This document discusses estimating asset price volatility using generalized autoregressive conditional heteroskedastic (GARCH) models. It begins with an introduction to modeling stock return volatility and the assumptions of non-constant variance. It then presents the GARCH model for estimating variance in the univariate case. Next, it discusses estimating the GARCH model parameters using maximum likelihood estimation. Finally, it discusses extending the GARCH model to the multivariate case to simultaneously estimate the volatilities and correlations of a portfolio of stocks.
"Correlated Volatility Shocks" by Dr. Xiao Qiao, Researcher at SummerHaven In...Quantopian
Commonality in idiosyncratic volatility cannot be completely explained by time-varying volatility. After removing the effects of time-varying volatility, idiosyncratic volatility innovations are still positively correlated. This result suggests correlated volatility shocks contribute to the comovement in idiosyncratic volatility.
Motivated by this fact, we propose the Dynamic Factor Correlation (DFC) model, which fits the data well and captures the cross-sectional correlations in idiosyncratic volatility innovations. We decompose the common factor in idiosyncratic volatility (CIV) of Herskovic et al. (2016) into the volatility innovation factor (VIN) and time-varying volatility factor (TVV). Whereas VIN is associated with strong variation in average returns, TVV is only weakly priced in the cross section
A strategy that takes a long position in the portfolio with the lowest VIN and TVV betas, and a short position in the portfolio with the highest VIN and TVV betas earns average returns of 8.0% per year.
Motivated by the problem of computing investment portfolio weightings we investigate various methods of clustering as alternatives to traditional mean-variance approaches. Such methods can have significant benefits from a practical point of view since they remove the need to invert a sample covariance matrix, which can suffer from estimation error and will almost certainly be non-stationary. The general idea is to find groups of assets which share similar return
characteristics over time and treat each group as a single composite asset. We then apply inverse volatility weightings to these new composite assets. In the course of our investigation we devise a method of clustering based on triangular potentials and we present associated theoretical results as well as various examples based on synthetic data.
Pricing vulnerable European options when the option’s payoff can increase the risk of financial distressPeter Klein, Michael InglisJournal of Banking & Finance
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Method of solving multi objective optimization problem in the presence of unc...eSAT Journals
Abstract
Multi-objective formulations are realistic models for many complex engineering optimization problems. In many real-life problems
considered objectives conflict with each other and optimization of one target solutions can lead to unacceptable results for other
purposes. A reasonable solution of the multi-objective problem is to study the set of solutions, each of which satisfies the objectives at
an acceptable level with no dominance of any of the solutions. The article provides a brief overview of multi-objective optimization
methods (by Pareto criteria) and their improvement.
Keywords: A fuzzy set, forecasting, risk, weakly formalized process for Pareto multi-objective optimization.
"Correlated Volatility Shocks" by Dr. Xiao Qiao, Researcher at SummerHaven In...Quantopian
Commonality in idiosyncratic volatility cannot be completely explained by time-varying volatility. After removing the effects of time-varying volatility, idiosyncratic volatility innovations are still positively correlated. This result suggests correlated volatility shocks contribute to the comovement in idiosyncratic volatility.
Motivated by this fact, we propose the Dynamic Factor Correlation (DFC) model, which fits the data well and captures the cross-sectional correlations in idiosyncratic volatility innovations. We decompose the common factor in idiosyncratic volatility (CIV) of Herskovic et al. (2016) into the volatility innovation factor (VIN) and time-varying volatility factor (TVV). Whereas VIN is associated with strong variation in average returns, TVV is only weakly priced in the cross section
A strategy that takes a long position in the portfolio with the lowest VIN and TVV betas, and a short position in the portfolio with the highest VIN and TVV betas earns average returns of 8.0% per year.
Motivated by the problem of computing investment portfolio weightings we investigate various methods of clustering as alternatives to traditional mean-variance approaches. Such methods can have significant benefits from a practical point of view since they remove the need to invert a sample covariance matrix, which can suffer from estimation error and will almost certainly be non-stationary. The general idea is to find groups of assets which share similar return
characteristics over time and treat each group as a single composite asset. We then apply inverse volatility weightings to these new composite assets. In the course of our investigation we devise a method of clustering based on triangular potentials and we present associated theoretical results as well as various examples based on synthetic data.
Pricing vulnerable European options when the option’s payoff can increase the risk of financial distressPeter Klein, Michael InglisJournal of Banking & Finance
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Method of solving multi objective optimization problem in the presence of unc...eSAT Journals
Abstract
Multi-objective formulations are realistic models for many complex engineering optimization problems. In many real-life problems
considered objectives conflict with each other and optimization of one target solutions can lead to unacceptable results for other
purposes. A reasonable solution of the multi-objective problem is to study the set of solutions, each of which satisfies the objectives at
an acceptable level with no dominance of any of the solutions. The article provides a brief overview of multi-objective optimization
methods (by Pareto criteria) and their improvement.
Keywords: A fuzzy set, forecasting, risk, weakly formalized process for Pareto multi-objective optimization.
Monitoring of concrete structures by electro mechanical impedance technique IEI GSC
By Dr. S.N.Khante Associate Professor & Bhagyashri Sangai
at 31st National Convention of Civil Engineers
organised by
Gujarat State Center, The Institution of Engineers (India) at Ahmedabad
Small is the new big, and for good reason. The benefits of microservices and service-oriented architecture have been extolled for a number of years, yet many forge ahead without thinking of the impact the users of the services. Consuming on micro services can be enjoyable as long as the developer experience has been crafted as finely as the service itself. But just like with any other product, there isn’t a single kind of consumer. Together we will walk through some typical kinds of consumers, what their needs are, and how we can create a great developer experience using brains and tools like Docker.
