Journal of Banking & Finance xxx (2011) xxx–xxx

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Option-implied volatility factors and the cross-section of market risk premia q Junye Li ⇑ ESSEC Business School, Paris-Singapore, 100 Victoria Street, 188064 Singapore, Singapore

a r t i c l e

i n f o

Article history: Received 31 January 2011 Accepted 19 July 2011 Available online xxxx JEL classification: G12 G13 C32 Keywords: Stochastic discount factor Volatility components Volatility risk premia Value and size effects Unscented Kalman filter

a b s t r a c t The main goal of this paper is to study the cross-sectional pricing of market volatility. The paper proposes that the market return, diffusion volatility, and jump volatility are fundamental factors that change the investors’ investment opportunity set. Based on estimates of diffusion and jump volatility factors using an enriched dataset including S&P 500 index returns, index options, and VIX, the paper finds negative market prices for volatility factors in the cross-section of stock returns. The findings are consistent with risk-based interpretations of value and size premia and indicate that the value effect is mainly related to the persistent diffusion volatility factor, whereas the size effect is associated with both the diffusion volatility factor and the jump volatility factor. The paper also finds that the use of market index data alone may yield counter-intuitive results. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction The presence of time-varying volatility and jumps in the market index are well documented. Furthermore, it has been shown that jumps arrive at different rates and tend to be clustered (Maheu and McCurdy, 2004; Huang and Wu, 2004; Christoffersen et al., 2009). Therefore, aggregate market volatility should arise from different sources including diffusion and jump risks. Different components of market volatility behave differently and influence expected returns in different ways. The determination of volatility components and the ways in which they are priced have important implications for pricing options, interpreting implied volatility, and understanding investors’ preferences. Even though empirical investigations of the aggregate market volatility pricing have been undertaken, the pricing of diffusion and jump volatility factors has not received enough attention. The CAPM predicts that the expected return of an asset is proportional to the covariance of its return with the market return. However, Fama (1970) shows that the CAPM is not applicable when the investment opportunity set of investors changes over time in q I would like to thank Ike Mathur (the editor), an anonymous referee, Laurent Calvet, Patrice Poncet, Jun Yu, Peng Xu, and participants at the HEC Paris finance seminar, the SMU Institute for Financial Economics seminar, the ESSEC Business School finance seminar, and the Financial Management Association European Conference for helpful comments. ⇑ Tel.: +65 6884 9780; fax: +65 6884 9781. E-mail address: [email protected]

the intertemporal setting. Merton’s ICAPM (1973) shows that any state variables that change the investment opportunity set of investors should be taken as pricing factors and be regarded as systematic risks. Time-varying market return and volatility are two fundamental factors that affect the investment opportunity set of investors (Campbell, 1993, 1996; Chen, 2003). Both are related to the consumption risk. An increase in the expected market return allows investors to consume more today through intertemporal consumption smoothing, while an increase in market volatility leads investors to consume less today by increasing their precautionary savings. Ang et al. (2006) investigate the cross-section of aggregate market volatility and expected returns by taking the VIX as a proxy of market volatility. Adrian and Rosenberg (2008) follow a component GARCH model to study how the short-run and long-run volatility components are priced in the cross-section of stock returns. Yan (2011) uses a semi-parametric approach to investigate how the jump risk is priced cross-sectionally. This paper is related to previous studies on the cross-sectional market pricing of volatility risks. However, it differs in a number of respects. First, the model specification problem is taken into account when modeling market volatility. Jump is a necessary component in dynamics of the market index. Furthermore, recent nonparametric studies provide even more convincing evidence of infinite activity jumps in asset prices (Cont and Mancini, 2008; Ait-Sahalia and Jacod, 2009; Lee and Hannig, 2010). The specification makes jumps serially correlated and introduces jump clustering. It even allows for the time-varying/stochastic higher moments.

0378-4266/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jbankfin.2011.07.005

Please cite this article in press as: Li, J. Option-implied volatility factors and the cross-section of market risk premia. J. Bank Finance (2011), doi:10.1016/ j.jbankfin.2011.07.005

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J. Li / Journal of Banking & Finance xxx (2011) xxx–xxx

Second, market volatility has two components: one arises from the diffusion shock and the other from the jump shock. These components behave differently and demand different market compensations. Diffusion volatility is persistent and has little variation. It plays an important role in the pricing of long-maturity options and affects return variance and investors’ consumption in the long run. Jump volatility introduces the time-varying jump arrival rate, which reverts to the mean very fast. It is critical in the pricing of short-maturity out-of-the-money options and in generating the short-maturity smile. Jump volatility contributes to aggregate market volatility on the short horizon. Third, diffusion and jump volatility factors are unobservable and must be estimated. Under the absence of arbitrage, options and their underlying asset should have the same risk factors. Furthermore, market incompleteness implies that options are not redundant instruments. Traded options are forward-looking and implicitly embody all available information. Therefore, joint use of options and returns should result in better estimates of volatility factors. Surprisingly, this fact has not been attached enough importance in the extant literature, and most studies still rely on the underlying asset alone. In this paper, I use an enriched dataset that includes not only S&P 500 index returns but also index options and VIX to estimate market volatility factors. I first investigate how the diffusion and jump volatility factors are priced in the cross-section of stock returns. Using the FamaFrench 25 size and book/market portfolios, I find that the diffusion and jump volatility risks have negative market prices, which are 0.78% and 0.22% per month, respectively. However, although the market price of diffusion volatility is highly statistically significant, that of jump volatility is not. Sensitivities to the two volatility factors vary dramatically across different stock portfolios, with the sensitivity to diffusion volatility ranging from 0.93 to 0.22, and the sensitivity to jump volatility ranging from 0.91 to 0.03. Negative prices for volatility factors indicate that an increase in volatility has an adverse influence on the investment opportunity set and that investors need to be paid for holding extra diffusion and/or jump volatility. Assets with high sensitivity (in absolute value) to market volatility factors command high risk premia because they reduce investors’ hedging ability. Second, I investigate how the value and size effects are related to the two volatility factors. In particular, I examine whether the historically observed high returns in value and small-size stocks can be explained as risk compensation for adverse changes in the investment opportunity set. There are broadly two competing interpretations of the value and size effects: risk-based interpretations and behavioral interpretations. Risk-based interpretations indicate that the existence of value and size premia reflects some systematic risks that cannot be captured by the market factor (Fama and French, 1995, 1996; Perez-Quiros and Timmermann, 2000; Liew and Vassalou, 2000; Petkova and Zhang, 2005). Behavioral interpretations argue that these effects are manifestations of market irrationality (Hong and Stein, 1999; Chen et al., 2001, Chen et al., 2002; Chan et al., 2003). In this paper, I find that the value and size effects are closely related to the two volatility factors. In particular, the value effect is mainly related to the diffusion volatility factor, whereas the size effect is associated with both the diffusion and jump volatility factors. For instance, on average, the diffusion volatility risk premium is 0.52% per month on value stocks and 0.21% per month on growth stocks, while the diffusion volatility risk premium is 0.36% per month on small-size stocks and 0.02% per month on big-size stocks. However, even though the jump volatility risk premium is 0.19% per month on small-size stocks and is 0.05% per month on big-size stocks, the value–growth premium spread cannot be unambiguously observed on the jump volatility factor. For example, on average, the jump volatility risk premia on value stocks and growth stocks are nearly the same.

