detecting abnormal market behavior using …

Working Paper 03-13 Departamento de Economía de la Empresa Business Economics Series 03 Universidad Carlos III de Madrid March 2003 Calle Madrid, 126 28903, Getafe (Spain) Fax +34 91 6249608

DETECTING ABNORMAL MARKET BEHAVIORDETECTING ABNORMAL MARKET BEHAVIOR USING RESAMPLI USING RESAMPLING TECHNIQUESNG TECHNIQUES

Lucía Cuadro-Sáezô, J. Ignacio Peña‡ and Juan J. Romo‡

AbstractAbstract

This article improves the assessment of market sentiment through risk neutral implicit density functions estimates. The main finding is an indicator of “abnormal market behavior”, which allows us to measure the degree to which the market is behaving normally and is not expecting bad news. We test the results on two kind of events in four out of the five main crises in emerging markets of the last decade: defaults on debt and devaluations or strong depreciations. We find evidence that, in many cases, this new indicator could have anticipated the main critical episodes of the crises shortly ahead. We use confidence intervals based on bootstrap percentiles for the skewness and kurtosis of the implicit risk neutral density functions to detect whether the market behaves normally or shows excessive risk aversion.

Keywords: Market sentiment, risk aversion, implicit risk neutral density function, bootstrap, skewness and kurtosis.

JEL Codes: G14, G15, F34.

ô Banco de España and Universidad Carlos III de Madrid. ‡ Universidad Carlos III de Madrid. We gratefully acknowledge Miguel de Las Casas for his help and suggestions. Also we thank Jose Maria Cuadro for interesting discussions at early stages of preparing this paper; and Jose Manuel Campa, Daniel Navia and Alicia García-Herrero for useful comments. All remaining errors are our own. The ideas expressed in the paper represent the personal opinion of the authors, which do not necessarily coincide with those of Banco de España. Corresponding address: [email protected]

Abstract

This article improves the assessment of market sentiment through risk

neutral implicit density functions estimates. The main …nding is an indica-

tor of “abnormal market behavior”, which allows us to measure the degree

to which the market is behaving normally and is not expecting bad news.

We test the results on two kind of events in four out of the …ve main crises in

emerging markets of the last decade: defaults on debt and devaluations or

strong depreciations. We …nd evidence that, in many cases, this new indi-

cator could have anticipated the main critical episodes of the crises shortly

ahead. We use con…dence intervals based on bootstrap percentiles for the

skewness and kurtosis of the implicit risk neutral density functions to detect

whether the market behaves normally or shows excessive risk aversion.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Data and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2.1 Implicit risk neutral density functions . . . . . . . . . . . . 12

3.2.2 The bootstrap method . . . . . . . . . . . . . . . . . . . . 19

4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1. Introduction

Financial markets have experimented a deep transformation throughout the last

two decades. Financial development and new technologies allow for more trans-

parency on trading operations, lower costs and an greater information availability.

There is no doubt about the growing in‡uence the stock market exerts on …nancial

3

development, not just in terms of size but also in terms of e¢ciency. Neverthe-

less, one should be aware of the drawbacks associated to this development since,

we have witnessed severe crashes on the stock markets during this period. Fi-

nancial market drops sometimes lead to market crashes or …nancial crises, which

may be dependent on market sentiment. “Self-ful…lling expectations” are critical

determinants of market crashes, specially when the country is dependent on for-

eign investment and therefore, vulnerable to external shocks1 . This is the case of

emerging market economies, which have su¤ered more dramatically the impact

of …nancial crises, either due to problems within the country or caused by conta-

gion among countries. This paper contributes to the literature by examining the

role of expectations during crises events using an indicator of imminent negative

event. This new methodology aimed at predicting “abnormal market behavior”

lays on the implicit risk neutral density function (PDF), as a measure of market

expectations, on the skewness and kurtosis of the PDF, as indicators of the risk

aversion. Our main contribution, consists on the identi…cation of the risk aversion

limits within “normal market behavior”. The idea is to determine the risk aver-

sion limit, at a given con…dence level, over which the market behaves abnormally.

Therefore, we can detect the abnormality by examining those observations out of

the con…dence intervals.

We test this new approach in four crises su¤ered by non industrial countries

1See García-Herrero (2001).

4

during the nineties2 : the Asian in 1997, the Russian in 1998, the Brazilian in 1999

and the Argentinean crisis in 2001. The results show that our indicator is able to

foresee “abnormal market behavior”.

The rest of the article is organized as follows: the next section presents a

literature review on implicit risk neutral density functions, the third one describes

the data and the methodology used, the fourth section presents the results and

the last one concludes.

2. Literature Review

After the 1987 stock market Crash, the Black and Scholes model presents a de-

viation on the volatility curve, named as volatility smile. When comparing the

moneyness degree with the implicit volatility, the function presents a smile-shape,

pro…le indicating an excessive volatility of the very in the money options and of

the very out of the money options, with lower volatility for the at the money

options. The deviation from the constant volatility assumed by the Black and

Scholes model becomes signi…cant after the crash, and re‡ects a di¤erent pricing

rule: the market behavior seems to be risk averse3.

Another way of measuring the same fact is to look at the implicit risk neutral

2See García-Herrero, A. (2001)3Peña, Rubio and Serna (1999) …nd that time to expiration is a main determinant of the

volatility smile. This is consistent with the idea that the implicit distribution function may varywith the time to expiration.

5

probability function, which also presents a deviation that is interpreted as the risk

averse behavior of the market. After the 1987 Crash, the implicit risk neutral

distributions became leptokurtic and left skewed4. This excessive skewness and

kurtosis are re‡ecting the risk averse behavior, provided that the left skewness

(for the call options) and the higher kurtosis are indicators of the probability

assigned to a market drop. The point is that before the Crash, the market did

not take into account the possibility of a crash. However, it happened, the market

learned, including this knowledge in the option valuation we observe thereafter. In

the following paragraphs we review the literature on implicit risk neutral density

functions.

The seminal paper on the implicit density functions is Breeden and Litzen-

berger (1978) where they study the relation between the price of an European call

option and the probability of gain when executing the contract. In other words,

they look for the relationship of the call option price and the probability of the

strike being lower than the price of the underlying asset at the expiration date.

This literature on implicit risk neutral probability functions can be classi…ed

into two research trends depending on the parametric or non-parametric approach.

The parametric vs. non-parametric distinction adopted in this paper is similar to

that presented in previous studies5. A parametric approach assumes that the

4See Jackwerth and Rubinstein (1996)5 Ibid

6

distribution function of the PDF is well known and that its coe¢cients need to

be estimated. The non-parametric approach, does not assume any distribution

function, therefore it allows for the estimation of any probability function. There

is no consensus in the literature on the best method to be applied. Furthermore,

it seems that there is a proper method for each empirical study.