Statistical Arbitrage
Pairs Trading, Long-Short Strategy
Cyrille BEN LEMRID

1 Pairs Trading Model 5
1.1 Generaldiscussion ................................ 5 1.2 Cointegration ................................... 6 1.3 Spreaddynamics ................................. 7
2 State of the art and model overview 9
2.1 StochasticDependenciesinFinancialTimeSeries . . . . . . . . . . . . . . . 9 2.2 Cointegration-basedtradingstrategies ..................... 10 2.3 FormulationasaStochasticControlProblem. . . . . . . . . . . . . . . . . . 13 2.4 Fundamentalanalysis............................... 16
3 Strategies Analysis 19
3.1 Roadmapforstrategydesign .......................... 19 3.2 Identificationofpotentialpairs ......................... 19 3.3 Testingcointegration ............................... 20 3.4 Riskcontrolandfeasibility............................ 20
4 Results
22
2
Contents

Introduction
This report presents my research work carried out at Credit Suisse from May to September 2012. This study has been pursued in collaboration with the Global Arbitrage Strategies team.
Quantitative analysis strategy developers use sophisticated statistical and optimization techniques to discover and construct new algorithms. These algorithms take advantage of the short term deviation from the ”fair” securities’ prices. Pairs trading is one such quantitative strategy - it is a process of identifying securities that generally move together but are currently ”drifting away”.
Pairs trading is a common strategy among many hedge funds and banks. However, there is not a significant amount of academic literature devoted to it due to its proprietary nature. For a review of some of the existing academic models, see [6], [8], [11] .
Our focus for this analysis is the study of two quantitative approaches to the problem of pairs trading, the first one uses the properties of co-integrated financial time series as a basis for trading strategy, in the second one we model the log-relationship between a pair of stock prices as an Ornstein-Uhlenbeck process and use this to formulate a portfolio optimization based stochastic control problem.
This study was performed to show that under certain assumptions the two approaches are equivalent.
Practitioners most often use a fundamentally driven approach, analyzing the performance of stocks around a market event and implement strategies using back-tested trading levels.
We also study an example of a fundamentally driven strategy, using market reaction to a stock being dropped or added to the MSCI World Standard, as a signal for a pair trading strategy on those stocks once their inclusion/exclusion has been made effective.
This report is organized as follows. Section 1 provides some background on pairs trading strategy. The theoretical results are described in Section 2. Section 3
Improving Returns from the Markowitz Model using GA- AnEmpirical Validation o...idescitation
Portfolio optimization is the task of allocating the investors capital among
different assets in such a way that the returns are maximized while at the same time, the
risk is minimized. The traditional model followed for portfolio optimization is the
Markowitz model [1], [2],[3]. Markowitz model, considering the ideal case of linear
constraints, can be solved using quadratic programming, however, in real-life scenario, the
presence of nonlinear constraints such as limits on the number of assets in the portfolio, the
constraints on budgetary allocation to each asset class, transaction costs and limits to the
maximum weightage that can be assigned to each asset in the portfolio etc., this problem
becomes increasingly computationally difficult to solve, ie NP-hard. Hence, soft computing
based approaches seem best suited for solving such a problem. An attempt has been made in
this study to use soft computing technique (specifically, Genetic Algorithms), to overcome
this issue. In this study, Genetic Algorithm (GA) has been used to optimize the parameters
of the Markowitz model such that overall portfolio returns are maximized with the standard
deviation of the returns being minimized at the same time. The proposed system is validated
by testing its ability to generate optimal stock portfolios with high returns and low standard
deviations with the assets drawn from the stocks traded on the Bombay Stock Exchange
(BSE). Results show that the proposed system is able to generate much better portfolios
when compared to the traditional Markowitz model.
Monitoring of concrete structures by electro mechanical impedance technique IEI GSC
By Dr. S.N.Khante Associate Professor & Bhagyashri Sangai
at 31st National Convention of Civil Engineers
organised by
Gujarat State Center, The Institution of Engineers (India) at Ahmedabad
Small is the new big, and for good reason. The benefits of microservices and service-oriented architecture have been extolled for a number of years, yet many forge ahead without thinking of the impact the users of the services. Consuming on micro services can be enjoyable as long as the developer experience has been crafted as finely as the service itself. But just like with any other product, there isn’t a single kind of consumer. Together we will walk through some typical kinds of consumers, what their needs are, and how we can create a great developer experience using brains and tools like Docker.
Statistical Arbitrage
Pairs Trading, Long-Short Strategy
Cyrille BEN LEMRID

1 Pairs Trading Model 5
1.1 Generaldiscussion ................................ 5 1.2 Cointegration ................................... 6 1.3 Spreaddynamics ................................. 7
2 State of the art and model overview 9
2.1 StochasticDependenciesinFinancialTimeSeries . . . . . . . . . . . . . . . 9 2.2 Cointegration-basedtradingstrategies ..................... 10 2.3 FormulationasaStochasticControlProblem. . . . . . . . . . . . . . . . . . 13 2.4 Fundamentalanalysis............................... 16
3 Strategies Analysis 19
3.1 Roadmapforstrategydesign .......................... 19 3.2 Identificationofpotentialpairs ......................... 19 3.3 Testingcointegration ............................... 20 3.4 Riskcontrolandfeasibility............................ 20
4 Results
22
2
Contents

Introduction
This report presents my research work carried out at Credit Suisse from May to September 2012. This study has been pursued in collaboration with the Global Arbitrage Strategies team.
