Out-of-sample density forecasts with affine jump diffusion models

Journal of Banking & Finance 47 (2014) 74–87

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Journal of Banking & Finance journal homepage: www.elsevier.com/locate/jbf

Out-of-sample density forecasts with affine jump diffusion models Jaeho Yun ⇑ Department of Economics, Ewha Womans University, 52, Ewhayeodae-gil, Seodaemun-gu, Seoul 120-750, Republic of Korea

a r t i c l e

i n f o

Article history: Received 21 March 2013 Accepted 29 June 2014 Available online 9 July 2014 JEL classification: C14 C22 C53 G13 Keywords: Density forecasts Time-series consistency Affine jump diffusion Time-varying jump risk premia Particle filters Beta transformation

a b s t r a c t We conduct out-of-sample density forecast evaluations of the affine jump diffusion models for the S&P 500 stock index and its options’ contracts. We also examine the time-series consistency between the model-implied spot volatilities using options & returns and only returns. In particular, we focus on the role of the time-varying jump risk premia. Particle filters are used to estimate the model-implied spot volatilities. We also propose the beta transformation approach for recursive parameter updating. Our empirical analysis shows that the inconsistencies between options & returns and only returns are resolved by the introduction of the time-varying jump risk premia. For density forecasts, the time-varying jump risk premia models dominate the other models in terms of likelihood criteria. We also find that for medium-term horizons, the beta transformation can weaken the systematic effect of misspecified AJD models using options & returns. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction Under the no-arbitrage assumption, a stock return model is specified as both the dynamics of a physical measure (henceforth, P-measure) and a risk-neutral measure (henceforth, Q-measure). As is well known, the Q-dynamics are used to price stock-related derivatives at a given point of time, while the P-dynamics capture the time-series properties of stock returns over time. Given correct model specifications, both dynamics can offer useful information to one another. For example, one can extract a model-implied spot volatility from the options data, and use it to predict the future path of stock returns over time. Extracting information from the options data is very useful; whereas historical stock return data is only backward-looking, options data is, by nature, forward-looking. Market participants trade options under their expectations of the future behavior of stock returns. Thus, options data is very informative regarding the future distribution of stock returns, if it is combined with a correctly specified stock return model. In this sense, it is important to find a correctly specified stock return model across P - and Q-measures.

⇑ Tel.: +82 2 3277 4468; fax: +82 2 3277 4010. E-mail address: [email protected] http://dx.doi.org/10.1016/j.jbankfin.2014.06.024 0378-4266/Ó 2014 Elsevier B.V. All rights reserved.

In this paper, we conduct out-of-sample density forecast evaluations for various affine jump diffusion (hereafter AJD) stock return models for the S&P 500 stock index, investigating the informational contents in the options data for density forecasts. For comparative purposes, we also use the realized volatility (hereafter, RV) models. For AJD models, we focus on the role of the time-varying jump risk premia in forecasting real-world densities. Different forecast horizons (e.g., 1 day, 1 week, 2 weeks, 3 weeks, and 4 weeks) are considered, because options data may have different informational content for density forecasts across different horizons. We employ the particle filters proposed by Johannes et al. (2009) in order to estimate model-implied spot volatilities from two different informational sources: (i) options & returns and (ii) only returns. Furthermore, for the various AJD models, we examine the issue of time series consistency between model-implied spot volatilities using options & returns versus only returns. Furthermore, we propose the so-called ‘‘beta transformation method’’ as a tool to recursively update density parameters. Due to high computational costs, it is sometimes difficult to frequently update model parameters in the AJD models. Thus, beta transformation is proposed as an alternative approach to updating parameters. In studies by Taylor (2005), Liu et al. (2007) and Shackleton et al. (2010), the beta transformation method was originally used in order to transform risk-neutral densities into real-world

J. Yun / Journal of Banking & Finance 47 (2014) 74–87

densities. However, we use this method to save computational costs of parameter updating. We study the density forecasts of stock returns models, because density forecasts have been an important research agenda in finance and economics; hence, finance theories, such as asset pricing, portfolio selection and option valuation, have modeled the uncertainty via a parametric distribution function (e.g., Bao et al., 2007). Moreover, density forecast is particularly important in the context of risk management. For instance, the value-at-risk (VaR), an important risk management tool in the financial industry, keeps track of certain aspects of the (conditional) distribution of asset returns. Needless to say, an accurate risk measurement is crucial for the efficient capital allocation in financial institutions. For density forecast models, we employ the AJD models because of their many advantages. First, the AJD models provide an analytically tractable option pricing method. Hence, one can easily combine option market information with the AJD models. Furthermore, they are highly flexible in describing both the P- and Q-dynamics, particularly depending on the restrictions imposed upon their jump specifications (for either return or volatility). Although more sophisticated nonparametric stock return models can be considered as alternatives, these parametric AJD models still have merit in their parsimonious, flexible and tractable model specifications.1 Using the high flexibility of the AJD models, as used by Bates (2000) and Pan (2002), we model the time-varying risk premia by incorporating the time-varying jump arrival intensity under the Q-measure. We should mention that our study is greatly influenced by three past studies: Yun (2011), Johannes et al. (2009) and Shackleton et al. (2010). Yun (2011) examined out-of-sample option pricing performances for the AJD models by using the S&P 500 stock index and its option contracts. He found strong evidence in favor of the time-varying jump risk premia in the cross-sectional option pricing during 2001–2007. He also discovered that during a period of low volatility, the role of jump risk premia becomes less pronounced. This finding can possibly explain the poor pricing performances of jump diffusion models in the studies whose sample periods are characterized by low volatility. However, his study focused only on the Q-dynamics of the AJD models. A natural follow-up study would be to relate the time-varying jump risk premia to P-dynamics: for instance, density forecasts for stock returns over time with forward-looking option-based information. We attempt to fill this gap through the research presented in this paper. Next, Johannes et al. (2009) proposed a particle filtering method for the AJD models by combining time-discretization schemes with Monte Carlo methods. As mentioned above, we use their method to estimate model-implied spot volatilities. Johannes et al. (2009) found a serious inconsistency in the AJD models across the ‘‘options & returns’’ and ‘‘returns only’’ informational sources. They discovered that for each model they considered, the model-implied spot volatilities estimated by using only returns tend to be lower than the volatilities using options & returns, particularly during times of high volatility. They considered only constant jump risk premia models; as a future research topic, they suggested timevarying jump risk premia in order to solve this time-series inconsistency. In our paper, we address this issue by introducing the time-varying jump risk premia. In terms of the method used for analyzing density forecasts, our study is close to Shackleton et al. (2010). They compared multihorizon density forecasts of the S&P 500 index from 1991 to

75

2004. Their density forecast models used either options data or realized volatilities data. They estimated risk-neutral densities by using cross-sectional options data, and applied various transformation schemes in order to obtain real-world densities (e.g., risk-premia transformations and calibration transformations). For a one-day horizon, it turned out that in density forecasts, realized volatility is more informative than options data. Meanwhile, they found that for horizons of two and four weeks, option prices are the most informative for density forecasts. They argued that incorporating risk-neutral jumps does not lead to improved real-world densities. They attributed this superior performance of option-based forecasts over medium term horizons to the forward-looking property of the option prices. Interestingly, similar to Shackleton et al. (2010), we also obtain the result that, with the help of the timevarying jump risk premia, options are very informative for the medium-term horizon density forecasts. However, in contrast to the results of Shackleton et al. (2010), our analysis provides empirical evidence in favor of the jump specification with time-varying jump risk premia, and furthermore, supports the important informational content of options, even in the one-day horizon forecasts. For density forecast evaluations of stock return models, we use two different criteria: the likelihood and the probability integral transform (henceforth, PIT) evaluations. The likelihood evaluations utilize model-implied conditional density functions for density forecast evaluations. For this approach, we use both log likelihood ratios and sequential likelihoods. For an additional evaluation method, PIT-based evaluations are employed. Unlike the likelihood-based evaluations, they use model-implied cumulative distribution functions (CDF), so they may possibly capture different aspects of the density forecast performances. For the PIT-based evaluations, we first use the popular Kolmogorov–Smirnov test, and also use the more recently developed Hong and Li’s (2005) test. To compute these test statistics, we exploit closed-form conditional moment generating functions for the AJD models. We are able to obtain the likelihood or probability integral transform for any forecast horizon by Fourier inversion of the conditional moment generating functions. Otherwise, we would have to have used the computationally expensive simulation methods of Pedersen (1995), Elerian et al. (2001), or Brandt and Santa-Clara (2002). One of the virtues of using conditional moment generating functions is that they do not produce any discretization bias or simulation error, while they can be applied to any forecast horizon, without additional computational costs. Our empirical analysis finds the following: First, for the AJD models, the time-series inconsistency between the model-implied spot volatilities using options & returns and only returns seems to be resolved by the introduction of the time-varying jump risk premia. Second, our density forecast evaluations indicate that, in terms of likelihood criteria, the time-varying jump risk premia models dominate over the other models. It turns out that the options’ information improves the density forecast ability of the AJD models; however, it has to be performed with the correct specifications of Q-dynamics, such as the time-varying jump risk premia. Lastly, for the medium term density forecasts (e.g., 1–4 week forecasting horizons), the beta transformation is able to improve the density forecast abilities of all the option-based models. This indicates that we can remove the systematic effect of misspecified density forecast models by means of our proposed beta transformation. 2. Models

1 The AJD models have solved many anomalies in asset pricing. For example, Hull and White (1987) indicated that stochastic volatility in the SV model is able to explain the ‘‘volatility smile (or volatility smirk)’’ in option prices, which cannot be validated by the Black–Scholes model. Moreover, the jump component in the AJD models can provide additional flexibility, in capturing some important features of asset return dynamics, such as conditional skewness and leptokurtosis.

