Global COE Hi-Stat Discussion Paper Series 264

Research Unit for Statisticaland Empirical Analysis in Social Sciences (Hi-Stat)

Investigating Impacts of
Self-Exciting Jumps in Returns and Volatility:
A Bayesian Learning Approach

Hi-Stat Discussion Paper

Andras Fulop
Junye LiJun YuDecember 2012

Hi-Stat
Institute of Economic Research
Hitotsubashi University
2-1 Naka, Kunitatchi Tokyo, 186-8601 Japan
http://gcoe.ier.hit-u.ac.jp

Investigating Impacts of Self-Exciting Jumps in Returns andVolatility: A Bayesian Learning Approach∗Andras Fulop†, Junye Li‡, and Jun Yu§
First Version: May 2011; This Version: October 2012.

Abstract
The paper proposes a new class of continuous-time asset pricing models where
whenever there is a negative jump in asset returns, it is simultaneously passed
on to diﬀusion variance and the jump intensity, generating co-jumps of prices
and volatility and jump clustering. To properly deal with parameter uncertainty
and hindsight bias, we employ a Bayesian learning approach, which generates
all quantities necessary for sequential real-time model analysis. Empirical study
using S&P 500 index returns shows that volatility jumps at the same time as
negative jumps in asset returns mainly through jumps in diﬀusion volatility. We
ﬁnd weak evidence of jump clustering. Learning and parameter uncertainty are
shown to have important implications for risk management, option pricing andvolatility forecasting.
Keywords: Self-Excitation, Volatility Jump, Jump Clustering, Parameter Learning, Sequential Bayes Factor, Risk Management, Option Pricing, Volatility Forecasting
JEL Classification: C11, C13, C32, G12∗

We are grateful for comments of Nick Polson, Yacine Ait-Sahalia, Jin-Chuan Duan, Nicolas Chopin,
Laurent Calvet, and seminar participants at National University of Singapore, Singapore Management
University, Xiamen Univeristy, SMU-ESSEC Symposium on Empirical Finance and Financial Econometrics, 2012 China International Conference in Finance, and Princeton/QUT/SMU Tripartite Conference on Financial Econometrics. We thank the Risk Management Institute at National Universityof Singapore for providing us the computing facility.
†
ESSEC Business School, Paris-Singapore. fulop@essec.fr‡
ESSEC Business School, Paris-Singapore. li@essec.edu§
Singapore Management University. jun.yu@smu.edu.sg

1

1

Introduction

The ﬁnancial meltdown of 2008 and the recent European debt crisis in 2011 raise questions about how likely extreme events are and how extreme events can be modeled
as they have already impacted ﬁnancial markets worldwide and have had far-reachingconsequences for the world economy. Understanding dynamics of extreme events thus
becomes crucial to many ﬁnancial decision makings, including investment decision,
hedging, policy reaction and rating. An important class of models is the continuoustime diﬀusions. Studies have successfully identiﬁed some important stylized facts suchas stochastic volatility and volatility clustering (Hull and White, 1986; Heston, 1993).
Furthermore, empirical explorations ﬁnd that a jump component is necessary to capture
extreme movements in asset prices1 .
However, more recently, it is found that only stochastic volatility and jumps in asset
returns may not capture the real dynamics of asset prices and therefore cannot generateenough probability of extreme events. It has been recognized that a big jump, in particular a big negative jump in asset prices, tends to be associated with an abrupt movein asset volatility, i.e., co-jumps of prices and volatility. A further intriguing empiricalobservation is that market turmoils seem to tell that an extreme movement in markets
tends to be followed by another extreme movement, resulting in jump clustering. To
document these facts, Table 1 reports the S&P 500 index returns and the corresponding
standard deviations computed using the previous 22-day returns during the four turbulent periods, covering the Black Monday in 1987, the crash of the Internet Bubble in2002, the bankruptcy of Lehman Brothers during the global ﬁnancial crisis in 2008, andthe European debt crisis in 2011. In all turbulent periods, extreme price movements
are accompanied by high volatility and extreme events seem to be clustered.
— Table 1 around here —Obviously, diﬀusion-based multi-factor volatility models are not good enough for
1

Parametric studies include Bates (1996, 2000), Bakshi, Cao, and Chen (1997), Andersen, Benzoni, and Lund (2002), Chernov et al. (2003), among others, whereas nonparametric works include
Barndorﬀ-Nielsen and Shephard (2007), Aı̈t-Sahalia and Jacod (2009, 2011), and Lee and Hannig(2010)

2

capturing co-jumps of prices and volatility and jump clustering as there is no mechanism to trigger extreme movements in asset returns and volatility during a turmoil.
Not surprisingly, Bates (2000) and Chernov et al. (2003) have found that the two-factor
volatility model does not oﬀer substantial improvements over the single-factor volatility
model. In the literature, two strands, which have pursued to accommodate co-jumps
of prices and volatility and jump clustering, co-exist. One strand uses synchronized
Poisson process to model asset returns and diﬀusion volatility (Duﬃe, Pan, and Singleton, 2000; Eraker, Johannes, and Polson, 2003; Eraker, 2004). In this framework,when a Poisson jump arrives, it not only moves asset price but also pushes up diﬀusion
volatility. Since the process for diﬀusion volatility is persistent, another large volatility
value is expected in the next period. Consequently, another extreme movement in asset
price is highly likely to be followed, even if there is no jump arrival. Another strand
proposes a mechanism whereby jumps in asset returns feedback to the jump intensity,
leading to self-excitation (Aı̈t-Sahalia, Cacho-Diaz, and Laeven, 2011; Carr and Wu,
2010). Here, large jumps in asset returns increase the likelihood of extreme events infuture asset returns and generate aggregate volatility jumps through the jump intensity.
While both approaches can generate co-jumps of prices and volatility and a correlation structure in extreme price movements, the implications of them are diﬀerent.
First, the propagating mechanism of extreme events is diﬀerent. In particular, the correlation in extreme events is driven by the correlation in the high volatility regime in
the former approach but by the correlation in the intensity in the latter. Second, theamount of conditional kurtosis generated by the two approaches is diﬀerent. As the
sampling interval gets smaller, it is expected that the impact of diﬀusion volatility on
the kurtosis is smaller than that of jumps. Thus, the amount of short-term tail risk
after a market crash is likely to diﬀer in the two cases. These diﬀerences inevitably
have implications for short-term option pricing and risk management. Therefore, it
is important to empirically examine the relative importance of these two alternative
mechanisms.
In the present paper, we propose a new class of continuous-time asset pricing mod3

els where both channels of co-jumps of prices and volatility and jump clustering areallowed. In our speciﬁcation, negative jumps play crucial roles. Whenever there isa negative jump in asset returns, it is simultaneously passed on to diﬀusion varianceand the jump intensity. Therefore, the likelihood of the future extreme events can beenhanced through jumps in diﬀusion volatility or jumps in the jump intensity or both.
The importance of negative jumps can be motivated from empirical observations inTable 1 where in all cases turmoils start with negative jumps. It is also consistent
with our understanding of ﬁnancial markets where investors are more sensitive to extreme downside risk. Our model has closed-form conditional expectation of volatility
components, making it easy to use in volatility forecasting and risk management.
The new model contains multiple dynamic unobserved factors including diﬀusion
volatility, the jump intensity, and (negative and positive) jumps. Since a richer model
framework is adopted here, we are naturally concerned about parameter uncertainty
and in-sample over-ﬁtting inherent in batch estimation2 . To deal with these issues, we
introduce a Bayesian learning approach for the proposed model. In practice, sequential
estimation of both parameters and latent factors is much more relevant than batch
estimation as we cannot obtain future information and need to update our belief onparameters and models in real time whenever new observations arrive. First, to ﬁlter theunobserved states and obtain the likelihood estimate for a given set of model parameters,
we develop an eﬃcient hybrid particle ﬁlter. It eﬃciently disentangles the diﬀusioncomponent and the positive and negative jumps. The algorithm performs much better
than the conventional bootstrap sampler for outliers that are an integral part of the
ﬁnancial data and of our model. Second, we turn to a sequential Bayesian procedure to
conduct joint inference over the dynamic states and ﬁxed parameters. In particular, we
employ the marginalized resample-move algorithm developed in Fulop and Li (2011),
2

Batch estimation of multifactor jump-diﬀusion stochastic volatility models includes Bayesian estimation with MCMC methods (Eraker, Johannes, and Polson, 2003; Li, Wells, and Yu, 2008; Li, 2011a),
maximum likelihood method (Bates, 2006, 2012), eﬃcient method of moments (Andersen et al., 2002;
Chernov et al., 2003), and non/semi-parametric methods (Todorov, 2009, 2011), among others. Stateﬁltering and sequential model comparison have been done by Johannes, Polson, and Stroud (2009) for
Poisson-Jump stochastic volatility models and by Li (2011b) for the time-changed Lévy models.

4

which is robust and eﬃcient and needs little design eﬀort from users3 . The algorithm
provides marginal likelihoods of individual observations that are crucial for sequential
model analysis with respect to information accumulation in real time. It is importantto point out that the simulated samples obtained at any time only depend on past dataso the approach is free from hindsight bias.
We use S&P 500 index returns ranging from January 2, 1980 to October 30, 2011
(in total, 8,033 observations) to empirically investigate our self-exciting models. Thisdataset is long enough and contains typical market behaviors: the 87’s market crash,
the 98’s Asian ﬁnancial crisis, the 02’s dot-com bubble burst, the 08’s global ﬁnancial
crisis, the 11’s European debt crisis, and calm periods in between. We ﬁnd that theevidence of co-jumps in volatility and returns through diﬀusion volatility is robust
ever since the 1987 market crash. The parameter driving the feedback from negative
return jumps to diﬀusion volatility is well identiﬁed and less than one. In contrast,
the self-exciting jump intensity is less important and we ﬁnd weak evidence of jump
clustering. The parameters driving the jump intensity dynamics are hard to identify.
The substantial uncertainty about the jump dynamics is mirrored in large uncertainty
about the magnitude of jump intensities during the recent ﬁnancial crisis.
As diﬀusion volatility jump is a necessary component in modeling the stock market
index and volatility co-jumps at the same time as negative jumps in asset returns,
traditional hedging strategies such as only using the underlying assets and/or using
both underlying and derivatives are no longer workable. We show that diﬀerent models
have diﬀerent risk management implications. Models with diﬀusion volatility jumps arecapable of generating high enough values of (conditional) Value-at-Risk (VaR/CVaR)
when extreme events happen and are ﬂexible to produce strong enough implied volatility
smiles and skews.
Furthermore, parameter uncertainty and learning may lead to important risk man3
Joint sequential state and parameter estimation is still an open issue. There has already been
progress towards tackling parameter learning in general state-space models. See Liu and West (2001),
Gilks and Berzouini (2001), Storvik (2002), Flury and Shephard (2009), and Carvalho et al. (2010).
For discussion of these methods, see Fulop and Li (2012). For a similar and concurrent contribution,
see Chopin et al. (2012).