These results provide an empirical underpinning of risk-based explanations of the value and size effects. Third, I explore information difference contained in options and its implications for the cross-sectional variations in market risk premia. Options contain rich information on volatility dynamics. I find that if only market index data are used to extract volatility factors, the market price of diffusion volatility is counter-intuitively smaller (in absolute value) than that of jump volatility (0.29% vs. 1.05% per month) and that its average risk premium is unreasonably smaller than that of jump volatility (0.15% vs. 0.26%). The value–growth premium spread is too small to explain the value effect. It averages 0.10% per month, while the option-implied value– growth spread is as high as 0.31% on average per month. The rest of paper is organized as follows. Section 2 presents the modeling framework for market volatility, which has two components: diffusion volatility and jump volatility. Section 3 introduces the econometric methodology. Section 4 discusses estimation results. Section 5 empirically investigates the market pricing of volatility factors and the cross-section of market risk premia. Finally, Section 6 concludes the paper. 2. Model and option valuation 2.1. Stock price and volatility dynamics Under a given probability space ðX; F ; PÞ and the complete filtration fF t gtP0 , the market index St has the following dynamics under the objective measure:

ln St =S0 ¼

Z 0

t

    us ds þ W T ð1Þ  kW ð1ÞT ð1Þ þ J T ð2Þ  kJ ð1ÞT ð2Þ ; t t t

t

ð1Þ

where ut captures its instantaneous mean, and kW(1) and kJ(1) are convexity adjustments for the Brownian motion and the jump process, respectively, and can be computed from their cumulant exponents: kðuÞ  1t lnðE½euLt Þ, where Lt is either Wt or Jt. The market index jumps due to the jump component, J, in (1). The inclusion of jumps is important for capturing the large discontinuous movements in asset returns. Many studies have shown that it can help generate the return non-normality and can explain the implied volatility smile/skew at short horizons. Recent nonparametric studies by Ait-Sahalia and Jacod (2009), Cont and Mancini (2008), and Lee and Hannig (2010) provide strong and convincing evidence of infinite activity jumps in asset prices. Therefore, in this paper, I model the jump component using the Variance Gamma process (Madan et al., 1998), which is an infinite activity stochastic process and has the following Lévy density:

v J ðxÞ ¼ c

ex=kþ ejxj=k 1x>0 þ c 1x<0 ; x jxj

ð2Þ

where c > 0 measures the overall and relative frequency of jumps, and k+ > 0 and k- > 0 govern how fast tails decay and lead to a skewed distribution when they are different. Its characteristic function has the form of:

/J ðuÞ  E½eiuJt  ¼ etwJ ðuÞ ; wJ ðuÞ ¼ c½lnð1  iukþ Þ þ lnð1 þ iuk Þ;

ð3Þ

  from which we can derive that Var½J t  ¼ c k2þ þ k2 t. ðiÞ T t defines a stochastic business time (Clark, 1973; Carr et al., 2003), which captures the randomness of the diffusion variance (i = 1) or of the jump intensity (i = 2) over a time interval [0, t]:

T ðiÞ t ¼

Z 0

t ðiÞ V s ds;

ð4Þ

which is finite almost surely V tðiÞ , which should be nonnegative, is the instantaneous variance rate (i = 1) or the jump arrival rate

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J. Li / Journal of Banking & Finance xxx (2011) xxx–xxx

(i = 2), both of which reflect the intensity of economic activity and information flow. Stochastic volatility and/or stochastic jump intensity are generated by replacing calendar time t with business time T ðiÞ t . The instantaneous variance rate and jump arrival rate are modeled using the square-root processes of Cox et al. (1985): ð1Þ

dV t

ð2Þ

dV t

qffiffiffiffiffiffiffiffi ð1Þ ¼ j1 ðh1  V tð1Þ Þdt þ r1 V tð1Þ dZ t ; qffiffiffiffiffiffiffiffi ð2Þ ¼ j2 ð1  V ð2Þ V ð2Þ t Þdt þ r2 t dZ t ;

ð5Þ ð6Þ

where ji is the mean-reverting parameter, hi is the long-run mean parameter, ri is the volatility of volatility parameter, Z ðiÞ t ’s are two standard Brownian motions that are independent each other, Z ð1Þ t is allowed to be correlated to Wt with a correlation parameter q in order to accommodate the diffusion leverage effect and independent of Jt, and Z ð2Þ is independent of Wt and Jt. The long-run mean of t V ð2Þ is normalized to one as Jt has a non-unit variance. t Eq. (6) implies that jumps are serially correlated and the model introduces jump clustering. Through the stochastic time change, the higher moments (i.e., skewness and kurtosis) of the timetransformed jump component J T ð2Þ become state-dependent and t are time-varying. Similar models have also been investigated by Huang and Wu (2004), Bakshi et al. (2008), Li (2011), among others. The conditional instantaneous return variance, Vt, of the market index return has two sources contributed by the diffusion shock and the jump shock, respectively: ð2Þ V t ¼ V ð1Þ t þ Var½J 1 V t :

ð7Þ V ð1Þ t

These two risks are fundamentally different. is usually persistent and represents a long-run volatility component. V ð2Þ represents t a short-run component, as jumps play a role at the short horizon and their effects are transient. Therefore, they may have different market prices and demand different compensation. In the following, V ð1Þ and V tð2Þ are called diffusion volatility and jump volatility, t respectively. If we set V ð2Þ ¼ 1 and r2 = 0, we obtain the frequently t used constant jump intensity model, which implies i.i.d jumps. We can see from (7) that at market crash, the large jump arrival rate V tð2Þ can contribute to an abrupt move in market volatility. Eraker et al. (2003) and Todorov (2010) directly model aggregate market volatility with a diffusion component and a jump component in order to capture sudden increase in aggregate volatility. 2.2. Pricing kernel and option pricing The no-arbitrage condition indicates that there exists at least one almost surely positive process, Mt, with M0 = 1 such that the discounted gains process associated with any admissible trading strategy is a martingale (Harrison and Kreps, 1979). Mt, which is assumed to be a semimartingale, is called the stochastic discount factor or the pricing kernel. If the market is complete, Mt is unique; otherwise, there may exist many different pricing kernels. By introducing stochastic volatility and jumps, the market is no longer complete. This feature may produce extra difficulty and complexity in change of measure since the objective dynamics could be extremely different from the risk-neutral ones. We are interested in the structure-preserving change of measure because it preserves tractability and the same structure under both measures. Thus, I propose a class of models for the stochastic discount factor Mt as follows:

  1 M t ¼ expðrf tÞ exp cW W T ð1Þ  c2W T tð1Þ t 2   ð2Þ  exp cJ J T ð2Þ  kJ ðcJ ÞT t ; t

where the risk-free rate rf is assumed to exist and to be constant, the market prices of risks (cW and cJ) are assumed to be constant, and the stochastic time-changed diffusion and jump are both regarded as systematic risks. This stochastic discount factor defines a Radon-Nikodym derivative for the market index as follows:

    dQ 1 2 ð1Þ ¼ exp  exp cJ J T ð2Þ  kJ ðcJ ÞT ð2Þ c W  c T ; ð1Þ W W t t T t t dP 2

which transforms the objective measure P to the risk-neutral measure Q. We then have the following risk-neutral model under the measure Q:

    Q Q þ J Qð2Þ  kJ ð1ÞT ð2Þ ; ln St =S0 ¼ rf t þ W Qð1Þ  kW ð1ÞT ð1Þ t t Tt Tt qffiffiffiffiffiffiffiffi ð1Þ ð1ÞQ dV t ¼ ðj1 h1  jQ1 V tð1Þ Þdt þ r1 V tð1Þ dZ t ; qffiffiffiffiffiffiffiffi ð2Þ ð2ÞQ dV t ¼ ðj2  jQ2 V ð2Þ V ð2Þ ; t Þdt þ r2 t dZ t

ð10Þ ð11Þ ð12Þ

where pV i ¼ jQi  ji defines the market price of the diffusion or jump volatility factor. Under the change of measure, W Qt and Z tðiÞQ are still Brownian motions. To guarantee the absolute continuity between Jt and JQt , the coefficients c should remain unchanged and only the tail parameters k+ and k- can differ (Sato, 1999; Cont and Tankov, 2004). The intuition behind this change of measure is consistent with our understanding of financial market movements. Large jumps, which determine the tail behaviors of the return distribution, play critical roles in option pricing and risk management. Following Duffie et al. (2000) and Carr and Wu (2004), we can find the analytical conditional characteristic function of the log return rt = ln (St/Sts) under the risk-neutral model specification (see Appendix A). This can be used to compute the European-type derivative price using the fast Fourier transform methods (Carr and Madan, 1999; Chourdakis, 2005). 2.3. Volatility factors and VIX VIX is a CBOE (Chicago Board Options Exchange) volatility index that measures the implied volatility of index options and is constructed from out-of-the-money European options (both calls and puts) of S&P 500 index with maturity of 30 days. The introduction of VIX provides a premier benchmark for U.S. stock market volatility. In essence, the stochastic volatility factors are directly linked to the VIX index. Ang et al. (2006) directly use VIX as a proxy for aggregate market volatility in their investigation of volatility risk pricing. VIX squared is the one-month (30-day) S&P 500 index variance swap rate (Carr and Wu, 2009). A variance swap has zero market value at entry, and the payoff at maturity of the long position is equal to the difference between realized variance and a predetermined fixed value that is called the variance swap rate. Under the no-arbitrage condition, the variance swap rate should be the expected value of realized variance under the risk-neutral measure Q:

VSt;T ¼

1 Q E ½RV t;T jF t : T t

ð13Þ

The market index process (1) implies that realized variance, which can be approximated by quadratic variation, is equal to:

RV t;T ¼

Z

T

V sð1Þ ds þ

t

¼

Z

t

ð8Þ

ð9Þ

þ

T

V sð1Þ ds þ Z t

T

Z R0

Z

T

R0

t

Z

t

Z

T

Z

R0

V sð2Þ x2 lðds; dxÞ V sð2Þ x2 v ðdxÞds

2~ V ð2Þ s x lðds; dxÞ

ð14Þ

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J. Li / Journal of Banking & Finance xxx (2011) xxx–xxx

over time horizon [t, T], where l(dt, dx) is a random jump measure ~ ¼ l  v is the compensated random with the density v(dx) and l jump measure. In our model, v(dx) is the Variance Gamma Lévy density. Taking expectation to Eq. (14) under the risk-neutral measure Q and plugging it into Eq. (13), we have the following variance swap pricing formula:

VSt;T ¼

1 T t

Z

T t

h i EQ V sð1Þ jF t ds þ

1 VarQ ½J 1  T t

Z

T

t

h i EQ V sð2Þ jF t ds;

LOt ðHÞ ¼ 

ð15Þ ~ is a where the expectation of the last term in (14) is zero because l ðiÞ compensated measure. The square-root process specifications of h i Vt Q ðiÞ result in the tractable conditional expectations of E V s jF t , for i = 1, 2. We therefore have the following result: Proposition. Under the risk-neutral specifications of the market index (15), the variance rate (11) and the jump arrival rate (12), the VIX is priced as follows: ð2Þ VIX 2t ¼ AðsÞ þ B1 ðsÞV ð1Þ t þ B2 ðsÞV t ;

ð16Þ

where A(s) = A1(s) + A2(s), and

A1 ðsÞ ¼

hQ1

1

1e

jQ 1

s

A2 ðsÞ ¼ VarQ ½J 1 hQ2 1 

;

1e

B1 ðsÞ ¼ jQ s 2

jQ2 s

1e

jQ1 s

! ;

jQ 1

s

;

B2 ðsÞ ¼ VarQ ½J1 

jQ s 2

1e

!

jQ2 s

 LOt ðHÞ þ LRt ðHÞ ;

ð22Þ

t¼1

where N is a compact parameter space and T is the length of total observations of data. Because the log likelihood function is misspecified for non-Gaussian models, a robust estimate of the variance– covariance matrix of parameter estimates can be derived using the approach proposed by White (1980):

1 T

ð23Þ

where

As volatility factors are unobservable, I transform the model into a dynamic state-space form and estimate it with a filtering method. By assuming that options and VIX data are collected with measurement errors, we have the following measurement equations:

yt ¼ f ðV tð1Þ ; V tð2Þ ; HRN Þ þ Ot ;

ð17Þ

ð2Þ V VIX 2t ¼ AðsÞ þ B1 ðsÞV ð1Þ t þ B2 ðsÞV t þ t :

ð18Þ

n

In Eq.(17), yt 2 R are the market observed option prices at time t, ð2Þ are the model-implied prices that depend on and f V ð1Þ t ; Vt ; H volatility components V tð1Þ and V tð2Þ , parameters HRN and other contractual variables. For Eq. (18), A, B1 and B2 are defined in the Proposition. Both market and model-implied option prices are normalized by the Black-Scholes Vega. Ot are measurement errors of option prices and assumed to be i.i.d normal with a mean of zero and a covariance matrix of r2O In , where In is an n  n identity matrix. Vt are measurement errors of VIX and also assumed to be i.i.d normal with a mean of zero and a variance of r2V . rO and rV measure the degree of mispricing of options and VIX, respectively. We have two state variables, whose objective dynamics are given by (5) and (6). After discretizing with a time interval s, the state equations are given by:

qffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ ð1Þ ð1Þ V tð1Þ ¼ j1 h1 s þ ð1  j1 sÞV t s þ r1 sV ts zt ; qffiffiffiffiffiffiffiffiffiffiffiffi ð2Þ ð2Þ ð2Þ V tð2Þ ¼ j2 s þ ð1  j2 sÞV t s þ r2 sV ts zt ;

T X 

b H ¼ ½AB1 A1 ; R

3. Estimating volatility factors

ð21Þ

 ^ where y t contains the predicted values of options and VIX, and P yt its predicted covariance matrix. In order to improve the identification of volatility risk premium parameters, I also use market index data h in estimation. With the filtered volatility factors i b ð1Þ ; V b ð2Þ , we can construct the likelihood function of the re^xt ¼ V t t turn at time t by inverting its objective characteristic function, which can be derived similarly to (31) under the objective measure. A similar approach has been used by Bakshi et al. (2008). Denote LRt ðHÞ as the time t log likelihood of the return. Parameter estimates can be obtained by maximizing the joint log likelihood of options, VIX and returns:

H2N

with hQ2 ¼ j2 =jQ2 and s  T  t = 30/365.

ðiÞ zt ’s

   1  1  ^t 0 ðP y Þ1 yt  y ^t ; ln P yt  yt  y t 2 2

b ¼ arg max H

!

jQ1 s

factors. I thus implement estimation using the unscented Kalman filter (Julier and Uhlmann, 1997), which overcomes pitfalls inherent in other nonlinear Kalman filters to a large extent, and improves estimation accuracy and robustness without increasing computational cost. Assuming that the predictive errors are normally distributed in the unscented Kalman filter, we can construct the log likelihood function of options and VIX at time t:

ð19Þ ð20Þ

where the are two mutually independent standard normal random variables. For a small time interval s, the discretization error can be negligible. The above state-space model is Gaussian but nonlinear, as the option pricing formula is a highly non-linear function of volatility

A¼

T bÞ 1X @ 2 Lt ð H ; T t¼1 @ H@ H0

B¼

T b Þ @Lt ð H bÞ 1X @Lt ð H : T t¼1 @ H @ H0

ð24Þ

b , the diffusion and jump volatility facWith parameter estimates H b ðiÞ can be extracted using the unscented Kalman filter. Furtors V t thermore, the unscented Kalman filter also produces the expected ðiÞ volatility factors V tjt1 . 4. Estimation results In this section, empirical results of model estimation are presented. Models are estimated with the maximum likelihood estimation method discussed in Section 3. The volatility factors model (LTS-2SV) discussed in Section 2 and one of its nested model, the constant jump intensity model (LTS-SV), are investigated. SubSection 4.1 presents the data. SubSection 4.2 offers a discussion of dynamics of risk factors and their economic implications. SubSection 4.3 presents a model performance analysis and discusses necessity and extra values of introducing options and/or VIX in estimation. 4.1. Data Data used in this paper are the S&P 500 index and index options traded on the Chicago Board Options Exchange (CBOE) from January, 1997 to September, 2008. They are obtained from OptionMetrics. The data are in weekly frequency and are those traded on Wednesday. If Wednesday is a holiday, Thursday options are used. There are 608 weeks in total. The dataset contains the following information on options: trading date, expiration date, spot price, strike price, best bid and ask prices, trading volume, open interest, BS implied volatility, and other Greeks. The interest rate used by

Please cite this article in press as: Li, J. Option-implied volatility factors and the cross-section of market risk premia. J. Bank Finance (2011), doi:10.1016/ j.jbankfin.2011.07.005