Within the non-parametric literature, Rubinstein (1994) develops an optimiza-

tion criterion to extract the PDFs from options prices. He obtains the PDFs from

a binomial tree, composed by the strikes of the options and their probability of

gain. In the same line, Jackwerth and Rubinstein (1996) base their investiga-

tion on the paper by Rubinstein (1994) and develop an alternative optimization

criterion that is tested it in the American S&P500 options market. They prove

that the PDFs look like a lognormal in the period prior to the Crash of 1987

and that it becomes leptokurtic, left skewed and sometimes multimodal, after it.

Finally, Manzano and Sanchez (1998) apply the results of Breeden and Litzen-

berger (1978) to derive the implicit risk neutral probability function of the Spanish

option market on interest rates.

By contrast, the other research trend applies parametric methods to compute

the PDFs. This literature is represented by Melick and Thomas (1997) who

assume a mixture of three lognormal distributions for the PDFs of oil prices, and

by Malz (1994), who obtains the probability of realignment of the exchange rates

during the European Monetary Crisis by assuming the lognormal shape for the

7

PDFs and a jump di¤usion process for the option prices. Bahra (1997) compares

some distributions proposed in the existing literature and selects the mixture of

two lognormal distributions to characterize the PDFs. Finally, Söderlind and

Svensson (1996) select a mixture of n lognormal distributions for the shape of the

PDFs.

This paper adopts a non-parametric approach, by computing the PDFs as

Jackwerth and Rubinstein (1996) propose. We go one step further transforming

the put options prices call option prices using the put-call parity. This transfor-

mation permits the observation of a wider range of moneyness given the limited

liquidity of the Spanish out of the money option market. Therefore, the range of

moneyness for which we compute the PDFs is wider than that we could obtain

by simply using the call options or the put options data.

3. Data and Methodology

3.1. Data

The empirical research presented in this paper is based on the data provided by

Mercado Español de Futuros Financieros de Renta Variable (MEFF RV) for the

European type options and futures on IBEX 35 index, and by Mercado Continuo

de Bolsa de Madrid for the underlying index, IBEX 35. The database includes

daily closing prices from December 9th, 1996, to October 7th, 2002.

8

The PDFs are calculated using both call and put options so we can complete

the database including all the information available on option prices. No dis-

tinction is made on the type of option, since both put and call prices provide

information on the risk aversion considered by the market. Put option prices are

transformed into call option prices using the put-call parity:

c = p+ St ¡X(1 + r)¡t

Where c is the call option price, p is the put option price, St is the asset value

at t and X(1 + r)¡t is the actual value of the strike price.

We exclude all those options presenting null volatility or their price being equal

to zero or not satisfying the put-call parity.

3.2. Methodology

The objective of the paper is the assessment of the market feeling on the extremes.

We identify the point up to which the market can be considered as behaving nor-

mally in terms of risk aversion. The graphs in Appendix 1 show the di¤erent

shapes of the PDF depending on the market sentiment that we catalog as “nor-

mal”, “rare” and “abnormal market behavior”. The classi…cation re‡ects the level

of excess risk aversion detected by our indicator: “normal” days do not present

any excess risk aversion, “rare” days do not have enough information to con…rm

the excess risk aversion, although we think that these are non- normal days; and

9

…nally an “abnormal market behavior” day, in which we can ensure at a 90%

con…dence level that there exist an excess of risk aversion in the market.

The PDF is a measure of market sentiment. This …eld has been largely studied

and we use the approach of Jackwerth and Rubinstein (1996) to compute the

distributions. This is a non-parametric method in the sense that the resulting

PDF may adopt any shape, either lognormal or any other distribution function.

Once we have the assessed market feeling, we evaluate risk aversion by looking

to the deviation from the risk neutral assumption. Under a risk neutral behavior

the PDFs should not present any excess of skewness or kurtosis. The excess of

(left) skewness observed in the PDFs of the (call) option prices is interpreted as the

market’s consideration of the possibility of a large drop in the future (the option

expiration date). In other words, the greater drop considered by the market, the

higher left skewness, other things left equal. The excess kurtosis indicates the

probability mass assigned to the drop. This means not only that the market

considers the possibility of a drop, but that it may become extremely worrying

when the market assigns a high probability to a large size drop. “Self-ful…lling

expectations” constitute a key factor in determining whether a large drop in the

market may become a crisis.

At any point in time, we can have the skewness and kurtosis of the PDF,

but we need a benchmark to determine whether the observation responds to an

abnormal behavior or not. Indeed, we should have many PDFs on the same day

10

to establish a reliable con…dence interval, but the market is not liquid enough to

compute them. Thus, we establish a benchmark con…dence interval based on the

bootstrap percentiles.

Within each year, for each expiration date, we take its 39 previous days, and

we compute the con…dence intervals for those PDFs satisfying certain conditions

that will be later explained. Therefore, we have con…dence intervals based on

the bootstrap percentiles with an average of 7830 observations for skewness and

kurtosis (see table in Appendix 2 for detailed statistics on the number bootstrap

replications per year and time to expiration) . The results are used to characterize

each day prior to the expiration by a certain con…dence interval within each year.

These con…dence intervals are the benchmark model to detect “abnormal market

behavior”.

The con…dence intervals are computed on the basis of year and days to expi-

ration. The skewness and kurtosis tend to decrease as the expiration date gets

closer, what makes necessary a con…dence interval for each day prior to expiration.

Besides, the market behavior evolves over time. Liquidity, information release and

market size among other aspects point out the need to consider the market’s abil-

ity to learn as was clearly demonstrated after the 1987 Crash. Therefore, we

consider subperiods of one year provided that the con…dence intervals computed

under certain market’s knowledge, also evolve over time.

In sum, we characterize the “normal market behavior” using the con…dence

11

intervals for the skewness and kurtosis of the PDFs, which are the proxies of the

market risk aversion. Note that the excess skewness indicates the size of the drop

discounted by the market, since it valuates those options very far from the at the

money options (that is a very low moneyness degree). On the other hand, the

excess kurtosis means a higher probability than the assumed under the “normal

market behavior” on the tails of the distribution, then it is interpreted as the

excess probability assigned to a drop in the market, when looking at the left side

of the distribution

On the following subsection we summarize the theoretical basis for the method-

ology applied in this paper. First, we present a detailed explanation on the im-

plicit risk neutral probability functions, and later on, we describe the bootstrap

resampling technique.

3.2.1. Implicit risk neutral density functions

Following Breeden and Litzenberger (1978) and Jackwerth and Rubinstein (1996),

we apply the following model for extracting the PDF from the option prices.