5

agement and option pricing implications in the form of substantially high tail risk
measures. We show that when ignoring learning, the resulted VaR/CVaR numbers aresubstantially decreased, especially for the 0.1% values. We also ﬁnd that impacts oflearning display a U-shaped pattern with respect to moneyness, that is, learning has
much stronger inﬂuence on out-of-the-money and in-the-money options than on at-themoney options. Furthermore, for out-of-the-money and in-the-money options, the eﬀectof learning is in general decreasing with respect to maturity, while it is increasing withrespect to maturity for at-the-money options. Updating of beliefs on jump parameters
may cause long-lasting shifts in patterns of implied volatility.
The paper makes several contributions to the literature. On the theoretical aspect,ﬁrst, we propose a new class of continuous-time asset pricing models, which provides anice framework to investigate where volatility jump is from and how it interacts withjumps in asset returns. Second, a generic econometric learning approach is developedthat can disentangle positive and negative jumps and allows us to perform joint sequential inference over states and parameters with respect to information accumulationin real time. On the empirical aspect, ﬁrst, we compare alternative self-exciting jumpmodels using S&P 500 daily data. We ﬁnd strong evidence of jumps in diﬀusive volatility but weak evidence of jump clustering. In addition, we provide new insights toextreme price movements and important implications of learning in risk management,
option pricing and volatility forecasting.
Our work is related to previous studies. Jacod and Todorov (2010) and Bandi and
Reno (2011) ﬁnd that asset returns and their volatility jump together. Their results are
based on high frequency data that become available only after 1990s. The use of daily
data allows us to go much further back into the history. One advantage of using a longer
time span is that we can have more jumps and hence potentially more episodes of jump
clustering. Focusing completely on the volatility dynamics, Wu (2011) and Todorov andTauchen (2011) show that volatility does jump. Our results are in accord with thembut we go further and allow two diﬀerent channels through diﬀusion volatility and thejump intensity. This also diﬀerentiates us from existing papers where either only the6

diﬀusion channel is present (Eraker, Johannes, and Polson, 2003; Eraker, 2004) or only
the jump intensity is aﬀected by return jumps (Aı̈t-Sahalia, Cacho-Diaz, and Laeven,
2011; Carr and Wu, 2010). We ﬁnd that the diﬀusion channel is more important andremains signiﬁcant even when the jump channel is allowed. As of the jump channel,
our results are weaker than in Aı̈t-Sahalia, Cacho-Diaz, and Laeven (2011) and Carrand Wu (2010). This may be either due to the diﬀerences in the samples, or to the factthat our speciﬁcation is more general.
The rest of the paper is organized as follows. Section 2 builds the self-exciting Lévy
asset pricing models. Section 3 develops an eﬃcient hybrid particle ﬁlter and introduces
our Bayesian learning algorithm. Section 4 presents the data used for model estimation.
Section 5 presents the estimation results. Section 6 discusses empirical and economic
implications of self-exciting jumps and learning. Finally, Section 7 concludes the paper.

2

Self-Exciting Asset Pricing Models

Under a probability space (Ω, F, P ) and the complete ﬁltration {Ft }t≥0 , the asset price
St has the following dynamics∫ t
ln St /S0 =() ()µs ds + WT1,t − kW (1)T1,t + JT2,t − kJ (1)T2,t ,(1)0where µt is the instantaneous mean, W is a Brownian motion, J is a jump component,
and kW (1) and kJ (1) are convexity adjustments for the Brownian motion and the jump()
process and can be computed from their cumulant exponents: k(u) ≡ 1t ln E[euLt ] ,where Lt is either Wt or Jt .
The dynamics (1) indicates two distinct types of shocks to asset returns: small
continuous shocks, captured by a Brownian motion, and large discontinuous shocks,
modeled in this paper by the Variance Gamma process of Madan, Carr, and Chang(1998), a stochastic process in the class of inﬁnite activity Lévy processes. The jumpcomponent is important for generating the return non-normality and capturing extreme
events. The empirical study by Li, Wells, and Yu (2008) shows that the inﬁnite activity
7

Lévy models outperform the aﬃne Poisson jump models. Furthermore, the recent
nonparametric works by Aı̈t-Sahalia and Jacod (2009, 2011) and Lee and Hannig (2010)provide strong evidence on inﬁnite activity jumps in asset returns.
The Variance Gamma process can be constructed through subordinating a Brownian
motion with drift using an independent subordinator

Jt = ωSt + η W̃ (St ),(2)

where W̃t is a standard Brownian motion, and St is a Gamma subordinator St =Γ(t; 1, v) with unit mean rate and variance rate of v. Alternatively, it can be decomposed into the upside component, Jt+ , and the downside component, Jt− , such that
Jt = Jt+ + Jt− ,= Γu (t; µu , vu ) − Γd (t; µd , vd ),(3)

where Γu is a Gamma process with mean rate µu and variance rate vu , Γd is a Gamma
process with mean rate µd and variance rate vd , and

µu
µd

)
1 (√ 22
ω + 2η /v + ω ,
=
2
(
)
1 √ 2
2
=
ω + 2η /v − ω ,2vu = µ2u v,(4)vd = µ2d v.

(5)

Ti,t deﬁnes a stochastic business time (Clark, 1973; Carr et al., 2003; Carr and Wu,
2004), which captures the randomness of the diﬀusion variance (i = 1) or of the jumpintensity (i = 2) over a time interval [0, t]∫ t
Ti,t =

Vi,s− ds,0

which is ﬁnite almost surely. Vi,t , which should be nonnegative, is the instantaneous
variance rate (i = 1) or the jump arrival rate (i = 2), both of them reﬂecting theintensity of economic activity and information ﬂow. Stochastic volatility or stochastic
8

jump intensity is generated by replacing calendar time t with business time Ti,t . The
time-changed jump component has the decomposition of JT2,t = JT+2,t + JT−2,t and its()+−
convexity adjustment term is kJ (1)T2,t = kJ (1) + kJ (1) T2,t .
The instantaneous variance rate and the jump arrival rate are modeled with thefollowing stochastic diﬀerential equations

dV1,t = κ1 (θ1 − V1,t )dt + σ11√
V1,t dZt − σ12 dJT−2,t ,dV2,t = κ2 (θ2 − V2,t )dt − σ2 dJT−2,t .

(6)
(7)

Equation (6) captures stochastic variance of the continuous shocks, where Z is a standard Brownian motion and is allowed to be correlated to W with a correlation parameterρ in order to accommodate the diﬀusion leverage eﬀect. Diﬀusion variance also depends
on the negative jumps J − , indicating that there will be an abrupt increase in V1,t once
there is a negative jump in asset price. If κ1 is positive and small, Equation (6) suggests a persistent autoregressive structure in V1,t . An abrupt increase in V1,t would then
imply that the future diﬀusion variance tends to be high and decays exponentially at
the speed κ1 . Equation (7) models the stochastic intensity of jumps. When κ2 > 0, it isa mean-reverting pure jump process. The speciﬁcation implies that the jump intensity
relies only on the negative jumps in asset returns.
The conditional expectation of the jump intensity (7) can be found as follows4)
κ2 θ2 (−(κ2 −σ2 µd )t
E[V2,t |V2,0 ] =1−e+ e−(κ2 −σ2 µd )t V2,0 ,κ2 − σ2 µd(8)from which its long-run mean can be obtained by letting t → +∞,

V̄2 =
4

κ2 θ2
.
κ2 − σ2 µd

Deﬁne f (t) = eκ2 t E[V2,t |V2,0 ]. f (t) can be analytically found by solving the ODE
f ′ (t) = σ2 µd f (t) + κ2 θ2 eκ2 t ,from which we obtain the conditional expectation (8).

9(9)

Solutions (8) and (9) indicate that the conditional expectation of the jump intensity isa weighted average between the current intensity, V2,0 , and its long-run mean, V̄2 . Using(8) and (9), the conditional expectation of diﬀusion variance (6) can also be found

E[V1,t |V1,0 ] = e

−κ1 t

(

V1,0 + θ1 1 − e

−κ1 t

)
+ σ12 µd

[ 1 − e−κ1 t)]e−(κ2 −σ2 µd )t − e−κ1 t (+V̄2 − V2,0 ,κ2 − σ2 µd − κ1

κ1

V̄2
(10)

and its long-run mean is given byV̄1 = θ1 +σ12µd V̄2 .
κ1(11)

The conditional expectation of diﬀusion variance composes of two parts, one arisingfrom the square-root diﬀusion part (the ﬁrst two terms on the right-hand side in (10))
and the other from negative return jumps (the last term on the right-hand side in(10)). If the jump intensity is constant, the contribution of jumps to the conditional
diﬀusion variance becomes constant over time. In what follows, we normalize θ2 to be
one in order to alleviate the identiﬁcation problem because the jump component, J,
has non-unit variance.
Dependence of diﬀusion variance and the jump intensity only on negative jumps inasset returns is consistent with the observation in Table 1 where turmoils always startwith negative jumps. It is also consistent with the well documented empirical regularity
in ﬁnancial markets that react more strongly to bad macroeconomic surprises than togood surprises (Andersen et al., 2007). This is because the stability and sustainability of
future payoﬀs of an investment are largely determined by extreme changes in economic
conditions, and investors are more sensitive to the downside movements in the economy.
The above model (hereafter SE-M1 ) indicates that time-varying aggregate volatility
is contributed by two sources: one arises from time-varying diﬀusion volatility andthe other from the time-varying jump intensity. Whenever there is a negative jumpin asset return, diﬀusion volatility and the jump intensity move up signiﬁcantly and10simultaneously. Consequently, aggregate volatility jumps. The self-exciting behavior iscaptured through two channels: (i) a negative jump in asset return pushes up the jumpintensity, which in turn triggers more jumps in future asset returns; (ii) a negative jumpin asset return makes diﬀusion volatility jump, and this high diﬀusion volatility tends toentertain big movements in future asset returns. In contrast, existing literature allows
only one of these channels at a time. In particular, Eraker, Johannes, and Polson (2003)
and Eraker (2004) allow co-movement of return jumps and diﬀusion volatility througha synchronized Poisson process, while Aı̈t-Sahalia, Cacho-Diaz, and Laeven (2011) andCarr and Wu (2010) link only the jump intensity to jumps in asset returns.
The central questions we are concerned about in the present paper are the dynamic
structure of extreme movements and how asset return jumps aﬀect total volatility. In
order to explore these issues, we also investigate the following nested models: (i) SEM2 : the self-exciting model where diﬀusion volatility does not jump, and the total
volatility jump and the jump clustering are from the time-varying jump intensity; (ii)
SE-M3 : the model where the jump intensity is constant, and the total volatility jump
and the self-exciting eﬀect are only from the diﬀusion volatility process; and (iii) SEM4 : no volatility jumps and no self-exciting eﬀects. Obviously, the SE-M4 model isnested by the SE-M2 model and the SE-M3 model. However, the SE-M2 model andthe SE-M3 model do not nest each other.