J. Li / Journal of Banking & Finance xxx (2011) xxx–xxx

OptionMetrics is calculated from a collection of continuouslycompounded zero-coupon interest rates at various maturities, collectively referred to as zero-curve, which is derived from LIBOR rates and settlements prices of Eurodollar futures. For each option, the corresponding zero-curve rate with the closest maturity to that of the option is selected. The following filters are applied to the dataset. First, I only consider call options. Second, in order to ensure that options are liquid enough, I select call options with maturity less than 1.5 years and with moneyness greater than 0.85 and less than 1.06. Furthermore, I rule out options with zero trading volume and with open interest of less than 100 contracts. Third, I exclude call options with maturity less than 10 days and best bid prices less than 3/8 dollar to mitigate market microstructure problems. After applying these filters, the dataset contains 40,304 call options and on average, 66 options at each time instant. For the purpose of model estimation, I construct four sets of options based on concerns of liquidity and term structures of options. Out-of-the-money options are in general more liquid than in-the-money options. Term structures of options are important for identifying the risk-premium parameters. Short-maturity options are useful for identifying the short-run volatility factor, and long-maturity options are helpful for estimating the long-run volatility factor. Further, short-maturity at-the-money options are the most liquid products in the market. Therefore, the four sets of options are constructed as follows: (1) at-the-money short maturity (ATM-SM) calls with maturity greater than 10 days and less than 60 days and with moneyness (S/K) larger than 0.97 and smaller than 1.03; (2) out-of-the-money short maturity (OTMSM) calls with maturity greater than 10 days and less than 60 days and with moneyness (S/K) less 0.97; (3) out-of-the-money medium maturity (OTM-MM) calls with maturity [60, 180] days and moneyness [0.85, 1.00]; and (4) out-of-the-money long maturity (OTM-LM) calls with maturity longer than 180 days and moneyness [0.85, 1.00]. For each of these four sets of options, I choose the option with the largest trading volume at each time instant. In addition to options, we also have access to the CBOE volatility index, VIX, which is constructed from the most liquid out-of-themoney short maturity (30 days) options, and contains rich information on jumps and jump volatility. As all available options are not used in estimation, VIX is incorporated in the dataset in order to improve estimation efficiency. Table 1 reports the descriptive statistics of S&P 500 index returns, VIX, and the constructed four sets of call options. In Fig. 1, the upper panel plots the time-series of index returns and the lower panel presents the time series of VIX obtained from the CBOE website.

Table 1 Descriptive statistics of data for model estimation. Mean

St. Dev.

Skewness

Kurtosis

Min

Max

0.198 0.627

5.139 3.449

0.108 0.100

0.102 0.449

Std Mn.

Mean Mt.

Std Mt.

Mean IV

Std IV

0.013 0.016 0.043 0.053

27.53 31.84 99.14 2.93e2

12.79 12.43 28.04 90.05

0.187 0.167 0.179 0.180

0.063 0.056 0.051 0.045

A. S&P 500 index returns and VIX Returns 0.043 0.167 VIX 0.210 0.067 Mean Mn. B. Constructed calls ATM-SM 0.994 OTM-SM 0.952 OTM-MM 0.953 OTM-LM 0.917

Note: The table presents the descriptive statistics of the data used for model estimation. The data are from January 1997 to September 2008 and have weekly frequency. There are 608 weeks in total. In panel A, mean and standard deviation are annualized. In panel B, Mn stands for moneyness, Mt for maturity (in days), and IV BS for implied volatility. ATM-SM represents at-the-money short maturity, OTM-SM out-of-the-money short maturity, OTM-MM out-of-the-money medium maturity, and OTM-LM out-of-the-money long maturity.

5

4.2. Dynamics of volatility components Table 2 presents parameter estimates and their standard deviations for the LTS-2SV model and the LTS-SV model. It also reports the average maximized likelihoods and the overall option pricing errors measured as means of absolute difference between the model-implied and the market quoted BS implied volatility in different models. The likelihood-ratio test using the maximized likelihoods strongly rejects the LTS-SV model. Furthermore, the option pricing error indicates that the LTS-2SV model reduces the pricing error by about 20% with comparison to the LTS-SV model (0.81% vs. 1.01%). The superiority of the LTS-2SV model can also been seen from the parameter estimates rO and rV, which measure the mispricing of options and VIX. The smaller values of these two parameters in the LTS-2SV model indicate that the LTS-2SV model-implied option prices and VIX values are more capable of explaining variations of real data. Therefore, I now focus on the LTS-2SV model. The jump parameter estimates indicate that both the objective and the risk-neutral distributions of index returns are left-skewed   and have fatter left tails because the left-tail parameters k =kQ   Q are bigger than the right-tail parameters kþ =kþ . The risk-neutral distribution is even more left-skewed, as kQ > k , indicating that markets are more concerned about extreme events under the risk-neutral measure. The parameter estimates kQþ and kQ are highly statistically significant, whereas their objective counterparts k+ and k- are not, implying that the objective tail behaviors are hard to determine and that longer time series of index returns may be needed to pin down these two parameters. The estimate of j1 (1.02) shows that diffusion volatility, V ð1Þ t , is persistent. The value of the volatility of volatility parameter r1 is small (0.21), indicating that V ð1Þ cannot have a large change at a t small time horizon. These two estimates imply that diffusion volatility plays an important role in the pricing of long-maturity options and affects return variance in the run. The long-run pffiffiffiffiffilong  mean estimate of diffusion volatility h1 is around 13.4% and the correlation estimate is high with a value of 0.91. The latter captures the leverage effect and also contributes to the left-skewed distribution of stock returns. When compared with the constant jump intensity model, diffusion volatility from the LTS-2SV model is more persistent and less volatile. In contrast, the jump arrival rate, V ð2Þ t , has a large mean-reverting parameter j2 (14.5), which means that the impact of the jump component lasts only for the short run. The jump arrival rate shows very high instantaneous volatility of volatility (r2 = 5.57). High values of j2 and r2 indicate that the jump arrival rate can have a sudden increase over a short time interval and that it can push aggregate return volatility to move up abruptly during a market crash. According to the model specification, index return variance is governed by two random sources: W and J, whose variance rates, V ð1Þ and V ð2Þ t t , are stochastic and are governed by separate squareroot processes. Dynamics of these state variables determine how shocks to W and J dissipate across the return variance term structure. A transient shock mainly affects short-term return variance and hence short-term options, whereas a persistent shock affects return variance at both short and long horizons. We note that V ð1Þ is persistent. Thus, shocks on W will have a long-lasting impact t on return variance. On the other hand, the jump arrival rate, V ð2Þ t , has a large mean-reverting parameter, indicating that shocks to J tend to quickly dissipate as the horizon increases. Fig. 2 plot the estimated volatility factors. It is clear that diffusion volatility and jump volatility play very different roles. The jump arrival rate suddenly moves to high levels during financial crises. The estimates pV 1 and pV 2 reveal negative market prices of the diffusion and jump variance risks, respectively. Bakshi and Kapadia (2003a,b) and Carr and Wu (2009) have documented the negative

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Index Return 0.1 0.05 0 −0.05 −0.1 1997

2000

2003

2006

2009

2006

2009

VIX 0.45 0.35 0.25 0.15 0.05 1997

2000

2003

Fig. 1. S&P 500 index returns and VIX. Note: The figure plots S&P 500 index returns and the VIX. The data are from January 1996 to September 2008 and have a weekly frequency. There are 608 weeks in total.

