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: nkapadia@som.umass.edu.Website: 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

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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

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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