The market for crash risk - New York...

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Journal of Economic Dynamics & Control 32 (2008) 2291–2321

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The market for crash risk

David S. Bates�

Tippie College of Business, University of Iowa, Iowa City, IA 52242-1000, USA

Received 22 March 2006; accepted 3 September 2007

Available online 10 October 2007

Abstract

This paper examines the equilibrium when stock market crashes can occur and investors

have heterogeneous attitudes towards crash risk. The less crash averse insure the more crash

averse through options markets that dynamically complete the economy. The resulting

equilibrium is compared with various option pricing anomalies: the tendency of stock index

options to overpredict volatility and jump risk, the Jackwerth [Recovering risk aversion from

option prices and realized returns. Review of Financial Studies 13, 433–451] implicit pricing

kernel puzzle, and the stochastic evolution of option prices. Crash aversion is compatible with

some static option pricing puzzles, while heterogeneity partially explains dynamic puzzles.

Heterogeneity also magnifies substantially the stock market impact of adverse news about

fundamentals.

r 2007 Elsevier B.V. All rights reserved.

JEL classification: G12; G13

Keywords: Stock index options; Heterogeneous agents; Dynamic equilibria; Complete markets

0. Introduction

The markets for stock index options play a vital role in providing a venue forredistributing and pricing various types of equity risk of concern to investors.Investors who like equity but are concerned about crash risk can purchase portfolio

see front matter r 2007 Elsevier B.V. All rights reserved.

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D.S. Bates / Journal of Economic Dynamics & Control 32 (2008) 2291–23212292

insurance, in the form of out-of-the-money put options. Direct bets on or hedgesagainst future stock market volatility are feasible; most simply by buying or sellingstraddles, more exactly by the options-based bet on future realized varianceproposed by Britten-Jones and Neuberger (2000) and analyzed further by Jiang andTian (2005). By creating a market for these risks, the options markets should inprinciple permit the dispersion of these risks across all investors, until all investorsare indifferent at the margin to taking on more or less of these risks given theequilibrium pricing of these risks. This idealized risk pooling underpins ourtheoretical construction of representative-agent models, and our pricing of risksfrom aggregate data sources – for instance, estimating the consumption CAPMbased on aggregate consumption data.

How well do the stock index option markets operate? Empirical evidence onoption returns suggests that the stock index options markets are operatinginefficiently. Such evidence is based on observed substantial divergences betweenthe ‘risk-neutral’ distributions compatible with observed post-1987 option prices,and the conditional distributions estimated from time-series analyses of theunderlying stock index. Perhaps most important has been the fact that implicit(risk-neutral) standard deviations (ISDs) inferred from at-the-money options havebeen substantially higher on average than the volatility subsequently realized overthe lifetime of the option. Furthermore, regressing realized volatility upon ISDsalmost invariably indicates that ISDs are informative but biased predictors of futurevolatility, with bias increasing in the ISD level.

While the level of at-the-money ISDs is puzzling, the shape of the volatility surfaceacross strike prices and maturities also appears at odds with estimates of conditionaldistributions. It is now widely recognized that the ‘volatility smirk’ that emergedafter the 1987 crash1 implies substantial negative skewness in risk-neutraldistributions, and various correspondingly skewed models have been proposed:implied binomial trees, stochastic volatility models with ‘leverage’ effects, and jumpdiffusions. And although these models can roughly match observed option prices,the associated implicit parameters do not appear especially consistent with theabsence of substantial negative skewness in post-1987 stock index returns. Toparaphrase Samuelson, the option markets have predicted nine out of the past fivemarket corrections, generating surprisingly large returns from selling crash insurancevia out-of-the-money put options.2 A further puzzle is that implicit jump riskassessments are strongly countercyclical. As shown below in Fig. 1, implicit jumprisk over 1988–1998 was highest immediately after substantial market drops, andwas low during the bull market of 1992–1996.

It is of course possible that the pronounced divergence between objective and risk-neutral measures represents risk premia on the underlying risks. The fundamentaltheorem of asset pricing states that provided there exist no outright arbitrageopportunities, it is possible to construct a ‘representative agent’ whose preferencesare compatible with any observed divergences between the two distributions.

1See Rubinstein (1994, pp. 774–775) or Bates (2000, Fig. 2).2See, e.g., Coval and Shumway (2001) or Bakshi and Kapadia (2003).

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0.00

0.04

0.08

0.12

0.16

88 90 92 94 96 98

vari

ance

/yea

r

0

2

4

6

8

10

jum

ps/y

earVt (left scale)

λt (right scale)*

Fig. 1. Implicit variance and jump intensity estimates from S&P 500 futures options, 1988–1998. Vt is the

implicit instantaneous total variance, including jump risk; l�t is the instantaneous jump intensity. Average

jump size: �6.6%; jump standard deviation: 11.0%. Parameter estimates are from a 2-factor stochastic

volatility/jump model; see Bates (2000, model SVJDC2) for estimation details.

D.S. Bates / Journal of Economic Dynamics & Control 32 (2008) 2291–2321 2293

However, Jackwerth (2000) and Rosenberg and Engle (2002) have pointed out thatthe preferences necessary to reconcile the two distributions appear rather oddlyshaped, with sections that are locally risk loving rather than risk averse.Furthermore, the post-1987 Sharpe ratios from writing put options or straddlesseem extraordinarily high – 2–6 times that of investing directly in the stock market.These speculative opportunities appear to have been present in the stock indexoptions markets for almost 20 years.

It may be the stock index options markets are functioning more as insurancemarkets, rather than as genuine two-sided markets for trading financial risks.Viewing options markets as an insurance market for crash risk may be able toexplain some of the option pricing anomalies – especially if the number of insurers isconstrained. If crash risk is concentrated among option market makers, calibrationsbased upon the risk-taking capacity of all investors can be misleading.3 Speculativeopportunities such as writing more straddles become unappealing when the marketmakers are already overly involved in the business. Furthermore, the dynamicresponse of option prices to market drops resembles the price cycles observed ininsurance markets: an increase in the price of crash insurance caused by thecontraction in market makers’ capital following losses.4

This paper represents an initial attempt to model the dynamic interaction betweenoption buyers and sellers. A two-agent dynamic general equilibrium model isconstructed in which relatively crash-tolerant option market makers insure crash-averse investors. Heterogeneity in attitudes towards crash risk is modeled viaheterogeneous state-dependent utility functions – an approach roughly equivalent toheterogeneous beliefs about the frequency of crashes. Crashes can occur in the

3Basak and Cuoco (1998) make a similar point regarding calibrations of the consumption CAPM when

most investors do not hold stock.4Froot (2001, Fig. 3) illustrates the strong, temporary impacts of Hurricane Andrew in 1992 and the

Northbridge earthquake in 1994 upon the price of catastrophe insurance.

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model, given occasional adverse jumps in news about fundamentals. Derivatives areconsequently not redundant in the model and serve the important function ofdynamically completing the market. Given complete markets, equilibrium can bederived using an equivalent central planner’s problem, and the correspondingdynamic trading strategies and market equilibria are identified. Those equilibria arecompared to styled facts from options markets.

There have been previous papers exploring heterogeneous-agent dynamicequilibria, some of which have explored implications for option pricing. Thesepapers diverge on the types of investor heterogeneity, the sources of risk, and thechoice between production and exchange economies. Back (1993) and Basak (2000)focus on heterogeneous beliefs. Grossman and Zhou (1996) explore the general-equilibrium implications of heterogeneous preferences (in particular, the existence ofportfolio insurers) in a terminal exchange economy, given only one source of risk(diffusive equity risk). Options are redundant in this framework, but the paper doeslook at the implications for option prices. Weinbaum (2001) has a somewhat similarmodel, in which power utility investors differ in risk aversion. Bardhan and Chao(1996) examine the general issue of market equilibrium in exchange economies withintermediate consumption, with heterogeneous agents and jump diffusions withdiscrete jump outcomes. Dieckmann and Gallmeyer (2005) use a special case of theBardhan and Chao structure to explore the general-equilibrium implications ofheterogeneous risk aversion.

This paper assumes a terminal exchange economy, and sufficient sources of riskthat options are not redundant. Perhaps the major divergence from the above papersis this paper’s focus on options markets. Whereas Bardhan and Chao (1996) andDieckmann and Gallmeyer (2005) assume there are sufficient financial assets todynamically complete the market, this paper focuses on the plausible hypothesisthat options are the relevant market-completing financial assets. The paper developssome tricks for computing competitive equilibria using the short-dated options withoverlapping maturities that we actually observe. Finally, the hypothesized sourceof heterogeneity – divergent attitudes towards crash risk – is plausible for motiva-ting trading in option contracts that offer direct protection against stock marketcrashes.

