How Crashes Develop: Intradaily Volatility and Crash Evolution

• Author: DAVID S. BATES
• DOI: http://doi.org/10.1111/jofi.12732
• Date: 01 February 2019
• Published date: 01 February 2019

THE JOURNAL OF FINANCE •VOL. LXXIV, NO. 1 •FEBRUARY 2019

How Crashes Develop: Intradaily Volatility

and Crash Evolution

DAVID S. BATES∗

ABSTRACT

This paper explores whether affine models with volatility jumps estimated on in-

tradaily S&P 500 futures data over 1983 to 2008 can capture major daily outliers

such as the 1987 stock market crash. Intradaily jumps in futures prices are typi-

cally small; self-exciting but short-lived volatility spikes capture intradaily and daily

returns better. Multifactor models of the evolution of diffusive variance and jump

intensities improve fits substantially, including out-of-sample over 2009 to 2016. The

models capture reasonably well the conditional distributions of daily returns and re-

alized variance outliers, but underpredict realized variance inliers. I also examine

option pricing implications.

WHAT IS A CRASH?INTHEjump-diffusion model of Merton (1976), a crash is a

rare event—a single adverse draw from a Poisson counter, with a vanishingly

small probability of multiple adverse draws within a single day. While this

model may be successful at capturing outliers in daily returns, it does not

appear to capture the intradaily evolution of major market downturns. The

28% drop in the December 1987 S&P 500 futures price (23% drop in the S&P

index) on Monday, October 19, 1987, from the preceding Friday’s closing level

did not occur within five minutes, for instance; it took all day to achieve the

full decline. Indeed, papers such as Tauchen and Zhou (2011) that use the

bipower variation approach of Barndorff-Nielsen and Shephard (2004,2006)

to decompose realized variance into diffusive and jump components suggest

there were no jumps at all on October 19. Instead, it was a draw of roughly

two standard deviations on a day that happened to have an unusually high

intradaily realized volatility of 12%.

While the increasing availability of high-frequency data has led to explo-

ration of intradaily volatility evolution, including in stock markets, there has

been little direct estimation of dynamic models with stochastic volatility and

∗David Bates is with the University of Iowa and the National Bureau of Economic Research.

I am grateful for comments on earlier versions of the paper from seminar participants at Iowa,

Northwestern, Houston, Lugano, and the Collegio Carlo Alberto and from conference participants

at the 2012 IFSID Conference on Structured Products and Derivatives, McGill University’s 2014

Risk Management Conference, the 2016 FMA/CBOE Conference on Volatility and Derivatives,

and the 2017 annual conferences of the Midwest Finance Association and Society for Financial

Econometrics. I have read the Journal of Finance’s disclosure policy and have no conflicts of

interest to disclose.

DOI: 10.1111/jofi.12732

193

194 The Journal of Finance R

jumps using intradaily data. Papers such as Andersen and Bollerslev (1997)

focus on volatility dynamics; in particular,on reconciling GARCH-based volatil-

ity evolution estimates from daily versus intradaily data. As described by

Andersen (2004), the recognition that realized variance effectively summa-

rizes intradaily volatility information and sidesteps the challenges in fitting

pronounced diurnal volatility patterns and announcement effects has led in-

tradaily research to shift focus to realized variance. Whether jumps are im-

portant has been assessed indirectly in this literature, with either the bipower

variation approach of Barndorff-Nielsen and Shephard (2004,2006)orthe

threshold approach of Mancini (2009) used to assess intradaily jump contri-

butions to realized variance. These approaches maintain the Merton (1976)

presumption that jumps are rare.

This indirect evidence and more direct parametric estimates by Stroud and

Johannes (2014) on intradaily data point to a fundamental mismatch between

jump magnitudes from intradaily versus from daily stock market data, let

alone those inferred from option prices. Stroud and Johannes (2014) find that

the standard deviation of unexpected jumps in five-minute returns is between

0.2% and 0.4%, and that magnitudes for predictable announcement effects are

similar. The jump magnitudes estimated by Bates (2012, Table VI) on daily

data over the 1926 to 2006 period using a double exponential jump distribution

are an order of magnitude higher: −2.1% on average for negative jumps and

+1.6% for positive jumps. The double exponential jump parameters inferred

from stock index options by Andersen, Fusari, and Todorov (2015) are even

larger: −3.9% on average for risk-neutral negative jumps and +2.7% for risk-

neutral positive jumps. Of course, one must be wary of parameter inferences

from option prices, as standard equity and volatility risk premia imply that

the frequency and magnitude of negative jumps are greater under the risk-

neutral than under the actual distribution. However, those effects are reversed

for positive jumps, implying that one should observe even larger (and more

frequent) positive jumps on average than the +2.7% estimate in Andersen,

Fusari, and Todorov (2015).