We assume risk neutral agents. This implies that risk preferences do not have

any impact in the price of the option when it is expressed in terms of the underlying

asset price. In this sense, all the agents are assumed to behave according to the

same utility function, so that we can consider an unique utility function for the

market.

12

Given the risk neutrality assumption, the price of an European call option can

be interpreted as a function of the probability of gain considered by the trader

and the time needed to execute the contract (note that the time to expiration is

the time needed to realize the gain).

c = f[P (ST > X); T ¡ t]

where P (¢) is the distribution function implicit in the option price, ST is the

underlying asset price at the expiration date, X is the strike price and T ¡ t is

the time to expiration.

We can isolate the e¤ect of the implicit distribution function by …xing a given

time to expiration, ¿ = T ¡ t:

c = f[P (ST > X )] f or ¿ = T ¡ t

The research on PDF analyzes the following relation:

P (ST > X) = g(c)

where P (¢) is the implicit distribution function, ST is the underlying asset price

at the expiration date, X is the strike price and g(c) is a function that relates the

price of the option with the probability of the option being in the money at the

13

expiration date.

This paper adopts a non parametric method for extracting information, as the

seminal paper of Breeden and Litzenberger (1978), where they prove that options

with the same underlying asset and expiration date and with di¤erent prices, may

be combined to replicate an asset whose returns depend on the market value at

the expiration date (what they call “state”). The price of this asset re‡ects the

expectation on the market taking a given value (“state”) at the expiration date.

For instance: an Arrow-Debreu (A-D) asset gives 1 unit under a given state, and

0 otherwise. Applied to our model:

output(A ¡D) =

8>>><>>>:

1 if ST = X

0 otherwise

where ST is the price of the underlying asset at the expiration date, and X is

the strike price. The price of an A-D asset is directly proportional to the risk

neutral probability of the state, that is, the price of the asset is directly related

to probability of the market reaching the given state (taking a given value) at the

expiration date.

The problem is that A-D prices can not be observed in the market. The

solution is to replicate the A-D asset using a butter‡y spread. The price of a

butter‡y spread, centered at a given market value, is the price of an A-D asset

for that state (market value). We will explain this idea carefully on the following

14

paragraphs.

A butter‡y spread is composed by 4 call options: a short position of 2 calls

with strikeX = ST , and a long position of two calls; one with strikeX = ST¡4ST

and the other one, with strike X = ST +4ST .

Assume that the market value at the expiration date has a discrete probability

distribution with possible values: M (T) = 1; : : : ; N . The payment vector of the

European call option with strike X and expiration at T is denoted by c(X;T ).

The payment matrix is de…ned by:

0BBBBBBBB@

c(ST ¡ 4ST ; ¿ ) c(ST ; ¿) c(ST +4ST ; ¿)MT = ST ¡ 24ST 0 0 0MT = ST ¡ 4ST 0 0 0MT = ST 4ST 0 0MT = ST +4ST 24ST 4ST 0MT = ST +24ST 34ST 24ST 4ST

1CCCCCCCCA

The payment vector for a butter‡y spread is the combination of the payment

vectors of the options that form the portfolio. The price of the A-D asset (centered

at X = ST ), in terms of the European call option prices, is given by the following

equation:

pAD(ST ; ¿ ;4ST) =[c(ST +4ST; ¿ ) ¡ c(ST ; ¿)]¡ [c(ST ; ¿) ¡ c(ST ¡ 4ST ; ¿ )]

4ST¤e¡r(t¡t)

This pricing function is analogous to a density function since the A-D asset

15

only pays under a given “state”. Then, the butter‡y spread (replication of the

A-D asset) makes a gain if, and only if, the market takes the value of the strike at

the expiration date. Therefore, the butter‡y spread value at each of the possible

market values and strikes is a measure of the probability of obtaining bene…ts.

The moneyness degree of any option is de…ned as the proportion of the strike

price relative to the price of the underlying asset at the end of the contract. In

the money call options present a strike price lower than the …nal price of the

underlying asset, and therefore, a moneyness degree lower than one. On the

contrary, out of the money options have higher strike price than the …nal price of

the underlying asset and …nally, the at the money options present a moneyness

degree equal to one. Note that we ignore the …nal price of the asset, denoted by

ST ; so we replace it by the present price of the future, Ft; with the same expiration

date than the option.

moneyness =X

Ft

8>>>>>>>><>>>>>>>>:

< 1 ¡! in the money

= 1! at the money

> 1 ¡! out of the money

We are interested in in the money options since their price contains the proba-

bility of the strike being lower than the price of the underlying asset at the end of

the contract (i.e. the probability of the market being lower than the strike price).

The probability measure computed through the butter‡y spread is quite sim-

16

ilar to the smoothness criterion developed by Jackwerth and Rubinstein (1996).

This last method behaves better than other optimization methods they test, for

instance, those using quadratic objective functions or maximum entropy objective

functions among others.

The PDF computation requires the following process: The …rst step is the es-

timation of the relation between the implicit volatility and the moneyness degree,

we assume a quadratic approximation:

¾ = a+ bm+ sm2

where ¾ is the volatility computed as Black and Scholes (1973) and m is the

moneyness degree observed in the market. The second step takes homogenized

series of moneyness (m) for each day. The homogenization consists on computing

a series with equal distance between two consecutive levels of moneyness. The

maximum and minimum moneyness observed in the market (m) are taken as

bounds for the estimated moneyness (m) to compute a new series for the volatility.

The technique used to approximate the volatility smile is not unique. In this paper

we use a similar method to those used by Shimko (1993) or by Campa, Chang and

Refalo (1999), …tting the implied volatility smile by a quadratic function on the

moneyness degree. This procedure can re‡ect that the Black and Scholes (1973)

model does not hold and even more, allows for the variation of volatility along the

17

moneyness degree. We estimate the implicit volatility for the homogenized series

of moneyness:

¾ = a+ bm+ sm2

With the estimated volatilities, ¾; we compute their corresponding option prices

with Black (1976) model.

c = e¡r(T¡t) ¤ [F ¤N (d1) ¡X ¤ N(d2)]

Where:

N(d1) =ln( F

X) + ¾2

2(T ¡ t)

¾ ¤ pT ¡ t

N(d2) = 1¡ ¾ ¤pT ¡ t

Finally, we use the result by Jackwerth and Rubinstein (1996) to compute the

implicit probabilities through the equation:

Pr(m) =cj¡1 ¡ 2 ¤ cj ¡ cj+1

±m¤ e¡r(T¡t)

Where Pr(m) is the probability assigned to the level of moneyness, m; cj is

18

option price corresponding to the jth market value, ±m is the distance between

two levels of moneyness and, e¡r(T¡t) is the discount factor. When the range of

moneyness considered is wide enough, the PDF can be completely characterized.

