^{*}

^{**}

^{***}

This paper's aim is to extend the Durham and Park's (2012) model by incorporating the market fractional behavior. The extension examines the stochastic dynamics of stock indexes for several of the world's main economies (US, Eurozone, UK and Japan), as well as emerging markets (China, Brazil and Mexico) during 1994-2017. The proposed model assumes that the returns are driven by fractional Brownian motions combined with Poisson processes and modulated by Markov chains. Risk factors such as: idiosyncratic volatility, market volatility, volatility of volatility were incorporated. To accomplish the purpose of the extension, Jump-GARCH and Markov regime-switching models were estimated, the Hurst coefficient was calculated and jumps behaviour was analysed during crisis periods. It was considered that the model accurately describes the stochastic dynamics of the stock indexes returns. The main empirical findings are that the USA stock market remains in high volatility most of the time, that the Brazil Stock Market has the highest intensity of jumps, and that the mean return is the highest for the IPC and the lowest for Nikkei Index during the period understudy.

Esta investigación tiene como propósito extender el modelo de Durham y Park (2012) al incorporar la conducta fraccional del mercado. La extensión examina la dinámica estocástica de los índices bursátiles de economías desarrolladas como en EUA, Eurozona, Reino Unido y Japón, así como los mercados emergentes de China, Brasil y México durante 1994-2017. Nuestro modelo supone que los rendimientos son conducidos por movimientos brownianos fracciónales, combinados con procesos de Poisson y modulados por cadenas de Markov. Se incorporaron factores de riesgo como: volatilidad idiosincrática, volatilidad del mercado y volatilidad de la volatilidad. Para ello, se estimaron los modelos Jump-GARCH y de cambio de régimen Markoviano, se calcularon los coeficientes de Hurst y se analizó el comportamiento de los saltos en periodos de crisis. Se encontró que el modelo propuesto describe adecuadamente la dinámica estocástica de los rendimientos de los índices bursátiles estudiados. Los principales hallazgos empíricos son que el mercado de valores de EUA, se mantiene en alta volatilidad la mayor parte del tiempo, que el mercado de valores de Brasil tiene la mayor intensidad de saltos, que el rendimiento medio más alto lo presenta el IPC y el más bajo el índice Nikkei durante el período de estudio.

Over a long period of time, several researchers have dealt with the stochastic dynamics and return probability distributions of the stock markets; however, there are still irregularities and stylized facts that need to be explained. The development of the stock market indexes has followed complex dynamics derived from intricate global investment strategies, requiring more sophisticated models and tools. Most of the models in the specialized literature can be broadly classified into two large groups: Models seeking to. explain the fundamental value of stock and models describing stock prices dynamics (

Several studies focused on return distributions with time-varying moments, Carr and Wu (2007) proposed a stochastic skew model for foreign exchange rates;

Another factor that has been relevant when examining returns dynamics is the volatility of volatility.^{1}

An important characteristic of the stock markets is the presence of unexpected and sudden jumps.

There are other studies addressing options and futures markets, such as

The above investigations have highlighted the importance of including the effect of volatility, volatility of volatility, unexpected jumps, and regimeswitching on stock returns. The hypothesis of this paper establishes that the returns of stock indexes are properly driven by fractional Brownian motion implying long-term memory. This article mainly extends current studies from

This paper is organized as follows: the first section presents the extended stochastic model of stock index returns; section two describes the data and defines the endogenous and exogenous variables; section three calibrates the proposed model; and finally the conclusions are provided.