2.1. Affine jump diffusion models This section describes the AJD stock return models studied in this paper. Consider the stock index ðSt Þ and stochastic volatility ðV t Þ processes under P-measure:

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pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffi S V d ln St ¼ ldt þ 1  q2 V t dW t þ q V t dW t þ Z St dNt ; pffiffiffiffiffi V dV t ¼ jðh  V t Þdt þ rv V t dW t þ Z Vt dNt ; W St

ð1Þ ð2Þ

W Vt

where and are independent Brownian motions under P; N t  PoiðkPt Þ is a Poisson distributed jump timing with a possibly time-varying kPt ; Z St is a jump-in-return size with Z St i.i.d.  P 2 intensity  V P NðlS ; rS Þ; Z t is a jump-in-volatility size with Z Vt  i.i.d. Exp lPV , and both Z St and Z Vt are independent of each other. We consider three popular specifications: the SV, SVJ, and SVCJ. The SV has no jump component (so, N t ¼ 0, almost surely); the SVJ has only the jump-in-return (so, Z vt ¼ 0, almost surely); and the SVCJ has both the contemporaneously arriving jump-in-return and jump-in-volatility. Since many existing studies have, for identification purposes, set kPt to be constant (e.g., kPt ¼ k ) under P, we also set kPt ¼ k . It is well known from the asset pricing literature that there is equivalence between no arbitrage and the existence of the risk-neutral probability measure Q. We consider the following Q-dynamics:

pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffi 1 S Q 1  q2 V t dW t ðQÞ d ln St ¼ ðrt  dt  V t  kQ t lS Þdt þ 2 pffiffiffiffiffi V þ q V t dW t ðQÞ þ Z St ðQÞdNt ðQÞ; pffiffiffiffiffi V V dV t ¼ ðjðh  V t Þ þ gQ v V t Þdt þ rv V t dW t ðQÞ þ Z t ðQÞdN t ðQÞ;

ð3Þ

ð4Þ

where rt is the short-term risk-free rate, dt is the dividend rate, and gQv is the diffusive volatility risk premium. Similar to Eqs. (1) and (2), W St ðQÞ and W Vt ðQÞ are uncorrelated under Q. We also have that   S  Q 2 V N t ðQÞ  Poi kQ i.i.d. NðlQ Þ; Z t ðQÞ i.i.d. ExpðlQ t ; Z t ðQÞ V Þ, S ; rS and both Z St ðQÞ and Z Vt ðQÞ are independent of each other. According to the well known Girsanov theorem, the parameters, rv ; q; j , and h are identical across P and Q. Moreover, for practicality, many P studies have assumed that rQ S ¼ rS . For the risk-neutral jump arrival intensity kQ t , we consider two different specifications. The first specification sets kPt ¼ kQ t ¼ k ; we denote the AJD models with kQ t ¼ k by the ‘‘constant jump risk premia models.’’ The other specification involves the time-varying jump arrival intensity under Q. To be specific, like Bates (2000) and Pan (2002), we set kQ t ¼ fV t . Thus, the state-dependent risk-neutral jump arrival intensity is assumed to be linear in spot volatility. We denote the AJD models with kQ t ¼ fV t by the ‘‘time-varying jump risk premia models.’’ Note that the time-varying jump risk premia models have a constant jump arrival intensity under P, but have a state-dependent jump arrival intensity under Q. In contrast, the constant jump risk premia models have a common and constant jump arrival intensity under both P and Q. The specification of the risk-neutral time-varying jump arrival intensity is possible, since the assumption of no arbitrage imposes much weaker restrictions for the change of measure on the jump process than on the diffusion process. For example, kPt and kQ t may differ in their current level (e.g., Singleton, 2006). Given Q-parameters, a European call option at time t (denoted by C t ) with maturity date T and strike price K can be priced by the following formula:

  Z C t ¼ EQ exp 

T

  r u du Max½ST  K; 0jF t

t

¼ C Model ½St; V t; K; T  t; r t ; dt :

ð5Þ

Bates (1996) and Duffie et al. (2000) provided a tractable option pricing method for various AJD specifications. The advantage is due to the fact that the AJD class has a closed-form conditional moment generating function. For a detailed closed-form option pricing formula for AJD models, refer to the Appendix in Yun (2011). Using this option pricing formula, given an AJD specification, we are able to extract model-implied spot volatilities from the options data.

In order to forecast densities using the AJD models, we need model-implied spot volatility estimates. We use particle filters to extract the daily model-implied spot volatilities. From the volatility particles at a given point of time, we take medians as the volatility estimates, because they are invariant to a monotonic transformation (e.g., transformation from variance to standard deviation). We checked to make sure that there is only a slight difference in magnitude between the mean and median estimates. When conducting the particle filters, we use two different information sources: ‘‘only returns’’ and ‘‘options & returns.’’ For the ‘‘only  returns,’’ we use only the historical returns It ¼ fys gts¼1 with yt ¼ 100  lnðSt =St1 ÞÞ to estimate model-implied spot volatilities. We place the suffix ‘‘-R’’ to identify these models. For these models, the Q -parameters are not necessary, because options data is not used to estimate their model-implied volatilities. For the other   models using ‘‘options & returns’’ information It ¼ fys ; cs gts¼1 , we place the suffix ‘‘-OR’’ after the names of the models. For each date, one representative liquid option is used. The formula in Eq. (5) above is then used to estimate the model-implied spot volatilities. These ‘‘options & returns’’ models need to assume either a constant or time-varying jump risk premia. The latter models are indicated by the addition of the suffix ‘‘-TV.’’ Our eight AJD models are summarized as follows:  SV-R, SVJ-R, SVCJ-R: Model-implied spot volatilities are esti  mated using only returns data It ¼ fys gts¼1 . Thus Q -dynamics do not have to be considered.  SV-OR, SVJ-OR, SVCJ-OR: Model-implied spot volatilities  are estimated using options & returns data It ¼ fys ; cs gts¼1 . Constant jump arrival intensities under P and Q are assumed (that is, kPt ¼ kQ t ¼ kÞ. We need the Q-parameter estimates since options data is used.  SVCJ-TV-OR, SVCJ-TV-OR: Model-implied spot volatilities are   estimated using options & returns data It ¼ fys ; cs gts¼1 . Timevarying jump arrival intensity under Q is assumed. We need the Q-parameter estimates since options data is used. 2.2. Realized volatility models Realized volatilities have been documented as a valuable informational source for stock return densities, especially for future volatilities. In order to use realized volatilities for density forecasts, a proper time-series model must be specified. For this purpose, we employ a GJR-type model (e.g., Glosten et al., 1993), which can take into account the well-known ‘‘leverage’’ effect. We use two RV models, depending on their distributional assumptions for innovative terms: the RV-N (normal distribution) and RV-T (t-distribution) models. Our RV models are similar to the HEAVY-r model used by Han and Park (2013) and the Intra models used by Shackleton et al. (2010). The model specifications are as follows:

yt ¼ l þ rt et ;

ð6Þ

r2t ¼ x þ ða1 þ a2 dt1 ÞRV t1 þ bt1 r2t1 ; where yt ¼ 100  lnðSt =St1 Þ; et  ð0; 1Þ, either normal (for RV-N) or t-distributed (for RV-T); dt ¼ 1 when et < 0, and is 0 otherwise; and RV t is a realized volatility measure at time t. Since we use five different forecast horizons, we estimate the RV models separately for each forecast horizon. 3. Empirical methods 3.1. Particle filters As noted, in order to forecast densities via an AJD model, estimation of the model-implied spot volatilities is required. We use

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the particle filtering method proposed by Johannes et al. (2009). Their approach, called the ‘‘optimal filtering,’’ combines time-discretization schemes with Monte Carlo methods. According to Pitt and Shephard (1999), the particle filters are defined as the class of simulation filters that recursively approximate the filtering random variable Lt jIt (Lt is a latent variable at time t, and It is a past history of observable variables up to time t2) by ‘‘particles’’ L1t ; . . . ; LNt , whereby a continuous random variable is approximated by a discrete one with random support. These discrete particles are viewed as samples from the conditional density, pðLt jIt Þ. It has been shown in the associated literature that as N!1 , the particles can approximate the density of Lt jIt increasingly well. There are two algorithms that can be used to conduct particle filtering: the sampling-importance resampling (Gordon et al., 1993) and the auxiliary particle filter (Pitt and Shephard, 1999) algorithms. Although the sampling-importance resampling algorithm is simple and easy to code, it may incur a well-known ‘‘sample impoverishment’’ problem, particularly during periods with large movement driven by outliers and rare events. To avoid sample impoverishment, Pitt and Shephard (1999) proposed the auxiliary particle filter algorithm.3 In our analysis, we adopt the auxiliary particle filter algorithm, since the stock market is sometimes driven by infrequently occurring ‘‘jumps.’’ Following Johannes et al. (2009), we adopt the Euler-discretization scheme in order to approximate both the return and the volatility processes. In addition, we use Bernoulli approximation to approximate the compound Poisson distributed jump occurrences. For example, the approximated SVCJ model is as follows:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ytþ1 ¼ a þ q V tþ1 e1;tþ1 þ ð1  q2 ÞV tþ1 e2;tþ1 þ J tþ1 Z Stþ1 ; pffiffiffiffiffi V tþ1 ¼ V t þ jðh  V t Þ þ rv V t e1;tþ1 þ J tþ1 Z Vtþ1 ;

ð8Þ

ln C tþ1 ¼ ln C Model ½Stþ1; V tþ1; K; T  t  1; rtþ1 ; dtþ1  þ rc e3;tþ1 ;

