI) as benchmarks for heir decisions-making. Based on Black-Scholes Model, he opion price is a funcion of he curren price of he underlying asse (S), he srike price (K), he volailiy of he underlying asse price (s), he ime o mauriy (T), he risk-free rae (R) and he dividend yield (Q) (if have). Because of several independen variables used in he opion price funcion, many differen opions wih differen prices can be wrien on he same underlying asse, and will expire wihin limied ime in he fuure, and many new opions will be creaed periodically on he new marke condiions and exis only for cerain ime period. To rack he opions prices movemen over he ime for an underlying asse, we need o rea all opions available for he ime being collecively as he consiuens o consruc an opion index o follow heir price movemens. Like Dow Jones Indusrial Average Index, by applying divisor-adjusing-mehodologies (DAM), i is feasible o build a valid and consisen opion price index on a coninuous ime base. No maer how many new opions will be wrien a new prices on he new marke condiions for an underlying asse; no maer how many opions for he same underlying asse are going o expire, his DAM mehod will allow us o eliminae all biases o keep he opion index of validiies. Go a furher sep, given opions prices indices for all required underlying asses such as all consiuen socks of Dow Index, we will be able o build a benchmark opion price index by reaing all individual opions prices indices collecively as consiuen indices (Index-on-Indices). The key ool, divisor-adjusing-mehodology, for building opion price indices, will be illusraed in deails in he following secion. 2 USING DIVISOR-ADJUSTING METHODOLOGY (DAM) IN BUILDING OPTION PRICE AVERAGE INDEX (OPAIS) Le s make a simple sample o illusrae he divisor-adjusing-mehodology. Supposed ha an underlying asse (S) has only 3 differen opion (A, B, C) a ime T, and one of hem (B) will expire a ime T+, and 2 new opions (E and F) will be creaed a ime T+. All are equally-weighed, and he prices are shown in Table : Table Opion Adjused A B C E F Num(N) Toal (S) Avg Index Price Divisor (d) T T Because he prices of he remaining opions (A and C) unchanged, he opion price average index should remain he same, no maer wha prices for hese 2 new - -

4 Building Opion Price Index opions (E and F). Assuming he adjused divisor (d ) equal o, a he ime T he index (I ) is compued by he price average divided by he adjused divisor (d ), or ha I S 6 = = = 2 dn (3) Similarly, a he ime T+ he index (I + ) is calculaed as follows: I S = = = d+ N+.25(4) 2 Here he adjused-divisor is deermined by he following formula : S N + d d S N = + / Here, So S = S ( Expired) + ( New ) + () ( ) S = (5+ ) = d + = () / = Le s change he prices of he new opion (E an F), say prices of hem are 2.00 and respecively (see Table 2). Table 2 Opion Price A B C E F Num(N) Toal (S) Avg Adjused Divisor (d) T T So i jus needs o change he adjused divisor o 2.00 as calculaed as follows: S N d = d = = S N I + S + + = = = d+ N+ 2(4) / () / From above, we could also see ha no maer how he prices of he newly creaed opions in he new marke condiions are, hey will no lead o any biases for he index s coninuiy and consisency by adjusing he divisor. Of course, when A or C and or boh change prices, he index could change accordingly. To generalize he calculaion of he adjused divisor for a weighed price index, i is assumed ha: Index See mahemaical derivaion Appendix : Proof () - 2 -

