Download The nature of Jumps in Brazil`s stock market

The nature of Jumps in Brazil’s stock market
Fabio Ramos, Italo Waddington, Luis G.L. Mussili,
Yuri L. O. Pinto, Alexia Pimentel, Lucas M. Carvalho
03/31/2016
Abstract. In this work, we investigate the causes and several statistical properties of jumps in
the Brazilian Stock Market Index, Ibovespa, by following the methodology suggested in [2]. We
also propose a new measure of local volatility change. We show that the so-called leverage effect
holds for Ibovespa with this newly defined measure. We also show that jumps caused by different
reasons may have different impacts concerning the local volatility change.
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Introduction
What causes market jumps in Brazil? A simple straightforward answer could be: a myriad
of things. But many questions still stand. How many of them are caused by the FED’s interest
rate policy? What is the weight of national macroeconomic news in these dramatic events? Are
political transitions more destabilizing than Chinese growth policies? How frequent are they? Are
they isolated events or are they sudden reactions amid a sea of high volatility? Do they increase
or decrease volatility?
Large movements in stock markets are certainly related to herding phenomena, and behavioral
psychologists and economists have studied a wide spectrum of cognitive biases related to these
extreme episodes, see, e.g., [3]. Although cognitive biases and herding effects surely play important
roles, and although many of these dramatic events are latter blamed on elusive causes, such as
market sentiment, it is indisputable that investors respond primarily to some specific classes
of more tangible news. Some obvious classes are: macroeconomic news and outlook, monetary
policy and central banking, political transitions and elections, corruption scandals and political
turmoil, commodities market news, and many others. Of course, the responses may differ in
time-scales, frequencies and amplitudes, and may, in fact, depend on the nature of the driving
news. Nonetheless, despite of the diversity of the possible causes, it is reasonable to assume that
if the market move is large enough, it will attract the attention of major national newspapers.
In [2], the authors have established a method to investigate the triggers of market jumps
based on readings of different newspapers by different readers. Because different newspapers
and readers may disagree among themselves, the authors established a protocol of how to define
the cause of each jump, controlling for this uncertainty. The work [2] is actually one more step
of the Economic Policy Uncertainty (EPU) Project, led by Steven J. Davis, Nick Bloom and
Scott R. Baker. One of their main questions is: what is the role of policy uncertainty in market’s
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moves? Theis seminal work is [1], and the project’s website contains many papers related to these
questions.
In this paper, we have partially followed the proposal established in [2] to investigate the causes
of market jumps above the threshold +3.5% and below −3.5%. The threshold is considerably
higher than the ±2.5% threshold in the american stock market because of the more volatile
nature of the Ibovespa. In Section 3, we describe, in details, the setup of our newspaper’s reading
research. In Section 4, we analyze some statistical signatures of the jumps, such as waiting
time distributions, and we propose a newly defined measure of local volatility change. With
this new concept, we exhibit some evidence that large price drops increase volatilty, and large
price increases decrease volatility, a very controversial phenomenon known in the literature as
the leverage effect. We also show evidences that the leverage effect may depend on the cause of
jumps, and in particular, we show that policy uncertainty jumps are responsible for the largest
changes in volatility in recent days.
Market jumps are facts of every investor’s lives. Nonetheless, most celebrated models in
financial economics assume a very gentle behavior for price moves. There are many models
including jumps in the literature, but most of them assume some statistical regularity for these
events. Jumps only happen because of a huge disparity in supply and demand in the current price
level. Frequent sudden changes of opinion concerning current prices suggest that the commonly
used hypothesis of market efficiency may be far from correct. Therefore, knowing the nature of
jumps, and some of their statistical signatures may help financial practitioners and theorists to
develop more robust models, for example by including leverage effects. In the coming section, we
discuss some of the common assumptions in the literature concerning the arrival of information
in the markets, and related hypotheses of how it changes prices.
2
Uncertainty and the role of jumps in some classical
financial models.
