tail and volatility indices from option prices...vix is an accurate measure of quadratic variation....
TRANSCRIPT
-
Tail and Volatility Indices from Option Prices
Jian Du and Nikunj Kapadia 1
Current Version: May 2014
Previous version: August 2012
1University of Massachusetts, Amherst. This paper has benefited from suggestions and conversa-tions with Sanjiv Das, Peter Carr, Rama Cont, Dobrislav Dobrev, Paul Glasserman, Scott Murray,Sanjay Nawalkha, Emil Siriwardane, Liuren Wu, Hao Zhou, and workshop and conference discussantsand participants at Baruch College, ITAM (Mexico City), Federal Reserve Board, Morgan Stanley(Mumbai and New York), Yale IFLIP, Villanova University, China International Conference in Finance,Singapore International Conference in Finance, and the European Finance Association Conference inCambridge. All errors are our own. We thank the Option Industry Council for support with optiondata. Please address correspondence to Nikunj Kapadia, Isenberg School of Management, 121 Pres-idents Drive, University of Massachusetts, Amherst, MA 01003. E-mail: [email protected]: http://people.umass.edu/nkapadia. Phone: 413-545-5643.
-
Abstract
Both volatility and the tail of stock return distributions are impacted by discontinuities or
large jumps in the stock price process. We show that the Bakshi, Kapadia, and Madan (2003)
measure of the variance is more accurate than the Chicago Board Options Exchange’s VIX
index in the presence of jumps. By measuring the jump-induced bias in the VIX, we construct
a model-free jump and tail index using 30-day index options. Our jump and tail index is a
specific portfolio of risk-reversals, and measures the time variation in the intensity of return
jumps. The tail index predicts one-year excess returns with an out-of-sample R2 of 25% over
the post-crisis period of 9/1998 to 12/2012.
JEL classification: G1, G12, G13
-
1 Introduction
Understanding time variation in volatility is important in asset pricing since it impacts the
pricing of both equities and options. It is also of interest to understand whether time variation
in tail risk —the possibility of an extreme return from a discontinuity or large jump in the
stock price process– should be considered an additional channel of risk.1 However, jumps in
the stock price process not only determine the tail of the distribution but also impact stock
return variability. Before one can determine whether there are potentially distinct roles for
stock return variability and tail risk, respectively, it is essential to differentiate one from the
other. In this paper, we address this issue by constructing model-free volatility and tail indices
from option prices that allow researchers to distinguish between the two channels.
Volatility and jump/tail indices constructed from option prices already exist in the liter-
ature. The most widely used option-based measure of stock return variability is the Chicago
Board Options Exchange’s VIX. However, the VIX is not model free and is biased in the pres-
ence of discontinuities, which makes it difficult to distinguish between volatility and tail risk.
As formalized by Carr and Wu (2003), a model-free measure of jump risk can, in principle,
be constructed from the pricing of extreme returns using close-to-maturity, deep out-of-the-
money (OTM) options. But because short-dated deep OTM options are illiquid, in practice,
the theory needs to be combined with extreme value statistics. Bollerslev and Todorov (2011)
construct an “investor [jump] fear index” using this approach.
Our first contribution is to show that the Bakshi-Kapadia-Madan (2003) (BKM, hereafter),
measure of the variance of the holding period return is more accurate than the VIX for mea-
suring quadratic variation —a measure of stock return variability—when there is significant
jump risk, as, for example, for the entire class of Lévy models (e.g., Merton, 1976). Moreover,
if the stock price process has no discontinuities (e.g., Hull and White, 1987; Heston, 1993), it
is about as accurate as the VIX when measured from short-maturity (e.g., 30-days) options
even though the VIX is designed specifically to measure the quadratic variation of a jump-free
process.2
1A number of papers have related jump or tail risk to asset risk premia. For example, Naik and Lee (1990),Longstaff and Piazzesi (2004), and Liu, Pan, and Wang (2005) model jump risk premia in equity prices, whileGabaix (2012) and Wachter (2012), extending initial work of Rietz (1988) and Barro (2006), relate equity riskpremia to time-varying consumption disaster risk.
2The quadratic variation of a jump-free process is referred to as the “integrated variance”. VIX was con-structed to measure the integrated variance using the log-contract based on the analysis of Carr and Madan(1998), Demeterfi, Derman, Kamal, and Zou (1999a, 1999b), and Britten-Jones and Neuberger (2000) and ear-lier work by Neuberger (1994) and Dupire (1996). Jiang and Tian (2005) argue in their Proposition 1 that theVIX is an accurate measure of quadratic variation. Carr and Wu (2009) show that the VIX is biased, but usenumerical simulations to argue that the bias is negligible. Consequently, a number of recent papers have used
1
-
Next, we construct a jump and tail index based on the fact that the accuracy of the VIX
deteriorates rapidly when a larger proportion of stock return variability is determined by fears
of jumps. Building on analysis by Carr and Wu (2009), we show that the bias in the VIX is
proportional to the jump intensity. By comparing the integrated variance to the BKM variance,
we can measure the jump-induced bias and construct a model-free tail index. In contrast to the
existing literature, our index can be constructed from standard 30-day maturity index options.
Moreover, instead of relying only on deep OTM options, the index uses the entire continuum
of strikes.
Technically, our jump and tail index (“JTIX”) measures time variation in the jump inten-
sity process. It is statistically distinguished from the quadratic variation because the index
comprises of the third and higher-order moments of the jump distribution. Economically, the
difference between these two measures of stock return variability maps into a short position in
an option portfolio of risk reversals. This option portfolio is the hedge that a dealer in variance
swaps should engage in to immunize a short swap position from risks of discontinuities. When
downside jumps dominate, the price of the risk reversal portfolio is negative.
Based on the theory, we construct the volatility and tail indices from Standard & Poor’s
500 index option data over 1996–2012. We document that the time variation in tail risk is
driven primarily by downside jump fears. At the peak of the financial crisis, fears of jumps in
the market were an extraordinary 50-fold those of the median month. The tail index allows us
to precisely compare the severity of crises over our sample period and reveals that the Long
Term Capital Management (LTCM) crisis had more jump risk than the Asian currency crisis,
the 2001–2002 recession, or the prelude to the Iraq war. On the other hand, the peak of the
Eurozone sovereign debt crisis in August and September of 2011 had about the same tail risk
as the LTCM crisis.
We use the volatility and tail indices to examine the two potential channels of jump risk.
To do so, we employ predictability regressions following the setups of Bollerslev, Tauchen,
and Zhou (2009) (BTZ hereafter),3 and consider both in-sample and out of sample results as
suggested by Welch and Goyal (2006).
BTZ demonstrate that the variance spread between the VIX and the historical variance
–also referred to as the variance risk premium– predicts index returns. We first use their
the VIX as a model-free measure of quadratic variation (e.g., Bollerslev, Gibson, and Zhou, 2011; Drechsler andYaron, 2011; Wu, 2011).
3There is a long-standing tradition of using predictability regressions (e.g., see Cochrane, 2007, and referencestherein). Papers that are closely related to our motivation and also use predictability regressions are Bakshi,Panayotov and Skoulakis (2011), Bollerslev, Gibson and Zhou (2011), Drechsler and Yaron (2011), and Kellyand Jiang (2013).
2
-
framework to examine the economic impact of the jump-induced bias in the VIX. We document
that in the financial crisis, the VIX underestimates the true stock return variability by as much
as 6.2% (annualized volatility points) and underestimates the variance risk premium by 30%.
Using the VIX to construct the spread results in anomalous findings, including one where the
predicted return in the financial crisis of 2007–2009 is lower than in the previous recession of
2001–2002. The use of the BKM variance eliminates these anomalies. Using the BKM variance
instead of the VIX refines and improves the BTZ results, and underscores the importance of
correctly accounting for jumps when estimating stock return variability.4
Our primary empirical focus is on ascertaining whether the jump-induced tail should be
considered an additional channel of risk. In in-sample results over our entire sample period,
we find that the tail index, JTIX, significantly predicts S&P 500 excess returns over horizons
of one and two years, after controlling for the variance spread. The key evidence of the
importance of tail risk is in out of sample (“OOS”) results. Over the entire sample period,
on a univariate basis, JTIX predicts one-year and two-year excess returns with OOS R2s of
3.9% and 7.4%, respectively. Importantly, in the aftermath of the bankruptcy of Lehman
Brothers when tail risk is a likely driver of asset prices, the index achieves OOS R2s of 25%
and 39.9% for the one-year and two-year horizons, respectively. Overall, the volatility and tail
indices recommended in this paper, dominate the variance risk premium based on VIX, as well
as fourteen traditional predictors used in previous literature. The sum total of the evidence
indicates that fears of jumps operate through two distinct channels of stock return volatility
and tail risk, respectively.
In related literature, Bakshi, Cao, and Chen (1997), Bates (2000), Pan (2002), and Broadie,
Chernov, and Johannes (2007), among others, demonstrate the significance of jump risk using
parametric option pricing models. Naik and Lee (1990), Longstaff and Piazessi (2004) and Liu,
Pan, and Wang (2005) develop equilibrium option pricing models that specifically focus on
jump risk. Broadie and Jain (2008), Cont and Kokholm (2010), and Carr, Lee, and Wu (2011)
observe, as we do, that jumps bias the VIX. Bakshi and Madan (2006) note the importance of
downside risk in determining volatility spreads.
Empirical evidence that jump risk contributes significantly to the variability of observed
stock log returns under the physical measure has been found by Aı̈t-Sahalia and Jacod (2008,
2009a, 2009b, 2010), Lee and Mykland (2008), and Lee and Hannig (2010), amongst others.
Bakshi, Madan, and Panayotov (2010) use a jump model to model tail risk in index returns4This caution also applies to the power claim of Carr and Lee (2008), applied in Bakshi, Panayotov and
Skoulakis (BPS) (2010); see our Appendix A.
3
-
and find that downside jumps dominate upside jumps. Kelly and Jiang (2013) demonstrate
that tail-related information can be inferred from the cross sections of returns. We add to this
literature by providing, under a risk-neutral measure, a model-free methodology for measuring
quadratic variation and time variation in jump intensity.
