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call options - r; 1 AWVW ECON 388 — Intro to Econometrics...

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Unformatted text preview: r; 1 AWVW ECON 388 — Intro to Econometrics ~Vol. 1, Issue No. 1 2009 pp. 1—15 Pricing a Call-Option Premium: A ‘ Regression Approach, with an Analysis of Resulting “Greeks” David J Mauler Brigham Young University ABSTRACT: Extensive efforts to develop models which appropriately price options, both from scholars as well as market participants, have led to increasingly complex approaches. I develop an alternative empirical method by using least squares regression to model a call option premium. I briefly review prior literature regarding option-pricing. I then discuss the theoretical foundations behind the model and develop a hypothesis pertaining to its comparative statics. After reporting results and various statistical tests performed to verify the model’s validity, and suggest alternative methodologies to OLS, I present an analysis of the estimators’ implications— particularly the indicated “Greeks”. The model’s accuracy is also discussed. Keywords: Empirical Option Pricing; Econometric Option Model Data Availability: Wharton Database, OptionMeirics I. INTRODUCTION Financial Engineers and Applied Mathematicians employ rather sophisticated assumptions and techniques in developing explanatory models for call-option premiums observed in markets. For instance, the paramount Black-Scholes formula (1973), relatively basic in comparison to its numerous extensions and improvements, is the solution to a PDE formulated by numerous assumptions such as the underlying equity having lognorrnal returns. In fact, the B- S formula bears striking resemblance to a well-known heat transfer equation (Rubash). The point being made, although startlingly accurate under its assumptions, the B-S formula and its descendants (briefly discussed later) are purely theoretical in nature. The motivation behind the model I develop is to compare what raw historical market data illustrates using regression techniques— in essence, a purely empirical approach—to the theoretical models found throughout the literature. In particular, I compare what has been coined the “Greeks,” derivatives widely estimated and used by market-participants regarding how a call premium changes with respect to its endogenous variables. II. LITERATURE REVIEW The Black—Scholes Option Pricing Model (1973), a formula from which much of modern ' finance has stemmed from, is most prominent in valuing an option’s theoretical value and is shown below— (21.53:) = 5%,) — Ira—t’iT—ti‘mdg) Where I _ rum/m + (r + {72/ng -- t) — 0m = d1 4— W 'T —- t. with C referring to the Call Price, S the stock price, N() the standard normal CDF, K the strike a}, price, (T-t) the time until expiry, r the annualized risk-free interest rate, and o the stock’s volatility—measured by the square root of the quadratic variation of the stock's log price process. The following “Greeks” result from this form: Delta: = N (d1) Vega: = 34Mf(d1)x/T -— t (it? _ 30 = ‘ SMWCdji)g 8T 2.. ET ._ t _ , Rho: T KIT - flfi—T'fiT-t’lflmfi} Theta: — rKeTflT‘fi) N However, subsequent improvements have been made in relaxing assumptions maintained . by the B-S Model. As a brief overview, Merton relaxed the assumptions of no dividends (1973) and constant interest rates (1976), Ingerson relaxed the assumption of no taxes or transaction costs (1976), and modifications have continued since then, extending the scope and applicability of the model. Throughout the evolution of the B-S model, countless empirical analyses have been conducted, evaluating its accuracy and the extent to which it is robust. To mention a few, Macbeth and Merville (1979) concluded disparities between predicted B-S and market prices primarily arose from fluctuation in the variance