Quantitative analysis strategy developers use sophisticated statistical and optimization techniques to discover and construct new algorithms. These algorithms take advantage of the short term deviation from the ”fair” securities’ prices. Pairs trading is one such quantitative strategy - it is a process of identifying securities that generally move together but are currently ”drifting away”.
Pairs trading is a common strategy among many hedge funds and banks. However, there is not a significant amount of academic literature devoted to it due to its proprietary nature. For a review of some of the existing academic models, see [6], [8], [11] .
Our focus for this analysis is the study of two quantitative approaches to the problem of pairs trading, the first one uses the properties of co-integrated financial time series as a basis for trading strategy, in the second one we model the log-relationship between a pair of stock prices as an Ornstein-Uhlenbeck process and use this to formulate a portfolio optimization based stochastic control problem.
This study was performed to show that under certain assumptions the two approaches are equivalent.
Practitioners most often use a fundamentally driven approach, analyzing the performance of stocks around a market event and implement strategies using back-tested trading levels.
We also study an example of a fundamentally driven strategy, using market reaction to a stock being dropped or added to the MSCI World Standard, as a signal for a pair trading strategy on those stocks once their inclusion/exclusion has been made effective.
This report is organized as follows. Section 1 provides some background on pairs trading strategy. The theoretical results are described in Section 2. Section 3
Improving Returns from the Markowitz Model using GA- AnEmpirical Validation o...idescitation
Portfolio optimization is the task of allocating the investors capital among
different assets in such a way that the returns are maximized while at the same time, the
risk is minimized. The traditional model followed for portfolio optimization is the
Markowitz model [1], [2],[3]. Markowitz model, considering the ideal case of linear
constraints, can be solved using quadratic programming, however, in real-life scenario, the
presence of nonlinear constraints such as limits on the number of assets in the portfolio, the
constraints on budgetary allocation to each asset class, transaction costs and limits to the
maximum weightage that can be assigned to each asset in the portfolio etc., this problem
becomes increasingly computationally difficult to solve, ie NP-hard. Hence, soft computing
based approaches seem best suited for solving such a problem. An attempt has been made in
this study to use soft computing technique (specifically, Genetic Algorithms), to overcome
this issue. In this study, Genetic Algorithm (GA) has been used to optimize the parameters
of the Markowitz model such that overall portfolio returns are maximized with the standard
deviation of the returns being minimized at the same time. The proposed system is validated
by testing its ability to generate optimal stock portfolios with high returns and low standard
deviations with the assets drawn from the stocks traded on the Bombay Stock Exchange
(BSE). Results show that the proposed system is able to generate much better portfolios
when compared to the traditional Markowitz model.
POSSIBILISTIC SHARPE RATIO BASED NOVICE PORTFOLIO SELECTION MODELScscpconf
This paper uses the concept of possibilistic risk aversion to propose a new approach for portfolio selection in fuzzy environment. Using possibility theory, the possibilistic mean,
variance, standard deviation and risk premium of a fuzzy number are established. Possibilistic Sharpe ratio is defined as the ratio of possibilistic risk premium and possibilistic standard deviation of a portfolio. The Sharpe ratio is a measure of the performance of the portfolio compared to the risk taken. The higher the Sharpe ratio, the better the performance of the portfolio is and the greater the profits of taking risk. New models of fuzzy portfolio selection
considering the possibilistic Sharpe ratio, return and skewness of the portfolio are considered. The feasibility and effectiveness of the proposed method is illustrated by numerical example extracted from Bombay Stock Exchange (BSE), India and is solved by multiple objective genetic
algorithm (MOGA).
Express measurement of market volatility using ergodicity conceptJack Sarkissian
Don't we want to base our trading decisions on current market conditions? Then why should we rely on time averages only because they are simple to comprehend? We can get current market volatility a lot faster by applying the ERGODICITY concept to financial markets. Ensemble averaging allows to measure market volatility quickly, based on only two points in time and is as relevant to volatility measurement as the traditional measures.
Value-at-Risk (VaR) has been adopted as the cornerstone and commonlanguage of risk management by virtually all major financial institutions and regulators. However, this risk measure has failed to warn the market participants during the financial crisis. In this paper, we show this failure may come from the methodology that we use to calculate VaR and not necessarily for VaR measure itself. we compare two different methods for VaR calculation, 1)by assuming the normal distribution of portfolio return, 2)
by using a bootstrap method in a nonparametric framework. The Empirical exercise is implemented on CAC 40 index, and the results show us that the first method will underestimate the market risk - the failure of VaR measure occurs. Yet, the second method overcomes the shortcomings of the first method and provides results that pass the tests of VaR evaluation.
Risk valuation for securities with limited liquidityJack Sarkissian
Everything seems simple with liquid securities - price is known, risks are more or less known too. It becomes a lot harder when we get illiquid instruments in the book. This is why we developed this model to enable modeling of securities with low liquidity and evaluate impact of risk sources associated with liquidity. And in order to do that we had to demonstrate that price formation has quantum chaotic character.