Finally, an important segment of the literature focuses on consumption disasters. This
literature, starting with Rietz (1988) and further elaborated by Barro (2006) and Gabaix
(2012), focuses on how infrequent consumption disasters affect asset risk premiums, potentially
providing a channel through which returns are predictable. By assuming that dividends and
consumption are driven by the same shock and by modeling tail risk as a jump process, Wachter
(2012) shows that the equity premium will depend on time-varying jump risk. Although a
precise correspondence between consumption tail risk and the tail risk inferred from index
options is yet incomplete (see Backus, Chernov and Martin, 2011, for a recent effort), our
findings regarding the importance of downside jump risk broadly support this literature.
The remainder of the paper is organized as follows. Section 2 illustrates our approach using
the Merton jump diffusion model. Section 3 collects our primary theoretical results. Section 4
describes the time-variation in the indices. Sections 5 present our empirical results. The last
section concludes the study.
2 An illustration using the Merton jump diffusion model
To provide a guide to our analysis of the next section, we provide an illustration using the
familiar jump-diffusion model of Merton (1976) (although the section can skipped entirely
without loss of information). Let the stock price ST at time T be specified by a jump diffusion
model under the risk-neutral measure Q,
ST = S0 +∫ T
0(r − λµJ)St− dt+
∫ T0σSt− dWt +
∫ T0
∫R0St−(e
x − 1)µ[dx, dt], (1)
where r is the constant risk-free rate, σ is the volatility, Wt is standard Brownian motion, R0
is the real line excluding zero, and µ[dx, dt] is the Poisson random measure for the compound
Poisson process with compensator equal to λ 1√2πσ2J
e−12(x−α)2 , with λ as the jump intensity.
4
-
From Ito’s lemma, the log of the stock price is
lnST = lnS0 +∫ T
0
1St−
dSt −∫ T
0
12σ2dt+
∫ T0
∫R0
(1 + x− ex) µ[dx, dt], (2)
= lnS0 +∫ T
0(r − 1
2σ2 − λµJ) dt+
∫ T0σ dWt +
∫ T0
∫R0
xµ[dx, dt]. (3)
Denote the quadratic variation over the period [0, T ] as [lnS, lnS]T . The quadratic variation
of the jump diffusion process is (e.g., Cont and Tankov, 2003)
[lnS, lnS]T =∫ T
0σ2dt+
∫ T0
∫R0x2µ[dx, dt]. (4)
First, we derive a relation between EQ0 [lnS, lnS]T and varQ0 (lnST /S0). From Ito’s lemma, the
square of the log return, (lnSt/S0)2 is,
(lnST /S0)2 =
∫ T0
2 ln(St−/S0) d lnSt + [lnS, lnS]T . (5)
The expected value of the stochastic integral in equation (5) is (see Appendix A),
EQ0
∫ T0
2 ln(St−/S0) d lnSt =(
(r − 12σ2 − λµJ) + λα
)2T 2,
= (EQ0 ln(St−/S0))2. (6)
Substituting equation (6) in equation (5), and rearranging,
EQ0 [lnS, lnS]T = EQ0 (lnST /S0)
2 −(
EQ0 (lnST /S0))2,
= varQ0 (lnST /S0) . (7)
Equation (7) states the quadratic variation is equal to the variance of the holding period
return for the Merton model. From BKM, varQ0 (lnST /S0) can be estimated model-free from
option prices. Therefore, using the BKM variance, we can measure the quadratic variation in
a model-free manner.
Next, with some abuse of notation, denote EQ0 [lnS, lnS]cT as the estimate of the expectation
of integrated variance, EQ0∫ T0 σ
2dt, under the assumption that the stock return process has nodiscontinuities. Carr and Madan (1998), Demeterfi, Derman, Kamal and Zou (1999a, 1999b),
5
-
and Britten-Jones and Neuberger (2000) demonstrate that, for a purely continuous process,
EQ0 [lnS, lnS]cT = 2 E
Q0
[∫ T0
1StdSt − lnST /S0
]. (8)
The RHS of equation (8) can be replicated using options and, therefore, we can precisely
estimate the integrated variance of a process with no discontinuities.
But, in the presence of discontinuities, from equation (2),
2EQ0
[∫ T0
1St−
dSt − lnST /S0
]= EQ0
∫ T0
σ2dt− 2EQ0∫ T
0
∫R0
(1 + x− ex)µ[dx, dt],
= EQ0 [lnS, lnS]T − 2EQ0
∫ T0
∫R0
(1 + x+
x2
2− ex
)µ[dx, dt]. (9)
Thus, EQ0 [lnS, lnS]cT ≡ 2 E
Q0
[∫ T0
1St−
dSt − lnST /S0]
is a biased estimator of both the quadratic
variation and the integrated variance.5
But, for the Merton model, we can measure the quadratic variation precisely by the BKM
variance. Replacing EQ0 [lnS, lnS]T by varQ0 (lnST /S0) in equation (9), we obtain,
varQ0 (lnST /S0)− 2 EQ0
[∫ T0
1St−
dSt − lnST /S0]
= 2EQ0
∫ T0
∫R0
(1 + x+
x2
2− ex
)µ[dx, dt].
(10)
Equation (10) states that the difference between the variance of the holding period return
and the integrated variance measure is determined solely by discontinuities in the stock price
process. We use the time-variation in this difference to construct a model-free jump and tail
index.
3 Measuring stock return variability and tail risk
3.1 Quadratic variation and variance of the holding period return
Let the log stock price lnSt at time t, t ≥ 0, be a semimartingale defined over a filteredprobability space (Ω,F , {Ft},Q), with S0 = 1. Denote the quadratic variation over a horizon
5Our analysis differs from that of Carr and Madan (1998), Demeterfi, Derman, Kamal, and Zou (1999a,1999b), and Britten-Jones and Neuberger (2000) because we consider a contract that pays the square of the logreturn (equation 5), as opposed to a contract that pays the log return (equation 8). Earlier literature focusedon the log-contract because, under the assumption of no discontinuities, the analysis indicated how a varianceswap could be replicated. Unfortunately, in the presence of discontinuities, the log-contract does not hedge norprice the variance swap.
6
-
T > 0 as [lnS, lnS]T and the variance of the holding period return as varQ0 (lnST /S0). Our
first result characterizes the relation between these two measures of stock return variability.
Proposition 1 Let EQ0 [lnS, lnS]T and varQ0 (lnST /S0) be the expected quadratic variation and
variance of the holding period return, respectively, over a horizon T
-
Example 1 Merton (1976) model:
Continuing the illustration of Section 2, observe that the log return for the Merton model can
be decomposed as
lnSt/S0 =(
(r − 12σ2) + λα
)t+Mt,
= At +Mt, (14)
where Mt is the sum of a continuous and pure jump martingale. The drift At is deterministic
for this model and therefore the quadratic variation and variance are equivalent. We now arrive
at the conclusion without explicitly evaluating the stochastic integral in equation (6).
The drift of the log return is stochastic in a stochastic volatility model such as Heston’s
(1993). Because the variance of the holding period return accounts for the stochasticity of the
drift, it differs from the quadratic variation by D(T ). The second part of the proposition shows
that the annualized D(T ) is O(T ) for small T for stochastic volatility models. In the special
case when the volatility process is uncorrelated with the log return process, as in Hull and
White (1987), D(T ) reduces with T even faster. Indeed, for ρ = 0, we can further characterize
D(T ) (see Appendix A) as
D(T ) =14
varQ0
∫ T0σ2t dt. (15)
When an analytical solution is available for varQ0∫ T0 σ
2t dt, D(T ) can be estimated precisely.
Example 2 Heston (1993) model:
For the Heston model, dσ2t = κ(θ − σ2t ) + ησtdW2,t, Bollerslev and Zhou (2002) demonstratein equation (A.5) in their appendix that varQ0
∫ T0 σ
2t dt = A(T )σ
20 +B(T ), where
A(T ) =η2
κ2
(1κ− 2e−κTT − 1
κe−2κT
), (16)
B(T ) =η2
κ2
(θT(1 + 2e−κT
)+
θ
2κ(e−κT + 5
) (e−κT − 1
)). (17)
Expanding around T = 0 and simplifying yields
varQ0
∫ T0σ2t dt = η
2(θ − σ20)T 3 +O(T 4).
such as the variance gamma process of Madan and Seneta (1990), the normal inverse Gaussian process ofBarndorff-Nielson (1998), and the finite-moment stable process of Carr, Geman, Madan, and Yor (2002). Forthese models, by the Lévy-Khintchine theorem, the characteristic function is determined by (γ, σ2, ν), so thatE0e
iu lnSt/S0 = etψ(u), where ψ(u) = iγu− 12σ2u2 +
RR0
`eiux − 1− iux1[|x|
-
When ρ = 0 and σ20 = θ,1TD(T ) = O(T
3).
The example illustrates that, depending on parameter values, D(T ) may reduce with T even
faster than noted in Proposition 1. Below, we will use numerical simulations to provide precise
estimates of D(T ) for the commonly used stochastic volatility and jump model of Bates (2000).
In their Proposition 1, BKM demonstrate that the variance of the holding period return
can be estimated model free from option prices. Thus, using the BKM methodology, we can
precisely estimate the quadratic variation for Lévy processes and, to a very good approximation,
using short-maturity options, for models with stochastic volatility.
3.2 Quadratic variation, integrated variance, and jump risk
To derive a measure of jump risk, we put more structure on the stock return process. Let the
log of the stock price be a general diffusion with jumps:
lnST = lnS0 +∫ T
0(at −
12σ2t ) dt+
∫ T0σt dWt +
∫ T0
∫R0
xµ[dx, dt], (18)
where the time variation in σt is left unspecified while at is restricted to ensure that the
discounted stock price is a martingale. Let the Poisson random measure have an intensity
measure µ[dx, dt] ≡ νt[dx]dt. Given our focus on tail risk and because we have already includeda diffusion component, we assume that
∫R0ν[dx] < ∞ and that all moments
∫R0xnν[dx],
n = 1, 2, ..., exist.
As in Carr and Madan (1998), Demeterfi, Derman, Kamal and Zou (1999a, 1999b), and
Britten-Jones and Neuberger (2000), we define EQ0 [lnS, lnS]cT as the (VIX) measure of inte-
grated variance under the assumption that the process is continuous:
EQ0 [lnS, lnS]cT ≡ E
Q0
[2(∫ T
0
dStSt− ln ST
S0
)]. (19)
In the absence of discontinuities in the stock return process, EQ0[2(∫ T
0dStSt− ln STS0
)]= EQ0
∫ T0 σ
2t dt.