Table 2 Parameter estimates.

j

h

r

q

pV

ð1Þ

2.182

0.025

0.306

0.875

0.749

ð1Þ

(0.936) 1.318

(0.008) 0.018

(0.014) 0.209

(0.027) 0.906

(0.366) 0.814

ð2Þ

(0.327) 14.47

(0.002) 1.000

(0.010) 5.571

(0.041) –

(0.381) 0.296

(2.096)

(–)

(0.294)

(–)

B. Jump

c

k+

k

kQþ

LTS-SV

0.415 (0.077) 0.913 (0.043)

0.021 (0.041) 0.022 (0.044)

0.064 (0.061) 0.069 (0.054)

C. Others

rO

rV

l

LTS-SV

0.014 (0.001) 0.012 (0.001)

0.013 (0.001) 0.008 (0.000)

0.044 (0.027) 0.043 (0.021)

A. Volatility LTS-SV

Vt

LTS-2SV

Vt Vt

LTS-2SV

LTS-2SV

0.063 (0.008) 0.038 (0.001) PE

(0.897) kQ 0.253 (0.022) 0.184 (0.002) LLF/T

1.01

19.84

0.81

22.17

Note: The table reports parameter estimates and standard deviations (in brackets) using all available data for estimation discussed in Section 4.1. LTS-SV is the constant jump intensity model and LTS-2SV is the volatility factors model. In last panel, PE represents the overall option pricing error measured as the mean of the absolute difference (in percentage) between the market quotes of BS implied volatility and the model-implied ones; and LLF/T denotes the average maximized log likelihood.

market price of variance risk in stock indexes. Our volatility factors model further decomposes return variance into two components and shows that both components have negative market prices. In general, these two parameters are hard to estimate and are often

found to be insignificant in time-series data (Pan, 2002; Broadie et al., 2007). Here we also meet this problem for the jump volatility risk. I will continue to investigate this issue in the next section from the cross-sectional perspective, which can allow us to price volatility factors market-widely and investigate different crosssectional effects, such as the value and size effects, and their relations with the volatility factors.

4.3. Model performance analysis using different datasets In this subsection, using four different datasets in estimating the LTS-2SV model, I investigate whether the joint use of index returns, options and VIX can result in more accurate and reliable volatility estimates. The first dataset contains all available data for estimation discussed in SubSection 4.1. The second one excludes the VIX index from the whole data. The third excludes both the VIX index and index returns and only contains options. And the last dataset includes only VIX and index returns. Using parameter and volatility estimates obtained from these different datasets, I implement option pricing and examine pricing errors. Table 3 presents option pricing errors resulting from using these four different datasets. As before, the option pricing errors are defined as means of absolute differences of BS implied volatility between the market quotes and the model-implied values. I investigate in total 15 groups divided according to maturity and moneyness. For these different datasets, the overall option pricing error from using the whole data is the smallest (0.81%), whereas it becomes very large when only index returns and VIX are used (1.98%). This is because the only use of VIX loses term structure of options, which is very important to correctly identify the riskpremium parameters (Pan, 2002; Broadie et al., 2007). Duan and

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(1)

Diffusion Volatility: Vt

0.08

0.06

0.04

0.02 0 1997

2000

2003

2006

2009

2006

2009

(2)

Jump Volatility: Vt 20

15

10

5 0 1997

2000

2003 Fig. 2. Filtered volatility factors.

Yeh (2010) also find that the only use of VIX in estimation results in counter-intuitive results and unreliable volatility estimates. The option pricing performance when using index returns and options and using only options is very similar each other with the overall pricing errors being 0.92% and 0.93%, respectively. A further examination of Table 3 finds that the inclusion of VIX mainly improves the pricing of in-the-money options as VIX is constructed from short-maturity out-of-the-money calls and puts, the latter of which correspond to in-the-money call options and contains rich information on jumps and jump volatility. 5. Market risk premia variations In this section, I empirically study how the market volatility factors are priced in the cross-section of stock returns. SubSection 5.1 presents an empirical factor model where the market return, diffusion volatility, and jump volatility are fundamental factors that change the investment opportunity set of investors. SubSection 5.2 discusses the cross-sectional risk premia variations. SubSection 5.3 briefly investigates the informational difference contained in options and returns. 5.1. An empirical factor model The stochastic discount factor (8) implicitly internalizes two fundamental risk factors: the time-varying market return and the time-varying market volatility. The latter has two components, one arising from the diffusion risk and the other from the jump risk. We can rewrite this stochastic discount factor in a general form as follows:

      Mt ¼ expðr f tÞE cR RM E cV1 V ð1Þ E cV2 V ð2Þ ; t t t

ð25Þ

where EðÞ denotes the stochastic (Doleans-Dade) exponential martingale operator, and RM t is the time-varying market factor. In a Lucas-type exchange economy (Lucas, 1982), the stochastic discount factor is also often called the marginal rate of substitution, which is the ratio of marginal utilities of aggregate consumption 0 Þ over the two time horizons: M tþ1 ¼ q uuðc0 ðctþ1 , where ct denotes tÞ consumption at time t, u() is a utility function, and q is called the subjective discount factor. For any asset i with a total return Rit , no-arbitrage indicates that the Euler equation must hold:

h i E M tþ1 Ritþ1 jF t ¼ 1:

ð26Þ

After applying a second-order Taylor expansion to Eq. (26), we arrive at the following log-Euler equation:

     1 E r ietþ1 jF t ¼  Var t r ietþ1 þ Cov t r ietþ1 ; mtþ1 ; 2

ð27Þ

where r ie t denotes the log-return of asset i in excess of the risk-free rate rf, mt+1 is the logarithm of Mt+1, and  12 Vart ðr ie tþ1 Þ is the Jensen’s inequality term that arises when the log-return is used rather than the simple return. According to the stochastic discount factor (25), the log-Euler Eq. (27) implies the following expected Return-Beta representation for asset i:

 E r ietþ1 jF t ¼ at þ bit;M cR þ bit;V1 cV1 þ bit;V2 cV2 ;

ð28Þ

where bi’s are regression coefficients of returns ri.e. on risk factors, and c’s are interpreted as prices of risks, which are the same for all assets (Cochrane, 2005). Eq. (28) indicates that the expected

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Table 3 Option pricing errors. Maturity

Data

Moneyness (S/K)

Sub-total

<0.94

0.94–0.97

0.97–1.00

1.00–1.03

>1.03

<60

All data No VIX Only options No options

1.16 1.07 1.05 2.95 (18.99) [2731]

0.68 0.66 0.64 2.08 (15.50) [4564]

0.65 0.72 0.73 1.96 (15.51) [6537]

1.06 1.24 1.27 1.66 (17.36) [5014]

1.88 2.35 2.40 2.08 (20.49) [2443]

0.96 1.06 1.07 2.06 (16.96) [21,289]

60–180

All data No VIX Only options No options

0.61 0.65 0.65 2.39 (16.55) [3649]

0.43 0.49 0.50 1.96 (15.78) [2315]

0.60 0.60 0.59 1.61 (16.95) [2760]

0.89 0.89 0.89 1.29 (18.01) [1824]

1.19 1.24 1.23 1.33 (19.61) [838]

0.66 0.68 0.69 1.86 (16.95) [11,386]

P180

All data No VIX Only options No options

0.47 0.75 0.77 2.19 (15.83) [3203]

0.60 0.87 0.87 1.96 (17.00) [1233]

0.73 1.00 0.97 1.72 (17.80) [1544]

0.84 1.12 1.13 1.65 (17.99) [1179]

0.94 1.15 1.17 1.82 (18.80) [470]

0.63 0.90 0.90 1.95 (16.93) [7629]

Sub-total

All data No VIX Only options No options

0.72 0.80 0.80 2.48 (17.01) [9583]

0.60 0.64 0.64 2.03 (15.81) [8112]

0.65 0.72 0.73 1.84 (16.20) [10,845]

0.99 1.14 1.16 1.58 (17.60) [8019]

1.61 1.95 1.98 1.88 (20.08) [3751]

0.81 0.92 0.93 1.98 (16.95) [40,304]

Note: The table reports option pricing errors, which are measured as means of the absolute difference (in percentage) between the market quotes of BS implied volatility and the model-implied ones. There are 40,304 call options in total and on average, 66 options each day. Small brackets present the average BS implied volatility of each group, and square brackets present the total number of options in each group. Four different datasets are taken into account. The first dataset contains all available data discussed in SubSection 4.1; the second one excludes the VIX index from the first one; the third excludes both the VIX index and index returns and only contains options; and the last dataset includes only VIX and index returns.