ð9Þ

ð7Þ

where yt ¼ 100  D ln St ; ðe1;t e2;t Þ0 i.i.d. Nð0; IÞ, Jt i.i.d. BerðkÞ, Z St i.i.d. NðlS ; r2S Þ; Z vt i.i.d. Expðlv Þ; e3;t i.i.d. Nð0; 1Þ for all t; C t is the market price for the call option, and C Model ½Stþ1 ; V tþ1 ;  is the model-implied call option price in Eq. (5). In this paper, we use P and Q parameter estimates from Yun (2011). Moreover, following Johannes et al. (2009), we set rc ¼ 0:1, consistent with the observed bid-ask spreads of at-themoney index options. Note that for the models using ‘‘only returns’’ information (e.g., SV-R, SVJ-R, and SVCJ-R), Eq. (9) does not need to be considered. We set the number of particles to be 2500. The detailed algorithm can be obtained from the Appendix in Johannes et al. (2009).4 3.2. Likelihood evaluations We employ likelihood evaluations for density forecast evaluations. In the past, Bao et al. (2007) and Shackleton et al. (2010) used likelihood evaluations to compare density forecast models applied to equity indices. Our explanation below is based on their discussion. As mentioned above, the likelihood criteria use a model-implied conditional density function for density forecast evaluations. 2 As noted, under ‘‘returns only’’ information, we set It ¼ fys gts¼1 , and under ‘‘options & returns’’ information, we set It ¼ fys ; cs gts¼1 . 3 In order to avoid sample impoverishment, Pitt and Shephard (1999) suggested two stages of resampling, which was also used by Johannes et al. (2009). 4 Johannes et al. (2009) explained the algorithm for the AJD models using only returns. For the ‘‘options & returns’’ model, the second stage weights in the ðiÞ;mod ðiÞ Appendix in Johannes to be modified to ptþ1 / ptþ1    et al. (2009) have   ðiÞ / ln C tþ1  ln C Model V tþ1 0; r2C , where / yjl; r2 indicates the normal density tþ1 function with mean l and variance r2 .

Given a model m (e.g., m ¼ 1; . . . ; M), for a fixed forecast horizon

s, let us consider the log-likelihood of realized stock returns yts;t (ln St  ln Sts Þ (Following usual conventions, Y ts;t denotes an unrealized stock return). For simplicity, we suppress the model parameters, and write yt for yts;t when there is no confusion. Then we have ðmÞ

LT

¼

T X

ln f ðmÞ ðyt jIt1 Þ;

ð10Þ

t¼1

where f ðmÞ ðyt jIt1 Þ is a conditional density function of yt with It1 . Even if all the models considered are misspecified, maximizing Lm among the M models will select for the model whose densities are nearest to the true densities, according to the information criterion of Kullback and Leibler. In addition to the full-sample likelihood, we also consider the sequential likelihood method suggested by Johannes et al. (2009). Using this method, one can detect the exact time periods in which the models fail, in a relative sense. According to Johannes et al. (2009), the sequential likelihood method allows researchers to discriminate between abrupt failures and those that accumulate slowly, providing a deeper understanding of how the models fit the data. For example, one can compare the models SV-R and SVJ-R via the sequential likelihood at time t as follows5: t X

LRðSVR;SVJRÞ ðtÞ ¼

ln

s¼1

 ðSVJRÞ  f ðys jIs1 Þ ; f ðSVRÞ ðys jIs1 Þ

t ¼ 1; . . . ; T:

ð11Þ

Interestingly, Johannes et al. (2009) found that model fits are primarily driven by large returns and periods of market stress, conveying the importance of accurate models for these periods. In their exercise, the SVJ-R model provided a much higher likelihood than the SV-R model. They found that the abrupt divergence of the two models occurs around the crash of 1987. That is because the SV-R model cannot generate a large movement, such as that on Black Monday, when the predicted volatility is only modest. In this paper, as in Johannes et al. (2009), by observing the sequential likelihood, we also examine when and how a model divergence occurs, if ever. Unfortunately, it is not always possible to obtain a closed-form likelihood function for a density forecast model. However, we can obtain the density function of an AJD model for any forecasting horizon from Fourier inversion of its conditional characteristic function. In contrast to other approximation methods, the Fourier inversion method does not incur any discretization bias, making it possible to forecast densities at any forecast time horizon without additional computational costs. Suppose that we have the conditional moment generating function (hereafter CMGF) for an AJD model. At date t with a filtration F t , one can define the CMGF of a stock return at time t þ s with the forecast horizon s(i.e., Y t;tþs ):

/t;tþs ðuÞ ¼ E½euY t;tþs jF t ;

ð12Þ

where u 2 C. It is well known that CMGF of the AJD model has the exponential affine form, as follows:

/t;tþs ðuÞ ¼ expð/0 ðs; uÞ þ /1 ðs; uÞV t Þ;

ð13Þ

where /0 ðs; uÞ and /1 ðs; uÞ are complex functions of sand u. The exact form of the above CMGF is provided in the Appendix in Yun (2011). Given the CMGF, it can be shown that the conditional density function can be derived as follows:

f ðyt;tþs jF t Þ ¼

1

p

Z 0

1



Re /t;tþs ðiuÞeiuyt;tþs du:

ð14Þ

5 Johannes et al. (2009) considered only ‘‘returns only’’ models. Therefore, in their study, the SV-R and SVJ-R models were simply denoted by the SV and SVJ, respectively.

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If we take the log, and plug a realized return into Eq. (14), we can obtain a log likelihood value. The integrations in Eq. (14) (moreover, integration in Eq. (16) below) can be quickly evaluated by the Gauss–Legendre quadrature for suitably chosen intervals. In our analysis, we use three intervals for the integration: [0, 10], [10, 100], and [100, 1000]. 3.3. Probability integral transform evaluations In this subsection, we describe PIT-based testing methods that use a time series of the realized conditional CDFs. As before, for expositional convenience, we can set s ¼ 1, and write yt for yts;t without loss of generality. Now, let us define the dynamic probability integral transform (PIT) as follows:

Z t ðhÞ ¼

Z

yt

f ðyjIt1 ; hÞdy;

t ¼ 1; . . . ; T;

ð15Þ

1

where his p an ffiffiffi unknown finite-dimensional parameter vector. In practice, a T -consistent estimator, say b h, is used in implementing the test. When the model is correctly specified in the sense that there exists some h0 2 Hsuch that f ðyjIt1 ; h0 Þ coincides with the transition density of fyt g , then the sequence fZ t ðh0 Þg is i.i.d. U½0; 1 (Diebold et al., 1998). The series fZ t ðhÞg is referred to as the ‘‘generalized residuals’’ of the transition density model f ðyjIt1 ; hÞ. The i.i.d. U½0; 1 property provides a basis for testing the model. If fZ t ðhÞg is not i.i.d. U½0; 1 for all h 2 H , then the model is not correctly specified. For simplicity, we will suppress h as before. The key step in the PIT-based tests is the dynamic probability integral transform, which requires integration of the modelimplied conditional density. Although the AJD models provide no closed-form conditional density, we can obtain the generalized residuals via the Fourier inversion, as in the case of the likelihood criteria, as follows:

zt;tþs ¼

1 1  2 p

Z 0

1



Im /t;tþs ðiuÞeiuyt;tþs du u

ð16Þ

  where /t;tþs ðuÞ ¼ E eiuY t;tþs jF t (the conditional moment generating function of Y t;tþs Þ. 3.3.1. Kolomogorov & Smirnov test If a model is correctly specified, the above generalized residuals follow i.i.d. U½0; 1. The Kolomogorov & Smirnov (KS) test assumes that the i.i.d. property is already satisfied; then tests the uniform property of the generalized residuals. Concretely, the KS test uses the maximum difference between the empirical and theoretical CDF functions. Let us denote the empirical unconditional CDF function by CDFðuÞ. Clearly, we have 0 6 u 6 1. The KS statistic can then be expressed as follows:

KS ¼ sup jCDFðuÞ  uj: 06u61

The KS test is widely used because it is intuitively appealing and simple to implement. However, when interpreting the testing results, one needs to be careful. The KS test checks U½0; 1 under the i.i.d. assumption, rather than testing the i.i.d. and U½0; 1 jointly. Furthermore, the KS test does not take into account the impact of the uncertainty of parameter estimation on the asymptotic distribution of the test statistic. 3.3.2. Hong & Li test Hong and Li (2005) proposed a specification testing method for a time series model using a discretely observed random sample fY t gTt¼1 . In contrast to the KS test, Hong & Li’s test considers the departure from i.i.d. and U½0; 1 jointly, and also takes into account the impact of parameter estimation uncertainty on the asymptotic

distribution of the test statistic. Hong & Li’s test has some advantages over other specification testing methods. This test is a nonparametric specification test using the transition density, which can capture the full dynamics of the underlying process. Hong et al. (2007) employed this testing method in order to evaluate the density forecast performances for various univariate foreign exchange rate models. Similar to Hong and Li (2005), Hong et al. (2004) also developed a testing method using the probability integral transform in order to evaluate the out-of-sample density forecasting performances of discrete-time spot interest rate models. Since Hong & Li’s testing method exploits the full dynamics of the underlying process, it has high power against almost any misspecified model. Furthermore, this testing method can be used to compare the relative performances of non-nested models. Hong & Li’s test measures the distance between a modelimplied transition density and the true transition density by comparing a kernel estimator b g j ðz1 ; z2 Þ for the joint density of the pair fZ t ; Z tj g with unity (the product of two U½0; 1 densities), where j is the lag order. The kernel estimator of the joint density is, for any integer of j > 0 as follows: 1 b g j ðz1 ; z2 Þ ¼ ðn  jÞ

n X

b t ÞK h ðz2 ; Z b tj Þ; K h ðz1 ; Z

ð17Þ

s¼jþ1

b t Þ is a boundary-modified kernel,6 defined below. where K h ðz1 ; Z Moreover hats are placed over the generalized residuals because pffiffiffi they are evaluated at a n -consistent estimator for the true parameter h0 . For x 2 ½0; 1, the boundary-modified kernel is defined as follows:

8 1 xy R 1 h k h = ðx=hÞ kðuÞdu; if x 2 ½0; hÞ; > > < 1 xy K h ðx; yÞ  h k h ; if x 2 ½h; 1  hÞ; > > : 1 xy R ð1xÞ=h h k h = 1 kðuÞdu; if x 2 ½1  h; 1;

ð18Þ

where the kernel kðÞ is a prespecified symmetric probability density, and h  hðnÞ is a bandwidth such that h ! 0; nh ! 1 as n ! 1 . One example of kðÞ is the quartic kernel 2 kðuÞ ¼ 15 ð1  u2 Þ 1ðjuj 6 1Þ, where 1ðÞ is the indicator function. In 16 practice, the choice of h is more important than the choice of kðÞ. 1 Like Scott (1992), we choose h ¼ b S Z n6 , where b S Z is the sample stanT b dard deviation of f Z t gt¼1 . This simple bandwidth rule attains the optimal rate for the estimation of bivariate density. Hong & Li’s test statistic is based on a properly standardized version of the quadratic form between b g j ðz1 ; z2 Þ and 1, the product of two U½0; 1 densities as follows:

b ðjÞ  ½ðn  jÞh Q

Z

1 0

Z 0

1

2

0

1=2

½b g j ðz1 ; z2 Þ  1 dz1 dz2  hAh =V 0 ;

ð19Þ

where the nonstochastic centering and scale factors are

 Z 1 A0h  ðh  2Þ "Z

1

2

k ðuÞdu þ 2

Z

1

1

V0  2

1

Z

1

kðu þ v Þkðv Þdv

1

Z

0

2

1

1

2 2 kb ðuÞdudb  1;

ð20Þ

;

ð21Þ

#2 du

1

Rb and kb ðÞ  kðÞ= 1 kðv Þdv . Under the correct model specifications, d b Q ðjÞ! Nð0; 1Þ for any fixed lag order j > 0 as n ! 1 . b ðjÞ statistics with different j’s reveals Although the use of the Q information on the lag orders at which there are significant departures from i.i.d. U[0,1], it is more convenient to construct a single 6 The modified kernel is used because the standard kernel density estimator produces biased estimates near the boundaries of the data, due to asymmetric coverage of the data in the boundary regions. The denominators of K h ðx; yÞ for x 2 ½0; hÞ [ ð1  h; 1 ensure that the kernel density estimator is asymptotically unbiased uniformly over the entire support [0,1] (Hong and Li, 2005).

J. Yun / Journal of Banking & Finance 47 (2014) 74–87

79

test statistic when comparing two different models. In this regard, Hong et al. (2007) suggested the following portmanteau evaluation test statistic:

4. Empirical results

X b ðjÞ c ðpÞ ¼ p1ffiffiffi W Q p j¼1

For parameter calibration, we import the P- and Q -parameter estimates from Yun (2011). It appears that the parameter estimates are quite similar to those in other existing studies on options. The parameter estimates are reported in Table 1. The following is a brief description of his two-step calibration method. In the first step, P-parameters were estimated via the Bayesian MCMC by using the S&P 500 daily returns from January 1987 to December 2000. He adopted some equality restrictions between the P- and Q-parameters, based on the existing studies.7 For those restricted parameters, he used the P-parameters estimated at the first step. In the second step, the remaining free Q-parameters were estimated via the nonlinear least squares by using the closing prices of crosssectional options on every Wednesday during the period of 1990– 2000. Due to the restrictions above, only a few Q-parameters (e.g., gQv ; lQS , and lQV Þ were estimated for each model. For the time-varying jump risk premia models, such as SVJ-TV-OR and SVCJ-TV-OR, he used a linear jump intensity specification, kQ t ¼ fV t . Rather than estimating f, he calibrated the value by dividing k(constant jump arrival intensity under P) by h(long-run average volatility under P). Consequently, when the spot volatility is at its long-run average level ðV t ¼ hÞ, the time-varying jump intensities kQ t for the SVJ-TV-OR and SVCJ-TV-OR models are, by construction, the same as k. For our out-of-sample density forecast analysis, we use the daily S&P500 stock index return data from January 2001 to December 2007. The beginning period, 2001–2003, and the later period, 2004–2007, are distinguished by different volatility levels: in terms of the VIX index, the average volatility levels are 25.0 for the former period (2001–2003),8 and 14.7 for the subsequent lowvolatility period (2004–2007). To estimate model-implied spot volatilities via particle filters, we use two separate information sources: (i) options & returns data and (ii) only returns data. For the spot volatilities using options & returns data, a representative call option is selected for each date.9 The idea of selecting one representative option was also employed by Pan (2002) and Shackleton et al. (2010). Among all available call options for each date, we select a call option contract with a time-to-maturity as close as possible to 30 calendar days, and moneyness (the ratio of the strike to the underlying price) as close as possible to 1. This scheme guarantees the selection of the most liquid option for each date.10 The remaining maturities of these

p

ð22Þ

where p is the lag truncation order. They showed that the above statistic converges to Nð0; 1Þ. This is a one-tailed test; hence, the larger c ðpÞ statistic, the larger the departure from the true model. For the W more details about this testing method, refer to Hong and Li (2005), or Hong et al. (2007). 3.4. Beta transformation We now explain the beta transformation method used as an alternative approach to updating the density parameters. Taylor (2005), Liu et al. (2007) and Shackleton et al. (2010) originally introduced beta-transformation in order to transform risk-neutral densities into real-world densities. In this paper, we employ the beta-transformation for the recursive update of density parameters. In contrast to direct parameter updating for the AJD models, using beta transformation incurs much lower computational costs. The explanation that follows is based on Taylor (2005). Let us denote ft;tþs and F t;tþs by the model-implied conditional density and cumulative distribution functions for stock returns Y t;tþs , respectively, under static density parameters. If F t;tþs were a correctly specified distribution, then F t;tþs ðY t;tþs Þ would be uniformly-distributed. We now define a random variable U t;tþs ¼ F t;tþs ðY t;tþs Þ. We introduce a calibration function C t;tþs that is used for updating the density parameters. Among many alternatives for the calibration function C t;tþs , we employ the beta distribution function as our calibration function. Therefore, its density ct;tþs ðuÞ is specified as follows:

ct;tþs ðuÞ ¼

ua1 ð1  uÞb1 ; Bða; bÞ

06u61

ð23Þ

with Bða; bÞ ¼ CðaÞCðbÞ=Cða þ bÞ. f t;tþs and e Let e F t;tþs be conditional density and cumulative distribution functions, respectively, updated by the calibration function. Then we have the following equations:

e F t;tþs ðxÞ ¼ C t;tþs ðF t;tþs ðxÞÞ

ð24Þ

and

ef t;tþs ðxÞ ¼ ft;tþs ðxÞct;tþs ðF t;tþs ðxÞÞ:

ð25Þ

Thus, the updated conditional density is as follows:

ef t;tþs ðxÞ ¼ F t;tþs ðxÞ

ð1  F t;tþs ðxÞÞb1 ft;tþs ðxÞ: Bða; bÞ

a1

ð26Þ

From Eq. (26), one can derive a log-likelihood function for an individual observation (Eq. (27)).

ln ef t;tþs ðxÞ / ða  1Þ ln F t;tþs ðxÞ þ ðb  1Þ lnð1  F t;tþs ðxÞÞ  ln Bða; bÞ:

4.1. Data and parameter calibration

ð27Þ

Eq. (27) is used for maximum likelihood estimation of the beta coefficients a and b. Note that in the maximum likelihood estimation, as a input data, a model-implied generalized residual under fixed density parameters is plugged into the term F t;tþs ðxÞ.

7 According to the well-known Girsanov theorem, parameters such as rv ; q; j , and Q hare identical across both P- and Q-measures. Yun (2011) also set rP S ¼ rS , and, for Q the constant jump risk premia models, kP t ¼ kt ¼ k . The latter restriction is due to identification purposes (e.g., Broadie et al. (2007)). For the models with a jump component, he also sets gQ v ¼ 0, which is from the consideration that the diffusive volatility risk premium will have only a second-order effect on the jump models. For all the restricted parameters, the P-parameter estimates are used. The remaining Q Q unrestricted Q-parameters, such as gQ v ; lS , and lV , are estimated by nonlinear least squares in the second step. 8 During the high-volatility period, there had been the bursting of the Dot-Com bubble, 9.11 attacks, and the Iraqi war. 9 Source: DeltaNeutral. 10 Unfortunately, there were 47 dates when options data were not available. For each date without options data, we estimated the model-implied spot volatility by using the lagged model-implied spot volatility and contemporaneous VIX index. Specifically, in order to approximate a model-implied spot volatility at date t, we ran the following regression:

pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi V s ¼ b0 þ b1 V s1 þ b2 VIX s þ b3 VIX 2s ; s ¼ 1; . . . ; t  1; pffiffiffiffiffi where V s is the square root of the model-implied spot volatility (standard deviation), and VIX s is the VIX index at date s. Given the estimated regression coefficients, pffiffiffiffiffi we predicted the model-implied spot volatility at date t (i.e., V t ). Note that for those dates without options data, model-implied spot volatilities are still predicted using the information up to the corresponding dates.