If there are discontinuities, Carr and Wu (2009) show in their Proposition 1 that the difference
between EQ0 [lnS, lnS]T and EQ0 [lnS, lnS]
cT is determined by the jump distribution,
EQ0 [lnS, lnS]T − EQ0 [lnS, lnS]
cT = 2E
Q0
∫ T0
∫R0ψ(x)µ[dx, dt], (20)
where ψ(x) = 1 + x+ 12x2 − ex.
9
-
From iterated expectations, the right-hand side of equation (20) is determined by the
compensator of the jump process. Tail risk is also determined by the expectation of jump
intensity because in the presence of jumps, the tail of the stock return distribution is determined
by (large) jumps.
Proposition 2 Let the log price process be specified as in equation (18). Let the intensity
measure νt[dx] be of the form
νt[dx] = λtf(x)dx,
where λt is the jump arrival intensity of a jump of any size with jump size distribution f(x).
Then,
EQ0 [lnS, lnS]T − EQ0 [lnS, lnS]
cT = 2 Ψ(f(x)) Λ0,T , (21)
where
Λ0,T = EQ0
∫ T0λtdt, (22)
with Ψ(f(x)) =∫R0 ψ(x)f(x)dx and ψ(x) = 1 + x+
12x
2 − ex.
Proof: See Appendix A.
Proposition 2 states that the difference between quadratic variation and the measure of
integrated variance over an interval T is proportional to the expectation of the number of
jumps over that interval. Because Ψ(·) is determined by higher-order moments (n ≥ 3) of thejump distribution, it is statistically distinct from the quadratic variation.
Proposition 2 can be generalized to allow for upside and downside jumps. Defining the in-
tensity measures for upside and downside jumps as ν+[x] = λtf+(x)dx and ν−[x] = λtf−(x)dx,
we now obtain
EQ0 [lnS, lnS]T − EQ0 [lnS, lnS]
cT = 2
(Ψ+ + Ψ−
)Λ0,T , (23)
where Ψ+ ≡∫R+ ψ(x)f
+(x)dx and Ψ− ≡∫R− ψ(x)f
−(x)dx. Under this specification, the dom-
inance of downside versus upside jumps determines the sign of EQ0 [lnS, lnS]T −EQ0 [lnS, lnS]
cT .
To illustrate, consider the model of Bakshi and Wu (2010) with a double exponential jump
size distribution (Kou, 2002):
f+(x) =
{e−β+|x|, x > 0,
0, x < 0;(24)
f−(x) =
{0, x > 0,
e−β−|x|, x < 0;(25)
10
-
We can explicitly evaluate Ψ(f+(x)) and Ψ(f−(x)) to observe that EQ0 [lnS, lnS]T−EQ0 [lnS, lnS]
cT
is positive (negative) when β+ >> β− (β− >> β+). Intuitively, from its definition in Propo-
sition 2, the magnitude of Ψ is determined to first order by the negative of the third moment
of the jump size distribution. When downside jumps dominate, the third moment is negative
and EQ0 [lnS, lnS]T − EQ0 [lnS, lnS]
cT > 0; that is, the integrated variance underestimates the
true quadratic variation.
3.3 Numerical analysis for the stochastic volatility and jump (SVJ) model
From Proposition 1, although varQ0 (lnST /S0) measures quadratic variation precisely for Lévy
processes, its accuracy for stochastic volatility models depends on the maturity of options
chosen to estimate the variance. In developing the tail index, we are especially interested in
using options of maturity 30 days. Therefore, before proceeding further, we conduct numerical
simulations to compare the accuracy with which the BKM variance and integrated variance
(measured by the VIX), respectively, estimate quadratic variation.
Let V be the annualized variance, V = 1T varQ0 (lnST /S0) =
1T
(EQ0 (lnST /S0)
2 − µ20,T)
,
where µt,T = EQ0 lnST /St. From BKM, the price of the variance contract is estimated from
OTM calls and puts of maturity T . Denoting C(St;K,T ) and P (St;K,T ) as the call and putof strike K and T as the remaining time to expiration, BKM demonstrate that
e−rTV =1T
[∫K>S0
2(1− ln(K/S0))K2
C(S0;K,T )dK +∫KS0
1K2
C(S0;K,T )dK +∫K
-
of the jump size distribution (α), we calibrate the parameters to those empirically estimated
by Pan (2002). The initial volatility and the mean of the jump size distribution are adjusted
to vary the contribution of the variability from jumps to the total variance from zero to 90%.
Panel A provides the comparison for the Merton model. For the Merton jump diffusion
model, V measures the quadratic variation perfectly, but IV does so with error because of jumprisk. The error increases as jumps contribute a larger fraction to the total variance.
Panel B provides numerical results for the SVJ model. Because the drift is stochastic, Vmeasures the quadratic variation with a (small) error. The maximum error is when there are
no jumps (i.e., for the Heston (1993) model), and is only 0.61% in relative terms. That is, while
the true√
[lnS, lnS]T is 20%, the BKM volatility estimate is 20.05%. As the contribution of
jumps increases, V is more accurate. In contrast, IV becomes less accurate as the contributionof jumps to return variability increases. When the contribution of jumps to the variance is
below 20%, the relative error is less than 1% of the variance, but it increases manifold as jump
risk increases. For example, when jumps contribute over 70% to the total variance,√
IV is19% instead of the correct 20%, an economically significance bias.
Proposition 1 notes that an increase in the magnitude of ρ makes V less accurate. To check,we consider a correlation of ρ = −0.90. Even for this extreme case, the error is negligible. Atworst,
√V gives an estimate of 20.09% instead of the correct 20%. The bias is well within the
bounds of accuracy with which we can estimate risk-neutral densities using option prices.
In summary, in the presence of jumps, the BKM variance measures quadratic variation
more accurately than the integrated variance since it correctly accounts for jumps, while the
stochasticity of the drift adds negligible error at short maturities. When jumps contribute
less than 20% to the variance, both V and IV accurately estimate the quadratic variation.8
However, with increasing jump risk, IV gets progressively less accurate.
3.4 Formalizing the jump and tail index, JTIX
From Propositions 1 and 2, the difference between the variance and the integrated variance
is the sum of two components, the first determined by the stochasticity of the drift and the8Here, our analysis concurs with that of Jiang and Tian (2005) and Carr and Wu (2009): The parameteri-
zations chosen in their numerical experiments correspond to (low) jump contributions of 14% and 11.7% to thequadratic variation, respectively. When jump risk is of low economic importance, we can accurately measurequadratic variation using either the VIX or the BKM variance.
12
-
second determined by jump risk. On an annualized basis,
1T
(varQ0 (lnST /S0)− E
Q0 [lnS, lnS]
cT
)=
1T
(EQ0
∫ T0
lnST /S0 d lnSt − EQ0 (lnST /S0)2
)+
2T
ΨΛ0,T .
(28)
Proposition 1 combined with our simulation evidence indicates that the impact of the stochas-
ticity of the drift (the first term) can be neglected for standard jump diffusion models for
short-maturity options, that is,
1T
(varQ0 (lnST /S0)− E
Q0 [lnS, lnS]
cT
)≈ 2T
ΨΛ0,T . (29)
Equation (29) states that the time variation in the difference between the variance and in-
tegrated variance is determined by the time variation in jump intensity. Our jump and tail
index, JTIX, is motivated by equation (29). Defining ĴTIX = V− IV using equations (26) and(27), we obtain
ĴTIX = V−IV = 2TerT
[∫KS0
ln(K/S0)K2
C(S0;K,T ) dK]+ᾱ,
(30)
where ᾱ = 2T (erT − 1− rT )− 1T µ
20,T . Because ᾱ is small, ĴTIX is primarily determined by the
OTM option portfolio represented by the first two terms of equation (30). Economically, the
option portfolio comprising the tail index is a short position in a risk reversal and the hedge
that a dealer in (short) variance swaps would buy to protect against the risk of discontinuities.
Our jump and tail index, JTIX, is constructed as the 22-day moving average
JTIXt =122
22∑i=1
ĴTIXt−i+1. (31)
As in Bollerslev and Todorov (2011), we use a 22-day moving average to reduce estimation
errors. To validate and investigate the economic significance of JTIX, we pose the following
questions.
1. Is JTIX = 0 (V ≈ IV)?The first question we consider is whether jump risk is significant. This question is equivalent
to asking whether the VIX is an accurate estimator of quadratic variation. When jump risk is
negligible, V and IV are approximately equal and JTIX is close to zero.
2. Is there time variation in JTIX and is the time variation related to jump risk?
If the difference between V and IV is related to λt, then it should increase in periods when
13
-
fears of jumps are higher. If downside jumps are more prevalent than upside jumps, we expect
the time variation in the jump and tail index to be countercyclical and JTIX to be especially
high in times of severe market stress.
3. What are the channels for jump risk?
Our primary question centers on whether jump risk is important for predicting market returns.
Assuming it is, we are interested in understanding the channel through which jump risk is sig-
nificant. Does jump risk impact market returns through the contribution of jumps to volatility
or through the impact of jumps on the tail of the distribution?
In investigating these questions, we follow BTZ, who demonstrate that the variance spread,
VS, defined as the difference between the integrated variance and the past realized variance,VSt = IVt − RVt−1, predicts index returns.9 First, we investigate whether the contributionof jumps to stock return variability is economically important. If it is important to correctly
account for jump risk, then V − RV will be more significant than IV − RV in predictingindex returns. Second, we consider whether tail risk is important in addition to stock return
variability for predicting index returns. If tail risk is a separate channel, then JTIX should also
be significant. Together, the exercises allow us to understand whether jump risk is significant
through one or both channels of volatility and tail risk.
4 The jump and tail index
To construct the volatility indices and the tail index, we use option prices on the S&P 500
(SPX) over the sample period of January 1996 to December 2012. The options data, dividend
yield for the index, and zero coupon yield are from OptionMetrics. We clean the data with
the usual filters, the details of which are provided in Appendix B.