excess return on an asset is explained by its sensitivity to three factors: (i) the market factor, (ii) a market volatility factor arising from the diffusion risk, and (iii) a market volatility factor arising from the jump risk. 5.2. Cross-sectional analysis For the cross-sectional investigation, the weekly volatility factors are aggregated to the monthly frequency. I first take difference b ðiÞ and the expected volatility V ðiÞ between the filtered volatility V t tjt1 for each series, and then sum the resulting innovations over the weeks in each month. The CAPM and the Fama-French three-factor model are used as benchmarks. Ang et al. (2006) investigate the volatility pricing by taking VIX as a proxy of market volatility. I also consider this factor model. The VIX factor is constructed by first taking the first-order difference of the daily VIX time series and then aggregating the daily innovations to the monthly frequency. Panel A of Table 4 presents the summary statistics for pricing factors including the market factor, the Fama-French SMB and HML factors, the constructed volatility factors, and the VIX factor. The volatility factors have very small mean values and autocorrelations. I use the Fama-French size and book-to-market 25 portfolios, which are good representatives for studies of cross-sectional variations of volatility risk premia, and the puzzling value and size effects. Panel B of Table 4 shows that that on average, value/ small-size stocks earn higher average returns than growth/bigcap stocks. The value premium and the size premium apparently exist. The Fama-French pricing factors and 25 portfolios data are downloaded from French’s online data library. With these cross-sectional data, I implement the Fama-MacBeth procedure (Fama and MacBeth, 1973). First, I find estimates of factor loadings (betas) from time-series regressions:

r eit ¼ ai þ bi1 MKT t þ bi2 DVF t þ bi3 JVF t þ it ;

ð29Þ

Table 4 Summary statistics of pricing factors and portfolios. MKT A. Pricing factors Mean 0.003 Std. dev. 0.045 Skewness 0.655 Kurtosis 3.506 q1 0.055 q2 0.064 q3 0.002 q4 0.072 q5 0.009 Small

VIX

DVF

JVF

SMB

HML

0.000 0.000 1.309 15.11 0.048 0.238 0.143 0.106 0.123

0.000 0.000 0.156 7.461 0.215 0.197 0.051 0.075 0.027

0.000 0.037 0.073 7.963 0.217 0.137 0.112 0.071 0.133

0.003 0.041 0.810 9.896 0.088 0.044 0.172 0.051 0.097

0.005 0.037 0.043 5.423 0.079 0.083 0.122 0.098 0.026

Size 2

Size 3

Size 4

Big

Average

B. Average returns on 25 size-BM portfolios Growth 0.306 0.483 0.474 BM 2 1.060 0.878 0.851 BM 3 1.154 1.104 1.049 BM 4 1.374 1.071 0.868 Value 1.371 1.128 1.342 Average 1.053 0.933 0.917

0.746 0.868 0.791 1.111 0.801 0.863

0.501 0.708 0.555 0.631 0.634 0.606

0.502 0.873 0.931 1.011 1.055 0.874

Note: The table reports summary statistics of pricing factors and Fama-French 25 size and book-to-market portfolios. Pricing factors include the market return (MKT), the VIX factor (VIX), the diffusion volatility factor (DVF), the jump volatility factor (JVF), the small-minus-big factor (SMB), and the high-minus-low factor (HML). The VIX factor is constructed by first taking the first-order difference of the daily VIX time series and then aggregating daily innovations to the monthly frequency. DVF and JVF are constructed by first taking difference between the filtered values and the expected values and then aggregating the weekly innovations to the monthly frequency. qi is the autocorrelation of each factor. The average monthly return of each portfolio is given in percentage.

for each of the 25 portfolios. In Eq. (29), r ei t is the return of portfolio i in excess of the risk-free rate, MKTt represents the market factor constructed as the difference between value-weighted market returns and the risk-free rate, and DVFt and JVFt are the diffusion

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and jump volatility factors, each of which is normalized by first subtracting the sample mean and then dividing by the sample standard deviation. Second, I estimate the market prices of risks by running a cross-sectional regression at each time period t:

r eit ¼ bi kt þ ait ;

ð30Þ

where bi = [bi1, bi2, bi3] are from the first-stage (29). Market prices k are estimated as the average of the cross-sectional regression estiP P ^ it represents the cross-sectional mates ^ k ¼ 1T Tt¼1 ^ kt . ai ¼ 1T Tt¼1 a pricing error. Standard deviations are adjusted for autocorrelation and heteroskedasticity. Finally, market risk premia on risk factors for each portfolio are calculated by multiplying the corresponding factor loadings (b) with the market prices of risks (k). The fourth row of Table 5 (VV3F) reports market prices of the market factor, the diffusion volatility factor, and the jump volatility factor, and Table 6 presents estimates of market risk premia and factor loadings (in square brackets) on these three factors. As expected, the market price of the market factor is positive (0.44% per month), and all the factor loadings on the market factor are positive and highly significant, indicating that the market factor positively changes the investment opportunity set of investors. An increase in the expected market return can increase investors’ consumption through intertemporal consumption smoothing. The market factor loadings vary from 0.63 to 1.56, with a mean of 0.96 and a standard deviation of 0.24. However, both the diffusion and jump volatility risk factors have negative market prices of 0.78% and 0.22%, respectively. The market price of the diffusion volatility risk factor is highly statistically significant, whereas that of the jump volatility risk is not. The difference in significance of the market prices of the diffusion and jump volatility factors can be explained by their roles in governing return variance. We have seen that diffusion volatility affects return variance in the long-run and its effect can last for a long time. Jump volatility plays role only in the short-run and dissipates very quickly. The market price of the diffusion volatility factor is higher (in absolute value) than that of the jump volatility factor, indicating that investors are more risk-averse to the persistent factor. Aversion to persistence often generates a stronger concern about future consumption. Investors particularly dislike assets that have high sensitivity to the persistent risk factor, which deteriorates the investment opportunities. Nearly all the diffusion volatility factor loadings have negative values (only three exceptions), and most of the jump volatility factor loadings are negative. The negative market prices of volatility risks and negative factor loadings indicate that an increase in volatility represents a deterioration of the investment opportunity set and that investors want to hedge against changes in market volatility (Campbell, 1996;

Chen, 2003). Another possible explanation of the negative market prices of volatility factors is that an increase in volatility coincides with downward market movement (French et al., 1987; Glosten et al., 1993) and assets with high sensitivities to market volatility risks provide hedges against the market downside risk (Bakshi and Kapadia, 2003a). In the cross-section of stock returns, Ang et al. (2006) find a negative price of the aggregate market volatility. Adrian and Rosenberg (2008) report negative prices for the longrun and short-run volatility components using a component GARCH model. In option pricing, Bakshi and Kapadia (2003a,b), Carr and Wu (2009), Todorov (2010), and many others have found a negative market price of volatility on equity indexes. Volatility factor loadings vary notably across portfolios. For example, diffusion volatility factor’s loadings range from 0.88 to 0.22, while jump volatility factor’s loadings vary between 0.78 and 0.29. For value stocks, nearly all factors loadings on the diffusion volatility factor are statistically significant, indicating that value stocks are more sensitive to long-run volatility. For small stocks, all factors loadings on the jump volatility factor are statistically significant, indicating that small stocks are highly sensitive to jump shocks. Furthermore, we find that value stocks are more sensitive to diffusion volatility and small stocks are more sensitive to both diffusion and jump volatility. Panel B of Table 6 shows that after controlling the size, the value stock has a higher (in absolute value) sensitivity to the diffusion volatility factor than the growth stock. For example, the sensitivity of the small-size value stocks is 0.66, whereas the sensitivity of the small-size growth stocks is only 0.37. However, this is undetermined in relation to the jump volatility factor as there is no consistent evidence that value stocks have higher sensitivities than growth stocks (Panel C). By controlling for the book-to-market value, panels B and C show that the small portfolios have higher sensitivities to both volatility factors than the big portfolios (for value stocks, 0.66 vs. 0.07 on the diffusion volatility, and 0.91 vs. 0.44 on the jump volatility). These different sensitivities to market volatility factors are consistent with risk-based interpretations of the value and size effects if we do think of volatility as a measure of risk. The value and small-size stocks have higher betas because of differences in risk embedded in these stocks. There are several theoretical justifications for why the value/ small stocks are riskier than the growth/big stocks. Fama and French (1995, 1996) show that the value effect is associated with relative distress. High book-to-market companies tend to have persistently low earnings, whereas low book-to-market firms tend to be strong firms with persistently high earnings. Size is also related to earnings and profitability. Perez-Quiros and Timmermann (2000) show that returns on small firms, which tend to be value firms, are more volatile during recessions when investors become