80

J. Yun / Journal of Banking & Finance 47 (2014) 74–87 Table 1 P- and Q-measure parameters estimated by Yun (2011). The parameter values correspond to daily percentage changes in the index value. By construction, Qparameter estimates are not required for the SV-R, SVJ-R, and SVCJ-R models. SV-OR & SV-R

SVJ-OR & SVJ-R

SVCJ-OR & SVCJ-R

SVJ-TV-OR

SVCJ-TV-OR

lS rS lV gQv

0.0401 0.0288 1.021 0.1813 0.496 – – – – 0.003

0.0432 0.0204 0.9856 0.1450 0.5912 0.0066 3.349 3.704 – –

0.0475 0.0362 0.7026 0.1459 0.5712 0.0049 4.646 2.71 2.104 –

0.0432 0.0204 0.9856 0.1450 0.5912 0.0066 3.349 3.704 – –

0.0475 0.0362 0.7026 0.1459 0.5712 0.0049 4.646 2.71 2.104 –

kQt

–

0.0066

0.0049

0.0067 Vt

0.0070 Vt

lQS lQv

–

3.71

5.38

9.20

9.59

–

–

1.34

–

1.24

l j H

rV P K

selected representative options provide an average of 30 days, a maximum of 58 days, a minimum of 11 days, and a standard deviation of 9 days. As an alternative, one may prefer to use more options for a given date in order to improve the estimation efficiency. However, using multiple options per day is computationally expensive. Furthermore, in order to use the particle filter, one needs to assume a covariance structure among pricing errors of the options. It is possible that some misspecified covariance structures may distort the estimation results. Considering the costs and benefits, like Pan (2002) and Shackleton et al. (2010), we choose a single liquid option for one day. Our calibration scheme is equivalent to assuming fixed density parameters. In a practical sense, one may be able to improve density forecast performance by recursively updating model parameters. Unfortunately, since our estimation of the AJD model incurs high computational costs, such recursive parameter updating is not an easy task.11 As an alternative to tackle this problem, we employ the beta transformation method (in Section 3.4). Originally, this method was used to transform risk-neutral densities into realworld densities by Taylor (2005), Liu et al. (2007) and Shackleton et al. (2010). However, we use this approach to save the computational costs of updating density parameters. We estimate two coefficients in the beta transformation (in Eq. (26)) from 1996 to a forecasting date in a recursive way, separately for different forecasting horizons.12 For example, suppose that at the end of the second week in 2002, we are to forecast one-week-ahead density for the SVJ-OR model. For this purpose, we estimate the beta coefficients a and b by maximizing the log-likelihood function implied by Eq. (27): this estimation uses both the model-implied generalized residual series and the non-overlapping weekly stock return series from the first week in 1996 to the second week in 2002. With the estimated beta coefficients, one-week-ahead density can be forecast via Eq. (26). Because we use information up to the end of the second week in 2002, our density forecast model still uses only ex-ante parameters. The same parameter updating procedure is applied to the RV-N and RV-T models (Section 2.2), letting both the AJD-type and the RV-type models be fairly treated. To be specific, for each forecast horizon, we separately estimate the RV-N and RV-T models in Eq. (6) from stock returns and realized volatilities data during the 1996–2000 period. For density forecasts under beta transforma11 According to Yun (2011), it takes a few days to estimate P -parameters for an AJD model with jumps via the Bayesian MCMC. 12 Instead of our approach which fixes the starting date of estimation in betatransformation, one may prefer to use a moving window of the estimation period. We compared our method with a 5-year moving window method. Our analysis indicates that in terms of density forecast performances, our method performs slightly better than the 5-year moving window method for most models.

tion, we then apply the beta transformation method in the same way as for the AJD models. For a realized volatility measure, we use the realized kernel data provided by Heber et al. (2009), which is available from the database, ‘‘Oxford-Man Institute’s Realized Library.’’13 The realized kernel, first introduced by BarndorffNielsen et al. (2008), is known to be robust to the noise of market microstructure effects. Han and Park (2013) also used the realized kernel data from the same database for their volatility forecasting study. 4.2. Time-series consistency Option prices are very informative about market expectations on stock return volatilities. This is because, contrary to historical returns, options are forward-looking. However, as argued by Johannes et al. (2009), the information content in options may be distorted due to model misspecifications under P- and Q-measures. Johannes et al. (2009) considered three constant jump risk premia AJD models: the SV, SVJ and SVCJ models. In their empirical analysis, they found that options are very informative for extracting model-implied spot volatilities because options greatly reduce the posterior standard deviations of the filtered volatilities. However, for all the models they considered, inconsistencies were obtained between the spot volatilities extracted from options & returns and those from only returns. Each model tends to provide higher volatilities when both options and returns are used than when only returns are used. Johannes et al. (2009) expected that using time-varying jump risk premia may be able to resolve this problem this time-series inconsistency. That issue is addressed in this section. Our empirical results for time-series consistency are reported in both Table 2 and Fig. 1. In Table 2, the fourth and fifth columns indicate that after options are used, posterior standard deviations fall dramatically. Thus, our results about informativeness are consistent with Johannes et al. (2009). As expected by Johannes et al. (2009), we find that the use of the time-varying jump risk premia are successful for resolving the inconsistency between the volatilities extracted from options & returns and from only returns. For all the AJD models considered, Fig. 1 presents the 20-day moving averages of the differences between the model-implied spot volatilities filtered from options & returns and those from only returns. For example, in Fig. 1, ‘‘SV-OR - SV-R’’ indicates the 20-day moving average of the difference between the model-implied spot volatilities for SV-OR and those for SV-R. First, let us examine the constant jump risk premia models (e.g., ‘‘SV-OR - SV-R’’, ‘‘SVJ-OR - SVJ-R’’, and 13

http://realized.oxford-man.ox.ac.uk/.

81

J. Yun / Journal of Banking & Finance 47 (2014) 74–87

Table 2 Filtering results for model-implied spot volatilities. The RMSD in the second and third columns indicates the root mean squared distances between ‘‘option & returns’’ and ‘‘returns’’ filtered volatilities for a given period. The SD in the fourth and fifth columns indicates the average of the posterior standard deviations of the model-implied spot volatilities. The probabilities of non-overlapping in the sixth column indicate the ratios that the two 90% confidence intervals for each model do not overlap. The numbers in parentheses represent the number of observations with no overlapping. Models

RMSD (%p) (2001–2003)

RMSD (%p) (2004–2007)

SD for ‘‘Options & Returns’’ (%p)

SD for ‘‘Returns only’’ (%p)

Probability of Non-overlapping (% and number)

SV-OR vs. SV-R SVJ-OR vs. SVJ-R SVCJ-OR vs. SVCJ-R SVJ-TV-OR vs. SVJ-R SVCJ-TV-OR vs. SVCJ-R

3.93 3.31 4.44 2.17 2.14

2.08 2.27 2.44 2.07 1.95

1.46 1.30 1.39 1.04 1.07

3.20 2.80 2.89 2.80 2.89

3.6 2.6 7.1 0.3 0.1

‘‘SVCJ-OR - SVCJ-R’’). We can see that during the earlier high volatility period, the model-implied spot volatilities extracted from options & returns are systematically higher than those from only returns; however, the two volatilities were broadly consistent during the later period of low volatility. On the other hand, the volatility inconsistency is only marginal for the time-varying jump risk premia models throughout the entire sampling period (e.g., see ‘‘SVJ-TV-OR - SVJ-R’’ and ‘‘SVCJ-TV-OR - SVCJ-R’’ in Fig. 1). This is because, with the help of the larger risk-neutral jump arrival intensity during a time of high volatility, the SVJ-TV-OR and SVCJ-TV-OR models are able to extract lower spot volatilities than the other constant jump risk premia models. As statistical evidence for the time-series consistency, Table 2 presents the probabilities (i.e., frequency ratios) that the 90% confidence intervals from ‘‘options & returns’’ and ‘‘returns only’’ volatilities do not overlap one another. For the SV, SVJ, and SVCJ models with constant jump risk premia, those probabilities are 3.6%, 2.6%, and 7.1%, respectively. In contrast, the probabilities calculated for both SVJ-TV-OR (vs. SVJ-R) and SVCJ-TV-OR (vs. SVCJ-R) are as small as 0.3% and 0.1%, respectively; hence, almost all of the observed confidence intervals overlap with each other. As further evidence, we provide the RMSD (the root mean squared distances between the two annualized model-implied spot standard deviations) between ‘‘options & returns’’ and ‘‘only returns’’ volatilities. These statistics also show that, for the constant jump risk premia models, the inconsistency between the two volatilities was more serious during the earlier period of high volatility: the constant jump risk premia models provide RMSDs of about 4%p and 2%p, during the early (high) and later (low) volatility periods, respectively; whereas the time-varying jump risk premia models indicate RMSDs of about 2%p for both periods. 4.3. Density forecast evaluations We now evaluate density forecasts of the eight AJD models (e.g., SV-R, SVJ-R, SVCJ-R, SV-OR, SVJ-OR, SVJ-TV-OR and SVCJ-TV-OR) and the two RV models (e.g., RV-N, and RV-T). Forecasting intervals do not overlap for each forecast horizon (e.g., 1 day, 1 week, 2 weeks, 3 weeks and 4 weeks); hence, longer-horizon forecasts involve a lesser number of forecasts.14 As mentioned above, for each model and each forecast horizon, we use two density forecast schemes: (i) fixed density parameters and (ii) beta transformation. We also employ two evaluation criteria, the likelihood and the probability integral transform criteria. 4.3.1. Evaluations from likelihood criteria Table 3 presents the log-likelihood ratios of density forecasts for each model under both fixed density parameters (Panel (a)) and