The objective of the paper is not to develop a better option pricing model. Thatcan be done better with ‘reduced-form’ option pricing models tailored to thatobjective; e.g., multi-factor option pricing models such as the Bates (2000) affinemodel or the Santa-Clara and Yan (2005) quadratic model. Furthermore, this paperignores stochastic volatility, which is assuredly relevant when building option pricingmodels. Rather, the objective of this paper is to build a relatively simple model of therole of options markets in financial intermediation of crash risk, in order to examinethe theoretical implications for prices and dynamic equilibria. Key issues include:what fundamentally determines the price of crash risk? Can we explain the sharpshifts we observe in the price of crash risk?

Section 1 of the paper recapitulates various stylized facts from empirical optionsresearch that influence model construction. Section 2 introduces the basic frame-work, and identifies a benchmark homogeneous-agent equilibrium. Section 3

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explores the equilibrium when agents are heterogeneous, while Section 4 exploresassociated option pricing implications. Section 5 concludes.

1. Empirical option pricing anomalies and stylized facts

Three categories of discrepancies between objective and risk-neutral probabilitymeasures will be kept in mind in the theoretical section of the paper: volatility, highermoments, and the implicit pricing kernel that in principle reconciles the twomeasures. Furthermore, each category can be decomposed further into averagediscrepancies, and conditional discrepancies.

The unconditional volatility puzzle is that ISDs from stock index optionsare typically higher than realized stock market volatility. For instance, ISDs from30-day at-the-money put and call options on S&P 500 futures over 1988–1998 havebeen on average 2% higher than the subsequent annualized daily volatility ofstock market returns over the options’ lifetime.5 This discrepancy has generatedsubstantial post-1987 profits on average from writing at-the-money puts or straddles,with Sharpe ratios roughly double that of investing in the stock market. See, e.g.,Fleming (1998) or Jackwerth (2000).

The conditional volatility puzzle is that regressing realized volatility upon ISDsgenerally yields slopes that are significantly positive, but significantly less than one.For instance, the regressions using the 30-day ISDs and realized volatilitiesmentioned above yield volatility and variance resultsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

365

T

XT

t¼tþ1

ðD ln F tÞ2

vuut ¼ :0160ð:0142Þ

þ :756 ISDtð:102Þ

þ�tþT ; R2 ¼ :45 (1)

365

T

XT

t¼tþ1

ðD ln F tÞ2¼ :0027ð:0033Þ

þ :681 ISD2t

ð:161Þ

þ�tþT ; R2 ¼ :33 (2)

with heteroskedasticity-consistent standard errors in parentheses.6 Since interceptsare small, the regressions imply that ISDs are especially poor forecasts of realizedvolatility when high. Straddle-trading strategies conditioned on the ISD levelachieved Sharpe ratios almost triple that of investing directly in the stock marketover 1988–1998.

The skewness puzzle is that the levels of skewness implicit in stock index optionsare generally much larger in magnitude than those estimated from stock indexreturns – whether from unconditional returns (Jackwerth, 2000) or conditional upona time-series model that captures salient features of time-varying distributions(Rosenberg and Engle, 2002). Furthermore, implicit skewness remains pronounced

5The puzzle is slightly exacerbated by the fact that at-the-money ISDs are downwardly biased predictors

of the (risk-neutral) volatility over the lifetime of the options.6Jiang and Tian (2005) find similar results from regressions using the ‘model-free’ implicit variance

measure of Britten-Jones and Neuberger (2000).

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for longer maturities of stock index options of, e.g., 3–6 months.7 By contrast, thedistribution of log-differenced stock indexes or stock index futures converges rapidlytowards near-normality as one progresses from daily to weekly to monthly holdingperiods.

A further puzzle is the evolution of distributions implicit in option prices. Fig. 1summarizes that evolution using updated estimates of the Bates (2000) 2-factorstochastic volatility/jump-diffusion model with time-varying jump risk. Sharpchanges are occasionally observed both for total variance and for the instantaneousrisk-neutral jump intensity l�t . The graph indicates that the sharp market declinesobserved over 1988–1998 (in January 1988, October 1989, August 1990, November1997, and August 1998) were accompanied by sharp increases in implicit jump risk.The puzzles here are the abruptness of the shifts (Bates (2000) rejects the hypothesisthat implicit jump risk follows an affine diffusion), and the magnitudes of implicitjump risk achieved following the market declines. Since affine models assume therisk-neutral and objective jump intensities are proportional, these models implyobjective crash risk is highest immediately following crashes. And while assessing thefrequency of rare events is perforce difficult, Bates (2000, Table 9) finds no evidencethat the occasionally high implicit jump intensities over 1988–1993 could in factpredict the intensity of subsequent stock return jumps.

Finally, there is the implicit pricing kernel puzzle discussed in Jackwerth (2000)and Rosenberg and Engle (2002). If the level of the stock index is viewed as areasonably good proxy for the overall wealth of the representative agent, the implicitmarginal utility function of the representative agent can be extracted directly fromthe divergence between the risk-neutral distribution inferred from prices of stockindex options and the objective conditional distribution estimated from stock marketreturns. However, Jackwerth finds these implicit functions can appear oddly shaped,with marginal utility of wealth locally increasing in areas – risk loving, rather thanrisk averse.8

There are currently three leading explanations for the above anomalies: a volatilityrisk premium, a jump risk premium, or demand pressures. Coval and Shumway(2001) and Bakshi and Kapadia (2003) attribute the substantial speculativeopportunities from writing stock index options to a volatility risk premium. Pan(2002), by contrast, finds that the volatility risk premium necessary to reconcile

7In options research, implicit skewness is roughly measured by the shape of the volatility ‘smirk,’ or

pattern of ISDs across different strike prices (‘moneyness’). The skewness/maturity interaction can be seen

by examined by the volatility smirk at different horizons, conditional upon rescaling moneyness

proportionately to the standard deviation appropriate at different horizons. See, e.g., Bates (2000, Fig. 4).

Tompkins (2001) provides a comprehensive survey of volatility surface patterns, including the maturity

effects.8Jackwerth’s results are disputed by Aıt-Sahalia and Lo (2000), who find no anomalies when comparing

average option prices from 1993 with the unconditional return distribution estimated from overlapping

data from 1989 to 1993. The difference in results perhaps highlights the importance of using conditional

rather than unconditional distributions, as in Rosenberg and Engle (2002). For instance, both conditional

variance and implicit standard deviations are time varying; and a substantial divergence between the two

because of mismatched data intervals can produce anomalous implicit marginal utility functions even in a

lognormal environment.

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objective and risk-neutral volatilities implies an excessively upward-sloped termstructure of ISDs, while a substantial risk premium on time-varying jump risk fits theterm structure better. Bates (2000) finds that this model can also match the maturityprofile of implicit skewness better than models with constant implicit jump risk. Thejump risk premium explanation sometimes appears in the guise of expectationalerror; e.g., Jackwerth’s (2000, p. 446) conjecture that OTM puts were overpricedbecause market participants overestimated the frequency of stock market crashes.Demand pressure explanations have appeared in Figlewski (1989), Jiang (2002),Bollen and Whaley (2004), Hodges et al. (2004), and Garleanu et al. (2005),sometimes accompanying the hypothesis that options markets are partly segmentedfrom equity markets. These models attribute the overpricing of OTM puts toexcess demand for those options, while Hodges et al. attribute the Jackwerthanomaly to excess demand for the long-shot positively skewed gambles provided byOTM calls.

The challenge for these explanations is in devising theoretical models ofcompensation for risk consistent with the magnitude of the speculative opportunities.The stochastic evolution of implicit jump risks from option prices also appearsdifficult to explain. This paper will focus on the jump risk premium explanation ofoption pricing anomalies, in an equilibrium model that also considers repercussionsfor equity markets. The apparent magnitude and evolution of the crash risk premiumare the two central stylized facts that I will attempt to match.

2. A jump-diffusion economy with homogeneous agents

I consider a simple continuous-time endowment economy over [0,T ], with a singleterminal dividend payment DT at time T. News about this dividend (or, equivalently,about the terminal value of the investment) arrives as a univariate Markov jumpdiffusion of the form

dln Dt ¼ md dtþ sd dZt þ gd dNt, (3)

where Zt is a standard Wiener process, Nt is a Poisson counter with constantintensity l, and gdo0 is a deterministic jump size or announcement effect, assumednegative. Dt ¼ EtDT is the current signal about the terminal payoff and follows amartingale, implying md ¼ �

12s2d � lðegd � 1Þ.

