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This paper consolidates and compares the applicability and practicality of Black-Litterman model versus traditional Markowitz Mean-Variance model. Although well-known model such as Mean-Variance is academically sound and popular, it is rarely used among asset managers due to its deficiencies. To put the discussion into context we shed light on the improvement made by Fisher Black and Robert Litterman by putting the performance and practicality of both Black- Litterman and Markowitz Mean-Variance models into test. We will illustrate detailed mathematical derivations of how the models are constructed and bring clarity and profound understanding of the intuition behind the models. We generate two different portfolios, composing data from 10-Swedish equities over the course of 10-year period and respectively select 30-days Swedish Treasury Bill as a risk-free rate. The resulting portfolios orientate our discussion towards the better comparison of the performance and applicability of these two models and we will theoretically and geometrically illustrate the differences. Finally, based on extracted results of the performance of both models we demonstrate the superiority and practicality of Black-Litterman model, which in our particular case outperform traditional Mean- Variance model.

PageRank update refers to the process of computing new PageRank values after change(s) (addition or removal of links/vertices) has occurred in real life networks. The purpose of the updating is to avoid recalculating the values from scratch. To efficiently carry out the update, we consider PageRank as the expected number of visits to target vertex if multiple random walks are performed, starting at each vertex once and weighing each of these walks by a weight value. Hence, it might be looked at as updating non-normalised PageRank. In the proposed approach, a scaled adjacency matrix is sequentially updated after every change and the levels of the vertices being updated as well. This enables sets of internal and sink vertices dependent on their roots or parents, thus vector-vector product can be performed sequentially since there are no infinite steps from one vertex to the other.

A PageRank update refers to the process of computing new PageRank valuesafter a change(s) (addition or removal of links/vertices) has occurred in real-lifenetworks. The purpose of updating is to avoid re-calculating the values from scratch.To efficiently carry out the update, we consider PageRank to be the expected numberof visits to a target vertex if multiple random walks are performed, starting at eachvertex once and weighing each of these walks by a weight value. Hence, it mightbe looked at as updating a non-normalized PageRank. We focus on networks of treegraphs and propose an approach to sequentially update a scaled adjacency matrix afterevery change, as well as the levels of the vertices. In this way, we can update thePageRank of affected vertices by their corresponding levels.

In this article, we study how PageRank can be updated in an evolving tree graph. We are interested in finding how ranks of the graph can be updated simultaneously and effectively using previous ranks without resorting to iterative methods such as the Jacobi or Power method. We demonstrate and discuss how PageRank can be updated when a leaf is added to a tree, at least one leaf is added to a vertex with at least one outgoing edge, an edge added to vertices at the same level and forward edge is added in a tree graph. The results of this paper provide new insights and applications of standard partitioning of vertices of the graph into levels using breadth-first search algorithm. Then, one determines PageRanks as the expected numbers of random walk starting from any vertex in the graph. We noted that time complexity of the proposed method is linear, which is quite good. Also, it is important to point out that the types of vertex play essential role in updating of PageRank.

Summary Updating PageRank refers to a process of computing new PageRank values after changes have occurred in a graph. The main goal of the updating is to avoid recalculating the values from scratch. This chapter focuses on updating PageRank of an evolving tree graph when a vertex and an edge are added sequentially. It describes how to maintain level structures when a cycle is created and investigates the practical and theoretical efficiency to update PageRanks for an evolving graph with many cycles. The chapter discusses the convergence of the power method applied to stochastic complement of Google matrix when a feedback vertex set is used. It also demonstrates that the partition by feedback vertex set improves asymptotic convergence of power method in updating PageRank in a network with cyclic components.

The paper is devoted to studies of perturbed Markov chains commonly used for description of information networks. In such models, the matrix of transition probabilities for the corresponding Markov chain is usually regularised by adding  a special damping matrix multiplied by a small damping (perturbation) parameter ε. In this paper, we present results of the detailed perturbation analysis of Markov chains with damping component and numerical experiments supporting and illustrating the results of this perturbation analysis.

This chapter is devoted to studies of perturbed Markov chains, commonly used for the description of information networks. In such models, the matrix of transition probabilities for the corresponding Markov chain is usually regularized by adding aspecial damping matrix, multiplied by a small damping (perturbation) parameter ε. In this chapter, we present the results of detailed perturbation analysis of Markov chains with damping component and numerical experiments supporting and illustrating the results of this perturbation analysis.

