Jump risk and option liquidity in an incomplete market

DOIhttp://doi.org/10.1002/fut.21934
Published date01 November 2018
Date01 November 2018
Received: 31 December 2016
|
Revised: 21 April 2018
|
Accepted: 1 May 2018
DOI: 10.1002/fut.21934
RESEARCH ARTICLE
Jump risk and option liquidity in an incomplete market
PeiLin Hsieh
1,2
|
QinQin Zhang
2
|
Yajun Wang
3
1
The Wang Yanan Institution for Studies
of Economics (WISE), Xiamen University,
Fujian, China
2
Department of Finance, School of
Economics, Xiamen University, Fujian,
China
3
China GuangFa Bank, Guangzhou,
China
Correspondence
PeiLin Hsieh, Wang Yanan Institute for
Studies in Economics (WISE), School of
Economics, D316, Economics Building,
Xiamen University, Xiamen,
Fujian 361005, China.
Email: HsiehPeilin@xmu.edu.cn
Funding information
China National Science Foundation,
Grant/Award Number: 71571153;
Fundamental Research Funds for the
Central Universities, Grant/Award
Number: 20720181004
This study investigates the effect of a jump risk on optionsbidask implied
volatility (IMV) spreads. We introduce theoretical models assuming market
makers encounter a Bernoullitype jump atnd optimize the meanvariance
utility by choosing the optimal hedging delta and price. We find, at a low jump
arrival rate, the BlackScholesMerton dynamic hedging for diffusion volatility
outperforms static hedging for both diffusion and jump risks. If dynamic
hedging is implemented, the jump components nonlinearly affect bidask
spreads. Our regression supports our theoretical conclusions, and for modelfree
IMV, jump risk factors are characterized by tstatistics above 7 with adjusted R
2
above 70%.
KEYWORDS
bidask spread, implied volatility, jump, options
JEL CLASSIFICATION
G12, G13, G17, G19
However, looking on the bright side, if we believe in jumps (as we must given the empirical evidence), options
are no longer redundant assets that may be replicated using stocks and bonds and by extension, option traders
can be seen to have genuine social value.
Jim Gatheral, Volatility surfaceA practitioners guide
1
|
INTRODUCTION
Although the options market relies heavily on market makers (MMs) to provide liquidity, the market microstructure in the
derivatives market remains surprisingly underexplored. A limited amount of research has been done so far to empirically
study the characteristics of optionsliquidity; those papers include Chong, Ding, and Tan (2003), Cao and Wei (2010), Wei
and Zheng (2010), and Norden and Xu (2012). Theoretical studies on the relationship between the fundamental feature
(i.e., pricing based on duplication cost) and optionsliquidity are even more sparse. For example, the maturity effect,
referring to an increasing bidask implied volatility (IMV) spread when the time approaches the expiration date, was first
documented in 2003 but did not receive much attention from the filed of theory. Not until recently did Hsieh and Jarrow
(2018) apply modelfree IMV methods to confirm the maturity effect and use the hedging uncertainty, which traders
encounter in an incomplete market, to explain the pattern of increasing bidask IMV spread.
1
J Futures Markets. 2018;38:13341369.wileyonlinelibrary.com/journal/fut1334
|
© 2018 Wiley Periodicals, Inc.
1
The bidask IMV spreadin this paper is always defined as an index by subtracting best bidding IMV from best asking IMV.
Hsieh and Jarrow (2018) actually brought a new argument toward the explanation of liquidity at options market.
Unlike options, stock pricing traditionally involves nothing with duplications, so the hedging (duplication) uncertainty
is much less concerned. Instead, information risk (adverse selection) and inventory and processing cost have been the
main issues to analyze the stocksliquidity. Whereas the prices of options are fundamentally and practically determined
by the duplication (delta hedging) cost; therefore, hedging uncertainty is also an important risk other than prevailing
costs and risks in stocksmarket.