We construct the volatility and tail indices as follows. First, we obtain option prices across a
continuum of strikes. Following Jiang and Tian (2005) and Carr and Wu (2009), we interpolate
the Black–Scholes implied volatility across the range of observed strikes using a cubic spline,
assuming the smile to be flat beyond the observed range of strikes. Next, we linearly interpolate
the smiles of the two near-month maturities to construct a 30-day implied volatility curve for
each day. The interpolated implied volatility curve is converted back to option prices using the
Black–Scholes formula. The daily volatility indices, V and IV, are computed using the BKM9BTZ suggest that the variance spread is important in predicting index returns because it measures the
variance risk premium. Technically, the variance risk premium (Bakshi and Kapadia, 2003; Carr and Wu, 2009)is the negative of the difference between the risk-neutral variance and the variance realized over the remainingmaturity of the option, that is, −(Vt − RVt) or -(IVt − RVt). To avoid confusion, we call it a variance spread.
14
-
variance and VIX formulas, equations (26) and (27), respectively. Finally, the tail index JTIX
is constructed as the 22-day moving average of V− IV, as noted in equation (31).
Figure 1 compares the two constant maturity volatility indices, the holding period volatility,
and the integrated volatility by plotting√
V−√
IV at a daily frequency. The plot demonstratesthat V is always greater than IV. On average, over this period,
√V is 23.4%, which is 0.5%
(volatility points) higher than√
IV on an annualized basis. The results are consistent with ourexpectation that when downside jumps dominate, the integrated variance will underestimate
the quadratic variation.
Figure 2 plots the tail indices, JTIX, and JTIX/V. The plot of JTIX shows that jumprisk is intimately associated with times of crisis and economic downturns, with a manifold
increase in jump risk. The most prominent spike occurs in October of 2008, with a more than
50-fold increase in jump risk compared with that of the median day over the sample period.
Additional spikes occur in the period leading up to the Iraq war, the dot-com bust of 2001, the
Russian bond and LTCM crises in 1998, and the Asian currency crisis in 1997. Unlike the two
volatility indices (especially the integrated variance measure), the jump and tail index clearly
differentiates between the 2001 recession and the LTCM crisis: The LTCM crisis has twice
the tail risk as the aftermath of the dot-com bubble. The tail index also captures the sharp
increase in tail risk in November of 1997, coincident with the crash in the Seoul stock market
during the Asian currency crisis. Interestingly, excepting the Iraq war, all the sharp increases
in tail risk correspond to the handful of financial crises that occurred in our sample period.
Extending the popular analogy of the Chicago Board Options Exchange VIX to a fear index,
JTIX can be viewed as an index of extreme fear.
To sharpen the distinction between the holding period variance and the integrated variance,
we also plot JTIX/V. The plot indicates that IV underestimates market variance by over 15%at the peak of the recent financial crisis. If we use the numerical analysis of Table 1 as a guide,
this magnitude of underestimation suggests that jump risk may comprise over 80% of stock
return variability at the peak of the crisis.
The discussion in Section 3.2 noted that if the intensity measure differs for downside and
upside jumps, ν+[x] = λtf+(x)dx and ν−[x] = λtf−(x)dx, then an increase in λt skews the
jump size distribution further to the left or right, respectively, depending on whether downside
or upside jumps dominate. The jump and tail index captures (to first order) the left (right) skew
through the relative pricing of OTM puts and calls—the short risk reversal option portfolio
in equation (30). To further grasp the relative importance of downside and upside jumps, we
decompose JTIX ≈ JTIX−− JTIX+, where JTIX− and JTIX+ correspond to the put and call
15
-
portfolios, respectively, of the risk reversal portfolio:
JTIX− =2TerT
∫KS0
(ln(K/S0)
K2
)C(S0;K,T ) dK. (33)
Figure 3 plots JTIX+ and JTIX−. Comparing Figures 3 and 2 confirms that the time variation
in JTIX is driven by OTM put prices, that is, downside jump risk. Economically, the price of
the risk reversal portfolio is driven by time variation in OTM put prices. Interestingly, using
risk-reversals in currency options, Bakshi, Carr, and Wu (2008) also find evidence of one-sided
jump risk.
In Figure 4, we compare JTIX with the (negative of) Bollerslev and Todorov’s (2011) fear
index (-BT).10 The BT index measures the difference between the variance risk premiums
associated with negative and positive jumps, respectively, and is estimated from short-dated
deep OTM options. Not only are both indices highly correlated, but they also show similar
peaks identifying the major financial crises. Not surprisingly, there is a close link between
investor fears noted in the jump variance risk premium and time-varying jump intensity Λ0,T .
5 Channels of jump risk
We undertake two exercises in this section. First, we ascertain whether it is economically
important to account for jump risk in estimating stock return variability. Second, we consider
whether the tail should be considered an additional channel for jump risk after accounting for
stock return variability. We first provide in-sample results following BTZ, and then provide
out-of-sample results as suggested by Welch and Goyal (2008).
5.1 Setup for predictability regressions
Our basic setup follows BTZ, who demonstrate that the spread between the integrated variance
and historical variance, IVt −RVt−1, predicts market excess returns after controlling for a setof traditional predictors. The predictive regression is specified as
Rt+j − rft+j = α+ β1 · (VSt) + β2 · (Tail Index) + Γ′ · Z̄t + �t, (34)
10 We thank Tim Bollerslev and Viktor Todorov for sharing their fear index.
16
-
where Rt+j denotes the log return of the S&P 500 from the end of month t to the end of month
t + j, rft+j denotes the risk-free return for the same horizon, and Z̄t denotes the commonly
used predictors, discussed further below. We alternatively define VSt as either Vt − RVt−1 orIVt −RVt−1. We use this basic specification, appropriately estimated, for both in-sample andout-of-sample tests. All variables are sampled at the end of the month. We use four different
horizons of three months, six months, one year and two years, respectively.
We also include a parsimonious set of traditionally used predictors, Z̄t. These include the
dividend yield, the term spread, and the default spread. The term spread and default spread
are included to control for any predictable impact of the business cycle, and because JTIX
is highly correlated with the default spread (correlation of 0.67) .11 Quarterly price–earnings
ratios and dividend yields for the S&P 500 are from Standard & Poor’s website. When monthly
data are not available, we use the most recent quarterly data. The term spread (TERM) is
defined as the difference between 10-year T-bond and three-month T-bill yields. The default
spread (DEF) is defined as the difference between Moody’s Baa and Aaa corporate bond yields.
Data needed to calculate the term spread and the default spread are from the Federal Reserve
Bank of St. Louis. To be consistent with BTZ, we download the monthly realized variance
RV calculated from high-frequency intra-day return data from Hao Zhou’s website.
Table 2 describes the data. Panel A reports the summary statistics of these variables and
Panel B reports their correlation matrix. The tail index JTIX is contemporaneously negatively
correlated with index excess returns (-0.30), and positively correlated with historical realized
volatility (0.88). With the variance spreads, the tail index JTIX has lower correlations. The
correlations of JTIX with V−RV and IV−RV are 0.19 and 0.00, respectively. JTIX is highlycorrelated with the default spread DEF, with a correlation of 0.67. Dividing the tail index
by V reduces the correlation with the historical volatility from 0.88 (for JTIX) to 0.57 (forJTIX/V) suggesting that the relation between JTIX and historical volatility is non-linear.
The last row of Panel A of Table 2 reports first-order autocorrelations for the control
variables. As noted in previous literature, autocorrelations are extremely high for many of the
control predictors — 0.97 for the price-to-dividend ratio and 0.98 for the term spread — raising
concerns about correct inference. In contrast, Panel C documents that V−RV, IV−RV, andJTIX have much lower autocorrelations, and, in Panel D, the Phillips–Perron unit root test
rejects the null hypothesis of a unit root in the tail index as well as in both variance spreads.11An earlier version of this paper included the quarterly CAY , as defined by Lettau and Ludvigson (2001) and
downloaded from their website, and RREL, the detrended risk-free rate, defined as the one-month T-bill rateminus the preceding 12-month moving average. The results were similar to the more parsimonious regressionused in the current version.
17
-
5.2 Variance spreads: V− RV vs. IV− RV
In (unreported) univariate predictive regressions, both V− RV and IV− RV are consistentlysignificant within sample. Corroborating the results of BTZ, on a univariate basis, the variance
spread has higher in-sample predictive power than most of the traditional variables used in
previous literature.
To examine the importance of accounting for jump risk in the estimation of stock return
variability, we focus on the multivariate specification that includes the variance spread and
the traditional predictors but excludes, for now, the tail index. To ascertain statistical signif-
icance, we use t-statistics based on Hodrick’s (1992) 1B standard errors under the null of no
predictability. Table 3 reports the results.
Both V − RV and IV − RV are significant at the usual levels, with the exception of thetwo-year horizon where V−RV is significant at the 95% level while the significance of IV−RVis slightly lower at 90%. Anticipating results below for the jump and tail index, accounting for
jumps appears to be especially important at longer horizons (although the adjusted R-squared
values are higher for V−RV at all horizons). The main difference between the results of V−RVand IV − RV is, however, in their relative economic importance. For every horizon, V − RVis economically more significant than IV − RV. For example, an increase of one standarddeviation in V − RV increases one-year expected returns by 4.35%, while the correspondingincrease in IV− RV increases expected returns by 5.21%.
To further quantify the economic impact of the bias, we consider the return predicted by
each measure of variance spread using the coefficients estimated in the regression. The return
predicted by IV−RV is lower not only because the economic impact (i.e.,the magnitude of theslope coefficient β1) of IV−RV is lower, but also because IV−RV is biased downwards. Table4 estimates the combined impact of the bias. First, Panel A tabulates the variance spread in
terms of both V − RV and IV − RV. Over the entire sample period, on average, V − RV is0.3% (variance points) higher than IV − RV and 0.9% (variance points) higher in recessionmonths. The measure of integrated variance particularly underestimates the variance spread
in the recent financial crisis because of the extraordinarily high degree of jump risk in this
period; V− RV is 42% greater than IV− RV.
Panel B of Table 4 reports the magnitude of the excess return predicted by V − RV andIV− RV, respectively. The returns are predicted using the coefficients estimated for the one-year horizon in Table 3 for each of the variance spreads for the one-year horizon. Over the
entire sample period of 1996-2012, the one-year excess return predicted by V − RV is 5.71%,
18
-
compared with 4.34% as predicted by IV−RV. As noted earlier, jump risk reached its highestlevel in the most current recession. The excess return predicted by V−RV in the most recentrecession is 7.42%, compared with 4.46% that is predicted by IV−RV. The return predicted byIV−RV during the financial crisis is lower than that predicted for the prior recession. This lastset of results underscores the economic importance of jump risk. Not correctly accounting for
jump risk in the measurement of stock return variability leads to the extraordinary conclusion
that the financial crisis period had lower risk than the 2001–2002 recession.