We consider a market model of financial engineering with three factors represented by three correlated Brownian motions. The volatility of the risky asset in this model is the sum of two stochastic volatilities. The dynamic of each volatility is governed by a mean-reverting process. The first stochastic volatility of mean-reversion process reverts to the second volatility at a fast rate, while the second volatility moves slowly to a constant level over time with the state of the economy.

The double mean-reverting model by Gatheral (2008) is motivated by empirical dynamics of the variance of the stock price. This model can be consistently calibrated to both the SPX options and the VIX options. However due to the lack of an explicit formula for both the European option price and the implied volatility, the calibration is usually done using time consuming methods like Monte Carlo simulation or the finite difference method.

To solve the above issue, we use the method of asymptotic expansion developed by Pagliarani and Pascucci (2017). In paper A, we study the behaviour of the implied volatility with respect to the logarithmic strike price and maturity near expiry and at-the-money. We calculate explicitly the asymptotic expansions of implied volatility within a parabolic region up the second order. In paper B we improve the results obtain in paper A by calculating the asymptotic expansion of implied volatility under the Gatheral model up to order three. In paper C, we perform numerical studies on the asymptotic expansion up to the second order. The Monte-Carlo simulation is used as the benchmark value to check the accuracy of the expansions. We also proposed a partial calibration procedure using the expansions. The calibration procedure is implemented on real market data of daily implied volatility surfaces for an underlying market index and an underlying equity stock for periods both before and during the COVID-19 crisis. Finally, in paper D we check the performance of the third order expansion and compare it with the previous results.

We study the double-mean-reverting model by Gatheral. Our previous results concerning the asymptotic expansion of the implied volatility of a European call option, are improved up to order 3, that is, the error of the approximation is ultimately smaller that the 1.5th power of time to maturity plus the cube of the absolute value of the difference between the logarithmic security price and the logarithmic strike price.

The double-mean-reverting model by Gatheral [1] is motivated by empirical dynamics of the variance of the stock price. No closed-form solution for European option exists in the above model. We study the behaviour of the implied volatility with respect to the logarithmic strike price and maturity near expiry and at-the- money. Using the method by Pagliarani and Pascucci [6], we calculate explicitly the first few terms of the asymptotic expansion of the implied volatility within a parabolic region.

The double-mean-reverting model by Gatheral is motivated by empirical dynamics of the variance of the stock price. No closed-form solution for European option exists in the above model. We study the behaviour of the implied volatility with respect to the logarithmic strike price and maturity near expiry and at-the-money. Using the method by Pagliarani and Pascucci, we calculate explicitly the first few terms of the asymptotic expansion of the implied volatility within a parabolic region.

The Gatheral double stochastic volatility model is a three-factor model with mean-reverting stochastic volatility that reverts to a stochastic long-run mean. Our previous paper investigated the performance of the first and second-order implied volatilities expansions under this model. Moreover, a simple partial calibration method has been proposed. This paper reviews and extends previous results to the third-order implied volatility expansions under the same model. Using Monte-Carlo simulation as the benchmark method, extensive numerical studies are conducted to investigate the accuracy and properties of the third-order expansion. 

 This thesis aims to address the Kusuoka approximation (K-approximation) within the Gatheral model using Yamada’s method, also known as the Gaussian K-approximation. Our approach begins by transforming the original Gatheral model into a model with independent Wiener processes through Cholesky decomposition. Subsequently, the system is reformulated into its Stratonovich form, facilitating the definition of vector fields and their exponentials. We will assess whether the system satisfies the Uniformly Finitely Generated (UFG) condition. Additionally, based on our calculations, a simulation code will be developed to compare our results with those obtained by Yamada.

This article presents a survey of techniques for ranking results in search engines, with emphasis on link-based ranking methods and the PageRank algorithm. The problem of selecting, in relation to a user search query, the most relevant documents from an unstructured source such as the WWW is discussed in detail. The need for extending classical information retrieval techniques such as boolean searching and vector space models with link-based ranking methods is demonstrated. The PageRank algorithm is introduced, and its numerical and spectral properties are discussed. The article concludes with an alternative means of computing PageRank, along with some example applications of this new method.