1. ESTIMATING ASSET PRICE VOLATILITY
by
Joel DeJesus
A Senior Honors Thesis Submitted to the Faculty of
the University of Florida
in Partial Fulfillment of the Requirements for the
Honors Degree of Bachelor of Science
in
Mathematics
April 20, 2005
2. ABSTRACT
The covariance matrix of a portfolio of stocks is used in finance to produce various
risk metrics. Thus, it is important to find a way to extract the covariance matrix
from stock time series. In this paper we assume that financial time series exhibit
generalized autoregressive conditional heterskedastic effects and proceed to estimate
the parameters of the GARCH equation. Then, we enforce specific conditions on
the parameter matrices to ensure consistency. Finally, we use a sample importance
resampling filter to filter out the sample noise.
i
3. I would like to thank my advisor, Professer Liqing Yan, for giving me guidance
and insights into the subject I have written about. I also thank my family for always
supporting me.
5. Chapter 1
Introduction
1.1 Background
In the last two decades the finance industry has seen a flurry of innovations that
has changed the way markets interact. Today, financial institutions increasingly find
themselves employing increasingly sophisticated quantitative models. The reasons
for the increased need for quantitative modeling are the creation of new financial
instruments, the development of computers, and the increasing size and complexity
of financial markets.
With the increasing complexity of financial instruments, financial managers must
find ways to model the risks of their market positions. For equity managers, the asset
that has the most significance is the stock security. A stock security is a document
granting partial ownership of a company to the holder of the security. An equity
manager, more than likely, also has to evaluate their derivatives positions, where a
derivative is a security whose value is derived from another asset. While measuring
the riskiness of derivative securities is an important topic, it is beyond the scope of
this paper.
For a financial institution, the item of interest is their portfolio. A portfolio is a
collection of securities that the financial institution owns. In the case of the equity
manager, a simple portfolio usually constitutes several stock securities. The goal of the
manager is to maximize the returns from these securities, either by price appreciation
or through dividend payments, while being risk averse. To be risk averse means if
there are two securities with the same expected return but one is riskier than the
other, the investor will be pick the least risky of the two. While the primary purpose
of the manager is to make money, the manager must also be able to control the risks.
In order to control risk the manager must have a firm grasp on what exactly are the
risk profiles of the portfolio. In this paper, the primary focus is on the riskiness of
stock securities, which have the simplest risk profile.
While stock is the simplest security in an equity portfolio, the concept of risk, on
1
6. the other hand, is rather abstract. Risk, to a financial manager, is the possibility
of losing money in a given portfolio. Hence, the most popular measure of risk given
a portfolio of stocks is Value at Risk (VAR). Value at Risk is the most money a
portfolio could lose in a given time period within a given confidence interval, usually
95-99 percent. Given a portfolio of stocks, it is clear that the more the stock fluctuates,
the greater the potential loss can be. Therefore, a measurement of how much a stock
fluctuates, or more commonly referred to as volatility, is needed in order to determine
the VAR of a portfolio. In addition to volatility, the degree to which stocks move
together in tandem, referred to as correlation, is also a concern to VAR. If all the
stocks are correlated with one another, in effect, the portfolio behaves like a single
security. If the stocks are not correlated with one another, then the rates of returns
on different stocks tend to cancel each other out, resulting in a lower volatility for the
portfolio as a whole.
In this paper, the purpose is to simultaneously estimate the volatilities and corre-
lation of a portfolio of stocks. The single object of interest is the covariance matrix of
log of returns of a portfolio of stocks. From the covariance matrix, one can derive the
statistical correlation between any two stocks and the individual variances, which can
be used as a proxy for volatility, of each stock. We assume that not only are financial
time series not homoskedastic, that is the covariance matrix is not constant over time,
but that they exhibit generalized autoregressive conditional heterskedasticity effects.
It has been found that this is a fair approximation of reality, see [7]. Based on this
assumption we first estimate the Garch parameter matrices following from [11]. Then
we use a particle filter to filter out the sample noise inherent in financial time series.
1.2 Notation
The purpose of this section is to establish the basic notation that will be used through-
out this thesis and to describe the underlying assumptions used by the theorems.
Following from [1] we will use the following definition for the return of a particular
stock.
Definition 1.2.1. Let yt be the return at time t and St be the stock price at time t.
Then define
yt = log
St
St−1
(1.2.1)
Throughout the paper we shall assume that yt is a random variable that evolves
over time. With this in mind we extend this notation to the general case of N stock
securities in a portfolio P.
Definition 1.2.2. Let yi
t represent the rate of return of the ith stock security in a
2
7. portfolio P with N stocks. Then define the following
Yt =
y1
t
y2
t
...
yN
t
(1.2.2)
Also, following along the same lines as [1] we make the assumption that Yt ∼ N(µt, Σt),
where µ is the mean vector of Yt , Σ is the covariance matrix of Yt, and N(0, I) signifies
the normal distribution with mean 0 and the identity matrix as the covariance.
3
8. Chapter 2
GARCH
2.1 Preliminaries
In this chapter, we assume portfolio returns are heteroskedastic, and further more,
that the evolution of the covariance matrix over time can be explained by a mul-
tivariate Garch model. In this chapter, we first introduce the Garch model for the
univariate case. Then we present a multivariate version that can be used to model
the covariance matrix. Finally, we focus on estimating the parameters of this specifi-
cation. However, before we proceed with presenting Garch, it is convenient to define
sample variance and covariance.
Definition 2.1.1. Let (xt)N
t=1 and (yt)N
t=1 be sequences of independently, identically,
distributed random variables. Then define the variance:
σx =
N
t=1(xt − ¯x)2
N − 1
(2.1.1)
where the mean, ¯x, is defined as:
¯x =
N
t=1 xt
N
and similarly, define the covariance as:
σxy =
N
t=1(xt − ¯x)(yt − ¯y)
N − 1
(2.1.2)
Using the above equations the question arises as to whether these formulas can
be used to estimate variance and covariance. A key assumption in these formulas
is that yt are independently, identically distributed, which implies E[yt] = µ and
E[(yt − µ)2
] = ν, where µ is a constant mean and ν is a constant variance. It is safe
to assume µ = 0 because µ
√
ν is true most of the time. The assumption that
4
9. 100 200 300 400 500
-0.05
0.05
0.1
0.15
Figure 2.1: The graph represents Yahoo’s stock price for the last two years.
