The estimation and usefulness of volatility risk premiums implicit in index options caused by asset price jumps

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The estimation and usefulness of volatility risk

premiums implicit in index options caused by asset

price jumps

Bachelor Thesis

Econometrics and Operations Research

Supervisor:

Prof. dr. H. Peter Boswijk

Student:

Daan Olivier Rotsteege

Student number: 10259384

Date final draft: 20-12-2013

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ABSTRACT

This paper tries to investigate whether a volatility risk premium implicit in options on the S&P 500 index really exists. The implied volatility is computed with the Black-Scholes model, the Merton model and is derived from the VIX, the realized volatility

of the S&P 500 index is calculated from 5-minutes returns. A significant positive volatility risk premium is found by all methods for implied volatility. Furthermore,

macro-finance variables can be used in predicting the next month volatility risk premium.

A relation with the S&P 500 volatility risk premiums, computed with the Black-Scholes and the Merton model, and the next month return of the Dow Jones Industrial

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3 3.1 3.2 3.3 THE MEASUREMENT OF IMPLIED VOLATILITY……. The Black-Scholes model………... The Merton jump-diffusion model………. The VIX index……… 7 8 8 10 4 4.1 4.2 4.3 THE VOLATILITY RISK PREMIUMS……… The relation between implied and realized volatility………. Descriptive statistics………... The cause of the volatility risk premiums……….. 10 10 12 13 5 PREDICTABILITY OF DJIA INDEX RETURNS BY MONTHLY S&P 5OO VOLATILITY RISK PREMIUMS.. ₁₅

6 CONCLUSION……….. 21

APPENDIX A ……… 23

APPENDIX B ……… 31

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ACKNOWLEDGEMENTS

Before the serious material is touched I want to thank some people, where my gratitude goes out to. For this reason, I want to pay special tribute to my supervisor and director of the Amsterdam School of Economics Research Institute: Prof. dr. H. P. Boswijk, for his great advice, lessons and time. Also not to forget are dhr. drs. R. van Hemert and dr. K. J. van Garderen, which made it possible to write my paper in a

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LIST OF TABLES

TABLE p.

S1 Ordinary OLS from the realized volatility on the Black-Scholes

and the Merton jump-diffusion implied volatility………. 11 S2 Descriptive statistics of the implied volatility from the

Black-Scholes model and the Merton jump-diffusion model………….. 12 S3 Descriptive statistics of the volatility risk premiums from the

Black-Scholes model, the Merton jump-diffusion model and the

VIX……… 14

S4 Mean = Zero test for the volatility risk premiums from the

Black-Scholes and the Merton jump-diffusion model………….. 14 S5 Ordinary OLS of the volatility risk premium from the

Black-Scholes and the Merton jump-diffusion model on the realized

volatility and one month lagged macro-finance variables……… 16 S6 TSLS with White-Standard Errors of the volatility risk premium

from the Black-Scholes and the Merton jump-diffusion model on the realized volatility and one month lagged macro-finance

variables……… 18

S7 OLS with White-Standard Errors of the volatility risk premiums from the Black-Scholes and the Merton jump-diffusion model

on the one month lagged macro-finance variables……… 19 S8 Ordinary OLS of the next month return of the Dow Jones

Industrial Averages on the volatility risk premium from the

Black-Scholes and the Merton jump-diffusion model………….. 20 S9 OLS with White-Standard Errors of the next month return of the

Dow Jones Industrial Averages on the volatility risk premium from the Black-Scholes and the Merton jump-diffusion

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1 Introduction

Trading options is basically trading in volatility. This is because an option gives the right to sell or to buy an underlying asset, but it is not an obligation. For this reason, there has been a lot of interest in previous studies as to whether implied volatility in an option’s price is an unbiased and efficient estimator for future realized volatility of the underlying asset. If implied volatility is an unbiased and efficient estimator then it is a good forecaster of the future return volatility over the remaing life of an option and that means that the option market is information efficient (Christensen & Prabhala, 1998, p. 126). Because prices of options are based on the prices of the underlying assets, it can be expected that the volatility implied from those options has a positive relationship with the volatility of the underlying assets. This intuition is confirmed by the study of Christensen and Prabhala (2003), which shows that

volatility implied by monthly S&P 100 index option prices is a biased estimator of the future realized volatility of the S&P 100 index, but that it does predict the future realized volatility. Other results are concluded by Canina and Figlewski (1993), who find that implied volatility for the S&P 100 index contains no information of future realized volatility of this index. The results of Canina and Figlewski are in sharp contrast with the results of Christensen and Prabhala, but the outcomes of Christensen and Prabhala are confirmed by other studies (Jorion, 1995; Day and Lewis, 1992).

Because the relationship between implied and realized volatility can be affected by the measurement of both volatilities and the data used, conclusions made in previous studies need to be approached carefully. For example, Christensen and Prabhala (1996) show that nonoverlapping data gives implied volatility more

predictive power for the realized volatility, than when overlapping data is used. Also they discuss the measurement errors in their implied volatility, which occur because the Black-Scholes option pricing model is used to derive the implied volatility of American options. However, even studies who take special account for these kind of measurement errors (e.g., Fleming, 1993) conclude that the implied volatility is a biased estimator for the future realized volatility. But Fleming finds like Christensen and Prabhala that the implied volatility is an upward biased estimator for the future realized volatility.

Alhough the exact relation between implied and realized volatility differs among studies, most of the existing literature shows that implied volatility is an

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upward biased estimator for the future realized volatility of the underlying asset. For this reason, the purpose of this study is to investigate the possible existence of a systematic volatility risk premium, i.e., a systematic difference between the implied volatility and the realized volatility. Option prices are positive correlated with the risk (volatility), thus an expected positive systematic risk premium means that option buyers pay a premium above the fair price, that is the price of the option calculated with the ex post volatility of the underlying asset.

Traditionally, implied volatility is computed with the Black-Scholes model. But over the years it has become clear that the market does not price all options with the Black-Scholes model, since different values of implied volatility are extracted from options on the same underlying asset with various strike prices and the same exercise date (Mayhem, 1995, p. 14). This contradicts the constant volatility assumption of the Black-Scholes model. The phenomenon that implied volatiliy varies with the strike price is also know as the “volatility smile”, which is possibly caused by stochastic volatility of the underlying asset is (Heston, 1993, p. 328). If the underlying asset follows a pattern with stochastic volatility, there maybe are better models in estimating the future realized volatility than the Black-Scholes options pricing model. One can think at the Heston stochastic volatility model or models with other, less static assumptions than the Black-Scholes model. The Merton jump-diffusion model is an example of the latter case, as it is an extension of the Black-Scholes model and allows for jumps in the price of the underlying asset. Today it is well known that the prices of assets can make unexpected discontinuous jumps (Tankov & Voltchkova, 1992, p. 2), so the Merton model can possibly give better close to real world estimates of the future realized volatility than the Black-Scholes model.