The efficient market hypothesis (EMH), in spite of the severe criticisms on its validity, forms
the basis of several of the most important models in modern economics and finance, such as the
Black-Scholes model and the CAPM. The hypothesis has been stated in several ways (weak-form,
semi-strong form, strong-form), and like any other model of the social and physical sciences, it is
an attempt to reduce the complexity of the investigated phenomena. Along the years, since its
popularization in the mid-1960s, several market anomalies have been observed which ultimately
lead to its rejection, at least in its strongest forms in many markets. Indeed, the EMH is an
assumption on human behavior, and since human behavior is complex and imprecise, so is the
EMH. We refer the reader interested in the EMH and related issues to reference [4], but it
essentially states that it is impossible to beat the market systematically because the current prices
of equities contain all the information we have about the economy and the market. It is only
new arriving information that may cause prices to change, and since the future is unknown, it is
impossible to predict changes of prices.
Quantitatively, an important assumption of the EMH is that bits of information arrive
uniformly every ∆t seconds. It is hypothesized that arriving good news causes the company’s
value to increase by a fixed percentage, and bad news to decrease. The stock price may grow
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in average (drift) but it randomly moves up and down about the mean. No wilder deviations
exist in addition to these fixed fluctuations uniform in time. More precisely, it is assumed that
stock returns are memoryless brownian
processes growing as µt, where µ is called the drift, and
√
fluctuating about the mean as σ t, where the fixed coefficient σ is the volatility, and it measures
the intrinsic size of the yearly fluctuations, see [5, 6].
Developed in the early 1960s, and one of the greatest accomplishments of the modern portfolio
theory, the capital asset pricing model (CAPM) is an extension of the EMH that takes into
account not only the risk of single stocks, but also of the entire stock market, see [4]. While the
EMH assumes that stocks move like air molecules in a diffusive process, CAPM recognizes that
stocks are constituents of a larger market, and they tend to move up or down when the whole
market moves up or down, respectively. This herding tendency is a classical research topic in
Economics, since, at least, Keynes’ observation of the required “Animal Spirits” to take risks.
CAPM assumes the existence of two kinds of risk, the stock’s idiosyncratic risk, and the whole
market risk. CAPM’s main idea is that because idiosyncratic risks are diversifiable, one should
not expect to be rewarded for holding this type of risk. It states that the only true risk is the
market risk, and it prescribes a method to produce a market portfolio, which is regarded as a
good representative of the whole market, based on its variance and expected return. As a matter
of fact, many stock’s indices rely on this idea. CAPM states furthermore that for every level of
risk, one can maximize the expected return by holding a combination of risk-free bonds and the
market portfolio.
CAPM assumes some sort of equilibrium in risk, which is usually estimated with long-term
historical prices. There is not much room for different types of time-scales for trading strategies,
such as high-freguency trading, daily trading, and so on. Every investor should look at the long
run volatility and expected return. It also lacks the possibility of sectorized investment strategies,
such as investing only in technology companies, or small cap companies, etc. Indeed, one of the
main measures of a trader’s strategy efficiency is its α, a measure of the difference of the actual
return and the theoretical one obtained via statisitcal estimates of β from CAPM. The very fact
that α may differ from zero implies that CAPM is not a very exact model of reality.
The efficiency hypothesis is in fact a statement about the equivalence between price and value.
Similarly, CAPM’s hypothesis on volatilities is an assumption on the equivalence of actual and
estimated volatility. Ultimately, EMH would imply that stock trades should cease to exist since
everyone should agree on every asset’s values, and CAPM implies that every investor should
hold the same investment portfolio. These conclusions are obviously flawed, and one of the
solutions for these apparent paradoxes is the imprecision of the estimates of value and risk. In his
groundbreaking work, “Noise", see [7], Fisher Black commented on the efficient markets:
“All estimates of values are noisy, so we can never know how far away price is from value.
However, we might define an efficient market as one in which price is within a factor of 2 of value,
i.e., the price is more than half of value and less than twice value. The factor of 2 is arbitrary,
of course. Intuitively, though, it seems reasonable to me, in the light of sources of uncertainty
about value and the strength of the forces tending to cause price price to return to value. By this
definition, I think almost all markets are efficient almost of the time. “Almost all" means at least
90%."
Black’s definition of efficiency is probably more correct, but on the other hand, it is so
imprecise that can not be very useful for building a consistent theory. Indeed, F. Black, together
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with M. Scholes, built on the EMH and the CAPM, and developed the Black-Scholes model
for pricing options. Of course, all the inconsistencies of the EMH and the CAPM are directly
inherited by the Black-Scholes model, and many modifications have been proposed. For example,
the hypothesis of constant volatility has been removed in many stochastic volatility models, and
the presence of statistically prescribed large price deviations (jumps) have been include in SVJ
and SVJJ models. Although more realistic, the assumptions in these extended models are still
very simplistic, since they cannot fully take into account many of the observed stylized facts in
the market.