5.3 Tail risk as an additional channel
Having ascertained the significance of accounting for the contribution of jumps to stock return
variability, we next consider whether the jump-induced tail should be considered an additional
channel for jump risk. As in the previous section, we focus on in-sample results over our entire
sample period. In (unreported) univariate regressions, the R2 for JTIX is consistently lower
than the R2 for V − RV for every horizon. For example, the R2 for the one-year horizon forJTIX is 4.65% versus 8.21% for V − RV. This is not surprising as over much of our sampleperiod, there is little time variation in the tail index (see Figure 2). The important question
is whether JTIX helps explain returns after controlling for V− RV.
Table 5 reports the results for each of the four horizons when JTIX is included in the
regression along with V− RV. As in previous tables, V− RV is statistically significant for allhorizons. Including the tail index does not impact the economic significance of V − RV; thesigns and magnitudes of the estimated coefficients remain about the same. Similarly, with the
exception of the shortest 3-month horizon, excluding V−RV also does not have a large impacton the coefficient of JTIX.
Should tail risk be considered as an additional channel for jump risk? JTIX is significant
at the 95% level for the one-year horizon and the two-year horizon. Including JTIX to the
regression with V−RV and the control variables increases the adjusted R2 from 26.5 to 35.1%for the 12-month horizon, and from 54.1 to 60.8% for the 24-month horizon. The coefficients
for the longer horizons are economically significant; one standard deviation increase in JTIX
increases expected one-year excess return by 6.75%. In comparison, a one standard deviation in
V− RV increases annual expected return by 5.05%. However, note that such a large increasein the tail index occurs rarely. In our sample set, a one-standard deviation increase in the
tail index occurs only at the implosion of Long Term Capital Management in September and
October of 1998, during the financial crisis from October 2008 to March 2009, the flash crash
in May 2010, and the peak of the Euro sovereign debt crisis in August and September of 2011.
19
-
If we consider a one-standard deviation change in JTIX in a sample period that excludes these
dates, then the economic impact of JTIX reduces but is still economically significant at 1.93%.
The tail index is not significant for the shortest horizons. However, this is not entirely a
surprise as short horizon returns can be noisy, making it difficult to discern the impact of tail
risk. The primary concern, however, is whether the statistical inference at longer horizons is
impacted by the auto-correlation in JTIX and the overlapping structure of returns. Although
JTIX is not as highly auto-correlated as the traditional predictors such as log(D/P), it is
non-zero. To understand whether the in-sample predictability of JTIX is indeed economically
significant, we turn to out-of-sample results.
5.4 Out of Sample Predictability
Welch and Goyal (2008) argue that predictability regressions should be evaluated on an out of
sample basis. A high in-sample fit with poor out of sample R2 may imply an overfitting of the
data. On the contrary, even a small level of out of sample predictability can help discern the
relative have a large impact on investor welfare (Campbell and Thomson, 2008). Given the
importance of out of sample predictability, we provide a comprehensive analysis that includes
the variables provided by Amit Goyal on his website.12 Following Welch and Goyal (2008),
the out-of-sample R2 is computed as,
OOSR2 = 1−∑t
(rt+j − r̂t+j|t
)2/(rt+j − r̄t+j|t
)2where rt+j is the excess log return from end of month t to month t + j, r̂t+j is the predicted
excess return for the same period, and r̄t+j is the historical j-month average return to t on the
S&P 500, respectively. The data used for predicting r̂t+j is in the information set. Specifically,
the coefficients used for estimating the r̂t+j return are estimated from x-variable observations
upto time t − j. For example, the 12-month predicted return for end of December 2008 iscomputed using coefficients estimated from the regression with x-variable data upto December
2007 (with the last y-variable being the observed 12-month return upto December 2008) in
conjunction with the realizations of the x-variables as of December 2008.
Table 6 reports the results for several sets of variables on an univariate basis. First, we
report our main indicators of interest, JTIX, and V−RV. In addition, we report for (i) VIX-based measures (IV and IV − RV), (ii) higher order risk-neutral moments (SKEW, KURT)and (iii) a set of 14 traditional variables examined in Welch and Goyal (2008). First, over
12see http://www.hec.unil.ch/agoyal/. We use the data updated through the end of 2012.
20
http://www.hec.unil.ch/agoyal/
-
the entire sample period (with the first forecast beginning as of the end of December 2000),
JTIX has an OOS R2 of 3.9% and 7.4% for the 12-month and 24-month horizons, respectively.
These OOS R2 are impressive because, as Campbell and Thomson (2008), a mean-variance
investor can use the information in the predictability regression to substantially increase the
portfolio return.13 Consistent with the in-sample results, JTIX does not demonstrate out of
sample predictability over the shortest horizons. Both V−RV and IV−RV have positive outof sample R2 over all horizons. Except for the shortest horizon of 3-months, V−RV has higherout of sample predictability than IV− RV.
Next, we consider whether the VIX-based spread IV−RV dominates V−RV, and whetherIV dominates JTIX. The former attests to the importance of accounting for jumps in measuringquadratic variation. The later is important because although the tail index is economically
different from the VIX —JTIX is approximately a short position in a portfolio of risk reversals
while stock return variability is a position in a portfolio of strangles— economic stress affects
both indices similarly. In times of stress, the prices of all OTM options increase (increasing
IV) and OTM puts increase more than OTM calls (increasing JTIX). If IV dominates JTIXthen it would suggest that the tail is not as economically significant as stock return variability.
The out of sample indicates that, first, V − RV, on average, dominates IV − RV, consistentwith the in-sample results. Except for the shortest maturity of 3-months, the OOS R2 are
higher for V−RV than for IV−RV. Second, JTIX dominates IV. The OOS R2 for 12-monthhorizon and 24-month horizon for IV is 1.2% and 5.1%, lower than the OOS R2 for JTIX.
We also consider higher order risk-neutral moments of Bakshi, Kapadia and Madan (2003).
These moments have been shown to explain cross-section of equity returns (Conrad, Dittmar
and Ghysels, 2009; Chang, Christoffersen and Jacobs, 2013). However, neither of these vari-
ables demonstrate that they can predict market excess returns out of sample. Finally, as noted
by first by Welch and Goyal (2008), the traditional variables perform poorly. For our sample
period, none of the 14 variables examined show positive out of sample predictability over any
of the four horizons considered.
A more powerful test can be constructed by focusing on the period where tail risk is
significant as in the financial crisis period that immediately follows the bankruptcy of Lehman
Brothers. Panel B of Table 6 reports the out of sample R2 over the period October 2008 to
December 2012. The OOS R2 increase dramatically. JTIX has out of sample R2s of 25%13Campbell and Thomson (2008) demonstrate that the increase in return would be approximately equal to
the ratio of the OOS R2 to the square of the Sharpe ratio. If we assume a annualized Sharpe ratio of about0.33, then the investor can use the OOS R2 of 3.9% to increase portfolio return by about (0.039)/(0.33)2, orabout 36%.
21
-
and 39.9% for the 12- and 24-month horizons, respectively. Across all the horizons, V − RVhas an OOS R2s ranging from 18.5% to 34.9%. Interestingly, over the longest horizon of 2-
years, JTIX dominates the variance spread consistent with the observation that this period
had extraordinary levels of tail risk.
A number of traditional variables also demonstrate positive out of sample R2s over the
crisis period, including log(P/D), the term spread, the book to market ratio, and the Treasury
bill rate. However, the majority of the traditional variables do not demonstrate out of sample
predictability.
In summary, across the entire sample period, V − RV and JTIX together dominate VIX-based measures of stock return variability and variance risk premium, risk neutral higher order
moments, and traditional predictors. Moreover, V−RV and JTIX also demonstrate extremelyhigh out of sample predictability over the crisis period, precisely when we expect them to
perform well. Overall, the variance spread based on V and JTIX dominate 19 alternativepredictors. The results are consistent with the notion that jump risk plays an important and
economically significant role through both channels of stock return variability and the tail of
the distribution.
6 Conclusion
When the risk-neutral stock return process incorporates fears of discontinuities, both stock
return variability and the tail of the return distribution are determined by fears of jumps.
To distinguish between the two channels, it is important to have model-free volatility and tail
indices that clearly distinguish between the contribution of jumps to stock return volatility and
the impact of jumps on the tail of the distribution. We demonstrate how volatility and tail
indices can be constructed in a model-free manner. The volatility index - based on the BKM
variance of holding period return - accurately measures quadratic variation in the presence of
jumps, unlike the VIX. The tail index is novel, based on the measurement of the jump-induced
bias on the VIX, is easily constructed from a continuum of options of standard maturity, and
is economically interpretable as a short portfolio of risk reversals.
We demonstrate that accounting for jump risk is economically important. First, when
the volatility index is constructed without full consideration of jump risk (as the VIX is con-
structed), stock return variability is significantly underestimated. In the immediate aftermath
of the financial crisis, the VIX-measure of stock return variability underestimates the stock
return variability by as much as 6.2% (annualized volatility points). When the BKM vari-
22
-
ance measure is used to construct the volatility index and jump risk is accounted for in the
volatility index, the variance spread predicts index excess returns with higher in-sample and
out-of-sample R2s. Second, we demonstrate that the tail index is also economically important.
Over the entire sample period from 1996 to 2012, the tail index achieves out of sample R2s of
3.9% and 7.4% over 12-month and 24-month horizons. Even more impressively, the tail index
achieves out of sample R2s of 25% and 39.9% in the aftermath of the Lehman bankruptcy, a
period distinguished by heightened tail risk. In essence, index excess returns are predicted by
tail risk.
Our empirical results highlights the importance of jump risk, and indicate a role for both
volatility and tail risk. It would be of interest for future research to develop a model that pro-
vides a single framework to understand these risks. It would also be interesting to understand
whether the risk premium associated with jump risk is best understood as arising from risk of
consumption disasters, or in terms of wealth disasters that result in a higher marginal utility
of aggregate wealth.