5 Building Opion Price Index (i) (ii) (iii) (iv) (v) (vi) The prices will be weighed by somehing, and le s called i volume ; Toal number of differen opions a ime T is N (), of which N (r, ) opions remain a ime T+ and N(e, ) expires a ime T+ A ime T+, N (n, +) new opions will be creaed for he underlying asse wih oal new prices and volumes. All prices and heir volumes for he N (r, ) remaining opions unchanged; Opions wih differen mauriies are reaed equally in calculaions; The adjused divisor (d ) for he firs day of launching he index always se o. Because all prices and heir volumes for he N (r, ) remaining opions are unchanged, he weighed price index should be remaining he same from ime T o T+. Since a ime he weighed price average index (WPAI) is compued as he weighed price average (WPA ) divided by he adjused divisor (d ), or ha, WPAI WPA d = = N i= d V i= (,) i (,) i N V P (,) i N N Le V(,) i P(,) i = S, V(,) i = V i= i= So WPAI S = dv A ime T+, we could simply make adjusmen for he divisor by he following formula 2 S V d+ = d / S V Here, N( e, ) N( n, + ) = exp ired ( j, ) epired ( j, ) + new( j, + ) new( j, + ) j= j= S S V P V P N( e, ) N( n, + ) = exp ired( j, ) + new( j, + ) j= j= V V V V Wih his adjusmen, he opion price average index will no change, or ha, WPAI + N+ V P V P WPA WPA = = = = = WPAI d (, i+ ) (, i+ ) (,) i (,) i + i= i= N+ N + d d+ V( i, + ) d V( i, ) i= i= N 3 2 See mahemaical derivaions in Appendix : Proof (2) 3 See mahemaical derivaions in Appendix : Proof (2) - 3 -

6 Building Opion Price Index By applying his divisor mehodology (DAM) on a daily base, we will able o achieve he coninuiy, comparabiliy and consisency for he opions price average indices (OPAI) on a daily fashion, no maer how many differen opions creaed periodically for he same underlying asse and no maer a wha prices hey creaed as new opions in new marke condiions, and no maer when hey expires. Therefore, we can build opion price average indices (OPAI) for boh call opion price average indices () and pu opion price average () for any kind of underlying asses. The above secion is illusraed o build Opions Price Average Indices (OPAI) for individual underlying asse. Go a furher sep, for Benchmark Opions Price Average Indices (BOPAI), firsly we can calculae opion price indices for all consiuen underlying asses consising a benchmark index such as Dow Jones Indusrial Average, S&P500, Nasdaq 00, ec., hen simply ake he average (equally-weighed) or weighed average of he opion price indices for all consiuen underlying asses o creae BOPAIs (like average-on-average or Index-on-Indices ). 3 OPTION PRICE AVERAGE INDEX (OPAIS) FOR ALL 30 CONSTITUENT STOCKS OF DJIA INDEX (FEBRUARY, 2005 TO MARCH 3, 2008) BY USING REAL DAILY CLOSE OPTIONS DATA Now I use he real opion daily close daa (Feb., 2005 Mar. 3, 2008) 4 for all socks of DJIA Index, by applying he above mehods and processes o calculae Call Opion Price Average Indices () and Pu Opion Price Average Indices () for all of hese underlying asses based on he following assumpions: (i) when an opion has no ransacion during a day, i is assumed ha is opion price unchanged during ha day; (ii) when a newly creaed opion has no ransacion a he very beginning, which has no price or zero price, in his case i is deemed as no exising, and will no be aken ino calculaions unil i has he firs ransacion. I jus show he resuls in graphs (Char -2) for 2 socks (underlying asses) wih a summary beneah each char here for illusraion purposes. These socks Ticks include AA, AIG, C, DD, GE, HD, IBM, JPM, KO, MSFT, PFE, WMT. For making an easy vision comparisons, he chars for daily close prices 5 of all of hese socks during he same period also illusraed ogeher. 4 Daa source: 5 Daa source: Yahoo Finance - 4 -

7 Building Opion Price Index Char : Tick-AA Opion Price Average Index Tick: AA Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May-08 Daily Close Price Tick: AA Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May-08 Daily Close Sock Price Alcoa: Opion Price Average Type Call Pu Hisorical 6 Low (Dae) $.32 (08/3/2008) $0.35 (07/6/2007) Hisorical high (Dae) $9.43 (07/3/2007) $7.28 (0/3/2005) 52 weeks range $.32-$9.43 $0.35-$.35 Currenly 7 a $2.97 $0.69 Char 2: Tick- AIG Opion Price Average Index Tick: AIG Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May-08 Daily Close Sock Price Tick: AIG Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May-08 Daily Close Sock Price AIG: Opion Price Average Type Call Pu Hisorical Low (Dae) $0.82(03/7/2008) $3.97 (06/8/2007) Hisorical high (Dae) $.04 (02//2005) $9.3 (03/7/2008) 52 weeks range $0.82-$5.20 $3.97-$9.3 Currenly a $0.94 $ Hereafer hisorical refers o he period of Feb., 2005 o Mar.3, Hereafer currenly refers o he dae of March 3,