This section presents the theoretical background needed to model the dynamics of stock indexes returns by using fractional Brownian motion combined with Poisson process modulated with Markov switching-regime stochastic volatility. Most of the empirical studies suggest that market volatility varies over time and stocks with high sensitivity to both jump and volatility risks have low expected returns (^{2}

In the proposed multifactor risk model, stock returns are driven by the fractional Brownian motion, _{
t
} ,and modulated by Markov regime switching,

where _{
t
} is a dependent variable determining the dynamics of the stock index return, _{
t
} is a Poisson jumps, _{
t
} is the idiosyncratic volatility, _{i} is the volatility state, _{
t
} is the volatility of volatility, α is the speed adjustment parameter, ^{3}

A Markov regime-switching process (

The fractional Brownian motion _{
t
} ε_{
tε [0,T]
}
^{4}

It is important to point out that

The ARCH model is briefly review, which is useful to explain the large residuals’ trend to cluster together (Engle, 1982). The ARCH model is given by:

where ^{
2
}
_{
t
} is the conditional variance,

where ^{
2
}
_{
t
} is the conditional variance, _{
2
} are unknown parameters, _{
t,1
} and _{
t,2
}
_{
t,1
} and _{
t,2
} are independent and satisfy:

The first innovation refers to market stability with no jumps, thus:

and

The second innovation describes an unexpected jump when an unusual event occurs. The returns of the stock market are impacted by an unexpected event. The distribution of jumps follows a Poisson distribution and ( is the parameter of the jump intensity, hence:

and

where _{
t
} stands for the dynamics of the return, _{
t
} is the jump component, _{
Yt,k
} is the jump size, λ_{
t
} is the jump intensity, _{
t
} denotes the number of jumps. The Poisson process _{
t
} with intensity parameter

Hence,

Then,

This section, aims to find out how well the proposed model captures and describes the dynamics of the stock index returns under study. The data for the US (S&P 500), Eurozone (EuroStoxx50), United of Kingdom (FTSE100), Japan (Nikkei), China (Hang Seng), México (IPC) and Brazil (Bovespa) were obtained from Bloomberg and includes daily returns of each stock index. The USA is considered as a benchmark since it is the world's largest economy and it has the biggest financial market.^{5}

The analysis sample period begins in January 1994 and ends in December 2017 (5980 daily returns for each stock index). The purpose of this study is to capture in the extension proposed the dynamics of stock market indexes before, during and after crisis periods. The most relevant extreme events are the Asian financial crisis, the bubble dot com in 2001, the subprime mortgage recession in 2008, the Eurozone debt crisis in 2011, the Brexit in June 2016, and the power takeover of president Trump in December 2016. The idiosyncratic volatility is represented by the standard deviation. The market volatility is calculated through the VIX index, which is a measure of the expected volatility of the US stock market during 30 days, calculated from real-time mid quote prices of S&P500 call and put options index (CBOE). Finally, the volatility of volatility is the square return of measure by the VIX index. The parameters for Markov regime switching were estimated using

The results of the Markov regime-switching models describing the degree of volatility of the previous period of the returns of S&P500, Eurostoxx50, FTSE100, Nikkei, Hang Seng, IPC and Bovespa indexes are shown in

Probabilities
S&P500
EuroStoxx50
FTSE100
IPC
Bovespa
Nikkei
Hang Seng
p11
0.41424
0.52744
0.54349
0.58354
0.56479
0.46037
0.48052
p12
0.58576
0.47255
0.45650
0.41646
0.43521
0.53963
0.51949
p21
0.29279
0.40675
0.50486
0.38694
0.44869
0.54721
0.37474
p22
0.70721
0.59324
0.49513
0.61306
0.55131
0.45279
0.62526

Source: Prepared by authors with Bloomberg data and E-views software.

The highest jump for S&P 500 is at the Subprime Mortgage Recession (

Event
S&P 500
EuroS- toxx50
FTSE 100
Nikkei
Hang Seng
IPC
Bovespa
Asian Financial Crisis
End of 1997 and 1998
End of 1997 and 1998
No impact
End of 1997 and 1998
End of 1997 and 1998
End of 1997
End of 1997
Dot Com Bubble
Middle of 2000 and 2001
2001 and beginning of 2002
Slightly impact
From 2000 to 2002
First semester of 2000
Middle of 2000 and 2001
End of 2001
S u b p r i m e Mortgage Recession
End of 2008 and beginning of2009
End of 2008 and beginning of 2009
Second half of 2008
Second half of 2008
From 2007 to 2009
End of 2008 and beginning of2009
End of 2008 and beginning of2009
Eurozone Debt Crisis
Slightly impact in August 2008
2010 and 2011
Slightly impact
Beginning of 2011
Beginning of2011
Slightly impact in August 2008
Slightly impact in July 2008
Brexit
Slightly impact in June 2016
June 2016
June 2016
Slightly impact in June 2016
Slightly impact on June 2016
Slightly impact in June 2016
Slightly impact in June 2016
Power takeover President Trump
Slightly impact in November 2016
Slightly impact in November 2016
Slightly impact in November 2016
2016
Slightly impact in November 2016
November 2016
November 2016

Source: Prepared by authors.

