GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.242

Option-Implied Risk Aversion Anomalies: Evidence fromJapanese Market

Nattapol TAKKABUTR

January 2012; revised September 2012GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Option-Implied Risk Aversion Anomalies: Evidence from Japanese Market

Nattapol TAKKABUTR
Graduate School of Economics, Osaka University, Japan
Email: jgm042tn@mail2.econ.osaka-u.ac.jp
September 2012

Abstract
This paper empirically studied the relative risk aversion (RRA) implied from the options and historicalreturns of the Nikkei 225 index around the 2007-2008 subprime loan crisis. The extended use of Japanese option
data and an estimation method of physical density are innovations introduced in this study. The RRA are
typically downward sloping across the options’ moneyness but show a clear U-shape and become negativearound the at-the-money (ATM) level. Also, the RRA level decreases substantially during the crisis. Previous
studies have explained these anomalies as the result of a change in the investor mix or a mispricing of options.

JEL Classification Code: G01, G130, G140
Key words: Crisis, Financial Crisis, Option, Option-implied, Risk Aversion

Acknowledgments
I gratefully acknowledge the comments and opinions provided by participants—in particular, Teruo Nakatsuma—
at the 35th meeting of the Japanese Association of Financial Econometrics and Engineering at Keio University in October
2011. Any remaining errors are of course solely my responsibility.

11.

Introduction
This paper chiefly aims to empirically study the relative risk aversion (RRA) of a representative

stock market investor in Japan to determine how the subprime loan crisis affected risk aversion. In
particular, the study examined the shape of the risk-neutral density derived from a cross section of Nikkei
225 options data from 2006 to 2010 and its relationship to a real-world density, derived from a time seriesof historical returns. The relationship between risk-neutral and real-world densities, as proposed by AitSahalia and Lo (2000) and Jackwerth (2000), is then used as a basis to estimate the implied risk aversion
measure. I found that the average RRA level rose in response to major financial events and when thecrisis began to build up. However, it fell below the pre-crisis level when the Lehman shock hit theJapanese stock market. In the post-crisis period, the RRA rose again but not as high as the pre-crisis level.
Previous studies that used option prices to derive empirical risk aversion measures can be divided
into two broad groups. The first group assumes a representative agent’s utility function in the form of
power and exponential functions that have one parameter as a constant relative risk aversion measure.
Studies in this group usually begin with an estimation of risk-neutral density from option prices, and thenuse risk aversion coefficients as risk preference adjustments to derive real-world densities, for example,
Liu et al. (2007), Bliss and Panigirtzoglou (2004), and Anagnou et al. (2002). The risk aversion measures
can then be estimated conveniently using a closed-form solution based on the assumed utility function.
However, the estimated risk aversion measures are either time-invariant or constant across price levels of
the underlying assets because of the pre-assumed utility function.
The second group of studies does not specify a representative agent’s utility functional form, butrather derives a risk aversion measure directly from the relationship between risk-neutral and real-world
densities, such as Ait-Sahalia and Lo (2000) and Jackwerth (2000). The risk aversion measure estimatedin this group is flexible and time-variant, which may reflect the current aggregate risk aversion better thanthe measures estimated from the first group. However, the estimated risk aversion measures generally
form a U-shape curve across the values of the underlying assets and turn negative for certain asset prices.
This paper can be categorized into the latter group. The risk aversion measures of the Japanesestock market shows a downward sloping trend across a range of underlying indexes; however, they also
show a U-shape pattern and partially fall into the negative region. This ill-behaved risk aversion measurecontradicts the economic theory that a representative agent is risk averse and has a concave utility2

function. Jackwerth (2000) offers a credible explanation for this behavior: the options used to derive riskneutral densities are mispriced. The results of Ait-Sahalia et al. (2001) and Bondarenko (2003) as well
support this mispricing explanation.
This paper further aims to propose a simple approach to solving a dilemma in the estimation ofreal-world density between the validity of a stationary assumption and the reliability of estimatedparameters. Real-world density estimation usually requires a long historical return series for theidentification of reliable distribution parameters. However, risk-neutral density parameters are estimatedfrom a cross section of option data at a single point in time. Ideally, data of the same period should be
used to estimate both risk-neutral and real-world densities to derive an exact implied risk aversion
measure. Moreover, the use of a long past sampling period implies that the return series is stationary overthe entire period, a questionable assumption in the real world. In addition, a non-overlapping series ofreturn should be used in density estimation to avoid the autocorrelation problem. With a 30-day target
horizon, one year data will provide only 12 non-overlapping monthly return observations. This leads to adilemma of whether to use a non-overlapping-return series observed over a long period or employ
overlapping returns from the shortest possible sampling period but subject to the autocorrelation problem.
To address this dilemma, I apply a parametric rather than a non-parametric method to reduce theamount of data required for reliable parameter estimation. Moreover, instead of using a target multiperiod return series, a single-period model is used to forecast multi-period distribution parameters. In
particular, I estimate real-world density using GARCH models both with and without leverage effects.
The GARCH(1,1) model is used for non-leverage and the Glosten−Jagannathan−Runkle [GJR(1,1)]
model (Glosten et al., 1993) for leverage situations. Daily Nikkei 225 returns were used for parameter
estimation of a daily GARCH(1,1) or GJR(1,1) model, and then the 30-day-ahead conditional mean andvariance of the distribution were forecasted using the estimation model. Therefore, with only past-oneyear data, the number of returns increased to around 250 observations. The multi-period conditional mean
and variance are estimated based on the simple law of expectations that does not alter the properties of theassumed model.
According to the likelihood ratio test proposed by Berkowitz (2001), in which the same GARCH
model specification with both normal and Student’s t innovation distributions are compared, models withnormally distributed innovations are better ex-ante forecasts of realized returns density. This contradicts3