3

Econometric Methodology

In this section, we present our Bayesian learning method. Section 3.1 develops aneﬃcient hybrid particle ﬁlter, which provides us more accurate likelihood estimate andseparates the diﬀusion component, positive jumps and negative jumps. Section 3.2
brieﬂy presents the parameter learning algorithm for model estimation.

11

3.1

An Eﬃcient Particle ﬁlter

Our model can be cast into a state-space model framework. After discretizing the returnprocess for a time interval τ using the Euler method, we have the following observation
equation()√
1ln St = ln St−τ + µ − V1,t−τ − k(1)V2,t−τ τ + τ V1,t−τ wt + Ju,t + Jd,t ,2(12)

where wt is a standard normal noise, and Ju,t and Jd,t are the upside and downside
jumps.
We take the diﬀusion variance V1,t , the jump intensity V2,t , and the upside/downside
jumps Ju,t /Jd,t as the hidden states. Diﬀusion variance and the jump intensity follow(6) and (7), and the upside/downside jumps are gammas. After discretizing, we havethe state equations as follows

V1,t = κ1 θ1 τ + (1 − κ1 τ )V1,t−τ + σ11√τ V1,t−τ zt − σ12 Jd,t ,(13)

V2,t = κ2 θ2 τ + (1 − κ2 τ )V2,t−τ − σ2 Jd,t ,(14)

Ju,t = Γ(τ V2,t−τ ; µu , vu ),(15)

Jd,t = −Γ(τ V2,t−τ ; µd , vd ),(16)

where zt is a standard normal noise, which is correlated to wt in (12) with the correlation
parameter ρ.
For a given set of model parameters, ﬁltering is a process of ﬁnding the posterior distribution of the hidden states based on the past and current observations, p(xt |y1:t , Θ),
where xt = {V1,t , V2,t , Ju,t , Jd,t }, and y1:t = {ln Ss }ts=1 . Because this posterior distribution in our model does not have analytical form, we turn to particle ﬁlters to approximate it. Particle ﬁlters are simulation-based recursive algorithms where the posterior
distribution is represented by a number of particles drawn from a proposal densityp̂(xt |y1:t , Θ) =

M
∑

)
(
(i)
(i)
w̃t δ xt − xt ,

i=1

12

(17)

(i)

where w̃t(i)

= ωt /∑M(j)
j=1 ωtwith ωt and w̃t being the importance weight and the(i)normalized importance weight, respectively, xt is the state particle, and δ(·) denotes
the Dirac delta function.
Particle ﬁlters provide an estimate of the likelihood of the observationsp̂(y1:t |Θ) =t∏p̂(yl |y1:l−1 , Θ),(18)

1 ∑ (i)w .
M i=1 l

(19)

l=1

where

M

p̂(yl |y1:l−1 , Θ) =Importantly, the estimate (18) is unbiased: E[p̂(y1:t |Θ)] = p(y1:t |Θ), where expectation
is taken with respect to all random quantities used in particle ﬁlters (Del Moral, 2004).
The most commonly used particle ﬁlter is the bootstrap ﬁlter of Gordon, Salmond,and Smith (1993), which simply takes the state transition density as the proposal
density. However, the bootstrap ﬁlter is known to perform poorly when the observation
is informative on hidden states. Our model has this feature because when we observe alarge move in asset price, the jump can be largely pinned down by this observation. On
the other hand, when the return is small, it is almost due to the diﬀusion component
and contains little information on the jump. Hence, to provide an eﬃcient sampler, we
use an equally weighted two-component mixture as the proposal on the jump: the ﬁrst
component is a normal draw, equivalent to sampling from the transition density of thediﬀusion component, and the second component involves drawing from the transition
law of the jump. We need this second component to stabilize the importance weights
for small returns. Otherwise, we would compute the ratio of a normal and a gamma
density in the importance weights which is unstable around zero. When the return ispositive, we use this mixture as the proposal for the positive jump and the transition
density for the negative jump, and vice-versa. See Appendix A for the algorithm.

13

3.2

A Parameter Learning Algorithm

While particle ﬁlters make state ﬁltering relatively straightforward, parameter learning,i.e., drawing from p(Θ|y1:t ) sequentially, remains a diﬃcult task. Simply including thestatic parameters in the state space and applying a particle ﬁlter over p(Θ, xt |y1:t ) does
not result in a successful solution due to the time-invariance and stochastic singularityof the static parameters that quickly leads to particle depletion. In what follows, we use
a generic solution to the parameter learning problem proposed by Fulop and Li (2010).
The key to this algorithm is that particle ﬁlters provide an unbiased estimate of the true
likelihood, so that we can run a recursive algorithm over the ﬁxed parameters using the∏sequence of estimated densities, p̂(Θ|y1:t ) ∝ tl=1 p̂(yl |y1:l−1 , Θ)p(Θ), for t = 1, 2, . . . , T .
Deﬁne an auxiliary state space by including all the random quantities produced by
the particle ﬁltering algorithm. In particular, denote the random quantities produced
(i)

(i)

by the particle ﬁlter in step l by ul = {xl , ξl ; i = 1, . . . , M }. Then at time t, the ﬁlter
will only depend on the population of the state particles in step t − 1, so we can writeψ(u1:t |y1:t , Θ) =t∏

ψ(ul |ul−1 , yl , Θ),

(20)

l=1

where ψ(u1:t |y1:t , Θ) is the density of all the random variables produced by the particle
ﬁlter up to t. Furthermore, the predictive likelihood of the new observations can be
written asp̂(yt |y1:t−1 , Θ) ≡ p̂(yt |ut , ut−1 , Θ).

(21)

We then construct an auxiliary density, which has the formp̃(Θ, u1:t |y1:t ) ∝ p(Θ)

t∏p̂(yl |ul , ul−1 , Θ)ψ(ul |ul−1 , yl , Θ).

(22)

l=1

The unbiasedness property in likelihood approximation means that the original target,p(Θ|y1:t ), is the marginal distribution of the auxiliary density. If we can sequentiallydraw from the auxiliary density, we automatically obtain samples from the original
target.
14

Assume that we have a set of weighted samples that represent the target distribution,
{(
)
}
(n)
(n)
(n)
(n)
p̃(Θ, u1:t−1 |y1:t−1 ), at time t−1:
Θ , ut−1 , p̂(y1:t−1 |Θ), st−1 ; n = 1, . . . , N , where(n)st−1 denotes the sample weight. Notice that for each n, the relevant part of ut−1
(i,n)

are M particles after resampling representing the hidden states {xt; i = 1, . . . , M }.

Therefore, in total we have to maintain M × N particles of the hidden states. The
following recursive relationship holds between the target distributions at t − 1 and t,p̃(Θ, u1:t |y1:t ) ∝ p̂(yt |ut , ut−1 , Θ)ψ(ut |ut−1 , yt , Θ)p̃(Θ, u1:t−1 |y1:t−1 ),(23)from which we can arrive to a set of samples representing the target distribution,p̃(Θ, u1:t |y1:t ), at time t through the marginalized resample-move approach developedby Fulop and Li (2012).
The marginalized resample-move approach has a natural byproduct of the marginal
likelihood of the new observation∫p(yt |y1:t−1 ) ≡=≈∫p(yt |y1:t−1 , Θ)p(Θ|y1:t−1 )dΘp̂(yt |ut , ut−1 , Θ)ψ(ut |ut−1 , yt , Θ)p̃(Θ, ut−1 |y1:t−1 )d(Θ, ut , ut−1 )

N
∑

(k)

(k)

(k)

πt−1 p̂(yt |ut , ut−1 , Θ(k) ),

(24)

k=1

(n)

(n)

(n)

(n)

where the new weight can be updated by st = st−1 × p̂(yt |ut , ut−1 , Θ(n) ) and the
(k)
(n)
(n) ∑
normalized weight is given by πt = st / N
k=1 st . From (24), a sequential Bayes
factor can be constructed for sequential model comparison. For any models M1 andM2 , the Bayes factor at time t has the following recursive formula
BFt ≡p(yt |y1:t−1 , M1 )p(y1:t |M1 )=BFt−1 .
p(y1:t |M2 )p(yt |y1:t−1 , M2 )(25)

Our particle learning algorithm naturally has the marginal likelihood estimate (24),which can be used in (25) for model assessment and monitoring over time. For moredetails, we refer readers to Fulop and Li (2012) and Chopin et al. (2012).
15

4The Data

The data used to estimate the models are the S&P 500 stock index ranging from January
2, 1980 to October 30, 2011 in daily frequency, in total 8,033 observations. This dataset
contains typical ﬁnancial market behaviors: the recent European debt crisis, the global
ﬁnancial crisis in the late 2008, the market crash on October 19, 1987 (-22.9%), thevolatile market and relatively tranquil periods. Table 2 presents descriptive statistics
of index returns. The annualized mean of index returns in this period is around 7.8%
and the annualized historical volatility is about 18.4%. A striking feature of the datais high non-normality of the return distribution with the skewness of -1.19 and the
kurtosis of 29.7. The Jarque-Bera test easily rejects the null hypothesis of normalityof returns with a very small p-value (less than 0.001). The index returns display very
weak autocorrelation. The ﬁrst autocorelation is about -0.03, while the sixth one is as
small as 0.008.
— Table 2 around here —Figure 1 plots S&P 500 index returns and standard deviations computed from theprevious 22-day returns at each time. The companion of abrupt moves in volatility
to extreme events in returns is very clear, and turbulent periods tend to be realized
through many consecutive large up and down return moves. What is hard to gauge isthe extent to which these are due to high diﬀusion volatility or persistent fat tails. The
model estimates that follow will shed more lights on this issue.
— Figure 1 around here —5Model Estimation and Learning

In this section, we present estimation and learning results. Models are estimated using
the Bayesian learning approach discussed in Section 3. In implementation, we set the
number of state particles to be 10,000 and the number of parameter particles to be 2,000.
The thresholds N1 and N2 are equal to 1,000. These tuning-parameters are chosen such16

that the acceptance rate at the move step is relatively high and the computational costis reasonable. Subsection 5.1 discusses sequential model comparison and monitoring;
subsection 5.2 investigates the information ﬂow and parameter learning; and subsection
5.3 presents the volatility and jump ﬁltering results.