ν is constant is known as homoskedasticity. However, there are problems with this
assumption. Just by inspection of 2.1 it is clear that stock returns do not exhibit
homoskedasticity, as is evident by the differences in the amplitudes of the returns
over time, known as ”volatility clustering” in [7]. Therefore, stock returns exhibit
heteroskedasticity, or non-constant variance.
Now, given that stock time series are heteroskedastic, what would be the best
way to proceed? The simplest method suggested is to divide the time period of
interest into small intervals and assume constant variance for each of these intervals,
see [5]. However, according to Engle [5], this seems to be an implausible assumption
with regards to how much weight is allocated to each data point, where variance can
be considered a weighted average of squared log returns. In definition 2.1.1, equal
weights are given to each data point in the time interval and zero weights are given
to data points outside of the time interval. With this in mind, Engle [5] suggests a
new way to weight data points.
2.2 Estimating Variance Univariate Case
The first model suggested for heteroskedastic timeseries is the autoregressive condi-
tional heteroskedastic model. Following from [5] we define the following dynamics for
the ARCH model of order q:
Definition 2.2.1 (ARCH(q)). Let yt be the return of a stock at time t, xt be en-
dogenous, nonstochastic vector of variables, β = [β0, . . . , βm] and α = [α0, . . . , αq] ,
5
10. t = yt − xtβ (generally β = 0), ˜t = [1, t−1, . . . , t−q] and ν represent the variance of
yt at time t.
yt ∼ N(xtβ, νt) (2.2.1)
νt = ˜tα (2.2.2)
The ARCH model deals with heteroskedasticity fairly well, however, when it is
applied, according to [2], a fairly high q is required for a reasonable fit, which calls
for a large number of parameters. To avoid this problem Bollerslev [4] proposed the
generalized autoregressive conditional heteroskedastic model. Following [4] we define
the following for GARCH(p,q):
Definition 2.2.2 (GARCH(p,q)). Let yt be the return of a stock at time t, xt be an
endogenous, nonstochastic vector of variables, β = [β0, . . . , βm] and α = [α0, . . . , αq] ,
t = yt − xtβ (generally β = 0), ˜t = [1, t−1, . . . , t−q] , δ = [δ1, . . . , δp] , ˜νt =
[νt−1, . . . , νt−p] and ν represent the variance of yt at time t.
yt ∼ N(xtβ, νt) (2.2.3)
νt = ˜tα + ˜νtδ (2.2.4)
In order for the variation of yt to be finite and positive the following constraints
must be imposed on α and δ:
α 0 (2.2.5)
δ 0 (2.2.6)
q
i=1
δi +
p
j=1
αj < 1 (2.2.7)
Now, given yt and the evolution of the variance over time the question is how to
estimate the parameters in 2.2.2. Since we are assuming β = 0 there is no reason to
use least squares. Instead, the maximum likelihood estimator is the most efficient,
unbiased way to estimate α and β. Maximum likelihood estimation involves assigning
probabilities to each of the data point observations multiplying them together to find
the total probability of the observing the given timeseries. The total probability is a
function of the parameters α and β, and the goal is to maximize this function.
Definition 2.2.3. Let L ∈ C2
(Rp+q+1
), where C2
(R)p+q+1
denotes the set of second
order, differentiable functions over Rp+q+1
, θ = [α , δ ] , define L as the following:
L(θ) =
T
t=1
exp (−
y2
t
2νt
)
√
2πνt
(2.2.8)
6
11. Usually, its easier to find the maximum of the log of L(θ) than it is to find the
maximum of L(θ) itself. In that case the log likelihood is:
ln L(θ) =
T
t=1
−
1
2
ln 2π − ln νt −
y2
t
νt
(2.2.9)
Later in the chapter, we will use a variation of the quasi-newton method to find the
maximum. But for now we derive the gradient, which is essential to the algorithm.
Let t = − ln νt −
y2
t
νt
, then for i = 1, . . . , p + q + 1:
∂ t
∂θi
= (
1
νt
)(
y2
t
νt
− 1)
∂νt
∂θi
(2.2.10)
∂ ln L(θ)
∂θi
=
T
t=1
∂ t
∂θi
(2.2.11)
The derivatives of νt with respect to θ are as follows:
∂νt
∂α0
= 1 +
p
i=1
δi
∂νt−i
∂α0
(2.2.12)
∂νt
∂δm
= νt−m +
p
i=1
δi
∂νt−i
∂δm
, m = 1, . . . , p (2.2.13)
∂νt
∂αn
= αny2
t−n +
p
i=1
δi
∂νt−i
∂αn
, n = 1, . . . , q (2.2.14)
where the values of the derivatives of νt when t < 0 is equal to 0.
An important question that arises is how accurate are the parameter estimates ˜θ
to the real values θ. For this we first define A as the following:
A = −T− 1
2 E[
T
t=1
∂2
t
∂θ∂θ
]
Then by [2], when θ0 is the true parameter and ˜θ is the sample estimate, the following
holds:
T
1
2 (˜θ − θ0)−→N(0, A−1
) (2.2.15)
Hence, we can use 2.2.15 as a proxy for the accuracy of a set of parameter estimates.