the common belief that in comparison with the normal distribution the Student’s t distribution is preferredin modeling equity returns because the t distribution has a fat-tailed property
For risk-neutral density estimation, I also chose a parametric rather than a non-parametric method.
Ait-Sahalia and Lo (2000) and Ait-Sahalia et al. (2001) used pooled cross sections of option data for theobservation period to non-parametrically estimate risk-neutral density. However, as Bliss andPanigirtzoglou (2004) pointed out, pooling cross sections of options data over the observation period
implies that risk-neutral density is stationary across periods, which is a questionable assumption because
the Nikkei 225 varied substantially during the sampling period and the number of available strikes used inestimation changes every observation day. Even if the stationary assumption is satisfied, Rosenberg andEngle (2002) suggested that the obtained risk-neutral density is an average of risk-neutral density over theobservation period, not the risk-neutral density that reflects the current preference of the representative
agent.
I used three different models to estimate risk-neutral density: the Heston (1993) density, thegeneralized beta density of the second kind (GB2), and a mixture of two lognormal densities (MLN).
Though these three models produce similar results, the MLN density is prone to producing erratic RRA
functions, although it is a direct extension of the basic Black-Scholes option pricing models. The Heston
density is a representative of the stochastic volatility models in option pricing but requires complex
numerical integration. The GB2 density has a flexible function form, but this, as well, sometimes
produces erratic results. However, this paper does not aim to determine which risk-neutral density modelis the best.
Section 2 discusses the estimation method in detail. Section 3 then describes the data and section 4
presents an empirical estimation of the results. Finally, I conclude in section 5.

2.
2.1.

Methodology
Risk Aversion Measures
This study builds on the assumptions that there exist a representative investor who is rational andrisk averse or has a concave utility function, and other general characteristics such as a complete andfrictionless market. With these assumptions, Ait-Sahalia and Lo (2000) showed that market-wide or4and the real-world density

aggregate risk-neutral density

representative investor’s utility function U

where

is the current price andkernel function, andare related through theas follows:

is a pricing

the end-of-period price of the underlying asset;

a constant independent of the price level of the underlying asset,. Jackwerth(2000) and Ait-Sahalia and Lo (2000) further illustrated that instead of using a utility function to derive arisk-aversion measure, which requires knowledge of thevalue, the pricing kernel can be written as.

Then, the Arrow-Pratt relative risk aversion measure can be derived from
′′

RRA

′

′

.

′

To obtain the Arrow-Pratt relative risk aversion measuredensity′and the real-world density(1)

defined above, the risk-neutralneed to be estimated. This risk aversion measure is also

called option-implied risk aversion because the estimation of risk-neutral density is usually based on
observed option data. Note that it is also possible for a researcher to predefine thefunction andone of the two densities, and then derive the remaining density. In other words, knowledge of any two of
the elements

,

, and

enables the researcher to estimate the remaining element.

For example, to derive the risk aversion function, Jackwerth (2000) fitted option-implied

volatility to the volatility from his model to derive risk-neutral density
to estimate the real-world density

and used a Gaussian kernel. Ait-Sahalia and Lo (2000) used a non-parametric model with aGaussian kernel to estimate both the risk-neutral density

and the real-world density. On theother hand, Bliss and Panigirtzoglou (2004) aimed to derive the implied real-world density

based

on predefined models of the risk neutral density

2.2.

and the risk aversion function.

Risk-Neutral Density Estimation
I used three different parametric models to estimate risk-neural density: the Heston (1993) density,

the GB2, and a mixture of two lognormal densities (MLN).

5

With regard to the Heston (1993) density, I followed the method applied by Gatheral (2006). Under
follow the process:the Heston density, stock prices

,
̅
〈
where

,
〉

is the conditional variance at time t andto its long-term mean ̅ .

is the speed of reversion of

The return density under Heston can be written as

,

where

∞

1
2

,

∙ exp

,

̅

,

log

/

(2)

∞

is the exercise price of the option,functions of the model parameters

is an integral variable, and

,

and

,

are

, . For details of the parameters and the derivation of the model,

refer to Gatheral (2006) or to the original work, Heston (1993). It should be noted that since risk neutral
densities are estimated for each observation date, the parameter set

implies that the parameters are,

dependent on time t. However, for simplicity, the subscript t is omitted in the distribution parameters.
With regard to the GB2 introduced by Bookstaber and MacDonald (1987), I follow the estimation
method used by Liu et al. (2007). The GB2 density is defined as
1
2,

where

2,

, , ,

,

parameters

,

1

/

,

0

(3)
,

are the parameters of the distribution andis a beta function withand . Risk neutrality imposes a condition that the forward price of the underlying asset

should be equal to
1/ ,
,

,

1/

.

The two-lognormal-density mixture, as well, follows the Liu et al. (2007) method, defined as
,

where

,
,

,

1,

,

,

,

1,

1

2,

2,

is a set of distribution parameters andparameters associated with the individual lognormal density

6,

defined as(4), , j = 1 to 2 is a set of

,

,

2

2

0.5 2

1

.

2 2

2

A European call option under each assumed risk-neutral density is computed as follows:
,

,

where

,

,

,

∞

,

,

,

0

is a theoretical European call option price at strikecomputed under the assumed density models
MLN in (4), given the set of parameters

, .

|

,
,

(5)

,

with time to maturity

with m equal to Hest in (2), GB2 in (3), and,is the risk-free rate andasset at maturity date . All assumed densities

|

,

,

is the price of the underlyingin (5) satisfied the relationship found by

Ross (1976), Breeden and Litzenberger (1978), and Banz and Miller (1978):
2
,

,

,

,

,
.

