Modelling Financial Contagion: A Theoretical Analysis

By

Dinu Mathew Panampunna

PGPDCM – II

(2004-06)

Submitted to

Mudra Institute of Communications, Ahmedabad

In partial fulfillment of the requirements for the Postgraduate Programme

Diploma in Communications Management

Dissertation Guide:

Dr. Rasananda Panda

Faculty, MICA

Mudra Institute of Communications, Ahmedabad

Ahmedabad

March 2006

2

Notice of Copyright

Mudra Institute of Communications

Copyright Dinu Mathew Panampunna 2006 and Mudra

Institute of Communications, Ahmedabad (MICA)

3

Executive Summary

The study attempts to sketch financial contagion as essentially a parametric

technique capable of being disseminated through mathematical equations.

Contagion is defined purely in terms of its empirical validity and the models

proposed henceforth are dependent on that definition. The literature on the

subject though not in any terms exhaustive has been examined for its robustness

and accessibility of rigour. The macro and micro economic factors of financial

distress has been touched upon with adequate explanations being provided

mathematically by different scenario testing. These have helped in attempting to

trace the evolution of the “world factor” to represent a variable that often goes

unobserved. Some testing models like bivariate and multivariate testing models

have been introduced to explain the factor model. The study attempts to sketch

the mathematical ideation behind time series analysis extending it into stochastic

processes. Therein an attempt has been made to model trends in financial data

by using Random Walk Hypothesis and its myriad variants. An S- Variant point

process model has been laid down to explain the effects that time and the type of

event has on the stock price. The stochastic intensity approach and its diagnostic

tool, the generalized Hawkes model has been introduced to further strengthen

the process. Itos Lemma has been introduced at this stage to test out the

deterministic and random component of a stock price and to analyse whether

one could explain any deterministic character out of the random component in

the Generalized Hawkes Model. Finally the study tries to look into the real world

and takes through a path of empirical studies on contagion throughout the

world. The study is highly quantitative and to facilitate easy understanding, a

brief synopsis is given at the beginning of each chapter.

4

ACKNOWLEDGEMENT ACKNOWLEDGEMENT

I am happy to place on record my gratitude towards numerous persons who have made valuable contributions academically and non-academically towards the completion of this work.

It is difficult for me to express in words the deep sense of gratitude I owe to Dr. Rasananda Panda for his generous assistance, constant supervision, and patient forbearance to the numerous problems and difficulties encountered during the present study. It would have been impossible for me to do this study but for the inspiration I received from Prof. Panda at different stages of my work.

I am thankful to Prof. Mathew, whom I constantly troubled with mallu expletives, Prof.

Neeraj Amarnani, Prof. Naval Bhargav, Dr. Shubhra Gaur, and Prof Anita Basalingappa for their valuable suggestions as my dissertation Committee. I place my sincere gratitude for them for spending their valuable time for discussion and help rendered to me for shaping the thesis.

I express my sincere thanks to my best friend Gujju, Shubs, Smita, Mohit and also Abhijit

who sat as a patient audience in my intellectual outbursts and for the encouragement they have extended during my study. I also sincerely express my heart felt gratitude to my junior Gaurav Bose who helped me in the type setting and also donned the role of a competitive evaluator.

I would fail in my duty if I forget to place my heartfelt gratitude to my fiancée, Lija

Ramachandran whose constant criticism and encouragement has helped me to improve the work significantly.

Last but not least, I would fail in my duty if I forget to mention my heartfelt thanks to

my parents who have been my continuous source of inspiration and encouragement over the years. Without their blessing, I would not have been able to start on and complete my dissertation.

Dinu Mathew Panampunna Dinu Mathew Panampunna

5

CONTENTS

Page No. Notice of Copy right

Executive Summary Acknowledgement

Chapter 1 Modelling Financial Contagion

1.1. Introduction 1.2. Review of Literature 1.3. Objective of the Study 1.4. Methodology 1.4 (a) Generalized Theoretical Models 1.4 (b) Empirical framework for intensity ana lysis

1.5. Contagion and Financial Crisis 1.6. Unanticipated Shock Models of Contagion 1.7. Bivariate Testing 1.8. Multivariate Testing 1.8 (a) Using Just Crisis Data 1.9. Databases 1.10 Scope of the Study 1.11 Limitations of the Study

1- 9

2 2 3 4 4 4 5 6 7 7 8 9 9 9

Chapter 2 Theoretical Modelling of Contagion: A New Perspective

2.1. A Mathematical Exposition into Time Series Analysis

2.2. Stochastic Process 2.3. Non-Stationary Time Series and Random Walk

Hypothesis 2.4. Modeling Different Trends 2.5. S-variate point process 2.6. Stochastic Intensity Approach 2.7. Generalised Hawkes Model 2.8. The Theoretically Conceived Model 2.9. Ito's Lemma 2.10. Looking Ahead: Analysis of the Model and

Randomness

10 – 21

11 11 11

12 14 15 17 18 18 20

Chapter 3 Financial Contagion: Models and Perspectives

3.1. Financial Contagion: An Introspective Analysis 3.2. Market Interdependence and Crisis Thresholds 3.3. Conclusions and Findings

22 – 26

22 23 25

Reference

Appendix Glossary

6

“Extreme, synchronized rises and falls in financial markets occur infrequently but they do occur. The problem with the models is that they did not assign a high enough chance of occurrence to the scenario in which many things go wrong at the same time- the “perfect storm” scenario” (Business Week, September 1998). “That which is static and repetitive is boring. That which is dynamic and random is confusing. In between lies art." —John A. Locke “Regulators have criticized LTCM and banks for not “stress-testing” risk models against extreme market movements... The markets have been through the financial equivalent of several Hurricane Andrews hitting Florida all at once. Is the appropriate response to accept that it was mere bad luck to run into such a rare event - or to get new forecasting models that assume more storms in the future?”