One of the most studied stylized facts was noticed by F. Black himself. In [7], he observed that
the volatility of stocks tends to increase after large price drops. This phenomenon is known as
the “leverage effect" in the literature because Black’s explanation argues that as asset prices drop,
companies become more leveraged because the relative value of their debt increases relative to
that of their equity. Consequently, it is expected that their stock becomes riskier, and, therefore,
more volatile. Jumps are, therefore, not only a volatility factor when it happens, it disturbs
future volatility behavior.
As a matter of fact, the presence of jumps is regarded as one of the main reasons that the
EMH, CAPM, and all of their more sophisticated extensions can not be a very precise model of
reality. EMH is not wild enough. Actual stock prices are more complex and vary more wildly, as
it is apparent in the vicinity of a large market jump. Volatility may rise to unimaginable levels,
and liquidity may disappear, as observed in the last 2007-2008 financial crisis.
The leverage effect has been studied by many authors see [8] for a list of references. There is
some controversy on the causes of the volatility increases after large price drops. Many authors
disagree on the leverage explanation. It has been related to a retarded response of investors
concerning arriving news in [9]. In [8], the authors claim that it is related to a Gain/Loss
asymmetry in stock markets. Despite all of the controversy concerning the leverage effect for
stocks, there is wide evidence of the negative correlation of volatility/return for indices, which
may seem a little surprising, since diversification should decrease risk. The common explanation is
a herding behavior following very bad news, ensuing panic or very good news, ensuing “irrational
exuberance". These news not only cause large market adjustments, but also change the market’s
perception on future uncertainty, leading to highly speculative behavior in the vicinity of the
events. In Section 4 we propose a measure of local volatility and one of volatility change, and
show strong evidence of leverage effect in Brazil’s stock markets. We also show evidences that
the amplitude of volatility change depends on the nature of the jump. In the coming section we
describe the methods and the results of our newspaper reading research.
3
What triggers IBOVESPA’s jumps?
In this section, we follow the method proposed in [2] to uncover the reasons for stock market
jumps by reading the following day newspapers. If the chosen threshold for the jumps is set high
enough, then it is indeed expected that it will attract the attention of major national newspapers.
In this work, we investigate the causes of jumps for the major Brazilian stock market’s index,
the Ibovespa, from 01/04/2002 to 30/03/2016. We set the threshold of ±3.5%, which results in
167 jumps in a period of 3458 trading days, corresponding to about 4.89% of the trading days.
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This percentage is comparable to the thresholds chosen in [2], and this explains why we have
chosen the much higher threshold ±3.5% instead of the ±2.5% threshold suggested in [2] for the
american and german markets, for example. Let us first explain the reading method.
3.1
The Method
The method for analyzing equity market jumps developed in [2], and used in this paper is:
1. Set daily jump threshold (±3.5% in this work)
2. Pull dates with market moves > threshold
3. Use newspapers articles to characterize jumps
A. Go to online newspaper archive
B. Enter newspaper, date range (next day) and search criteria.
C. Select article
4. Read article
5. Record the reason for the jump, its geographic source, confidence level, ease of coding, etc.
Figure 1: Table of Daily Jumps, Ibovespa. Selected Policies and Categories.
To control for reading and source biases, for each event, we have assigned a group of 5 trained
readers for each event. Each reader read two different newspapers, such that at least three
5
newspapers have been read for each event. We have chosen four different newspapers for this task:
Jornal Valor Econômico, Jornal O Globo, Jornal O Estado de São Paulo, Jornal A Folha de
São Paulo. We have thus guaranteed a total amount of 10 readings for each event. The average
agreement was about 85% with standard deviation of 18.6%. Figure 11 displays the agreement
rate for each event from 04/2002 to 03/2016.
The different categories of reasons for jumps found in this process can be found in Table 3.1.