2

A closed-form solution for the theoretical option prices in (5) under the Heston density in (2) isbased on Gatheral (2006), whereas the GB2 density in (3) and the MLN density in (4) are based on Liu etal. (2007).
The least-squares method is used to estimate the parameters of densities in (2), (3), and (4) byminimizing the sum of the squared differences between the theoretical pricemarket price,,at all strike prices

∗

, ,

min

,

,

,

and the observed

according to the equation

,

,

,

,

,

2

.

Previous studies that estimated risk-neutral density using Nikkei 225 option data include Nakamura
and Shiratsuka (1999) and Shiratsuka (2001) which used the finite difference method documented in
Breeden and Litzenberger (1978) and Neuhaus (1995).
Ait-Sahalia and Lo (2000) applied a non-parametric method to estimate risk-neutral density, but
their method is subject to the following assumption: the representative investor’s preference is constantacross the observation periods, and the derived risk-neutral density reflects the average density over7

multiple periods rather than the current period density, as highlighted by Bliss and Panigirtzoglou (2004)
and Rosenberg and Engle (2002).
A parametric approach offers many candidate models to choose from. Previous studies seem to bein favor of a variation of the spline functions estimation, a mixture of the lognormal density, thegeneralized beta density, and the Heston (1993) density. Examples of relevant studies are Shackleton et al.
(2010), Liu et al. (2007), Moodley (2005), Anagnou et al. (2002), and Melick and Thomas (1997).
For comparability with previous studies, I chose Heston as a representative risk-neutral density
based on the stochastic volatility model. A mixture of two lognormal densities should be more flexible
than one lognormal density, while a mixture of three lognormal densities requires a larger cross section of
option data for reliable parameter estimation. The GB2 is relatively quick to estimate compared with theother two models, with Heston being the most time-consuming.

2.3.

Real-World Density Estimation
Ideally, estimation of real-world density should use the shortest possible observation period of past

returns because risk-neutral density can be estimated with a cross section of option data for the current
observation period. However, estimation of real-world density generally requires the use of a long periodof observed past returns, which in turn may not reflect the true distribution of assets at the current
observation date. Consequently, risk aversion measures derived from risk-neutral and real-world densities
based on different observation periods may be inaccurate.
Any researcher can increase the sample of observed returns by using an overlapping-returns series.
However, the overlapping-returns series would be plagued by autocorrelation problems that would, inturn, result in unreliable parameter estimation. To estimate reliable real-world density parameters using
the shortest possible observation period, I use a daily parametric model to forecast multi-period
distribution parameters. Therefore, a daily returns series can be used in model parameter estimation that,in turn, results in an instant increase in observation data with a minimum autocorrelation problem.
Specifically, I use a three-step estimation process summarized as follows.
1.

Estimate GARCH(1,1) and GJR (1, 1) model parameters based on a daily returns series overthe past one year from the current observation date.

2.

Select the model with the lowest AIC statistic.

8

3.

Use estimated parameters of the selected model to forecast multi-period distribution
parameters matching the target maturity horizon of risk-neutral density.

In the first step, daily stock returns are assumed to follow either a symmetric GARCH(1,1) modelor an asymmetric GARCH model, proposed by Glosten et al. (1993), known as GJR (1, 1). Under the
GJR (1, 1) model, the daily returns of the Nikkei 225 index follow the process:
/

1

where

2

2

1

1 with1

is the daily return ln

,

/

~
1 if

1

at time t,

0,1 ,
1

, otherwise

is the conditional variance at time t, andasymmetric dummy variable. The innovation process

(6)

is an

is assumed to follow either a normal distribution

or a standardized Student’s t distribution. Under step 1, parameters
freedom

1

0

,

,

, and

and the degree ofin the case of a t distribution were estimated. The GARCH(1,1) model is a restricted version of(6) with the coefficient of leverage term

equal to zero.

is assumed to be constant, or it follows thefirst-order autoregressive [AR (1)] process that is automatically selected according to the AIC statistic
criterion in step 2.
Under step 1, a total of eight models are estimated. Four models are estimated under each
; GJR(1,1) with AR(1), GJR(1,1) without AR1, GARCH(1,1) with AR(1) andassumed distribution of

GARCH(1,1) without AR(1). An AIC statistic is then used to select the most appropriate model undereach assumed distribution.
Next, the estimated parameters of the selected model are used to forecast the next single-period
conditional mean

and variancet+2 are then forecasted based on. The single-period conditional mean and variance of periodand. The process is repeated until the forecast period reaches

the target horizon, in this case, 30 days ahead. All of the 30-single-period-forecasts are then summed to∑obtain the multi-period conditional mean∑

and the conditional variance. The relationship between single-period and multi-period parameters is based on theproperties of expectation.

9

3.

Data
This study applies the Nikkei 225 from December 2005 to December 2010 as it represents the broadindex of the Japanese stock market. Option data on the Nikkei 225 index, obtained from NEEDSFinancialQUEST2.0, are used to estimate risk-neutral density. Nikkei 225 options are European-style
options that can be exercised on the second Friday of the expiration month. The settlement price is basedon special quotation (SQ) calculated from the total opening prices of each component stock of the Nikkei
225 on the business day following the last trading day. Strike prices are multiples of ¥500 intervals basedon the Nikkei 225, but for the nearest three expiration months the strike prices are multiples of ¥250
intervals. On any trading day, strike prices were set such that 17 strike prices were available for any
maturity month, eight below and eight above the at-the-money (ATM) strike price. Of a total of 7,155
call and put options at the end of each month during the sampling period, only 4,223 options, or around60%, passed the screening process.
The screening process began by eliminating options with maturities less than seven days. Then, foreach day, the pair of ATM call and put options with the least difference in the closing transaction prices
was identified for each maturity. If a pair could not be identified, the options in that maturity wereentirely disregarded. During this step, an implied forward index level,,, was calculated based on theclosing transaction prices of ATM call and put options from the relationship described in Ait-Sahalia andLo (2000),
,

where

,

,

,

,

,

(7)

is the risk-free rate, which is an average of both buying and selling rates of new issues of three-month certificate of deposits (CDs). is the remaining time-to-maturity of the options, and,are the prices of ATM call and put options, respectively.