Financial assets are claims on terminal outcomes. Given the simple specification ofnews arrival, any three nonredundant assets suffice to dynamically span thiseconomy; e.g., bonds, stocks, and a single long-maturity stock index option.However, it is analytically convenient to work with the following three fundamentalassets:

(1)
a riskless numeraire bond in zero net supply that delivers one unit of terminalconsumption in all terminal states of nature;
(2)
an equity claim in unitary supply that pays a terminal dividend DT at time T, andis priced at St at time t relative to the riskless asset; and

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(3)
a jump insurance contract in zero net supply that costs an instantaneous andendogenously determined insurance premium l�t dt and pays off 1 additional unitof the numeraire asset conditional on each jump. The terminal payoff of oneinsurance contract held to maturity is NT �
R T

0 l�t dt.

Other assets such as options are redundant given these fundamental assets, and arepriced by no arbitrage given equilibrium prices for the latter two assets.Equivalently, the jump insurance (or crash insurance) contract can be synthesizedfrom the short-maturity options markets with overlapping maturities that weactually observe. The equivalence between options and crash insurance contracts isdiscussed below in Section 4.2.

Agents are assumed to have possibly state-dependent preferences over terminaloutcomes of the form

UðW t;Nt; tÞ ¼ Et½UðW T ;NT Þ�, (4)

where W T ¼ DT is terminal wealth, NT is the number of jumps over [0,T ], andUðW T ;NT Þ is assumed increasing and concave in WT. Particular specifications willbe discussed below.

Asset prices are determined by the terminal marginal utility of wealth ZT �

UW ðDT ;NT Þ and its current expectation Zt � EtZT – the marginal utility of currentwealth. In particular, the price St of equity (in riskless bond units) is determined bythe Euler condition associated with exchanging St riskless bonds for an uncertainterminal equity payoff DT:

Et½UW ðDT ;NT ÞðDT � StÞ� ¼ 0 (5)

implying

St ¼EtZT DT

Zt

. (6)

The instantaneous equity premium can be derived from the martingale propertiesof Zt and ZtSt, yielding

Et

dSt

St

� �¼ �Et

dSt

St

dZt

Zt

� �. (7)

Crash insurance can be priced comparably. Since crash insurance withinstantaneous cost l�t dt pays off 1 unit of the numeraire conditional upon a jumpoccurring in ðt; tþ dt�, its price is

Ztl�t dt ¼ Et½Ztþdt1jdN¼1� ¼ ldtZtþdtjdN¼1. (8)

This can be rearranged to yield a crash risk premium of the form

l�tl¼ 1þ

dZt

Zt

��dN¼1

. (9)

Thus, the precise evolution dZt=Zt (or, equivalently, d ln Zt) is of key importancefor determining equity and crash risk premia. The nature of that evolution, andits dependency upon the functional form of UW ð�Þ, can be clarified by writing ln Zt in

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the form

ln ZðDt;Nt; t;TÞ ¼ ln Et½UW ðDteDd ;Nt þ nÞ�, (10)

where Dd � lnðDT=DtÞ and n � NT �Nt are future shocks with distributionsindependent of the current values of ðDt;NtÞ. If terminal utility depends solely onterminal wealth DT, then both Zt and St are monotonic functions of current Dt butdo not otherwise depend upon Nt. The most popular utility specification has beenpower utility, which is the only wealth-dependent utility function consistent withstationary equity returns when Dt follows a geometric process such as (3) above.

This paper will explore a state-dependent expansion of power utility, of the form

UðW t;Nt; tÞ ¼ Et eYNTW 1�R

T � 1

1� R

� �; R40. (11)

Associated with this ‘crash-averse’ utility specification is a current marginal utility

Zt ¼ Et½eYNT D�R

T � ¼ eYNt D�Rt Et½e

Yn�RDd �. (12)

As this specification has not previously appeared explicitly in the financeliterature, some motivation is necessary.

First, this specification makes explicit in utility terms what is implicit in the affinepricing kernels routinely used in the affine asset pricing literature. A typical affineapproach for the pricing kernel Zt specifies a linear structure in the underlyingsources of risk:

d ln Zt ¼ mZ dtþ sZ dZt þ gZ dNt. (13)

See, e.g., Ho et al. (1996); or Wu (2006, Eq. (8)) for a recent application involvingLevy processes. Since Zt is nonnegative, such specifications are consistent withabsence of arbitrage. For analytical tractability, affine models place functional-formconstraints on how sZ

2 and the jump intensity can depend on any underlying statevariables, but do not otherwise restrict the magnitudes of sZ and gZ. However, by thejump-diffusion version of Ito’s lemma, any purely wealth-dependent utilityspecification severely constrains the sensitivities of ln ZðDt; tÞ to diffusion and jumpshocks. For instance, the power utility specification (12) with Y ¼ 0 implies

d ln Zt ¼q ln Zt

qtdt� Rd ln Dt ¼ OðdtÞ � Rsd dZt � Rgd dNt (14)

implying relative pricing kernel sensitivities to large versus small shocks areconstrained by the ratio gZ=sZ ¼ gd=sd . Any deviation of postulated relativesensitivities from this ratio is equivalent to introducing state dependency into themarginal utility function (12) of the form Y ¼ gZ þ Rgd .

Perhaps the most intuitive justification is that the crash aversion parameter Y canbe viewed a utility-based proxy for subjective beliefs about crash risk. Investors withcrash-averse preferences (Y40) are equivalent to investors with state-independent

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preferences and a subjective belief that the jump intensity is leY:

E0½eYNT uðW T Þ� ¼

X1N¼0

e�lT ðlTeY ÞN

N!E0½uðW T ÞjN jumps�

¼ elTðeY�1ÞE�0½uðW T Þjl�¼ leY �. ð15Þ

This reflects the general proposition that preferences and beliefs are indistinguish-able in a terminal exchange economy. It should be recognized, however, that thisinterpretation involves very strong subjective beliefs, in that investors do not updatetheir subjective jump intensities leY based on learning over time, or based on tradingwith other investors in the heterogeneous-agent equilibrium derived below.

A final and related justification is provided by Liu et al. (2005), who derive themarginal utility specification (12) from robust-control methods given uncertaintyaversion to imprecise knowledge of the jump intensity. In the deterministic-jump special case of their model, investors consider alternate possibilities lx forthe jump intensity parameter, and trade off the adverse utility consequences ofhigher lx against the divergence of lx from the benchmark l – presumably theempirical point estimate. The outcome of that trade-off (Eqs. (28) and (3) inLiu et al.) is that cautious investors use an upwardly biased jump intensityassessment lea�– an approach observationally equivalent to crash-averse preferencesfor a� ¼ Y .

It is also worth noting that crash-averse preferences (11) possess convenientproperties: they retain the homogeneity of standard power utility, and the myopicinvestment strategy property of the log utility subcase (R ¼ 1). Furthermore, it willbe shown below that crash-averse preferences generate stationary equity returns in ahomogeneous-agent economy.

2.1. Equilibrium in a homogeneous-agent economy

The following lemma is useful for computing relevant conditional expectations.

Lemma. If dt � lnDt follows the jump-diffusion in (3) above, then

EteFdTþcNT ¼ expfFdt þ cNt þ ðT � tÞ½Fmd þ

12F2s2d þ lðeFgdþc � 1Þ�g. (16)

Proof. For t � T � t, there is a probability wn ¼ e�ltðltÞn=n! of observing n �

NT �Nt jumps over (t,T ]. Conditional upon n jumps, Dd � ln DT=Dt is normallydistributed with mean mdtþ ngd and variance sd

2t. Consequently,

EteFdTþcNT ¼ eFdtþcNt Et exp½FDd þ cn�

¼ eFdtþcNt Et exp½FDdjn¼0 þ nðFgd þ cÞ�

¼ eFdtþcNt exp½ðFmd þ12F2s2dÞtþ ltðeFgdþc � 1Þ�. ð17Þ

The last line follows from the independence of the Wiener and jump components,and from the moment generating functions for Wiener and jump processes. &

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Using the lemma and Eqs. (12), (6), and (9) yield the following asset pricingequations:

l� ¼ leY�Rgd , (18)

Zt ¼ D�Rt eYNteðT�tÞ½�Rmdþ

12

R2s2dþðl��lÞ�, (19)

St ¼ Dt expfðT � tÞ½ðmd þ12s2d � Rs2dÞ þ l�ðegd � 1Þ�g. (20)

The last equation implies that the price of equity relative to the riskless numerairefollows roughly the same i.i.d. jump-diffusion process as the underlying news aboutterminal value, with identical instantaneous volatility and jump magnitudes:

dSt=St ¼ mdtþ sd dZt þ kðdNt � ldtÞ (21)

for k ¼ egd � 1. The instantaneous equity premium

m ¼ Rs2d þ ðl� lnÞk � Rðs2d þ lg2dÞ þ ð�lgdÞY (22)

reflects required compensation for two types of risk. First is the required compensationfor stock market variance from diffusion and jump components, roughly scaled bythe coefficient of relative risk aversion. Second, the crash aversion parameter YX0increases the required excess return when stock market jumps are negative.