Table 5 Market prices of risks and pricing errors. Model

MKT

(1)

CAPM

(2)

FF3F

(3)

VV2F

(4)

VV3F

(5)

VV4F

(6)

VV3FS

0.823 (4.651) 0.442 (1.137) 0.628 (1.583) 0.441 (1.130) 0.446 (1.142) 0.423 (1.072)

VIX

DVF

JVF

SMB

HML

RMSE 0.032

0.379 (1.082)

0.016 0.024

0.543 (1.679)

0.263 (0.789)

0.493 (1.541)

0.784 (2.634) 0.796 (2.722) 0.288 (0.970)

0.219 (0.645) 0.208 (0.630) 1.047 (2.304)

0.018 0.018 0.025

Note: The table reports market prices of different pricing factors obtained using the Fama-MacBeth approach. In brackets are the absolute values of t-ratios which take into account the autocorrelation and heteroskedasticity. FF3F represents the Fama-French three-factor model; VV2F represents the VIX factor model; VV3F represents the optionimplied volatility factors model; VV4F is a factor model taking the market return, the VIX factor, and the option-implied volatility factors as inputs; and VV3FS is the indeximplied volatility factors model. RMSE is the root-mean-squared pricing error of each model.

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J. Li / Journal of Banking & Finance xxx (2011) xxx–xxx

Table 6 The cross-section of factor risk premia and factor loadings. Small A. Market factor Growth 0.687 BM 2 0.537 BM 3 0.402 BM 4 0.361 Value 0.379 Average 0.473 H–L 0.307

[1.558]⁄⁄⁄ [1.218]⁄⁄⁄ [0.911]⁄⁄⁄ [0.819]⁄⁄⁄ [0.861]⁄⁄⁄

Size 2 0.634 0.455 0.374 0.363 0.380 0.441 0.254

[1.438]⁄⁄⁄ [1.031]⁄⁄⁄ [0.848]⁄⁄⁄ [0.824]⁄⁄⁄ [0.862]⁄⁄⁄

Size 3 0.606 0.447 0.377 0.338 0.350 0.424 0.256

B. Diffusion (long-run) volatility Growth 0.294 [0.375] BM 2 0.380 [0.485]⁄ BM 3 0.334 [0.426]⁄⁄⁄ BM 4 0.257 [0.328]⁄⁄ Value 0.515 [0.657]⁄⁄ Average 0.356 H–L 0.221

0.314 0.403 0.416 0.411 0.687 0.446 0.373

[0.401]⁄ [0.515]⁄⁄⁄ [0.531]⁄⁄⁄ [0.525]⁄⁄ [0.876]⁄⁄⁄

0.322 0.342 0.317 0.460 0.586 0.405 0.263

C. Jump (short-run) volatility Growth 0.234 [1.064]⁄⁄ BM 2 0.171 [0.779]⁄⁄ BM 3 0.177 [0.805]⁄⁄⁄ BM 4 0.193 [0.878]⁄⁄⁄ Value 0.199 [0.908]⁄⁄⁄ Average 0.195 H–L 0.034

0.095 0.091 0.115 0.144 0.175 0.124 0.080

[0.432] [0.414] [0.525]⁄⁄ [0.654]⁄⁄⁄ [0.795]⁄⁄

0.023 0.067 0.068 0.076 0.005 0.048 0.018

Size 4

[1.375]⁄⁄⁄ [1.015]⁄⁄⁄ [0.854]⁄⁄⁄ [0.767]⁄⁄⁄ [0.794]⁄⁄⁄

[0.411] [0.436]⁄⁄ [0.405]⁄ [0.586]⁄ [0.747]⁄⁄

[0.104] [0.303] [0.308]⁄ [0.346]⁄ [0.021]

0.584 0.418 0.382 0.358 0.350 0.417 0.239 0.111 0.150 0.262 0.584 0.729 0.367 0.618 0.035 0.004 0.036 0.031 0.006 0.020 0.041

Big

Average

S–B

0.591 0.446 0.379 0.340 0.358 0.423 0.233

0.244 0.164 0.040 0.083 0.044 0.115

[1.324]⁄⁄⁄ [0.946]⁄⁄⁄ [0.865]⁄⁄⁄ [0.813]⁄⁄⁄ [0.781]⁄⁄⁄

0.443 0.373 0.361 0.279 0.336 0.358 0.107

[1.004]⁄⁄⁄ [0.846]⁄⁄⁄ [0.819]⁄⁄⁄ [0.632]⁄⁄⁄ [0.762]⁄⁄⁄

[0.141] [0.192] [0.334] [0.745]⁄⁄ [0.930]⁄⁄

0.013 0.173 0.096 0.108 0.056 0.023 0.069

[0.016] [0.220] [0.122] [0.139] [0.072]

0.206 0.221 0.247 0.364 0.515 0.310 0.309

0.306 0.552 0.429 0.148 0.459 0.379

0.076 0.006 0.042 0.038 0.097 0.052 0.022

[0.345] [0.029] [0.192] [0.172] [0.443]

0.093 0.068 0.087 0.096 0.094 0.087 0.002

0.158 0.165 0.135 0.155 0.102 0.143

[0.161] [0.016] [0.163] [0.142] [0.027]

Note: The table reports the market risk premia and factor loadings (in square brackets) from regressions of each size and book-to-market portfolio on the market return, the normalized DVF, and the normalized JVF. t-ratios are calculated by taking the autocorrelation and heteroskedasticity into account. H–L and S–B represent the value–growth and small–big premium spreads. ⁄ The significance of the factor loadings at the 10% level. ⁄⁄ The significance of the factor loadings at the 5% level. ⁄⁄⁄ The significance of the factor loadings at the 1% level.

more risk averse. Liew and Vassalou (2000) argue that returns on value or size portfolios seem to predict GDP growth, and thus may capture some aspects of business cycle risk. Petkova and Zhang (2005) and Zhang (2005) argue that time-varying risk goes in the right direction in explaining the value premium, and they try to provide an explanation based on irreversible investments and the countercyclical price of risk. Chen et al. (2011) explain the value and size effects with an investment factor, which is the difference between returns on a portfolio of low-investment stocks and a portfolio of high-investment stocks, and a return-on-assets factor, which is the difference between returns on a portfolio of stocks with high returns on assets and on a portfolio of stocks with low returns on assets. The results also indicate that the value premium is mainly related to diffusion (long-run) volatility, whereas the size premium is related both to diffusion (long-run) and jump (short-run) volatility. In a rational market, short-term variation of risk should have little effect on the stock price and the book-to-market ratio, and long-term investors should care more about long-lasting shocks that change investment opportunities. The value effect should be associated with long-run volatility and long-term differences in profitability (Fama and French, 1995). The diffusion volatility is highly persistent and hence affects investors’ long-run consumption. Bansal and Yaron (2004) argue that investors worry about the long-run risk, which they define as persistent changes in expected consumption growth. Parker and Julliard (2005) and Hansen et al. (2008) show that the exposure of value firms to the long-run consumption risk accounts for the value premium. Bansal et al. (2005) argue that average returns of value and growth stocks can be understood by different covariances with the long-run consumption growth. Table 6 also reports the cross-sectional variation of risk premia of different pricing factors. On average, the market factor earns 0.42% per month, taking nearly 52% of the total risk premium. We find different directions for the value and size effects. While small stocks have higher risk premium than the big stocks on the