It also implies that as T−→∞ the estimate ˜θ approaches θ0. The underlying assump-
tion, though, when using a maximum likelihood estimator is that the disturbance
distributions are correctly specified in the likelihood function. In this paper, we as-
sume the disturbances are normally distributed and so we use the gaussian density
as the likelihood function. The optimization technique used to find the maximum of
the likelihood function are discussed in detail at the end of the chapter.
7
12. 2.3 Estimating GARCH Multivariate Case
In this section we will present a way to estimate the correlations between stocks, then
we will focus on the conditions needed to ensure validity of the covariance matrix.
The main problem, assuming that returns are normally distributed, is finding the
covariance matrix as a function of time. Or in other words, given Yt ∼ N(0, Σt)
what is Σt. A necessary constraint on Σt is that it must be positive semi-definite. In
mathematical terms, for any vector w of weights the following must hold:
w Σtw 0 (2.3.1)
In [3], Wooldridge proposes a multivariate extension of the GARCH(p,q) model. The
following definition defines the GARCH-M model.
Definition 2.3.1. Let Xt be endogenous, nonstochastic variables, Ai, Bi, C are N2
×
N2
parameter matrices, and . Then define GARCH-M(p,q) as the following:
Yt ∼ N(Xtβ, Σt) (2.3.2)
vech(Σt) = C +
q
i=1
Aivech( t−i t−i) +
p
j=1
Bjvech(Σt−j) (2.3.3)
where for a K × K matrix D:
vech(D) =
D1,1
...
DK,1
D1,2
...
DK,K
(2.3.4)
Just as we previously assumed earlier in the paper for the univariate case, we will
assume β = 0. The log likelihood function for 2.3.2, where
θ = [vech(A1) , . . . , vech(Aq) , vech(B1) , . . . , vech(Bp) , vech(C) ] , is as follows:
L(θ) =
T
t=1
−
N
2
ln 2π −
1
2
ln |Σt| −
1
2
Yt Σ−1
t Yt (2.3.5)
Various constraints are proposed to ensure the positive, semidefiniteness of Σt. In this
paper we will follow along the lines of [3], which assumes that A, B, C are diagonal
matrices. If that is the case then the multivariate problem is stripped down into
1
2
N(N − 1) bivariate problems and N univariate problems. Also, before we proceed
further, we will only consider the case where p = 1 and q = 1. For simplicity, denote
8
13. the following dynamics for Σt, where A, B, C are N × N matrices and ⊗ signifies
element by element multiplication:
Σt = C + B ⊗ Σt + A ⊗ Yt−1Yt−1 (2.3.6)
In [6], Engle proposes a two step estimation of the log likelihood function. First,
we maximize with respect to the univariate parameters, then we maximize, using
the estimated variances from the univariate equations, with respect to the bivariate
equations. The set of equations can be written as follows:
Yt ∼ N(Xtβ, Σt)
(Σt)i,j = Ci,j + Ai,j(Yt−1Yt−1)i,j + Bi,j(Σt−1)i,j
i = 1, . . . , N
The matrices A, B, C are symmetric. We already defined the likelihood function
for the univariate case, now we turn our attention towards the bivariate case. The
likelihood function for the bivariate case is as follows:
Definition 2.3.2. Let Xt represent the 2 × 1 vector of returns, hij,t = (Σt)i,j, φ =
[ci,j, ai,j, bi,j] , then define the following:
Xt =
(Yt)i
(Yt)j
Ht =
hii,t hij,t
hij,t hjj,t
(2.3.7)
L(φ) =
T
t=1
1
2π |Ht|
exp −
1
2
XtH−1
t Xt (2.3.8)
The log likelihood function that has to be optimized is 2.3.5, where N = 2. The
values hii,t are the estimated variances from the univariate parameters. After A, B, C
is estimated pairwise, a problem that must be confronted is whether the estimated
Σt are positive semidefinite. In [11], Ledoit et al. present three conditions must be
satisfied in order for Σt to be positive semidefinite.
Proposition 2.3.3. If C ÷( −B) 0, where is a column of ones and ÷ denotes
element by element division, B 0, and A 0, then Σt 0 almost surely.
Proof. For the proof see Ledoit et al. [11].
The initial estimates ˜A, ˜B, ˜C are usually not positive, semidefinite. Ledoit et al.
[11] has a way to find the nearest positive, semidefinite matrix with respect to the
Frobenius Norm. The algorithm is stated in the appendix. Now equipped with the
parameter estimates, we can now calculate the covariance matrix at each time step t.
9
14. 100 200 300 400 500 600 700
0.001
0.002
0.003
0.004
0.005
0.006
Figure 2.2: The graph represents the estimated volatility of Yahoo’s stock price for
the last two years.
2.4 Summary
In this section, we conclude the chapter by doing numerical examples. We calculate
the parameter matrices, where the model portfolio consists of Yahoo and Ford stocks,
then we plot the estimated volatility of Yahoo over a time period of approximately 2
years. The parameter matrices for Yahoo and Ford are as follows:
α0 δ1 α1
YHOO 5.17853e-5 .878123 .0880492
F 8.37845e-5 .707522 .209021
YHOO & GM 1.70347e-5 .905386 .069641
Table 2.1: Garch(1,1) Parameters
10
15. Chapter 3
Sequential Monte Carlo Methods
3.1 Preliminaries
The purpose of this chapter is to deal with the problem of measurement noise. As
was evident in 2.2, Garch volatility estimates suffer from significant variability. As
such, there are ways to mitigate this problem in order to arrive at a much closer
approximation to the true covariance matrix. The problem itself, is known as Bayesian
state estimation, and the problem of Bayesian estimation is to estimate the underlying
value of an unobservable quantity using indirect observations. The dynamics of the
underlying variable, in this case the covariance matrix, is known before hand, while
the observation model is also known before hand. Using Bayesian Estimation, one
can filter the ”noise” from the Garch volatilities.