,

andis the ATM strike price.

For maturities that ATM options were identified, only options with existing closing bid and ask
prices and ask prices within two times the bid prices were retained. Next, mid-prices were calculated forall options. The risk-neutral densities estimated were based on out-of-the-money options that have more
liquidity than in-the-money options and, hence, are less subject to pricing error. Out-of-the-money puts
were converted to calls using the put-call parity. Only options data that implied volatility could be
estimated and only values less than 200% per year were retained. Table 1 shows summary statistics of the10screened options for 61 months, from the end of December 2005 to the end of December 2010. Only
4,223 options, or around 60% of the total, passed the screening process; of these, 41% were out-of-themoney call and 59% were out-of-the-money put data. An ATM option is defined as a call option withstrike price directly below the implied forward index level estimated from (7). By definition, this option isregarded as an in-the-money call option. Consequently, the call price implied from out-of-the-money put
price was used instead of the real in-the-money call price at ATM strike.
[Table 1 around here]

Real-world densities were also estimated at the end of every month from December 2005 to
December 2010. The estimation window was one year prior to each end-of-month date. For example, areal-world density estimated on the last trading day of December 2005 requires one-year daily Nikkei 225
data from December 2004. Non-overlapping one-month empirical RRA measures are then derived using
the estimated real-world and risk-neutral densities.

4.
4.1.

Empirical Results
Real-World Density Estimation
At the end of each month from December 2005 to December 2010, two real-world densities withnormal and t-distributed innovations were estimated based on the selected models either GARCH(1,1) or
GJR (1, 1) using (6). Parameters were estimated from the daily returns series from one year before theobservation date. The estimation used AIC statistic as a basis to automatically select among fourcandidate models, either a constant mean or AR(1) conditional mean GJR(1,1) or GARCH(1,1),
separately for assumed normal and t-distributed innovations. Table 2 shows average AIC statistics of thefour candidate models as well as how many times each model was selected during an observation period.
Models with the lowest AIC statistic were selected separately under each assumed distribution of
innovations, with Panel (a) showing the results for Student’s t distribution and Panel (b) for the normal
distribution.
[Table 2 around here]
The observation periods were divided into three sub-periods, the pre-crisis period from December
2005 to June 2007, the crisis period from July 2007 to March 2009, and the post-crisis period from April

11

2009 to December 2010. I divided the periods based on the observed movement of the Nikkei 225 index.
Although the Lehman shock actually hit the Japanese stock market in October 2008, the Nikkei 225
continuously declined from July 2007 and did not pick up until April 2009.
It should be noted that the numbers presented in Table 2 are the average AIC statistics of anobservation period. The statistics for any observation month can be quite different from the numbers
shown in Table 2. To provide a meaningful conclusion, Table 2 shows, in parentheses, how many times
each model achieved the lowest AIC statistic and was selected in an observation period. Regardless of theassumed distribution and observation periods, symmetric GARCH(1,1) models dominated GJR(1,1)
models with leverage effects, which were not selected at all.
Under the Student’s t-distributed innovation assumption in Panel (a), the GARCH(1,1) model with aconstant mean but without a leverage term was the dominant model, and was selected 56 times out of 61
observations. The GARCH(1,1) model with conditional mean AR(1) was selected three times in the precrisis period and two times in the post-crisis period.
Under the normal innovation assumption, the symmetric GARCH(1,1) model with a constant mean
was selected 59 times out of 61 observations. The GARCH(1,1) with conditional mean AR(1) was
selected once during the pre-crisis period, in September 2009, and once during the post-crisis period, inNovember 2010.
Table 3 shows averages of the estimated parameters under the GARCH(1,1) models by observation
period. Numbers in parenthesis are average robust standard errors of each coefficient. The numbers in theAR(1) coefficientcolumn are averages of applicable sampling dates only. N/A indicates that models

with the coefficients were not selected at all during the observation period. Again, it should be noted that
average estimated coefficients and standard errors can be used only as reference values for the averagemagnitude of each observation period. An inference about the statistical significance of coefficientsshould not be made because the estimated coefficients on any observation date may be considerably
different from the number shown in Table 3.
[Table 3 around here]

While practitioners often use AIC statistics as a tool for model selection, it cannot be usedrigorously to compare models across different distribution assumptions of innovation series. To compare12between the assumptions under normal and Student’s t-distributed innovation, I used the likelihood ratio(LR) statistic in Berkowitz (2001), which provides a reliable test of the entire distribution even with asmall sample size of forecasted distributions. I could compare the Berkowitz LR statistic across different
estimated densities because it is robust against the functional form of densities.
Nevertheless, it should be noted here that the Berkowitz LR statistic is conceptually different fromthe AIC statistic. The AIC statistic indicates how well a model fits a given data set, but the Berkowitz LR
statistic shows how well a model predicts future realization. It can be seen from Table 2 that the
GARCH(1,1) with a constant mean is the dominant model under both the assumed distributions.
Consequently, to estimate the Berkowitz LR statistic, I only took samples from an observation date when
the constant mean GARCH(1,1) model was simultaneously selected under both normal and t-distributed
innovations. The final observation inputs for the Berkowitz LR statistic is reduced to only 55 observations
under this setting.
According to Berkowitz (2001), an ex-ante forecasted density function is considered appropriate if
the following transformed variable,, is iid N (0, 1):
Ф 1

(8)
∞

is an ex-ante forecasted density function of the realization

where. In this study,

is therealization of the 30-days-ahead Nikkei 225 index level, which is the actual index level on observation
value is calculated and then the series

day t + 30. On every end-of-month date t, the

,

,…,

over

the entire observation period is used to estimate the following AR (1) model with a constant mean andvariance:zz

The maximum-likelihood estimated ̂ ,

, and

(9)

.