of the asset’s return. Bhattacharya (1980) showed how B-S demonstrated systematic bias which was operationally insignificant in all but rare instances, and Cetin, J arrow, Protter, and Warachka (2006) undertook an empirical analysis of options on illiquid assets which presented evidence of a significant liquidity cost intrinsic to every option. The brief overview above mentioning these various papers only skims the literature on option-pricingwthe point being emphasized is simply that the literature using empirical evidence is almost exclusively devoted towards validating or refuting previous financial models, with commentary on alternative approaches which relax or extend assumptions. However, here I pursue an empirical approach which is model deterministic, rather than responsive or qualifying. ~ III. DESCRIPTION OF THE MODEL For purposes of simplicity, I chose to model a call option on Microsoft stock, an extremely liquid asset, for a given strike price. The chosen functional form for the call option pricing model is presented below, pl ’0 f3 05 rt -<‘ \/ C=B1+B2Sffi382+B486+Bsst+BGSF+B76+B862+896t+l3100T+B11t+B12t2+B13tr+B14r+B151‘2 (1) with notation as follows: C indicating the Call Premium (in USD), [3i unknown constants, S the price of the underlying equity (adjusted for dividends and splits) minus the option’s strike price (in USD), 6 the equity’s volatility (measured using the standard deviation of the previous forty daily closing prices), t the time until expiry (days), and r being representative of the risk—free interest rate (indicated by 30-day US Treasury Bills). The data for each variable is taken from each trading day between 8/20/2007 and 4/18/2008, the period in which April 2008 MSFT Call Options were publicly traded. A strike price of $30.00 was chosen because of its proximity to the corresponding stock price. I reasoned that such a strike price will best capture the influence of endogenous variables, as opposed to a more extreme strike price considerably “in—the-money” (below the current stock price) or “out-of—the-money” (above the current stock price). Note: Data References found in Appendix I J ustification for the included eiifigofisavéiables above is founded upon what economic theory suggests a call option premium is influenced by. It is readily apparent that a call premium is primarily dependent upon the price of its underlying asset, in this case MSFT Stock (S). The increase in an asset’s market value causes the right to purchase the asset at a given strike price to also increase in value. Because volatility (a) impacts the likelihood of the option being exercised, it also influences the call premium market participants are willing to pay. An increase in an asset’s volatility causes the call option to have an increased likelihood of fiiture profitable exercise. With similar reasoning, I include time until expiry (t)—the further an option is from its expiry, the higher the likelihood of a favorable stock movement. Lastly, since the asset of interest is of reference to a future right, that right to purchase at the strike price must be discounted by the risk-free interest rate (r) to account for opportunity cost. An increased risk-free rate indicates a higher opportunity cost borne by the option holder. Therefore, expectations regarding the comparative statics behind the model are as follows: I have reason to expect— using the economic rational discussed above— that resulting Bi will show 6C/6S, 6C/6c, 0C/0t as being positive, and 0C/0r as being negative. Justification for the functional form (1) centers upon its flexibility. BLlack-Scholes and its descendents suggest an exact dependent relationship reliant upon specified assumptions M (discussed earlier). However, the equation here (1) is a second order Taylor Series expansion ~—*.