5.1

Sequential Model Comparison and Monitoring

In a Bayesian context, model comparison can be made by Bayes factors, deﬁned as
the ratio of marginal likelihoods of models5 . Table 3 presents the overall Bayes factors(in log) for all models investigated using all available data. We ﬁnd that the SE-M1
model and the SE-M3 model, both of which allow negative return jumps to aﬀect
diﬀusion volatility, outperforms the SE-M2 model and the SE-M4 model that exclude
this channel. For example, the log Bayes factors between the SE-M1 model and theSE-M2/SE-M4 models are about 12.5 and 13.3, respectively, and the log Bayes factors
between the SE-M3 model and the SE-M2/SE-M4 models are about 11.7 and 12.5,
respectively. Thus, there is decisive evidence in the data for negative return jumps
aﬀecting diﬀusion volatility and co-jumps of returns and volatility. Furthermore, there
seems to be evidence for return jumps aﬀecting the jump intensity. Comparing theSE-M1 model where both self-exciting channels are allowed to the SE-M3 model where
only diﬀusion volatility is inﬂuenced by return jumps, the former is weakly preferredwith a log Bayes factor of 0.84.
— Table 3 around here —The above batch comparison does not tell us how market information accumulates
and how diﬀerent models perform over time. Does one model outperform the other one
at a certain state of economy, but underperform it at another state of economy? Our
Bayesian learning approach has a recursive nature and produces the individual marginal
5

In Bayesian statistics, Jeﬀreys (1961) gave a scale for interpretation of Bayes factors. For twomodels M1 and M2 , if the value of the log Bayes factor of M1 to M2 is between 0 and 1.1, M1 is barely
worth mentioning; if it is between 1.1 and 2.3, M1 is substantially better than M2 ; if it is between 2.3
and 3.4, M1 is strongly better than M2 ; if it is between 3.4 and 4.6, M1 is very strongly better thanM2 ; and if it is larger than 4.6, M1 is decisively better than M2 .

17

likelihood of each observation over time. One can then construct the sequential Bayes
factors and use them for real-time model monitoring and analysis.
Figure 2 presents the sequential Bayes factors (in log) that gives us a richer picture
on model performance over time. We notice from the upper panels that in the beginningwhen market information is little, both the SE-M1 model and the SE-M3 model, which
are the two best models according to the Bayes factor in Table 3, perform nearly thesame as the SE-M2 model and the SE-M4 model. As market information accumulates
over time, in particular after the 87’s market crash, the SE-M1 model and the SE-M3
model begin to outperform the other two models. The lower left panel of Figure 2
shows that the SE-M1 model with the time-varying jump intensity hardly diﬀerentiatesfrom the constant jump-intensity model, the SE-M3 model as the log Bayes factors arewithin [-1.1, 1.1]. Interestingly, as for the SE-M2 model and the SE-M4 model, bothof which shut down the diﬀusion volatility jump component, in the beginning the twomodels perform nearly the same as log sequential Bayes factors vary around zero. At
the 87’s market crash the log Bayes factor of the SE-M2 model to the SE-M4 model
moves abruptly to a level above 2 and almost stays there till 2002 dot-com bubble
burst. Afterwards, the Bayes factor decreases gradually to a value around 1. This
result indicates that the diﬀusion volatility jump is a necessary component in modeling
S&P 500 index and models shutting down this component are clearly misspeciﬁed.
— Figure 2 around here —5.2

Information Flow and Parameter Learning

Table 4 presents the parameter estimates (5%, 50%, and 95% quantiles) of all models
using all available data6 . Focusing on parameter estimates in the SE-M1 model and theSE-M3 model that are two best models according to the sequential Bayes factors, we ﬁndthat the jump-size related parameters and the diﬀusion volatility-related parameters
The priors for the SE-M1 model are given by Θ0 ∼ N (Θ∗0 , Σ20 ), where Θ∗0 = (0.10, −0.05, 0.10, 1.00,
3.00, 0.03, 0.30, −0.50, 0.50, 20, 5.0) and Σ0 = diag(0.20, 0.08, 0.15, 3.00, 5.00, 0.05, 0.30, 0.50, 1.00, 20, 10).
Any non-positive values of η, v, κ1 , θ1 , σ11 , σ12 , κ2 and σ2 , and any ρ ∈/ (−1, 1) are automaticallydiscarded. The priors for other models are also similar.
618

have narrow 90% credible intervals and small standard deviations, indicating that it
is easy to identify these parameters using all available data. In particular, the selfexciting eﬀect parameter σ12 has the posterior mean of about 0.52, the 90% credibleinterval of 0.35-0.66 and the posterior standard deviation of 0.09 in the SE-M1 model,
and it has the posterior mean of about 0.47, the 90% credible interval of 0.33-0.60 andthe posterior standard deviation of 0.09 in the SE-M3 model. These narrow credibleintervals and small standard deviations imply that diﬀusion volatility does jump at
the same time as negative jumps in returns. However, the parameter estimates of thejump intensity-related parameters in the SE-M1 model have large credible intervals,
especially the mean-reverting parameter κ2 , indicating that it is hard to identify theseparameters only using the time-series of underlying index data.
— Table 4 around here —In the SE-M1 and SE-M3 models, we ﬁnd that the posterior mean of ω is negative(about -0.07) and its 90% credible interval is narrow and in negative side, indicatingthat index returns jump downward more frequently than jump upward. The jump
structure parameter v has a posterior mean of about 0.95 and a 90% credible intervalof [0.40, 1.83] in the SE-M1 model, and it has a posterior mean of about 1.07 and a 90%
credible interval of [0.45, 2.11] in the SE-M3 model, implying that in general, small/tiny
jumps happen with a very high frequency and large/huge jumps occur only occasionally.
The mean-reverting parameter estimate κ1 of the diﬀusion volatility process is a little
bit larger in the SE-M1 model than in the SE-M3 model (4.23 vs. 3.89), but both
estimates of θ1 and σ11 are very similar in both models. The negative estimate of ρ,which is about -0.6 in both models, reveals existence of the diﬀusion leverage eﬀect.
The long-run means of diﬀusion volatility and the jump intensity are given by (11) and(9), respectively, in the SE-M1 model. Using the estimates in Table 4, they are 0.028
and 1.127, respectively. In the SE-M3 model, the long-run mean of diﬀusion volatility
is given by V¯1 = θ1 + σ12 µd /κ1 , which is about 0.029 using the corresponding parameter
estimates in Table 4. Thus, the model-implied unconditional return volatility in both19the SE-M1 and SE-M3 models is 18.4%, which is nearly the same as the historicalreturn volatility (18.4%).
Our Bayesian learning approach provides us more than parameter estimates themselves. It gives us the whole picture of how parameters evolve over time with respect toaccumulation of information. Figure 3 presents the sequential learning of the jump anddiﬀusion volatility-related parameters in the SE-M1 model. Clearly, all parameters havebig variations at the beginning when market information is very little. With respect
to accumulation of information, the credible intervals become narrower and narrower.
We ﬁnd that the jump-related parameters, the mean-reverting parameter, and the selfexciting eﬀect parameter usually take long time to reach reliable regions, indicating
information on these parameters accumulates very slowly, and in practice we need longdataset to obtain accurate estimates. Very similar results can also be obtained fromthe parameter learning in the SE-M3 model as shown in Figure 5. Figure 4 presents
the jump intensity-related parameter learning (κ2 and σ2 ) in the SE-M1 model. We
ﬁnd from the upper panels that credible intervals of these two parameters are barely
narrowing down over time. Only from the recent European debt crisis on, we observe
a little narrowing-down of their credible intervals. The lower panels plot the prior andposterior distributions (solid and dashed lines, respectively) of these two parameters.
The dispersions of both priors and posteriors are very big. This result indicates thatthe information we have is not enough to well identify these two parameters.
— Figure 3 around here —— Figure 4 around here —— Figure 5 around here —5.3

Volatility and Jump Filtering

Embedded in our learning algorithm is an eﬃcient hybrid particle ﬁlter. One meritof this particle ﬁlter is that it can separate positive jumps and negative jumps. This
separation is important from both the statistical and the practical perspectives. Statistically, it makes our self-exciting models feasible to estimate since both diﬀusion
20

volatility and the jump intensity depend only on the negative jump. Practically, investors are mostly concerned about negative jumps. The ability to disentangle negative
jumps provides us an important tool for risk management.
Figure 6 presents the ﬁltered diﬀusion volatility and the ﬁltered jump intensity. We
can see that whenever there is a big negative jump, diﬀusion volatility in the SE-M1 andSE-M3 models and the jump intensity in the SE-M1 model abruptly move up to a highlevel. However, there are some important diﬀerences between the two state variables.
Diﬀusion volatility in both the SE-M1 and SE-M3 models is well identiﬁed with a tight90% credible interval. In contrast, our ability to pin down the jump intensity in the
SE-M1 model is much more limited as we can see that its credible intervals are wide
during crisis periods. Further, there seems to be an abrupt change in the behavior ofthis latent factor since the 2008 crisis. Prior to this episode, after widening the credibleintervals of jump intensities during crisis periods, they quickly revert to their long-run
mean, whereas they have remained consistently high and wide since the 08’s ﬁnancial
crisis. It suggests that as far as the tails are concerned, the recent crisis is special, witha sustained probability of large extreme events going forward.
— Figure 6 around here —Figure 7 presents the ﬁltered positive and negative jumps in the SE-M1 and SE-M3
models. In both models, the ﬁltered negative jumps can eﬀectively capture all market
turmoils such as the 87’s market crash, the 98’s Asian ﬁnancial crisis, the 08’s ﬁnancial
crisis and the 11’s European debt crisis. However, as shown in the upper panels, theﬁltered positive jumps are quite small, especially after the 87’s market crash. Thisis a new and potentially important empirical result, suggesting that whenever jumps
in volatility are taken into account, the positive jump component in index returns is
not so important and the positive movements in index returns can be captured by thediﬀusion component. This ﬁnding reinforces our choice of giving negative jumps more
prominence.
— Figure 7 around here —21