1

are used to calculate the log-likelihood valueof (9). The likelihood ratio test statistic can now be given with a restricted likelihood value obtained by
setting

0,

1, and0. Specifically, the Berkowitz LR statistic is in the form
LR

2

0,1,0

13

̂,

2

,

.

(10)

Table 4 shows the Berkowitz LR statistic and log-likelihood values obtained from the real-world
density under the constant mean GARCH(1,1) model with the assumption of normal and Student’s tdistributed innovations. The LR statistics show that a real-world density under normally distributed
innovations is appropriate to forecast a 30-days-ahead density of Nikkei 225 returns. The p-value of aStudent’s t-distributed innovations model implies that the model was rejected at a 1% significance level.
This finding contradicts the general assumption that the Student’s t distribution is a more appropriate
choice than the normal distribution to model equity returns because of its fat-tail property.
[Table 4 about here]

4.2.

Risk-Neutral Density Estimation
At the end of each observation month, a cross section of the options data that passed the screening

process described in section 3 was used to estimate the 30-days-ahead risk-neutral density. An option
series with a maturity nearest to 15 days from 30 days to maturity is used as a proxy for 30-day maturity
options. However, if two option series were within

15 days from 30 days to maturity, risk-neutral

densities were estimated from both near-term and next-term option series and then interpolated in thesame manner as the Chicago Board Options Exchange (CBOE) does for the VIX index. Before
interpolation, each density is normalized by the respective implied forward level,, .

The number of available strikes must meet the minimum strikes requirement of each density model
to ensure the estimation of density parameters. Because Heston’s (1993) density has five parameters, I set
the required minimum number of strikes equal to eight. The GB2 has four parameters with one parameter
restricted to be equal to the forward index level under risk neutrality. The minimum number of strikes for
the GB2 density was set to seven. Because the mixture of two lognormal densities has five parameters,
the minimum number of strikes required was set to 10. However, if the minimum number of strikes for
the Heston and GB2 densities were not attained, the cross section option data of the previous trading day
were used instead. The mixture of two lognormal densities reduced to a lognormal density with aminimum strike requirement of five. If any of the models did not pass the minimum strike requirement,
all models used the same previous trading day data.

14

4.3.

Risk-Aversion Measure Estimation
With both the risk-neutral density

aversion measure RRA

and the real-world density, the relative risk

on each observation month can be estimated using (1). The following results

include only RRA measures that are calculated based on real-world density under the normally distributed
innovations assumption as suggested by Berkowitz’s LR statistics. All RRA measures based on threedifferent models of risk-neutral densities specified in section 2.2 are presented.
The RRA measures defined by (1) were estimated only over the observable strike ranges on eachmonth end. Consequently, RRA measures are defined over a different domain of underlying assets. To
compare the RRA measures across observation periods, I normalized the domain by using moneyness
defined as a ratio of striketo the forward index level,

instead of the gross level of theunderlying index.
Figure 1 shows the commonly observed 30-days-ahead option-implied risk-neutral densities andthe matching horizon real-world density estimated from the GARCH(1,1) model with normal innovation.
The three models of risk-neutral density include the Heston (1993) density, the GB2, and a mixture of two
lognormal densities. The upper rows of Panels (a), (b), and (c) show three typical shapes of densities
found during the pre-crisis period, the crisis period, and the post-crisis period, respectively. The lower
rows indicate the associated relative risk aversion functions

estimated from each pair of real-world and risk-neutral densities using (1).
[Figure 1 around here]

During the pre-crisis period, from December 2005 to June 2007, the mode of real-world densities(labeled RWD-N) is located to the right of the mode of risk-neutral densities (labeled Hest, GB2, and
MLN). A well-behaved downward sloping RRA function in line with economic theory can be observed
when the mode of the real-world density is located further to the right and higher than the mode of therisk-neutral density [Panel (a), left]. A U-shaped pattern can be observed around the ATM level as foundin previous studies, for example, Jackwerth (2000) and Ait-Sahalia and Lo (2000). The RRA function
exhibits a sharp U-shaped curve when the mode of the real-world density falls below and approaches themode of the risk-neutral density [Panel (a), middle]. When news about the subprime mortgage crisis in15the U.S. began to emerge in 2007, the mode of the real-world density moved toward the left of the modeof risk-neutral density, resulting in steep downward sloping RRA functions with sharp U-shape curves[Panel (a), right]. The RRA functions also partially become negative around ATM levels.
In the crisis period, risk-neutral densities become more left skewed, showing expectations offurther decline in asset prices in the future. The relationships between real-world and risk-neutral
densities are the opposite of those in the pre-crisis period. All the modes of real-world densities across theobservation months are located to the left of the risk-neutral densities’ modes. The same downward
sloping RRA function and U-shaped curve around the ATM level can be observed, but the RRA function
turned more negative than the pre-crisis level. The steep downward sloping RRA function can beobserved when the modes of both real-world and risk-neutral densities are close to each other [Panel (b),left]. When the mode of the real-world density fell below those of risk-neutral densities, sharp U-shaped
curves are observed again [Panel (b), middle]. When the Lehman shock hit the Japanese market in
October 2008, an obvious upward sloping risk aversion function implied a perverse risk preference of therepresentative investor [Panel (b), right]. In other words, the representative investor became risk seeking
instead of risk averse.
During the post-crisis period, the mode of real-world density began to move right toward the modeof risk-neutral density. A downward sloping risk aversion function is observed, and the RRA function
becomes less negative around the ATM level. In late 2009, the relationship between real-world density
and risk-neutral densities was similar to the crisis period [Panel (c), left]. In the beginning of 2010, theRRA function around the ATM level began to recover and shift into the positive region when the mode ofreal-world density moved toward the right of the mode of risk-neutral densities [Panel (c), middle].
However, the RRA function around the ATM level fell back a little into the negative region and remained
around -5 till the end of December 2010 [Panel (c), right].
All of the sub-periods show that the RRA functions, on average, are downward sloping in line witheconomic theory. However, the functions sometimes show a sharp U-shaped curve, partially become
negative, and then increase with the level of expected asset prices in the future. These three anomalies
suggest that the standard assumptions in economic theory may be wrong and representative investors may
not be rational or may have a convex utility function. However, as pointed out in Jackwerth (2000), thestandard assumptions may hold true, but the investors’ expectation about future asset prices may not be