14 The numbers of forecasts are 1758 (1 day), 365 (1 week), 183 (2 weeks), 126 (3 weeks) and 91 (4 weeks), respectively.

(63) (46) (125) (5) (2)

beta transformation (Panel (b)). The SV-R model under fixed density parameters is set as a benchmark model; thus, the statistics in Table 3 are the log-likelihoods of each model in excess of the log-likelihood of the fixed-parameter SV-R model. Additionally, in order to examine the effectiveness of beta transformation for each model, Panel (c) presents the differences in the log-likelihood ratios between fixed density parameters and beta transformation. Therefore, a positive number in Panel (c) implies that beta transformation is an improvement over the fixed density parameters for the corresponding model. We also present the sequential likelihood results in Fig. 2 (oneday-ahead densities) and Fig. 3 (two-week-ahead densities). As noted, Johannes et al. (2009) argued that the investigation of sequential likelihood reveals the exact timing when a model fails to fit the data. Different from the log-likelihood ratios in Table 3, for better illustration purposes, we use the SVJ-TV-OR under fixed density parameters as a benchmark model. Hence, if the sequential likelihood of a given density forecast model exceeds zero at a given date, it is implied that up to this date, the given model outperforms the fixed-parameter SVJ-TV-OR model. To save space, we provide the results only for the one-day and two-week forecast horizons. We should mention that the results for the other medium-term horizons (e.g., 1-, 3-, and 4-week horizons) are similar to the results for the two week horizon. We first discuss the results of density forecasts under fixed density parameters (Panel (a) in Table 3). Remarkably, across all forecast horizons, the two time-varying jump risk premia models (i.e., SVJ-TV-OR and SVCJ-TV-OR) dominate over the other models. Between the two models, the SVJ-TV-OR model exhibits slightly better performance. Hence, it seems that the time-varying jump risk premia play an important role in exploiting the forward-looking information of the options data for density forecasts. It is also noteworthy that these option-based jump models forecast densities well even for the one-day forecast horizon. Their remarkable performances in the short horizon are, as noted above, in contrast to the results in Shackleton et al. (2010). For the one-day forecast horizon, the RV-T model (i.e., realized volatility model with t-distribution assumption) also shows similar performance to the time-varying jump risk premia models. As can be expected, the realized volatility models works better with t-distribution than with normal distribution. However, for the longer horizons (e.g., 1–4 week horizon), the RV models perform very poorly, regardless of their distributional assumptions. Our empirical findings are consistent with other studies, such as Jiang and Tian (2005) and Shackleton et al. (2010): they showed that realized volatilities provide valuable information only in short horizon forecasts (e.g., 1 day); whereas the realized volatilities provide less informational content in medium horizon forecasts than options data. In order to identify when and how different models diverge in their performances, we examine the sequential likelihood ratios. Panel (a) in Fig. 2 plots one-day-horizon sequential likelihood

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J. Yun / Journal of Banking & Finance 47 (2014) 74–87

Fig. 1. 20-Day moving averages of differences between filtered volatilities extracted from options & returns and from only returns for a given AJD model. For example, ‘‘SVOR–SV-R’’ indicates 20-day moving average of filtered volatilities from SV-OR minus those from SV-R.

Table 3 Log-likelihood ratios of density forecasts for all models under both fixed density parameters (Panel (a)) and beta transformation (Panel (b)). The numbers tabulated in Panels (a) and (b) are the log-likelihoods of each model in excess of the SV-R benchmark value under fixed density parameters. Panel (c) reports the pairwise differences in log-likelihood ratios between the beta transformation and fixed density parameters (Panels (b) and (c)). The number in each row shown in bold font indicates the highest ranked model for the corresponding forecast horizon. Forecast horizon

SVJ-R

SVCJ-R

SV-OR

SVJ-OR

(a) Fixed density parameters 1 day 0.0 1 week 0.0 2 weeks 0.0 3 weeks 0.0 4 weeks 0.0

SV-R

43.3 0.9 0.6 0.5 0.9

36.7 1.6 0.6 0.3 0.1

44.3 1.5 2.1 0.6 1.4

52.5 7.4 1.5 2.7 3.5

(b) Beta transformation 1 day 1.4 1 week 0.6 2 weeks 2.1 3 weeks 2.6 4 weeks 0.5

37.6 2.8 2.4 4.2 1.6

34.9 1.4 0.5 2.2 0.6

52.8 13.3 7.5 8.2 5.2

(c) Differences in log-likelihood ratios (Panels (b) and (c)) 1 day 1.4 5.7 1.7 1 week 0.6 1.8 0.2 2 weeks 2.1 3.0 1.0 3 weeks 2.6 3.7 1.9 4 weeks 0.5 0.8 0.5

8.6 11.7 9.7 7.6 3.8

SVJ-TV-OR

SVCJ-TV-OR

RV-N

RV-T

36.0 6.0 0.9 2.0 3.0

70.2 11.8 5.7 5.0 4.0

67.5 11.6 5.5 4.6 3.3

50.7 3.6 1.9 4.4 5.2

67.5 2.8 1.7 3.7 9.2

44.8 13.6 8.8 8.7 6.4

25.6 12.4 8.4 8.1 5.9

75.2 12.2 7.6 7.3 4.0

74.6 11.6 6.7 6.5 3.0

57.8 2.3 1.0 2.2 3.9

72.4 3.9 0.8 1.2 0.5

7.6 6.1 7.3 6.0 2.9

10.4 6.4 7.6 6.0 2.9

4.9 0.4 1.9 2.3 0.1

7.1 0.0 1.2 1.9 0.3

7.1 5.9 3.0 2.1 1.2

4.9 6.7 2.5 2.5 8.8

ratios under fixed parameters. It is observed that during the high volatility period, the sequential likelihood ratios for most models are around zero. However, starting from 2004, the ratios for some of the models begin to decline, due to poor performances during the low volatility period. In particular, the models without a jump component, such as the SV-R and SV-OR models, could not avoid a large decrease in their likelihoods when faced with a sharp rise in the stock price during the tranquil time: this happened when there was a 3.5% decrease in the S&P 500 stock index on February 27, 2007, with the VIX index of as low as 11.15 on the previous date. Panel (a) in Fig. 2 also illustrates that while the RV-T model outperforms the benchmark SVJ-TV-OR during the high volatility period, the RV-T model exhibits relatively poor density forecast performance during the latter period of low volatility. Overall, the RV-T model performs very similarly to the SVJ-TV-OR model. On the other hand, in the case of the RV-N model, as is similar

SVCJ-OR

for the AJD models without a jump component, an abrupt decrease in the likelihood is observed. This implies that the assumption of tdistribution allowing a fat tail plays a role similar to the jump specifications in the AJD models. Meanwhile, Panel (a) in Fig. 3 presents two-week-horizon sequential likelihood ratios (also under fixed density parameters). We can see that, unlike one-day-ahead densities, no sudden decrease in the likelihood ratios occurs, because a large change in stock prices over a medium-term horizon can possibly be generated even with low volatility. It seems that the SVJ-TV-OR model (i.e., benchmark model) outperforms the other models persistently throughout the entire sample. Next, we turn to the discussion about the effects of beta transformation on density forecast performances (Panels (b) and (c) in Table 3). Note that in Panels (b) and (c) in Table 3, the benchmark model is still The SV-R model under fixed density parameters.

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Fig. 2. Sequential likelihoods for one-day-ahead density forecasts of selected models under both fixed density parameters (Panel (a)) and beta transformation (Panel (b)). The benchmark model is the SVJ-TV-OR under fixed density parameters; thus, every ratio is the log-likelihood of a given model in excess of the SVJ-TV-OR model.

Importantly, the effectiveness of the beta transformation differs across the forecast horizons. First, for one-day-ahead forecasts, beta transformation is effective only for the density forecast models that have also performed well under fixed parameters: they are the SVJ-TV-OR, SVCJ-TV-OR, and RV-T models. According to Panel (c) in Table 3, with the help of beta transformation, their log-likelihood ratios increase by 4.9, 7.1, and 4.9, respectively. However, little change is observed by beta transformation in all of the other models, and for some models, beta transformation even makes the density forecast performances worse.15 The sequential likelihood ratios in Panel (b) in Fig. 2 (the benchmark is the SVJ-TV-OR under fixed density parameters) show the reason why the beta transformation is not effective for some models. They show that for the SV-R, SV-OR, and RV-N models, beta transformation fails to remedy an abrupt decrease in the likelihoods, on February 27, 2007, in particular. Note that before this date, the SV-OR model provides remarkable 15 Exceptionally, the log-likelihood ratio of the SV-OR model increases by 8.6 (Panel (c) in Table 3). However, its forecast performance after beta transformation (e.g., the log-likelihood ratio of 44.8 in Panel (b)) is still much worse than the SVJ-TB-OR, SVCJTV-OR, and RV-T models.

improvement to the density forecasts due to beta transformation. However, the model could not avoid a huge decrease in the likelihood ratio on this date. In contrast, the beta transformation for medium-term density forecasts (e.g., 1–4 week horizons) produces totally different results. For these longer horizons, the SVJ-OR model demonstrates the best performance, followed by the SV-OR and SVCJ-OR models. It is also notable that as in Panel (c) in Table 3, the biggest improvements via beta transformation occur in the SV-OR model, which is an ‘‘options & returns’’ model without a jump component. The sequential likelihood ratios in Panel (b) in Fig. 3 also illustrate that for the medium-term forecasts, the beta transformation remarkably improves the density forecast performances for the ‘‘option & returns’’ models. In particular, it is clearly seen that the sequential likelihood ratios between ‘‘options & returns’’ and ‘‘returns only’’ models begin to diverge around 2004. Importantly, our results imply that the beta transformation used in our study can weaken the systematic effects of model misspecifications on the density forecast model. Our results are similar to the finding by Shackleton et al. (2010) who showed empirical evidence against jump components in density forecasts. One

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Fig. 3. Sequential likelihoods for two-week-ahead density forecasts of selected models under both fixed density parameters (Panel (a)) and beta transformation (Panel (b)). The benchmark model is the SVJ-TV-OR under fixed density parameters; thus every ratio is the log-likelihood of a given model in excess of the SVJ-TV-OR model.

important feature of their density forecast study is a frequent update of the parameters. Our empirical findings also illustrate that, after beta transformation, there are little difference across the ‘‘options & returns’’ models in the medium-term density forecasts. Thus, parameter updating may be able to make a jump component less substantial to forecasting densities. However, it should be kept in mind that beta transformation works well only for medium-term density forecasts, not for the one-day horizon. For the one-day horizon, our results support the jump specification, even after beta transformation. 4.3.2. Evaluations from the probability integral transform criteria We now consider the PIT-based density forecast evaluations. First, Table 4 reports the popular Kolmogorov–Smirnov (KS) testing results. The results are different across the various forecast horizons. Unlike the previous likelihood results for the one-day forecast horizon under fixed density parameters (Panel (a) in Table 4), there seems to be no satisfactory models for density forecasts. At a 5% significance level, all models are rejected under the null hypothesis that generalized residuals are U½0; 1. Meanwhile, for the other medium-term horizons, the two time-varying jump risk premia models and some of the other models are not rejected at the 5% significance level.

