This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: r; 1 AWVW ECON 388 — Intro to Econometrics ~Vol. 1, Issue No. 1 2009
pp. 1—15 Pricing a CallOption 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 optionpricing. 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 calloption premiums
observed in markets. For instance, the paramount BlackScholes 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 wellknown heat transfer equation (Rubash). The point
being made, although startlingly accurate under its assumptions, the BS 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 marketparticipants 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, (Tt) the time until expiry, r the annualized riskfree 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'ﬁTt’lﬂmﬁ} Theta: — rKeTﬂT‘ﬁ) N However, subsequent improvements have been made in relaxing assumptions maintained .
by the BS 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 BS 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 BS and market prices
primarily arose from ﬂuctuation in the variance of the asset’s return. Bhattacharya (1980)
showed how BS 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 optionpricingwthe 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 30day 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—themoney”
(below the current stock price) or “outof—themoney” (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 riskfree interest rate (r) to account for opportunity cost. An increased riskfree 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. BLlackScholes 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 CanPrice 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*** (4.86eO6)
.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 chisquare 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 tstatistics 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 BreuschPagan (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 varianceweighted 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**'* ' (4.86e06) (4.41e—06) (8.80e06) (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 inthemoney and out—ofthemoney. It
appears that when the stock price is outof—themoney delta is less signiﬁcant than when the
stock price is in—themoney, 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 outof—themoney and conversely lower when the
option is in—themoney. Again, this is somewhat intuitive—namely, a calloption has more positive responsiveness to increased volatility when outofthemoney than when inthemoney. 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 outof themoney 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 inthemoney or out—of—themoney 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
paneldata approach could be used, including varying strikeprices 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 nonequity 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 BlackScholes Call Option Pricing Model” The Journal of Finance
Bhattacharya, Mihir. 1980 “Empirical Properties of the BlackScholes 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
30Day 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 .05...
View
Full Document
 Winter '08
 Mcdonald,J
 Econometrics

Click to edit the document details