The way to compute realized volatility has also been the subject of various studies, because different ways in computing it can cause different conclusions on the predictability of implied volatility. Zhu and Shang (2003, p. 88) conclude that the forecast ability of implied volatility on the realized volatility can be improved by using the most accurate measurement of realized volatility. Realized volatility that is created from 5- minutes returns gives in the study of Zhu and Shang the best relation between implied- and realized volatility in comparison with other estimators of the realized volatility. The realized volatility calculated from high frequency data, like the

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5-minutes returns, is also called “model-free” realized volatility. This study will use the model-free realized volatility in computing the risk premium.

Under the hypothesis that implied volatility is an upward biased estimator of the future realized volatility of the option’s price, option traders are systematically estimating the future realized volatility higher than what was expected from the ex post realized volatility over the same period. The study of Bollerslev, Gibson and Zhou (2011) examines the causes of risk premium derived from the VIX, which is a volatility expectation index of the S&P 500 index, and find that this premium is for one part predicted by macro variables. Furthermore, they find that the volatility risk premium has a relation with the next month’s return of the S&P 500 index and suggest for future research to investigate this same relation between the volatility risk premium implied for the S&P 500 index and other markets return. This study goes further on the research from Bollerslev, Gibson and Zhou and the volatility risk premiums are therefore estimated from S&P 500 index options. To avoid that the estimated risk premium becomes significant through misspecifications in the implied volatility, two methods are used in computing them: the Black-Scholes and the Merton jump-diffusion model. The implied volatility and hence the volatility risk premium from both models are compared to their estimates and will be studied in further depth throughout. The final goal of this paper is to show a possible relation between the next month return of the Dow Jones Industrial Average (DJIA) and both risk premiums from the S&P 500 index.

The remainder of this paper is organized as follows. The next Section

describes the data and the sampling method used to compute the implied volatility and the realized volatility. Section 3 describes the Black-Scholes option pricing model and the Merton jump-diffusion model, subsequently the implied volatilities and the

volatility risk premiums are computed and analysed in Section 4. The relation between the risk premiums of the S&P 500 index and the DJIA is examined in Section 5, Section 6 concludes.

2 Data and sampling method

This study investigates whether there is any significant risk premium in the implied volatility of S&P 500 index call options, with a remaining lifetime of exactly one

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month at the measurement date. S&P 500 options are European style, which makes the interpretation of the implied volatility simpler than American options because there is no possibility of early exercise. Christensen and Prabhala (1996) point out that the early exercise problem can lead to misspecifications in the exact relation between implied and realized volatility (pp. 131-133).

The sample period used in this study is from April 21, 2006 to May 17, 2013. The reason that not the beginning of the month is taken as start is because S&P 500 index options can expire or arise once a month, on the third Friday. The implied volatility is computed from the nearest at-the-money (ATM) call option on the first Wednesday following the expiration date of an option. As said, this option ends in the upcoming month. This is because the monthly sample data needs to come from non-overlapping time series. Canina and Figlewski (1993) show that non-overlapping time series may lead to truly no correlation between the implied and realized volatility. The realized volatility for this monthly S&P 500 index ATM option is based on the 5-minutes returns of the S&P 500 index during the life of the last “option-month” until expiring. The option-month is referred to the period between the third Fridays among two following months.

3 The measurement of implied volatility

Implied volatility can be derived from a various number of models. Undoubtedly the best known model is the Black-Scholes option pricing model – henceforth BSM – , but it has been shown in previous research that it has some disadvantages in

comparison with the newer and more refined models for implied volatility. This is because the the BSM makes the assumption that the return of the underlying asset follows a normal distribution with known mean and variance. But Fama (1965) observes that in general stock market returns does not follow a normal distribution, because the observed returns are skewed and have a higher kurtosis than a normal distribution. Therefore it is interesting to investigate the possible difference in implied volatility when this is derived from other more sophisticated models, which loses some of the assumptions of the Black-Scholes model. One can think of the Merton jump-diffusion model – henceforth MJD – , which is an extension of the BSM and it assumes that the market information can be splitted into two groups and modeled differently. The first group is the arrival of normal market information and this is

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modeled with a diffusion process. The second group is the arrival of abnormal market information, which are the amount of sudden changes in an asset’s price, also known as jumps, and this group is modeled with a Poisson process (Nafas, 2003, p. 2).

3.1 The Black-Scholes model

If the price of an asset is lognormally distributed with constant volatility, !, then the return of this asset is normally distributed. This means that the price of a current European call option on this asset can be computed with the Black-Scholes formula. Because the BSM also assumes that de asset does not pay any dividends prior the expiration date of the option, one needs to adjust the price of the asset, so that it is dividend exclusive.

Let S0 be the current price of the asset, K the strike price of the option and T

be the time to maturity. Furthermore, let r be the risk-free interest rate and C0 the

unknown price of the option, then:

!_!"(!, !, !, !, !) = ! !_!!! !_! − !!!!!!!!_!!(! !) ! ! = ! 1 !2!! ! !!! _! ! !! !!" !_! = !!log!(!! !) + !!! ! ! +! ! ! 2 !!!!!!!!,!!!!!!!!! = ! !!− !! !

To compute the implied volatility from the Black-Scholes pricing model, the price of the option !_! is taken from the market and ! is computed by setting !_!− !_!" to zero.

3.2 The Merton jump-diffusion model

Because the Merton jump-diffusion model allows for jumps in an asset’s price, it may be expected that this model can fit the returns of an asset better than the BSM. But the possibility of stochastic jumps in the price implies that the market is not complete anymore, because the payoff of the option cannot be fully replicated. To deal with this problem, Merton (1976) makes the assumption that the risks of the jumps are

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with unknown mean, !, and variance, !, then the Merton option pricing formula can be expressed as a sort of adjusted infinite sum of the BSM.

Let S0 be the price of an underlying asset, K the strike price of the option, T the time

to maturity and r the risk-free interest rate. Then the Merton jump-diffusion model takes the following form:

!_!(!, !, !, !, !; !, !, !) = ! !!! !_! !(!!_!)!_! !! !!"(!!, !, !!, !!, !) ! !!!!! !_! = ! +!! ! ! +!!! − !!"!! !_!! _{= ! !}!_+!! !!!! !! _{= !! 1 + ! = !!!} !!! ! ! !

!!" is the Black-Scholes option pricing model defined before. The Merton model has

three unknown parameters in addition to the Black-Scholes option pricing model: !, ! and !, which are the parameters of the jump process. The parameter ! is the mean amount of the jumps and the jump sizes are lognormally distributed with parameters ! and !. If the volatility, !, is also taken as unknown then the model has four

parameters to be estimated. If four prices of options with the same time to maturity are observed from the market, then these parameters can be solved by four equations. Subsequently, the implied volatility can be computed from ! with the following formula.