Unfortunately, sometimes there is not enough liquidity in the market, specially

when the market sentiment is not clear, and this provokes that the PDFs can

not be completely characterized. These days, present a cumulative probability

signi…cantly lower than 1. Our interpretation is that the market is not behaving

normally although we can not con…rm any assumption about the market sentiment

when this occurs. This problem can be partially mitigated by transforming put

option prices into call option prices although, as will be shown later, this solution

is not very rewarding.

3.2.2. The bootstrap method

The bootstrap is a resampling technique proposed by Efron (1979)6. It is used for

assigning measures of accuracy to statistical estimates. The idea of the bootstrap

consists on drawing with replacement the observations from the original sample

to create one independent bootstrap sample, which can have any size. The boot-

strap algorithm generates a large number B of independent bootstrap samples

of size n, x¤b; that replicate the original sample, by drawing with replacement

from the original dataset. Corresponding to each bootstrap sample, x¤b, there

6See Efron and Tibshirani (1993) for a complete description of the Bootstrap.

19

is a bootstrap replication for any statistic, s(x¤b). The main application of this

technique, which is applied in this paper, is that the distribution of the statis-

tic, s(x); can be inferred from the bootstrap replications of the statistic, s(x¤b).

Provided with these replications, s(x¤b), we can construct the con…dence intervals

for the skewness and kurtosis statistics. We characterize the “normal market be-

havior” through these con…dence intervals and we use them as a benchmark to

compare the new observations coming from the market, those values of skewness

and kurtosis exceeding the con…dence intervals are considered as an evidence of

“abnormal market behavior”.

The procedure applied to build the con…dence intervals for skewness and kurto-

sis is the following: For each day prior to expiration, we take the original sample,

which contains the trading date, the expiration date, the strike price, and the

corresponding price for each option traded in the market7. From this original

sample we draw with replacement 1000 bootstrap samples. The size of the boot-

strap samples equal the size of the original sample8. Each bootstrap sample is

processed as an independent trading date so that we obtain, for each day prior

to expiration, one observation for skewness and one observation for kurtosis from

the original sample and also. Besides, we have 1000 bootstrap replications for the

7Remember that put option prices were transformed into call option prices through the put-call parity.

8That is, for each day prior to expiration we have an original sample of size n and 1000replications of the original sample of the same size, n.

20

skewness and for the kurtosis statistics from the bootstrap sample replications.

We use the 1000 bootstrap replications for the statistics to compute the con…dence

intervals at a 90% con…dence level. The discussion on the con…dence level will

be presented in the next section. Note that the large number of the bootstrap

replications ensures the accuracy of the con…dence intervals estimations.

The following section presents the results of our empirical study, in which

we apply the described methodology to build a leading indicator for imminent

negative event.

4. Results

We a¢rm these con…dence intervals constitute a benchmark for “normal market

behavior” because all bootstrap replications of the PDFs are …ltered to ensure their

“normality”. In this sense, we erase from the sample all those PDFs that have

a cumulative probability smaller than 0.99 or greater than 1. The histograms of

skewness and kurtosis are available in Appendix 3. The distributions for skewness

during the six years is right skewed while the distribution of kurtosis is left skewed.

This re‡ects nothing but the lack of liquidity. When the PDF can not be properly

characterized, the skewness tends to be larger than one and the skewness tends

to be below 2.8. We need to keep in mind that sometimes, the market lacks a

su¢cient degree of liquidity, as the narrow range of moneyness shows in some

critical days. and this is shown on the narrow range of moneyness during some of

21

the critical days. This was especially notorious in 1997, when the Spanish options

and futures market was still very young. In occasions, we have two estimations

of the PDF on the same trading day, one for each of the next two expirations

dates. Needless to say that the longer the expiration date, the smaller the market

liquidity and therefore, the greater the anomalies when examining the “normal

market behavior”.

For each trading date, we compare the observed skewness and kurtosis with

their corresponding con…dence interval to test the excess risk aversion in the mar-

ket behavior. Recall that the con…dence interval depends on the trading year and

on the days to expiration. Appendix 4 shows the graphs of the observed skewness

and kurtosis and their corresponding con…dence interval.

When the observed skewness is below the lower limit of its con…dence interval,

we …nd evidence of the market considering a drop. On the same vein, when the

kurtosis observed is larger than the upper limit, we …nd evidence of the market

assigning a probability of large movement higher than the corresponding to the

“normal market behavior”. Therefore, the trading days in which the estimates for

skewness and kurtosis exceed the bootstrap con…dence interval are interpreted as

the existence of an “abnormal market behavior”. As a result, any time the market

is expecting or su¤ering a crisis episode, the shape of the distribution function

should become more left-skewed and leptokurtic.

We identify three types of trading days. First, the “normal market behav-

22

ior” days, where skewness and kurtosis do not exceed the con…dence interval

limits. Second, the “rare days”, where the PDF can not be properly character-

ized. Finally, the “abnormal market behavior” days, where we can con…rm that

the market is expecting bad news releases. We show the PDF shape for the three

types Appendix 1.

The main contribution of this paper is a new methodology to build a leading

indicator of excessive risk aversion, through the detection of “abnormal market

behavior”. This methodology provides a benchmark for the “normal market be-

havior” in terms of the risk aversion observed in the market. Our method focuses

on the market considering the possibility of a large drop, that is, our benchmark

is able to predict the market feeling. The implications of applying this procedure

to the prediction of self-ful…lling expectations during pre-crisis periods are quite

relevant as has been demonstrated by testing this indicator on days previous to

unexpected default announcements.

The events we study include some of the main episodes of …nancial crises:

defaults and devaluations. We support of the usefulness of this new methodology

for monetary authorities and international organizations, who would be able to

anticipate crises.

² Defaults: The Spanish options market was expecting negative news a few

days prior to the announcements of the two cases analyzed. The Russian

default was released on August 17th,1998, along with the devaluation of

23

the Russian Rouble. We detect a signi…cant “abnormal market sentiment”

from August 10th ,1998, to August 14th the last trading date before the

default. The Argentinean default was released on November 6th, 2001, and

our indicator signs to an “abnormal market sentiment” from November 1st

until November 5th. Data for default is available in Appendix 5, tables

1 and 2. The indicator we have constructed is specially e¤ective for debt

defaults but this e¤ectiveness is reduced when testing excessive risk aversion

in devaluations or strong depreciations. In fact, as explained in detail in the

next paragraph, in devaluation events the market showed a “rare” behavior

that, however, cannot be considered as “abnormal market behavior”. The

reason may be found in the type of asset analyzed. We think that looking at

the derivatives on exchange rates would be better than looking at the options

on indices to explore the expectations on devaluations/strong depreciations.