23
-
7 Appendix A: Proofs of results
Proof of Proposition 1Given that lnSt is a semimartingale, the quadratic variation exists and is defined by stochasticintegration by parts:
(lnST /S0)2 = 2
∫ T0
lnSt−/S0 d lnSt + [lnS, lnS]T . (A-1)
From the definition of the variance,
varQ0 (lnST /S0) ≡ EQ0 (lnST /S0)
2 − (EQ0 lnST /S0)2. (A-2)
From equations (A-1) and (A-2), it follows that D(T ) =(
EQ0 (lnST /S0))2−EQ0
[∫ T0 2 lnSt−/S0 d lnSt
]and that D = 0 if and only if 2EQ0
∫ T0 lnSt−/S0 d lnSt =
(EQ0 lnST /S0
)2.
Proof of Proposition 1, part i.Next, when lnSt/S0 = At +Mt with At deterministic and Mt a martingale,
EQ0
∫ T0
2 lnSt−/S0 d lnSt = EQ0
∫ T0
lnSt−/S0 Et d lnSt, (A-3)
=∫ T
02At dAt, (A-4)
= (AT )2 =(
EQ0 lnST /S0)2. (A-5)
Therefore, D = 0. In the above equations, the first equality follows from the law of iterativeexpectations and the second because Mt is a martingale. The third equality follows becausethe drift is of finite variation with continuous paths (Theorem I.53 of Protter, 2004). Thisproves Proposition 1, part i. �
Proof of Proposition 1, part ii.For the stochastic volatility diffusion model defined by equations (12) and (13),
D(T ) = EQ0
[∫ T0
2 lnSt/S0 d lnSt
]−(
EQ0 (lnST /S0))2
(A-6)
= I + II.
To study the speed of convergence of D(T ) when T → 0, we apply the Ito–Taylor expansion(Milstein, 1995) to each term of D(T ) in equation (A-6). We define the operators L ≡ ∂∂t +(r− 12σ
2t )
∂∂(lnSt)
+θ[σ2t ]∂
∂(σ2t )+ 12σ
2t
∂2
∂(lnSt)2+ 12η
2[σ2t ]∂2
∂(σ2t )2 +ρσtη[σ2t ]
∂2
∂(lnSt)∂(σ2t ), Γ1 ≡ σt ∂∂(lnSt) ,
and Γ2 ≡ η[σ2t ] ∂∂(σ2t ) . Noting that applying the Ito–Taylor expansion on a deterministic func-
24
-
tion g(lnSt, σ2t , t) yields g(lnSt, σ2t , t) = g(lnS0, σ
20, 0)+
∫ t0 (L[g]du+ Γ1[g]dW1,u + Γ2[g]dW2,u),
applying the expansion twice to the integral in term I of equation (A-6) yields∫ T0
lnSt/S0 d lnSt =∫ T
0lnStS0
(r − 12σ2t )dt+
∫ T0
lnStS0σtdW1,t,
=∫ T
0
∫ t0
[(r − 1
2σ2u)
2 − 12
lnSuS0θ[σ2u]−
12σuη[σ2u]ρ
]du dt
+∫ T
0
∫ t0
(r − 12σ2u)σudW1,udt−
12
∫ T0
∫ t0
lnSuS0η[σ2u]dW2,udt+
∫ T0
lnStS0σtdW1,t,
=∫ T
0
∫ t0
[(r − 1
2σ20)
2 − 12σ0η[σ20]ρ
]du dt
+∫ T
0
∫ t0
(r − 12σ20)σ0dW1,u dt−
12
∫ T0
∫ t0
lnS0S0η[σ20]dW2,u dt+
∫ T0
lnStS0σtdW1,t
+A0, (A-7)
where A0 consists of terms such as∫ T0
∫ t0
∫ v0 L
2[·]dv du dt,∫ T0
∫ t0
∫ v0 Γ1 [L [·]] dW
1v du dt, and so
on. With repeated applications of the Ito-Taylor expansion and using the martingale propertyof the Ito integral, EQ0 (A0) =
∑∞n=0 h
n(σ20)Tn+3
(n+3)! , for deterministic functions hn, n ∈ {0, 1, 2...},
and, therefore, EQ0 [A0] = O(T3). Taking expectations and integrating, it follows that
EQ0
∫ T0
2 lnSt/S0 d lnSt =[(r − 1
2σ20)
2 − 12σ0η[σ20]ρ
]T 2 +O(T 3). (A-8)
We can similarly proceed to evaluate term II of equation (A-6) by applying the stochasticTaylor expansion to the integral defining the log return process:
lnStS0
=∫ T
0(r − 1
2σ2t )dt+
∫ T0
√σ2t dW
1t ,
= (r − 12σ20)T −
∫ T0
∫ t0
12θ[σ2u]du dt+
∫ T0
∫ t0−1
2η[σ2u]dW2,u dt+
∫ T0
√σ2t dW
1t ,
= (r − 12σ20)T −
T 2
4θ[σ20] +
∫ T0
∫ t0−1
2η[σ2u]dW2,u dt+
∫ T0
√σ2t dW
1t +A1, (A-9)
where A1 = O(T 3). Taking expectations,
EQ0 (lnST /S0) = (r −12σ20)T −
T 2
4θ[σ20] +O(T
3). (A-10)
Combining equations (A-8) and (A-10), we have
D(T ) = EQ0∫ T
02 lnSt/S0 d lnSt −
(EQ0 ln
StS0
)2= −1
2σ0η[σ20]ρT
2 +O(T 3). (A-11)
It follows that 1TD(T ) is O(T ) and, when ρ = 0,1TD(T ) = O(T
2).
�
25
-
Proof of Proposition 2Given that jumps of all sizes have the same arrival intensity, it follows that EQ0 2
∫ T0
∫R0 ψ(x)µ[dx, dt] =
2∫ T0 E
Q0
[∫R0 ψ(x) f(x)dx
]λtdt = 2Ψ(f(x))E
Q0
∫ T0 λtdt , where Ψ(·) is determined by the jump
size distribution. �
Proof of Equation 6To evaluate the integral, first observe from equation (3) that
EQ0 ln(St−/S0) = (r −12σ2 − λµJ)t+ λαt. (A-12)
Therefore,
EQ0
∫ T0
2 ln(St−/S0) d lnSt = 2 EQ0
[∫ T0
ln(St−/S0) (r −12σ2 − λµJ) dt+
∫ T0
∫R0
ln(St−/S0)xµ[dx, dt]
],
= 2
[∫ T0
EQ0 (ln(St−/S0)) (r −12σ2 − λµJ) dt
+
[.
∫ T0
∫R0
EQ0 (ln(St−/S0))xλ√
2πσ2Je−
(x−α)22 dxdt
],
=(
(r − 12σ2 − λµJ) + λα
)2T 2,
=(
EQ0 ln(St−/S0))2. (A-13)
Therefore, for the Merton model, EQ0∫ T0 2 ln(St−/S0) d lnSt =
(EQ0 ln(St−/S0)
)2. �
Proof of equation 15, D(T ) = 14var∫ T0 σ
2t dt for a stochastic volatility diffusion with ρ = 0
Let lnST /S0 be a continuous semimartingale,
lnST = lnS0 +∫ T
0(r − 1
2σ2t )dt+
∫ T0σtdW1,t, (A-14)
where σ2t is another continuous semimartingale, orthogonal to W1,t. By the law of total vari-ance,
varQ0 (lnST /S0) = varQ0
(EQ0 lnST /S0 | {σ
2t }0≤t≤T
)+ EQ0
(varQ0 (lnST /S0) | {σ
2t }0≤t≤T
).
(A-15)Now, from equation(A-14),
varQ0(
EQ0 lnST /S0 | {σ2t }(0≤t≤T )
)=
14
varQ0
∫ T0σ2t dt, (A-16)
26
-
and
EQ0(
varQ0 (lnST /S0) | {σ2t })
= EQ0
∫ T0σ2t dt (A-17)
from Ito isometry. Putting this together into (A-15) and given that the quadratic variation forthe diffusion process is the integrated variance,
∫ T0 σ
2t dt, we obtain
D(T ) = varQ0 (lnST /S0)− EQ0 [lnS, lnS]T =
14
varQ0
∫ T0σ2t dt. (A-18)
Thus, if the stochastic volatility process is independent of W1,t, then D(T ) is proportional tothe variance of the integrated variance. �
Power claim of Carr and Lee (2008) in presence of jumpsFirst, following Carr and Lee (2008), assume there are no discontinuities and that the volatilityprocess, σt, is independent of the diffusion determining the log stock process, W1,t. Withoutloss of generality, we can also assume that the risk-free rate is zero. If so,
lnST /S0 =∫ T
0−1
2σ2t dt+
∫ T0σtdW1,t (A-19)
and, therefore, the power claim,
EQ0 exp(p lnST /S0) = EQ0 exp
(∫ T0−p
2σ2t dt+
∫ T0pσtdW1,t
), (A-20)
= EQ0 exp(
(p2
2− p
2)∫ T
0σ2t dt
). (A-21)
Bakshi, Panayotov and Skoulakis (BPS, 2010) use this theory to estimate EQ0(
exp(∫ T0 σ
2t dt)
from option prices.14 Now relax the assumption of no discontinuities by adding Merton-stylejumps, following Section 2,
EQ0 exp (p lnST /S0) = EQ0 exp
(∫ T0
−p2σ2dt−
∫ T0
pλµJ dt+∫ T
0
pσ dWt +∫ T
0
∫R0pxµ[dx, dt]
).
(A-22)
14 Because the volatility process is independent of W1,t, the log return is normally distributed with mean and
variance equal to − 12
R T0σ2t dt and
R T0σ2t dt, respectively, leading to equation (A-21). We can rewrite equation
(A-21) as
exp
„λ̄
Z T0
σ2t dt
«= EQ0 (ST /S0)
12±√
14 +2λ̄.
Because the power claim (ST /S0)p can be synthesized from options, we can estimate the price of a claim paying
the exponential of the integrated variance. In addition, Carr and Lee (2008) demonstrate that a properlychosen portfolio of power claims is not sensitive to a non-zero correlation. Following the Carr–Lee analysis, BPSconsider the case for λ̄ = −1.
27
-
We can re-write equation A-22 using the definition of quadratic variation to get,
EQ0 exp (p lnST /S0) = EQ0
[exp
(−p
2[lnS, lnS]T +
∫ T0
pσ dWt
)exp
(−∫ T
0
pλµJ dt+∫ T
0
∫R0
(px+p
2x2)µ[dx, dt]
)].
