Sergio M. Focardi

Pôle Universitaire Léonard de Vinci · Finance Group ESILV EMLV

Advanced economies are increasingly forced to act in order to mitigate climate changes and to reduce the impact of human activity on the environment. The key strategy promoted by the OECD and other international organizations is green growth. The key tenet of green growth is the ability of science and technology to let economies grow without damagi...

In this paper we analyze the existence of cointegrating relationships between Bitcoin, S&P 500, and the quantity of money M2. We perform our analysis with and without applying time warping pre-processing. In all cases we find strong evidence that, in the period 2016-2021 the three time series show two cointegrating relationships and therefore share...

In this paper we test for regime changes and possible regime commonalities in the price dynamics of Bitcoin, Ethereum, Litecoin and Monero, as representatives of the cryptocurrencies asset class. Several parametric models are considered for the joint dynamics of the basket price where parameters are modulated through a Hidden Markov Chain with fini...

In this paper, we apply dynamic factor analysis to model the joint behaviour of Bitcoin, Ethereum, Litecoin and Monero, as a representative basket of the cryptocurrencies asset class. The empirical results suggest that the basket price is suitably described by a model with two dynamic factors. More precisely, we detect one integrated and one statio...

In this paper we show that persistence and switching of trends are phenomena that appear in most long-lived stock return series. We model stock returns using a family of models based on hidden Markov models with duration-dependent transition probabilities. Trends are correlated so that aggregates such as indexes exhibit the same persistence and swi...

By enabling the storage and transfer of purchasing power, money facilitates economic transactions and coordinates economic activity. But what is money? How is it generated? Distributed? How does money acquire value and that value change? How does money impact the economy, society? This book explores money as a system of "tokens" that represent the...

Brownian motion as the source of randomness and uncertainty is used for most applications to real-world problems encountered in financial economics. The Black-Scholes-Merton framework for pricing options is the best known example of the application of Brownian motion. However, there is a preponderance of empirical evidence that fails to support Bro...

There are different approaches for studying the behavior of a stochastic process. A stochastic process can be studied as a stochastic differential equation, a partial integro-differential equation, and a fractional partial differential equation. The efficiency of these different approaches depends on the dynamics of the asset price process and the...

Since the early 1960s, there have been a good number of papers related to heavy tail distributions. These papers support the view that the heavy tail property is a stylized fact about financial time series. Stable distributions have infinite variance, a property which is not found in empirical samples where empirical variance does not grow with the...

In this monograph we discuss how fractional calculus and fractional processes are used in financial modeling, finance theory, and economics. We begin by giving an overview of fractional calculus and fractional processes, responding upfront to two important questions:

In this chapter, our objective is twofold. First, we establish a connection between the stable distributions with fractional calculus. This is accomplished by defining appropriate fractional diffusion equations, the fundamental solution of which provides the PDF for the univariate and multivariate stable distributions.1 Second, by using some analyt...

Geometric stable (geo-stable) distributions are suitable alternatives for the normal distribution, which suffers from lack of heavy-tail and asymmetric property. These distributions share the same problem with stable distributions, which do not have a closed-form formula for either their probability density function (PDF) and cumulative distributio...

Continuous-time random walk is an extension of the random walk. More specifically, it is constructed by introducing a new source of randomness to the random walk. This new source of randomness is waiting time. In this chapter, we first discuss the continuous-time random walk, and then move on to its applications in financial economics.

Chapter 2 described how fractional calculus can be applied to generate fat-tailed distributions; discussed how to apply fractional processes to the pricing of derivatives. As fractional processes are not semi martingales, violations of the no-arbitrage condition might occur. We have seen how to circumvent this problem.

Fractional Calculus and Fractional Processes with Applications to Financial Economics presents the theory and application of fractional calculus and fractional processes to financial data. Fractional calculus dates back to 1695 when Gottfried Wilhelm Leibniz first suggested the possibility of fractional derivatives. Research on fractional calculus...

In this article, we review the critical issues associated with applying financial econometrics to building factor models. Although factor models are widely used in the industry to engineer investment and trading strategies and control risk, there are many issues associated with their implementation. We focus on three issues of importance to asset m...

We introduce an original application of Suprathreshold Stochastic Resonance (SSR). Given a noise-corrupted signal, we induce SSR in effort to filter the effect of the corrupting noise. This will yield a clearer version of the signal we desire to detect. We propose a financial application that can help forecast returns generated by big orders. We as...