16

accurate. The wrong expectation of future returns results in mispricing of options, which in turn distortthe estimation of risk-neutral densities.
Bliss and Panigirtzoglou (2004) explained that the risk preference of the representative investor
might actually change during a period of high volatility of underlying assets. The mix of market
participants changed because investors with greater risk aversion left the market during the high volatile
period. However, I found that Jackwerth’s (2000) option mispricing interpretation might be a moreplausible explanation. Figure 2 shows the effective strike range introduced by Andersen et al. (2011) tomeasure the coverage of options used to compute the volatility index (VIX). The effective range on any
observation date is defined as
1, /
, √

where

,

,

,

, /

,

, √

,

is the Black-Scholes implied volatility of ATM options andthe underlying asset.

,

and

,

(11)

,

is the forward price level ofare, respectively, the lowest and highest strikes observed. The effective

range can be considered as the range of highest and lowest returns at the future date, T.
[Figure 2 around here]

The square shaded area in Figure 2 indicates the crisis period. The minus region represents thecoverage of out-of-the-money put data as, . The positive region shows the coverage of out-of-the-money call data. Notice how the effective range of out-of-the-money put reduced to nearly zero when the
Lehman shock hit the Japanese market in October 2008, during which daily returns varied from +13% to−13%. Options are subject to a high degree of mispricing during such a volatile period. The cross sectionof options data used in the risk-neutral density estimation of October 2008 observations was from theprevious two trading days, and not the precise last trading day of the month. The estimation failed toobtain valid parameters from the data of the previous day and the last trading day because of a lack ofvalid options data.
Figure 3 shows the average relative risk aversion measures
defined byat each level of moneyness,/ . The range of moneyness was limited to 0.90−1.05 levels, which are available ranges

shared by all sub-sampling periods. Results for the pre-crisis, crisis, and post-crisis periods are shown in17

Panels (a), (b), and (c), respectively. Across all observation periods, the RRA function shows a U-shaped
pattern around the ATM level, as in Jackwerth (2000). Figure 3 also confirms the findings in Figure 1 thaton average the RRA functions were positive during the pre-crisis period [Panel (a)], become negative
during the crisis period [Panel (b)], and recover a little in the post-crisis period [Panel (c)].
[Figure 3 around here]

The U-shape pattern found in Figures 1 and 3 suggest that, after the prices of the underlying asset
increase to a certain level, the representative investor’s risk aversion level again increases. Instead of
welcoming the higher returns from rising asset prices, the representative investor prefers lower returns.
As Jackwerth (2000) suggested, the U-shape pattern and negative risk aversion may be the result of
option mispricing. Bondarenko (2003) studied the anomaly of the so-called overpriced puts puzzle of
S&P 500 put options and explained that excessive weighting by investors of the probability of negative
S&P 500 returns results in biased subjective future returns density and, hence, a biased risk-neutral
distribution. Jackwerth (2000) and Ait-Sahalia et al. (2001) showed that an option trading strategy canearn positive returns, which supports the mispricing explanation.

4.4.

Relative Risk Aversion Over TimeTo study the structure of risk aversion over time, the definition of relative risk aversion in (1) isacross levels of strike

modified. On each observation month t, I calculate the sample average ofpricesto obtain a representative relative risk aversion level
over various observation periods. The average levels of. Table 5 shows the average, regardless of the risk-neutral density

model, decreased during the crisis period and recovered in the post-crisis period.
[Table 5 around here]

Figure 4 shows the movement of relative risk aversion ,, over the observation periods

together with Nikkei 225 index levels and returns, as well as effective strike ranges. The shaded area
represents the crisis period. Figure 4 suggests that the RRA function fluctuates over time. During the precrisis period, themoves around 5 to 20 points. During the crisis period, it varies substantially from18

a peak of approximately 28 points in September 2007 to approximately −2 points in September 2008. The
September 2007 peak may be the result of bankruptcies of distressed financial institutions in the U.S. andEurope. In the post-crisis period,rose again, but not until 2010, when it began to recover to the pre-crisis level.
One finding from Figure 4 is in contrast to the general belief that the risk aversion level will
skyrocket when an external shock hits the stock market the hardest. The risk aversion level bottomed out
in September 2008 when Lehman Brothers filed for bankruptcy protection on September 15, and the
Nikkei 225 closed 5% lower than the previous day. Even in October 2008, when the Daiwa Life
Assurance bankruptcy sent the Nikkei 225 index down 10% in one day, the RRA measure was still well
below the non-crisis level. The change in investor mix explanation provided by Bliss and Panigirtzoglou(2004) can also be applied to this anomaly. Only investors with a high degree of risk tolerance remainedin the market, and those with a higher risk aversion left the market during the highly volatile period.
Although their explanation is rational and perhaps accurate, they did not formally test this hypothesis.
Nevertheless, option mispricing is a more plausible explanation, as evident from the lower panel of Figure
4, which shows the effective range of out-of-the-money put options contracted substantially during
September and October 2008. With highly volatile underlying assets, options are subject to a high degreeof mispricing.
[Figure 4 around here]

5.