Our research studies the construction and estimation of copula-based semi parametric Markov model for the processes, which involved in water flows in the hydro plants. As a rule analyzing the dependence structure of stationary time series regressive models defined by invariant marginal distributions and copula functions that capture the temporal dependence of the processes is considered. This permits to separate out the temporal dependence (such as tail dependence) from the marginal behavior (such as fat tails) of a time series. Dealing with utility company data we have found the best copula describing data - Gumbel copula. As a result constructed algorithm was used for an imitation of low probability events (in a hydro power industry) and predictions.

In graph theory, centrality measures are very crucial in ranking vertices of the graph in order of their importance. Alpha and eigenvector centralities are some of the highly placed centrality measures applied especially in social network analysis, disease diffusion networks and mechanical infrastructural developments. In this study we focus on recalculating alpha and eigenvector centralities using graph partitioning techniques. We write an algorithm for partitioning, sorting and efficiently computing these centralities for a graph. We then numerically demonstrate the technique on some sample small-sized networks to recalculate the two centrality measures

When working with a network, it is often of interest to locate the “most important”nodes in the network. A common way to do this is by using some graph centralitymeasures. Since what constitutes as an important node varies from one network toanother, or even in applications on the same network, there is a large number ofdifferent centrality measures proposed in the literature. Due to the large amount ofdifferent centrality measures proposed in different fields, there is also a large amountof very similar or equivalent centrality measures (in the sense that they give the sameranks). In this chapter, we focus on the centrality measures based on powers of theadjacency matrix and those based on random walk. In this case, we show how someof these centrality measures are related, as well as their lazy variants.We will performsome experiments to demonstrate the similarities between the centrality measures.

When working with a network it is often of interest to locate the "most important" nodes in the network. A common way to do this is using some graph centrality measures. Since what constitutes an important node is different between different networks or even applications on the same network there is a large amount of different centrality measures proposed in the literature. Due to the large amount of different centrality measures proposed in different fields, there is also a large amount very similar or equivalent centrality measures in the sense that they give the same ranks. In this paper we will focus on centrality measures based on powers of the adjacency matrix or similar matrices and those based on random walk in order to show how some of these are related and can be calculated efficiently using the same or slightly altered algorithms.

The development of financial markets imposes more complex models on the option pricing problems. On the previous papers by the authors, we consider a model under which the underlying asset is driven by two independent Heston-type stochastic volatility processes of multiscale (fast and slow) mean-reverting rates and we compute an approximate solution for the option pricing problem, using asymptotic expansion method. In the present paper, we aim to calibrate the model using the market prices of options on Euro Stoxx 50 index and an equity stock in the European market. Our approach is to use the market implied volatility surface for calibrating directly a set of new parameters required in our second-order asymptotic expansion pricing formula for European options. This secondorder asymptotic expansion formula provides a better approximation formula for European option prices than the first-order formula, as explained in an earlier work of the authors.

PageRank is an algorithm for ranking web pages. It is the first and best known webgraph-based algorithm in the Google search engine. The algorithm is simple, robust and reliable to measure the importance of web pages. This chapter presents a comparative review of three variants of PageRank, namely ordinary PageRank (introduced by Brin and Page as a measure of importance of a web page), lazy PageRank and random walk with backstep PageRank. It compares the variants in terms of their convergence and consistency in rank scores for different graph structures with reference to PageRank’s parameters, damping factor and backstep parameter. The chapter also shows that ordinary PageRank can be formulated from the other two variants by some proportionality relationships.

PageRank was initially defined by S. Brin and L. Page for the purpose of measuring the importance of web pages (nodes) based on the structure of links between them. Due to existence of diverse methods of random walk on the graph, variants of PageRank now exists. They include traditional (or normal) PageRank due to normal random walk and Lazy PageRank due to lazy random walk on a graph. In this article, we establish how the two variants of PageRank changes when complete graphs are connected to a line of nodes whose links between the nodes are in one direction. Explicit formulae for the two variants of PageRank are presented. We have noted that the ranks on a line graph are the same except their numerical values which differ. Further, we have observed that both normal random walk and lazy random walk on complete graphs spend almost the same time at each node.

PageRank is a widely-used hyperlink-based algorithm to estimate the relative importance of nodes in networks [11]. Since many real world networks are large sparse networks, this makes efficient calculation of PageRank complicated. Moreover, one needs to escape from dangling effects in some cases as well as slow convergence of the transition matrix. Primitivity adjustment with a damping (perturbation) parameter ε(0,ε₀] (for fixed ε₀ 0.15) is one of the essential procedure that is known to ensure convergence of the transition matrix [24]. If ε is large, the transition matrix looses information due to shift of information to teleportation matrix [27]. In this paper, we formulate PageRank problem as the first and second order Markov chains perturbation problem. Using numerical experiments, we compare convergence rates for the two problems for different values of ε on different graph structures and investigate the difference in ranks for the two problems.