Some categories are straightforward to be coded. There are, however, some details concerning
the following categores which needs to be explained. Let us first transcript how the authors in
[2] suggest to separate Monetary Policy and Central Banking from Macroeconomic News and
Outlook
3.1.1
Monetary Policy and Central Banking (from [2])
Actions, possible actions, and concerns related to the conduct and policies of the central bank
or other chief monetary authority. Such actions and policies pertain to interest rate changes and
monetary policy announcements, inflation control, liquidity injections by the monetary authority,
changes in reserve requirements or other bank regulations used by the monetary authority to
exercise control over monetary conditions, lender-of-last resort actions, and extraordinary actions
by the monetary authority in response to bank runs, systemic financial crisis and threats to the
payments system.
3.1.2
Macroeconomic News and Outlook (from [2])
News relating to macroeconomic forecasts or reports such as inflation, housing prices, unemployment or employment, personal income, industrial production, manufacturing activity,
etc.
Also included in this category: News about financial crisis developments that does not fall
into another category such as Monetary Policy and Central Banking. • Trade and exchange rate
news NOT attributed to policy (e.g., news about trade deficits or currency movements).
The 2008 financial crisis was the period with largest amount of registered jumps, as we will
see below. It was also the period with the least agreement among readers. The main reason
is that many news in the period cited both Monetary Policy and Central Banking reasons and
Macroeconomic News and Outlook, because at that turbulent period, the FED was highly active
in promptly responding to dire macroeconomic news. We show in figure 11 a measure of agreement
among readers for all jumps.
We should also mention that in our research, we have found that all jumps with no clear
source cited some sort of “bad mood" in foreign stock markets, and this is the reason why we
have decided to merge the category Foreign Stock Markets (No specific reason) with the category
Unknown. This is how this category is defined:
3.1.3
Foreign stock markets (No specific reason) & Unknown (from [2])
News reports that attribute the large domestic market move directly to foreign stock-market
moves without any further explanation. For example, an article blaming stock price moves on
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“Wall Street” without any further reason.
3.1.4
Commodities
News reports that attribute the large domestic market move directly to commodities moves
without any explanation relating to macroeconomic news. For example, an article blaming
oil prices without any further reason. These news are usually international by definition, and,
therefore, they do not add to the National Source category. Furthermore, we remark that most of
these news cite some fear of slower growth in China, among other reasons. Nonetheless, every
time that this fear was related to some macroeconomic news, such as the official publishing of the
current Chinese growth rate, this news was classified as Macroeconomic News and Outlook.
Figure 2: Waiting time between two consecutives jumps. Red line in the bottom illustrates the
5-days threshold. Notice the striking volatile streak during the financial crisis. Notice also some
large periods without any jumps, including the period of the 2010’s presidential race.
In this work, we have chosen 9 different categories of reasons for jumps, as shown in the
table in Figure 1, and we have further subdivided them in two subcategories: national policytriggered, international policy-triggered and non policy-triggered. The national policy triggered
sub-category contains: Domestic Monetary Policy & Central Banking, Elections & Political
Transitions and Corruption Scandals& Political Turmoil. The international policy triggered
sub-category contains: International Monetary Policy & Central Banking, Elections & Political
Transitions and Corruption Scandals(only 2 american events) and International Military. The
non policy-triggered sub-category contains: Macroeconomic News & Outlook (both national
and international), Commodities, Corporate Earnings & Profits, and Foreign stock markets (No
specific reason) & Unknown.
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3.1.5
Analysis
There are several important features in the table presented in Figure 1. First, we notice that
the two-year period 04/2008 − 03/2010 concentrates 38.9% of all jumps since 04/2002. This is
obviously due to the financial crisis. As one can observe in Figure 2, in the period of six months
from September, 2008 to March, 2009, there were astonishing 47 jumps (∼ 28% of all jumps),
with at least one jump registered in almost every week. This turbulent period displays such a
high volatility, that no financial models can hope to perform well. This, in fact, can be itself
another source of mounting volatility. As one becomes used to calculate risks with some standard
models, he may insist in a sort of unbearable futility of modelling, since he will be, in fact, fitting
noise, adding up even more uncertainty to markets, as stressed by E. Dermann in [6].