—___.. with respect to each exogenous variable, therefore presenting a model independent of previous assumptions. It should be noted that the strike price does not independently appear in the functional form, seeing as the model is specific to a given strike price. IV. MODEL ESTIMATION NOTE: A summary of estimation procedures is given in Table F Table A: Summary Statistics servation C Can-Price 2.842976 2.219941 ik 6 2(stidflev1atlon or 7 i ' 168 1.421894 .7692368 .444831 2.760051 previous 40 daily Stock Prices _ _ '3't-- da 06 2 786488 1 016899 is 6:8 .0. .0- 0. 2.609986 2.502346 .1978746 7.617879 8.780351 5.233867 .0625 21.8089 ’ ; 63.32202 380.8019 -669.75 1023.4 6:8 160.5796 111.5532 .6667979 445.1995” n 168 405.1001 300.1214 “ 1102.12 L I began estimation by performing OLS regression according to the model mentioned previously (1). The results were as follows: Table B: Determinants of MSF T Call Premium (K=30) _ .0041195*** (.0008361) —.0543o79 (0495214) -.000373 (0009112) .7154455*** (.124982) F(14, 153) = 6316.07 Prob > F = 0.0000 Adjusted R2: 0.9981 Notes: Standard Deviations are in parenthesis. * * * Significant at the .01 level; **Significant at the .05 level; *Significant at the .10 level (STATA Printout available in Apdx II) Because of the several statistically insignificant resulting coefficients, I chose to modify the model by removing those terms from the regression. This allows for a more rigorous evaluation of resulting Greeks when partial derivatives are taken (differentiating with insignificant coefficients would create unwanted noise). The restricted model is as follows: C:[31+BZS+B3SZ+B4SG+B5St+B6Sr+B762+Bgct+B9t+Blotz (2) An OLS regression was performed upon this model with results below: Table C: Determinants of MSF T Call Premium (K=30) Resultin - Coefficient .043553*** (0019337) -.0000637*** (4.86e-O6) -.0010022*** (0001443) S*r .0480189*** (.007906) 6*1: — .0032826*** (.000369) .8697919*** (0443734) F( 9, 158) = 9278.01 Prob > F = 0.0000 Adjusted R2: 0.9980 Notes: Standard Deviations are in parenthesis. * * * Significant at the .01 level; * * Significant at the .05 level; (STATA Printout available in Apdx III) Also, to justify using the restricted model (2) I performed a Likelihood Ratio test to see if statistically significant explanatory power is lost. The resulting chi-square statistic indicated a failure to reject the null hypothesis (at the 99% level) of each model having similar explanatory power (however, it did in fact reject the null at the 95% level). Even though perhaps questionable as to whether the restricted model is appropriate in terms of how much explanatory power is lost, I proceed in my analysisnsing—it (see STATA Printout available in Apdx IV). I now qualify the model’s validity by investigating the assumptions maintained by OLS. I begin by noting possible theoretical multicollinearity—it would be expected that sharp changes in stock price would correspond with changes in volatility—however the measure of volatility (std. deviation of previous 40 daily closing prices) ensures the volatility variable will only change with persistent fluctuation in prices. This largely eliminates the presence of multicollinearity, as seen by significant t-statistics for variables independently involving a and S. I next proceed in testing for normality in the model’s residuals. Shown below is a table summarizing the characteristics of the residuals. Table D: Residuals The severe kurtosis indicates an absence of normality—I confirmed this by performing a JB test for normality. The resulting statistic rejects the assumption of normality at a 99% level (STATA Printout available in Apdx V). I also explore the possibility of the residuals having a Laplace distribution. Included in Table F are the results for performing a LAD regression on the model (2). 