!"# !log! !

! 0 ! = !! !!+ !!!(!!+!!!)

So an extra component is added to the !!_{from the the first four equations, in order to}

take into account for the variance of the jump process. The result can be seen as the total variance of the return of the underlying asset under a jump-diffusion process and thus the implied volatility is the square root of the right hand side of this equation.

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3.3 The VIX Index

In this research, the risk premiums are derived from the BS and the MJD model, but for comparison also from the VIX. This index is provided by the Chicago Board of Options Exchange (CBOE) and is simply a volatility expectation index of the S&P 500 index. The base date of the VIX is January 1990, but the way of calculation of the volatility has changed in 2006. Because the time period of this research starts in April 2006, only the newer computation method of the VIX index is used. This method computes the 30 days volatility of the S&P 500 index from a various number of put- and call-options and therefore incorparates the “volatility smile”. This volatility smile is named after the interesting smile patern of implied volatility which arises when implied volatility is computed from different options with different strike prices, but with the same maturity. The VIX is therefore a beautiful complement to the Black-Scholes model and the Merton jump-diffusion model, because of the three different ways of computation the implied volatility.

4 The volatility risk premiums

In this study, the volatility risk premium is defined as the difference between the volatility implied from S&P 500 index options and the ex post realized volatility of this index over the same timeline. A monthly sample period is used for the Black-Scholes implied volatility and the Merton jump-diffusion implied volatility, hence the implied volatilities are based on options with a remaining life of one month at the beginning of this period. This sample month is not a calendar month, but the period between two consecutively third Fridays, because options on the S&P 500 index can only expire only on the third Friday of the month. The volatility risk premium

constructed from the VIX is based on daily differences between the opening values of this index (divided by 100) and the realized volatility from the S&P 500 index over the same trading day.

4.1 The relation between implied and realized volatility

Before the risk premiums are studied it is helpful to get an intuitive idea of the implied volatilities itself. For this reason the realized volatility is regressed on a constant and the implied volatility from the Black-Scholes and the Merton model. From this regressions it is possible to see whether the used methods for implied

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volatility are good estimates of the realized volatility. The VIX will be excluded in this regression, because it is commonly known that this index is a good predictor of the realized volatility of the S&P 500 index for the next 30 days and research in the relation among the two is therefore superfluous.

Table S1: Ordinary OLS from the realized volatility of the S&P 500 index on the Black-Scholes and the Merton implied volatility from the most ATM S&P 500 index options. Both implied volatilities are signifcant at a 5% level.

If the implied volatility is an unbiased estimator for the realized volatility, then the coefficients for the constant and the relative implied volatility should be 0 and 1 respectively, in the above regression. However, both estimates of the implied

volatilities are less than 1. The striking part of this regression is that the coefficient of the Merton implied volatility is less than the coefficient of the Black-Scholes implied volatility, thus the presumption that a model with the possibility of stochastic jumps in the underlying asset’s price is a better forecast for the future realized volatility is invalidated in this case. Such kind of result was also found by the research of Shu and Zhang (2003), which found that even a stochastic volatility model did not improve the relation of the implied and realized volatility in comparison with the Black-Scholes model. The Merton jump-diffusion model does not have stochastic volatility, but it is in someway an stochastic extension of the Black-Scholes model.

The implied volatilities from both models are highly significant with a p-value of 0.0000 and because the coefficients are less than 1, it should be clear that the implied volatilities overestimate the future realized volatility. Further tests on the relation between realized volatility and implied volatility will not be done, because there has been extensive research on this subject in the past. These studies have shown that the implied volatility is in general an upward biased estimator for the realized volatility and the outcomes from the regression above confirm this conclusion.

OLS Realized(Volatility Realized(Volatility

C .0.0056 (0.0134) 0.0031 (0.0119) Black.Scholes(Implied(Volatility a_0.8813 _(0.0596) _. _. Merton(Implied(Volatility . . a_0.7420 _(0.0459) R.Squared 0.7275 0.7612 F.Statistic((Prob.) 0.0000 0.0000 a_{(p.value(<(5%}

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4.2 Descriptive statistics

It is helpful to look at the descriptive statistics of all the implied volatilities together with the realized volatility before the descriptive statistics of the risk premiums. I start by describing the statistics of the Black-Scholes model and the Merton jump-diffusion model, which are both described in Table S2. The describing statistics of the VIX index are found in the Appendix (Table A1). Some comments need to be made about the use of the Black-Scholes model, directly but also indirectly through the usage in the Merton model. This is because the Black-Scholes model applies to European call options on an asset that does not pay any dividends before the expiration date. The options on the S&P 500 index are indeed European, but the index does pay dividends. Because dividends reduce the value of a call option, it is a misspecification to use the market value of this option without reducing the future dividends. So when the Black-Scholes model is used to compute the implied volatility from options where the underlying asset does pay dividend, it underestimates the true implied volatility. In this research, the S&P 500 index is not adjusted with the present value of the dividends, over the remaining life of the option. This is because the source used for the option data has no dividend adjusted option prices available for the period 2006-2008. The found implied volatilities are for this reason not correct, but assuming that the dividend payments on the S&P 500 index are relatively uniform, the difference between the non-dividend adjusted implied volatilities with the real values should be more or less constant (Christensen and Prabhala, 1998, pp. 131-133). However, implied volatilities in this research and hence volatility risk premiums are upward biased1.

Table S2: Statistics of the implied volatility from the Black-Scholes and the Merton jump-diffusion model are shown together with the realized volatility, both annualized.

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

1_!_{Dividends on the S&P 500 index in the sample period are mostly around 2% on annual basis,}

reaching its peak in the beginning of 2009 with more than 3%. !

Statistic Implied(Volatility Implied(Volatility Realized(Volatility

Black3Scholes Merton 5(Minutes

Mean 0.201 0.227 0.172 Median 0.167 0.182 0.140 Standard(Deviation 0.103 0.125 0.107 Skewness 2.089 2.472 2.350 Kurtosis 6.447 9.817 6.890 Min 0.078 0.085 0.071 Max 0.702 0.908 0.631

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The mean of the Black-Scholes implied volatility is around 15% larger than the realized volatility, but the standard deviation from the mean is nearly the same for both. Another remarkable observation is that the Black-Scholes implied volatility and the realized volatility are both are highly skewed with almost the same magnitude, furthermore they both are leptokurtic. The implied volatility of the Merton jump-diffusion model has a larger sample mean and standard deviation than the realized volatility but approximately the same skewness. The implied volatility from the Merton model is also leptokurtic. The Merton implied volatility is more upward biased for the realized volatility than the Black-Scholes implied volatility, as can be seen from the previous regression and the descriptive statistics.