PageRank is a widely used hyperlink-based algorithm for estimating the relative importance of nodes in networks. Since many real-world networks are large sparse networks, efficient calculation of PageRank is complicated. Moreover, we need to overcome dangling effects in some cases as well as slow convergence of the transition matrix. Primitivity adjustment with a damping (perturbation) parameter is one of the essential procedures known to ensure convergence of the transition matrix. If the perturbation parameter is not small enough, the transition matrix loses information due to the shift of information to the teleportation matrix. We formulate the PageRank problem as a first- and second-order Markov chains perturbation problem. Using numerical experiments, we compare convergence rates for different values of perturbation parameter on different graph structures and investigate the difference in ranks for the two problems.

We propose a statistical method for analyzing key comparisons with two transfer standards measured in two petals. The new approach is based on an extension of the established random effects model. A full Bayesian analysis based on the reference prior is developed and analytic expressions for the results are derived. One benefit of the suggested approach is that it provides a comprehensive assessment of the laboratory biases in terms of their posterior distributions. Another advantage is that it can easily be applied in practice. The approach is illustrated for the CCM.M-K7 key comparison data.

Random-effects meta-analysis has become a well-established tool applied in many areas, for example, when combining the results of several clinical studies on a treatment effect. Typically, the inference aims at the common mean and the amount of heterogeneity. In some applications, the laboratory effects are of interest, for example, when assessing uncertainties quoted by laboratories participating in an interlaboratory comparison in metrology. We consider the Bayesian estimation of the realized random effects in random-effects meta-analysis. Several vague and noninformative priors are examined as well as a proposed novel one. Conditions are established that ensure propriety of the posteriors for the realized random effects. We present extensive simulation results that assess the inference in dependence on the choice of prior as well as mis-specifications in the statistical model. Overall good performance is observed for all priors with the novel prior showing the most promising results. Finally, the uncertainties reported by eleven national metrology institutes and universities for their measurements on the Newtonian constant of gravitation are assessed.

Weighted least-squares estimation is commonly applied in metrology to fit models to measurements that are accompanied with quoted uncertainties. The weights are chosen in dependence on the quoted uncertainties. However, when data and model are inconsistent in view of the quoted uncertainties, this procedure does not yield adequate results.

When it can be assumed that all uncertainties ought to be rescaled by a common factor, weighted least-squares estimation may still be used, provided that a simple correction of the uncertainty obtained for the estimated model is applied. We show that these uncertainties and credible intervals are robust, as they do not rely on the assumption of a Gaussian distribution of the data. Hence, common software for weighted least-squares estimation may still safely be employed in such a case, followed by a simple modification of the uncertainties obtained by that software. We also provide means of checking the assumptions of such an approach.

The Bayesian regression procedure is applied to analyze the CODATA values for the Planck constant published over the past decades in terms of three different models: a constant model, a straight line model and a spline model. Our results indicate that the CODATA values may not have yet stabilized

The Guide to the Expression of Uncertainty in Measurement (GUM) includes formulas that produce an estimate of a scalar output quantity that is a function of several input quantities, and an approximate evaluation of the associated standard uncertainty.

This contribution presents approximate, Bayesian counterparts of those formulas for the case where the output quantity is a parameter of the joint probability distribution of the input quantities, also taking into account any information about the value of the output quantity available prior to measurement expressed in the form of a probability distribution on the set of possible values for the measurand.

The approximate Bayesian estimates and uncertainty evaluations that we present have a long history and illustrious pedigree, and provide sufficiently accurate approximations in many applications, yet are very easy to implement in practice. Differently from exact Bayesian estimates, which involve either (analytical or numerical) integrations, or Markov Chain Monte Carlo sampling, the approximations that we describe involve only numerical optimization and simple algebra. Therefore, they make Bayesian methods widely accessible to metrologists.

We illustrate the application of the proposed techniques in several instances of measurement: isotopic ratio of silver in a commercial silver nitrate; odds of cryptosporidiosis in AIDS patients; height of a manometer column; mass fraction of chromium in a reference material; and potential-difference in a Zener voltage standard.