Crash aversion also directly affects the price of crash insurance relative to theactual arrival rate of crashes:

logðln=lÞ ¼ �Rgd þ Y . (23)

Finally, derivatives are priced as if equity followed the risk-neutral martingale

dSt=St ¼ sd dZ�t þ kðdN�t � ln dtÞ, (24)

where N�t is a jump counter with constant intensity l*. The resulting (forward)option prices are identical to the deterministic-jump special case of Bates (1991),given the geometric jump diffusion.

2.2. Consistency with empirical anomalies

The homogeneous crash aversion model can explain some of the stylized factsfrom Section 1. First, unconditional bias in implied volatilities is explained by thepotentially substantial divergence between the risk-neutral instantaneous variances2d þ l�g2d implicit in option prices, and the actual instantaneous variance s2d þ lg2d oflog-differenced asset prices. Second, the difference between the l* inferred fromoption prices and the estimates of l from stock market returns is consistent with theobservation in Bates (2000, pp. 220–221) and Jackwerth (2000, pp. 446–447) of toofew observed jumps over 1988–1998 relative to the number predicted by stock indexoptions. The extra parameter Y permits greater divergence in l* from l than isfeasible under standard power utility models.

To illustrate this, consider the following calibration: a stock market volatilitysd ¼ 15% annually conditional upon no jumps, and adverse news of gd ¼ �10%

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that arrives on average once every 4 years (l ¼ .25).9 From Eqs. (22) and (23), theequity and crash risk premia are

m � :025Rþ :025Y ; ln ðl�=lÞ ¼ :10Rþ Y . (25)

For R ¼ 1 and Y ¼ 1, the equity premium is 5%/year, while the jump risk l*implicit in option prices is three times that of the true jump risk. Thus, the crashaversion parameter Y is roughly as important as relative risk aversion for the equitypremium, but substantially more important for the crash premium. Achieving theobserved substantial disparity between l* and l using risk aversion alone (Y ¼ 0)would require levels of R that most would find unpalatable, and which would implyan implausibly high equity premium.

Since returns are i.i.d. under both the actual and risk-neutral distribution, thehomogeneous-agent model is not capable of capturing the dynamic anomaliesdiscussed in Section 1. The standard results from regressing realized on implicitvariance cannot be replicated here, because neither is time varying in this model.Second, the model cannot match the observed tendency of l�t to jump contempor-aneously with substantial market drops. Finally, the i.i.d. return structure implies thatimplicit distributions should rapidly converge towards lognormality at longermaturities, which does not accord with the maturity profile of the volatility smirk.

Furthermore, Jackwerth’s (2000) anomaly cannot be replicated under homo-geneous crash aversion. As discussed in Rosenberg and Engle (2002), Jackwerth’simplicit pricing kernel involves the projection of the actual pricing kernel upon assetpayoffs. E.g., stock index options with terminal payoff V ðStÞ have an initial price

v0 ¼E0½ZtV ðStÞ�

Z0¼ E0 V ðStÞ

E0½ZtjSt�

E0½Zt�

� �� E0½V ðStÞMðStÞ�, (26)

where MðStÞ has the usual properties of pricing kernels: it is nonnegative, andE0½MðStÞ� ¼ 1.

It is shown in Appendix A that for crash-averse preferences, this projection takesthe form

MðStÞ ¼ kðtÞS�Rt

pðStjleY Þ

pðStjlÞ, (27)

where k(t) is a function of time and p(Stjl) is the probability density function of St

conditional upon a jump intensity of l over (0, t). Implicit relative risk aversion isgiven by �q ln MðSÞ=q ln S. For Y ¼ 0, one observes the strictly decreasing pricingkernel and constant relative risk aversion associated with power utility. For Y40, it isproven in Appendix A that ln MðStÞ is a strictly decreasing function of ln St that isillustrated below in Fig. 2. The result is relatively intuitive. The ratio pðStjleY Þ=pðStjlÞin (27) is the change of measure from the crash-averse model to an equivalent economy

9As Dt is the signal regarding terminal stock market valuation, sd and gd are appropriately calibrated

from stock market movements. By construction, this paper is using wealth- rather than consumption-

based calibration, for two reasons. First, the empirical option pricing anomalies of Section 1 use wealth-

based criteria. Second, stock market jumps are identified using high-frequency daily data, for which there

do not exist comparable consumption data.

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-0.5

0

0.5

1

1.5

2

2.5

-30% -20% -10% 0% 10% 20% 30%ln (St /S0)

ln M (St)

Fig. 2. Log of the implicit pricing kernel conditional upon realized returns. Calibration: t ¼ 112, sd ¼ .15,

R ¼ Y ¼ 1, gd ¼ �.10, l ¼ .25.


discussed above in Eq. (15), in which homogeneous investors have strictly wealth-dependent preferences uðW T Þ and a subjective belief that the jump intensity is leY.Pricing kernels in this equivalent economy take the form Mt ¼ E�t ½u

0ðDteDdÞ� for

Dd ¼ lnðDT=DtÞ. As this kernel is a strictly decreasing function of Dt (or of St), itcannot replicate the negative implicit risk aversion (positive slope) estimated byJackwerth (2000) and Rosenberg and Engle (2002) for some values of St.

10

Crash-averse preferences (or biased subjective beliefs) do replicate the higherimplicit risk aversion (steeper negative slope) for low ln St values that was estimatedby those authors and by Aıt-Sahalia and Lo (2000). Correspondingly, crash aversioncan generate apparently favorable empirical investment opportunities from putwriting strategies. For instance, the instantaneous annualized Sharpe ratio onwriting crash insurance is

l� � Et½1jdN¼1�=dtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVart½1jdN¼1�=dt

p ¼l� � lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilð1� ldtÞ

p ¼l� � lffiffiffi

lp . (28)

This can be substantially larger than the instantaneous Sharpe ratio m=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 þ lk2

pon equity given investors’ aversion to this type of risk.11 The put selling strategiesexamined in Jackwerth implicitly involve a portfolio that is instantaneously longequity and short crash insurance. Since adding a high Sharpe ratio investment to amarket investment must raise instantaneous Sharpe ratios, this model is consistentwith the substantial profitability of option-writing strategies reported in Jackwerth(2000), Coval and Shumway (2001), and Bakshi and Kapadia (2003).12

10A corollary is that distorted subjective beliefs and the Liu et al. (2005) robust-control approach are

equally incapable of explaining the Jackwerth anomaly.11Ex post Sharpe ratio estimates use instead of , where is the frequency of jumps observed over the data

sample.12Coval and Shumway (2001, Table IV) also explicitly reject the hypothesis that option and stock index

returns are jointly compatible with a power utility pricing kernel.

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3. Equilibrium in a heterogeneous-agent economy

As this model is dynamically complete, equilibrium in the heterogeneous-agentcase can be identified by examining an equivalent central planner’s problem inweighted utility functions. The solution to that problem is Pareto-optimal, and canbe attained by a competitive equilibrium for traded assets in which all investorswillingly hold market-clearing optimal portfolios given equilibrium asset priceevolution. Section 3.1 below outlines the central planner’s problem, while Section 3.2discusses the resulting asset market equilibrium. Section 3.3 identifies the supportingindividual wealth evolutions and associated portfolio allocations, while Section 3.4confirms the optimality of the equilibrium.

3.1. The central planner’s problem

For tractability, I assume all investors have common risk aversion R, but differ incrash aversion Y. Under homogeneous beliefs about state probabilities, the centralplanner’s problem of maximizing a weighted average of expected state-dependentutilities is equivalent to constructing a representative state-dependent utility functionin terminal wealth (Constantinides, 1982, Lemma 2):

UðW T ;NT ;oÞ � maxfW YT g

XY

oY f YðNT Þ

W 1�RYT � 1

1� R; R40

s: t: W T ¼X

Y

W YT ;W YTX0 8Y ð29Þ

for fixed weights o � foY g that depend upon the initial wealth allocation in afashion determined below in Section 3.3. Since the individual marginal utilityfunctions UW ðW YT ;NT ;Y Þ ¼ þ1 at W YT ¼ 0 and the horizon is finite, theindividual no-bankruptcy constraints W YTX0 are nonbinding and can be ignored.Optimizing the Lagrangian

maxfW YT g;ZT

XY

oY f YðNT Þ

W 1�RYT � 1

1� Rþ ZT W T �

XY

W YT

" #(30)

yields a terminal state-dependent wealth allocation

wY ðNT ;T ;oÞ �W YT

W T

¼½oY f Y

ðNT Þ�1=RP

Y ½oY f YðNT Þ�

1=R(31)

and a Lagrangian multiplier

ZT ¼W�RT

XY

½oY f YðNT Þ�

1=R

( )R

�W�RT f ðNT ;oÞ, (32)

where f ð�Þ is a CES-weighted average of individual crash aversion functions f Yð�Þ’s.