² Devaluations and severe depreciations: 1997 constitutes a period of spe-

cially numerous devaluations and hard depreciations. The a¤ected curren-

cies were: Brazilian Real (February 18th), Thailand Bath (July, 2nd), Philip-

pine Peso(July, 11th), Singapore Dollar (July 17th), Malaysian Ringgit (July

24th), Indonesian Rupiah (August 14th), Vietnam Dong and Taiwan Dol-

lar (October 14th), and Korean Won (November 17th). We identify “rare”

days in virtually all these devaluations or severe depreciations. Our indica-

tor is able to predict devaluations since the Korean Wong case. The Bank

24

of Korea defended its currency in the market until November 17th, when

decided this policy was no longer sustainable and requested IMF …nanc-

ing to avoid default. In our view, the risk of default faced by Korea was

re‡ected in options on IBEX-35 prices and, thus, shown by our indicator.

At the beginning of 1998 (January 5th) the Asian currencies su¤ered from

a serious attack, and we detect the abnormal market behavior since Jan-

uary, 1st. Later, on August 11th, China announced a temporal devaluation

of the Yuan, and on August 17th, Russia e¤ectively devaluated the Rou-

ble. Both cases are detected by the indicator. And …nally, devaluations of

Brazil (January 13th, 1999)and Argentina (January 7th, 2002) are preceded

by rare days. 1998 began with hard speculative attacks on Asian curren-

cies. Malaysian, Indonesian, Thai and Philippine currencies were seriously

damaged by the massive bought of US dollars on January 5th. Our indi-

cator shows the presence of “abnormal market behavior” since December

31st. China and Russia announced devaluation in August 11th and 17th re-

spectively. These coincide with the Russian default, and we …nd “abnormal

market behavior” since August 10th, probably anticipating the default. On

January 13th, next year, Brazil decided to stop defending its currency and

abandoned the crawling peg. This time what we observe are “rare days”.

The same happens on January 6th, 2002, when Argentina announced the

end of the currency board and a 29%devaluation of the Argentinean Peso

25

against the Dollar. Data on devaluations is available in Appendix 5 tables

3 to 5.

The main factor that prevents us from detecting the ”abnormal market be-

havior” in devaluation events is that we do not look at the assets that provide

this information. Exchange rate derivatives are the assets that include agents’

expectations on exchange rate evolution. In any case, our indicator provides in-

formation on the existence of greater risk aversion, manifested in the reduced

liquidity in the options market on stock indices.

The next step within this research in the application of this methodology to

the foreign exchange market, that su¤ered from liquidity limitations those dates.

As can be seen in Tables 3 to 5 in Appendix 5, the …ve days before each one of

the devaluations showed many cases of “rare” behavior. The out of the money

options were very illiquid, what leads to a not fully characterized PDF (the right

tale of the distribution is very short) and thus, we can not compute the statistics

accurately for our purposes. More speci…cally, those distributions with cumulative

probability lower than 0.99 produce, in general, unusual distributions that are

regarded as “rare” days, being impossible a more in depth analysis of skewness

and kurtosis. We assume that in “rare” days risk aversion is higher and investors

leave options market, following a “wait and see” strategy. As noted above, the

lack of liquidity in options is specially severe in those options with a longer time

to expiration. However, in the case of default events, markets do not show the

26

same degree of illiquidity, what allows for a evaluation of market sentiment on

the basis of skewness and kurtosis. It is important to note at this point that the

Spanish options and futures market is relatively young and that it is not one of

the main European markets of its type. The number of days catalogued as “rare”

would probably diminish if the data used were those of a more liquid market.

5. Conclusions

This paper provides a new indicator of market sentiment which is able to anticipate

critical crises episodes and defaults on debt. We improve the assessment of market

sentiment through risk neutral implicit density functions estimates by …nding a

new indicator of “abnormal market behavior”, which allows us to identify future

abnormal episodes in the short term.

We test the results on a six years period in which four main crises in emerging

markets took place. We support the view that this new indicator could have

anticipated the defaults shortly ahead. In relation to devaluations and severe

depreciations, many times the liquidity lack makes impossible to estimate PDFs

properly which only allows us to identify “rare” behaviors that seem to re‡ect

the adoption of “Wait and See” strategies. To improve de ability of this new

methodology in predicting this kind of events it would be more appropriate to use

the prices of exchange rate derivatives. By focusing on stock exchange indexes

27

we can only a¢rm that the market is not able to judge the situation with the

available amount of information.

Finally, recall that the methodology lays on the con…dence intervals based on

bootstrap percentiles for the skewness and kurtosis of the implicit risk neutral

density functions. In this sense, this article improves the existing literature by

providing a new accurate benchmark that permits to measure the degree to which

the market is behaving normally and is not expecting bad news that make the

risk aversion exceed normal levels.

28

References

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Theory and application. Bank of England Working Paper Series 66.

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of American option values. Journal of Finance 42, 301-320.

3. Bates, D. (1988). Pricing options on jump di¤usion processes. University

of Pennsylvania, Rodney L. White Center for Financial Research Working

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4. Bates, D. (1991). The Crash of ‘87: Was it expected? The evidence from

options markets. Journal of Finance 46, 1099-1044.

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6. Black, F. (1976). The pricing of commodity contracts. Journal of Financial

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29

published manuscript.

9. Cox, J. and S. Ross (1976). The valuation of options for alternative stochas-

tic processes. Journal of Financial Economics 3, 145-166.

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NY: Chapman&Hall, 1993.

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economías emergentes: Lecciones para prevenir y manejar estas crisis. Per-

spectivas del Sistema Financiero 73, 111-129.

12. Gonzalez, W., J. M. Prada and J. Romo (1994). The bootstrap: a review.

Computational Statistics 9, 165-205.

13. Hull, J. Options, Futures and Other Derivative Securities. Englewood Cli¤s.

NJ: Prentice Hall, 1993.

14. Jackwerth, J. C. and M. Rubinstein (1996). Recovering probability distri-

butions from option prices.The Journal of Finance 51, 1611-1631.

15. Malz, A. (1994). Using option prices to estimate realignment probabilities

in the European Monetary System, in C. Dunis(ed.), Forecasting Financial

Markets: Exchange Rates, Interest Rates and Asset Management. John

Wiley & Sons, 127

30

16. Melick, W. and C. Thomas (1997). Recovering an asset’s implied PDF

from option prices: An application to crude oil during the Gulf Crisis. The

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los tipos de interés a corto plazo: La información contenida en el mercado

de opciones. Banco de España. Documento de Trabajo: 9816.