(The Economist, October 1998, after the LTCM rescue)

7

Chapter 1

Modelling Financial Contagion

Abstract

This chapter attempts to sketch financial contagion as essentially a

parametric technique capable of being disseminated through mathematical

equations. Contagion is defined purely in terms of its empirical validity and the

models proposed henceforth are dependent on that definition. The literature on

the subject though not in any terms exhaustive has been examined for its

robustness and accessibility of rigour. The macro and micro economic factors of

financial distress has been touched upon with adequate explanations being

provided mathematically by different scenario testing. These have helped in

attempting to trace the evolution of the “world factor” to represent a variable

that often goes unobserved. Some testing models like bivariate and multivariate

testing models have been introduced to explain the factor model.

8

1.1 Introduction

Financial risk-taking is a concern of public policy because associated with the risk

taking actions of individuals there are externalities, i.e. costs and benefits accruing

to the society that are external to the calculations of the individual investor, and

not accounted for in the market place.1 1 In an economy where there are important

externalities, competitive markets will be socially inefficient. The task of public

policy, in this case of financial regulation, is to attempt to mitigate these

inefficiencies. Financial externalities are particularly potent because they are

transmitted macro economically. Yet despite all this talk of externalities, contagion

and panics, a peculiarity of market expectations is that they seem to be remarkably

stable (or tranquil) for substantial periods of time, even when underlying real

circumstances might be decidedly unpropitious.

Periods of tranquility defined by stable expectations and stable market confidence

may sustain the illusion that financial markets are truly reflecting a strong real

economy. The shattering of that illusion can be catastrophic.

1.2 Review of Literature

The externality of systemic risk is in large part manifest through what the

economist John Maynard Keynes called a “beauty contest”. In Keynes’s contest,

beauty is not in the eye of the beholder. Instead, the game is won by those who can

accurately assess what others think is beautiful. In financial markets, it is knowing

what others believe to be true that is the key to knowing how markets will behave.

The market is driven by participants’ belief about what average opinion believes

average opinion believes and so on, ad infinitum (Keynes 1936:; Eatwell and

Taylor 2000).

1 There are a number of other important market failures in the financial sector which attract the

concerns of public policy, most notably the asymmetry of information between individual savers and market professionals that is the motivation of consumer protection. This lecture deals solely with the market failure manifest in systemic risk.

9

Individual investors and traders must be highly heterogeneous, with different

financial objectives. In traditional economics this was described as the difference

between those seeking income certainty and those seeking wealth certainty, with

different patterns of risk aversion, different investment time horizons and so on

(Robinson 1951).

Liberalisation of financial markets that has taken place over the past 3 decades has

inevitably reduced the heterogeneity in financial markets. By definition

liberalisation has broken down market segmentation - cross-market correlations

have risen sharply. And with liberalisation has come a growing professionalisation

of financial management (BIS 1998: chpt. V), and extensive consolidation of

financial institutions (Group of Ten 2001). The professional investor is not only

subject to a continual pressure to maximize short–term returns, but also in a

competitive market myopic (i.e. short-time horizon) investment is an optimal

strategy (Kurz 1987). So whatever the preferences of the private investor might be,

convergence on myopic strategies by professional investors is homogenizing the

market.

Macroeconomic and microeconomic aspects of international regulation

Public policy also needs to take into account the fact that beliefs about average

opinion transmit externalities through macroeconomic variables – the interest rate

say, or the general level of stock prices, or the exchange rate. So effective

regulation of firms should be conceived in conjunction with macroeconomic

policy. This is particularly true in an international setting, where a major focus of

systemic risk is the exchange rate. In policy terms, macroeconomic action may be

a far more efficient means of reducing systemic risk than traditional

microeconomic regulation.

1.3 Objective of the Study

10

My main objective through this study would be to analyse the following,

1) To theoretically conceive a new perspective to capture multivariate

market event data based on real time econometric methods

1.4 Methodology

The methodological aspects of the study would focus on the following

1.4(a) Generalized Theoretical Models

To analyse the above stated objective the following methodology is proposed. The

study will briefly describe the aspects of PP theory that are central to the paper and

discuss an intensity based approach to inference for PPs. I adopt a different

approach in which the model is specified via the vector stochastic intensity. This

provides a natural and powerful modeling framework for multivariate market

event data. Each element of the stochastic intensity is a continuous time process

that may be interpreted as the conditional hazard for the particular type of market

event in question. The “σ ” field upon which the hazard is conditioned is updated

continuously as new information arrives, thus allowing other types of event to

influence the hazard as they occur in continuous time. My approach is closest to

that of Russell (1999), who also specfies a multivariate PP model via the stochastic

intensity.

Therein I would like to introduce a new class of models for financial market event

data (the generalized Hawkes) models. These allow the estimation of the nature of

the dependence of the intensity on the events of previous trading days rather than

imposing strong, a priori assumptions concerning this dependence. Continuing

from the generalized Hawkes model I would attempt to build on the suggestions of

Russell (1999) for the construction of diagnostic tests for parametric, multivariate

PP models.

11

1.4(b) Empirical framework for intensity analysis

Using the theoretical framework, an empirical framework shall be developed to

estimate the nature and magnitude of the intensity of variables occurring as

conditional hazards on the financial market

1.5 Contagion and Financial Crisis

There is now a reasonably large body of empirical work testing for the existence of

contagion during financial crises. A range of different methodologies are in use,

making it difficult to assess the evidence for and against contagion, and

particularly its significance in transmitting crises between countries.2 The origins

of current empirical studies of contagion stem from Sharpe (1964) and Grubel and

Fadner (1971), and more recently from King and Wadhwani (1990), Engle, Ito and

Lin (1990) and Bekaert and Hodrick (1992).