The Lagrangian multiplier ZT ¼ UW ðW T ;NT ;oÞ is the shadow value of terminalwealth, and therefore determines the pricing kernel when evaluated at W T ¼ DT .

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From the first-order conditions to (30), all individual terminal marginal utilities ofwealth are directly proportional to the multiplier:

UW ðW YT ;NT ;Y Þ ¼ZT

oY

. (33)

3.2. Asset market equilibrium

As in Eqs. (6)–(9) above, the pricing kernel ZT=Zt can be used to price all assets. Thatasset market equilibrium depends critically upon expectations of average crash aversion.Define

gðNt; t; l0Þ � Et½ f ðNt þ nÞjl0� ¼

X1n¼0

e�l0ðT�tÞ½l0ðT � tÞ�n

n!f ðNt þ nÞ (34)

as the conditional expectation of f ðNT Þ given jump intensity l0 over (t,T ] for future jumpsn � NT �Nt. It is shown in Appendix A that the resulting asset pricing equations are

Zt ¼ ekZðT�tÞD�Rt gðNt; t; le�Rgd Þ (35)

St

Dt

¼ ekSðT�tÞ gðNt; t; leð1�RÞgd Þ

gðNt; t; le�Rgd Þ� ekSðT�tÞmðNt; tÞ, (36)

l�ðNt; tÞ ¼ le�RgdgðNt þ 1; t; le�Rgd Þ

gðNt; t; le�Rgd Þ, (37)

where

kZ ¼ �Rmd þ12R

2s2d þ lðe�Rgd � 1Þ

and

kS ¼ ðmd þ12s2dÞ � Rs2d þ le�Rgd ðegd � 1Þ.

The equilibrium equity price follows a jump-diffusion of the formdSt

St

¼ mðNt; tÞdtþ sd dZt þ kðNt; tÞðdNt � ldtÞ, (38)

where

mðNt; tÞ ¼ �Et

dSt

St

dZt

Zt

� �¼ Rs2d þ ½l� l�ðNt; tÞ�kðNt; tÞ (39)

and

1þ kðNt; tÞ ¼ egdmðNt þ 1; tÞ

mðNt; tÞ(40)

for mðN; tÞ defined above in Eq. (36). The risk-neutral price process follows a martingaleof the form:

dSt

St

¼ sd dZt þ kðN�t ; tÞðdN�t � l�t dtÞ (41)

for N�t a risk-neutral jump counter with instantaneous jump intensity l�ðN�t ; tÞ, thefunctional form of which is given above in Eq. (37).

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Several features of the equilibrium are worth emphasizing. First, conditional uponno jumps the asset price follows a diffusion similar to the news arrival process Dt –i.e., with identical and constant instantaneous volatility sd. This property reflects theassumption of common relative risk aversion R, and would not hold in general underalternate utility specifications or heterogeneous risk aversion.13 A further implicationdiscussed below is that all investors hold identical equity positions.

Second, the equilibrium price process and crash risk premium depends criticallyupon the heterogeneity of agents. This is simplest to illustrate in the R ¼ 1 case, forwhich equilibrium values can be expressed directly in terms of the weighteddistribution of individual crash aversions. Define pseudo-probabilities

pYt �oY exp½YNt þ le�gd ðT � tÞðeY � 1Þ�PYoY exp½YNt þ le�gd ðT � tÞðeY � 1Þ�

(42)

as the Nt-dependent weight assigned to investors of type Y at time t, and define cross-sectional average ECSð�Þ, variance VarCSð�Þ, and covariance with respect to thoseweights. It is shown in Appendix A that the asset market equilibrium takes the form

lnðl�t =lÞ ¼ �gd þ ln ECS½eY � � �gd þ ECS½Y � (43)

lnðSt=DtÞ

T � t¼ � ks þ

ln ECS½eFðT�tÞðeY�1ÞjF ¼ le�gd ðegd � 1Þ�

T � t

� md þ12s2d þ le�gd ECS½e

Y �ðegd � 1Þ, ð44Þ

lnð1þ ktÞ � gd ½1þ le�gd ðT � tÞCovCSðY ; eY Þ�. (45)

Heterogeneity has a divergent impact on the instantaneous level of prices versusthe evolution of prices. To a first-order approximation, the crash risk premium in (43)and equity prices in (44) just replicate at any instant the homogeneous-agentequilibria of (18) and (20) at R ¼ 1, using wealth-dependent weighted average valuesfor Y and eY, respectively. By contrast, the change in equity prices conditional upon ajump has two components: the direct impact of adverse news, and the indirect impactof the change in relative weights as wealth is transferred from crash-tolerant tocrash-averse investors. The result is that relatively modest adverse news aboutterminal dividends can have a substantially magnified impact upon the stock market.

Fig. 3 below illustrates these impacts in the case of only two types of agents,conditional upon the initial wealth distribution and its impact on social weights o(given below in Eq. (47)) and conditional upon an adverse news shock gd ¼ �:03.Crash-tolerant agents (Y ¼ 0) can be viewed as knowing the true jump intensity l.They trade with crash-averse agents (Y ¼ 1), who can be viewed as having asubjective belief that crashes occur at e1 � 2:7 times the true frequency. The presenceof both types of agents in the economy has an extremely pronounced impact on thestock price response to jumps: a modest 3% drop in the terminal value signal caninduce a 3–18% drop in the log price of equity! Crashes redistribute wealth, making

13Weinbaum (2001) and Dieckmann and Gallmeyer (2005) find that heterogeneous risk aversion

increases stock market volatility relative to the underlying sources of risk.

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-20%

-15%

-10%

-5%

0%

0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1

0.5124

0%

5%

10%

15%

0 0.2 0.4 0.6 0.8

Log jump size ln (1 + kt) Crash premium λt /λ

R = 4

R = ½

R = 4

R = 2 R = 1, ½ (indistinguishable)

Equity premium μt = Rσ2 + (λ − λt) kt

R = 4

R = 2 R = 1

R = ½

1

∗

∗

Fig. 3. Impact of initial relative wealth share w1 upon initial equilibrium quantities. Two agents, with

crash aversion Y ¼ 0,1, respectively. Calibration: s ¼ .15, l ¼ .25, gd ¼ �.03; T ¼ 50, t ¼ 0. Common risk

aversion R ¼ 12, 1, 2, or 4 (separate lines).


the ‘average’ investor more crash-averse and exacerbating the impact of adversenews shocks. As indicated in Fig. 3, this magnification is also present for alternatevalues of the risk aversion parameter R.

The crash risk premium lnt =l is always between the e�Rgd value of the crash-tolerant investors (Y ¼ 0), and the eY�Rgd value of the crash-averse investors. Itsvalue depends monotonically upon the relative weights of the two types of investors.The equity premium mt varies with R and with the magnitude of crash risk, but takeson generally reasonable values.

A final observation is that the asset market equilibrium depends upon the number ofjumps Nt, and is consequently nonstationary. This is an almost unavoidable feature ofequilibrium models with a fixed number of heterogeneous agents. Heterogeneityimplies agents have different portfolio allocations, implying their relative wealthweights and the resulting asset market equilibrium depend upon the nonstationaryoutcome of asset price evolution.14 In this model, the number of jumps Nt and time t

are proxies for wealth distribution. Crashes redistribute wealth towards the more crash

14See Dumas (1989) and Wang (1996) for examples of the predominantly nonstationary impact of

heterogeneity in a diffusion context. An interesting exception is Chan and Kogan (2002), who show that

external habit formation preferences can induce stationarity in an exchange economy with heterogeneous

agents.

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averse, making the representative agent more crash averse. An absence of crashes hasthe opposite effect through the payment of crash insurance premia.