8 Building Opion Price Index Char 3: Tick-C Tick: C Tick: C Opion Price Average Index Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May-08 Daily Close Sock Price Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May-08 Daily Close Sock Price CITIGROUP: Opion Price Average Type Call Pu Hisorical Low (Dae) $0.22 (03/7/2008) $2.0 (05/30/2007) Hisorical high (Dae) $0.95 (2/27/2006) $3.96 (03/7/2008) 52 weeks range $0.22-$9.67 $2.0-$3.96 Currenly a $0.27 $2.90 Char 4: Tick-DD Tick: DD Tick: DD Opion Price Average Index Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May-08 Daily Close Sock Price Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May-08 Daily Close Sock Price DuPon: Opion Price Average Type Call Pu Hisorical Low (Dae) $.95 (08/4/2006) $.67 (07/23/2007) Hisorical high (Dae) $0.5 (03/09/2005) $8.5 (0/7/2005) 52 weeks range $2.8-$6.06 $.67-$3.0 Currenly a $2.87 $

9 Building Opion Price Index Char 5: Tick-GE Tick: GE Tick: GE Opion Price Average Index Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May-08 Daily Close Sock Price Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May-08 Daily Close Sock Price GE: Opion Price Average Type Call Pu Hisorical Low (Dae) $2.00 (03/0/2008) $.34 (0/0/2007) Hisorical high (Dae) $6.55 (0/02/2007) $5.05 (02/07/2006) 52 weeks range $2.00-$6.55 $.34-$3.5 Currenly a $2.96 $2.27 Char 6: Tick HD Tick: HD Tick: HD Opion Price Average Index Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May-08 Daily Close Sock Price Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May-08 Daily Close Sock Price Home Depo: Opion Price Average Type Call Pu Hisorical Low (Dae) $0.46 (03/0/2008) $.72 (03/23/2006) Hisorical high (Dae) $7.4 (02/7/2005) $5.94 (0/09/2008) 52 weeks range $0.46-$3.0 $2.06-$5.94 Currenly a $0.55 $

10 Building Opion Price Index Char 7: Tick IBM Tick: IBM Tick: IBM Opion Price Average Index Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May-08 Daily Close Sock Price Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May-08 Daily Close Sock Price IBM: Opion Price Average Type Call Pu Hisorical Low (Dae) $4.2 (07/8/2006) $3.02 (0/6/2007) Hisorical high (Dae) $5.92 (0/6/2007) $20.85 (05/3/2005) 52 weeks range $7.23-$5.92 $3.02-$8.02 Currenly a $2.54 $3.34 Char 8: Tick-JPM Tick: JPM Tick: JPM Opion Price Average Index Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May-08 Daily Close Sock Price Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May-08 Daily Close Sock Price JP Morgan: Opion Price Average Type Call Pu Hisorical Low (Dae) $2.82 (03/4/2008) $.07 (05/6/2007) Hisorical high (Dae) $.85 (05/09/2007) $5.95 (0/3/2005) 52 weeks range $2.82-$.85 $.07-$2.94 Currenly a $4.59 $

11 Building Opion Price Index Char 9: Tick - KO Tick: KO Tick: KO Opion Price Average Index Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May-08 Daily Close Sock Price Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May-08 Daily Close Sock Price Coca-Cola: Opion Price Average Type Call Pu Hisorical Low (Dae) $3.54 (0/20/2006) $0.48 (0/0/2008) Hisorical high (Dae) $8.29 (0/0/2008) $7.30 (02/02/2005) 52 weeks range $5.77-$8.29 $0.48-$2.69 Currenly a $4.20 $0.63 Char 0: Tick - MSFT Tick: MSFT Tick: MSFT Opion Price Average Index Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May-08 Daily Close Sock Price Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May-08 Daily Close Sock Price Microsof: Opion Price Average Type Call Pu Hisorical Low (Dae) $2.74 (06/3/2006) $.72 (0//2007) Hisorical high (Dae) $8.05 (02//2007) $7.97 (06/3/2006) 52 weeks range $3.03-$8.05 $.72-$5.23 Currenly a $3.3 $