Before developing a model of contagion, a model of interdependence of asset

markets during non-crisis periods is specified as a latent factor model of asset

returns. The model has its origins in the factor models in finance based on

Arbitrage Pricing Theory for example, where asset returns are determined by a set

of common factors and a set of idiosyncratic factors representing non-diversifiable

risk (Sharpe 1964; Solnik 1974). Similar latent factor models of contagion are used

by Dungey and Martin (2001), Dungey, Fry, Gonzalez Hermosillo and Martin

(2002a), Forbes and Rigobon (2002) and Bekaert, Harvey and Ng (2003).

2 The literature on financial crises themselves is much wider than that canvassed here and is

reviewed in Flood and Marion (1998) while more recent papers are represented by Allen and Gale (2000), Calvo and Mendoza (2000), Kyle and Xiong (2001) and Kodres and Pritsker (2002).

12

To simplify the analysis, the number of assets considered is three. Extending the

model to N assets is straightforward with an example given below. Let the returns

of three asset markets during a non-crisis period be defined as

{x1,t, x2,t, x3,t} ……………………………………. (1.1)

All returns are assumed to have zero means. The returns could be on currencies, or

national equity markets, or a combination of currency and equity returns in a

particular country or across countries. The following trivariate factor model is

assumed to summarise the dynamics of the three processes during a period of

tranquility.

xi,t = ëiwt + äiui,t, i= 1, 2, 3………………………..(1.2)

The variable wt represents common shocks that impact upon all asset returns with

loadings ëi. These shocks could represent financial shocks arising from changes to

the risk aversion of international investors, or changes in world endowments

(Mahieu and Schotman 1994; Rigobon 2003b). In general, wt represents market

fundamentals which determine the average level of asset returns across

international markets during normal, that is, tranquil, times. This variable is

commonly referred to as a world factor, which may or may not be observed. Much

would be talked about this factor model in the next chapter.

1.6 Unanticipated Shock Models of Contagion

The definition of the term contagion varies widely across the literature. In this

paper contagion is represented by the transmission of unanticipated local shocks to

another country or market. This definition is consistent with that of Masson

(1999a,b, c), who divides shocks to asset markets as either common, spillovers that

result from some identifiable channel, local or contagion, and as shown below that

of other approaches, such as Forbes and Rigobon (2002) where contagion is

13

represented by an increase in correlation during periods of crisis. The first model

discussed is based on the factor structure developed by Dungey, Fry, Gonzalez-

Hermosillo and Martin (2002a,b) amongst others, where contagion is defined as

the effects of unanticipated shocks across asset markets during a period of crisis.

To distinguish between asset returns in a non-crisis and crisis period, yi,t represents

the return during the crisis period and xi,t the return during the non-crisis period.

Consider the case of contagion from country 1 to country 2. The factor model is

now augmented as follows

y1,t = ë1wt + ä1u1,t

y2,t= ë2wt + ä2u2,t + ãu1,t ...................(1.3)

y3,t = ë3wt + ä3u3,t

where the xi,t are replaced by yi,t to signify demeaned asset returns during the

crisis period. The expression for y2,t now contains a contagious transmission

channel as represented by unanticipated local shocks from the asset market in

country 1, with its impact measured by the parameter ã. The fundamental aim of

all empirical models of contagion is to test the statistical significance of the

parameter ã.3

1.7 Bivariate Testing

Bivariate tests of contagion focus on changes in the volatility of pairs of asset

returns. From (1.3), the covariance between the asset returns of countries 1 and 2

during the crisis is

3 An important assumption underlying 3 is that the common shock and idiosyncratic shocks have

the same impact during the crisis period as they have during the non-crisis period.

14

E [y1,t ,y2,t] = ë1ë2 + ãä1……………………….(1.4)

Comparing this expression with the covariance for the pre-crisis period in (2)

shows that the change in covariance between the two periods is

E [y1,t,y2,t] . E [x1,tx2,t] = ãä1………………….(1.5)

If ã > 0, there is an increase in the covariance of asset returns during the crisis

period as ä1 > 0 by assumption. This is usually the situation observed in the data.

However, it is possible for ã < 0, in which case there is a reduction in the

covariance. Both situations are valid as both represent evidence of contagion via

the impact of unanticipated shocks in (1.3).

1.8 Multivariate Testing

The test for contagion presented so far is a test for contagion from country 1 to

country 2. However, it is possible to test for contagion in many directions provided

that there are sufficient moment conditions to identify the unknown parameters.

For example, (1.3) can be extended as

y1,t = ë1wt + ä1u1,t + ã1,2u2,t + ã1,3u3,t

y2,t = ë2wt + ä2u2,t + ã2,1u1,t + ã2,3u3,t …….(1.6)

y3,t = ë3wt + ä3u3,t + ã3,1u1,t + ã3,2u2,t,

In this case there are 6 parameters, ãi,j , controlling the strength of contagion across

all asset markets. This model, by itself, is unidentified as there are 12 unknown

parameters. However, by combining the empirical moments of the variance-

covariance matrix during the crisis period, 6 moments, from the empirical

15

moments from the variance-covariance matrix of the pre-crisis period, another 6

moments, gives 12 empirical moments in total which can be used to identify the 12

unknown parameters.

1.8 (a) Using Just Crisis Data

Identification of the unknown parameters in the factor model framework discussed

above is based on using information from both non-crisis and crisis periods.

However, there may be a problem for certain asset markets in using non-crisis data

to obtain empirical moments to identify unknown parameters, such as for example

in the move from fixed to floating exchange rate regimes during the East Asian

currency crisis. However, it is possible to identify the model using just crisis

period data, provided that the number of asset returns exceeds 3 and a limited

number of contagious links are entertained. For example, for N = 4 asset returns,

there are 10 unique empirical moments from the variance-covariance matrix using

crisis data. Specifying the factor model in for N = 4 assets, means that there are 4

world parameters and 4 idiosyncratic parameters. This suggests that 2 contagious

links can be specified and identified.

1.9 Databases

The study will be based on secondary data. It will be a purely theoretical

exposition into financial risk management. Empirical testing of the subsequent

models on the two principal stock exchanges should be the next phase of study,

though it’s beyond the scope at this juncture. Major sources of data would be from

the period bulletins from the two principal stock exchanges of the country. In

addition to these, annual reports from SEBI, RBI etc would be used as

supplementary tools for analysis.

