Shedding light on a dark matter: Jump diffusion and option‐implied investor preferences

• Published date: 01 March 2021
• Author: Hamed Ghanbari,Michael Oancea,Stylianos Perrakis
• Date: 01 March 2021
• DOI: http://doi.org/10.1111/eufm.12276

Eur Financ Manag. 2021;27:244–286.wileyonlinelibrary.com/journal/eufm244

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© 2020 John Wiley & Sons Ltd.

DOI: 10.1111/eufm.12276

ORIGINAL ARTICLE

Shedding light on a dark matter: Jump

diffusion and option‐implied investor

preferences

Hamed Ghanbari

1

|Michael Oancea

2

|Stylianos Perrakis

3

1

Dhillon School of Business, University of

Lethbridge, Lethbridge, Canada

2

Independent scholar, Toronto, Canada

3

John Molson School of Business,

Concordia University, Montreal, Canada

Correspondence

Stylianos Perrakis, RBC Distinguished

Professor of Financial Derivatives, John

Molson School of Business, Concordia

University, 1455 de Maisonneuve Blvd

West, Montreal, QC H3G 1M8, Canada.

Email: stylianos.perrakis@concordia.ca

Funding information

Canadian Social Sciences and Humanities

Research Council, Canada; RBC

Distinguished Professorship in Financial

Derivatives, Canada

Abstract

We compare equilibrium jump diffusion option

prices with endogenously determined stochastic dom-

inance (SD) option bounds. We use model parameters

from earlier studies and find that most equilibrium

model prices consistent with SD bounds yield econom-

ically meaningless results. Further, the implied dis-

tributions of the SD bounds exhibit a tail risk

comparable to that of the underlying return data, thus

shedding light on the dark matter of the inconsistency of

physical and risk‐neutral tail probabilities. Since the SD

bound assumptions are weaker, we conclude that these

bounds should either replace or be used to verify the

equilibrium model results.

KEYWORDS

incomplete markets, jump diffusion, option pricing, risk aversion,

stochastic dominance, tail risk

JEL CLASSIFICATION

G12; G13

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We thank John Doukas (the editor) and the anonymous referees, George Constantinides, Peter Christoffersen, Jin

Chuan Duan, Michal Czerwonko, Suzanne Lee, and participants at meetings of the Northern Finance Association, the

Midwest Finance Association, the European Financial Management Association, and the Multinational Finance Society

for helpful advice and comments. Financial support from the Canadian Social Sciences and Humanities Research

Council and the RBC Distinguished Professorship in Financial Derivatives is gratefully acknowledged.

1|INTRODUCTION

The late Ross (2015, p. 616) defined “dark matter”in finance as “the very low probability of a

catastrophic event and the impact that changes in that perceived probability can have on asset

prices. …When we extract the risk neutral probabilities of such [catastrophic] events from the

prices of options on the S&P 500 we find the probability of …a 25% drop in one month to be

higher than the probability calculated from historical stock return.”This inconsistency between

the implied rare event probabilities extracted from the risk‐neutral

Q

‐distribution of index

options and the observed historical frequency of such events in the underlying market's esti-

mated

P

‐distribution had also been observed earlier by Bates (2000, pp. 220–221) and Jackwerth

(2000, pp. 446‐447). Although the option‐implied return distribution is supposedly forward‐

looking and need not be the same as the one extracted from historical returns, this implied

distribution should not imply implausible statistical behavior that has never been observed in

the real world.

The inconsistency in left tail between observed and option‐implied probability distributions

is one of several failures of the models used in empirical research in the option markets. Such

models cannot account for persistently observed facts characterized as “anomalies,”even

though they have lasted for almost 20 years. These models have, almost in their entirety,

assumed frictionless markets for both underlying and option, ignoring the wide bid–ask spreads

prevailing in the index option market, especially for shorter maturities and out‐of‐the money

(OTM) options.

1

One anomaly in such a frictionless market is the claim that OTM puts are

overpriced,

2

which by put–call parity (PCP) should imply that in‐the‐money (ITM) calls should

also be overpriced. This latter claim has not been made by anyone and, in fact, segmentation of

the market for index puts and calls is admitted explicitly by Chen, Joslin, and Ni (2018, p. 31)

and Constantinides and Lian (2019, pp. 8–9) and contradicted by the anomaly claimants' own

data.

3

Therefore, this anomaly cannot be resolved within these frictionless models as, by con-

struction, they assume unique equilibrium prices for both index and option markets and cannot

relax the PCP assumption. Since frictionless equilibrium is unobservable, the empirical appli-

cations discard all ITM options to maintain PCP.