3.3. Supporting wealth evolution and portfolio choice

An investor’s wealth at any time t can be viewed as the value (or cost) of acontingent claim that pays off the investor’s share of terminal wealth WT ¼ DT

conditional upon the number of jumps:

W Yt ¼ Et

ZT

Zt

DT wY ðNT ;T ;oÞ� �

¼ StEt½ f ðNT ;oÞðo

1=RY eYNT=R=

PYo

1=RY eYNT=RÞjleð1�RÞgd �

Et½ f ðNT ;oÞjleð1�RÞgd �

� StwY ðNt; t;oÞ, ð46Þ

see Eq. (A.14) in Appendix A for details. The quantity wY ðNt; t;oÞ is the currentshare of current total wealth W t ¼ St, and appropriately sums to 1 across allinvestors. The weights o of the social utility function are implicitly identified up toan arbitrary factor of proportionality by the initial wealth distribution:

wY ð0; 0;oÞ ¼ kE0½o1=RY eYNT=Rf ðNT ;oÞ

1�ð1=RÞjleð1�RÞgd � (47)

for k � E0½ f ðNT ;oÞjleð1�RÞgd �: In the R ¼ 1 case the mapping between o and theinitial wealth distribution is explicit, and takes the form

wY ð0; 0;oÞ ¼ koYelTðeY�1Þ. (48)

The investment strategy that dynamically replicates the evolution of WYt can beidentified using positions in equity and crash insurance that mimic the diffusion- andjump-contingent evolution:

X Yt ¼qW Yt

qSt

¼ wY ðNt; t;oÞ

QYt ¼ ½DW Yt � X YtDSt�dN¼1 ¼ Sð1þ ktÞ½wY ðNt þ 1; t;oÞ � wY ðNt; t;oÞ�, ð49Þ

where kt ¼ kðNt; tÞ is the percentage jump size in the equity price given above inEqs. (40) and (45). Thus, each investor holds X Yt ¼W Yt=St shares of equity (i.e., is100% invested in equity), and holds a relative crash insurance position of

qYt �QYt

W Yt

¼ ð1þ ktÞwY ðNt þ 1; t;oÞ

wY ðNt; t;oÞ� 1

� �. (50)

The wealth-weighted aggregate crash insurance positionsP

Y wY ðNt; t;oÞqYt

appropriately sum to 0.Fig. 4 graphs the individual crash insurance demands (q0,q1) given crash aversions

Y ¼ 0 and 1, respectively, conditional upon the initial wealth share w1 of the crash-averse investors and its impact upon equilibrium ðlnt ; ktÞ at time t ¼ 0. The aggregatedemand for crash insurance w1q1 is also graphed, using the same calibration asin Fig. 3 above. At w1 ¼ 0, crash-tolerant investors (Y ¼ 0) set a relatively lowmarket-clearing price lnt ¼ le�gd and sell little insurance. Crash-averse investors(Y ¼ 1) insure heavily individually, but are a negligible fraction of the market. As w1

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-0.6

0

0.6

1.2

1.8

0 0.2 0.4 0.6 0.8

Crash-averse investors’ q1

Total demand w1 q1

Crash-tolerant investors’q0

1

w1

Fig. 4. Equilibrium crash insurance positions and aggregate demand for crash insurance, as a function of

wealth share w1. Calibration is the same as in Fig. 3, with R ¼ 1.


increases, l�t does as well (see Fig. 3 above) and the crash insurance positions of bothinvestors decline. Aggregate crash insurance volumes are heaviest in the centralregions where both types of investors are well represented. As w1 approaches 1, thehigh price of crash insurance induces crash-tolerant investors to sell insurance thatwill cost them 61% of their wealth conditional upon a crash.

3.4. Optimality

The individual’s investment strategy yields a terminal wealth WYT, and anassociated terminal marginal utility of wealth UW ðW YT ;NT ;Y Þ that from Eq. (33) isproportional to the Lagrangian multiplier ZT that prices all assets. Therefore, noinvestor has an incentive to perturb his or her investment strategy given equilibriumasset prices and price processes. Furthermore, as noted above, the markets for equityand crash insurance clear, so the markets are in equilibrium. Since all individualstate-dependent marginal utilities are proportional at expiration, the market iseffectively complete. All investors agree on the price of all Arrow–Debreu securities,so their introduction would not affect the equilibrium.

4. Option markets in a heterogeneous-agent economy

4.1. Option prices

At time 0, European call options of maturity t and strike price X are priced atexpected terminal value weighted by the pricing kernel:

cðS0; t;X Þ ¼ E0Zt

Z0maxðSt � X ; 0Þ

� �� En

0 ½maxðSt � X ; 0Þ�. (51)

Conditional upon Nt jumps over (0, t], Zt and St have a joint lognormaldistribution that reflects their common dependency on Dt given above in Eqs. (35)

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and (36). Consequently, it is shown in Appendix A that the risk-neutral distributionfor St is a weighted mixture of lognormals, implying European call option prices area weighted average of Black–Scholes–Merton prices:

cðS0; t;X Þ ¼X1n¼0

wn

ncBSMðS0; t;X ; bn; r ¼ 0Þ

¼X1n¼0

wn

n S0ebntNðd1nÞ � XNðd1n � sd

ffiffitpÞ

� , ð52Þ

where l0 � le�Rgd ,

w�n �e�l

0tðl0tÞn

n!

gðn; t; l0Þgð0; 0; l0Þ

,

bn ¼ � l0ðegd � 1Þ þ fngd þ ln½mðn; tÞ=mð0; 0Þ�g=t,

d1n ¼ ½lnðS0=X Þ þ bntþ 12s2d t�=sd

ffiffitp

and N( � ) is the standard normal distribution function. Put prices can be computedfrom call prices using put-call parity:

pðS0; t;X Þ ¼ cðS0; t;X Þ þ X � S0. (53)

Since jumps are always negative, the risk-neutral distribution of log-differencedequity prices implicit in option prices is always negatively skewed. Correspondingly,implicit standard deviations from options prices exhibit a substantial volatility smirkthat is illustrated below in Fig. 5. However, this model’s volatility smirk flattensout at longer maturities. This is inconsistent with empirical evidence from Tompkins(2001) and Bates (2000, Fig. 4), who find that longer-maturity volatility smirksin stock index options are at least as pronounced as those from short-maturityoptions, when moneyness lnðX=S0Þ is measured in maturity-specific standarddeviation units.

4.2. Option replication and dynamic completion of the markets

Options can be dynamically replicated using positions in equity and crashinsurance. Instantaneously, each call option has a price cðSt;Nt; tÞ, and can beviewed as an instantaneous bundle of cS units of equity risk, and ½Dc� cSDS�dN¼140units of crash insurance.

This equivalence between options and crash insurance indicates how investorsreplicate the optimal positions of Section 3.3 dynamically, using the call and/or putoptions actually available. Crash-averse investors choose an equity/options bundlewith unitary delta overall and positive gamma (e.g., hold 11

2stocks and buy one

at-the-money put option with a delta of �12), while crash-tolerant investors take

offsetting positions that also possess unitary delta (e.g., hold 12stock, and write 1 put

option). Equity and option positions are adjusted in a mutually acceptable andoffsetting fashion over time, conditional upon the arrival of news. As options expire,

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-3-2 -1

0 12

30.5

1215%

20%

25%

30%

Mat

urity

(mon

ths)

ln (X/S0) in standard deviation units

Fig. 5. Annualized ISDs, versus moneyness and maturity. Maturity ranges from 12to 12 months, while

moneyness ln(X/S0) ranges over �3ISDatm

ffiffiffiffiTp

. Calibration: w1 ¼ .3, R ¼ 1; other parameters are the same

as in Fig. 3.


new options become available and investors are always able to maintain their desiredlevels of crash insurance. All investors recognize that the price of crash insuranceimplicit in option prices will evolve over time, conditional on whether crashes do ordo not occur, and take that into account when establishing their positions.

A further implication is that the crash-tolerant investors who write optionsactively delta-hedge their exposure, which is consistent with the observed practice ofoption market makers. As l�t =l increases (e.g., because of wealth transfers to thecrash averse from crashes), the market makers respond to the more favorable pricesby writing more options as a proportion of their wealth.15 They simultaneouslyadjust their equity positions to maintain their overall target delta of 1. This strategyis equivalent to market makers putting their personal wealth in an index fund, andfully delta-hedging every index option they write.

4.3. Consistency with empirical option pricing anomalies

The heterogeneous-agent model explains unconditional deviations between risk-neutral and objective distributions analogously to the homogeneous-agent model.The divergence in the jump intensity lnt implicit in options and the true jump frequency

15As indicated above in Fig. 4, the aggregate positions (open interest) in crash insurance and therefore in

options can either rise or fall as the wealth distribution varies.

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l can reconcile the average divergence between risk-neutral and objective variance,and between the predicted and observed frequency of jumps over 1988–1998. Bothmodels generate volatility smirks that flatten out at longer maturities, contrary to thematurity profile of smirks observed in stock index options.

The advantage of the heterogeneous-agent model is that it partially explains someof the conditional divergences as well. First, the stochastic evolution of lnt isqualitatively consistent with the evolution of jump intensity shown above in Fig. 1.lnt depends directly upon the relative wealth distribution, which in turn follows apure jump process given above in (46). Market jumps cause sharp increases in lnt ; thecrash insurance (or options) contracts transfer wealth to crash-averse investors,increasing demand (and reducing supply) for crash insurance. An absence of jumpssteadily transfers wealth in the reverse direction, generating geometric decay in lnttowards the lower level of crash-tolerant investors.