1 next proceed in testing for homoskedasticity. A Breusch-Pagan (at 95 % level) and Modified White Test (at the 99% level) indicate a rejection of homoskedasticity (STATA Printout available in Apdx VI). I then use bootstrap simulation to correct for the OLS t- statistics—these results are given in Table F. In attempts to find the MLE estimators under this violation I explore variance-weighted least squares regression. I try two assumptions as to how to describe (it —first, that ot2=exp(5tt) (suggested by Wooldridge) and second, that ot2=et2 (suggested by White). I use the variable t in the Wooldridge’s methodology because of its relationship with the residuals as shown below. plat; e timeleft . 3232.0 (I) WWW? QI‘LUIII‘JJ .l._._..._____.._._...._..._u—___._.1. | .5 l!) [A (I) if! M 1 timeleft ' 2'23 The graph suggests that the variance of the residuals is correlated with t (time until expiry). Both assumptions are used in VWLS estimation—results are shown in Table F. (STATA Printout available in Apdx VII). The last Violation of assumptions maintained by OLS estimation I investigate is the absence of autocorrelation. A Durbin Watson, Breusch—Godfrey, and Wooldridge test indicated rejection of a null of no autocorrelation at a 99% level (STATA Printout available in Apdx VIII). I use a Newey correction (with a lag of 2) to obtain valid OLS t—statistics for the original estimators. In an attempt to estimate the MLE for the model’s coefficients Iperform iterative MLE assuming an AR(1) model—namely, the prais methodology. Both corrections are included in Table F. ' It should be noted that the discussed corrections are only reliable in correcting the relevant Violation, while maintaining all other OLS assumptions. Since several violations occur, this places the model’s validity in question. Alternatives such as using the log of included variables were explored without improvement. Table F : Summarizing Results for Various Regression Methodologies for Model (2) Original VWLS Newey- Prais OLS OLS OLS Bootstrap on Var OLS (Wooldridge) .5725841*** .629900*** .5725841*** .7416259*** .5857918*** (01999) (01836) (0343777) (.0180436) (.0084467) (0426939) (.0266978) . (001059) (.0009598) (.0019336) (.0011809) (0004543) (.0025166) (0034314) 043553*** .03813*** 043553*** .0327162*** .0419298*** 043553*** 0440744*** (0019337) (0017505) (0024793) (0011579) (0007423) (.0034615) (0020249) -.05346*** —077324*** .O315686*** -.073828*** —.048788*** (.0136328) (0122927) (.0169688) (0129135) (0052837) (.0229652) (.0311198) -.00005*** —.000063*** —.000026*** -.000060*** —.000063**'* ' (4.86e-06) (4.41e—06) (8.80e-06) (3.21e—06) (2.23e—06) (.0000116) (0000132) (.0080001) (0072409) (.0060794) (.0070498) (0019229) (.0079819) (.0111286) (0001443) (000131) (.0001697) (000075) (0000435) . (.0001819) (000207) .0480189*** .03606*** .0480189*** .0536398*** .0428135*** .0480189*** 0101375*** (.007906) (.0072436) (0091743) (.0020146) (.0026215) (.0107161) (.0082531) 0*t (.000369) (0003349) (.0003671) (0001974) (.0001162) (.0005216) (.000872) Cons .8697919*** .94308*** .8697919*** 1.177982*** .8990071*** .8697919*** .5172239*** ( 0443734) (.0399285) (.0869168) (.1123956) (.0188404) (1125005) (.1816025) Notes: Standard Deviations are in parenthesis. R2 and F—stat did not significantly deviate throughout the varying methodologies. * * * Significant at the .01 level I I V. ANALYSIS AND IMPLICATIONS Listed again in the model: C=l31+BZS+B3S2+B4SG+I35St+B68r+B762+B36t+B9t+B10t2 (2) From this model, it clearly follows that the “Greeks” are as follows: Delta: 0003 = fi2+2B3S+ 1346+ [55H [561‘ (3) Vega: dC/do = fi4S+2B7o+ list (4) Theta: —aC/at = -(|35S+[386+B9+2B10t) (5) Rho: aC/ar = [368 (6) Each graph below plots the estimated “greek” over the life of the option, evaluated at each day I with the same historical data used in estimating the various Bi. The following graphs use the Wooldrz'dge VWLS estimated fli: W MSFT April 2008 Call—Optiona K=$3O Delta(3): As seen from the chart, the model’s delta