In this article we show that the inconsistency in tail distribution can be addressed by the

stochastic dominance (SD) approach to the pricing of index options in the presence of rare‐

event risk. This approach has been applied to the valuation of catastrophe‐indexed financial

instruments and is the subject of a recently published book.

4

Since the SD model assumptions

are weaker than the equilibrium model ones, we examine the equilibrium model results within

the framework of SD. Unlike equilibrium models, in SD the

Q

‐distribution is endogenous, in the

form of upper and lower bounds on admissible option prices and the corresponding boundary

pricing kernels given the underlying return

P

‐distribution. As trading becomes more frequent,

the SD option bounds converge to two limits that define an interval within which the “correct”

equilibrium option price should lie. These limits contain all risk‐neutral option prices consistent

1

See Constantinides, Czerwonko, and Perrakis (2020).

2

See Jackwerth (2000), Driessen and Maenhout (2007), Gârleanu, Pedersen, and Poteshman (2009), Bondarenko (2014),

and Chambers, Foy, Liebner, and Lu (2014).

3

See Bondarenko (2014, table I): investing in long positions in at‐the‐money (ATM) puts yields a negative and highly

significant average monthly return of

−

0.39%, whereas long positions in ATM calls yield a positive but nonsignificant

average monthly return of 0.04%.

4

See Perrakis and Boloorforoosh (2013) and Perrakis (2019).

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245

with the observed

P

‐distribution. Although they are consistent with PCP, they can be estab-

lished separately for put and call options, thus avoiding the pitfalls of imposing the PCP

assumption on the observed data.

We consider these SD option bound limits to make inferences about investor preferences

and assess the dark matter and anomaly claims within equilibrium models, which must

be consistent with the weaker assumptions of SD. We find that the “dark matter”disappears in

all examined cases, insofar as the risk‐neutral jump probabilities imbedded in the SD bounds do

not correspond to implausibly large probabilities of catastrophic events. We also find that an

increase in the riskiness of the market, in the form of a higher risk premium on the index,

corresponds to a widening of the bounds by increasing the SD upper bound, consistent with the

adjustment of the option market to the higher perceived risk of the underlying. We show

theoretically that the width of the SD option bound interval, as a proportion of the bound

midpoint, increases for OTM options, as do the corresponding observed bid–ask spreads;

however, the SD option bound interval is tighter than the observed spreads for short‐term

options. For instance, given our base case parameters, the maximum spread between the

bounds is about 3.5% of the midpoint for one‐month ATM call options and 4.5% for 2% OTM

options; these values are lower than the observed bid–ask spreads for similar index options.

5

For the implied relative risk aversion (RRA) extracted from the SD bounds, we concentrate

on those models that assume the existence of a “representative investor”with a constant

relative risk aversion (CRRA) utility. We show that, in equilibrium models, there is a very

narrow range of RRA coefficients consistent with “reasonable”values of their endogenously

determined equilibrium mean return to the index. However, several RRA values in earlier

studies extracted from option market data do not correspond to this narrow interval. On the

other hand, we show that the SD implied bounds on RRA coefficients are consistent with RRA

estimates as high as those that have appeared in equity premium puzzle studies, the only option

price‐implied estimates to possess this property.

The main empirical advantage of the SD option pricing approach vis‐à‐vis the no‐arbitrage

and simultaneous equilibrium models that have dominated index option research is that it uses

information from the underlying asset market only and does not rely on option market data to

provide missing parameter estimates. This is a major advantage, because the large size of the

bid–ask spreads in the option market makes the identification of the “correct”equilibrium

option price a highly uncertain prospect, in addition to invalidating PCP. For instance, on

January 19, 2017, at around 13:30 EST, the S&P 500 index was trading at 2,267, and the close to

ATM February puts and calls with strikes at 2,265 and 2,270 were trading at spreads of 6.8% and

6.15% of their respective midpoints. These spreads escalated dramatically for OTM options in

the same cross‐section, which are the major source of information in calibration of no‐arbitrage

and equilibrium models. For instance, the 2,100 strike put (7.4% OTM) had an 18.2% spread,

and the spread for the 2,400 strike call (5.9% OTM) was a stunning 60.87%. By contrast, the

spreads for the index are very small, implying that the

P

‐distribution is observable almost

without error, thus validating the SD option pricing approach.

Since the

P

‐distribution can be extracted from historical data with very little error, it follows

that the SD bounds can also be estimated accurately without any model error. It is then possible

to assess the errors arising out of the attempts to estimate frictionless model prices from

observed data in a market with frictions. Many no‐arbitrage and simultaneous equilibrium

5

See Constantinides et al. (2020, figure 1).

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