One more important feature concerns the policy-driven causes. In the total period since
04/2003, around 56% of the news were related to policy-triggered causes. This share remains
essentially the same in and out of the critical two-year period 04/2008 − 03/2010. National
policy-triggered news were responsible for about 22% of jumps during the turbulent period
04/2008 − 03/2010, and for for about 33% of jumps out of this period. We remark, furthermore
that the more political categories: Elections & Political Transitions and Corruption Scandals&
Political Turmoil were responsible for about 42% of jumps during the period 04/2002 − 03/2014,
and for about 45% of them in the period 04/2014 − 03/2016.
Another important remark concerns the commodities’ driven news. We notice that they are
mainly concentrated in the highly speculative period 04/2008 − 03/2010, totalizing more than 50%
of this kind of event. This happens mostly because of the nature of our coding method. We only
code as commodities when there is no mention to other possible reason, such as a macroeconomic
news in China. Many of the news related to jumps driven by commodities blame on a “fear of
economic meltdown" eroding commodities’ prices. This kind of news were very present in this
complicated period, but not much out of it. Despite of this, the easy of coding was relatively
high for commodities-triggered jumps.
To conclude this section, we comment on the Foreign stock markets (No specific reason) &
Unknown category. Several articles in the literature state that there are no major agreement
on what causes large market moves and high periods of volatility, see, for example, the article
[10]. The present work contradicts this sort of statement for the Brazilian market. Only a very
small amount of jumps were attributed to this more uncertain category, about 8% of total. Even
considering together with the also very uncertain category, such as commodities, this still amounts
only to about 22%. Moreover, these categories do not display any unusually large jumps, as
can be seen in Figures 9 and 10. We will see, however, that days falling in the category Foreign
stock markets (No specific reason) & Unknown have an above average change of volatility in
their vicinity, as it can be seen also in Figures 9 and 10. In the coming section, we will define a
measure of local volatility, and define a measure of volatility change around a given day. We will
compute it for the days with jumps to conclude that there exists a sort of leverage effect, and we
will also see that different classes of jumps may have different volatility behavior.
8
4
Jumps and Volatility Change
In this section, we present some inverse statistical measures of returns, and discuss the so-called
leverage effect for the Ibovespa. Let us begin with some nomenclature. We denote by Hi , Li , and
ri the daily high, the daily low and the daily return at day i.
Let us define the daily range volatility as
RVi = log2 (
Hi
).
Li
(1)
Let 0 < λ < 1 represent a weight parameter (the value λ = 0.94 is commonly found in the
literature because of RiskMetrics, and this value is used in this paper), and n ≥ 1 represents a
number of days. We define the exponentially weigthed forward volatility at day i as
F Vi =
n
1X
λj RVi+j ,
w j=1
(2)
n
−λ
where w = λλ−1
.
Similarly, let us define the exponentially weighted past volatility at day i as
P Vi =
n
1X
λj RVi−j .
w j=1
(3)
Now, we define the volatility change around day i as the ration between past volatility, PV,
and forward volatility, FV:
Pi
(4)
V Ci = .
Fi
A small value of V C indicates that the forward volatility is larger than the past volatility,
implying a volatility increase after the jump (Notice thatg the definition does not include the day
i itself, only its vicinity). This very simple definition reflects a strong assumption on investor’s
behavior. For a given day, the investor we will give more weight to the return of days in its
immediate vicinity. So that an investor will feel that volatility has increased after a jump down
(leverage effect) if V Ci is way smaller than the average V Cj , j 6= i, and, therefore, the leverage
effect is related to small values of V C.
We have, however, to be very careful with the definition of the vicinity. If one takes a large
value of n, then V Ci will tend to a constant. Therefore, one has to choose a meaningful number
of days, compatible mainly with the jumps one wants to study. Essentially one single jump can
be dwarfed in a very long time-series. We are aiming to find a reasonable time-scale for this jump
scale.
For that reason, to study the volatility around a ±3.5%, we have decided to choose a window
of a specific size `, such that the prices typically do not move more than ±3.5% in a window of
that size. We will show now that ` = 5 is a reasonable choice, leading to n = 5 in Definitions (2),
(3) and (4) above.
In order to find this number `, we apply an inverse statistics method discussed in [8], related
to the so-called optimal investment horizon concept. We first filter out the drift from the index
9
price series with a waveletet filter, and then we capture the average amount of days required for
the index to increase 3.5% or to decrease below −3.5% of the current price. This amounts to the
concept of volatility around a drift discussed in Section 2 for the EMH and the CAPM. Figures 3
and 4 illustrate the method. The reader is reffered to [8] and [11], for more details. We have also
included a very brief description of this filtering method in the appendix.