Statistical arbitrage strategies are typically based on models of returns. We introduce a new statistical arbitrage strategy based on dynamic factor models of prices. Our objective in this paper is to exploit the mean-reverting properties of prices reported in the literature. We do so because, to capture the same information using a return-based fa...

In the aftermath of the 2007-2009 financial crisis, mainstream economics was criticized for having failed to either forecast or help prevent the market crash, which resulted in large losses for investors, although markets were back to pre-crisis levels by the end of the first quarter of 2013. Some have suggested that the crash itself was the result...

Today's mainstream economics, embodied in Dynamic Stochastic General Equilibrium (DSGE) models, cannot be considered an empirical science in the modern sense of the term: it is not based on empirical data, is not descriptive of the real-world economy, and has little forecasting power. In this paper, I begin with a review of the weaknesses of neocla...

In this article, the authors suggest how to think about a new framework for the analysis of financial bubbles and a possible vector of variables able to signal when an economy enters a state of disequilibrium. The working hypothesis is that market crashes are preceded by a bubble. The authors define a bubble as an anomalous increase in asset prices...

The two most commonly used statistical techniques used to reduce a large number of observed variables to a smaller number of factors are factor analysis and principal component analysis. These two statistical techniques are applied to a set of variables when there is interest in identifying which variables in the set of variables form subsets that...

A categorical variable is a variable that take on values that are names, attributes, or labels. For example, given a set of stocks, each stock may be categorized in terms of its investment style as a growth stock or a value stock. Thus investment style is a categorical variable that indicates to what category-growth or value-each stock belongs. Or...

In a regression model, as with all statistical models, it is necessary to make sure that the assumptions of the model are satisfied. Violation of any of the assumptions may have a minimal impact on the model's predictive ability or it may result in a poor predictive model that is due to the inclusion of explanatory variables that should not have be...

In financial time series the current value of a variable is often related to the prior or lagged value of the same variable. Two models that are used to characterize this behavior are the autoregressive model and the moving average model. The autoregressive model is appropriate to employ in situations where the current value is determined by the va...

In financial econometrics, model selection involves a trade-off between model complexity and the size of the data sample. To implement this trade-off, ensuring that models have forecasting power, the fitting of sample data is constrained to avoid fitting noise. Because financial data are generally scarce given the complexity of their patterns, ther...

The parameters of financial econometric models must be estimated. Model estimation is a set of methods employed to determine population parameters from a sample. Consequently, an estimator is a function of the sample data. Four estimation methods used in financial econometrics are the least-squares (ordinary least squares, weighted least squares, a...

The variance of an asset's price or return is an important measure in finance because it is one of the key metrics used to quantify an asset's risk. The simplest approach for measuring historical volatility involves calculating the variance from a sample of prices or returns observed over a recent short-time period. Historical volatility can be com...

A set of observations concerning financial data such as prices, returns, or interest rates made at different points in time is referred to as a time series. Each pair of observations then consists of time and value. In time series analysis, one seeks to ascertain what the values are decomposable into at each point in time. The most traditional deco...

Financial econometrics is used in academic research to empirically test various financial economic theories such as asset pricing theory and market price efficiency theory, as well as to investigate the potential impact of actual and proposed financial regulations. Asset managers use financial econometrics to formulate investment strategies while r...

In the classical linear regression model, the regression coefficient of some explanatory variable represents the change in the dependent variable produced by a one unit change in the explanatory variable associated with that coefficient. A quantile regression is a model that can be used for estimating the relation between the explanatory variables...

The relationships among nonstationary variables can be analyzed if they share a common stochastic trend. A way of capturing this common stochastic trend is the application of the tool of cointegration. That is, cointegration can be used to identify long-run relationships between variables. The two most-often employed methods to test for cointegrati...

Regression is the basic statistical tool in the financial econometrician's toolkit. It is employed to model the dependence of a variable (called the dependent variable) on one (or more) explanatory variables. In the basic regression, the functional relationship between the dependent variable and the explanatory variables is expressed as a linear eq...

The method for estimating the coefficient of the classical linear regression model is the ordinarily least squares method, a fairly easy computation methodology. However, the estimates of the regression coefficient can be quite sensitive to outliers in the dataset. Techniques developed in the field of robust statistics which addresses the problem o...

The purpose of a multiple linear regression is to estimate the dependence of the dependent variable on more than two explanatory variables. The same method for estimating the parameters in a simple linear regression model is used in estimating the parameters for a multiple linear regression. When estimating a multiple linear regression there are th...