In order to estimate the parameters of the Jump-GARCH model, the log likelihood was computed, the log likelihood of a GARCH model on the residuals was evaluated, jumps were examined, and the accumulated first and second moments were defined.

Parameters
S&P 500
EuroS- toxx50
FTSE 100
IPC
Bovespa
Nikkei
Hang Seng
M
0.0002940
0.0001540
0.0004290
0.0004958
0.0002748
0.0000286
0.0001576
0.0001365
0.0002031
0.0001737
0.0002323
0.0006977
0.0002410
0.0002626
0.011684
0.014252
0.013178
0.015241
0.026414
0.015523
0.016204
52.59522
49.21371
50.42269
49.55913
47.35933
50.06818
50.05000
A
0.000638
0.000569
0.000498
0.000664
0.000982
0.000365
0.000533
-0.037814
-0.013797
-0.036261
0.087828
0.065490
-0.109516
0.042823
γ
0.000001
0.000002
0.000003
0.000001
0.000016
0.000005
0.000002
0.094715
0.081803
0.117276
0.081836
0.117298
0.101665
0.071503
0.894829
0.911664
0.864243
0.917049
0.860306
0.880997
0.921362

Source: Prepared by authors with Bloomberg data with RATS.

The outcome of the Hurst coefficient

Hurst Coefficient
S&P 500
EuroStoxx50
FTSE100
IPC
Bovespa
Nikkei
Hang Seng
0.556061
0.533003
0.542380
0.516423
0.560925
0.539031
0.535857

Source: Prepared by authors with

For many decades, researchers have worked on describing the financial market behaviour since this is the thermometer of the economy. Relevant economics and political events (crisis) have occurred since 1929. Nowadays, the evolution of the stock market has posed new challenges such as the understanding of the behaviour of volatility of volatility and volatility clusters, thus it is required to apply new models that include sophisticated tools such as fractional Brownian motion modulated by Markov chains, in order to explain the market behaviour.

Until now, several researchers have worked on finding better models to explain the behaviour of stock returns. A substantial proportion of the variation of stock returns remains unexplained; this lack of knowledge generates uncertainty and instability, and therefore affects, not only financial markets but the economy as a whole. This paper seeks to contribute to reduce this gap by constructing a model that explains the behaviour of stock indexes volatility based on the

After calibrating the extension, it can be noticed that the stock indexes that have a probability over 60% to remain in high volatility are S&P 500 with 70%, and Hang Seng with 62%; while, IPC, Bovespa and FTSE100 have

a high probability to stay in low volatility, 58%, 56% and 54%, respectively. The percentage of changes from high to low volatility from one period to another is just 29% for S&P 500. Nikkei has the greatest probability to move from high to low volatility but it will not remain in low volatility for long time. S&P500 and Hang Seng were found to be more volatile than other indexes. Moreover, from the GARCH estimation is observed that Bovespa and FTSE100 have the highest lag random deviation (0.1172); Hang Seng has the highest lag variance, 0.9213; Bovespa has the largest jump in size and intensity; and S&P has the greatest amount of jumps in the period studied.

Volatility of volatility is a measure of volatility expected of the n-day forward price of the volatility and this drives nearby volatility options price.

See also Christoffersen et al. (2009) and Ang et al. (2006).

Other papers dealing with jump-diffusion processes are

Investors of financial markets often take decisions based on Eurozone and UK economic data.

The Mexican Index (IPC) showed an important jump at the beginning of 1995 due to a

The Brazilian