In addition, IMV is a riskneutral measure based on option prices. As a riskneutral measure contains the
information of risk premium, many studies investigate what kind of risk premium attributes to the difference between a
risk neutral and a statistical measure (e.g., Andersen, Fusari & Todorov, 2015; Bollerslev & Todorov, 2011; Carr & Wu,
2009; Duan & Zhang, 2014; Pan, 2002) or extract tail information relating to the jumps from option prices (e.g., Du &
Kapadia, 2012; Gao, Gao & Song, 2018). However, none of them investigates the risk premium by using the information
of optionsbidask spread; hence it is of great interest to understand, theoretically and empirically, what risk premiums
are contained in optionsbidask spreads. Furthermore, while riskneutral probabilities matter significantly in pricing
derivatives, it should be noted that the riskneutral probability is not unique when the assumption of complete market
does not hold. In this regard, the size of the possible sets of riskneutral probabilities could measure the pricing
uncertainty, and the bidask spread measured in IMV (bidask IMV spread) is here in this paper theoretically justified
as a ideal candidate to measure the pricing uncertainty. Understanding the bidask IMV spread helps to comprehend
the uncertainty of option pricing.
Numerous optionpricing models assuming jumpdiffusion process have been proposed; however, mosteither explicitly
or implicitly price the jump size itself rather than variation of the jump size (random components of the jump). Cox and
Ross (1976), Merton (1976), and Amin (1993) assume that the risk of jump size variation can be diversified, and thus the
randomness of jump size is not priced in physical measure. Duffie, Pan, and Singleton (2000) propose an option pricing
model of the affine jump diffusion (AJD) process, and later, Jarrow, Lando, and Yu (2005) show that AJD can be applied
with statistical probability to price derivatives if jump risk is diversifiable. On the other hand, some research also indicates
that the diversification is becoming hard to achieve. As noted by Campbell, Lettau, Malkiel, and Xu (2001), the number of
stocks needed to achieve a given level of diversification has increased such that the explanatory power of the ordinary
capital asset pricing model (CAPM) model has decreased.Given Campbell et al.s (2001) finding and Jarrow et al.s(2005)
theoretical conclusion, we are inspired to study whether the randomness of the jump size is priced or not. And our
research provides a solid and positive answerto this question. More important, as Hsieh and Jarrow (2018) did notdirectly
link the liquidity to any particular cause resulting in an incomplete market, this paper explores the effect of jump
occurrence, a typical case which hinders continuous trading and leads to an incomplete market, on the optionsliquidity.
To study the options bidask spread, we apply the equilibrium model analysis which is different from the approach
of Carr, Geman, and Madan (2001). They introduce the concept of acceptable opportunity and show that a
representative state price density function (RSPF) exists if the economy has no strictly acceptable opportunity. They
advance to derive the conditions for the uniqueness of RSPF, which can be used for pricing options even when the
market is incomplete. Compared to arbitragefree assumption, the acceptable opportunity allows traders to benefit from
taking risks, while it does not need to specify the endowments, return processes, utility functions, and wealth levels.
However, in their setup, the test measures are also subjective and exogenous. We consider our equilibrium model a
special case in their framework where hedging variance is endogenized to be a test measure. Our equilibrium model
tightens the hedging plans and uncertainty premiums and gains a better insight into the dynamic between liquidity and
jump risks.
This study first extends the framework of Amin (1993) to study the bidask spread in the options market. We assume
that diversification assumption does not hold and apply meanvariance utility function to incorporate hedging
uncertainty into the quoting of options. Unlike Kyle (1985), Glosten and Milgrom (1985), and Easley, Kiefer, OHara,
and Paperman (1996) who assume riskneutral MMs and apply Bayesprobability to model the information flows, we
focus on endogenizing hedging risk. Numerous research also assume riskaverse dealers to accommodate the risk other
than information risk. For example, Viswanathan and Wang (2002) endogenize the dealers belief of demand side into
equilibrium of quoting. Hsieh and Jarrow (2018) show some interesting observed bidask IMV spread patterns are
naturally resulted from optimal hedging decision by riskaverse MMs.