remains positive throughout the life of the option—confirming my expectations of dC/dS being positive, The significant fluctuation in delta is likely a result of the stock price varying between being in-the-money and out—of-the-money. It appears that when the stock price is out-of—the-money delta is less significant than when the stock price is in—the-money, where delta approaches 1.00. This is rather intuitive. (Note: A graph of the corresponding stock price is given later for comparison). Vegafiél): Vega is positive a majority of the option’s life—again, confirming the initial hypothesis. Interestingly, vega is higher when the option is out-of—the-money and conversely lower when the option is in—the-money. Again, this is somewhat intuitive—namely, a call-option has more positive responsiveness to increased volatility when out-of-the-money than when in-the-money. Theta(5 2: Theta is almost entirely negative over the life of the option—providing validity to the initial hypothesis. Furthermore, it appears that theta becomes much more influential the last third of the option’s life. This indicates that as an option nears expiry, time becomes more of a factor in how market participants value the option. Rho: The results here are probably the least valid and hence the least interesting. The restricted model i used (2), eliminated all terms involving r except for [3681: This, in effect, ties 6C/dr = [56S directly to the stock price, yielding a trivial rho. Theory and the literature suggest rho is the least influential of the greeks, so it comes as no surprise that an empirical regression approach is unable to detect this subtlety. As seen from an extremely large R2, above .99, the model’s predicted pricing closely follows the historical data, as shown below: 9 8 7 6 5 4 I—‘OI—‘NUJ The model loses its accuracy towards the very end of the option life, likely due to it being out-of- the-money and near expiration. MSFT Stock Price Strikeé VI. CONCLUDING THOUGHTS Even with its limitations due to violated assumptions, the resulting greeks’ general conformity to theoretical intuition provides a good measure of validity to the formulated model (2). Further work might be pursued in several directions. A comparison of using strike prices exclusively in-the-money or out—of—the-money with the results obtained here could be desirable in terms of resulting greeks’ characteristics. It would also be'informative to observe the model’s success in correctly explaining future years’ MSFT April K=$30 call option prices. Perhaps a panel-data approach could be used, including varying strike-prices over time, in attempts to produce a more general model only specific to the asset itself. I would also be interested to use this approach on options over longer periods, as well as options on other non-equity assets. Dr. McDonald, thank you for your help this semester- You are a tremendous example of kindness to all those around you! VII. REFERENCES Rubash, Kevin. “A Study of Option Pricing Models.” Bradley University. http://bradley.bradley.edu/~arr/bsm/pg05.html Macbeth, James D., Merville, Larry J. 1979 “An Empirical Examination of the Black-Scholes Call Option Pricing Model” The Journal of Finance Bhattacharya, Mihir. 1980 “Empirical Properties of the Black-Scholes Formula Under Ideal Conditions” The Journal of Financial and Quantitative Analysis I Cetin, U., I arrow, R., Protter, P., Warachka, M. 2006 “Pricing Options in an Extended Black- Scholes Economy with Illiquidity: Theory and Empirical Evidence” Review of Financial Studies VIII. APPENDIX 1. DATA REFERENCES Historical Option Prices: Wharton Database, OptionMetrics Stock—Prices: http://finance.yahoo.com 30-Day Treasury Bill Rates: www.ustreas. gov V II. Original Model OLS reg callprice SminasK SMERHEKE SmdnusKVolatility SminusKtimeleft SminuaKrate > Vhlatility Vblatilieyfi Vblatilitgtimeleft Volatilityrate timeleft time1eft2 > timeleftxate rate ratez Source i 55 df MS ' Number 0f obs = 168 ————————————— -+—————————————————————————————— Ft 1%, 153} = 6316.07 Model 3 321.5?TT53 1% 58.6341256 Prob > E = 0.06D0 Residual : 1.