market factor, growth stocks counterintuitivelly earn more risk premium than value stocks on the market factor, indicating that the market factor alone cannot explain the value effect. The diffusion volatility factor has a higher market price than the jump volatility factor (see Table 5) and earns a higher risk premium (on average, 0.31% vs. 0.09% per month). Consistent with the above discussion on the risk sensitivity and the value and size effects, value portfolios have higher risk premia than growth portfolios on the diffusion volatility factor, and small portfolios have larger risk premia than big portfolios on both the diffusion and the jump volatility factors. For example, the average value premium spread on the diffusion volatility factor is as large as 0.31% per month, but it is negligible on the jump volatility factor. The size premium spread is 0.38% on the diffusion volatility factor, larger than the spread on the jump volatility factor (0.14%). To compare the volatility factors model (VV3F) with the CAPM, the Fama-French model, and the VIX factor model (VV2F), we find from the last column (RMSE) in Table 5 that the CAPM fits the actual returns the worst (0.032), the option-implied volatility factors model performs similarly to the Fama-French model (0.018 vs. 0.016), and the VIX factor model performs better than the CAPM model but worse than the other two (0.024). We also find that in the VIX factor model the market price of the VIX factor is negative (0.54%) but only statistically significant at the 10% level (1.68). Whenever we combine the VIX factor and the option-implied volatility factors together (VV4F), the market price of the VIX factor becomes smaller (in absolute value, 0.26%) and statistically insignificant. However, market prices of the option-implied volatility factors are nearly the same as those in the VV3F model and the significance of the diffusion volatility factor becomes even stronger. 5.3. The role of options and VIX We have noted that even though two markets exist, most empirical studies still use stock price data alone. What roles do

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J. Li / Journal of Banking & Finance xxx (2011) xxx–xxx Table 7 Index-implied volatility factor loadings and risk premia. Small

Size 2

Size 3

Size 4

A. Diffusion (long-run) volatility Growth 0.410 [1.421]⁄⁄ BM 2 0.316 [1.097]⁄⁄ BM 3 0.231 [0.802]⁄⁄ BM 4 0.278 [0.965]⁄⁄ Value 0.386 [1.342]⁄⁄⁄ Average 0.324 H–L 0.023

0.121 0.144 0.149 0.174 0.211 0.160 0.089

[0.420] [0.498] [0.517] [0.604]⁄ [0.731]

0.104 0.097 0.080 0.166 0.181 0.126 0.077

[0.360] [0.338] [0.278] [0.578]⁄⁄ [0.628]⁄⁄

B. Jump (short-run) volatility Growth 0.274 [0.262] BM 2 0.314 [0.300] BM 3 0.356 [0.340] BM 4 0.193 [0.184] Value 0.356 [0.341]⁄ Average 0.298 H–L 0.082

0.254 0.483 0.544 0.391 0.570 0.448 0.316

[0.243] [0.462] [0.519]⁄⁄ [0.373] [0.545]⁄⁄

0.194 0.349 0.409 0.474 0.395 0.364 0.202

[0.185] [0.333] [0.391] [0.453] [0.378]⁄

Big

Average

S–B

0.044 0.104 0.144 0.127 0.116 0.107 0.072

[0.152] [0.361]⁄⁄ [0.499] [0.442]⁄⁄ [0.401]

0.116 0.013 0.064 0.083 0.147 0.033 0.263

[0.403]⁄⁄ [0.047] [0.221] [0.290] [0.510]⁄

0.112 0.129 0.134 0.166 0.208 0.150 0.096

0.526 0.329 0.167 0.194 0.240 0.291

0.118 0.271 0.290 0.702 0.330 0.163 0.212

[0.113] [0.259] [0.277] [0.670]⁄⁄ [0.315]

0.168 0.216 0.245 0.283 0.065 0.030 0.233

[0.161] [0.206] [0.234] [0.271] [0.062]

0.087 0.326 0.271 0.406 0.211 0.261 0.124

0.443 0.098 0.600 0.091 0.292 0.269

Note: The table reports factor loadings and risk premia under the VV3FS model, where volatility factors are estimated only from the market index. t-ratios are calculated by taking the autocorrelation and heteroskedasticity into account. H–L and S–B represent the value–growth and small–big premium spreads. ⁄ The significance of the factor loadings at the 10% level. ⁄⁄ The significance of the factor loadings at the 5% level. ⁄⁄⁄ The significance of the factor loadings at the 1% level.

options play? In this subsection, I implement the same procedures but use volatility factors obtained from the S&P 500 index alone. The model is estimated using MCMC methods proposed by Li (2011), which are the most efficient one when using only stock price data. I first take the first-order difference for each volatility series and then aggregate the daily innovations to the monthly frequency over the period from January 1997 to September 2008. The last row of Table 5 reports the root-mean-square-error and market prices of the volatility factors in the index-implied volatility factors model (VV3FS). Its pricing performance is the worst with comparison to other models (0.025). We find that although both are negative, the market price of diffusion volatility is smaller (in absolute value) than that of jump volatility (0.29% vs. 1.04%). Table 7 shows that diffusion volatility has an average market risk premium of 0.15% and jump volatility has an average market risk premium of 0.26%. This seems counter-intuitive in a rational market. As discussed previously, the diffusion volatility is very persistent and affects the aggregate market volatility in the long run, whereas the jump volatility dissipates quickly and contributes to the aggregate market volatility only in the short run. Therefore, diffusion volatility should require a higher market price and a higher risk premium than jump volatility. As shown in Table 7, few sensitivities to the jump volatility factor are statistically significant at the 5% level. This is in contrast to results obtained in SubSection 5.2. In general, option-implied jump factor loadings are higher than their index-implied counterparts, indicating that firms, especially small firms, are more sensitive to jump volatility when options are taken into account. Furthermore, the size effect is associated with both the diffusion and jump volatility factors, and the value effect is related to the diffusion volatility factor. However, the value– growth premium spread is too small to explain the historically observed high returns in value stocks. For example, the indeximplied average value–growth premium spread (H–L) in Table 7 on the diffusion volatility is only 0.096%, whereas the option-implied spread (H–L) in Table 6 is as high as 0.31%. The spreads of H–L and H–L on jump volatility are 0.12% and 0.002%, respectively. In total, the index-implied value–growth premium spread is 0.22%, which is smaller than the option-implied value–growth premium spread (0.31%). The average size premium spread is very similar under two cases: while the option-implied size premium spread (S–B) is larger than the index-implied one (S–B) on the diffusion volatility factor (0.38%

vs. 0.29%), it is smaller than the index-implied value on the jump volatility factor (0.14% vs. 0.27%).

6. Conclusion This paper develops a multifactor model by proposing a pricing kernel, where the market return, diffusion volatility, and jump volatility are fundamental factors that change the investment opportunity set of investors. Based on estimates of volatility factors using S&P 500 index returns, options and VIX, the paper finds negative market prices for volatility factors in the cross-section of stock returns. The findings are consistent with risk-based interpretations of value and size premia and indicate that the value effect is mainly related to the persistent diffusion volatility factor, whereas the size effect is associated with both the diffusion and jump volatility factors. The paper also finds that the use of the market index alone may lead to a loss of information on the evolution of volatility factors, especially information on the jump volatility factor, and may yield counter-intuitive results.

Appendix A. The conditional characteristic function The conditional characteristic function of log return rt = ln (St/ Sts) under the risk-neutral measure Q is given by:

/R ðu; s; V ts Þ  EQ ½eiurt jF ts  ¼ expðiur

f

s  Aðu; sÞ  Bðu; sÞV ts Þ;

h i0 ð2Þ where A ¼ A1 þ A2 ; B ¼ ½B1 ; B2 ; V ts ¼ V ð1Þ ts ; V ts , and







ð31Þ

  ðc  ji Þð1  eci s Þ ji hi þ ci  ji s ; 2 log 1  i 2 2ci ri 2ui ðuÞð1  eci s Þ   Bi ðu; sÞ ¼ ; 2ci  ci  ji ð1  eci s Þ Ai ðu; sÞ ¼

ci ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ji 2 þ 2r2i ui ðuÞ;

1 2 u2 ðuÞ  uQX ðuÞ ¼ wQX ðuÞ þ iukQX ð1Þ;

u1 ðuÞ  uQW ðuÞ ¼ ðiu þ u2 Þ;

j1 ¼ jQ1  iuqr1 ; j2 ¼ jQ2 :

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12

J. Li / Journal of Banking & Finance xxx (2011) xxx–xxx

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