There are many models for mortality rates. A well-known problem that complicates modeling of human mortality rates is the “accident hump” occurring in early adulthood. Here, two models of mortality rate based on power-exponential functions are presented and compared to a few other models. The models will be fitted to known data of measured death rates from several different countries using numerical techniques for curve-fitting with the nonlinear least-squares method. The properties of the model with respect to forecasting with the Lee–Carter method will be discussed.

In order for governments and demographers to, among other things, design policies and pensionplans, as well as for insurance companies to offer policies that serve general public, having reliable mortality data plays a crucial role. The academic world works actively in developing tools (models and methods) that can, based on collected mortality data, forecast future death rates in the observed population. Obviously, to be able to rely on the predicated data one needs a reliable source of existing mortality data. In the light of the ongoing COVID-19 pandemic, reliability of certain death-case reporting has been questioned. In this thesis, the Benford’s Law is used to evaluate how well countries with authoritarian regimes (Azerbaijan, Belarus),and with democratic regimes (Greece, Serbia, Sweden), report their COVID-19 cases to theworldwide public. Statistical tests such as the Chi-squared test, mean absolute deviation, and the distribution distance were used to obtain the results needed to form our conclusions. During our testing, we found that countries with democratic regimes do conform better to the Benford’s law than the authoritarian ones.

We consider a market model with four correlated factors and two stochastic volatilities, one of which is rapid-changing, while another one is slow-changing in time. An advanced Monte Carlo method based on the theory of cubature in Wiener space is used to find the no-arbitrage price of the European call option in the above model.

Multiscale stochastic volatilities models relax the constant volatility assumption from Black-Scholes option pricing model. Such model can capture the smile and skew of volatilities and therefore describe more accurately the movements of the trading prices. Christoffersen et al. [3] presented a model where the underlying priceis governed by two volatility components, one changing fast and another changing slowly. Chiarella and Ziveyi [2] transformed Christoffersen’s model and computed an approximate formula for pricing American options. They used Duhamel’s principle to derive an integral form solution of the boundary value problem associated to the option price. Using method of characteristics, Fourier and Laplace transforms, they obtained with good accuracy the American options prices. In a previous research of the authors (Canhanga et al. [1]), a particular case of Chiarella and Ziveyi [2] model is used for pricing of European options. The novelty of this earlier work is to present an asymptotic expansion for the option price. The present paper provides experimental and numerical studies on investigating the accuracy of the approximation formulae given by this asymptotic expansion. We present also a procedure for calibrating the parameters produced by our first-order asymptotic approximation formulae. Our approximated option prices will be compared to the approximation obtained by Chiarella and Ziveyi [2].

1. Canhanga B., Malyarenko, A., Ni, Y. and Silvestrov S. Perturbation methods for pricing European options in a model with two stochastic volatilities. 3rd SMTDA Conference Proceedings. 11-14 June 2014, Lisbon Porturgal, C. H. Skiadas (Ed.) 489-500 (2014).

2. Chiarella, C, and Ziveyi, J. American option pricing under two stochastic volatility processes. J. Appl. Math. Comput. 224:283–310 (2013).

3. Christoffersen, P.; Heston, S.; Jacobs, K. The shape and term structure of the index option smirk: why multifactor stochastic volatility models work so well. Manage. Sci. 55 (2) 1914-1932; (2009).

Multiscale stochastic volatilities models relax the constant volatility assumption from Black-Scholes option pricing model. Such models can capture the smile and skew of volatilities and therefore describe more accurately the movements of the trading prices. Christoffersen et al. Manag Sci 55(2):1914–1932 (2009) presented a model where the underlying price is governed by two volatility components, one changing fast and another changing slowly. Chiarella and Ziveyi Appl Math Comput 224:283–310 (2013) transformed Christoffersen’s model and computed an approximate formula for pricing American options. They used Duhamel’s principle to derive an integral form solution of the boundary value problem associated to the option price. Using method of characteristics, Fourier and Laplace transforms, they obtained with good accuracy the American option prices. In a previous research of the authors (Canhanga et al. 2014), a particular case of Chiarella and Ziveyi Appl Math Comput 224:283–310 (2013) model is used for pricing of European options. The novelty of this earlier work is to present an asymptotic expansion for the option price. The present paper provides experimental and numerical studies on investigating the accuracy of the approximation formulae given by this asymptotic expansion. We present also a procedure for calibrating the parameters produced by our first-order asymptotic approximation formulae. Our approximated option prices will be compared to the approximation obtained by Chiarella and Ziveyi Appl Math Comput 224:283–310 (2013).