%215538 153 .009291235 R—squared = 0.3933 ——————————— —-+————————————————————-———————— Adj Rfsquared = 0.9931 Total | 822.999317 16? é.92813962 RDDt.MSE = .09639 callprice 1 Beef Std Err t P>it1 [95% Conf. Interval] ___________ __+___________.______________..._.._________..___.__________________________ SminusK 1 .5660?58 .0293932 2?.76 0.009 .5257372 .6063644 SminusKZ l .QfiiQSGS .8020471 29.48 0.600 .DBTBBES .0459743 SminusKUDEMy | —.0332T39 .0106337 —3.11 0.082 —.0543953 —.012162% SminusKtimvt [ —.DQ08147 .0502041 —3.EQ H.600 —.0fl121?9 —.DOD€116 Sminusfirate 3 .0450fi12 .Dil?619 3.30 0.008 .0216Q68 .6684755 Vblatility 3 .133354? .1036932 1.00 6.320 —.16149&? .3082042 Unlatilityfi 3 —.G831187 .0296599 —2.97 8.003 —.1&6?145 —.0295223 'vnlatilitgat ; .0041195 .0008361 4.Q3 0.0Q0 .0029676 .UGB?71& $blati11ty~e 1 —.85430?9 .DéEBZEé —1.10 9.2T5 —.1523%19 .0435262 timeleft 1 .QEQOQDT .DDETSEE 10.73 D.QQG .0155185 .DZZQSES timelefEE 1 —.9000564 8.525—06 —6.62 U.QDO —.UGODTBE —.0069396 timeleftrate i —.0003T3 .DU09112 —D.91 0.683 —.DD£1T31 .0Q142T2 rate 3 .1133781 .1143252 1.04 0.30U —.165381T .3445379 ratez 1 —.012168 .OBélSfiQ ~fl.35 Q.722 -.OT96085 .0552726 _c0ns 1 .7154455 .124982 5.72 0.0fi0 .4635322 .9623533 III. Restricted Model OLS reg callprice SmimusK SminusKE SminusKVulatility 5mimusKtimeleft Sminusflrate > anatilityZ'Valatilitytimeleft timeleft Sminuus SminusKV01~y Sminusfitimwt Vblatilitgz Vblatility~t timeleft timeleftz _cons I i I E 3 SminusKrate ] I I I l i | O on 13 p.33. U1 _ LII 0) F" | C) D Ellie-’14 emu ‘00 [U y: (El II) D M D COM LQN | C1 h—J “J (J) N 1.5 to .0032326 .018?337 —.Qfl0063? .36B7919 IV. Likelihood Ratio Test Unrestricted Model: estat is Restricted Model: estat ic timaleftz df HS 9 91.2716fi72 158 .00883?%15 167 3 32313962 Std Err t P§|ti .QlSQB 28.6% 0.000 .UUZQEET 22.52 0.039 .Oflaflflfll —3.99 0.090 .Oflfliéés —6.94 0.006 .007986 6.07 0.030 .0136328 ~5.6? 0.890 .000369 8.39 0.0DU .001659 18.69 0.000 $.36e—06 —13.11 5.000 .0443734 18.60 0.0QU Elfinallz llfimndell —371.BE? 162.@8é lltnullj lltmmdelj —3T1.857 154.9846 Number of abs = 163 Ft 9, 1537 = 9278.01 Prob > E = 0.0000 R—squared = 0.9981 AdeR—squared = 0.9980 Root HSE = .09913 {95% Eunf. Interval} .5331019 .6120652 BEETSES .flé?3?2i —.Gé?683 ~.0160812 —.DDl§373 —.QDU?1?1 .0324036 .063634 —.104250% —.DSQ3982 .0025538 .0090114 .01?5871 .flElBEOfi —.DUDU?33 —.fl0005é1 .7821503 .9574335 REE BIC +2E%.BETH —2&E.1385 RIC EEC —28L.9691 —253.7295 Therefore, 2(162.484—154.9846)= 14.998 8~ch12(5). Hence since Prob>ch12=0.010371 and we fail to reject null at 99% level. V. JB Test predict e,resid Skewnessturtosis tests for Harmality ————— —— gain: —————— 'Variable l abs PriSkewmess} Prfifiurtnsisy adj chiZEE} Prab>ch12 e 5 168 Q 196 Q Q00 13.36 D ODUfi VI. Tests for Homoskedastici’gg Modified White Test: estat hettest Chat chatE, fatat Breusch—Eagan f flock—Weisberg test for heteroskedasticit; Ha:aflonstamt variance variables: chat chat? 3.?8 FEE , 165] F 0.02%3 Prob } H Breusch-Pagan Test: restat fiercest animus? Smimfiszz EmiflusKWDEEtility SminmsKtimeleft Sminusfirate 1 > Vblatiiityz'Vblatilitvtime aft timaleft timelef32, fatat Breusch—Pagan f Cook—Waisberg test far heternskedasticity fin: Constant variance I variables: SminusK SminusKE Sminusfivolatility Sminusxtimeleft Sminusfirate'flblatilitgfi Vblatilitytimeleft timeleft timeleftz ELB , 1531 = 2.92 Prob :> F = 0.0032 VII. VWLS 2 A roaches White’s Suggested Test: »ice SmiuusK SminusKZ SminusKVolatility Sminusxtimeleft SminusKrate r “tyz volatllitycimeleft timeleft timeleftz, sdfieabs) Variance—weighted least—squares regression Number of-obs = 168 Goodness—af—fit 0012(158) = 157.59 