In general, if we fix a return level, denoted by ρ, mathematically, the first passage time is
defined as

n
o
inf ∆t|r∆t ≥ +ρ ,
n
o
τ±ρ =
inf ∆t|r∆t ≤ −ρ .
In the inverse statistics literature, the distributions of the waiting times for gains τ+ρ , and
losses, τ−ρ , are denoted by p(τ±ρ ). Figure 5 displays the pdfs for ±3.5% for the Ibovespa. In
Figure 6, we plot the same functions for the Dow Jones Index. We notice that for both indices,
there is a gain/loss assymetry, which is related to the leverage effect in [8]. Notice that both
median and mean are much shorter for Ibovespa than for DJI. This reflects that much more
volatile nature of the brazillian stock market.
Now, back to the definition of V Ci , we have decided to define a meaningful size of window `
based on the median of the distribution. Since 13 is the median size of a ±3.5% move, we have
decide to choose a window of size 11, with the corresponding investigated day in the middle of
the window, such that we have n = 5 in the definitions (2), (3) and (4) of F Vi , P Vi and V Ci .
This guarantees that the typical move inside this 11 days window is not typically much larger to
the jump itself, which confers significance to the jump event in the window. Figure 7 displays a
13 -days window around a jump up.
One major advantage of this definition over the more common method of relating leverage
effect to some negative correlation between return and volatility, is the fact that this measure is
local, and, therefore, we can study leverage effects for different types of jumps. The correlation
approach is a global approach, and this is why we have not used it in this work.
Now, with the n = 5 empirical choice, we study the typical average behavior of the volatility
change, V C, around jumps compared to regular days. The idea is to group all days with returns
in a given range, and calculate the average V C for this group. In Figure 8, we display this
procedure by plotting (with blue markers) the averaged volatility change V C for each decile of
days ranked by return. More specifically, the first blue marker (from left ro right) is the average
volatility change for the worst 10% days ranked by return. The blue marker in his immediate
right is the group of 10% − 20% worst days, and so on. The last blue marker represents the
averaged V C for the best 10% days by returns. More precisely, x-axis represent the averaged
return for each quantile, while the y-axis represent the averaged V C in each quantile. The blue
markers represent 10%-quantiles.
The red dots represent the same quantities, however, restricted to the jumps above/below
±3.5%. Notice that the down jumps display a very pronounced change of volatility. Indeed, one
can see that future volatility is so much higher compared to the past volatilities in this group
that V C falls to about 70% of the averaged behavior of small return days.
Similarly striking is the behavior of the jumps up group. Future volatility falls so much after
a price increase that the V C measure increases to about 130% of the regular behavior of small
return days.
10
Although V C is a relatively crude measure of volatility change, it is very surprising that it
can actually capture such an important phenomenon. We are not aware of similar definitions in
the literature, and this is one of our original contributions in this work.
Figure 9 exhibits the behavior of averaged return and averaged volatility change by jump
categories for days with down jumps. We first notice that there are no major differences in
averaged returns by each category. We remark, however, that there are some major differences in
the volatility change behavior among the categories. For example, the domestic monetary policy
category has a considerably higher V C measure than the international monetary policy category
(34% points higher), and even higher than the Foreign Stocks & unknown category. It suggests
that domestic monetary policy triggered jumps propagate less uncertainty in the future than the
other categories.
Figure 9 exhibits the behavior of averaged return and averaged volatility change by jump
categories for days with up jumps. We can observe that jumps related to International Monetary
Policy have a relative large return compared to many other types of jumps. One may also notice
that the volatility drop for this category is substantially large compared to all the other categories
displayed, except for the class of days driven by Political Turmoil. We remark however that the
Political Turmoil category is not very statistically significant.
Another important observation is that for the classes driven by commodities and domestic
monetary policy, the volatility change measure does not differ much from the up days to the
down days. As a matter of fact, for these classes, volatility typically rises for both types of jumps.
5
Conclusions
In this work, we have investigated the causes of jumps by reading next-day newspapers,
following the methodology developed in [2]. We have observed that the main causes of jumps
may vary along time. For example, jumps triggered by macroeconomic news are much scarcer
out of the critical period of 2006 − 2008. We have also observed that policy-triggered jumps are
predominant. This corroborates the claim in [1] that policy may be a major driver in adding
uncertainty to markets.