Interested by the volatility behavior, different models have been developed for option pricing. Starting from constant volatility model which did not succeed on capturing the effects of volatility smiles and skews; stochastic volatility models appearas a response to the weakness of the constant volatility models. Constant elasticity of volatility, Heston, Hull and White, Schöbel-Zhu, Schöbel-Zhu-Hull-Whiteand many others are examples of models where the volatility is itself a random process. Along the chapter we deal with this class of models and we present the techniques of pricing European options. Comparing single factor stochastic volatility models to constant factor volatility models it seems evident that the stochastic volatility models represent nicely the movement of the asset price and its relations with changes in the risk. However, these models fail to explain the large independent fluctuations in the volatility levels and slope. Christoffersen et al. in [4] proposed a model with two-factor stochastic volatilities where the correlation between the underlying asset price and the volatilities varies randomly. In the last section of this chapter we introduce a variation of Chiarella and Ziveyi model, which is a subclass of the model presented in [4] and we use the first order asymptotic expansion methods to determine the price of European options.

The celebrated Black–Scholes model made the assumption of constant volatility but empirical studies on implied volatility and asset dynamics motivated the use of stochastic volatilities. Christoffersen in 2009 showed that multi-factor stochastic volatilities models capture the asset dynamics more realistically. Fouque in 2012 used it to price European options. In 2013 Chiarella and Ziveyi considered Christoffersen's ideas and introduced an asset dynamics where the two volatilities of the Heston type act separately and independently on the asset price, and using Fourier transform for the asset price process and double Laplace transform for the two volatilities processes, solved a pricing problem for American options. This paper considers the Chiarella and Ziveyi model and parameterizes it so that the volatilities revert to the long-run-mean with reversion rates that mimic fast(for example daily) and slow(for example seasonal) random effects. Applying asymptotic expansion method presented by Fouque in 2012, we make an extensive and detailed derivation of the approximation prices for European options. We also present numerical studies on the behavior and accuracy of our first and the second order asymptotic expansion formulas.

Financial models have to reflect the characteristics of markets in which they are developed to be able to predict the future behavior of a financial system. The nature of most trading environments is characterized by uncertainties which are expressed in mathematical models in terms of volatilities. In contrast to the classical Black-Scholes model with constant volatility, our model includes one fast-changing and another slow-changing stochastic volatilities of mean-reversion type. The different changing frequencies of volatilities can be interpreted as the effects of weekends and effects of seasons of the year (summer and winter) on the asset price.

We perform explicitly the transition from the real-world to the risk-neutral probability measure by introducing market prices of risk and applying Girsanov Theorem. To solve the boundary value problem for the partial differential equation that corresponds to the case of a European option, we perform both regular and singular multiscale expansions in fractional powers of the speed of mean-reversion factors. We then construct an approximate solution given by the two-dimensional Black-Scholes model plus some terms that expand the results obtained by Black and Scholes.

After Black-Scholes proposed a model for pricing European Option in 1973, Cox, Ross and Rubinstein in 1979, and Heston in 1993, showed that the constant volatility assumption in the Black-Scholes model was one of the main reasons for the model to be unable to capture some market details. Instead of constant volatilities, they introduced non-constant volatilities to the asset dynamic modeling. In 2009, Christoffersen empirically showed "why multi-factor stochastic volatility models work so well". Four years later, Chiarella and Ziveyi solved the model proposed by Christoffersen. They considered an underlying asset whose price is governed by two factor stochastic volatilities of mean reversion type. Applying Fourier transforms, Laplace transforms and the method of characteristics they presented an approximate formula for pricing American option.The huge calculation involved in the Chiarella and Ziveyi approach motivated us to investigate another approach to compute European option prices on a Christoffersen type model. Using the first and second order asymptotic expansion method we presented a closed form solution for European option, and provided experimental and numerical studies on investigating the accuracy of the approximation formulae given by the first order asymptotic expansion. In the present chapter we will perform experimental and numerical studies for the second order asymptotic expansion and compare the obtained results with results presented by Chiarella and Ziveyi.