18. Mati, A. M. (2001). Extreme returns: News and patterns of contagion.

Unpublished manuscript.

19. Peña, I., G. Rubio and G. Serna (1999). Why do we smile? On the deter-

minants of the implied volatility function. Journal of Banking and Finance

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21. Shimko, D. (1993). Bounds of probability. Risk 6, 33-37.

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1997.

31

23. Tompkins, R. (1998). Implied volatility surfaces: Uncovering regularities for

options …nancial futures. Non published manuscript. Vienna U. of Technol-

ogy.

32

33

Appendix 1: PDF graphs and statistics depending on the market behaviour

0.7 0.8 0.9 1 1.1 1.2 1.3 1.40

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05PDF for 7 days to expiration normal trading date (Apr 13, 2000)

moneyness

Type normalYear 2000

Trading date apr-13Obs. in PDF 137Cum. Prob. 1.0000

Observed skewness -0.2597Skewness: 90% lower limit -0.4768Skewness: 90% upper limit -0.0838

Observed Kurtosis 3.1539Kurtosis: 90% lower limit 3.0497Kurtosis: 90% upper limit 3.4150

0.6 0.7 0.8 0.9 1 1.10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045PDF for 7 days to expiration rare trading date (Feb 11, 2000)

moneyness

Type rareYear 2000

Trading date feb-01Obs. in PDF 127Cum. Prob. 0.7969

Observed skewness 1.2579Skewness: 90% lower limit -0.4293Skewness: 90% upper limit 0.2167


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.30

0.005

0.01

0.015

0.02

0.025

0.03

0.035

moneyness

PDF for 7 days to expiration abnormal trading date (Mar 10, 2000)

Type abnormalYear 2000

Trading date mar-10Obs. in PDF 147Cum. Prob. 0.9999

Observed skewness -0.4781Skewness: 90% lower limit -0.4768Skewness: 90% upper limit -0.0838


34

Appendix 2: Summary of the bootstrap results

Days to expiration 1997 1998 1999 2000 2001 2002

1 9985 10997 12000 12000 11986 70002 10930 11800 12000 12000 10914 90003 10863 11352 10999 12000 12925 79744 9283 9536 13000 10846 11000 74357 8606 10394 11999 10000 8915 79198 9557 10989 11980 11979 13000 70289 8823 10759 10000 11996 10765 7993

10 6533 7786 4000 13000 11297 799911 4602 9940 2410 10999 11932 819514 5958 7989 12000 11999 11997 831915 4519 7986 9790 10968 10921 700016 4954 8999 12999 11995 11880 700017 2987 8314 11997 11907 10422 700018 3703 6589 11000 8993 9443 692021 3210 7982 9994 10418 10016 840322 2772 6982 9947 10978 10086 899823 2387 6998 12977 10974 9837 600024 2437 7520 9703 10990 8400 640825 2131 4865 8997 6994 7316 599828 2276 4433 7995 8664 9930 999229 2140 4000 7920 10094 7960 773830 1404 3972 8841 6763 8697 670031 1799 4051 7895 7384 7944 632732 2972 3032 6529 7000 6889 599135 2612 2994 5679 7119 5998 579736 1937 2983 4861 5934 8998 499537 1955 2977 3963 5000 5996 499738 1804 2987 3000 5000 8916 502839 1803 2847 4367 4986 7184 5766

Number of bootstrap samples per year and time to expiration

Statistics for trading date classification

1997 337 195 1 1961998 344 139 24 1631999 335 67 25 922000 336 60 29 892001 342 64 13 772002 261 51 4 55

Year of

trading

Trading dates "Rare days" "Abnormal market

behavior"

Total "non normal

trading days"

35

Appendix 3: Histograms for skewness and kurtosis per year

-1 -0 .5 0 0 .5 1 1 .50

0 .5

1

1 .5

2

2 .5

3

3 .5

4

4 .5x 1 0

4 H i s t o g r a m f o r S k e w n e s s . P e r i o d : 1 9 9 7

1 .5 2 2 .5 3 3 .5 40

0 .5

1

1 .5

2

2 .5

3

3 .5

4

4 .5

5x 1 0

4 His tog ram fo r Ku to r s i s . Pe r i od : 1997

36

-1 .5 -1 -0 .5 0 0 .5 1 1 .50

0 .5

1

1 .5

2

2 .5

3

3 .5

4x 1 0


1 1 .5 2 2 .5 3 3 .5 4 4 .50

0 .5

1

1 .5

2

2 .5

3

3 .5

4

4 .5x 1 0


37

-1 -0 .5 0 0 .5 1 1 .50

1

2

3

4

5

6x 1 0


2 2 .5 3 3 .5 4 4 .5 50

0 .5

1

1 .5

2

2 .5

3

3 .5

4x 1 0


38

-0 .6 -0 .4 -0 .2 0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1 .40

1

2

3

4

5

6

7x 1 0


1 .8 2 2 .2 2 .4 2 .6 2 .8 3 3 .2 3 .4 3 .60

1

2

3

4

5

6

7x 1 0


39

-1 -0 .5 0 0 .5 1 1 .5 20

1

2

3

4

5

6

7

8x 1 0


1 .6 1 .8 2 2 .2 2 .4 2 .6 2 .8 3 3 .2 3 .4 3 .60

1

2

3

4

5

6

7x 1 0


40

-1 -0 .5 0 0 .5 1 1 .5 20

1

2

3

4

5

6x 1 0


2 .2 2 .4 2 .6 2 .8 3 3 .2 3 .4 3 .6 3 .80

0 .5

1

1 .5

2

2 .5

3

3 .5x 1 0


41

APPENDIX 4. Part I: Skewness confidence intervals

Skewness, 1 day to expiration

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

19970116 19970813 19980219 19980917 19990318 19990916 20000419 20001019 20010419 20011018 20020418Note: Rare days excluded

Skewness, 7 days to expiration

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

19970110 19970711 19980109 19980814 19990108 19990709 20000114 20000714 20010209 20010914 20020412

Note: Rare days excluded

42

Skewness, 14 day to expiration

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

19970103 19970704 19971205 19980703 19990205 19990903 20000204 20000804 20010202 20010803 20020201 20020802



-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

19961227 19970724 19971226 19980626 19990129 19990730 20000225 20000825 20010223 20010928 20020426


43


-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

19961220 19970523 19971121 19980522 19981120 19990521 19991119 20000721 20001222 20010622 20020118 20020719



-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

19961213 19970613 19980116 19980814 19990212 19990813 20000211 20000811 20010209 20010914 20020412