(A-23)A comparison of equation (A-23) with equation (A-20) demonstrates that the power claim canmeasure the quadratic variation without bias only if jumps are absent. �
28
-
8 Appendix B: Description of Data
In line with previous literature, we clean the data using several filters. First, we exclude optionsthat have (i) missing implied volatility in OptionMetrics, (ii) zero open interest, and (iii) bidsequal to zero or negative bid–ask spreads. Second, we verify that options do not violate the no-arbitrage bounds. For an option of strike K maturing at time T with current stock price St anddividend D, we require that max(0, St−PV [D]−PV [K]) ≤ c(St;T,K) ≤ St−PV [D] for Euro-pean call options and max(0, PV [K]− (St−PV (D))) ≤ p ≤ PV [K] for European put options,where PV [·] is the present value function. Fourth, if two calls or puts with different strikeshave identical mid-quotes, that is, c(St;T,K1) = c(St;T,K2) or p(St;T,K1) = p(St;T,K2),we discard the one furthest away from the money quote. Finally, we keep only OTM optionsand include observations for a given date only if there are at least two valid OTM call and putquotes. Table B.1 provides summary statistics of the final sample.
Table B.1. Option dataThis table reports the summary statistics of all options used to construct a 30-day JTIX with a dailyfrequency. The variable Implied volatility is the Black–Scholes implied volatility; Range of Moneynesson a certain date for a given underlying asset is defined as (Kmax −Kmin)/Katm, where Kmax, Kmin,and Katm are the maximum, minimum, and at-the-money strikes, respectively; near term refers tooptions with the shortest maturity (but greater than seven days) on each date; and next term refers tooptions with the second shortest maturity on each date. The sample period is from January 1996 toOctober 2010.
Near Term Next TermMEAN SD MIN MAX MEAN SD MIN MAX
# of options per date 50 26 13 146 53 34 11 151Range of moneyness per date 37% 15% 10% 137% 50% 18% 13% 153%Option price 5.37 7.32 0.08 68.1 9.82 11.29 0.08 84.05Implied volatility 28.10% 15.28% 4.88% 183.33% 26.73% 13.12% 6.69% 181.76%Maturity 21 9 7 39 51 9 14 79Trading volume 2239 5389 0 200777 976 2986 0 120790Open interest 20292 31648 1 366996 12170 23673 1 348442
29
-
References
[1] Aı̈t-Sahalia, Y., Jacod, J., 2008. Fisher’s information for discretely sampled Lévy pro-cesses. Econometrica 76, 727-761.
[2] Aı̈t-Sahalia, Y., Jacod, J., 2009a. Estimating the degree of activity of jumps in highfrequency data. Annals of Statistics 37, 2202-2244.
[3] Aı̈t-Sahalia, Y., Jacod, J., 2009b. Testing for jumps in a discretely observed process.Annals of Statistics 37, 184-222.
[4] Aı̈t-Sahalia, Y., Jacod, J., 2010. Analyzing the spectrum of asset returns: Jump andvolatility components in high frequency data. Working paper. Princeton University,Princeton.
[5] Ang, A., Bekaert, G., 2007. Stock return predictability: Is it there? Review of FinancialStudies 20 (3), 651-707.
[6] Backus, D., Chernov, M., Martin, I., 2011. Disasters implied by equity options. Journalof Finance 66, 1969-2012.
[7] Baker, Malcolm and Jeffery Wurgler, 2006. Investor sentiment and the cross-section ofstock returns. Journal of Finance 61 (4), pp. 1645-1680.
[8] Bakshi, G., Cao, C., Chen, Z., 1997. Empirical performance of alternative option pricingmodels. Journal of Finance 52(5), 2003-2049.
[9] Bakshi, G., Kapadia, N., 2003. Delta-hedged gains and the negative market volatility riskpremium. Review of Financial Studies 16 (2), 527-566.
[10] Bakshi, G., Kapadia, N., Madan, D., 2003. Stock return characteristics, skew laws, andthe differential pricing of individual equity options. Review of Financial Studies 16 (1),101-143.
[11] Bakshi, G., Carr, P., Wu, L., 2008. Stochastic risk premiums, stochastic skewness incurrency options, and stochastic discount factors in international economies. Journal ofFinancial Economics 87, 132-156.
[12] Bakshi, G., Madan, D., 2006. A theory of volatility spreads. Management Science, 52 (12),1945-1956.
[13] Bakshi, G., Madan, D., Panayotov, G., 2010. Deducing the implications of jump modelsfor the structure of crashes, rallies, jump arrival rates and extremes. Journal of Businessand Economic Statistics, 28 (3), 380-396.
[14] Bakshi, G., Panayotov, G., Skoulakis, G., 2010. Improving the predictability of real eco-nomic activity and asset returns with forward variances inferred from option portfolios.Journal of Financial Economics, 100, 475-495.
[15] Bakshi, G., Wu, L., 2010. The behavior of risk and market prices of risk over the Nasdaqbubble period. Management Science 56 (12), 2251-2264.
30
-
[16] Bali, T. G., Hovakimian, A., 2009. Volatility spreads and expected stock returns. Man-agement Science 55 (11), 1797-1812.
[17] Barndorff-Nielsen, O. E., 1998. Processes of normal inverse Gaussian type. Finance andStochastics 2, 41-68.
[18] Barro, R., 2006. Rare disasters and asset markets in the twentieth century. QuarterlyJournal of Economics 121, 823-866.
[19] Bates, D. S., 2000. Post-’87 crash fears in the S&P 500 futures option market. Journal ofEconometrics 94 (1-2), 181-238.
[20] Black, F., Scholes, M., 1973. The pricing of options and corporate liabilities. Journal ofPolitical Economy 81 (3), 637-654.
[21] Bollerslev, T., Tauchen, G., Zhou, H., 2009. Expected stock returns and variance riskpremia. Review of Financial Studies 22 (11), 4463-4492.
[22] Bollerslev, T., Gibson, M., Zhou, H., 2011. Dynamic estimation of volatility risk pre-mia and investor risk aversion from option implied and realized volatilities. Journal ofEconometrics 160 (1), 235-245.
[23] Bollerslev, T., Todorov, V., 2011. Tails, fears and risk premia. Journal of Finance 66,2165-2211.
[24] Bollerslev, T., Zhou, H., 2002. Estimating stochastic volatility diffusion using conditionalmoments of integrated volatility. Journal of Econometrics 109, 33-65.
[25] Boudoukh, J., M. Richardson,and R. Whitelaw,
[26] Britten-Jones, M., Neuberger, A., 2000. Option prices, implied price processes, andstochastic volatility. Journal of Finance 55 (2), 839-866.
[27] Broadie, M., Chernov, M., Johannes, M., 2007. Model specification and risk premia.Journal of Finance, 62, 1453-1490.
[28] Broadie, M., Jain, A., 2008. The effect of jumps and discrete sampling on volatility andvariance swaps. International Journal of Theoretical and Applied Finance, 11, 761-797.
[29] Campbell, John Y. and Samuel B. Thomson, 2008. Predicting excess stock returns outof sample: Can anything beat the historical average? Review of Financial Studies 21(4):1509-1531.
[30] Carr, P., Madan, D., 1998. Towards a theory of volatility trading. Volatility, Risk Publi-cation, 417-427.
[31] Carr, P., Geman, H., Madan, D., Yor, M., 2002. The fine structure of asset returns: Anempirical investigation. Journal of Business 75 (2), 305-332.
[32] Carr, P., Lee, R., 2008. Robust replication of volatility derivatives. Working paper. NewYork University and University of Chicago (http://www.math.uchicago.edu/rl/rrvd.pdf).
31
http://www.math.uchicago.edu/ rl/rrvd.pdfhttp://www.math.uchicago.edu/ rl/rrvd.pdf
-
[33] Carr, P., Lee, R., Wu, L., 2011. Variance swaps on time-changed Lévy Processes. Financeand Stochastics, 10.1007/s00780-011-0157-9, 1-21.
[34] Carr, P., Wu, L., 2003. What type of process underlies options? A simple robust test.Journal of Finance 58, 2581-2610.
[35] Carr, P., Wu, L., 2009. Variance risk premiums. Review of Financial Studies 22, 1311-1341.
[36] Chang, Bo, Peter Christoffersen and Kris Jacobs, 2013. Market skewness risk and thecross section of stock returns. Journal of Financial Economics 107 (1), 46-48.
[37] Cochrane, J., 2008. The dog that did not bark: A defense of return predictability. Reviewof Financial Studies 21 (4), 1533-1575.
[38] Conrad, Jennifer Robert Dittmar, and Eric Ghysels, 2009. Ex ante skewness and expectedstock returns. Working paper. University of North Carolina-Chapel Hill.
[39] Cont, R., Kokholm, T., 2010. A consistent pricing model for index options and volatilityderivatives. Mathematical Finance, forthcoming.
[40] Cont, R., Tankov, P., 2003. Financial Modeling with Jump Processes. Chapman andHall/CRC Press, Boca-Rator, Florida.
[41] Demeterfi, K., Derman, E., Kamal, M., Zou, J., 1999a. A guide to volatility and varianceswaps. Journal of Derivatives 6 (4), 9-32.
[42] Demeterfi, K., Derman, E., Kamal, M., Zou, J., 1999b. More than you ever wanted toknow of volatility swaps. Quantitative Strategies Research Note, Goldman Sachs workingpaper.
[43] Drechsler, I., Yaron, A., 2011. What’s vol got to do with it. Review of Financial Studies24 (1), 1-45.
[44] Dupire, B., 1996. A unified theory of volatility. Working paper. BNP Paribas. Reprintedin Carr., P. (Ed.), Derivatives Pricing: The Classic Collection, Risk Books.
[45] Gabaix, X., 2012. Variable risk disasters: An exactly solved framework for ten puzzles infinance. Quarterly Journal of Economics 127, 645-700.
[46] Heston, S., 1993. A closed-form solution for options with stochastic volatility with appli-cations to bond and currency options. Review of Financial Studies 6, 327-343.
[47] Hodrick, R., 1992. Dividend yields and expected stock returns: Alternative procedures forinference and measurement. Review of Financial Studies 5 (3): 357-386.