Conclusion
This study extends previous studies on option-implied risk-neutral density in the Japanese market

by making use of additional information from a real-world density data to derive a relative risk aversion
measure.
A simple approach that uses a daily model to estimate multi-period distribution parameters worked
well in deriving real-world density, as evidenced by the likelihood ratio test of Berkowitz (2001).
However, contrary to the assumption that the Student’s t distribution models stock returns better, theBerkowitz LR test suggested a model with normally distributed innovations. With regard to risk-neutral
density estimations, three different models produced similar results. However, the GB2 and MLN density
models, though flexible and easy to estimate, are prone to producing erratic RRA functions, while the19Heston model involves complex estimation but is less likely to produce an erratic result. I suggest the useof a model that fits the available data on hand. An objective selection from these three candidate models
is beyond the scope of this paper.
In general, the relative risk aversion in the Japanese market shows downward sloping
characteristics consistent with the economic theory that the representative agent is risk averse. However,
the market also shares the ill-behaved risk aversion function as documented in previous S&P 500 studies.
Through an examination of density shapes and the associated RRA function patterns, I showed that thedesired relationship between the real-world and risk-neutral densities occurs when the mode of the realworld density

stays further to the right and higher than the mode of the risk-neutral density.

U-shape and partially negative patterns are also observed in the Japanese market when RRA
measures at each price level of the underlying assets were averaged across observation periods, similar to
Jackwerth (2000). The RRA curves stayed in the negative region at almost all levels of moneyness during
the crisis period, but recovered in the post-crisis period.
On each observation date, the market-wide level of relative risk aversion measureis an average of, whichacross price levels of underlying assets, showed that risk aversion levels

dropped below the normal level during a market turmoil characterized by high-return volatility. Previous
literature preferred the option mispricing explanation for both the U shape and the sinking RRA level
during the crisis period, and showed that a simple option trading strategy can result in abnormal positive
returns. I agree with the mispricing explanation, particularly for the bottomed-out phenomenon, as
evident from a very volatile effective strike range of a cross section of option data. However, whether
some arbitragers could exploit a constant mispricing of options and correct the prices is arguable. Further
research on the mispricing explanation is needed in the Japanese market.

20

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Bond and Currency Options," Review of Financial Studies 6, pp.327–343.

21

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22

Panell (a): Typical shaapes of densitiess and RRA funcctions during prre-crisis period frrom December 20052
to June 20007

Panell (b): Typical shaapes of densitiess and RRA funcctions during criisis period from July 2007 to Maarch 200923

Panell (c): Typical shaapes of densities and RRA funcctions during po st-crisis period fromf
April 2009 to December 20010

Figure 1: Density eestimation andd associated RR
RA functions
The upperu
row in eeach panel shows three typicaal shapes founnd from estimatted real-world densities
innov
vations and riskk-neutral densitties
risk aversionafunctioons

under GARC
CH(1,1) with normalnn (2), GB2 in (33), and MLN inn (4). The loweer rows show reelative
under assumed moodels: Heston ineestimatedfrom each pair of reaal-world and risk-neutral densities using (1).

Figure 2: Effective strike range
The effectiveestrike range on each observation daate

is definned in (11). And
dersen et al. (20011) introducedd the effective ranger
to measuure thecoverrage of options used in the vollatility index (VVIX) calculationn.

24

P
Panel (a):Panel (b):Panel (cc):Figure 3: Average RRA at each priceplevel of th
he underlying asset.
Curv
ves labeled GB22, Heston, and MLN
M
are averaage relative rissk aversion measuresat each leveel of moneynesss defined, by

calcu
ulated from riskk-neutral densitiies under assum
med models andd real-world dennsity under GAR
RCH(1,1) with normal innovations.

25

/

,

Nikkei 2255 index and daaily returns

Reelative risk aveersion measure

based on real-world density under normal innovation models aand three risk--neutral densitties

Effective ranges of optiion strikes

Figure 4: Average relative risk avversion measure
The middle chart sshowsmoneeyness defined as/

and tthe Nikkei 225 index

accross observation periods.
,

is a sample average of relative risk aaversion

for eacheobservationn month t.

26

across levels ofTable 1: Summary statistic of Nikkei 225 options
Options

Options

Average

Averagebeforeafter

beforeafter

screening

screening

screening

screening

OTM call

3,731

2,512

61

OTM put

3,424

1,711

Total

7,155

4,223

Minimum

Maximum

41

23

77

56

28

13

63

117

69

Out-of-the-money is abbreviated as OTM. Options before and after screening refer to the total number of options before
and after the screening process at the end of the month from December 2005 to December 2010 (61 months). Averages
before and after screening refer to the average numbers of options before and after the screening process on eachobservation. Minimum and maximum are the minimum and maximum number of options that passed the screening
process on each observation.