A great deal of ecological theory is based on simple Lotka-Volterra-type of unstructured population models in the study of complex population dynamics and communities. The main reason is to obtain important information for predicting their future evolution. In these unstructured models, it is assumed that all individuals in the population are identical, with the same birth and death rates, and consume equally from shared resources in a homogeneous environment. In reality, these assumptions are not biologically true but still forms a basis for modeling population ecology. We apply this paradigm on the grazing system consisting of coupled ordinary differential equations describing the dynamics of forage resource and livestock population in a grassland ecosystem. We do this by investigating the dynamics of the individuals at different life-history stages of juvenile and adult livestock. The mathematical derivation of the model is carried out to show how the physiologically structured population model can be approximated using a three stage-structured population model. Thus the resulting system of ordinary differential equations can be solved to predict density-dependent properties of the population since it provides a somewhat close-to-reality description of the natural and traditional grazing system. This model therefore certainly contains the needed information in the modeling methodology and accommodates the necessary amount of biological details about the population.

In this thesis, we propose a new and simple approach of extending the single-factor Heston stochastic volatility model to a more flexible one in solving option pricing problems.  In this approach, the volatility process for the underlying asset dynamics depends on the time to maturity of the option. As this idea is inspired by the Heath-Jarrow-Morton framework which models the evolution of the full dynamics of forward rate curves for various maturities, we name this approach as the HJM-type stochastic volatility (HJM-SV)  model. We conduct an empirical analysis by calibrating this model to real-market option data for underlying assets including an equity  (ABB stock) and a market index (EURO STOXX 50), for two separated time spans from Jan 2017 to Dec 2017 (before the COVID-19 pandemic) and from Nov 2019 to Nov 2020 (after the start of COVID-19 pandemic). We investigate the optimal way of dividing the set of option maturities into three classes, namely, the short-maturity, middle-maturity, and long-maturity classes. We calibrate our HJM-SV model to the data in the following way, for each class a single-factor Heston stochastic volatility model is calibrated to the corresponding market data. We address the question that how well the new HJM-SV model captures the feature of implied volatility surface given by the market data.

Mälardalen University (Sweden), Riga Technical University (Latvia) and University of Tartu (Estonia) organised courses “Exploring the world of financial engineering" for teachers and students of the above higher education institutions under financial support of the Nordplus Framework mobility project HE-2010_1a-21005. These courses take place in the city of Västerås (Sweden) on May 9–May 13, 2011. In this book, we present the material of the courses’ lectures.

In this paper, we present a comprehensive survey of continuous stochastic volatility models, discussing their historical development and the key stylized facts that have driven the field. Special attention is dedicated to fractional and rough methods: without advocating for either roughness or long memory, we outline the motivation behind them and characterize some landmark models. In addition, we briefly touch on the problem of VIX modeling and recent advances in the SPX-VIX joint calibration puzzle.

This thesis presents advancements in the valuation and modeling of financial derivatives, with a focus on American and Bermudan options. Traditional models such as Black–Scholes assume constant volatility, often leading to inaccurate pricing during periods of high market turbulence. This research addresses these limitations by considering more flexible regime–switching and stochastic volatility models.

The first part of the thesis focuses on the pricing of American options under a Markovian regime–switching model, extending previous works by addressing assumptions of asset returns across economic states. By incorporating the Totally Positive of Order 2 (TP2) property for Transition Probability Matrix (TPM) and Conditional Probability Matrix (CPM), the model ensures probabilistic progression between economic states. Extensive numerical experiments confirm the importance of TPM in maintaining the monotonicity of optimal exercise boundaries.

Secondly, the thesis investigates asymptotic expansions of implied volatility under the Gatheral model. Numerical analysis reveals the accuracy of first and second–order expansions, with a partial calibration method validated using market data from the COVID–19 pandemic.

Next, the thesis introduces a Backward Stochastic Differential Equation (BSDE)–θ scheme for pricing American options under the Heston model, simplifying the computational process and requiring only one parameter for pricing while also deriving schemes for Delta and Vega hedging strategies. Extensive numerical experiments validate the scheme’s accuracy and robustness, especially for in–the–money options.

Finally, the thesis develops an Almost–Exact Simulation (AES) scheme for Bermudan and American option pricing under Heston–type models. The AES scheme ensures non–negative variance and significantly improves simulation accuracy compared to the Euler scheme when the number of steps equals the number of exercise dates. Numerical experiments reveal that the AES scheme offers improvements in accuracy, efficiency, and memory usage, particularly for in–the–money and at–the–money options with minimal time steps.