1.10 Scope of the Study

16

Randomness has always been a critical aspect of study in financial markets.

Financial mathematics is a field that is currently in full growth. Financial

mathematics presents a source of interesting mathematical problems from the

modeling of financial phenomena to the development of high performance tools

that implement such models. Areas of research involving complex mathematical

models are gaining huge significance after the pricing of options successfully in

the Black-Scholes equation.

1.11 Limitations of the Study

Randomness as a theoretical concept has been relatively easy to interpret and

model. Extension analysis to the empirical phenomenon would be beyond the

scope of the present realm of study. This would also require a greater deal of

mathematical expertise involving complex time series analysis both discrete and

continuous. This can be considered to be a sever limitation imposed on the study

which restricts the empirical validity of the theoretical construct that is being

proposed.

The next chapter goes on to explain in detail the modeling of the hazards that

accompany a contagion and trying to make use of robust models to predict the

hazard variables.

Chapter 2

Theoretical Modeling of Contagion: A New Perspective

Abstract

17

This chapter attempts to sketch the mathematical ideation behind time series

analysis extending it into stochastic processes. Therein an attempt has been

made to model trends in financial data by using Random Walk Hypothesis and

its myriad variants. An S- Variant point process model has been laid down to

explain the effects that time and the type of event has on the stock price. The

stochastic intensity approach and its diagnostic tool, the generalized Hawkes

model has been introduced to further strengthen the process. Itos Lemma has

been introduced at this stage to test out the deterministic and random

component of a stock price and to analyse whether one could explain any

deterministic character out of the random component in the Generalized

Hawkes Model.

2.1 A Mathematical Exposition into Time Series Analysis

18

There are two main goals of time series analysis: (a) identifying the nature of the

phenomenon represented by the sequence of observations, and (b) forecasting

(predicting future values of the time series variable). Both of these goals require

that the pattern of observed time series data is identified and more or less formally

described. Once the pattern is established, one can interpret and integrate it with

other data. Mathematical ideation behind Time Series Analysis (TSA) is the

concept of an abstract probability space defined as (Ù, F, P) such that 0<Ti�Ti+1

Where, Ù à Sample space

F à Sigma Algebra defined in Ù

P à Probability on Ù

Ti à point process on (0, ��)

2.2 Stochastic Process

In the mathematics of probability, a stochastic process is a random function. In the

most common applications, the domain over which the function is defined is a

time interval (a stochastic process of this kind is called a time series in

applications) or a region of space (a stochastic process being called a random

field). Mathematically it can be defined as a family of random variables {x (t,w),

t�T, w�Ù defined on a probability space (Ù, F, P)}

2.3 Non-Stationary Time Series and Random Walk Hypothesis

If a time series is stationary, it’s mean, variance, and autocovariance (at various

lags) remain the same no matter at what point one measures them; that is they are

time invariant. Such a time series will tend to return to its mean and fluctuations

around this mean (measured by its variance) will have a broadly constant

amplitude. If a time series is not stationary in the sense just defined, it is called a

19

non-stationary time series. In other words, a non-stationary time series will have a

time- varying mean or a time-varying variance or both.

It’s very important to understand why a stationary time series is important over a

non-stationary one. If a time series is non-stationary, one can study its behaviour

only for the time period under consideration. Each set of time series data will be

therefore for a particular episode. As a reason it is not possible to generalize it to

other time periods. Therefore for the purpose of forecasting, such nonstationary

time series may be of little practical value.

Classical example of a nonstationary time series is the random walk model

(RWM). It is often said that asset prices, such as stock prices or exchange rates,

follow a random walk which is nonstationary. There are two types of the RWM,

namely the random walk without the drift (no constant or intercept term) and the

random walk with drift.

2.4 Modeling Different Trends

Economic data usually follow trends. One typically distinguishes deterministic and

stochastic trends. Consider the following model of time series Yt

Yt = â1 + â2.t + â3Yt-1+ ut ………………………………….(2.1)

ut à white noise error term and t is measured chronologically.

A stochastic process that is purely random is referred to as white noise.

All this throws open different possibilities,

a) Pure Random Walk

If â1�0; â2=0; â3=1; one gets

20

Yt = Yt-1 + ut which is nothing but a pure random walk without drift and hence is non-stationary

�Yt = (Yt – Yt-1) = ut ---- (2.2)

which then becomes a difference stationary process

If â1=0; â2=0; â3=1;

Yt = â1 + Yt-1 + ut

�Yt = â1+ut ----- (2.3)

Here Yt exhibits a stochastic trend (positive or negative) depending on the value of

âi. RWM with or without drift would be non-stationary and could be converted

into a stationary series by taking the first difference.

b) Deterministic Trend If â1 � 0; â2�0; â3=0 Yt = â1 + â2.t + ut Trend Stationary Process ---

(2.4)

Here though the mean of Yt is â1 + â2.t, its variance is. Once the means of â1and â2

are known, the mean can be forecast perfectly. Therefore if one subtracts the mean

of Yt from Yt, the resulting series will be stationary, hence the name trend

stationary.

c) Random walk with drift and deterministic trend If â1 � 0; â2�0; â3=1 Yt = â1 + â2.t + Yt-1 + ut ---------------(2.5) Here Yt is non-stationary

21

It becomes pertinent at this point after laying out all the possibilities of the RWM

and having determined its variability of its trends to look into more specific factors

and theoretical constructs that define and determine multivariant market event

data. (Multivariate refers to more than one event under consideration). A relevant

issue to be considered at this point is whether the obvious difficulties of modeling

Multivariate market event data could not be mitigated somewhat by adopting a

simpler approach based on time series methods, rather than attempting to set the

models in continuous time. I believe that the development of statistical models for

market event data set in continuous, real time to be an important challenge in

financial econometrics for the following reasons. First, models set in event time

may well ignore aspects of the evolution of the market that are economically

important. In addition, most practical applications of models of market event data

such as volatility measurement and the design of optimal order submission

strategies require that the models relate somehow to real time.