44

Part II: Kurtosis confidence intervals

Kurtosis, 1 day to expiration

2.950

2.975

3.000

3.025

3.050

3.075

3.100

3.125

3.150

19970116 19970813 19980219 19980917 19990318 19990916 20000419 20001019 20010419 20011018 20020418


Kurtosis, 7 days to expiration

2.50

2.75

3.00

3.25

3.50

3.75

4.00

19970110 19970711 19980109 19980814 19990108 19990709 20000114 20000714 20010209 20010914 20020412


45


2.50

2.75

3.00

3.25

3.50

3.75

4.00

4.25

4.50

4.75

19970103 19970704 19971205 19980703 19990205 19990903 20000204 20000804 20010202 20010803 20020201 20020802



2.50

2.75

3.00

3.25

3.50

3.75

4.00

4.25

4.50

4.75

5.00

5.25

19961227 19970724 19971226 19980626 19990129 19990730 20000225 20000825 20010223 20010928 20020426


46


2.50

2.75

3.00

3.25

3.50

3.75

4.00

4.25

4.50

19961220 19970523 19971121 19980522 19981120 19990521 19991119 20000721 20001222 20010622 20020118 20020719



2.50

2.75

3.00

3.25

3.50

3.75

4.00

4.25

4.50

4.75

5.00

5.25

19961213 19970613 19980116 19980814 19990212 19990813 20000211 20000811 20010209 20010914 20020412


47

Appendix 5: Results for default events

Table 1

Type Trading date Days to

expiration

Obs. in

PDF

Cum.

Prob.

Observed

skewness

Skewness: 90% lower

limit

Skewness: 90% upper

limit

Observed

Kurtosis

Kurtosis: 90% lower

limit

Kurtosis: 90% upper

limit Normal 19980807 14 87 0.999 -0.441 -0.895 -0.035 3.309 2.875 4.092Normal 19980810 11 87 0.999 -0.385 -0.833 -0.220 3.234 3.089 3.982

Abnormal 19980810 39 121 0.994 -0.416 -0.413 -0.201 3.234 2.842 3.228Normal 19980811 10 87 0.995 -0.004 -0.599 0.067 2.841 2.797 3.645

Abnormal 19980811 38 121 0.996 -0.322 -0.318 -0.215 2.993 2.859 2.989Normal 19980812 9 87 0.999 -0.315 -0.727 0.000 3.144 3.067 3.738

Abnormal 19980812 37 121 0.995 -0.394 -0.394 -0.247 3.142 2.922 3.138Normal 19980813 8 87 1.000 -0.342 -0.698 -0.032 3.199 2.910 3.677

Abnormal 19980813 36 121 0.995 -0.392 -0.401 -0.306 3.143 2.981 3.140Normal 19980814 7 87 1.000 -0.383 -0.677 -0.302 3.292 3.215 3.654

Abnormal 19980814 35 121 0.997 -0.410 -0.409 -0.340 3.164 2.983 3.161Normal 19980817 4 87 1.000 -0.293 -0.530 -0.175 3.190 3.088 3.418Normal 19980817 32 121 0.997 -0.391 -0.532 -0.338 3.150 3.053 3.280

Russian default on August 17, 1998

Table 2


expiration

Obs. in

PDF

Cum.

Prob.

Observed

skewness

Skewness: 90% lower

limit

Skewness: 90% upper

limit

Observed

Kurtosis

Kurtosis: 90% lower

limit

Kurtosis: 90% upper

limit Rare 20011026 21 79 0.975 0.267 -0.521 0.008 2.735 2.948 3.501

Normal 20011029 18 79 0.992 -0.266 -0.463 -0.089 3.172 3.024 3.390Normal 20011030 17 79 0.998 -0.381 -0.457 -0.031 3.212 2.988 3.388Normal 20011031 16 79 0.994 -0.297 -0.455 -0.036 3.207 3.004 3.396

Abnormal 20011101 15 79 0.996 -0.344 -0.303 -0.039 3.247 2.911 3.238Normal 20011102 14 79 0.998 -0.394 -0.410 -0.015 3.279 2.833 3.316

Abnormal 20011105 11 79 0.997 -0.316 -0.351 -0.027 3.241 2.875 3.230Normal 20011106 10 79 0.996 -0.252 -0.333 -0.048 3.182 3.022 3.205

Argentinean default on November 6, 2001

48

Table 3


expiration

Obs. in

PDF

Cum.

Prob.

Observed

skewness

Skewness: 90% lower

limit

Skewness: 90% upper

limit

Observed

Kurtosis

Kurtosis: 90% lower

limit

Kurtosis: 90% upper

limit Normal 19970625 23 57 0.991 0.082 -0.364 0.110 2.675 2.668 3.017Normal 19970626 22 57 0.994 -0.033 -0.367 0.104 2.749 2.678 3.018Normal 19970627 21 57 0.995 -0.065 -0.577 0.101 2.767 2.692 3.378Normal 19970630 18 57 1.000 -0.377 -0.610 0.094 3.002 2.675 3.395Normal 19970701 17 57 1.000 -0.390 -0.602 -0.313 3.019 2.960 3.416Normal 19970702 16 57 0.999 -0.364 -0.577 0.118 3.013 2.717 3.411Normal 19970707 11 57 0.999 -0.311 -0.350 0.307 3.044 2.677 3.137Rare 19970707 38 47 0.909 1.308 -0.059 0.240 1.910 2.763 2.893

Normal 19970708 10 57 1.000 -0.337 -0.361 0.189 3.066 2.735 3.260Rare 19970708 37 47 0.919 1.306 -0.083 0.147 1.981 2.791 2.965

Normal 19970709 9 57 0.999 -0.321 -0.332 0.412 3.056 2.673 3.231Rare 19970709 36 47 0.913 1.308 -0.160 0.177 1.932 2.806 2.952

Normal 19970710 8 57 1.000 -0.299 -0.313 0.135 3.041 2.810 3.194Rare 19970710 35 47 0.941 1.229 -0.220 0.128 2.118 2.712 2.968

Normal 19970711 7 57 1.000 -0.305 -0.482 0.110 3.065 2.824 3.495Normal 19970714 4 57 1.000 -0.238 -0.392 0.184 3.041 2.790 3.305Rare 19970714 31 47 0.922 1.303 -0.314 0.149 1.974 2.823 3.046

Normal 19970715 3 57 1.000 -0.187 -0.336 -0.032 3.025 2.936 3.224Rare 19970715 30 47 0.952 1.138 -0.321 0.216 2.194 2.769 3.018

Normal 19970716 2 57 1.000 -0.159 -0.273 -0.081 3.015 2.959 3.140Rare 19970716 29 47 0.926 1.303 -0.357 0.140 2.042 2.625 2.998