We have also proposed a volatility change definition, V C. The measure suggests that the
so-called leverage effect indeed holds, but that is dependent on the jump’s class. We have shown
that some classes of jumps may, indeed, display an anti-leverage effect. This new measure needs
further studies and validation, and we are currently conducting them.
As remarked in the introduction of this work, market jumps are facts of every investor’s lives.
They represent great opportunities, but more importantly, great risks. There has been a great
deal of work trying to incorporate them in financial models, but as remarked in Section 2, they are
usually prescribed as an ad-hoc tool to attenuate the imprecision of stochastic volatility models.
A great achievement would be to incorporate some of the characteristics presented in this work
to the classical theories of asset pricing, such as the EMH and the CAPM.
In any case, financial theorists need to compromise between realist hypotheses on jumps, and
useful pricing models. The aim of this work was to present a relatively detailed view on the
phenomenon of jumps.
Because much of the research was based on opinions of the readers, and on the newspapers’
11
Ibovespa Historic Prices from 02/01/2002 to 12/08/2015
Logarithmic Price
11.5
11
10.5
10
9.5
Drift
Ibovespa
9
0
500
1000
1500
2000
Trading Days
2500
3000
3500
Signal’s Noise
0.4
Logarithmic Price
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
Noise
-0.4
0
500
1000
1500
2000
Trading Days
2500
3000
3500
Figure 3: Ibovespa Price Series. Noise extracted using a Daubechies 4 wavelet filter as described
in Chpter 13 in [11].
views on the subject, it would be possible that different groups of readers would reach different
conclusions. The authors are currently conducting a larger scale extension of this research, and
we will communicate on this elsewhere.
Acknowledgment. This work was also partially supported by CNPq and CAPES Foundation.
F.R. has also benefited from a research grant from DNIT (Departamento Nacional de Infraestrutura
e Transportes).
A
Appendix: Optimal Investment Horizon
In this appendix, we concern ourselves with the study of the optimal investment horizon for
various return values for the Bovespa Index, which is defined to be the number of days one most
frequently has to wait in order to first attain a desired return. It is calculated by initially taking
12
Ibovespa, Year 2014
0.1
0.08
Logarithmic Price
0.06
ρ>3.5%
0.04
0.02
ρ<-3.5%
0
-0.02
-0.04
22/09
30/09
13/10
Trading Days
Figure 4: Schematic overview of inverse statistics on logarithmic prices. It calculates the number
of days until a prescribed level ±ρ is reached.
Return=3.5%,Median=13,Mean=25.0433
Return=-3.5%,Median=13,Mean=27.1633
0.08
0.07
0.06
Pdf(τρ)
0.05
Empirical Pdf for Return=3.5%
Empirical Pdf for Return=-3.5%
0.04
0.03
0.02
0.01
0
100
101
102
103
τρ (days)
Figure 5: Probability distribution function (pdf), p(τ±ρ ) of the waiting times τ±ρ , with a return
level ρ = 3.5%. Ibovespa series.
a day in the historical prices series and counting from that day on the number of days it takes to
first achieve a desired return, and then repeating this procedure for every day in the time series.
Afterwards one should be able to draw a relative frequency plot, which will show the optimal
investment horizon as the number of days with the highest frequency.
In order to execute the aforementioned process, we don’t actually use the Ibovespa time series,
13
Return=2.5%,Median=19,Mean=32.3018
Return=-2.5%,Median=18,Mean=31.0598
0.05
0.04
Pdf(τρ)
0.03
Empirical Pdf for Return=2.5%
Empirical Pdf for Return=-2.5%
0.02
0.01
0
100
101
102
103
τρ (days)
Figure 6: Probability distribution function (pdf), p(τ±ρ ) of the waiting times τ±ρ , with a return
level ρ = 2.5%. Dow Jones Series.