6Empirical and Economic Implications

6.1

Volatility Estimation and Variance Decomposition

We have seen that the parameters driving the jump intensity have large 90% credibleintervals. It is interesting to examine the extent to which these results are due tolearning. For this purpose, Figure 8 depicts the ﬁltered dynamic states when the fullsample posterior means of the ﬁxed parameters are plugged into the particle ﬁlter.
In the case of diﬀusion volatility, the picture does not change much, consistent withthe relatively tight posteriors on most diﬀusion-related parameters. The only notablediﬀerence is a smaller peak around the 1987 crash. This can be explained by thelarge uncertainty at this point on the parameter driving the volatility feedback, σ12 .
The real-time posterior contains larger values that give rise to a more pronounced
volatility feedback phenomenon. However, when looking at the lower panel, we observemuch larger diﬀerence. First, ﬁxing the parameters considerably shrinks the credibleintervals, suggesting that a large part of the uncertainty in jump intensities observed
before in Figure 6 is the result of parameter uncertainty from learning. Second, thepeak in jump intensities in 1987 is bigger than before, a mirror image of what we haveobserved for diﬀusion volatility. Finally, when parameters are ﬁxed, even after 2008,
the jump intensities revert back to their long run mean fairly quickly and the credibleinterval does not stay wide. Thus, the large uncertainty about the tails in the futureseems mainly related to the lack of precise knowledge about the parameters driving the
dynamics of the jump intensity.
— Figure 8 around here —Learning and parameter uncertainty also have large impact on variance contributions of diﬀerent shocks to total return variance. The conditional instantaneous return
variance, Vt , has two sources in our models contributed by the diﬀusion shock and thejump shock, respectively,
Vt = V1,t + V ar(J1 )V2,t ,

22

(26)

where V ar(J1 ) = ω 2 v + η 2 is variance of the jump component at time t = 1. For the
SE-M3 and SE-M4 models, the jump intensity is constant with V2,t = 1. With learned
parameters on ω, η, and v, and ﬁltered diﬀusion variance and jump intensity over time,
we can investigate the percentage of total return variance contributed by the jumpcomponent. The upper panels of Figure 9 show the jump contribution to total return
variance in the SE-M1 and SE-M3 models, which is computed by V ar(J1 )V2,t /Vt overtime. In both models, we ﬁnd that in the beginning, the jump contribution to total
return variance is pretty high, around 50%. However, whenever we ignore learningand parameter uncertainty and simply insert parameter and state estimates using all
available data, the contribution of the jump component to return variance is in general
reduced. The middle panels of Figure 9 plot the instantaneous variance contributionof the jump component when leaning and parameter uncertainty are ignored. With
comparison to the upper panels, we noticed that for the initial period before the 87’s
market crash, the jump contribution is reduced dramatically. This is largely because
we have little information in the beginning, and parameter uncertainty plays dominantrole. The lower panels plot the diﬀerence of the jump contribution to return variancewith and without parameter uncertainty. Clearly, in the beginning, the diﬀerence isvery big. However, with respect to information accumulation over time, this diﬀerence
becomes smaller and smaller. Overall, the jump contributions to total return varianceare 34.3% and 34.6% when taking into account learning, whereas they are 24.6% and23.4% when ignoring learning in the SE-M1 and SE-M3 models, respectively. These
values are larger than those obtained by Huang and Tauchen (7%, 2005) and Andersen,
Bollerslev and Diebold (15%, 2007).
— Figure 9 around here —6.2

VaR Computing and Risk Management Implications

The results we have found have important implications for risk management. As diffusion volatility jump is a necessary component in modeling the stock market index
and volatility co-jumps at the same time as negative jumps in asset returns, traditional23

hedging strategies such as only using the underlying assets and/or using both underlying and derivatives are no longer workable. Furthermore, the big uncertainty in thejump intensity may lead to important risk management implications in the form of
substantially high tail risk measures.
Here we investigate diﬀerent Value-at-Risk (VaR) and conditional Value-at-Risk(CVaR) measures implied by our models and learning algorithm. The combinationof VaR and CVaR can provide us rich information for understanding normal risk andtail risk. Table 5 reports summary statistics of one-day and one-week VaR/CVaR
numbers both for the full sample and the recent ﬁnancial crisis period, i.e., the sample
after Lehmans’ bankruptcy on September 15, 2008. We have the following interestingﬁndings. First, look at the 1% VaR, a frequently used day-to-day measure of “normal”
risk. We ﬁnd that the diﬀerence across the diﬀerent models in the average VaR numbers
is moderate. They are about -0.024 for the one-day measure and are about -0.076 for the
one-week measure. However, we do observe that the minimum VaR’s in the full sample
are much more extreme for the models with the jump feedback to diﬀusion volatility(SE-M1/SE-M3). For example, the minimum one-day VaR’s implied by the SE-M1 andSE-M3 models are about -0.110, while those impled by the SE-M2 and SE-M4 models
are about -0.076. Similar results can also been noticed in one-week minimum VaR’s.
These mainly reﬂect the fact that the SE-M2 and SE-M4 models miss the peak involatility after the 1987 crash. Next, let us check what the 0.1% VaR numbers convey.
These can be interpreted as a measure of tail risk. Here for the full sample minimum
VaR’s we ﬁnd again large diﬀerences across the models but the division lies between themodels with and without self-exciting jumps, i.e., SE-M1/SE-M2 vs. SE-M3/SE-M4,
with the former exhibiting much larger tail risk. For example, the full sample one-day
minimum VaR’s implied by the SE-M1/SE-M2 models are about -0.30, much larger
than those implied by SE-M3/SE-M4 models. Last, CVaR numbers reveal very similar
implications to those implied by 0.1% VaR no matter 1% or 0.1% CVaR is considered.
This is because CVaR calculates the average loss larger than the corresponding VaR,
and put greater emphasis on the presence of extreme downside events and tail risk.
24

Overall these results suggest that both feedback channels are important but their risk
management implications are somewhat diﬀerent, with self-exciting jumps exerting theirinﬂuence deeper in the left tail.
— Table 5 around here —However, when the learning eﬀect is ignored, the resulted Var/CVaR numbers arevery diﬀerent. Here we focus on CVaR and recalculate the one-day/one-week 0.1%/1%
CVaR numbers using the posterior means of parameters from all available data. Figure
10 reports ratios of CVaR numbers with and without learning in the SE-M3 model.
One prominent ﬁnding is that most of CVaR ratios are larger than one, indicating the
CVaR numbers with learning are in general bigger than those without learning. The
ratios are larger for the 0.1% case than for the 1% case, indicating that learning does
really have strong impacts on probability that extreme downside events happen. This
impact is even stronger for the one-week measures. However, we also notice that withrespect to information accumulation, the learning eﬀect is less and less remarkable.
— Figure 10 around here —6.3

Option Pricing Implications

What roles do self-exciting jumps play in option pricing? Figure 11 plots the BS
implied volatility smiles and skews under the four models considered for diﬀerent maturities and strikes. We consider the near-the-maturity options with maturity 7 days,
the short-maturity options with maturity 30 days, the medium-maturity options withmaturity 90 days, and the long-maturity options with maturity 250 days. Options are
priced using the Monte-Carlo simulation method. We ﬁx the risk-free rate at 5% andchoose the parameter set for the SE-M1 model as (ω, η, v, κ1 , θ1 , σ11 , ρ, σ12 , κ2 , σ2 ) =(−0.06, 0.05, 1.00, 3.50, 0.02, 0.30, −0.60, 0.60, 10.0, 15.0). For other nested models,
the relevant parameters have the same values as the above.
The ﬁgure shows that when considering the near-the-maturity options (7 days),
we ﬁnd that the SE-M1 and SE-M2 models, which allows for the time-varying self25

exciting jump intensity, can generate stronger smile eﬀect than the SE-M3 and SE-M4
models, which assume the constant jump intensity. However, when we move to longer
maturity options (30, 90, and 250 days), we clearly see a separation between the SE-M1
and SE-M3 models, which allows for the self-exciting diﬀusion variance jump, and theSE-M2 and SE-M4 models, which shut down the diﬀusion volatility jump component.
The SE-M1 and SE-M3 models can not only generate stronger skew eﬀect, but also
highly price these options than the SE-M2 and SE-M4 models. We then conclude thatthe self-exciting jump intensity plays a role in option pricing only for the very shortmaturity options; however, for other options, the diﬀusion volatility jumps play much
more important role.
— Figure 11 around here —Now we begin to investigate how learning aﬀects the implied volatility surface. As
here we only use the underlying data to estimate the models, we ignore the risk premia
and assume that the jump and volatility parameters remain the same under the changeof measure. At each time starting from January 1981 in learning, whenever we obtain
the parameter and volatility particles, we use the Monte-Carlo method to price call
options with maturity 7, 30, 90, and 250 days and with moneyness (deﬁne as K/S)
0.85, 0.90, 0.95, 1.00, 1.05, 1.10, and 1.15. We price the same cross-section of options
again only using the posterior means of the parameters from all available data andﬁltered distribution of volatility obtained from these estimates. Thus, in the latter, thelearning eﬀect is ignored.
Table 6 presents the time-series mean of the implied volatility ratios with and without learning for each option in the SE-M1 and SE-M3 models. For the SE-M1 model,
almost all ratios are larger than one, and for the SE-M3 model, except some cases inthe 7-day maturity options, the rest is larger than one, indicating that learning does
(positively) aﬀect the option pricing. In particular, we ﬁnd that (1) no matter whichmaturity is concerned, the ratios display a U-shaped pattern, that is, the ratios for inthe-money and out-of-the-money options are larger than those for at-the-money options.