Introduction The Intuition behind Stochastic Differential Equations Itô Processes Stochastic Differential Equations Generalization to Several Dimensions Solution of Stochastic Differential Equations Derivation of Itô's Lemma Derivation of the Black-Scholes Option Pricing Formula Key Points

Introduction Riemann Integrals Lebesgue-Stieltjes Integrals Indefinite and Improper Integrals The Fundamental Theorem of Calculus Integral Transforms Calculus in More Than One Variable Key Points

Introduction Sets and Set Operations Distances and Quantities Functions Variables Key Points

Introduction Maxima and Minima Lagrange Multipliers Numerical Algorithms Calculus of Variations and Optimal Control Theory Stochastic Programming Application to Bond Portfolio: Liability-Funding Strategies Key Points

In this paper, we analyze factor uniqueness in the S&P 500 universe. The current theory of approximate factor models applies to infinite markets. In the limit of infinite markets, factors are unique and can be represented with principal components. If this theory would apply to realistic markets such as the S&P 500 universe, the quest for proprieta...

Focardi and Fabozzi argue that current mainstream economics is not a science in the sense of the physical sciences, and they draw some conclusions from the point of view of asset management. Their key point is that economics as embodied in the general equilibrium theories describes an idealized rational economic world as opposed to one based on emp...

What Is Time Series?Decomposition of Time SeriesRepresentation of Time Series with Difference EquationsApplication: The Price ProcessConcepts Explained in this Chapter

Tests for LinearityAssumed Statistical Properties about the Error TermTests for the Residuals Being Normally DistributedTests for Constant Variance of the Error Term (Homoskedasticity)Absence of Autocorrelation of the ResidualsConcepts Explained in this Chapter

Pie ChartsBar ChartStem and Leaf DiagramFrequency HistogramOgive DiagramsBox PlotQQ PlotConcepts Explained in this Chapter

Parameters of LocationParameters of ScaleConcepts Explained in this ChapterAppendix: Parameters for Various Distribution Functions

The Problem of MulticollinearityIncorporating Dummy Variables as Independent VariablesModel Building TechniquesConcepts Explained in this Chapter

The Notion of Vector and MatrixMatrix MultiplicationParticular MatricesPositive Semidefinite Matrices

Conditional ProbabilityIndependent EventsMultiplicative Rule of ProbabilityBayes' RuleConditional ParametersConcepts Explained in this Chapter

Normal DistributionChi-Square DistributionStudent's t-DistributionF-DistributionExponential DistributionRectangular DistributionGamma DistributionBeta DistributionLog-Normal DistributionConcepts Explained in this Chapter

The Multivariate Linear Regression ModelAssumptions of the Multivariate Linear Regression ModelEstimation of the Model ParametersDesigning the ModelDiagnostic Check and Model SignificanceApplications to FinanceConcepts Explained in this Chapter

Binomial CoefficientMultinomial Coefficient

Call OptionsDeriving the Price of a European Call OptionIllustration

Higher Dimensional Random VariablesJoint Probability DistributionMarginal DistributionsDependenceCovariance and CorrelationSelection of Multivariate DistributionsConcepts Explained in this Chapter

Historical Development of Alternative Approaches to ProbabilitySet Operations and PreliminariesProbability MeasureRandom VariableConcepts Explained in this Chapter

Sample, Statistic, and EstimatorQuality Criteria of EstimatorsLarge Sample CriteriaMaximum Likehood EstimatorExponential Family and SufficiencyConcepts Explained in this Chapter

Data Tables and FrequenciesClass Data and HistogramsMarginal DistributionsGraphical RepresentationConditional DistributionConditional Parameters and StatisticsIndependenceCovarianceCorrelationContingency CoefficientConcepts Explained in this Chapter

Continuous Probability Distribution DescribedDistribution FunctionDensity FunctionContinuous Random VariableComputing Probabilities from the Density FunctionLocation ParametersDispersion ParametersConcepts Explained in this Chapter

The Role of CorrelationRegression Model: Linear Functional Relationship Between Two VariablesDistributional Assumptions of the Regression ModelEstimating the Regression ModelGoodness of Fit of the ModelLinear Regression of Some Nonlinear RelationshipTwo Applications in FinanceConcepts Explained in this Chapter

HypothesesError TypesQuality Criteria of a TestExamplesConcepts Explained in this Chapter