Ibovespa, Year 2002
0.06
0.04
Logarithmic Price
0.02
0
-0.02
-0.04
-0.06
-0.08
16/08 19/08 20/08 21/08 22/08 23/08 26/08 27/08 28/08 29/08 30/08 02/09 03/09
Figure 7: Illustration of the 13-days window centered on a day where the jump exceeded the 3.5%
threshold. The V C uses the shorter 11-days window.
instead we first take its natural logarithm and then put it through a filter that removes any
long-term tendencies the Ibovespa may be showing, its "drift", leaving behind only the log-price
variations around this drift, the "noise". It is the noise that is actually used to carry out the
calculations described in this section’s first paragraph , as it is shown along with the drift for the
Ibovespa from January 2, 2002 to August 12, 2015 in Figure 3.
14
Figure 8: The x-axis represent the average return for each quantile. The y-axis represent the
average V C for each quantile. The blue markers represent deciles. The red markers represent the
jumps. The smaller the measure V C, the larger the volatility increase in the future. Similarly,
the smaller the measure V C, the larger the volatility decrease in the future. The picture shows
that large jumps down increase future volatility, and large jumps up decrases future volatility.
Figure 9: Jumps Down: Blue bars represent averaged return per jump category. Red bars
represent averaged V C per jump category. 1 - International Monetary Policy and Central
Banking. 2-Domestic Monetary Policy and Central Banking. 3- Elections and Political Transitions
4- Macroeconomic News and Outlook. 5- Commodities. 6 Foreign Stock Markets (No specific
reason). The smaller the VC, the larger is the forward volatility jump.
The drift is to be understood as an average value the Ibovespa takes as time goes by, reflecting
the overall tendency of the economy, while the noise reflects the variations around this average
value, which are the result of the influence exerted by the random variables that influence the
market, as for example: news of oil discovery in the pre-salt layer, news about corruption amongst
Brasília’s top brass, and others.
In greater detail, the filter acts on the Ibovespa log-price time series taking it to the wavelet
15
Figure 10: Jumps Up: Blue bars represent averaged return per jump category. Red bars represent
averaged V C per jump category. 1 - International Monetary Policy and Central Banking.
2-Domestic Monetary Policy and Central Banking. 3- Elections and Political Transitions 4Macroeconomic News and Outlook. 5- Commodities. 6 Corruption Scandals and Political Turmoil
Figure 11: Agreement among Readers and Newspapers for each jump.
domain, by multiplying it by a suitable matrix, setting all wavelet coefficients corresponding to
scales larger than 128 trading days to zero, and finally transforming back to the time domain.
The choice of setting all wavelet coefficients corresponding to scales larger than 128 trading days
to zero tells the wavelets that they should only capture price variations that take place in less
than 128 days, meaning "ups and downs" that happen in less than 128 days will constitute the
noise.
The choice of scale of 128 days was considered to be optimal by this article authors, as different
choices lead to pairs drift-noise which are not reasonable for our analysis. Examples follow:
16
Ibovespa Historic Prices from 02/01/2002 to 12/08/2015
Ibovespa Historic Prices from 02/01/2002 to 12/08/2015
11.5
11
Logarithmic Price
Logarithmic Price
11.5
10.5
10
9.5
11
10.5
10
9.5
Drift
Ibovespa
Drift
Ibovespa
9
9
0
500
1000
1500
2000
Trading Days
2500
3000
3500
0
500
1000
Signal’s Noise
2500
3000
2500
3000
3500
Signal’s Noise
0.8
0.1
0.6
0.05
Logarithmic Price
Logarithmic Price
1500
2000
Trading Days
0.4
0.2
0
-0.2
-0.4
0
-0.05
-0.1
-0.15
-0.6
Noise
Noise
-0.8
-0.2
0
500
1000
1500
2000
Trading Days
2500
3000
3500
0
(a) The problem with this choice of scale, which
consists in capturing as noise price variations that
happen in less than 1024 days, is that it produces
a poor drift and consequently a poor noise, since it
considers, amongst other issues, the 2008 crisis, the
pronounced valley between 1500 and 2000 days, to
be a jump, and not a long-term tendency.
500
1000
1500
2000
Trading Days
3500
(b) The problem with this choice of scale, which
consists in capturing as noise price variations that
happen in less than 4 days, is that it produces
a poor drift and consequently a poor noise, since
it considers, amongst other issues, the drift to be
practically equal to the Ibovespa, leaving almost
no noise, which means that there are practically no
random events causing price variations, which we
know not to be true because of events like the ones
already mentioned in this section’s third paragraph.
Figure 12: Different choices of scale
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