The problem is very simple to state in mathematical terms, however, it is not a
trivial matter finding a solution. First, following from Gordon et al. [9], let Zt ∈ Rq
be the observation vector at time t, Θt ∈ Rn
be the unobservable quantity at time t,
Wt ∈ Rm
and Vt ∈ Rp
be random vectors, f : Rn
× Rm
−→Rn
, g : Rn
× Rp
−→Rq
. Then
the transition equation is
Θt = f(Θt, Wt) (3.1.1)
and the observation equation is
Zt = g(Θt, Vt) (3.1.2)
The random variables Wt, Vt are zero mean, white noises, and the distributions of Wt
and Vt are assumed to be known. Let Dt = {Z1, . . . , Zt}.
The goal of Bayesian estimation is to find the posterior density, p(Θt|Dt). This
probability density can be calculated recursively, assuming at time t − 1 the density
p(Θt−1|Dt−1) is known, by first the prediction stage
p(Θt|Dt−1) = p(Θt|Θt−1)p(Θt−1|Dt−1)dΘt−1
11
16. followed by the update stage via Bayes rule
p(Θt|Dt) =
p(Zt|Θt)p(Θt|Dt−1)
p(Zt|Θt)p(Θt|Dt−1)dΘt
The densities p(Θt|Θt−1) and p(Zt|Θt) are derived directly from 3.1.1 and 3.1.2, re-
spectively. After the updating stage, the filtered value of Θt is calculated from the
posterior distribution by taking the expectation
˜Θt = E[Θt|Dt] = Θtp(Θt|Dt)dΘt
The difficulty in computing these filtering equations lie in the evaluation of the
integrals in the prediction and updating stages. For most cases, no analytical so-
lution exists. Only when f(Θt, Wt) and g(Θt, Vt) are linear functions, and Wt and
Vt are gaussian white noises. If the model falls under those set of restrictions then
the solution is the famous Kalman Filter, which is very simple, yet elegant set of
difference equations [10]. For our situation, as we will be shown later in the chapter,
our f(Θt, Wt) and g(Θt, Vt) are nonlinear, which precludes it from a Kalman Filter
solution. One way of dealing with nonlinearity is to approximate 3.1.1 and 3.1.2 and
then use the Kalman filter, see [10]. On the other hand, Fearnhead [8] suggests that
approximating non-Gaussian densities with Gaussian ones has the potential to cause
the filter to diverge. Hence, a new method is needed to deal with this problem.
3.2 Particle Filter
The filter we use in this paper is a simple particle filter called the Sample Importance
Resampling filter, also known as the Bayesian bootstrap filter. The algorithm uses
random samples and a likelihood function to approximate the posterior distribution
p(Θt|Dt). The algorithm, from [8], is as follows
1. Initialization Initialize the filter by sampling N particles, {Θ
(i)
0 }N
i=1, from
p(Θ0).
2. Prediction(step t) Assuming that Θ
(i)
t−1 is distributed according to p(Θt−1|Dt−1),
generate the set of points {Θ
(i)
t|t−1}N
i=1 via the equation
Θ
(i)
t|t−1 = f(Θ
(i)
t−1, W
(i)
t )
where W
(i)
t are inpendently, identically distributed random variables with prob-
ability density p(Wt).
12
17. 3. Filtration Assign probability weights qi
t to each Θ
(i)
t according to the equation
qi
t =
p(Zt|Θ
(i)
t|t−1)
N
j=1 p(Zt|Θ
(j)
t|t−1)
Then sample {Θ
(j)
t }N
j=1 from the distribution
p(Θ
(j)
t = Θ
(i)
t|t−1) = qi
t
The theoretical justification for this algorithm can be found in [8]. The benefit in
using this specification is that there are no restrictions on 3.1.1 or 3.1.2, or on the
distributions of Wt and Vt. But, the disadvantage of using this algorithm is the high
computational cost necessary to generate a sufficiently close approximation to the
posterior distribution.
In this paper we use the particle filter to estimate Σt from 2.3.6, where the ob-
servation vector is the vector of returns of a portfolio of stocks. So, notation wise,
Θt = Σt and Zt = Yt. The transition equation is derived from 2.3.6. By hypothe-
sis E[YtYt ] = Σt, so if Wt ∼ N(0, I) and define σt such that σtσt = Σt, then the
transition equation for Σt is
Σt = C + B ⊗ Σt−1 + A ⊗ σt−1Wt−1Wt−1σt−1 (3.2.1)
The observation likelihood function, p(Σt|Yt), where Np is the number of stocks, is
p(Σt|Yt) = [(2π)Np
|Σt|]− 1
2 exp −
1
2
Yt Σ−1
t Yt (3.2.2)
We used the sample covariance matrix as the initial starting point for the filter.
3.3 Summary
Finally, given the historical returns of a portfolio of stocks, one can estimate the
covariance matrix using a combination of a multivariate GARCH specification and
bayesian filtering theory. The following is a brief synopsis of the procedure
1. Estimate the parameter matrices A, B, C using the methods in chapter 2
2. Then initialize the SIR filter at the sample covariance matrix
Using this procedure one can filter out the measurement noise from the observations,
a technique frequently used in the engineering disciplines. As long as the parameter
matrices are properly conditioned, the estimated covariance matrices will be positive
definite.