Information about the volatility risk premiums can be found in Table S3, where also the VIX risk premiums are included. In this table, the mean risk premiums are the same as the difference of the mean implied volatilities and the mean realized volatility, from Table S2. However, this table also includes new relevant information about the risk premiums, for example the standard deviations. It is striking that the standard deviations and the mean values from the Merton model and the VIX are relative the same, but that the Black-Scholes volatility risk premium is on average around 50% smaller than the Merton volatility risk premium. Another remarkable fact is that the kurtosis of the VIX risk premium is much higher than the other two models but that the skewness is more or less the same. This means that the VIX has a lot of extreme values in comparison with the realized volatility. The volatility risk

premiums from the VIX, the Black-Scholes and the Merton model are depicted in Figures A3-A4 of the Appendix. Intuitive can be concluded from this figures that the volatility risk premiums are positive over time, especially the risk premiums from the VIX are significant positive. But the question remains whether the risk premiums of the Black-Scholes and the Merton model are significant. A test if the means of these volatility risk premiums differs significantly from zero is for this reason done and test results are placed in Table S4. The t-statistics are highly significant for both models at a 5% level, even with the assumptions of dependent data. The null-hypothesis of a zero population mean can therefore be rejected and it may be assumed that significant positive volatility risk premiums in S&P 500 index options exists.

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Table S3: Statistics of the annualized volatility risk premiums, the VIX is also included. All risk premiums are negative skewed and have a high kurtosis. The mean and the standard deviation from the Black-Scholes volatility risk premiums are around 50% smaller than the other two volatility risk premiums.

Table S4: Above are tested whether the means of the Black-Scholes and the Merton volatility risk premium are significant different from zero. All t-statistics are significant at a 5% level and it may be assumed that the volatility risk premiums are significant positive.

4.3 The cause of the volatility risk premiums

From Figure A3 of the Appendix, it becomes clear that the risk premium follows an alternating pattern, where volatile periods are succeeded by periods with smaller fluctuations. As seen, the implied volatility is highly correlated with the realized volatility (Table S1) and thus it is expected that strong jumps in the realized volatility cause high volatility risk premiums. In Figure A1 of the Appendix, time series for the calculated implied volatilities are combined with the realized volatility. Large

negative risk premiums are from periods where the future realized volatility suddenly rises, so that the realized volatility is much greater than the volatility implicit in options at the beginning of the month. In general, the implied volatility is much larger than the realized volatility in the subsequent months after the sudden rise of the

Statistic Volatility(Risk(Premiums

Black3Scholes Merton VIX

Mean 0.029 0.055 0.065 Median 0.036 0.060 0.070 Standard(Deviation 0.057 0.061 0.066 Skewness 32.251 30.004 32.253 Kurtosis 12.190 6.101 18.274 Min 30.273 30.143 30.739 Max 0.140 0.292 0.375 Mean=0'Test

Black/Scholes T"Statistic Prob..T"Statistic

No.Autocorrelation 4.7361 0.0000

Consistent.Standard"Errors.for.Autocorrelation* 4.3837 0.0000

AR(1).process 4.2593 0.0001

Merton T"Statistic Prob..T"Statistic

No.Autocorrelation 8.2956 0.0000

Consistent.Standard"Errors.for.Autocorrelation* 8.4730 0.0000

AR(1).process 8.7045 0.0000

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realized volatility, which causes for high positive risk premiums in these periods after the large negative volatility risk premiums.

The monthly realized volatility of the S&P 500 index is quite dependent on macro events in the same month in U.S. (Table A4, Appendix). This is very logical considering that the S&P 500 index is an index with the most important firms of the U.S., such that this index mirrors the U.S. market in whole. The striking part of the time series of the volatility risk premium is that the first and last sample year have much lower volatility than the period between these years (Table A3, Appendix). The high volatile period falls together with the outbreak of the global financial crisis and the general unrest in the world markets, this creates the presumption of macro event correlated volatility risk premiums. Hence, regressions are done to predict the volatility risk premiums from the Black-Scholes and the Merton model, explained with the realized volatility and some important macro-finance variables of the

previous month. The reason that the macro-finance variables from the previous month are used together with the realized volatility from the next month as explanatory variables, is because in first instance I am interested in the direct effect of these macro-finance variables on the volatility risk premiums. Table S5 contains the ordinary OLS regressions (information about the macro variables can be found in Table A3 of the Appendix).

The realized volatility is highly significant for the Black-Scholes volatility risk premium at a 5% level, with a t-value of -6.332 and hence a p-value of 0.0000.

Furthermore, the P/E ratio of the S&P 500 index and the Moody’s AAA corporate yield are both also significant at a 5% level for the BSM, with a t-value of -4.470 and 2.427 respectively. The other variables are insignificant under the same level, but they show some relation with the Black-Scholes volatility risk premium. From the Merton jump-diffusion model risk premium, quite the same conclusions for the variables with respect to the significance can be made, the only difference between both models is that the significance is more clear from the Black-Scholes model. The realized volatility has a t-value of -2.741 with a p-value of 0.0077, the P/E ratio has a t-value of -3.918 with a p-value of 0.0002 and the Moody’s AAA corporate yield has a t-value of 2.477 and hence a p-t-value of 0.0155.

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Table S5: Ordinary OLS regression of the volatility risk premiums from the Black-Scholes and the Merton model on a constant, the realized volatility of the same month and several one month lagged macro-finance variables from the U.S.

The R-Squared values are pretty low for both volatility risk premiums, this is argumentative because the risk premium is the difference between the implied volatility and the realized volatility and of course is the implied volatility omitted as explanatory variable, for the reason of trivial solutions. The goal of the two above regressions is to see whether the height of the volatility risk premium is caused by macro events from the previous month, which I tried to be expressed in the macro-finance variables. But because the realized volatility is highly correlated with these lagged macro variables (Table A4, Appendix), it is also included in the explanatory variable set for the reason of the omitted variable bias in the estimators, when it was excluded.