[48] Hull, J., White, A., 1987. The pricing of assets on options with stochastic volatility.Journal of Finance 42 (2): 281-300.
[49] Jiang, G., Tian, Y., 2005. Model-free implied volatility and its information content. Reviewof Financial Studies 18, 1305-1342.
32
-
[50] Kelly, B., 2010. Risk premia and the conditional tails of stock returns. Working paper.University of Chicago, Chicago.
[51] Kou, S. G., 2002. A jump-diffusion model for option pricing. Management Science 48 (8),1081-1101.
[52] Lee, S. S., Hannig, J., 2010. Detecting jumps from Lévy jump diffusion processes. Journalof Financial Economics, 96(2), 271-290.
[53] Lee, S. S., Mykland, P. A., 2008. Jumps in financial markets: A new nonparametric testand jump dynamics. Review of Financial Studies 21, 2535-2563.
[54] Lettau, M., Ludvigson, S., 2001. Consumption, aggregate wealth, and expected stockreturns. Journal of Finance 56 (3), 815-849.
[55] Liu, J., Pan, J., Wang, T., 2005. An equilibrium model of rare-event premia and itsimplications for option smirks. Review of FInancial Studies 18 (1), 131-164.
[56] Longstaff, F. A., Piazzesi, M., 2004. Corporate earnings and the equity premium. Journalof Financial Economics 74, 401-421.
[57] Madan, D., Seneta, E., 1990. The variance gamma (VG) model for share market returns.Journal of Business 63, 511-524.
[58] Merton, R., 1976. Option pricing when underlying stock returns are discontinuous. Journalof Financial Economics 3 (1-2), 125-144.
[59] Milstein, G. N., 1995. Numerical integration of stochastic differential equations. KluwerAcademy Publishers: Boston.
[60] Naik, V., Lee, M., 1990. General equilibrium pricing of options on the market portfoliowith discontinuous returns. Review of Financial Studies 3, 493-521.
[61] Neuberger, A., 1994. The log contract. Journal of Portfolio Management 20 (2), 74-80.
[62] Pan, J., 2002. The jump-risk premia implicit in options: Evidence from an integratedtime-series study. Journal of Financial Economics 63 (1), 3-50.
[63] Protter, P., 2004. Stochastic Integration and Differential Equations, 2nd edition. Springer,New York.
[64] Rietz, T. (1988). The equity risk premium: A solution. Journal of Monetary Economics,22, 117-131.
[65] Wachter, J., 2012. Can time-varying risk of disasters explain aggregate stock marketvolatility? Journal of Finance forthcoming.
[66] Welch, I., and A. Goyal, 2008. A comprehensive look at the empirical performance ofequity premium prediction. Review of Financial Studies 21(4), 1454-1508.
[67] Wu, L., 2011. Variance dynamics: Joint evidence from options and high-frequency returns.Journal of Econometrics 160 (1), 280-287.
33
-
Table 1. Numerical comparison: EQ[lnS, lnS]T , V, and IVNumerical comparison of annualized expected quadratic variation EQ[lnS, lnS]T with V (holding periodvariance) and IV (integrated variance) for the jump diffusion model of Merton (1976) (Panel A) andthe jump diffusion and stochastic volatility (SVJ) model of Bates (2000) (Panel B). The specificationfor the Merton model is d lnSt = (r − µJλ − 12σ
2) dt + σdWt + x dJt, where x ∼ N(α, σ2J), andµJ = eα+
12σ
2J − 1. The specification for the SVJ model is: d lnSt = (r− 12σ
2t −λµJ)dt+σtdW 1t +x dJt,
dσ2t = κ(σ2t − θ)dt + ησtdW 2t , where corr(dW 1t , dW 2t ) = ρ, x ∼ N(α, σ2J) and λ = λ0 + λ1 · σ2t . The
parameters are from Pan (2002): λ0 = 0, λ1 = 12.3 (for Merton’s model, λ = λ1 · 0.04), σJ = 0.0387,κ = 3.3, θ = 0.0296, η = 0.3, and ρ = −0.53. In addition, the risk-free rate r = 0.03 and time tomaturity τ = 30/365. The remaining parameters are shown in the table. The first column denotesthe contribution of jumps to the total variance, defined as the holding period variance of the purejump component of the stochastic process divided by the holding period variance of the combined jumpdiffusion process.
Panel A. Merton modelIV− [lnS, lnS]T
varJumpvar σ
20 α E
Q[lnS, lnS]T V IV Absolute Relative (%)0% 0.0400 0.0000 0.0400 0.0400 0.0400 0.0000 0.0010% 0.0360 -0.0814 0.0400 0.0400 0.0399 -0.0001 -0.3620% 0.0320 -0.1215 0.0400 0.0400 0.0396 -0.0004 -0.9230% 0.0280 -0.1513 0.0400 0.0400 0.0393 -0.0007 -1.6340% 0.0240 -0.1761 0.0400 0.0400 0.0390 -0.0010 -2.4450% 0.0200 -0.1979 0.0400 0.0400 0.0387 -0.0013 -3.3660% 0.0160 -0.2174 0.0400 0.0400 0.0383 -0.0017 -4.3670% 0.0120 -0.2354 0.0400 0.0400 0.0378 -0.0022 -5.4380% 0.0080 -0.2521 0.0400 0.0400 0.0374 -0.0026 -6.5890% 0.0040 -0.2677 0.0400 0.0400 0.0369 -0.0031 -7.80
Panel B: SVJ modelIV− [lnS, lnS]T V− [lnS, lnS]T
varJumpvar σ
20 α E
Q[lnS, lnS]T V IV Absolute Relative (%) Absolute Relative (%)0% 0.0415 0.0000 0.0400 0.0402 0.0400 0.0000 0.00 0.0002 0.6110% 0.0369 -0.0868 0.0400 0.0402 0.0399 -0.0001 -0.37 0.0002 0.6120% 0.0323 -0.1372 0.0400 0.0402 0.0396 -0.0004 -1.01 0.0002 0.6030% 0.0278 -0.1826 0.0400 0.0402 0.0392 -0.0008 -1.89 0.0002 0.5940% 0.0232 -0.2296 0.0400 0.0402 0.0388 -0.0012 -3.04 0.0002 0.5850% 0.0186 -0.2825 0.0400 0.0402 0.0382 -0.0018 -4.54 0.0002 0.5760% 0.0141 -0.3471 0.0400 0.0402 0.0374 -0.0026 -6.52 0.0002 0.5570% 0.0095 -0.4338 0.0400 0.0402 0.0363 -0.0037 -9.24 0.0002 0.5280% 0.0049 -0.5690 0.0400 0.0402 0.0347 -0.0053 -13.34 0.0002 0.4790% 0.0004 -0.8545 0.0400 0.0401 0.0316 -0.0084 -21.04 0.0001 0.36
34
-
Tab
le2.
Pre
dic
tive
regr
essi
on:
Sum
mar
yst
atis
tics
Thi
sta
ble
repo
rts
the
sum
mar
yst
atis
tics
ofth
eva
riab
les
used
inpr
edic
tive
regr
essi
ons
over
the
sam
ple
peri
odfr
omJa
nuar
y19
96to
Dec
embe
r20
12.
All
vari
able
sar
esa
mpl
edm
onth
lyat
the
end
ofth
em
onth
.P
anel
Are
port
sth
eba
sic
stat
isti
cs.
Pan
elB
repo
rts
the
cros
s-co
rrel
atio
nm
atri
x.P
anel
Cre
port
sth
eau
toco
rrel
atio
nsup
toor
der
six.
Pan
elD
repo
rts
the
resu
lts
for
the
Phi
llips
–Per
ron
unit
root
test
.H
erer t
=Rt−rf t
isth
em
onth
lyex
cess
log
retu
rnon
the
S&P
500
inde
x;V
isth
ean
nual
ized
hold
ing
peri
odva
rian
ce;
IVis
the
annu
aliz
edin
tegr
ated
vari
ance
;R
Vis
the
annu
aliz
edre
aliz
edva
rian
cefr
omH
aoZ
hou’
sw
ebsi
te;l
og(P/D
)is
the
loga
rith
mof
the
pric
e–di
vide
ndra
tio
onth
eS&
P50
0;T
ER
Mis
the
diffe
renc
ebe
twee
nth
e10
-yea
ran
dth
ree-
mon
thT
reas
ury
yiel
ds;a
ndD
EF
isth
edi
ffere
nce
betw
een
Moo
dy’s
BA
Aan
dA
AA
bond
yiel
ds.
Pan
elA
.Sum
mar
ySta
tist
ics
r tJT
IXt
JTIX
t/Vt
RVt−
1Vt−
RVt−
1IV
t−
RVt−
1log(Pt/D
t)
TERMt
DEFt
Mea
n0.
0033
0.00
340.
0433
0.03
190.
0303
0.02
704.
0459
0.01
660.
0102
Med
ian
0.00
900.
0018
0.03
750.
0188
0.02
390.
0223
4.03
790.
0159
0.00
91St
d.D
ev.
0.04
660.
0065
0.02
250.
0507
0.02
830.
0277
0.22
810.
0122
0.00
46M
ax0.
1033
0.05
820.
1552
0.57
540.
1935
0.17
434.
4931
0.03
690.
0338
Min
-0.1
830.
0003
0.01
410.
0047
-0.1
12-0
.178
3.37
67-0
.006
0.00
54Sk
ewne
ss-0
.76
6.06
2.45
7.07
1.27
-0.6
4-0
.21
-0.0
22.
80K
urto
sis
4.22
46.7
811
.17
70.6
612
.11
21.1
43.
501.
8012
.64
AR
(1)
0.10
0.76
0.59
0.62
0.29
0.21
0.97
0.97
0.96
35
-
Pan
elB
.C
orre
lati
onM
atri
xr t
JTIX
JTIX/V
RVt−
1Vt−
RVt−
1IV
t−
RVt−
1lo
g(Pt/Dt)
TERMt
DEFt
r t1
-0.3
0-0
.01
-0.4
0-0
.15
-0.0
6-0
.06
-0.0
2-0
.11
JTIX
10.
740.
880.
19-0
.00
-0.3
20.
210.
67JT
IX/V
10.
570.
11-0
.00
-0.2
60.
190.
54R
Vt−
11
-0.0
5-0
.25
-0.2
40.
190.
58Vt−
RVt−
11
0.97
-0.1
2