13
18. Appendix A
Quasi-Newton Method
In this section, we present an iterative procedure to find a local extremum, θ∗
∈ Rm
,
of a function f : Rm
−→R. The method we use in this paper to find the minimum of
f(θ) is the variable metric method, otherwise known as the quasi-newton method.
More specifically, since we will not use the hessian, we will use the BFGS algorithm
detailed in [12]. Let Hi be the ith approximation of the inverse of the hessian of f(θ).
Let fi = f(θi), H0 be the identity matrix and θ0 be in the neighborhood of θ∗
,
where f(θ) is approximately quadratic. Then the algorithm is as follows:
θi+1 = θi − Hi fi (A.0.1)
Hi+1 = Hi +
(θi+1 − θi) ⊗ (θi+1 − θi)
(θi+1 − θi) ( fi+1 − fi)
− (A.0.2)
[Hi( fi+1 − fi)] ⊗ [Hi( fi+1 − fi)]
( fi+1 − fi) Hi( fi+1 − fi)
+ [( fi+1 − fi) Hi( fi+1 − fi)]u ⊗ u
where u is defined as the vector
u =
θi+1 − θi
(θi+1 − θi) ( fi+1 − fi)
−
Hi( fi+1 − fi)
( fi+1 − fi) Hi( fi+1 − fi)
As θi−→θ∗
, BFGS enjoys the quadratic convergence rate of newton’s method with the
known hessian. However, this algorithm requires a relatively accurate initial guess.
To find a decent approximation we use the steepest descent algorithm, which is just
newton’s method with the assumption that the hessian is equal to a constant times the
identity matrix. In this section, we present the pose the question of finding positive
semidefinite matrices given a suitable estimate that is not positive semidefinite.
14
19. Appendix B
Finding Nearest Correlation
Matrix
In this section, we summarize the algorithm used in Ledoit et al. [11] to find the
closest fitting positive, semdefinite matrix to the current estimated matrix. Given a
symmetric matrix A with the property diag(A) > 0, the algorithm finds a symmetric,
positive, semidefinite matrix M with diag(M) = diag(A), such that quantity A −
M F , where F is the Frobenius norm, is minimized. First, start with the 1st row and
column.
A =
a11 a
a ¯A
M =
a11 m
m ˜M
where diag(M) = diag(A) and M = M . Define P as the following
P =
ρ x
0 In−1
We iterate be setting
˘M = PMP =
ρ2
a11 + 2ρx m + x ˜Mx ρm + x ¯M
ρm + ˜Mx ˜M
(B-1)
For each iteration the quantity a − (ρm + ˜Mx) must be minimized subject to the
constraint that ρ2
a11+2ρx m+x ˜Mx = a11 Ledoit et al. [11] derive a simple algorithm
to this. Let λ, F, Fλ be scalars, then the algorithm is as follows
1. Initialize λ = 0 (starting point is arbitrary).
2. x = ( ˜M2
+ λ ˜M)−1
( ˜Mb − λρm)
3. Set F = ρ2
a11 +2ρx m+x ˜Mx−a11 and Fλ = −2(ρm+ ˜Mx) ( ˜M2
+λ ˜M)−1
(ρm+
˜Mx)
15
20. 4. λ = λ − F
Fλ
5. Repeat steps 2.-4. until satisfactory convergence.
Rho is usual set as a constant between zero and one, say .5. This way det( ˘M) =
ρ2
det(M), which means M converges to a singular matrix at the rate ρ2
.
16
21. Bibliography
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Journal of Political Economy 81 (1973), 637–654.
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Elsevier Science Pub Co, New York, NY, USA, 1999.
[3] T. Bollerslev, R. F. Engle, and J. M. Wooldridge, Capital asset pricing model
with time-varying covariances, Journal of Political Economy 96 (1988), 116–131.
[4] Tim Bollerslev, Generalized autoregressive conditional heteroskedasticity, Journal
of Econometrics 31 (1986), 307–327.
[5] Robert F. Engle, Autoregressive conditional heteroscedasticity with estimates of
the variance of united kingdom inflation, Econometrica 50 (1984), 987–1008.
[6] , Dynamic conditional correlation - a simple class of multivariate garch
models, Jul 1999.
[7] , Garch101: The use of arch/garch models in applied econometrics, The
Journal of Economic Perspectives 15 (2001), 157–168.
[8] P. Fearnhead, Sequential monte carlo methods in filter theory, Ph.D. thesis, Ox-
ford, 1998.
[9] N. J. Gordon, D. J. Salmond, and A. F. M. Smith, Novel approach to
nonlinear/non-gaussian bayesian state estimation, IEE Proceedings-F 140
(1993), 107–113.
[10] R. E. Kalman, A new approach to linear filtering and prediction problems, Trans-
action of the ASME, 1960, pp. 35–45.
[11] O. Ledoit, P. Santa-Clara, and M. Wolf, Flexible multivariate garch modeling
with an application to international stock markets, The Review of Economics
and Statistics 85 (2003), 735–747.
17
22. [12] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical
recipes in c: The art of scientific computing, Cambridge University Press, New
York, NY, USA, 1992.
18
23. Name of Candidate: Joel DeJesus
Birth Date: 10 October 1984
Birth Place: N¨urnburg, Germany
Address: 1000 SW 62blvd Gainesville, FL