A standard time series analysis of aggregated data using fixed intervals of real time

is also problematic. Since the data records the timing and characteristics of

individual market events, aggregation involves an undesirable loss of information.

Thus the characteristics and timing relations of individual transactions will be lost,

mitigating the advantages of moving to transactions data in the first place. The

considerations set out above suggest that models for market event data set in

continuous time are likely to provide important economic insights into the

functioning of financial markets.

2.5 S-variate point process

Let {Ti}i�(1,2….) be a simple point process on [0,�] defined on (Ù, F, P) and

{Zi}i�(1,2...) be a sequence of {1,2….. M} valued random variables. Then the

double sequence

22

{Ti, Zi}i�{1,2} is called a S-variate point process on [0,�]. Econometrically

speaking, market event data can be viewed as the realisation of a Marked Point

Process that is, as the realisation of a double sequence(Ti;Zi)i�(1;2;:::) of random

variables where Ti is the random occurrence time of the ith event and Zi is a vector

of additional variables (or `marks') associated with that event.

where Ti à Occurrence time of the ith market event

Zi à The event type

Whilst considerable progress has been made in modeling the univariate case using

time series models of durations, multivariate extensions of this work have been

slow to emerge in the econometrics literature. It is believed that approaching the

problem of modelling multivariate market event data by directly specifying the

stochastic intensity provides a powerful, flexible framework.

There is a family of models, the generalised Hawkes models for which analytic

likelihoods are available and diagnostic tests based on the integrated intensity can

be constructed. In contrast to previous work, the models are general enough to

allow one to estimate the nature of the dependence of the intensity on the events of

previous trading days rather imposing strong apriori assumptions concerning this

dependence.

This approach recognises a significant feature of financial markets: namely, that

for the majority of markets, the market does not operate continuously so that the

question of dependence between trading days is of considerable importance.

Furthermore, the models have intuitively appealing economic interpretations since

the stochastic intensity for events of a particular type can be interpreted as a

conditional hazard that is, as the conditional expectation in the limit of the number

of events of that type that will occur per unit time in the future given the current

(multivariate) information set.

23

Now with a plethora of activities affecting a single event it would be difficult to

analyse the impact of conditional hazards unless one goes in for a segregation

analysis. There should again be an intensity filter which would filter out the

contagion hazards. In contrast to previous work, the models (Here we are talking

of the Generalised Hawkes Model) are general enough to allow us to estimate the

nature of the dependence of the intensity on the events of previous trading days

rather imposing strong apriori assumptions concerning this dependence. This

approach recognises a significant feature of financial markets: namely, that for the

majority of markets, the market does not operate continuously so that the question

of dependence between trading days is of considerable importance.

2.6 Stochastic Intensity Approach

After determining the trend modeling process one should be able to proceed with

the earlier specification of the model via stochastic intensity.

Let N(t) be a simple point process on (0,Ù) defined on (0, Ù, P) that is adopted to

some filtration {Ft} and let Pt be a positive, Ft predictable process.

E [N(t) – N(s)/Fs] = E [s�t ë(n) dn/Fs]

.......................(2.6)

This is one of the most crucial assumptions that we could be making. The

conditional expectation of the point process N(t) after filtering out the effects

through stochastic intensity should very well be the standing ground on the basis

of which the whole theoretical construction can be accepted or rejected. The

stochastic filter should be able to identify conditional hazards after having

observed the historical pre determined predicted contingency effects. The

extensional analysis should go from here to a very explicit assumption of trying to

24

predict the additional events from hereon. This would form the very crux of what

the study started out to achieve.

The simple assumption could be as follows;

Standing at time ‘s’ having observed the history Fs and if one wishes to predict the

numbers of additional events that will occur by time ‘t’ using the conditional

expectation of (Nt – Ns)/Fs. Then one can equivalently use

E [s�t ë(u) du/Fs] …………………(2.7)

For all s, t such that 0�s�t,

ë(t) is the intensity of N(t) …………………(2.8)

One has to carefully look back at this point for introspection since this forms one

of the crucial building blocks for the analysis. This is also one of the inflexion

points where the study lacks the depth for overall understanding of the subject.

The intensity factor culled out based on the predicted Ft process could very well

have been empirically tested for its validity based on the past data and the financial

scams that have rocked the Indian financial market.

At the theoretical level, tracing the observed herding behavior to market

participant’s uncertain beliefs and information asymmetries is a key element for

understanding how contagious effects arise. It is argued that the recent focus on

better understanding of high-frequency financial returns data and decision making

at the market microstructure level are promising avenues for understanding the

transmission of shocks across markets and countries.

2.7 Generalised Hawkes Model

25

The model is defined by the stochastic intensity

ë(t) = ì (t) + j=1�k ëj(t) …………… (2.9)

ì(t) is a positive, deterministic function of time and for j = 1,2….. k

j=1�k ëj(t) represents k non-deterministic components

Stochastic intensity of the Hawkes model is thus the sum of the deterministic

component output ìt and k non-deterministic components (ëj(t)jk = 1)

The mathematical rigor needed for the complete explanation of the Hawkes model

is beyond the scope of the present study but at this juncture it gives us that much

needed theoretical support to continue to go along with the original idea of

separating the conditional hazards. It seems very much an empirical impossibility

to mathematically model these hazards but the purpose of the study has to be

looked upon the light of it providing a new perspective to the idea of modeling

financial contagion. To simplify the issue further strong assumptions needed to be

made at this point. The Hawkes model might well end up by assuming that the non

deterministic component as defined by the Hawkes model follows a martingale

process.

In probability theory a discrete time martingale is a discrete time stochastic

process X1, X2, X3,…that satisfies the identity

………………………(2.10)

i.e; the conditional expected value of the next observation, given all the past

observations, is equal to the last observation.