Normal 19970717 1 57 1.000 -0.101 -0.202 -0.060 3.003 2.991 3.066Rare 19970717 28 47 0.941 1.242 -0.373 0.115 2.161 2.602 3.017Rare 19970721 24 47 0.971 0.899 -0.327 0.093 2.462 2.658 3.018Rare 19970722 23 47 0.962 1.021 -0.364 0.110 2.298 2.668 3.017Rare 19970723 22 47 0.972 0.798 -0.367 0.104 2.390 2.678 3.018Rare 19970724 21 47 0.977 0.768 -0.577 0.101 2.538 2.692 3.378

Normal 19970811 3 47 1.000 -0.136 -0.336 -0.032 3.048 2.936 3.224Rare 19970811 39 83 0.961 0.698 -0.058 0.102 2.341 2.803 2.947

Normal 19970812 2 47 1.000 -0.116 -0.273 -0.081 3.034 2.959 3.140Rare 19970812 38 83 0.954 0.838 -0.059 0.240 2.286 2.763 2.893

Normal 19970813 1 47 1.000 -0.079 -0.202 -0.060 3.015 2.991 3.066Rare 19970813 37 83 0.974 0.418 -0.083 0.147 2.466 2.791 2.965Rare 19970814 36 83 0.981 0.256 -0.160 0.177 2.539 2.806 2.952Rare 19970814 36 83 0.981 0.256 -0.160 0.177 2.539 2.806 2.952

Episodes of the Asian crisis in 1997 (I)

49

Table 4


expiration

Obs. in

PDF

Cum.

Prob.

Observed

skewness

Skewness: 90% lower

limit

Skewness: 90% upper

limit

Observed

Kurtosis

Kurtosis: 90% lower

limit

Kurtosis: 90% upper

limit Normal 19971010 7 47 1.000 -0.209 -0.482 0.110 3.083 2.824 3.495Normal 19971013 4 47 1.000 -0.154 -0.392 0.184 3.046 2.790 3.305Rare 19971013 39 45 0.952 1.224 -0.058 0.102 2.365 2.803 2.947

Normal 19971014 3 47 1.000 -0.132 -0.336 -0.032 3.034 2.936 3.224Rare 19971014 38 45 0.963 1.066 -0.059 0.240 2.404 2.763 2.893

Normal 19971113 8 57 0.997 -0.302 -0.313 0.135 3.188 2.810 3.194Normal 19971113 36 95 0.998 -0.152 -0.160 0.177 2.911 2.806 2.952

Abnormal 19971114 7 57 1.000 -0.491 -0.482 0.110 3.520 2.824 3.495Normal 19971114 35 95 0.999 -0.211 -0.220 0.128 2.946 2.712 2.968Normal 19971117 4 57 1.000 -0.383 -0.392 0.184 3.292 2.790 3.305Normal 19971117 32 95 1.000 -0.286 -0.307 0.151 2.972 2.683 3.002Normal 19971230 17 67 0.997 -0.367 -0.723 -0.087 3.097 2.918 3.886

Abnormal 19971231 16 67 1.000 -0.819 -0.813 -0.118 3.894 2.955 3.991Abnormal 19980102 14 67 1.000 -0.901 -0.895 -0.035 4.084 2.875 4.092Abnormal 19980105 11 67 1.000 -0.857 -0.833 -0.220 4.010 3.089 3.982Abnormal 19980105 11 67 1.000 -0.857 -0.833 -0.220 4.010 3.089 3.982

Episodes of the Asian crisis in 1997 (II)

50

Table 5


expiration

Obs. in

PDF

Cum.

Prob.

Observed

skewness

Skewness: 90% lower

limit

Skewness: 90% upper

limit

Observed

Kurtosis

Kurtosis: 90% lower

limit

Kurtosis: 90% upper

limit Normal 19980806 15 87 0.997 -0.280 -0.729 0.026 3.040 2.853 3.942Normal 19980807 14 87 0.999 -0.441 -0.895 -0.035 3.309 2.875 4.092Normal 19980810 11 87 0.999 -0.385 -0.833 -0.220 3.234 3.089 3.982

Abnormal 19980810 39 121 0.994 -0.416 -0.413 -0.201 3.234 2.842 3.228Normal 19980811 10 87 0.995 -0.004 -0.599 0.067 2.841 2.797 3.645

Abnormal 19980811 38 121 0.996 -0.322 -0.318 -0.215 2.993 2.859 2.989Normal 19980812 9 87 0.999 -0.315 -0.727 0.000 3.144 3.067 3.738

Abnormal 19980812 37 121 0.995 -0.394 -0.394 -0.247 3.142 2.922 3.138Normal 19980813 8 87 1.000 -0.342 -0.698 -0.032 3.199 2.910 3.677

Abnormal 19980813 36 121 0.995 -0.392 -0.401 -0.306 3.143 2.981 3.140Normal 19980814 7 87 1.000 -0.383 -0.677 -0.302 3.292 3.215 3.654

Abnormal 19980814 35 121 0.997 -0.410 -0.409 -0.340 3.164 2.983 3.161Normal 19980817 4 87 1.000 -0.293 -0.530 -0.175 3.190 3.088 3.418Normal 19980817 32 121 0.997 -0.391 -0.532 -0.338 3.150 3.053 3.280Normal 19990108 7 117 0.999 -0.251 -0.618 -0.235 3.096 3.084 3.851Rare 19990111 4 117 1.000 -0.210 -0.451 -0.211 3.075 3.075 3.440Rare 19990111 39 77 0.837 1.382 -1.088 -0.393 2.037 3.086 5.157Rare 19990112 3 117 1.000 -0.172 -0.397 -0.172 3.052 3.052 3.317Rare 19990112 38 83 0.881 1.472 -1.077 -0.373 2.403 3.052 5.123Rare 19990113 2 117 1.000 -0.122 -0.326 -0.122 3.027 3.027 3.212Rare 19990113 37 123 0.965 0.561 -1.088 -0.370 2.438 3.097 5.185

Normal 20011228 21 77 0.999 -0.353 -0.411 0.028 3.261 2.831 3.394Normal 20020102 16 77 1.000 -0.318 -0.464 -0.227 3.232 2.960 3.432Normal 20020104 14 77 1.000 -0.364 -0.446 -0.050 3.337 2.926 3.422Normal 20020107 11 77 1.000 -0.317 -0.399 -0.149 3.271 3.020 3.341Rare 20020107 39 77 0.983 0.272 -0.471 0.022 2.618 2.799 3.470

Episodes of the Russian, Brazilian and Argentinean crises (1998- 2002)

detecting abnormal market behavior using …

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