12 Building Opion Price Index Char : Tick PFE Tick: PFE Tick: PFE Opion Price Average Index Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May-08 Daily Close Sock Price Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May-08 Daily Close Sock Price Pfizer: Opion Price Average Type Call Pu Hisorical Low (Dae) $0.67 (03/28/2008) $4.04 (06/04/2007) Hisorical high (Dae) $4.22 (05/24/2005) $8.55 (2/2/2005) 52 weeks range $0.67-$2.30 $4.04-$7.69 Currenly a $0.69 $7.50 Char 2: Tick - WMT Tick: WMT Tick: WMT Opion Price Average Index Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May-08 Daily Close Sock Price Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May-08 Daily Close Sock Price Mal-Mar: Opion Price Average Type Call Pu Hisorical Low (Dae) $0.94 (09/0/2007) $3.48 (03/24/2008) Hisorical high (Dae) $6.52 (02/08/2005) $8.42 (09/2/2005) 52 weeks range $0.94-$2.28 $3.48-$7.53 Currenly a $.76 $

13 Building Opion Price Index 4 CORRELATIONS AND VOLATILITIES OF OPTION PRICE INDICES AND UNDERLYING ASSETS Afer we have obained all opion price average indices for hese underlying asses, correlaions coefficiens beween he call opion average index () as well as he pu opion average index () and he underlying asse price during he same ime period have been calculaed. The resuls show ha s are highly posiively correlaed wih heir underlying sock prices wih average of , while s are on opposie, highly negaively correlaed wih heir underlying sock prices on average of , summarized in Table 3 (See he deails in Appendix 2) Table 3: Correlaion Coefficien Correlaion Coefficien Underlying Sock Price Underlying Sock Price Min Max Average These are very good resuls, as call opion prices heoreically increase wih he underlying price s going up, while pu opion prices work in he opposie way. These resuls also affirmed ha he mehods using in building he opions price average indices are valid and effecive. The volailiies of opions price indices and he underlying asse prices have also been compued, and he resuls for he annualized 8 volailiies for boh call opion average index and pu opion average index as well as underlying sock prices are summarized in Table 4 (See he deails in Appendix 2). Table 4: Annualized Volailiies Comparisons Annualized Volailiies Underlying Asses Min 26.% 29.% 2.46% Max 75.0% 96.5% 7.00% Average 50.2% 47.0% 25.20% Boh call and pu opion price average index are much more volaile (nearly double on average) han he underlying asse, which explains leverage effecs for opions invesmens. Nex by linearly regressing he correlaion coefficiens and volailiies for boh s and s wih respec o he volailiies of he underlying asse prices, I graph in Char 3. 8 The annualized volailiy is compued as σ = σ 250, and he daily volailiy is he sandard Annualized deviaion of he daily reurns, which is calculaed by r = ln( S / S ). Here S is he daily close price for underlying asse, or he opion price average index a ime (). - - daily

14 Building Opion Price Index Char 3: Regression on Volailiies s: Annualized Volailiies s: Annualized Volailiies Volailiies of s 80% 70% 60% 50% 40% 30% 20% 0% 0% y = x R 2 = Volailiies of s 20% 00% 80% 60% 40% 20% 0% y = x R 2 = % 20% 40% 60% 80% 0% 20% 40% 60% 80% Volailiies of Underlying Socks Volailiies of Underlying Socks Alhough R-squares are relaively low ( for s, and for s wih underlying asses), i is apparenly ha volailiies for boh s and s are posiively correlaed wih he volailiies of he underlying asses, because higher volailiy of an underlying asse could resul in higher volailiy for is opions prices index. The regression coefficiens for boh s and s are less han, which could reasonably explain ha on average boh call and pu opions price changes should be less han he changes of he underlying asse prices. I seems rue, since boh call and pu opions price changes would mos likely be less han he price changes of he underlying asse. Similarly, by regressing correlaion for boh s and s wih respec o he volailiies of he underlying asses, he graphs are shown in Char 4. Char 4: Regression on Correlaion s: Correlaion wih Underlying Sock s: Correlaion wih Underlying Sock Correlaion Coefficiens 00% 80% 60% 40% 20% y = x R 2 = Correlaion Coefficiens 00% 50% 0% -50% y =.6434x -.24 R 2 = % 0% 20% 40% 60% 80% Volailiies of Underlying Socks -00% 0% 20% 40% 60% 80% Volailiies of Underlying Socks I seems ha higher volailiies of he underlying asse end o reduce he posiive correlaion beween call opion price average index and he underlying asse price movemen. I could be rue because higher volailiy of he underlying asse leads o higher volailiy of call opion index, bu no o he same exen, which causes lower posiive correlaion coefficien for call opion index. I is also rue for he case of he pu opion price index, because i weakens he negaive correlaion (i.e., reducing he correlaion coefficien in absolue value)