2.8 The Theoretically Conceived Model

It may seem too far fetched at this point of time to introduce a theoretically

conceived model but for all its uncertainties. The construction may well be a

26

hazard in itself but the entire intellectual process should be viewed in the light of

creating an entirely new perspective to the randomness approach. Financial

markets and contingencies have always aroused the curiosity of the academia at

large though randomness still remains a topic to be studied at large.

The randomness of the non deterministic component still lurks in the dark and in

this situation another strong conceptual assumption is being made. The non

deterministic component as derived in the Hawkes Model may well follow Itos

Lemma (Proof derived in the appendix).

2.9 Ito's Lemma

Changes in a variable such as stock price involve a deterministic component which

is a function of time and a stochastic component which depends upon a random

variable. Let S be the stock price at time t and let dS be the infinitesimal change in

S over the infinitesimal interval of time dt. The change in the random variable z

over this interval of time is dz. The change in stock price is given by

dS = adt + bdz………………………………………………………….

(2.11)

dS- Change in stock price

dt- infinitesimal change in time

dz- infinitesimal change in randomness in time dt

where a and b may be functions of S and t as well as other variables.

The expected value of dz is zero so the expected value of dS is equal to the

deterministic component, adt.

The random variable dz represents an accumulation of random influences over the

interval dt. The Central Limit Theorem then implies that dz has a normal

27

distribution and hence is completely characterized by its mean and standard

deviation. The mean or expected value of dz is zero. The variance of a random

variable which is the accumulation of independent effects over an interval of time

is proportional to the length of the interval, in this case dt. The standard deviation

of dz is thus proportional to the square root of dt, (dt)1/2. All of this means that the

random variable dz is equivalent to a random variable w(dt)1/2, where w is a

standard normal variable with mean zero and standard deviation equal to unity.

One of the strongest assumptions in this study is being put forward. It can be put

forward that the non-deterministic component is a function of the change in stock

price and time. If a change in stock price is ultimately related to as explained in

(2.11) and if one makes the assumption that the random component is a function of

the stock price and time, it should naturally follow that this random component

should have a deterministic and a non-deterministic component.

Consider the following mathematical assumptions,

As mentioned earlier, dS = adt + bdz,

Now suppose that ëj(t)jk = 1, the random component as derived in the Hawkes

model is a function of stock price and time.

ëj(t)jk = f ( S,

T)………………………………………………(2.12)

Because ëj(t)jk is a function of the stochastic variable S, it will have a stochastic

component as well as a deterministic component. ëj(t)jk will have a

representation of the form:

dëj(t)jk = pdt +

qdz………………………………………………(2.13)

28

The crucial problem is how the functions p and q are related to the functions a and b in the equation

dS = adt + bdz.

Ito's Lemma gives the answer. The deterministic and stochastic components are given by:

p=�f/�t+(�f/�S)a +(1/2)(�2f/�S2) b2 ……………………………(2.14) q = (�f/�S)b…………………………………….(2.15)

2.9 Looking Ahead: Analysis of the Model and Randomness

Humankind has been concerned with randomness since prehistoric times, mostly

through divination (reading messages in random patterns) and gambling. The

opposition between free will and determinism has been a divisive issue in

philosophy and theology . Mathematicians focused at first on statistical

randomness and considered block frequencies (that is, not only the frequencies of

occurrences of individual elements, but also those of blocks of arbitrary length) as

the measure of randomness, an approach that extended into the use of information

entropy in information theory.

Some argue randomness should not be confused with practical unpredictability,

which is a related idea in ordinary usage. Some mathematical systems, for

example, could be seen as random; however they are actually unpredictable. This

is due to sensitive dependence on initial conditions (see chaos theory). Many

random phenomena may exhibit organized features at some levels. For example,

while the average rate of increase in the human population is quite predictable, in

the short term, the actual timing of individual births and deaths cannot be

predicted. This small-scale randomness is found in almost all real-world systems

29

Sensibly dealing with randomness is a hard problem in modern science,

mathematics, psychology and philosophy. Merely defining it adequately, for the

purposes of one discipline has proven quite difficult. Distinguishing between

apparent randomness and actual randomness has been no easier. In addition,

assuring unpredictability, especially against a well-motivated party (in

cryptographic parlance, the "adversary"), has been harder still.

Some philosophers have argued that there is no randomness in the universe, only

unpredictability. Others find the distinction meaningless.

All these lend credence to the model explained in the study. The theoretical

construction followed has shed some illuminating insights into the deterministic

randomness of the non-deterministic component. Once the values of the constant

are known it makes sense then to model out the deterministic nature of the

randomness associated with forecasting randomness. Randomness as a concept

looks beyond the inevitable and this study has been an attempt to model the

invisible within the explicit.

This theoretical construct may seem incredulous but it certainly sheds light into a

new perspective of looking at randomness. Though its empirical validity requires

complex spread sheet programs and voluminous data the theoretical underpinnings

gives us no reason to absurdity. Complex algorithms could be made based on these

models, though at the present time, the lack of it, severely acts as an impediment to

its credibility and subsequent validity.

30

Chapter 3

Financial Contagion: Models and Perspectives

3.1 Financial Contagion: An Introspective Analysis

Use of the word ‘contagion’ to describe the international transmission of financial

crises has become fraught with controversy, to the extent that some recent authors

have seen fit to avoid using the word entirely; see Favero and Giavazzi (2002) and

Rigobon (2003). The term evokes an emotive response among both producers and

consumers of research on international financial markets, and there is no general

agreement over its use.

Emotional responses stem in part from the borrowing of epidemiological

terminology – contagion is intrinsically associated with disease, and even more

dismally with death, as contagion was often used as a synonym for the Bubonic

Plague in Europe as late as the 19th century. The term also implies, at least to

some, that those who fall prey to financial crises do so through no fault of their

own. However, this is an idea that some analysts are inclined to strongly resist:

speculators appear to discriminate in choosing the countries they attack.