From the above regressions it becomes clear that the macro-finance variables from the previous month do say something about the volatility risk premiums. Moreover, it is evident that the realized volatility has a negative effect on the risk premium. In terms of the sample period, large negative risk premiums arose in months where the realized volatility suddenly increased. Because the realized volatility of the S&P 500 index is likely affected by many kind of global market events, the possibility exists, which is likely, that the realized volatility as explanatory variable is

endogenous and hence it is correlated with the random variation in the error terms. Normal OLS is not consistent in the case of endogenous variables and conventional tests, such as the t-tests, are not valid. To check this possible misspecification of the model, a Hausman LM-test on exogeneity of the realized volatility is done (Table A5, Appendix). One month lagged macro-finance variables are used as instruments, which are all assumed to be uncorrelated with the disturbances. The outcomes of the LM test

OLS Volatility(Risk Premiums

Black3Scholes Merton C 30.3016 (0.5970) 30.0424 (0.7427) Realized a_30.4357 _(0.0688) a_30.2346 _(0.0856) Housing(Start((31) 30.0004 (0.0002) 30.0004 (0.0003) P/E(Ratio(S&P(500((31) a_30.0136 _(0.0030) a_30.0148 _(0.0038) Employment(Growth(Rate((31) 311.4786 (8.4086) 34.3423 (10.4608) Moody's(AAA(Corporate(Yield((31) a_0.0248 _(0.0102) a_0.0315 _(0.0127) Producers(Price(Index((31) 0.9081 (1.4926) 0.8623 (1.8568) Average(Wage((31) 0.0519 (0.0564) 0.0250 (0.0702) R3Squared 0.4617 0.2761 F3Statistic((Prob.) 0.0000 0.0008 a_{(p3value(<(5%}

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statistics are for the Black-Scholes and the Merton volatility risk premium

respectively 4.209 and 9.317. The LM statistic is chi-squared distributed with one degree of freedom and both test statistics are significant under a 5% level. Therefore, the null-hypothesis of exogeneity of the realized volatility can be rejected for both models and we may assume that the realized volatility is indeed endogenous when the chosen instruments are valid. The instruments are tested with a Sargan test, where the LM test statistic is again chi-squared distributed with one degree of freedom and haves the value of 0.027 and 0.164 for the Black-Scholes and the Merton model respectively. Both are not significant at a 5% level and the null-hypothesis of exogenous and thus valid instruments can not be rejected. For this reason, the outcomes of the Hausman LM-test are reliable and hence OLS is not consistent. By the use of two-stage least squares (TSLS) one can get consistent estimators for a model with endogenous variables and therefore this method is used next (Table A6, Appendix).

The outcomes of TSLS are further investigated for other misspecifications in the model, because it is still assumed that the error terms have zero mean, constant variance and that they are uncorrelated with each other. When the error terms meets the second property, they are called homoskedastic. But when this property is not satisfied, TSLS is not efficient anymore and more accurate estimates can be achieved by the use of other estimation methods. For this reason, it is important to check whether the residuals of the normal TSLS gives rise to the presumption of

heteroskedastic error terms. The residuals are therefore investigated with a White-test for heteroskedasticity with cross terms (Table A7, Appendix). The null-hypothesis of this test assumes that the model does have homoskedastic error terms and the

alternative hypothesis supposes of heteroskedastic error terms. The White-test has a LM-test statistic of 77.149 for the Merton volatility risk premium model and 72.059 for the regression with the Black-Scholes volatility risk premium, both statistics are chi-squared distributed with 34 degrees of freedom and hence have a p-value of 0.0000 and 0.0002 respectively. The null-hypothesis is therefore in both cases

rejected at a 5% level in favour of the alternative hypothesis and thus there is enough reason to assume that the models have heteroskedastic error terms. One can adjust a model with heteroskedastic error terms when it allows for White standard errors instead of the ordinary TSLS standard errors. When this is done, efficient estimates will only be achieved when the disturbances are uncorrelated. For this reason, serial

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correlation of the error terms is checked with a Breusch-Godfrey LM-test. The test outcomes are placed in Table A7 of the Appendix, under the null-hypothesis of no serial correlated error terms. Both regression models have a p-value of more than 5% and are therefore not significant under the same level, it can therefore be assumed that the disturbances are not serial correlated.

Table S6: Two-stages least squares of the two volatility risk premiums on the realized volatility and a set of one month lagged macro-finance variables from the U.S. Information about the instruments can be found in Table A3, Appendix.

From the TSLS with White-Standard Errors it becomes clear that the macro-finance variables from the previous month do not have much predictive power for the next month volatility risk premium from both models. Certainly for the risk premium of the Merton jump-diffusion model, where all the explanatory variables are

insignificant at a 5% level. It even has a R-squared of nearly zero, whereas the Black-Scholes volatility risk premium has a R-squared that is lower than the one from ordinary OLS, but still is around 0.38.

It is also striking that the realized volatility is not significant anymore in both models. Because the macro-finance variables from the previous month are reasonably correlated with the realized volatility of the next month (Table A4, Appendix), only the direct effects of the macro-finance variables are estimated. This was done with purpose, because I was looking for the effects of macro explanatory variables on the risk premium that were not contained in the realized volatility of the same month. Because the volatility risk premium is the difference between the implied and the realized volatility, it is maybe better to remove the realized volatility from the estimation model, as the effect on the volatility risk premium is known and trivial.

TSLS$with$White+Standard$Errors Volatility(Risk Premiums

Black3Scholes Merton C 31.1458 (0.6979) 31.6050 (1.1440) Realized 30.1987 (0.2402) 0.2041 (0.3231) Housing(Start((31) 0.0000 (0.0003) 0.0002 (0.0004) P/E(Ratio(S&P(500((31) a_30.0103 _(0.0045) _30.0088 _(0.0063) Employment(Growth(Rate((31) 34.8952 (10.3366) 7.8435 (14.0712) Moody's(AAA(Corporate(Yield((31) 0.0157 (0.0116) 0.0147 (0.0152) Producers(Price(Index((31) 2.2686 (2.0016) 3.3806 (2.6563) Average(Wage((31) 0.1270 (0.0659) 0.1642 (0.1071) R3Squared 0.3766 0.0227 F3Statistic((Prob.) 0.0047 0.0299 a_{(p3value(<(5%} Instruments:(shouse(31),(pe(31),(employ(31),(aaa(31),(ppi(31),(awage(31),(cpi,(treas*(+(constant(term *See(the(list(of(all(variables(for(the(meanings(of(the(used(acronyms

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When this variable is removed from the model, one can estimate the total effect of the macro-finance variables on the next month volatility risk premiums. Because the macro-finance variables were exogenous instruments for the volatility risk premiums, TSLS is not needed in this regression model. Ordinary OLS is done, whence it

becomes clear that the model has heteroskedastic error terms, but no autocorrelation (Table A8, Appendix). Thus the final model for the effects of the macro-finance variables on the volatility risk premiums is an OLS model with White-error terms.

Table S7: The regression of the volatility risk premiums on the macro-finance variables estimated with OLS and White-Standard Errors, the realized volatility is excluded in this model.

Table S7 shows that when the realized volatility is excluded from the regression, some macro-finance variables have a solid effect on the volatility risk premiums from the next month. The P/E ratio is significant at a 5% level for both regression models and the Producers Price Index is also significant at the same level for the Black-Scholes regression model. Although, the total effect of the included macro-finance variables on the next month volatility risk premiums is not that big, there exists some relation and these explanatory variables do say something about the upcoming volatility risk premium.