Model 0012(9) = l 3e+06 Prob > 0012 = 0.4943 Prob > chi: = 0 0000 callprice i Coef. z P>lzl {05% Conf. Interval} SminusK 1 .5057018 .0084067 69.35 0.000 .5692366 .602347 SminusKZ 3 .0419290 .0007023 56.49 0.000 .040075 .0033846 SminusKV01~y 3 —.031%935 .0019229 —16.38 0.000 —.0352625 —.027?24T SminusKtimyt 3 —.000919 .0000035 ~21.12 0.000 -.0010043 —.0008337 SménusKrate 1 .0028135 .0026215 16.33 0.000 .0376750 .0é7QSZE Vblatilityz 1 —.0?38250 .0052337 —13.97 0.000 —.08€1843 —.0634725 'Voiatility~t 3 .00320é7 .0001162 27.59 0.000 .002977 .0030320 timeleft i .0191524 .0004543 42.16 0.000 .018202 .0200428 timeleftz 1 —.0000606 2.23e—06 —27.16 0.000 -.000065 —.0000562 cans : .8090071 .0108004 07.72 0.000 .3620307 .0359335 Wooldridge’s Suggested Test: gen predictedsig = (exp(—.033308*timeleftj}“.5 vwls callprice SminusK SminusKZ SminusKVblatility SminusKtimeleft 5minusKrate > Volatilityz valatilitycimeleft timeleft timelaftz, sd(predictedsig} variance—weighted least—squares regression Number of’obs = 168 Goodness—of—fic chiztlsa} = 303.13 Mude1 chi2[9} = 2.6e+05 Prob-> chi” = 0.0000 Prob > chi: = 0.0000 callprice 3 Coef Std 3:: z P>|zl [95% Covf interval: Smiausi.1 .7015259 .0130436 41.10 0.000 .7062612 .??6990? SminusKZ g .0327102 .00115?9 23.15 0.000 .0304060 .039905? SminflsKVOl~y : —.0423601 .0070493 —6.01 0.000 -.0561*66 —.0285517 Sminasfitime: 1 —.0015306 .000075 <20.52 0.000 —.001£855 —.0013917 SminusKrate 3 .0530398 .0020146 26.62 0.000 .0496911 .05?583% Vblatilityz 1 .0315606 .0129135 2.44 0.015 .0062506 .0568786 finiatility~t : .0010393 .0001070 5.27 0.000 .0006525 .0010262 timeleft 1 .0123?03 .0011003 10.43 0.000 .0100550 .0106349 timelefcz | —.0000266 3.21e—06 —0.29 0.000 —.0000329 -.0000203 _cons | 1.177932 .1123956 10.43 0.000 .9570902 .1.393273 VIII. Tests for Autocorrelation estat bgadfrey IBreusch—Godfrey LH test for autucorrelation lagsip] l ch12 ‘df Prob > 0012 1 l 63.996 1 0.0000 Durbin-Watson (before and after prais) Prais—Wiusten.fiR[1] regression —— iterated estimates Source 1 55 df HS Number of abs = 163 ———————————— —+-————————————————---—————————— F: 9, 150} — 791.45 Hadel i 26.4195769 9 2.03550855 Prob > F = 0 0000 Residual 1 .586029124 158 .003709005 3—squared = 0 0?33 ———————————— —+————————————-—-————--————————— .Adj R—squared = 0 BTTl Total 1 2T.0055061 16? .161710216 Root MSE — 0609 callprize I Coef Std Err t P>iti [95%EConf. interval] Sminusfi 3 .4335223 .0266378 18.20 0.000 .3807922 .485253€ SminflsKE 5 .00é0790 .0020269 21.77 0.000 .0000?5 .0080738 SminusKVbl~y 5 —.0175235 .0111286v —1.5? 0.117 —.0BBEQSE .ODQQEES SminusKtimrt ! .000?102 .000207 3.43 0.001 .0003015 .001119 SainusKrate 1 .01013?5 .0032531 1.23 0.221 —.0061&31 .026433 Vblatilit_2 l —.043T834 .0311198 —l.57 0.119 —.1102529 .012676 Volatility~t I .001055‘ .000572 1.67 0.005 —.0002641 .0031803 timeleft | .0243511 .0034314 7.10 0.000 .0175730 .0311284 timeleftz | —.0000639 .0000132 —&.03 0.000 —.00009 —.00003?8 _can5 | .5172238 .1816025 2.05 0.005 .1535022 .3750056 ____ _________+___________________________________________-____________________ rho I 8501621 Durbin—Watson statistic cariginall 0.609055 Durbin—Watson statistic {transformed} 1.920170 Wooldridge’s Suggested Test: reg e 1.e Scarce 3 SS df MS Number af abs = 15? ———————————— ~+——~————————-—-———e—————-———-—— E( i, 165] = 133.04 Model 5 .556139253 1 .535139253 Prob > F = 0.0000 Residual g .724?65TB& 165 .00639252 R—squared = 0.0%?1 ——————————— ——+————-————————————————————-———— Adj R—squazed = 0.0038 Total 3 1.31090505 166 .00788?018 Root ESE = .06628 e I *Coef. Std. Err. t P>Iti [95%'Canf. interval] 8 I L1. 3 .6201702 .0536808 11.55 0.000 .5141636 .7261719 cons l .0035903 .0051289 0.70 ‘0.43 -.006536& 013717 ...
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call options - r; 1 AWVW ECON 388 — Intro to Econometrics...

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