A variation on the above definition is whether contagion represents the

unanticipated transmission of shocks. When cross-country linkages countries are

anticipated – for instance through trade and financial flows or other a priori links -

then these represent fundamental linkages, hence they are not contagion. Arguably,

the particular channel through which contagion is transmitted is equally important,

such as through financial markets, trade relations, political linkages and

expectations. Researchers emphasizing the importance of identifying the channels

argue that this is a way of reassuring observers that underlying the estimated

correlations is really the international transmission of financial stress, and not

31

simply variables which are common across countries but omitted from the

specification.

The choice of fundamentals is not independent of the problem at hand. The

literature tends to adopt definitions of contagion specific to each application, and

given the difficulties inherent in defining the appropriate control variables this may

be appropriate. As a result, contagion may in fact be a concept that is defined

relative to a particular set of fundamentals, so its appropriate definition is the co-

movement of excess returns in one country with excess returns in another country

after controlling for the effects of specified fundamentals. Contagion is then

defined relative to the chosen fundamentals control group.

If financial contagion is associated with excess returns, then the problem of

defining these is immediately raised. To know when returns are ‘excessive’

requires an effective model of asset prices during normal times. At a minimum, the

statistical properties of financial markets data need to be accommodated in any

modeling exercise4.distributions of daily financial market returns are typically

non-normal and display volatility clustering (time-varying

heteroscedasticity/GARCH effects) and fat tails (leptokurtosis). Thus, researchers

proposing to model financial market processes should arguably be expected to

reproduce these characteristics. The production of data distributions with fat tails

requires some form of non- linearity, and introducing this constitutes an important

strand of contagion research.

3.2 Market Interdependence and Crisis Thresholds

4 One set of stylized facts for contagion is the existence of strong regional effects in equity and currency

market contagion, such as documented by Agenor, Miller, Vines and Weber (1999), Eichengreen (2002), Kaminsky and Reinhart (2002) and Krugman (2000). However, for bond market data this regionality does not seem to be present, a feature noted by Masson, Chakravarty and Gulden (2003) and confirmed in Dungey, Fry, Gonzalez-Hermosillo and Martin (2002) in a study of the Russian and LTCM crises.

32

In order to isolate contagious effects, the relative strength of market

interdependence and contagion need to be both modeled simultaneously and

separately identified. Interdependence, as opposed to contagion, occurs if cross-

market co-movement is not significantly bigger after a shock to one country, or

group of countries. Controlling for this is easier in a bivariate setting with two

countries and two asset markets than in a multivariate environment, although the

resulting dynamics are not as rich.

There has been extensive evidence of rising correlation between international

financial markets in recent decades, keeping pace with the trend towards capital

account liberalization; see, for example, Longin and Solnik (1995). More recently,

the bivaria te test for significant changes in the conditional correlation between

asset returns over noncrisis and crisis periods was popularised by Forbes and

Rigobon (2001, 2002). Applications include Baig and Goldfajn (1999) and Ellis

and Lewis (2000). This is the most common test in the literature; some of the

relevant issues in running it are covered in Corsetti, Pericoli and Sbracia (2001),

Loretan and English (2000) and Boyer, Gibson and Loretan (1999). The

correlation test is a variant of the World Bank’s ‘very restrictive’ definition of

contagion, although in the majority of applications there is some attempt to control

for (a limited set of) fundamentals5.

Leading indicators of financial crises serve to avert policy makers to looming

crises. They also present one means of identifying thresholds between non-crisis

and crisis periods. Unfortunately, very few fundamental indicators are found to be

statistically significant control variables in existing applications.

Perhaps more disturbingly, crisis indicators based on fundamental indicators have

also proved to have poor predictive power in forecasting financial crises; see, for

example, Edison (2000) and Berg and Patillo (1999). This is reflected in the

5 Butler and Joaquin (2002) conduct tests consistent with the ‘very restrictive’ definition of contagion,

although their paper is not directly addressing this issue.

33

heterogeneity of currency crises causes and features for different countries,

documented in Frankel and Rose(1996), as well as in the unpredictability of

reversals in short-term capital flows, emphasized by Calvo’s (1998) sudden stops.

Poor predictive power suggests that nonlinearity and breaks in the generating

processes of financial market data are pervasive, as discussed earlier.

Susceptibility to contagion is highly non- linear, and historical relationships –

however robust – are not useful in predicting future financial crises.

More generally, models of financial contagion can be classified as fundamental or

behavioural. In the first category the analysis is event-driven, where the event is

usually a financial crisis. Examples include the applications of Glick and Rose

(1999) and Van Rijikghem and Weder (2001) to shocks from a particular country

identified as ‘country zero’. On the other hand, behavioural models consider that

changing beliefs and ‘herding’ underlie the transmission of shocks between countries.

A good example is the situation presented by Miller, Thampanishvong and Zhang

(2003) investigating the turmoil in Brazilian financial markets in 2001. Although

turmoil existed in Brazilian financial markets, the feared event of sovereign debt

default did not materialize. Consequently, there was no identifiable crisis event –

rather, the turmoil was caused by fear of the potential cost of default. A useful

distinction between the biological and behavioural models is that biological models

tend to operate in a time series domain, following an event, whereas fundamental

models operate in both time series and cross-section dimensions.

Macro economic equilibria may also explain the contagion phenomenon.

Macroeconomic models with rational expectations generically have multiple

solutions. It follows that researchers’ different informational assumptions can have

different implications regarding the number of equilibria. If a given set of

fundamentals can give rise to multiple equilibria, then speculative attacks can be

self- fulfilling and contagion can also be ‘irrational’, that is unanticipated. Within a

cross-section, investors’ behavior can be interpreted as jumps between different

equilibria. The issue of multiplicity is critical for some models of contagion – for

example, see Shiller’s (2000) account of the U.S. stock market bubble in the late

34

1990s. The strength of the self- fulfilling mechanism for contagion may also be a

potent explanatory factor underlying the collapse of the ERM in 1992-93; see

Drazen and Masson (1994).