5 Predictability of the Dow Jones Index by the S&P 500 volatility

risk premiums

In this section, the volatility risk premium derived from the S&P 500 index is linked at the next-month return of the Dow Jones Industrial Averages (DJIA). Bollerslev, Gibson and Zhou (2011) showed that the S&P 500 risk premium is useful in

OLS$with$White+Standard$Errors Volatility(Risk Premiums

Black3Scholes Merton C 31.8536 (1.2023) 30.8782 (1.2095) Housing(Start((31) 0.0003 (0.0003) 30.0001 (0.0003) P/E(Ratio(S&P(500((31) a_30.0076 _(0.0036) a_30.0116 _(0.0042) Employment(Growth(Rate((31) 0.6245 (11.5838) 2.1752 (11.0071) Moody's(AAA(Corporate(Yield((31) 0.0082 (0.0088) 0.0225 (0.0121) Producers(Price(Index((31) a_3.4092 _(1.7110) _2.2092 _(2.6690) Average(Wage((31) 0.1901 (0.1150) 0.0994 (0.1162) R3Squared 0.1740 0.2036 F3Statistic((Prob.) 0.0210 0.0069 a_{(p3value(<(5%}

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predicting the next-month return of the S&P 500 index and indicated that this

conclusion can possibly be made for other markets aswell. The Dow Jones Industrial Averages embraces the 30 largest companies of the U.S., whereas the S&P 500 index contains the 500 largest companies of the U.S. Both indices are used by investors to indicate the general stock market trend in the U.S., it is therefore interesting to check whether the monthly volatility risk premiums from S&P 500 index have also

predicting power to the DJIA. Ordinary OLS of the next-month return of the DJIA on a constant and the volatility risk premium, both from the Black-Scholes and the Merton jump-diffusion model are placed below.

Table S8: Normal OLS of the Dow Jones Industrial Average on the volatility risk premiums. From the ordinary OLS regression a presumption is created that indeed the volatility risk premiums do say something about the next-month return of the DJIA. The Black-Scholes volatility risk premium is significant at a 5% level with a p-value of 0.0008, the Merton volatility risk premium is not significant at a 5% level with a p-value of 0.0651, but it does indicate a relation. Before firm conclusions are taken, some misspecifcation tests for both regression models will be discussed (comprehensive results can be found in Table B1, Appendix). The White-test for heteroskedastic error terms gives significant LM-test statistics, with values of 20.776 and 19.981 for the Black-Scholes and the Merton model respectively, which are chi-squared distributed with two degrees of freedom and the null-hypothesis of homoskedastic error terms is therefore rejected. On the other hand, the Breusch-Godfrey LM-test statistics are not significant at a 5% level, with values of 1.854 and 4.115 for the Black-Scholes and the Merton model respectively, which are chi-squared distributed with two degrees of freedom and thus there is no reason to assume autocorrelated error terms.

Endogeneity of the volatility risk premiums is also tested with a Hausman LM-test, from which it becomes clear that at a 5% significance level both volatility risk

OLS DJIA DJIA

C &0.0023 (0.0023) &0.0023 (0.0029)

Black&Scholes a_0.1236 _(0.0357) _& _&

Merton & & 0.0651 (0.0348)

R&Squared 0.1292 0.0414

F&StatisticH(Prob.) 0.0008 0.0651

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premiums are exogenous. By the previous results, the regression model is estimated with OLS with White-standard errors.

Table S9: Outcomes of OLS with White-Standard Errors for the model where the DJIA is predicted by the volatility risk premiums.

The outcomes of the above regression show that the Black-Scholes volatility risk premium is still a good predictor for the the next-month DJIA return, it has a p-value of 0.0199 and is therefore significant at a 5% level. However, the Merton volatility risk premium has become less significant in comparison with ordinary OLS and has therefore a very small relation with the next-month return of the DJIA.

6 Conclusion

The aim of this study was to show that there exists a significant volatility risk premium implicit in options. The volatility risk premium and hence the implied volatility were estimated by various methods, all with different assumptions on the future volatility of the analysed options; the most ATM S&P 500 index options with a remaining life of one month. It has turned out that there is a significant positive risk premium implicit in these options, however the Black-Scholes model shows to give the less significant volatility risk premium in comparison with the Merton model and the VIX. This is intuitive peculiar, because it is a relative simple model and hereby outperforms the more refined models (Merton, VIX) in estimating the future realized volatility. For future research I therefore recommend to investigate the risk premium when stochastic volatility estimation models are used, which I hereafter expect to give also more significant volatility risk premiums than the Black-Scholes model.

Regressions of the volatility risk premiums from the Merton and Black-Scholes model have indicated that some macro-finance variables, which are lagged one month, have a clear effect on the height of the next month risk premium. The P/E ratio of the S&P 500 index has in particular a clear relation with the volatility risk

OLS$with$White+Standard$Errors DJIA DJIA

C &0.0023 (0.0021) &0.0023 (0.0036)

Black&Scholes a_0.1236 _(0.0520) _& _&

Merton & & 0.0651 (0.0591)

R&Squared 0.1292 0.0414

F&StatisticF(Prob.) 0.0199 0.2740

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premiums. However, the relation of the volatility risk premiums with the one month lagged macro-finance variables dissapears when the realized volatility is included in the regression model. Thus the macro-finance variables do have a relation with the next month S&P 500 volatility risk premium, but this is more an indirect relation through the high correlation with the next month realized volatility. Because the realized volatility is only known over the past periods, the macro-finance variables can be used in estimating the next month volatility risk premium.

The final goal of this paper was to explore an eventual relation between the DJIA and the volatility risk premiums from the S&P 500 index. This idea was fueled by the paper of Bollerslev, Gibson and Zhou (2011), which found a relation between the risk premium derived from the VIX and the next month return of the S&P 500 index. This study has found a significant positive relation with next month return of the DJIA and the Black-Scholes volatility risk premium computed from S&P 500 index options. The Merton volatility risk premium does show a relation with the next month return of the DJIA, however this relation is not significant and the sign is not clear.

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Appendix A The Black-Scholes implied volatility plotted against the monthly realized volatility of the S&P 500 index, both annualized. The annualized implied volatility computed with the Merton jump-diffusion model plotted against the monthly realized volatility of the S&P 500 index. Daily implied volatility derived from the VIX is plotted against the daily realized volatility of the S&P 500 index.

Figure A1: The implied volatilities from the Black-Scholes model, Merton jump-diffusion model and the VIX against the realized volatility of the S&P 500 index (annualized).