In the simple economy with competitive MMs, we show that the equilibrium asking (bidding) price is the price of
Amin (1993) model plus (minus) the risk attitude factor multiplied by hedging variance. After we deliver the simplest
solutions for bidding and asking prices of a call option, we introduce a more general model to incorporate the liquidity
demand side into equilibrium. Given the meanvariance utility framework, the solutions are similar because the
HSIEH ET AL.
|
1335
variables of demand sides always become parts of a multiplier on the solution at the simplest model. Hence, the analysis
of jump componentseffect on bidask spread remains the same for both models, but the magnitude differs given
different parameter setups of the demand side.
Our model shows that MMs require a hedging risk premium above the fair option price, the price assuming a well
diversified economy, to write an option, while they subtract hedging risk premium from the fair option price to make an
offer price for taking the long position. Consequently, the bidask spread is determined by the hedging variance related
to the implemented hedging strategy. We analyze two different hedging schemes, including a dynamic hedging and a
static hedging, and compare the efficacy of these two hedging strategies. Our result indicates that, without considering
transaction cost, dynamic hedging is more efficient under sensible jump intensity (i.e., jumps are rare events with a low
occurrence probability of –ask spread
patterns and conclude that jump components have a nonlinear effect on bidask spread under a sensible jump intensity.
Finally, our paper supplements an empirical evidence to support the implications based on our theoretical analysis.
This paper is organized as follows: Section 2 describes a simple models construction and numerical results under
two hedging schemes; Section 3 generalizes the model to accommodate the demand side into our equilibrium; Section 4
illustrates our sample, estimations of the jump components, and regression results; and finally, Section 5 concludes our
findings and results.
2
|
MODEL
Merton (1976) is the first to employ a jump process to the pricing of options. By assuming that jumps are nonsystematic
such that portfolio jumps can be diversified away, Merton derives a partial differential equation formula for an option
pricing model under a Poisson jump process. Amin (1993) presents a counterpart version for the discrete option pricing
model under a Bernoulli jump process. Because both models assume that the jump risk is diversifiable, a unique
solution can be derived. As the diversification assumption may fail to describe the reality, an extended model that
assumes undiversifiable jump risk is constructed in this paper for an equilibrium analysis.
2.1
|
Setup
Following Amins setup, we use
i
,
∈…
i
T{0,1,2,3 }
to indicate asset trades that occur discretely and
j
,∈−− −
j
{ , , 2, 1,0,1,2, , }to represent price states. Corresponding to the diffusion and jump movements
in a continuous setup, discrete movements include local and nonlocal price changes. If the price at date
t
is
S
t()
j
, the
local price of the underlying asset moves either to +
+
t(1)
j1
or +
t(1)
j1
. Typically, a stock price undergoes local
changes, whereas stock price movements to +
+
S
t(1)
jk
and
∣≥k
2
, are considered a jump occurrence or simply a rare
event. If we focus only on local movements, the setup is the same as in the binomial option pricing model of Cox, Ross,
and Rubinstein (1979) model. As the number of steps increases to infinity, causing the time intervals for each step to
become infinitesimal, the binomial model converges to the BlackScholesMerton (BSM) model. In the binomial
options pricing model, a riskfree, selffinanced portfolio with zero initial value that determines the call option price can
be formed by using a unit of call option,
N
shares of stock and
B
dollars invested or borrowed in the money market at
the riskfree rate
r
. Assume the options are European type with the underlying asset being nondividend paying index.
2
The portfolio value
V
i(
)
j
at time
i
at the initial price state =
j
0is
=++=
V
iNSiCiB() () () 0
.
000 (1)
S
i()
0and Ci(
)
0are the underlying price and the option price at time
i
, respectively. The following condition at the
next step +
i1
must hold for the binomial model.
++ ++ += ++ ++ +=
++ −−
N
SiCiBrNSiCiBr( 1) ( 1) (1 ) ( 1) ( 1) (1 ) 0
.
11 11
(2)
Solving the equations yields solutions for
N
B,
, and Ci(
)
0. They are
1336
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HSIEH ET AL.
2
Our theoretical and empirical work only focus on European type options. For deep ITM call options, the αapproaches 0. This result is quite different from a conventional delta hedge.

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