26

The eﬀect is stronger on deep out-of-the-money options than on deep in-the-money options; (2) For deep out-of-the-money and in-the-money options, the learning eﬀect is in
general decreasing with respect to maturity; on the contrary, for at-the-money options,the learning has stronger eﬀect on long-maturity options; and (3) it seems that themain diﬀerence between the SE-M1 and SE-M3 models is at pricing the 7-day maturity
options.
— Table 6 around here —To further investigate the learning eﬀect, Figure 12 plots the time-series of implied
volatility computed from the SE-M1 model. The solid and dashed (red) lines plot implied volatility when learning is taken into account and when it is ignored, respectively.
Given that jumps are known to primarily aﬀect options with short maturities, here
we focus on contracts with 7 and 30-day maturities. In middle panels, we observethat implied volatility of at-the-money options closely follow the path of the diﬀusion
variance, depicted in Figure 6. Further, learning does not seem to have a ﬁrst-ordereﬀect on option prices except during the market crash in October 1987. The picturesare starkly diﬀerent in upper and lower panels, reporting implied volatility of deepin-the-money and deep out-of-the-money calls. These ﬁgures track the path of jumpintensities, reinforcing our intuition that jumps deﬁne the deep tails of the predictive
distribution at short maturities. Further, learning here does have a ﬁrst-order eﬀect,
especially at the ﬁrst half of the distribution. Naturally, as the sample size grows, theeﬀect of learning diminishes, but it begins to matter again during the recent crisis,
especially for the 7-day maturity deep in-the-money options. Moreover, for the deepin-the-money options, when learning is ignored, implied volatility is almost constantwith abrupt move-up and drop-back during the crisis periods, whereas when learning istaken into account, implied volatility moves up to high levels during the crisis periods
and stay there for a very long time. The deep in-the-money options are sensitive toextreme downside movements and the above observation seems to indicate that learning
lead to long-lasting shifts on beliefs of the left tail of predictive distribution. Benzoni,

27Collin-Dufresne, and Goldstein (2011) argue that updating of beliefs about jump parameters may cause permanent shifts in option prices. Our investigation reinforces thisintuition. Overall, the above results suggest that parameter uncertainty and learning
are likely to have important implications for pricing options that depend on tails of thepredictive distribution.
— Figure 12 around here —We now further compare the SE-M1 and SE-M3 models from time-series perspective,
and see what roles the self-exciting jump intensity plays. Figure 13 presents the timeseries ratios of implied volatility of call options with maturity 7 and 30 days between the
SE-M1 and SE-M3 models. For 7-day maturity options, we ﬁnd from the left-middle
panel that the SE-M1 model prices the at-the-money options quite similarly to the
SE-M3 model no matter learning is taken into account or not. However, they perform
diﬀerently when pricing deep in-the-money and deep out-of-the-money options. Whenpricing deep in-the-money options, the left-upper panel shows that implied volatility
from the SE-M1 model is very close to that from the SE-M3 model when learning
is ignored, whereas when learning is taken into account, these two models produce
very diﬀerent implied volatility. On the contrary, when we move to price the deep
out-of-the-money options, the SE-M1 and SE-M3 have similar implied volatility whenlearning is considered, but they have diﬀerent implied volatility when learning is ignored.
However, for options with maturity 30 days, the right panels indicate that both models
perform quite similarly no matter which moneyness is concerned. For other options
with maturity larger than 30 days, we again ﬁnd similarity between the SE-M1 andSE-M3 models (not reported). The ﬁndings here are consistent with what we havefound in Table 6 and Figure 11 and imply that the two models perform diﬀerently only
when pricing the near-the-maturity options.
— Figure 13 around here —

28

6.4

Volatility Forecasting Implications

It is known that superior in-sample performance does not necessarily lead to better
out-of-sample properties. In this section, we evaluate the relative performance of thefour jump models for predicting daily volatility. As the jump intensity dynamics is
not well identiﬁed, we focus on the diﬀusion volatility instead, that is, the integrated∫ t+1
daily volatility next period, t V1,t+s ds, using S1 , · · · , St . Following Barndorﬀ-Nielsen
and Shephard (2004), we approximate the integrated daily diﬀusion volatility by thestandardized realized bipower variation based on 5-minute returns,
π∑|rt+jh,h ||rt+(j−1)h,h |,2 j=2n

BVt+1 =

(27)

where h corresponds to a 5-minute interval within a trading day, n the number of 5minute returns within a day, and rt,h = log(St /St−h ). As shown in Barndorﬀ-Nielsen∫ t+1
and Shephard (2004), as h → 0, BVt+1 → t V1,t+s ds.
Our inferential framework allows us to obtain daily volatility forecast as a byproduct. This is achieved by using the formula (10) and the parameter and stateparticles at each time t. The posterior mean of V1,t+1 is then obtained. This is treated∫ t+1
as the one-day-ahead volatility forecast of t V1,t+s ds and matched against BVt+1 .
This volatility forecast obviously takes into account of parameter uncertainty. Table 7
reports the root mean square errors (RMSE) for the annualized square-root estimatesfrom the four candidate models over the three periods.
As expected, it is more diﬃcult to predict volatility when the market is more volatile(between January 2008 and October 2011). It is clear that in all three forecasting
periods, the SE-M1 model always performs the best and marginally beats the SE-M3
model. Interestingly, the SE-M2 model performs worst in all cases, consistent withwhat have found in subsection 5.1. Perhaps the most important result is that boththe SE-M1 and SE-M3 model perform much better than the SE-M2 and SE-M4 model.
For example, the improvement of the SE-M1 model over the SE-M2 model is 4.4%,
4.6% and 4.1% over the three periods, respectively. These results once again reinforce
29importance of allowing negative jumps in returns to lead to jumps in diﬀusive volatility.
— Table 7 around here —7Concluding Remarks

We introduce a new class of self-exciting asset pricing models where negative jumps
play important roles. Whenever there is a negative jump in asset return, this negative
jump is simultaneously passed on to diﬀusion variance and the jump intensity, generating co-jump of prices and volatility and jump clustering. We investigate the models by
employing a Bayesian learning approach. Using S&P 500 index returns ranging fromJanuary 2, 1980 to October 31, 2011, we ﬁnd that negative jumps in asset returns leadto jumps in total volatility mainly through diﬀusion variance. We ﬁnd weak evidenceof jump clustering. Parameter uncertainty and learning have been shown to have important implications for risk management, option pricing and volatility forecasting inpractice.
There are several interesting research directions that our results open up. First,
it would be interesting to examine what we can ﬁnd if option prices are included in
the dataset. This should have the potential to better identify the jump intensity process. Second, the sequential nature of our joint parameter and state estimation routine
promises several practical applications such as derivative pricing or portfolio allocation.

Appendix: A Hybrid Particle Filter
The algorithm of the proposed hybrid particle ﬁlter consists of the following steps:
{
(i)
(i)
(i)
Step 1: Initialize at t = 0: set initial particles to be V1,0 = θ1 ; V2,0 = 1; Ju,0 =}M(i)0; Jd,0 = 0
and give each set of particles a weight 1/M ;i=1

Step 2: For t = 1, 2, . . .
• If Rt = ln St − ln St−τ > 0,(i)

– draw Jd,t from its transition law (16);
30

(i)

– draw Ju,t both from its transition law (15) and its conditional posterior dis√tribution Ju,t = ln St −ln St−τ −(µ− 12 V1,t−τ −k(1)V2,t−τ )τ −Jd,t − τ V1,t−τ wt ,which is normally distributed. Equal weights are attached to particles obtained from the transition law and the conditional posterior;– compute the particle weight by
(i)
(i)
(i)
(i)
(i)
(i)
p(ln St |Ju,t , Jd,t , V1,t−τ , V2,t−τ )p(Ju,t |V2,t−τ )
(i)
wt =
,
(i)
(i)
0.5p(Ju,t |V2,t−τ ) + 0.5ϕ(µ̄, σ̄)where ϕ(·, ·) represents the normal density with mean µ̄ = ln St − ln St−τ −
√
(i)
(i)
(i)
1 (i)
(µ − 2 V1,t−τ − k(1)V2,t−τ )τ − Jd,t and standard deviation σ̄ = τ V1,t−τ ;• Otherwise, if Rt = ln St − ln St−τ < 0,(i)

– draw Ju,t from its transition law (15);(i)– draw Jd,t both from its transition law (16) and its conditional posterior dis√tribution Jd,t = ln St −ln St−τ −(µ− 12 V1,t−τ −k(1)V2,t−τ )τ −Ju,t − V1,t−τ wt ,which is normally distributed. Equal weights are attached to particles obtained from the transition law and the conditional posterior;– compute the particle weight by
(i)
(i)
(i)
(i)
(i)
(i)
p(ln St |Ju,t , Jd,t , V1,t−τ , V2,t−τ )p(Jd,t |V2,t−τ )
(i)
wt =
,
(i)
(i)
0.5p(Jd,t |V2,t−τ ) + 0.5ϕ(µ̄, σ̄)where ϕ(·, ·) represents the normal density with mean µ̄ = ln St − ln St−τ −
√
(i)
(i)
(i)
(i)
(µ − 21 V1,t−τ − k(1)V2,t−τ )τ − Ju,t and standard deviation σ̄ = τ V1,t−τ ;
(i)

(i)

• Normalize the weight: w̃t = wt /

∑M
j

(j)

wt ;

Step 3: Resample (Stratiﬁed Resampling)• Draw the new particle indexes by inverting the CDF of the multinomial charac(i)(i)

terized by w̃t at the stratiﬁed uniforms i+UM• reset the weight to 1/M ;31

where U (i) are iid uniforms;

Step 4: Update the diﬀusion variance and the jump intensity particles using (13) and√(14), where zt = ρwt + 1 − ρ2 z̃t with z̃ being an independent standard normal noise.
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35

36

28.4
82.6
84.4
89.3
89.8
89.8
94.4

Vol
-2.74
-3.91
-3.35
-2.74
5.57
-0.56
1.67

Period 2
ReturnJul 18, 2002
Jul 19, 2002
Jul 22, 2002
Jul 23, 2002
Jul 24, 2002
Jul 25, 2002
Jul 26, 2002

Date
28.0
31.0
32.5
33.5
38.0
38.0
38.1

Vol
-4.83
1.74
-4.83
4.24
3.95
-3.90
-1.58

Period 3
ReturnSep 15, 2008
Sep 16, 2008
Sep 17, 2008
Sep 18, 2008
Sep 19, 2008
Sep 22, 2008
Sep 23, 2008

Date
27.4
28.0
32.4
35.0
37.4
39.6
39.9

Vol

-4.90
-0.06
-6.90
4.63
-4.52
-4.53
0.52

Period 4
ReturnAug 4, 2011
Aug 5, 2011
Aug 8, 2011
Aug 9, 2011
Aug 10, 2011
Aug 11, 2011
Aug 12, 2011

Date

23.3
23.3
32.8
36.3
38.9
41.8
41.8

Vol

Note: The table reports the S&P 500 index returns and corresponding diﬀusive volatility, which is approximated by the standard deviation computed using
the previous 22-day returns at each time during the four turbulent periods including the Black Monday in 1987, the crash of the Internet Bubble in 2002, thebankruptcy of Lehman Brothers during the global ﬁnancial crisis in 2008, and the recent European debt crisis in 2011. Both returns and standard deviations
are in percentage.