Data TypesFrequency DistributionsEmpirical Cumulative Frequency DistributionData ClassesCumulative Frequency DistributionsConcepts Explained in this Chapter

Parameters vs. StatisticsCenter and LocationVariationMeasures of the Linear TransformationSummary of MeasuresConcepts Explained in this Chapter

CopulaAlternative Dependence MeasuresConcepts Explained in this Chapter

Continuous FunctionIndicator FunctionDerivativesMonotonic FunctionIntegralSome Functions

Generalized Extreme Value DistributionGeneralized Pareto DistributionNormal Inverse Gaussian Distributionα-Stable DistributionConcepts Explained in this Chapter

Confidence Level and Confidence IntervalConfidence Interval for the Mean of a Normal Random VariableConfidence Interval for the Mean of a Normal Random Variable with Unknown VarianceConfidence Interval for the Variance of a Normal Random VariableConfidence Interval for the Variance of a Normal Random Variable with Unknown MeanConfidence Interval fo...

Discrete LawBernoulli DistributionBinomial DistributionHypergeometric DistributionMultinomial DistributionPoisson DistributionDiscrete Uniform DistributionConcepts Explained in this Chapter

This paper proposes an extension of the inventory production model developed in Bak, Chen, Scheinkman, and Woodford (BCSW, 1993). We show how the Pareto-Levy type of aggregate distributions emerge in global models as well as in local models. We extend the BCSW model by allowing random connections between firms. The distribution of production in the...

A comprehensive look at how probability and statistics is applied to the investment process. Finance has become increasingly more quantitative, drawing on techniques in probability and statistics that many finance practitioners have not had exposure to before. In order to keep up, you need a firm understanding of this discipline. Probability and St...

Economic science is generally considered less viable than the physical sciences. Sophisticated mathematical models of the economy have been developed but their accuracy is questionable to the point that the present economic crisis is often blamed on an unwarranted faith in faulty mathematical models. In this paper, we claim that the mathematical ha...

Instead of assuming the distribution of return series, Engle and Manganelli (2004) propose a new Value-at-Risk (VaR) modeling approach, Conditional Autoregressive Value-at-Risk (CAViaR), to directly compute the quantile of an individual asset's returns which performs better in many cases than those that invert a return distribution. In this paper w...

It is often objected that we cannot use mathematical methods in finance because (1) finance is dominated by unpredictable unique events (the black swans), (2) there are qualitative effects that cannot be quantified, and (3) the laws themselves of finance keep on changing. In this paper we discuss these three objections, offering arguments to reject...

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 10 Abstract: Volatility is a key parameter used in manyfinancial applications, from deriva- tives valuation to asset management and risk management. Volatility measures the siz...

Common stock portfolio strategies can be classified as active and passive. The selection of a strategy depends on the risk tolerance of the investor and the investor's view of the efficiency of the stock market. Investors who believe the stock market is efficient would tend to favor a passive strategy such as indexing; investors who believe the sto...

Regressions are the probabilistic equivalent of functions in the deterministic domain. A regression expresses a functional link between a random variable Y and one or more independent variables Xi called regressors. The variables Xi can be either deterministic or random variables. The regression function is the expectation of the variable Y given t...

As quantitative techniques have become commonplace in the investment industry, the mitigation of estimation and model risk in portfolio management has grown in importance. Robust optimization, which incorporates estimation error directly into the portfolio optimization process, is typically used with conventional robust statistical estimation metho...

More and more quantitative models have started to find their way into the investment management industry. Typically, the quantitative efforts at firms start with risk management functions and portfolio optimization. Needless to say, there are many areas beyond these were quantitative methods are valuable. First, for example, the usage of equity der...

It is well known that applying classical portfolio optimization in practice may lead to problems; in fact, the “optimal” portfolio may not be optimal at all. The problems encountered in real-world portfolio optimization include issues such as unstable portfolio weights, corner solutions, and poor performance. Some portfolio managers are using Bayes...

Trading costs can be classified as fixed versus variable transaction costs and explicit versus implicit transaction costs. Portfolio managers and traders need to be able of effectively model the impact of trading costs of their portfolios and trades. In doing so, they seek to minimize the total transaction costs. There are several approaches for th...

There are many conflicting interpretations of security prices and price determination in financial markets. They range from academic theories based on efficient markets and rational expectations hypotheses, to more traditional methods of fundamental analysis, to theories of "value" and "growth" investing, to chart-reading and technical analysis, to...