0" 0.1" 0.2" 0.3" 0.4" 0.5" 0.6" 0.7" 0.8"

May/06" Dec/06" Jul/07" Feb/08" Sep/08" Apr/09" Nov/09" Jun/10" Jan/11" Aug/11" Mar/12" Oct/12" May/13"

Black/Scholes"Vol" Realized"Vol" 0" 0.1" 0.2" 0.3" 0.4" 0.5" 0.6" 0.7" 0.8" 0.9" 1"

May006" Dec006" Jul007" Feb008" Sep008" Apr009" Nov009" Jun010" Jan011" Aug011" Mar012" Oct012" May013"

Merton"Vol" Realized"Vol" 0" 0.2" 0.4" 0.6" 0.8" 1" 1.2" 1.4" 1.6" 01/05/2006" 01/05/2007" 01/05/2008" 01/05/2009" 01/05/2010" 01/05/2011" 01/05/2012" 01/05/2013" VIX" Realized"Vol"

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Appendix A

Figure A2: The graphs above show the Black-Scholes implied volatility together with the Merton implied volatility (annualized).

0.0 0.2 0.4 0.6 0.8 1.0 2006 2007 2008 2009 2010 2011 2012 2013 BSV OL MERTVOL 0.0 0.2 0.4 0.6 0.8 1.0 2006 2007 2008 2009 2010 2011 2012 2013 (BSV OL,MERTV OL) 0.0 0.2 0.4 0.6 0.8 1.0 2006 2007 2008 2009 2010 2011 2012 2013 BSV OL MERTVOL

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Appendix A The Black-Scholes model volatility risk premium implicit in S&P 500 index options with a remaining life of one month. Significant positive risk premiums can be seen. When monthly implied volatilities are computed with the Merton model on S&P 500 index options, then the risk premiums are looking like the figure on the left.

The volatility risk premiums derived from the VIX are to the left. Almost only positive risk premiums can be seen.

Figure A3: Three kinds of volatility risk premiums implicit in S&P 500 index options are plotted (annualized). !0.3% !0.25% !0.2% !0.15% !0.1% !0.05% 0% 0.05% 0.1% 0.15% 0.2%

May!06% Nov!06% May!07% Nov!07% May!08% Nov!08% May!09% Nov!09% May!10% Nov!10% May!11% Nov!11% May!12% Nov!12% May!13%

!0.2% !0.15% !0.1% !0.05% 0% 0.05% 0.1% 0.15% 0.2% 0.25% 0.3% 0.35%

May!06% Nov!06% May!07% Nov!07% May!08% Nov!08% May!09% Nov!09% May!10% Nov!10% May!11% Nov!11% May!12% Nov!12% May!13%

!0.8% !0.6% !0.4% !0.2% 0% 0.2% 0.4% 0.6%

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Appendix A

Figure A4: The Black-Scholes volatility risk premiums plotted against the Merton jump-diffusion volatility risk premiums in different ways (annualized).

-.3 -.2 -.1 .0 .1 .2 .3 2006 2007 2008 2009 2010 2011 2012 2013 BS MERT -.3 -.2 -.1 .0 .1 .2 .3 2006 2007 2008 2009 2010 2011 2012 2013 ( BS,ME RT) -.3 -.2 -.1 .0 .1 .2 .3 2006 2007 2008 2009 2010 2011 2012 2013 BS MERT

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Appendix A

Table A1: The statistics of the VIX implied volatility and the daily realized volatility of the S&P 500 index.

Table A2: The statistics of the estimated Merton model parameters, the above sigma is not the implied volatility as discussed in Section 3.

Table A3: The description of the used variables in this paper.

Statistic Implied(volatility Daily(realized(volatility

VIX 5(Minutes Mean 0.228 0.163 Median 0.202 0.131 Standard(Deviation 0.108 0.120 Skewness 1.968 2.979 Kurtosis 4.840 14.119 Min 0.097 0.034 Max 0.807 1.397

Statistic Lambda Mu Delta Sigma

Mean 1.8631 50.1068 0.0216 0.1778

Median 0.8189 50.0652 0.0057 0.1548

Standard=Deviation 2.1830 0.2089 0.0459 0.0898

Variable Acronym Description

Realized(Volatility REAL The(realized(volatility(of(the(S&P(500(index(

Housing(Start SHOUSE The(number(of(monthly(housing(construction(project(started(in(the(U.S.,(divided(by(10000

P/E(Ratio(S&P(500(Index PE The(Price/Earnings(ratio(of(the(S&P(500(index

Employment(Growth(Rate EMPLOY The(monthly(logarithmic(growth(rate(of(the(total(employees(in(the(U.S. Dow(Jones(Industrial(Average DJIA The(next(month(logarithmic(growth(rate(of(the(Dow(Jones(Industrial(Average Moody's(AAA(Corporate(Yield( AAA U.S.(Corporate(Bond(Yield,(multiplied(by(100

Producers(Price(Index PPI Producers(Price(Index(in(U.S.

Average(Wage AWAGE Average(hourly(real(earnings(of(all(employees(in(the(U.S.

Consumers(Price(Index CPI Consumers(Price(Index(in(U.S.

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Appendix A

Table A4: Ordinary OLS regressions of the realized volatility on the macro-finance variables from the same month and the macro-finance variables lagged one month.

OLS Realized(Volatility C 1.2594 (0.6881) Housing(Start a_@0.0011 _(0.0003) P/E(Ratio(S&P(500( a_@0.0199 _(0.0036) Employment(Growth(Rate @16.2021 (10.0515) Dow(Jones(Index a_@0.9784 _(0.3640) Moody's(AAA(Corporate(Yield a_0.0441 _(0.0119) Producers(Price(Index a_@6.7650 _(1.8649) Average(Wage @0.0877 (0.0656) R@Squared 0.7654 F@Statistic((Prob.) 0.0000 a_{(p@value(<(5%} OLS Realized(Volatility C 3.5109 (0.9413) Housing(Start((>1) >0.0015 (0.0004) P/E(Ratio(S&P(500((>1) >0.0136 (0.0050) Employment(Growth(Rate((>1) >27.4909 (13.7951) Dow(Jones(Index((>1) >0.1106 (0.4967) Moody's(AAA(Corporate(Yield((>1) 0.0378 (0.0166) Producers(Price(Index((>1) >5.5640 (2.5408) Average(Wage((>1) >0.3125 (0.0898) R>Squared 0.5683 F>Statistic((Prob.) 0.0000 a_{(p>value(<(5%}

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Appendix A

Table A5: Misspecification tests for the ordinary OLS regression model where the Black-Scholes and the Merton volatility risk premium are regressed on the realized volatility and the one month lagged macro-finance variables.

Table A6: Ordinary TSLS of the volatility risk premiums on the one month lagged macro-finance variables and the realized volatility of the same month.