The Gatheral model is a three factor model with mean-reverting stochastic volatility that reverts to a stochastic long run mean. This chapter reviews previous analytical results on the first and second order implied volatility expansions under this model. Using the Monte Carlo simulation as the benchmark method, numerical studies are conducted to investigate the accuracy and properties of these analytical expansions. The classical Black–Scholes option pricing model assumes that the underlying asset follows a geometric Brownian motion with constant volatility. The chapter discusses partial calibration procedure is proposed and synthetic and real data calibration. If a full calibration is desired, we can use the results from the partial calibration as inputs for the final local optimization over all model parameters. In implementing the calibration procedure, the effect of the Covid-19 pandemic on the model calibration is high.

In this paper, a numerical forward-backward stochastic differential equations (FBSDEs) approach is used for pricing American options under the Heston model. The motivation behind this approach lies in Heston model’s ability to capture significant market features, including stochastic volatility and the correlation between the underlying asset and the volatility. Yet, to the best of our knowledge, there is a lack of dedicated investigation on American option pricing using FBSDE approach under this model. In the present study, the FBSDE representation is obtained and utilized to develop a numerical scheme, which enables the approximation of option prices and hedge ratios. The primary focus lies in approximating American option price as the solution to the backward stochastic differential equation (BSDE) within the FBSDE representation using a θ discretization, which we name as the BSDE-θ Scheme. This BSDE-θ scheme offers computational advantages and flexibility for handling both lower and higher dimensions. Finally, numerical experimental studies are conducted to assess the accuracy and stability of the scheme across various choices of θ. Additionally, the impact of the choice of basis functions in the form of global polynomials on the estimation of conditional expectation functions is investigated. The findings indicate that the choice of θ influences the accuracy of the numerical scheme in particular cases. Moreover, the choice of basis functions under consideration does not exhibit any observable impact on the results.

In this article, a model under which the underlying asset follows a Markov regime-switching process is considered. The underlying economy is partially observable in a form of a signal stochastically related to the actual state of the economy. The American option pricing problem is formulated using a partially observable Markov decision process (POMDP). Through the article, a three-state economy is assumed with a focus on the threshold for the early exercise, hold regions, and its monotonicity. An extensive numerical experimental study is conducted in order to clarify the relationship between the monotonicity of the exercising strategy and the sufficient conditions which are obtained in Jin, Dimitrov, and Ni. In this article, the effect of sufficient conditions is confirmed. It was shown that sufficient conditions are not necessary for the monotonicity of the exercising strategy, and a discussion including milder conditions is presented based on the numerical studies.

In this thesis we study necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes, we describe the class of all possible limit distributions, and give the applications of the results to risk theory and queueing systems.

In paper A, we consider first-rare-event times for semi-Markov processes with a finite set of states, and give a summary of our results concerning necessary and sufficient conditions for weak convergence of first-rare-event times and their actuarial applications.

In paper B, we present in detail results announced in paper A as well as their proofs. We give necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes with a finite set of states in non-triangular-array mode and describe the class of all possible limit distributions in terms of their Laplace transforms.

In paper C, we study the conditions for weak convergence for flows of rare events for semi-Markov processes with a finite set of states in non-triangular array mode. We formulate necessary and sufficient conditions of convergence and describe the class of all possible limit stochastic flows. In the second part of the paper, we apply our results to the asymptotical analysis of non-ruin probabilities for perturbed risk processes.

In paper D, we give necessary and sufficient conditions for the weak convergence of first-rare-event times for semi-Markov processes with a finite set of states in triangular array mode as well as describing the class of all possible limit distributions. The results of paper D extend results obtained in paper B to a general triangular array mode.

In paper E, we give the necessary and sufficient conditions for weak convergence for the flows of rare events for semi-Markov processes with a finite set of states in triangular array case. This paper generalizes results obtained in paper C to a general triangular array mode. In the second part of the paper, we present applications of our results to asymptotical problems of perturbed risk processes and to queueing systems with quick service

In different applications (i.e., information retrieval, filteringor analysis), it is useful to detect similar terms and to provide the possibilityto use them jointly. Clustering of terms is one of the methods whichcan be exploited for this. In our study, we propose to test three methodsdedicated to the clustering of terms (hierarchical ascendant classification,Radius and maximum), to combine them with the semantic distance algorithmsand to compare them through the results they provide whenapplied to terms from the pharmacovigilance area. The comparison indicatesthat the non disjoint clustering (Radius and maximum) outperformthe disjoint clusters by 10 to up to 20 points in all the experiments.