27

Table 2: Average AIC statistics for candidate models of each sampling period
Panel (a): Average AIC statistics forStudent’s t distribution models
Full

BeforeCrisisAfter20051230 : 20101230

20051230 : 20070629

20070731 : 20090331

20090430 : 20101230

Full

BeforeCrisisAfter20051230 : 20101230

20051230 : 20070629

20070731 : 20090331

20090430 : 20101230

Without leverage

GJR(1,1)

GARCH(1,1)

AR meanConstant mean

AR mean

1,432.49

1,429.58

1,422.39

(0)

(0)

(5)

1,569.16

1,566.74

1,557.54

(0)

(0)

(3)

1,409.25

1,406.35

1,397.42

(0)

(0)

(0)

1,332.08

1,328.70

1,325.08

(0)

(0)

(2)

Panel (b): Average AIC statistics fornormal distribution models

With leverage

Constant mean

1,419.42(56)

1,555.37(16)

1,394.31(21)

1,321.52(19)

With leverage

Without leverage

GJR(1,1)

GARCH(1,1)

AR meanConstant mean

1,429.57

1,426.62(0)(0)

1,566.46

1,563.96(0)(0)

1,405.14

1,402.23(0)(0)

1,330.16

1,326.75(0)(0)

AR meanConstant mean

1,417.72

1,412.66

(2)

(59)

1,553.94

1,544.86

(0)

(19)

1,389.64

1,387.22

(1)

(20)

1,322.54

1,318.48

(1)

(20)

For each observation period, a model with the lowest AIC statistic is used to estimate parameters of real-world
density. Enclosed in parentheses is the number of times each model achieved the lowest AIC statistic and wasselected during an observation period.

28

Table 3: Average of estimated parameters for each sampling period
Panel (a): GARCH(1,1) - Student’s t distributed innovations
Observation period
*
Start
End
Full

BeforeCrisis20051230:

20051230:

20070731:

20101230

20070629

20090331

0.0003-0.0210

1.05e-05

0.09800.862775.9110(1.99e-05)(0.0004)(2.68e-07)(2.11e-03)(0.0188)(0.3468)0.0012-0.0223

3.25e-06

0.08220.897483.6669(1.99e-05)(0.0002)(5.27e-08)(7.91e-04)(0.0131)(0.1137)-0.0004

N/A

8.03e-06

0.10520.874530.2719(2.10e-07)(2.61e-03)(0.0181)(0.1836)(2.48e-05)

After20090430:

20101230

0.0002-0.0189

1.96e-05

0.10500.8196114.5328(1.49e-05)(0.0007)(5.20e-07)(2.80e-03)(0.0246)(0.7208)

Panel (b):GARCH(1,1) – normally distributed innovations
Observation period
Start
Full

BeforeEnd20051230:

20051230:

20101230

20070629*

0.0002-0.0352

1.14e-05

0.10960.8448(1.33e-05)(0.0007)(1.91e-07)(1.96e-03)(0.0134)0.0012

N/A

4.07e-06

0.09040.8821(3.60e-08)(1.60e-03)(0.0103)(1.49e-05)

CrisisAfter

20070731:

20090430:

20090331

20101230-0.0005-0.0648

9.67e-06

0.12760.8405(1.64e-05)(0.0013)(2.21e-07)(3.01e-03)(0.0206)

0.0002-0.0056

1.98e-05

0.10890.8153(8.71e-06)(0.0001)(3.01e-07)(1.23e-03)(0.0089)* Average of applicable observations only.
The GARCH(1,1) model is estimated using (6). The mean parameter is, which is automatically selected to be aconstant mean or an AR(1) mean based on AIC statistics. If the AR(1) model is selected
is a constant,

is a first-order autoregressive coefficient and0.

model implies
variance

andis the lag of the current return. The constant mean

is a constant of the conditional variance equation,is the coefficient of conditional

is the coefficient of squared innovation of the previous period

distributed innovations, the degree of freedom, where. In the case of Student’s t-

is an additional parameter. Numbers in parentheses are average

standard errors of coefficients.

29

Table 4: Berkowitz likelihood ratio test of real-world density models
Berkowitz likelihood ratio statistics of GARCH(1,1) with constant mean for full observation periodt-distributed innovations
Normally-distributed innovations
Berkowitz LR statistics

11.4686

4.3561p-value(0.0094)(0.2255)

LL unrestricted-61.5825-65.5777

LL restricted-55.8482-63.3996

Berkowitz LR statistics are calculated using (10) and distributed as chi-squared with three degrees of freedom with a null
hypothesis that the series of transformed variablein (8) are independent with zero mean and unit variance. LL

unrestricted is a maximized log-likelihood value from model (9) without restriction and LL restricted is a maximized
0,log-likelihood value under model (9) with the restrictions
Table 5: Average

1, andbased on real-world density with normally distributed innovations

Observation period
Start
Full

20051230

BeforeCrisisAfter20051230

20070731

20090430

Risk-neutral density models

End
:

:

:

:

20101230

20070629

20090331

20101230

Heston

GB2

MLN

Ave

10.0338

11.0699

11.1104Min-2.4926-1.4471-1.7866

Max

27.8457

28.441928.1556

Ave

13.1501

14.2775

14.3218

Min

4.8355

7.15205.4612

Max

20.3776

21.379122.3255

Ave

7.82718.68738.4533Min-2.4926-1.4471-1.7866

Max

27.8457

28.441928.1556Ave

9.421010.550310.8619Min

2.27523.4242

3.3852

Max

19.8608

20.8870

20.6085

The numbers under each column labeled Heston, GB2, and MLN are average levels of
under the assumed risk-neutral density models.
as

/

0.

is a sample average offor each observation month t.

30

across observation periods
across levels of moneyness defined