Tests%Ordinairy%OLS

Hausman'LM*test' Volatility'Risk Premiums

Black*Scholes Merton

LM*test'statistic 4.2089 9.3173

Prob.'Chi*Squared(1) 0.0402 0.0023

Sargan'LM*test' Volatility'Risk Premiums

Breusch*Godfrey'serial'correlation'LM*test Volatility'Risk Premiums Black*Scholes Merton

White*test'for'heteroskedasticity'with'cross*terms' Volatility'Risk Premiums Black*Scholes Merton LM*test'statistic 65.0356 73.4064 Prob.'Chi*Squared(34) 0.0011 0.0001 Instruments:'shouse(*1),'pe(*1),'employ(*1),'aaa(*1),'ppi(*1),'awage(*1),'cpi,'treas*'+'constant'term *See'the'list'of'all'variables'for'the'meanings'of'the'used'acronyms TSLS$ Volatility(Risk Premiums Black3Scholes Merton C 31.1458 (0.7936) 31.6050 (1.0659) Realized 30.1987 (0.1503) 0.2041 (0.2019) Housing(Start((31) 0.0000 (0.0003) 0.0002 (0.0004) P/E(Ratio(S&P(500((31) a_30.0103 _(0.0037) _30.0088 _(0.0050) Employment(Growth(Rate((31) 34.8952 (9.7511) 7.8435 (13.0978) Moody's(AAA(Corporate(Yield((31) 0.0157 (0.0121) 0.0147 (0.0162) Producers(Price(Index((31) 2.2686 (1.7730) 3.3806 (2.3816) Average(Wage((31) 0.1270 (0.0736) 0.1642 (0.0988) R3Squared 0.3766 0.0227 F3Statistic((Prob.) 0.0047 0.0299 a_{(p3value(<(5%} Instruments:(shouse(31),(pe(31),(employ(31),(aaa(31),(ppi(31),(awage(31),(cpi,(treas*(+(constant(term *See(the(list(of(all(variables(for(the(meanings(of(the(used(acronyms

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Appendix A

Table A7: Misspecification tests for the TSLS regression model of the Black-Scholes and the Merton volatility risk premium on the realized volatility and the one month lagged macro-finance variables.

Table A8: Misspecification tests for the ordinairy OLS regression models, without the realized volatility.

Tests%Ordinairy%TSLS

Breusch(Godfrey.serial.correlation.LM(test Volatility.Risk Premiums Black(Scholes Merton

LM(test.statistic 0.3471 4.8705

Prob..Chi(Squared(2) 0.8407 0.0876

White(test.for.heteroskedasticity.with.cross(terms. Volatility.Risk Premiums Black(Scholes Merton LM(test.statistic 72.0585 77.1490 Prob..Chi(Squared(34) 0.0002 0.0000 Instruments:.shouse((1),.pe((1),.employ((1),.aaa((1),.ppi((1),.awage((1),.cpi,.treas*.+.constant.term *See.the.list.of.all.variables.for.the.meanings.of.the.used.acronyms Tests%Ordinairy%OLS%without%realized%volatility

Breusch(Godfrey.serial.correlation.LM(test Volatility.Risk Premiums Black(Scholes Merton

Prob..Chi(Squared(2) 0.8607 0.0657

White(test.for.heteroskedasticity.with.cross(terms. Volatility.Risk Premiums Black(Scholes Merton

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Appendix B

Figure B1: The plots of the return of the DJIA and the volatility risk premiums of the Black-Scholes and the Merton model are depicted above. The Black-Black-Scholes volatility risk premium does have a significant relation with the next month return of the DJIA and the Merton volatility risk premium does not.

-.08 -.06 -.04 -.02 .00 .02 .04 .06 2006 2007 2008 2009 2010 2011 2012 2013 DJIA -.3 -.2 -.1 .0 .1 .2 2006 2007 2008 2009 2010 2011 2012 2013 BS -.2 -.1 .0 .1 .2 .3 2006 2007 2008 2009 2010 2011 2012 2013 MERT

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Appendix B

Table B1: Misspecification tests for the ordinairy OLS regression of the DJIA (the next month return of the DJIA index) on the volatility risk premiums from the Merton and the Black-Scholes model, together with a constant term.

Tests%Ordinairy%OLS%of%DJIA%on%the%volatility%risk%premiums

Hausman'LM*test' Volatility'Risk Premiums

Sargan'LM*test' Volatility'Risk Premiums

Breusch*Godfrey'serial'correlation'LM*test Volatility'Risk Premiums Black*Scholes Merton

White*test'for'heteroskedasticity'with'cross*terms' Volatility'Risk Premiums Black*Scholes Merton

Instruments:''cpi,'treas*'+'constant'term

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References

Bollerslev T., Gibson M., Zhou H. (2011), Dynamic estimation of volatility risk premia and investor risk aversion from option-implied and realized volatilities, Journal of Econometrics, Vol. 160, No. 1, pp. 235-245

Canina L., Figlewski S. (1993), The informational conten of implie volatility, Review of Financial Studies, Vol. 5, No. 3, pp. 659-681

Christensen B.J., Prabhala N.R. (1998), The relation between implied and realized volatility, Journal of Financial Economics, Vol. 50, No. 2, pp. 125-150

Day T., Lewis C. (1992), Stock market volatility and the information content of stock index options, Journal of Econometrics, Vol. 52, No. 1-2, pp. 267-287

Fama E.F. (1965), The Behavior of Stock-Market Prices, The Journal of Business, Vol. 38, No. 1, pp. 34-105

Fleming J. (1998), The quality of market volatility forecasts implied by S&P 100 index options, Journal of Empirical Finance, Vol. 5, No. 4, pp. 317-435

Heston S.L (1993), A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, The Review of Financial Studies, Vol. 6, No. 2, pp 327-343

Jorion P. (1995), Predicting volatility in foreign exchange market, Journal of Finance, Vol. 50, No. 2, pp. 507-528

Mayhem S. (1995), Implied Volatility, Financial Analysts Journal, Vol. 51, No. 4, pp. 8-20

Merton R.C. (1976), Option Pricing when the Underlying Stock Returns are Discontinuous, Journal of Financial Economics, Vol. 3, No. 1-2, pp. 125-144 Navas J.F. (2003), Calculation of Volatility in a Jump-Diffusion Model. Retrieved from http://upo.es/personal/jfernav/papers/Jumps_JOD_.pdf

Shu J., Zhang J.E. (2003), The Relation Between Implied and Realized Volatility of S&P 500 Index, WILMOTT magazine, No. 1, pp. 83-91

Tankov P., Voltchkova E. (1992), Jump-diffusion models: a practitioner’s guide. Retrieved from http://proba.jussieu.fr/pageperso/tankov/tankov_voltchkova.pdf

The estimation and usefulness of volatility risk premiums implicit in index options caused by asset price jumps