The objective of this paper is to bridge the gap between intradaily and daily

evidence on stock market returns and to explore continuous-time affine models

that might be compatible with both. The key feature of the models is “self-

exciting” synchronous and correlated jumps in intradaily stock returns and

volatility, which is essentially a stochastic-intensity version of the Duffie, Pan,

and Singleton (2000) constant-intensity volatility jump model. Every small

intradaily jump substantially increases the probability of more intradaily co-

jumps in volatility and returns, and these multiple price jumps can accumu-

late into the major outliers in daily returns that we occasionally observe. The

model is estimated on intradaily and overnight S&P 500 futures returns over

the 1983 to 2008 period using Bates’s (2006,2012) approximate maximum

likelihood (AML) filtration methodology, taking into account special features of

intradaily futures data. Estimates are then tested for compatibility with daily

returns—including movements exceeding 10% in 1987 and 2008. The 2009 to

2016 period is used for out-of-sample tests of the model.

How Crashes Develop 195

The two central mechanisms of the model are volatility feedback (via jumps)

and leverage; that is, a tendency of conditional volatility to become more volatile

at higher levels combined with negative correlations between price and volatil-

ity shocks. These mechanisms have previously been proposed and estimated

on daily data using a variety of models and estimation methodologies. The

diffusive affine stochastic volatility model of Heston (1993) has both, and is es-

timated on daily stock market data by various authors surveyed in Bates (2006,

Table 7). The nonaffine diffusive log variance models in Chernov et al. (2003)

have substantial volatility feedback; the diffusive power variance model in

Jones (2003) has even more. Models with jumps typically have leverage but not

volatility feedback through jump channels, for example, the price/volatility co-

jump model of Eraker,Johannes, and Polson (2003) estimated on daily data and

the cojump model of Stroud and Johannes (2014) estimated on intradaily data.

Both of these papers use the Monte Carlo Markov chain estimation method-

ology and have constant-intensity rather than self-exciting jumps. Calvet and

Fisher (2008) propose a tightly parameterized Markov chain model for daily log

variance evolution that also lacks volatility feedback. A¨

ıt-Sahalia, Cacho-Diaz,

and Laeven (2015) and Fulop, Li, and Wu (2015) employ affine models with

stochastic volatility and self-exciting volatility jumps, which they estimate on

daily stock market data. Andersen, Fusari, and Todorov (2015) have a model

of self-exciting price/volatility cojumps similar to this paper’s model, but their

estimation methodology differs in relying heavily on matching options data.

The nonparametric literature, of course, makes extensive use of intradaily

returns, typically at a five-minute horizon. That literature focuses primarily on

decomposing intradaily realized variance into diffusive and jump components,

and on developing tests of the null hypothesis of no jumps or cojumps.1Such

analyses can also be conducted in the affine parametric framework used here.

Indeed, as discussed below, any affine latent characteristic can be estimated

from observed data using Bayesian filtration methods: the number and size of

stock market jumps, quadratic variation and its diffusive variance and squared

jump components, and even the magnitude of volatility jumps. Nested models

without volatility jumps can be tested via standard likelihood ratio tests.

The key difference between this paper and prior realized variance papers

is its focus on the intradaily dynamics of diffusive variance and jump intensi-

ties. Nonparametric estimates have an aliasing problem: if integrated diffusive

variances are estimated each day from intradaily data by bipower variation or

threshold techniques, the approach can at best describe the daily dynamics of

the series. This paper,by contrast, estimates dynamic models on intradaily data

to see whether volatility feedback in the form of self-exciting volatility/price co-

jumps is present at intradaily frequencies. The sign and magnitude of every

15-minute return contains important information for the probability of future

1See Jacod and Todorov (2010) for statistical tests of price/volatility cojump models, and Bandi

and Ren`

o(2016) for nonparametric estimates of cojump models on S&P futures returns over the

1982 to 2009 period. The latter includes a model in which the mean and volatility of price jumps

are affected by the level of conditional volatility—another form of volatility feedback.