Interestingly, under beta transformation (Panel (b) in Table 4), most models are not rejected at the 5% significance level for the medium-term horizons. Thus, even if misspecified, our beta transformation improve the goodness-of-fit of conditional density for a given model, that is, by remarkably improving the U½0; 1 property rather than i.i.d. property. However, as aforementioned, one should be careful in interpreting the KS tests. The KS test checks U½0; 1 under the i.i.d. assumption rather than testing the i.i.d. and U½0; 1 jointly. Furthermore, the KS test does not take into account the impact of the uncertainty of parameter estimation on the asymptotic distribution of the test statistic. To overcome the problems associated with the KS test, we conduct Hong and Li’s (2005) test (Section 3.3.2), which also uses the PIT-based criteria. Table 5 presents Hong & Li’s WðpÞ statistics with p equal to 5. It turns out that the results for p ¼ 10 or 20 (not reported) are almost the same. Panel (a) in Table 5 reports the statistics for one-day-ahead densities under fixed density parameters. As are the results of the KS test with the same horizon, all models are far from satisfactory. The time-varying jump risk premia models, though the best in terms of likelihood, produce Wð5Þ statistics as large as 20. The RV-T model also performs poorly in terms of the Hong & Li test.

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Table 4 Kolmogorov–Smirnov test results for all models under both fixed density parameters (Panel (a)) and beta transformation (Panel (b)). The numbers in both tables are Kolmogorov– Smirnov test statistics. Forecast horizon

⁄ ⁄⁄

SVJ-R

SVCJ-R

SV-OR

SVJ-OR

SVCJ-OR

SVJ-TV-OR

SVCJ-TV-OR

RV-N

RV-T

(a) Fixed density parameters 1 day 0.040⁄⁄⁄ 1 week 0.071⁄⁄ 2 weeks 0.114⁄⁄ 3 weeks 0.119⁄ 4 weeks 0.143⁄⁄

SV-R

0.037⁄⁄ 0.074⁄⁄ 0.101⁄⁄ 0.110 0.137⁄

0.034⁄⁄ 0.065⁄ 0.107⁄⁄ 0.104 0.129⁄

0.056⁄⁄⁄ 0.093⁄⁄⁄ 0.121⁄⁄⁄ 0.143⁄⁄ 0.171⁄⁄⁄

0.035⁄⁄ 0.064 0.099⁄ 0.122⁄ 0.146⁄⁄

0.038⁄⁄ 0.069⁄ 0.102⁄⁄ 0.126⁄⁄ 0.147⁄⁄

0.041⁄⁄⁄ 0.056 0.086 0.105 0.128⁄

0.042⁄⁄⁄ 0.061 0.090 0.110 0.127⁄

0.055⁄⁄⁄ 0.099⁄⁄⁄ 0.131⁄⁄⁄ 0.124⁄⁄ 0.162⁄⁄

0.042⁄⁄⁄ 0.104⁄⁄⁄ 0.124⁄⁄⁄ 0.126⁄⁄ 0.240⁄⁄⁄

(b) Beta transformation 1 day 0.047⁄⁄⁄ 1 week 0.059⁄ 2 weeks 0.093⁄ 3 weeks 0.084 4 weeks 0.130⁄

0.050⁄⁄⁄ 0.062⁄ 0.103⁄⁄ 0.084 0.128⁄

0.051⁄⁄⁄ 0.061⁄ 0.099⁄ 0.086 0.129⁄

0.027 0.045⁄⁄ 0.073 0.078 0.107

0.038⁄⁄ 0.057⁄ 0.070 0.086 0.118

0.039⁄⁄⁄ 0.057⁄ 0.076 0.090 0.112

0.031⁄ 0.062⁄ 0.085 0.079 0.112

0.033⁄⁄ 0.062⁄ 0.087 0.082 0.108

0.039⁄⁄⁄ 0.071⁄ 0.096⁄ 0.081 0.137⁄

0.028 0.072⁄ 0.092⁄ 0.078 0.134⁄

Null hypothesis is rejected at the 10% significance level. Null hypothesis is rejected at the 5% significance level. Null hypothesis is rejected at the 1% significance level.

⁄⁄⁄

Table 5 Hong and Li’s testing results for all models under both fixed density parameters (Panel (a)) and beta transformation (Panel (b)). The numbers tabulated are Hong and Li’s W(p) statistics with p equal to 5. The number in each row shown in bold font indicates the highest ranked model for the corresponding forecast horizon. Forecast horizon

SVJ-R

SVCJ-R

SV-OR

SVJ-OR

(a) Fixed density parameters 1 day 13.83 1 week 5.54 2 weeks 5.29 3 weeks 3.32 4 weeks 6.71

SV-R

SVCJ-OR

SVJ-TV-OR

SVCJ-TV-OR

RV-N

RV-T

12.03 7.36 5.31 3.87 7.42

15.08 5.33 4.43 2.57 5.37

19.34 12.98 9.51 7.34 9.24

9.95 6.97 6.36 5.45 7.22

10.65 7.27 6.86 5.33 7.14

19.12 3.96 1.97 2.19 4.77

21.15 3.93 1.66 1.79 4.19

22.62 12.79 6.93 4.38 8.35

14.74 13.15 6.33 4.37 16.14

(b) Beta transformation 1 day 19.11 1 week 4.56 2 weeks 2.40 3 weeks 0.06 4 weeks 4.86

16.07 5.38 2.51 0.29 4.53

17.16 5.27 2.98 0.09 4.52

11.91 1.84 0.39 0.55 4.15

18.92 2.97 0.47 0.52 2.97

21.21 3.05 0.82 0.43 3.18

12.72 3.16 0.27 0.42 3.06

13.10 3.30 0.35 0.31 3.16

17.62 7.31 3.31 2.60 7.91

10.91 7.24 3.01 2.06 5.39

In fact, it can be concluded that, in terms of the Hong & Li test, all models performs poorly for one-day forecasts, because the Wð5Þ statistics for most models are above 10. Considering that all models are rejected from the previous KS test, it seems that all models fail to satisfy the U½0; 1 property rather than the i.i.d. property. For the other longer-term horizons, however, consistent with the results from likelihood evaluations, the time-varying jump risk premia models turn out to forecast densities well. This also supports the important informational content in options data for medium-term density forecasts. Next, let us consider the effect of beta transformation, reported in Panel (b) of Table 5. For the one-day horizon, it seems that there is still no satisfactory density forecast model. The beta transformation turns out to be a little effective only for the RV-T, SVJ-TV-OR, SVCJ-TV-OR, and SV-OR models. However, their Wð5Þ statistics are still too large. The performances of the remaining models become worse after beta transformation, which is similar to the results from likelihood evaluations. For 1–4 week horizons, the testing results are very similar to those in the likelihood evaluations. Most ‘‘options & returns’’ models show improvement in their density forecasts with the help of beta transformation. In particular, for 2–3 week horizons, they are improved to the extent that the null hypothesis of misspecification could not be rejected for all ‘‘options & returns’’ models. In summary, though the likelihood and PIT criteria focus on different aspects of density forecasts, both criteria seem to produce qualitatively similar results, except for the results of the oneday-ahead densities under fixed density parameters. The excellent

performances of ‘‘options & returns’’ models for the medium-term horizons, in terms of both likelihood and Hong & Li’s test, is consistent with the important findings of Jiang and Tian (2005) and Shackleton et al. (2010). Jiang and Tian (2005) showed that the informational content of options is more than that of daily and intraday index values when forecasting the volatility of the S&P 500 index over horizons from one to six months. The results of Shackleton et al. (2010) similarly imply that options are the most informative when the forecast horizon is similar to their remaining maturities; however, they are not so informative for very short horizons. Shackleton et al. (2010) argued that the superior performance of option-based forecasts over medium-term horizons (e.g., 2–3 weeks) can be attributed to the forward-looking property of the option prices, as they used only option contracts with medium-term maturities. They conjectured that the historical density estimated from realized volatilities is superior over the shorter horizon of one day, because an accurate forecast of tomorrow’s volatility is obtainable from high-frequency returns summarized by the daily measures of realized volatility. Similar arguments are applicable to our analysis. Recall that the remaining maturities of the representative options in our study range from 11 days to 58 days. This is close to 2–4 weeks, which are the forecast horizons for which the time-varying jump risk premia models provide the best forecast performances. However, our study also shows some departure from Jiang and Tian (2005) and Shackleton et al. (2010). Recall that in terms of the likelihood criteria, for the one-day horizon, the option-based models with the time-varying jump risk premia model outperform the realized volatility models. According to our study, it seems that the

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informational content of options is still important for short horizons. Also, for this short horizon, jump specifications are important.