15 Building Opion Price Index 5 BENCHMARK OPTION PRICE AVERAGE INDEX FOR DJIA INDEX (FEBRUARY, 2005 TO MARCH 3, 2008) The Benchmark Opion Price Average Index (BOPAI) is designed for a collecion of he underlying asses such as all consiuen socks of Dow Jones o follow heir opions prices changes collecively over he ime. BOPAI is simply se equal o he arihmeic (equally weighed) or weighed (such as marke-cap weigheed) average of opion price indices for all consiuen socks, which consruc he benchmark index such as DJIA (like Average-on-Average ). BOPAI is also classified as B (Benchmark Call Opion Price Average Index) and B (Benchmark Pu Opion Price Average Index). Here, jus BOPAI for DJIA Index has been consruced, as B.Dow and B.Dow defined as a Dow Benchmark Call Opion Price Average Index and Dow Benchmark Pu Opion Price Average Index respecively. Again based on s (Call Opion Price Average Indices) and s (Pu Opion Price Average Indices) for all 30 Dow socks obained, I simply ake an arihmeic average of hem o calculae B.Dow and B.Dow Indices during he period of Feb., 2005 o Mar. 3, The resuls are graphed in Char 5. Char 5: Benchmark Opion Price Index Benchmark Opion Price Average Index (Dow) DJIA Daily Close Index Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May Feb-05 Aug-05 Mar-06 Sep-06 Apr-07 Oc-07 May-08 B_Dow B_Dow DJIA Close Price The oucome shows ha B is nearly perfecly posiively correlaed wih DJIA during he same ime period, wih a correlaion coefficien up o , and B is also nearly perfecly negaively correlaed wih DJIA wih a correlaion coefficien of This also could provide an evidence for validiy and effeciveness for he mehodology I use o build he opions price average indices above. The annualized 9 volailiy for DJIA during he period is calculaed as 2.9%, while for B index he annualized volailiy is 23.3% and for B is 22.5%, which are nearly double of DJIA s. Similarly, by using he same process, Benchmark Call and Pu Opion Price Average indices can be easily esablished for oher major global sock indices such as S&P Indices series, NASDAQ Index, ec.. 9 The same mehod as described in foonoe 8 is used o calculae volailiies

16 Building Opion Price Index 6 CONCLUSION I is feasible o esablish call and pu opion price average indices boh for individual underlying asse and for a collecion of underlying asses such as all consiuen socks for a benchmark index (such as DJIA, S&P500, ec.) by using he above processes. The divisor adjusing mehodology (DAM) allows us o eliminae biases arising from change consiuen opions in order o mainain coninuiy, comparabiliy and consisency of he opion price indices. The above resuls obained from he real opion daa during February, 2005 o March 3, 2008 have furher provided enough proofs for he feasibiliy in building boh call and opion price average indices. The opion price average indices illusraed in his paper is using an equally-weighed average mehod (or arihmeic average), and of course, i could also be possible o creae opions price indices in a non-equally-weighed average such as marke-cap-weighed average o build benchmark opion price indices. Wih he valid and consisen opion price indices, i will be grea useful for invesors as well as all financial world for heir more-informed decision-making and researches as hey could easily rack and follow opion price changes or movemens horizonally, while hey could scan opions verically by using implied volailiies o idenify which opions are undervalued or overvalued