-5.30
-22.9
5.20
8.71
4.00
-0.01
-8.64

Period 1
ReturnOct 16, 1987
Oct 19, 1987
Oct 20, 1987
Oct 21, 1987
Oct 22, 1987
Oct 23, 1987
Oct 26, 1987

Date

Table 1: S&P 500 Index Returns and Volatility in Four Turbulent PeriodsTable 2: Summary Statistics of S&P Index Returns
Returns

ACF

Mean

Std.

Skewness

Kurtosis

Min

Max

0.078

0.184

-1.19329.73-0.229

0.110

ρ1

ρ2

ρ3

ρ4

ρ5

ρ6

-0.028-0.044-0.004-0.015-0.0160.008

Note: The table presents descriptive statistics of data for model estimation and empirical analysis.
Data are from January 2, 1980 to October 31, 2011 in daily frequency. In total, there are 8,033
observations. Mean and standard deviation are annualized. ρ’s stand for autocorrelations.

37

Table 3: Log Bayes Factors at Final Time T
SE-M1

SE-M2

SE-M3

SE-M4

SE-M1

0.000

—

—

—

SE-M2

12.58

0.000

—

—

SE-M3

0.842

-11.74

0.000

—

SE-M4

13.33

0.758

12.49

0.000

Note: The table presents the log Bayes factor of the column model to the row model using all available
S&P 500 index return data from January 2, 1980 to October 31, 2011. The interpretation of values inthe table is given in Footnote 5.

38

39

0.056
0.019
0.089
0.022

0.054
0.017
0.090
0.023

0.064
0.024
0.107
0.024

Mean
0.05 Qtl
0.95 Qtl
Std
C. SE-M3

Mean
0.05 Qtl
0.95 Qtl
Std
D. SE-M4

Mean
0.05 Qtl
0.95 Qtl
Std-0.045-0.068-0.0250.015-0.068-0.097-0.0430.016-0.040-0.063-0.0190.013-0.072-0.102-0.038
0.019

ω

0.027
0.005
0.052
0.014

0.024
0.003
0.046
0.013

0.038
0.018
0.061
0.014

0.024
0.007
0.044
0.011

η

1.219
0.453
2.280
0.591

1.070
0.453
2.113
0.540

1.415
0.471
2.486
0.650

0.952
0.400
1.834
0.424

v

3.743
2.936
4.460
0.472

3.890
3.004
4.770
0.554

3.461
2.701
4.216
0.469

4.232
3.357
5.122
0.505

κ1

0.028
0.025
0.032
0.020

0.021
0.018
0.024
0.002

0.029
0.026
0.033
0.002

0.019
0.016
0.023
0.002

θ1

0.340
0.308
0.368
0.019

0.300
0.267
0.328
0.019

0.321
0.290
0.349
0.018

0.289
0.263
0.320
0.017

σ11

-0.615-0.666-0.5630.031-0.592-0.661-0.532-0.038-0.596-0.656-0.5430.034-0.596-0.671-0.531
0.038

ρ

—
—
—
—

0.466
0.333
0.597
0.087

—
—
—
—

0.515
0.352
0.658
0.088

σ12

—
—
—
—

—
—
—
—

15.40
2.427
33.42
9.929

17.46
1.871
43.36
13.48

κ2

—
—
—
—

—
—
—
—

12.43
2.707
24.01
6.473

20.67
8.253
32.43
7.305

σ2

Note: The table presents the parameter estimates of the four models considered using all available S&P 500 index return data from January 2, 1980 to October
31, 2011. Models are estimated using the Bayesian learning method discussed in Section 3. 5% quantile, mean, 95% quantile, and standard deviation of eachparameter estimate are reported.

0.050
0.010
0.107
0.027

Mean
0.05 Qtl
0.95 Qtl
Std
B. SE-M2

A. SE-M1µ

Table 4: Parameter Estimates at Final Time T40-0.072(-0.163)-0.295(-0.591)-0.079(-0.129)-0.149(-0.220)-0.024(-0.045)-0.110(-0.183)-0.034(-0.051)-0.079(-0.107)

SE-M1-0.070(-0.164)-0.302(-0.589)-0.069(-0.128)-0.149(-0.223)-0.024(-0.044)-0.076(-0.148)-0.033(-0.049)-0.076(-0.097)-0.067(-0.160)-0.160(-0.429)-0.068(-0.115)-0.105(-0.149)-0.024(-0.044)-0.110(-0.142)-0.033(-0.048)-0.073(-0.088)-0.067(-0.164)-0.148(-0.402)-0.060(-0.105)-0.106(-0.147)-0.024(-0.043)-0.076(-0.091)-0.033(-0.046)-0.076(-0.091)

SE-M4-0.255(-0.432)-0.834(-1.310)-0.197(-0.265)-0.342(-0.422)-0.077(-0.148)-0.315(-0.509)-0.101(-0.140)-0.214(-0.265)

SE-M1-0.255(-0.439)-0.789(-1.262)-0.202(-0.288)-0.311(-0.398)-0.076(-0.147)-0.223(-0.452)-0.098(-0.140)-0.207(-0.251)-0.254(-0.429)-0.569(-1.071)-0.185(-0.252)-0.283(-0.339)-0.076(-0.146)-0.290(-0.370)-0.097(-0.133)-0.198(-0.234)

One Week
SE-M2
SE-M3

-0.257(-0.437)-0.572(-0.975)-0.173(-0.239)-0.282(-0.338)-0.075(-0.146)-0.202(-0.285)-0.096(-0.128)-0.202(-0.239)

SE-M4

Note: The table presents 1% and 0.1% one-day and one-week Value-at-Risk (VaR) and conditional Value-at-Risk (CVaR, in brackets) numbers for the diﬀerent
models. Statistics on both the full sample and the post-Lehman bankruptcy period are reported.

Minimum, After Lehman Bankruptcy

Average, After Lehman Bankruptcy

Minimum, Full Sample

Average, Full Sample

B. 0.1% VaR (CVaR)

Minimum, After Lehman Bankruptcy

Average, After Lehman Bankruptcy

Minimum, Full Sample

Average, Full Sample

A. 1% VaR (CVaR)

One Day
SE-M2
SE-M3

Table 5: VaR and CVaR Measures Implied by the Models

Table 6: Impacts of Learning on Option Pricing

K/S
0.85
0.90
0.95
1.00
1.05
1.10
1.15

7 Days
1.113
1.077
1.023
0.997
0.998
1.059
1.168

SE-M1
30 Days 90 Days
1.093
1.042
1.043
1.020
1.004
1.010
1.008
1.015
1.049
1.035
1.091
1.068
1.143
1.104

250 Days
1.033
1.025
1.022
1.024
1.029
1.039
1.052

7 Days
0.999
0.978
0.957
0.999
1.007
1.088
1.478

SE-M3
30 Days 90 Days
1.118
1.058
1.060
1.031
1.012
1.016
1.010
1.018
1.046
1.036
1.084
1.065
1.137
1.097

250 Days
1.040
1.031
1.026
1.026
1.031
1.039
1.051

Note: The table presents the mean ratios of BS implied volatility with and without learning. Option
prices are computed using the Monte-Carl simulation method.

41

Table 7: Volatility Forecasting
SE-M1

SE-M2

SE-M3

SE-M4

2001.01-2011.10

6.549

6.846

6.565

6.773

2001.01-2007.12

5.129

5.380

5.159

5.366

2008.01-2011.10

8.825

9.202

8.826

9.049

Note: The table presents RMSEs in percentage of one-day-ahead daily volatility forecasts. True daily
volatility is approximated by the bipower variation based on 5-minute returns within a day. All
volatility estimates are annualized.

42Index Returns
0.15

0.05

−0.05

−0.15

−0.25
1980

1984

1988

1992

1996

2000

2004

2008

2012

2000

2004

2008

2012

Realized Volatility
1
0.8
0.6
0.4
0.2
0
1980

1984

1988

1992

1996

Figure 1: S&P 500 Index Returns and Realized Volatility
Note:The ﬁgure plot S&P 500 index returns (upper panel) ranging from January 2, 1980 to October
31, 2011, and realized volatility (lower panel) which is computed using the previous 22-day returns at
each time.

43

M1 vs. M2/M4

M3 vs. M2/M4

15

15
M1 vs. M2
M1 vs. M4

M3 vs. M2
M3 vs. M4

10

10

5

5

0

0

−5
1980 1984 1988 1992 1996 2000 2004 2008 2012−51980 1984 1988 1992 1996 2000 2004 2008 2012

M1 vs. M3

M2 vs. M4

1.5

4
3
2

0
1
0
−1.5
1980 1984 1988 1992 1996 2000 2004 2008 2012−1
1980 1984 1988 1992 1996 2000 2004 2008 2012

Figure 2: Sequential Model Comparison
Note: The Figure plots the sequential log Bayes factors for recursive model comparison and monitoring.
The dashed lines in each panel represents 0, 1.1, 2.3, and 3.4, respectively, which determine how strong
one model outperforms the other. The statistical interpretation of these values is given in Footnote 5.