Fig. 6 illustrates the resulting evolution of instantaneous risk-neutral varianceRs2d þ lnt g

2t conditional on the five major shocks over 1988–1998, and conditional

on starting with w1 ¼ .1 at end-1987. This behavior is qualitatively similar to theactual impact of jumps on overall variance and on jump risk shown above in Fig. 1.However, the absence of major shocks over 1992–1996 and the resulting wealthaccumulation by crash-tolerant investors/option market makers implies that theshocks of 1997 and 1998 should not have had the major impact that was in factobserved. Furthermore, all simulated variance shocks are substantially smaller thanthe magnitudes seen in Fig. 1.

It is possible the heterogeneous model can explain the results from ISD regressionsas well. The analysis is complicated by the fact that instantaneous objective and risk-neutral variance are nonstationary, with a nonlinear cointegrating relationship fromtheir common dependency on the nonstationary variable Nt:

Vart½d ln S� ¼ ½s2d þ lg2t �dt; Varnt ½d ln S� ¼ ½s2d þ lnðNt; tÞg2t �dt (54)

0.02

0.025

0.03

0.035

0.04

88 90 92 94 96 98

Fig. 6. Simulated instantaneous risk-neutral variance conditional upon jump timing matching 5 jumps

observed over 1988–1998. Calibration: R ¼ 1, w1 ¼ :1; i.e., crash-averse investors initially own 10% of

total wealth at end-1987. Other parameters are the same as in Fig. 3.

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for gt � ln½1þ kðNt; tÞ� and lnt 4l. It is not immediately clear whether regressingrealized on implied volatility is meaningful under nonlinear cointegration. However,the fact that implicit variance does contain information for objective variance but isbiased upwards suggests that running this sort of regression on post-1987 data wouldyield the usual informative-but-biased results reported above in Eq. (2), withestimated slope coefficients less than 1 in sample.

It does not appear that the heterogeneous-agent model can explain the implicitpricing kernel puzzle. Using the same projection as in (26) above (in Appendix A,Section A.3), the projected pricing kernel is

MðStÞ �E0½ZtjSt�

Z0¼ kðtÞS�R

t

P1N¼0w

nnN pðStjNÞ

pðStÞwhere

wN ¼e�ltðltÞN

N!; wnn

N ¼wNmðN; tÞRgðN; t; leð1�RÞgd ÞP1

N¼0wNmðN ; tÞRgðN ; t; leð1�RÞgd Þð55Þ

and k(t) is a time-dependent scaling factor that does not affect implicit risk aversion.As illustrated in Fig. 7, this implicit pricing kernel appears to be a strictlydecreasing function of St – in contrast to the locally positive sections estimatedin Jackwerth (2000) and Rosenberg and Engle (2002). However, the aboveimplicit kernel can replicate those studies’ high implicit risk aversion for largenegative returns, as indicated by the steep line in Fig. 7 for Ds in the �10% to �15%range.

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-30% -20% -10% 0% 10% 20% 30%ln (St /S0)

ln M(St)

Fig. 7. Log of the implicit pricing kernel conditional upon realized asset returns. Calibration: w1 ¼ .3,

t ¼ 1/12, R ¼ 1 month. Other parameters are the same as in Fig. 3.

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5. Summary and conclusions

This paper has proposed a modified utility specification, labeled ‘crash aversion,’to explain the observed tendency of post-1987 stock index options to overpredictrealized volatility and jump risk. Furthermore, the paper has developed a complete-markets methodology that permits identification of asset market equilibriaand associated investment strategies in the presence of jumps and investorheterogeneity. The assumption of heterogeneity appears to have stronger con-sequences than observed with diffusion models. In particular, stock marketcrashes become partly endogenous. Relatively small adverse announcement effectsbecome substantially magnified by equilibrium wealth redistribution towards morecrash-averse investors. The model in this paper consequently offers an explanationwhy we occasionally observe substantial crashes or ‘corrections’ in the stock market(e.g., the 1987 crash) despite no correspondingly large news about firms’ futureprospects.

The model has been successful in explaining some of the stylized facts from stockindex options markets. The specification of crash aversion is compatible with thetendency of option prices to overpredict volatility and jump risk, while heterogeneityof agents offers an explanation of the stochastic evolution of implicit jump risk andimplicit volatilities. In this model, the two are higher immediately after market dropsnot because of higher objective risk of future jumps (as predicted by affine models),but because crash-related wealth redistribution has increased average crash aversion.Crash aversion is also consistent with the implicit pricing kernel approach’sassessment of high implicit risk aversion at low wealth levels, although the approachcannot replicate the local risk-loving behavior reported in Jackwerth (2000) andRosenberg and Engle (2002).

While motivated by empirical option pricing regularities, the heterogeneous-agentmodel in the paper is unfortunately not suitable for direct estimation. First, jumprisk is not the only risk spanned in the options markets. Stochastic variations inconditional volatility occur more frequently, and are also important to optionmarket makers. Second, the nonstationary equilibrium derived here and character-istic of almost all heterogeneous-agent models hinders estimation. The purpose ofthe paper is to provide a theoretical framework for exploring the trading of jump riskthrough the options market, as an initial model of the option market makingprocess.

The heterogeneous-agent model does, however, have some interesting implicationsfor empirical equity and options research. In particular, the model indicatesthat implicit stock market crash magnitudes should follow stochastic jump processes,and that the magnitudes are related to the extent of investor heterogeneity atany given time. Models with time-varying jump distributions have not beenextensively considered; perhaps they should be. And while there has beenconsiderable work on developing measures of heterogeneity in beliefs (e.g., usinganalysts’ forecasts) and examining implications for equity markets, little of this hasspilled over into options research – with the notable exception of Buraschi andJiltsov (2006).

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The framework in this paper can be expanded in various ways. For simplicity, thispaper has focused on deterministic jumps, but extending the model to random jumpswould be straightforward. A particularly interesting extension could be to explorethe implications of portfolio constraints on positions in options and/or crashinsurance. Selling crash insurance requires writing calls or puts – a strategy thatindividual investors cannot easily pursue. Further research will examine the impactof such constraints upon equilibria in equity and options markets.

Acknowledgments

I am indebted to comments on earlier versions of this article from seminarparticipants at Iowa, Missouri, Toronto, Turin, Wisconsin, the NBER Asset Pricingworkshop, and the 2003 AFA conference. Comments from the editor, CarlChiarella, and three anonymous referees greatly improved the paper.

Appendix A

A.1. Asset market equilibrium in a heterogeneous-agent economy (Section 3.2)

Lemma. If the log signal dt � ln Dt follows the jump-diffusion given above in Eq. (3)and f ðNT Þ is an arbitrary function, then

Et½DmT f ðNT Þjl� ¼ Dm

t ekðmÞðT�tÞEt½ f ðNt þ nÞjlemgd �, (A.1)

where n � NT �Nt, kðmÞ � mmd þ12m2s2d þ lðemgd � 1Þ, and Et½jl� denotes expecta-

tions conditional upon a jump intensity l over (t, T ].

Proof. Define Dd � lnðDT=DtÞ and t � T � t. Then

Et½DmT hðNT Þ� ¼ Dm

t Et½emDd f ðNT Þ� ¼ Dm

t Et½emðDdjn¼0þngd Þ f ðNt þ nÞ�

¼ Dmt e

t½mmdþ12

m2s2d�X1n¼0

e�ltðlemgd Þn

n!f ðNt þ nÞ

¼ Dmt eðT�tÞ½mmdþ

12m2s2

dþlðemgd�1Þ�Et½ f ðNt þ nÞjlemgd �.

ðA:2Þ

Define gðNt; t; l0Þ � E½ f ðNt þ nÞjl0�. The asset pricing equations (35)–(37) follow

directly from the lemma:

Zt ¼ Et½D�RT f ðNT Þ� � D�R

t ekð�RÞðT�tÞgðNt; t; le�Rgd Þ, (A.3)

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St ¼Et½D

1�RT f ðNT Þ�

Zt

¼ Dte½kð1�RÞ�kð�RÞ�ðT�tÞ gðNt; t; leð1�RÞgd Þ

gðNt; t; le�Rgd Þ

� DtekSðT�tÞmðNt; tÞ, ðA:4Þ

lnt ¼ lZtjjump

Zt

¼ e�RgdgðNt þ 1; t; le�Rgd Þ

gðNt; t; le�Rgd Þ: & (A.5)

A.1.1. Asset pricing in the R ¼ 1 subcase

In the special case R ¼ 1 and for arbitrary l0; f ðNT Þ is additively separable andg( � ) becomes

gðNt; t; l0Þ ¼ Et

XYoYe

YNT jl0h i

¼X

YoY exp½YNt þ l0ðT � tÞðeY � 1Þ�. ðA:6Þ

Define l0 � le�Rgd and l00 � leð1�RÞgd , and define pseudo-probabilities

pYt �oY exp½YNt þ l0ðT � tÞðeY � 1Þ�PYoY exp½YNt þ l0ðT � tÞðeY � 1Þ�

(A.7)

as the cross-sectional weight associated with investors of type Y. Using (A.6) forg( � ), the equity pricing equation (A.4) becomes