17 Building Opion Price Index VI. REFERENCES (i) Financial Risk Manager Handbook (GARP), Philippe Jorion, 2007 (ii) The Complee Guide o Capial Markes for Quaniaive Professionals, Alex Kuznesov, 2006 (iii) Saisical Analysis of Financial Daa in S-Plus, Rene A. Carmona, 2004 (iv) Sandard & Poor s 500 Index Calculaion, hp:// (v) The Uses and Limis of Volailiy, David Harper, February 2004 (vi) Index Design and Implicaion for Index Tracking: Evidence from S&P 500 Index Funds, Alex Frino, David R. Gallagher, Alber S. Neuber, Teddy N. Oeomo, 2003 (vii) Guide o Calculaion Mehods for he UK Series of he FTSE Acuaries Share Indices, Version 4. May 2005 (viii) SSE 80 and SSE 50 Index Mehodology, China Securiies Index, Sepember 2007 (ix) Rules for he HS60 Index, HOLT, Version 4, Ocober 2006 (x) Index Mehodology and Calculaion, Bakers Invesmen Group January

18 Building Opion Price Index Appendix : Formula for divisor adjusmen () Proof : Equally weighed opion price average index Assuming a he ime T+, prices of all remaining (or ha, no-ye-expired) opions unchanged, we should proof he index will also remain he same. Under his (here assumpion, we have (Re main ) = S ( Expired + ) ( ) opions prices a ime T), so S is he sum of S = (Re main) + ( New) = S ( Expired) + ( New) = S + + ( + ) ( ) ( + ) Since S N + d d S N + = /, we have I S S d S N S d S N S I = = / / = / / = = d+ N+ N+ S N N+ S N dn Therefore, in his case, he opion price average index a ime T+ remains a he same as a ime T. (2) Proof : non-equally-weighed opion price average index Le s suppose i is weighed by Volume (V). Similarly, assuming prices and volumes for all remaining opions a ime T+ no changed, we should proof he weighed opion price index also unchanged. In his case, we have N( e, ) V P = S V P (here, S = V(,) i P(,) i all ) ( k, + ) ( k, + ) remaining exp ired ( j, ) epired ( j, ) j= N( e, ) V = V V (here, V = V(,) i all ) So ( k, + ) remaining exp ired ( j, ) j= N( n, + ) S = V P + V P + ( k, + ) ( k, + ) remaining new( j, + ) new( j, + ) j= N( e, ) N( n, + ) exp ired ( j, ) epired ( j, ) new( j, + ) new( j, + ) j= j= = S V P + V P = S N( n, + ) N( e, ) N( n, + ) + = ( k, + ) remaining + new( j, + ) = exp ired ( j, ) + new( j, + ) = j= j= j= V V V V V V V Since WPAI Therefore, S S = =, and + + d+ V+ d+ V d S V = + d / S V - 6 -

19 Building Opion Price Index S S S S V S WPAI d WPAI + + = = = / / = = d+ V+ d+ V V S V dv - 7 -

20 Building Opion Price Index Appendix 2: Correlaions and Volailiies Tick Correlaion Coefficien wih Underlying Sock Annualized Volailiies s s s s Annualized Volailiies of Underlying Sock AA % 68.8% 29.9% AIG % 44.% 24.7% AXP % 40.9% 26.2% BA % 6.3% 22.5% C % 34.6% 25.8% CAT % 64.9% 45.9% DD % 36.8% 20.3% DIS % 36.4% 9.6% GE % 4.3% 6.6% GM % 50.9% 42.5% HD % 38.8% 24.3% HON % 53.0% 20.9% HPQ % 64.8% 26.% IBM % 38.8% 8.9% INTC % 33.6% 27.7% JNJ % 29.% 2.5% JPM % 45.2% 24.6% KO % 35.3% 3.2% MCD % 52.0% 20.9% MMM % 43.3% 8.3% MO % 96.5% 7.0% MRK % 59.6% 24.8% MSFT % 40.5% 2.0% PFE % 32.7% 20.5% PG % 29.7% 4.2% T % 58.2% 9.7% UTX % 37.5% 42.8% VZ % 42.5% 8.4% WMT % 40.0% 8.3% XOM % 59.6% 22.8% Average % 47.0% 25.2% - 8 -