44

ω

η

0.1

0.4
0.3
0.2
0.1
0
1980 1984 1988 1992 1996 2000 2004 2008 2012
κ1

0
−0.1
−0.2
1980 1984 1988 1992 1996 2000 2004 2008 2012
v
8
6
4
2
0
1980 1984 1988 1992 1996 2000 2004 2008 2012
θ1

15
10
5
0
1980 1984 1988 1992 1996 2000 2004 2008 2012
σ11

0.06

0.8
0.6
0.4
0.2
0
1980 1984 1988 1992 1996 2000 2004 2008 2012
σ12

0.04
0.02
0
1980 1984 1988 1992 1996 2000 2004 2008 2012
ρ
0.5

3

0

2

−0.5

1

−1
1980 1984 1988 1992 1996 2000 2004 2008 2012

01980 1984 1988 1992 1996 2000 2004 2008 2012

Figure 3: Parameter Learning in the SE-M1 Model: Jump size and Diﬀusion Volatility
Note: The ﬁgure presents the learning of the jump and diﬀusion volatility-related parameters in the
SE-M1 model using the S&P 500 index return starting from January 2, 1980. 5% quantile, mean, and
95% quantile are reported.

45

κ2

σ2

70

40

60
30

50
40

20
30
20

10

10
0
1980 1984 1988 1992 1996 2000 2004 2008 2012

01980 1984 1988 1992 1996 2000 2004 2008 2012

κ

σ

2

2

0.06

0.04

0.03
0.04
0.02
0.02
0.01

0

0

10

20

30

40

50

60

70

0

80

0

10

20

30

40

50

Figure 4: Parameter Learning in the SE-M1 Model: Jump Intensity
Note: The upper panels presents the learning of the jump intensity-related parameters in the SE-M1
model using the S&P 500 index return starting from January 2, 1980. 5% quantile, mean, and 95%
quantile are reported. The lower panels plot the kernel densities of the prior (solid line) and posterior(dashed line) distributions of each parameter.

46

ω

η

0.1

0.4
0.3
0.2
0.1
0
1980 1984 1988 1992 1996 2000 2004 2008 2012
κ1

0
−0.1
−0.2
1980 1984 1988 1992 1996 2000 2004 2008 2012
v
8
6
4
2
0
1980 1984 1988 1992 1996 2000 2004 2008 2012
θ1

15
10
5
0
1980 1984 1988 1992 1996 2000 2004 2008 2012
σ11

0.06

0.8
0.6
0.4
0.2
0
1980 1984 1988 1992 1996 2000 2004 2008 2012
σ12

0.04
0.02
0
1980 1984 1988 1992 1996 2000 2004 2008 2012
ρ
0.5

3

0

2

−0.5

1

−1
1980 1984 1988 1992 1996 2000 2004 2008 2012

01980 1984 1988 1992 1996 2000 2004 2008 2012

Figure 5: Parameter Learning in the SE-M3 Model
Note: The ﬁgure presents the learning of the jump and diﬀusion volatility-related parameters in the
SE-M3 model using the S&P 500 index return starting from January 2, 1980. 5% quantile, mean, and
95% quantile are reported.

47

M3: Diffusion Volatility
1
0.8
0.6
0.4
0.2
0
1980

1984

1988

1992

1996

2000

2004

2008

2012

2000

2004

2008

2012

2000

2004

2008

2012

M1: Diffusion Volatility
1
0.8
0.6
0.4
0.2
0
1980

1984

1988

1992

1996
M1:Jump Intensity

10
8
6
4
2
0
1980

1984

1988

1992

1996

Figure 6: Filtered Diﬀusion Volatility and Jump Intensity
Note:
The ﬁgure presents 5% quantile, mean, and 95% quantile of the ﬁltered diﬀusion volatility√( V1,t ) and the ﬁltered jump intensity (V2,t ) using the algorithm presented in Section 3.

48

M1: Positive Jumps

M3: Positive Jumps

0.04

0.04

0.03

0.03

0.02

0.02

0.01

0.01

0
1980 1984 1988 1992 1996 2000 2004 2008 2012

01980 1984 1988 1992 1996 2000 2004 2008 2012

M1: Negative Jumps

M3: Negative Jumps

0

0

−0.05

−0.05

−0.1

−0.1

−0.15

−0.15

−0.2

−0.2

−0.25
1980 1984 1988 1992 1996 2000 2004 2008 2012

−0.25
1980 1984 1988 1992 1996 2000 2004 2008 2012

Figure 7: Filtered Positive and Negative Jumps
Note: The ﬁgure presents the ﬁltered positive jumps (Jt+ ), and negative jumps (Jt− )using the algorithm
presented in Section 3.

49

M3: Diffusion Volatility
0.6
0.4
0.2
0
1980

1984

1988

1992

1996

2000

2004

2008

2012

2000

2004

2008

2012

2000

2004

2008

2012

M1: Diffusion Volatility
0.6
0.4
0.2
0
1980

1984

1988

1992

1996
M1: Jump Intensity

8
6
4
2
0
1980

1984

1988

1992

1996

Figure 8: Filtered Diﬀusion Volatility and Jump Intensity without LearningNote:
The ﬁgure presents 5% quantile, mean, and 95% quantile of the ﬁltered diﬀusion volatility√( V1,t ) and the ﬁltered jump intensity (V2,t ) when the full-sample estimates of the ﬁxed parameters
are plugged into the particle ﬁlter.

50M1: With Learning

M3: With Learning

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0
1980 1984 1988 1992 1996 2000 2004 2008 2012

01980 1984 1988 1992 1996 2000 2004 2008 2012

M1: Without Learning

M3: Without Learning

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0
1980 1984 1988 1992 1996 2000 2004 2008 2012

01980 1984 1988 1992 1996 2000 2004 2008 2012

M1: Contribution Difference

M3: Contribution Difference

0.6

0.6

0.4

0.4

0.2

0.2

0

0

−0.2
1980 1984 1988 1992 1996 2000 2004 2008 2012

−0.2
1980 1984 1988 1992 1996 2000 2004 2008 2012

Figure 9: Variance Decomposition: Jump Contribution
Note: The ﬁgure presents the variance contribution of the jump component to return variance. The
upper panels plot the jump contribution when learning is taken in account, whereas the middle panels
show its contribution when learning is ignored. The lower panels are the diﬀerence between upper andlower panels, revealing the impact of learning on variance decomposition.

51

One−Day 0.1% CVaR

One−Week 0.1% CVaR

4

5

4
3
3
2
2
1

1

1980 1984 1988 1992 1996 2000 2004 2008 20121980 1984 1988 1992 1996 2000 2004 2008 2012

One−Day 1% CVaR

One−Week 1% CVaR

2

3

1.5

2

1
1
1980 1984 1988 1992 1996 2000 2004 2008 20121980 1984 1988 1992 1996 2000 2004 2008 2012

Figure 10: Impacts of Learning on CVaR in the SE-M3 Model
Note: The ﬁgure reports the ratios of CVaR numbers with and without learning in the SE-M3 model.

52

Maturity: 7 Days

Maturity: 30 Days

0.4

0.26
M1
M2
M3
M4

0.35

0.24
0.22

0.3

0.2

0.25

0.18
0.16

0.2

0.14

0.15
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25

0.12
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25

Maturity: 90 Days

Maturity: 250 Days

0.24

0.24

0.22

0.22

0.2
0.2
0.18
0.18
0.16
0.14

0.16

0.12
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25

0.14
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25

Figure 11: Implied Volatility Smiles and Skews
Note: The ﬁgure presents the BS implied volatility for diﬀerent maturities and strikes under the fourmodels considered. We consider call options with maturity 7, 30, 90, and 250 days, and moneyness(K/S) 0.85, 0.90, 0.95, 1.00, 1.05, 1.10, 1.15, and 1.20. Option are priced using the Monte-Carlosimulation method.

53

Maturity: 30 Days
0.7

0.75

0.6

0.65

0.5

0.55

0.4

0.45

0.3

0.35
1980 1984 1988 1992 1996 2000 2004 2008 2012

0.2
1980 1984 1988 1992 1996 2000 2004 2008 2012

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0
1980 1984 1988 1992 1996 2000 2004 2008 2012

01980 1984 1988 1992 1996 2000 2004 2008 2012

Mn = 1.15

Mn = 1.00

Mn = 0.85

Maturity: 7 Days
0.85

0.65

0.65

0.45

0.45

0.25

0.25

0.05
1980 1984 1988 1992 1996 2000 2004 2008 2012

0.05
1980 1984 1988 1992 1996 2000 2004 2008 2012

Figure 12: Implied Volatility for Call Options with Maturity 7 and 30 Days
Note: The ﬁgure plots the time-series of implied volatility for call options with maturity 7 and 30
days from 1981 to 2011.10 computed from the SE-M1 model. Options are priced using the MonteCarlo simulation method. The solid line plots implied volatility when learning is taken into account,whereas the dashed (red) line plots implied volatility when learning is ignored. We consider options
with moneyness (K/S) equal to 0.85 in upper panels, 1.00 in middle panels, and 1.15 in lower panels.

54Mn = 0.85

1.1
1.05
1
0.95
0.9
1980 1984 1988 1992 1996 2000 2004 2008 2012

1.1

1.1

1.05

1.05

1

1

0.95
1980 1984 1988 1992 1996 2000 2004 2008 2012

0.95
1980 1984 1988 1992 1996 2000 2004 2008 2012

2.5

1.15

2

1.1

Mn = 1.15

Mn = 1.00

Maturity: 30 Days

Maturity: 7 Days
1.5
1.4
1.3
1.2
1.1
1
0.9
1980 1984 1988 1992 1996 2000 2004 2008 2012

1.05

1.5

1

1

0.95

0.5
1980 1984 1988 1992 1996 2000 2004 2008 2012

0.9
1980 1984 1988 1992 1996 2000 2004 2008 2012

Figure 13: Implied Volatility Ratios of the SE-M1 Model and the SE-M3 Model
Note: The ﬁgure plots the time-series ratios of implied volatility for call options with maturity 7 and
30 days between the SE-M1 and SE-M3 models from 1981 to 2011.10. Options are priced using theMonte-Carlo simulation method. The solid line plots implied volatility ratios when learning is taken
into account, whereas the dashed (red) line plots implied volatility ratios when learning is ignored. We
consider options with moneyness (K/S) equal to 0.85 in upper panels, 1.00 in middle panels, and 1.15
in lower panels.

55