3.3 Conclusions and Findings

Containing the likelihood of contagious financial crises is a pressing policy issue

at both national and international levels. As yet, there is no professional consensus

on the appropriate definitions of what constitutes a financial crisis or contagion,

despite substantial research progress towards these goals. We know that financial

crises and contagion are intrinsically linked, and that contagious effects arise when

crises are propagated across countries or markets after controlling for fundamental

linkages and interdependencies. We also know that these transmissions may spread

further through mechanisms such as cross-market hedging.

However, broad agreement can be obtained on the following points:

1. Crises are in some way associated with an increase in the conditional volatility

of financial market returns.

2. The association of excess returns in one country or market with excess returns

in another country after controlling for fundamentals (excess co-movement) is

consistent with financial market contagion.

3. The theoretical model outlined in the second chapter provides a new

perspective of using stochastic intensity as a filter

4. The theoretical insights into Ito’s lemma and its application into volatility

models sheds light into randomness

35

5. The association of randomness in non-deterministic models may help in making forecasts smoother leading to better

accuracy.

New models will undoubtedly be required with the advent of new crises. However,

some of the salient aspects outlined in this study are likely to recur. These include:

the fundamental linkages, the means of transmission across countries and asset

classes, the statistical properties of the data, the simultaneous identification of

contagion, interdependency and herding and the endogenous identification of crisis

and non-crisis periods from sample data. Each of these issues is extremely

important for assessing the appropriate policy response to prevent crises and

adequately managing those that occur.

36

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40

Appendix

Derivation of Ito’s Lemma

The Taylor series for f(S,t) gives the increment in ëj(t)jk as:

dëj(t)jk = (�f/�t)dt + (�f/�S)dS +(1/2)(�2f/�S2)(dS)2 +(�2f/�S�t)(dS)(dt) +

(1/2)(�2f/�t2)(dt)2 + higher order terms.

The increment in stock price dS is given by

dS = adt + bdz,

but dz=vw[dt]1/2, where w is a standard normal random variable. Substitution of

adt + bvw(dt)1/2 for dS in the above equation yields:

dëj(t)jk = (�f/�t)dt + (�f/�S)adt + �f/�S)bvw(dt)1/2+ 1/2(�2f/�S2)(adt + bvw(dt)1/2)2

+ (�2f/�S�t)(adt + bvw(dt)1/2)(dt) + 1/2(�2f/�t2)(dt)2 + higher order terms.

With the expansion of the squared term and the product term the result is:

dëj(t)jk = ( �f/�t)dt + (�f/�S)adt + �f/�S)bvw(dt)1/2+ (1/2)(�2f/�S2)(a2dt2 +

2abvw(dt)3/2 + b2v2w2dt)+ (�2f/�S�t)(a(dt)2 + bvw(dt)3/2) + 1/2(�2f/�t2)(dt)2+

higher order terms.

Taking into account the infinitesimal nature of dt so that dt to any power

higher than unity vanishes, reduces to:

dëj(t)jk = (�f/�t)dt + (�f/�S)adt + (�f/�S)bvw(dt)1/2+ 1/2(�2f/�S2)(b2v2w2dt)

41

Noting that the expected value of w2 is unity the expected value of dëj(t)jk

is:

[�f/�t + (�f/�S)a + 1/2(�2f/�S2)b2]dt.

This is the deterministic component of dëj(t)jk . The stochastic component is the

term that depends upon dz, which in (8) is represented as vw(dt)1/2. Therefore the

stochastic component is:

[(�f/�S)b]dz…………………………………………………………………………(

1)

From the above derivation it would seem that there is an additional stochastic term

that arises from the random deviations of w2 from its expected value of 1; i.e., the

additional term

(1/2)(�2f/�S2)(b2v2w2dt)……………………………………………………………

……(2)

However the variance of this additional term is proportional to (dt)2 whereas the

variance of the stochastic term given in (1) is proportional to (dt). Thus the

stochastic term given in (2) vanishes in comparison with the stochastic term given

in (1)

42

Glossary

Contagion – This follows Eichengreen and Rose (1995) and Eichengreen, Rose

and Wyplosz (1996), who propose that contagion refers to the association of

excess returns in one country with excess returns in another country after

controlling for the effects of fundamentals. This definition is closely related to

‘true’ contagion, as defined in Kaminsky and Reinhart (2000), arising in the

absence of, or after controlling for, common shocks and all possible

interconnection channels.

Stochastic process- Any variable whose value changes over a period of time in an

uncertain way is said to follow a stochastic process. They can be classified into

discrete time and continuous time, where in the latter the underlying va riable can

take any value within a certain range while only certain discrete value are possible

in the former

Sigma field- ó-algebra, a mathematical concept, is a collection of subsets of a

given set. It is a key concept necessary for the definition of measure, which is itself

a key concept in analysis. Mathematicians who research and study probability

often refer to ó-algebras as ó-fields.

Systematic Risk- Risk that cannot be diversified and arises due to the correlations

between the returns from the investment and the stock market as a whole. The

investor expects a rate of return higher than the risk-free interest rate for bearing

positive amounts of systematic risk.

Non-systematic Risk- Non-systematic risk should not be at all important to an

investor. It can be almost completely elimintated by a well diversified portfolio

43

Ito’s Lemma – This is derived from the Ito’s process. This is a generalised

stochastic process where the parameters a and b are functions of the value of the

underlying variable, x, and time, t. Both the expected drift and variance rate of an

Ito process are liable to change over time. This is used in the derivation of the

famous Black Scholes equation in the pricing of European options

Black-Scholes Pricing Formulae- In their path breaking paper, Black and Scholes

succeeded in solving their differential equation to obtain exact formulas for the

prices of European call and put options

Normal Distribution- A distribution with mean zero and standard deviation one