5. Conclusions This paper has evaluated out-of-sample density forecasts of various AJD models for the S&P 500 stock index. For each model and each forecast horizon considered, density forecasts have been separately conducted based on two different informational sources: ‘‘options & returns’’ and ‘‘only returns.’’ The particle filters proposed by Johannes et al. (2009) have been used to estimate the model-implied spot volatilities from each informational source. In particular, we have focused on the role of the time-varying jump risk premia. For comparison purposes, we have also considered two realized volatility models: the RV-N and RV-T models. Furthermore, as an alternative to the direct update of density parameters, we have proposed a computationally efficient beta transformation. Thus, density forecasts have been evaluated under both fixed density parameters and the beta transformation method. Besides investigating density forecasts, we have examined the issue of time-series consistency between model-implied spot volatilities extracted from options & returns data and from only returns data. Our empirical analysis supports the role of time-varying jump risk premia in reconciling the inconsistency between the modelimplied volatilities extracted from different sources. The findings of our density forecast analysis are as follows. First, we have found that in terms of the likelihood criteria, under fixed density parameters, the time-varying jump risk premia models (e.g., the SVJ-TV-OR and SVCJ-TV-OR) are able to forecast densities the best across all forecast horizons. Rather surprisingly, for oneday-ahead forecasts, these models marginally outperform the realized volatility models. Our results are different from those of Shackleton et al. (2010); they found that their realized volatility model outperforms option-based models for the one day horizon, and that a jump specification makes little contribution to density forecasts. In contrast, our results are in favor of the informational role of options even for the one-day horizon. Moreover, jump specifications have a large contribution via time-varying jump risk premia specification. The Hong & Li testing approach has also provided similar results, although it has failed to choose a satisfactory model to produce one-day-ahead densities; for the 1–4 week horizon, the time-varying jump risk premia models dominate over the other models. Our analysis implies that the information embedded in options data improves the density forecasting ability of the AJD models via the forward-looking property of the options; however, it has to be conducted through the correct specification of Qdynamics, such as time-varying jump risk premia. Next, we have proposed beta transformation as an alternative to direct parameter updating. This beta transformation method has been employed because the parameter estimation method is computationally too expensive for frequent parameter updating. It seems that many estimation methods used in this area may suffer from similar problems. It turns out that the effectiveness of beta transformation is mixed across different horizons. For a one-day horizon, the beta transformation improves only the models which have also performed well under fixed density parameters (e.g., the SVJ-TV-OR, SVCJ-TV-OR, and RV-T models). Meanwhile, it has been able to improve the performances of all option-based AJD models for the longer horizons. This implies that for these medium-term horizons, beta transformation is able to overcome the systematic effects of model misspecification in density forecast models. This partly explains the reason why Shackleton et al. (2010), who employed frequent parameter updating, found little contribution from jump specifications over the two and four-week horizons. Possibly, they may have removed the systematic effect of model

misspecifications via their proposed calibration transformations with frequent parameter updating. Although our suggested beta transformation has been able to improve the density forecasts for the medium-term horizons, we do not insist that it is a perfect substitute for direct parameter updating. Since our parameter estimation method is computationally too expensive, we could not apply our estimation method to direct parameter updating. A subsequent study may adopt a more computationally efficient method for parameter estimation and its updating. For example, the approximate maximum likelihood estimation by Ait-Sahalia and Kimmel (2007) may be among the candidates. If such an efficient method is used, a performance comparison between direct parameter updating and beta transformation should be possible. That will be an interesting and important future research topic. Finally, among the RV models using realized volatilities, the RV-T model forecasts densities well only for the one-day horizon. This is consistent with the important findings of Jiang and Tian (2005) and Shackleton et al. (2010) that the realized volatilities provide valuable information in short-term forecasts, but are less informative in the medium-term forecasts compared with options’ data. This paper has focused only on the AJD models with finitely arriving jumps. Yu et al. (2011) found that the Lévy-driven jump models with infinitely arriving jumps significantly outperform the AJD models in capturing both the P- and Q-dynamics of the S&P 500 index. Specifically, their best model was the variance gamma model of Madan et al., 1998 with stochastic volatility. However, they did not consider the time-varying jump risk premia models among their AJD models. Their pricing exercise utilized only a one-year sample (1993). It would be interesting to extend the sample period and compare the Lévy-driven jump models with the time-varying jump risk premia models in future research. Acknowledgments I would like to thank the anonymous refrees, Carol Alexander (Editor), and seminar participants at Sogang University, Kyung Hee University, and the 2012 and 2013 meetings of the Korean Econometric Society, for their valuable comments and suggestions on the earlier versions of this paper. This paper was motivated from some chapter in my Ph.D. dissertation. Thus, I am also indebted to my Ph.D. advisors, Yongmiao Hong, Nicholas Kiefer, and Morten Nielsen for their helpful comments and discussions. I am responsible for any remaining errors. References Ait-Sahalia, Y., Kimmel, R., 2007. Maximum likelihood estimation of stochastic volatility models. Journal of Financial Economics 83, 413–452. Bao, Y., Lee, T., Saltoglu, B., 2007. Comparing density forecast models. Journal of Forecasting 26, 203–225. Barndorff-Nielsen, O.E., Hansen, P.R., Lunde, A., Sherphard, N., 2008. Designing realized kernels to measure the ex-post variation of equity prices in the presence of noise. Econometrica 71, 1481–1536. Bates, D., 1996. Jumps and stochastic volatility: exchange rate process implicit in deutsche mark options. Review of Financial Studies 9, 69–107. Bates, D., 2000. Post-’87 crash fears in the S&P500 futures option market. Journal of Econometrics 94, 181–238. Brandt, M., Santa-Clara, P., 2002. Simulated likelihood estimation of diffusions with an application to exchange rate dynamics in incomplete markets. Journal of Financial Economics 63, 161–210. Broadie, M., Chernov, M., Johannes, M., 2007. Model specification and risk premia: the evidence from futures options. Journal of Finance 62, 1453–1490. Diebold, F.X., Gunther, T., Tay, A., 1998. Evaluating density forecasts with application to financial risk management. International Economic Review 39, 863–883. Duffie, D., Pan, J., Singleton, K., 2000. Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, 1343–1376. Elerian, O., Chib, S., Shephard, N., 2001. Likelihood inference for discretely observed nonlinear diffusions. Econometrica 69, 959–993.

J. Yun / Journal of Banking & Finance 47 (2014) 74–87 Glosten, L., Jagannathan, R., Runkle, D., 1993. On the relation between the expected value and the volatility of the nominal excess returns on stocks. Journal of Finance 48, 365–384. Gordon, N., Salmond, D., Smith, A., 1993. Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proceedings F-140, 107–113. Han, H., Park, M.D., 2013. Comparison of realized measure and implied volatility in forecasting volatility. Journal of Forecasting 32, 522–533. Heber, G., Lunde, A., Shephard, N., Shephard, K., 2009. OMI’s Realized Library, Version 0.1. Oxford-Man Institute, University of Oxford. Hong, Y., Li, H., 2005. Nonparametric specification testing for continuous-time models with applications to term structure of interest rates. Review of Financial Studies 18, 37–84. Hong, Y., Li, H., Zhao, F., 2004. Out-of-sample performance of discrete-time spot interest rate models. Journal of Business & Economic Statistics 22, 457–473. Hong, Y., Li, H., Zhao, F., 2007. Can the random walk model be beaten in out-ofsample density forecasts? Evidence from intraday foreign exchange rates. Journal of Econometrics 141, 736–776. Hull, J., White, A., 1987. The pricing of options on assets with stochastic volatilities. Journal of Finance 52, 281–300. Jiang, G.J., Tian, Y.S., 2005. The model-free implied volatility and its information content. The Review of Financial Studies 18, 1305–1342. Johannes, M., Polson, N., Stroud, J., 2009. Optimal filtering of jump diffusions: extracting latent states from asset prices. Review of Financial Studies 22, 2759–2799.

87

Liu, X., Shackleton, M.B., Taylor, S.J., Xu, X., 2007. Closed-form transformations from risk-neutral to real-world distributions. Journal of Banking and Finance 31, 1501–1520. Madan, D., Carr, P., Chang, E., 1998. The variance gamma process and option pricing. European Finance Review 2, 79–105. Pan, J., 2002. The jump-risk premia implicit in options: evidence from an integrated time-series study. Journal of Financial Economics 63, 3–50. Pedersen, A.R., 1995. A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations. Scandinavian Journal of Statistics 22, 55–71. Pitt, M., Shephard, N., 1999. Filtering via simulation: auxiliary particle filter. Journal of American Statistical Association 94, 590–599. Scott, D.W., 1992. Multivariate Density Estimation, Theory, Practice, and Visualization. John Wiley and Sons, New York. Shackleton, M.B., Taylor, S.J., Yu, P., 2010. A multi-horizon comparison of density forecasts for the S&P 500 using index returns and option prices. Journal of Banking & Finance 34, 2678–2693. Singleton, K., 2006. Empirical Dynamic Asset Pricing. Princeton University Press, Princeton, NJ. Taylor, S.J., 2005. Asset Price Dynamics, Volatility, and Prediction. Princeton University Press, Princeton, NJ. Yu, C.L., Li, H., Wells, M.T., 2011. MCMC estimation of Lévy jump models using stock and option prices. Mathematical Finance 21, 383–422. Yun, J., 2011. The role of time-varying jump risk premia in pricing stock index options. Journal of Empirical Finance 18, 833–846.