St

Dt

¼ ekSðT�tÞ

PYoY exp½YNt þ l00ðT � tÞðeY � 1Þ�PYoY exp½YNt þ l0ðT � tÞðeY � 1Þ�

¼ ekSðT�tÞX

YpYt exp½ðl

00� l0ÞðT � tÞðeY � 1Þ�

¼ ekSðT�tÞECS½eFðT�tÞðeY�1Þ� ðA:8Þ

for the cross-sectional expectation ECSð�Þ defined with regard to probabilities (A.7),and for F � l00 � l0 ¼ legd ðe�Rgd � 1Þ. From (A.5), the jump risk premium has asimilar representation:

lntl¼ e�Rgd

PYoY exp½Y ðNt þ 1Þ þ l0ðT � tÞðeY � 1Þ�P

YoY exp½YNt þ l0ðT � tÞðeY � 1Þ�

¼ e�Rgd

XY

pYteY ¼ e�Rgd ECSðe

Y Þ: ðA:9Þ

The approximation for the log jump size follows from the following approxima-tions:

ln mðNt; tÞ � lngðNt; t; l

00Þ

gðNt; t; l0Þ

� ��

q ln gðNt; t; l0Þ

ql0ðl00 � l0Þ, (A.10)

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lnð1þ ktÞ ¼ Rgd þ lnmðNt þ 1; tÞ

mðNt; tÞ

� �� Rgd þ

q ln mðNt; tÞ

qNt

� Rgd þq2 ln gðNt; t; l

0Þ

qNtql0 ðl00 � l0Þ. ðA:11Þ

From (A.6) and (A.7), the cross-derivative turns out to be

q2 ln gðNt; t; l0Þ

qNtql0 ¼ ðT � tÞ

XY

pYtY ðeY � 1Þ �

XY

pYtYX

Y

pYtðeY � 1Þ

( )

¼ ðT � tÞCovCSðY ; eY Þ. ðA:12Þ

Consequently (from (A.11)),

lnð1þ ktÞ � Rgd þ ðl00� l0ÞðT � tÞCovCSðY ; e

Y Þ

� Rgd þ legd ðe�Rgd � 1ÞðT � tÞCovCSðY ; eY Þ. ðA:13Þ

Section 3.3, Eq. (46)

Vt ¼ EtZT

Zt

DTeYNT

f ðNT Þ

� �1=R" #

¼ EtD1�R

T eYNT=Rf ðNT Þ1�1=R

Et½D�RT f ðNT Þ�

" #

¼D1�R

t

D�Rt

ekSðT�tÞ Et½eYNT=Rf ðNT Þ

1�1=Rjleð1�RÞgd �

Et½ f ðNT Þjle�Rgd �. ðA:14Þ

Substituting in St ¼ DtekSðT�tÞEt½ f ðNT Þjleð1�RÞgd �=Et½ f ðNT Þjle�Rgd � from (A.4)

yields (46).

A.2. Objective and risk-neutral distributions

From (36), gross stock returns are

St

S0¼ e�kSt Dt

D0

mðNt; tÞ

mð0; 0Þ(A.15)

for kS ¼ ðmd þ12s2dÞ � Rs2d þ le�Rgd ðegd � 1Þ. Since Dd � ln ðDt=D0Þ is an Nt-depen-

dent mixture of normals, log-differenced stock prices Ds � lnðSt=S0Þ are also amixture of normals:

pðDsÞ ¼X1N¼0

wNnðDsjmN ;s2dtÞ for wN �

e�ltðltÞN

N!, (A.16)

where nðzjm;s2Þ is the normal density function with mean m and variance s2, and

mN � ðR�12s2d Þtþ le�Rgd ðegd � 1ÞtþNgd þ ln½mðN; tÞ=mð0; 0Þ�.

Define 1ðDs ¼ zÞ as the delta function that takes on infinite value when Ds ¼ z,zero value elsewhere, and integrates to 1. The objective density function

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pðzÞ ¼ E0½1ðDs ¼ zÞ�, while the risk-neutral density function is

p�ðzÞ ¼ E0Zt

Z01ðDs ¼ zÞ

� �¼X1N¼0

wNE0½Zt1ðDs ¼ zÞjN jumps�

Z0. (A.17)

For any two normally distributed variables x and y and any arbitrary functionh(y),

E½exhðyÞ� ¼ E½ex�E½hðyþ sxyÞ�, (A.18)

where sxy ¼ Covðx; yÞ. Conditional upon N jumps, ln Zt and Ds are both normallydistributed with covariance �Rsd

2. Consequently, (A.17) can be re-written as

p�ðzÞ ¼X1N¼0

wNE0ðZtjN jumpsÞE0½1ðDs� Rs2d ¼ zÞjN jumps�

Z0

¼X1N¼0

wNE0ðZtjN jumpsÞnðzjmN � Rs2d ;s2d Þ

Z0

�X1N¼0

w�NnðzjmN � Rs2d ;s2dÞ. ðA:19Þ

Since Z0 ¼ E0Zt ¼P

NwNE0½ZtjN jumps�, the weights w�N sum to 1. Furthermore,since

Zt ¼ e�kZðT�tÞD�R0 exp½�RðDdjNt¼0 þNtgdÞ�gðNt; t; le�Rgd Þ, (A.20)

it is straightforward to show that

w�N ¼wNe

�Rgd NgðN; t; le�Rgd ÞP1N¼0wNe�Rgd NgðN; t; le�Rgd Þ

¼e�ltðl0tÞN

N!

gðN; t; l0Þgð0; 0; l0Þ

(A.21)

for l0 � le�Rgd .

A.3. Implicit pricing kernels (Eqs. (27) and (55))

Using Eqs. (19) and (20), the projection of the pricing kernel upon the asset pricein the homogeneous-agent case is

MðStÞ �E0½ZtjSt�

Z0¼ E0½D

�Rt eYNtk0ðtÞjSt�

¼ S�Rt E0

St

Dt

� �R

eYNtk0ðtÞjSt

" #¼ k1ðtÞS�R

t E0½eYNt jSt�, ðA:22Þ

where k0(t) and k1(t) capture time-dependent terms irrelevant to implicit riskaversion. The distribution of st � lnSt is an Nt-dependent mixture of normals:

pðstjNtÞ ¼ nðm0 þNtg2d ;s2dtÞ with probability wNt

¼e�ltðltÞNt

Nt!(A.23)

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for an appropriate choice of m0. Consequently, the conditional expectation in (A.22)can be evaluated using Bayes’ rule to evaluate the conditional probabilities

Prob½Nt ¼ njSt� ¼wnpðstjnÞP1n¼0wnpðstjnÞ

(A.24)

yielding an implicit pricing kernel

MðStÞ ¼ k1ðtÞS�Rt

P1n¼0wnpðstjnÞe

YnP1n¼0wnpðstjnÞ

¼ kðtÞS�Rt

pðstjleY Þ

pðstjlÞ, (A.25)

where pðstjlÞ denotes the unconditional density of st given a jump intensity of l over(0, t]. Taking partials with respect to st and using the fact that qpðstjnÞ=qst ¼

�pðstjnÞ½st � ðm0 þ ngd Þ=s2d � yields (after some tedious calculations) an implicit risk-

aversion value

�q ln MðStÞ

qSt

¼ Rþ�gd

s2dt

Cov��0 ðeY ~n; ~nÞ

E��0 ½Y ~n�, (A.26)

where E��0 and Cov��0 are defined with regard to the probabilities in (A.24). Since eYn

and n are both increasing functions of n, the covariance term is positive.Consequently, the implicit risk aversion is everywhere positive given gdo0.

The heterogeneous-agent case is similar. From (35) and (36), the Lagrangianmultiplier is

Zt ¼ ekZðT�tÞS�Rt

St

Dt

� �R

gðNt; t; le�Rgd Þ

¼ eðkZþRkSÞðT�tÞS�Rt mðNt; tÞ

RgðNt; t; le�Rgd Þ. ðA:27Þ

This is of the same form as (A.22), with mðNt; tÞRgðNt; Þ replacing eYNt .

Consequently, the implicit pricing kernel becomes

MðStÞ ¼ k1ðtÞS�Rt

P1N¼0wNpðstjNÞmðN; tÞ

RgðN; t; le�Rgd ÞP1N¼0wNpðstjNÞ

¼ kðtÞS�Rt

P1N¼0w��N pðstjNÞP1N¼0wNpðstjNÞ

ðA:28Þ

for

w��N �wNmðN ; tÞRgðN ; t; le�Rgd ÞP1

N¼0wNmðN; tÞRgðN; t; le�Rgd Þ.

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