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  • 1.
    Andrejs, Matveevs
    et al.
    Riga Technical University, Latvia.
    Fjodorovs, Jegors
    Riga Technical University, Latvia.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Algorithms of the Copula Fit to the Nonlinear Processes in the Utility Industry2017In: Procedia Computer Science, ISSN 1877-0509, E-ISSN 1877-0509, Vol. 104, p. 572-577Article in journal (Refereed)
    Abstract [en]

    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.

  • 2.
    Betuel, Canhanga
    et al.
    Faculty of Sciences, Dept of Mathematics and Computer Sciences, Eduardo Mondlane University, Mozambique.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ni, Ying
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Rancic, Milica
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Calibration of Multiscale Two-Factor Stochastic Volatility Models: A Second-Order Asymptotic Expansion Approach2018In: / [ed] Christos H Skiadas, ISAST: International Society for the Advancement of Science and Technology , 2018Conference paper (Refereed)
    Abstract [en]

    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.

  • 3.
    Canhanga, Betuel
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Murara, Jean-Paul
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ni, Ying
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Numerical Studies on Asymptotics of European Option under Multiscale Stochastic Volatility2015In: ASMDA 2015 Proceedings: 16th Applied Stochastic Models and Data Analysis International Conference with 4th Demographics 2015 Workshop / [ed] Christos H Skiadas, ISAST: International Society for the Advancement of Science and Technology , 2015, p. 53-66Conference paper (Refereed)
    Abstract [en]

    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).

  • 4.
    Canhanga, Betuel
    et al.
    DMI, Eduardo Mondlane University, Maputo, Mozambique.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Murara, Jean-Paul
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ni, Ying
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Numerical Studies on Asymptotics of European Option Under Multiscale Stochastic Volatility2017In: Methodology and Computing in Applied Probability, ISSN 1387-5841, E-ISSN 1573-7713, Vol. 19, no 4, p. 1075-1087Article in journal (Refereed)
    Abstract [en]

    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).

  • 5.
    Canhanga, Betuel
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Murara, Jean-Paul
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Pricing European Options Under Stochastic Volatilities Models2016In: Engineering Mathematics I: Electromagnetics, Fluid Mechanics, Material Physics and Financial Engineering / [ed] Sergei Silvestrov; Milica Rancic, Springer, 2016, p. 315-338Chapter in book (Refereed)
    Abstract [en]

    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.

  • 6.
    Canhanga, Betuel
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Faculty of Sciences, Department of Mathematics and Computer Sciences, Eduardo Mondlane University, Maputo, Mozambique.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ni, Ying
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Rancic, Milica
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Analytical and Numerical Studies on the Second Order Asymptotic Expansion Method for European Option Pricing under Two-factor Stochastic Volatilities2018In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 47, no 6, p. 1328-1349Article in journal (Refereed)
    Abstract [en]

    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.

  • 7.
    Canhanga, Betuel
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ni, Ying
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Perturbation Methods for Pricing European Options in a Model with Two Stochastic Volatilities2015In: New Trends in Stochastic Modelling and Data Analysis / [ed] Raimondo Manca, Sally McClean, Christos H Skiadas, ISAST , 2015, p. 199-210Chapter in book (Refereed)
    Abstract [en]

    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.

  • 8.
    Canhanga, Betuel
    et al.
    Faculty of Sciences, Department of Mathematics and Computer Sciences, Eduardo Mondlane University, Maputo, Mozambique.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ni, Ying
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Second Order Asymptotic Expansion for Pricing European Options in a Model with Two Stochastic Volatilities2015In: ASMDA 2015 Proceedings: 16th Applied Stochastic Models and Data Analysis International Conference with 4th Demographics 2015 Workshop, 30 June – 4 July 2015 University of Piraeus, Greece / [ed] C. H. Skiadas, ISAST: International Society for the Advancement of Science and Technology , 2015, p. 37-52Conference paper (Refereed)
    Abstract [en]

    Asset price processes with stochastic volatilities have been actively used by researchers in financial mathematics for valuing derivative securities. This type of models allows characterizing the uncertainties in the asset price process in financial markets. In a recent paper Chiarella and Ziveyi analyzed a model with two stochastic volatilities of mean reversion type with one variable changing fast and the other changing slowly. They used method of characteristics to solve the obtained partial differential equation and determine the price of an American option. Fouque et al presented also a similar model in which the volatility of the underlying asset is governed by two diffusion processes which are not of mean reversion type. They developed a first-order asymptotic expansion for the European option price via a perturbation method.

    In this chapter we consider the model given in Chiarella and Ziveyi. Instead of pricing American options we price European options by generalizing the techniques presented in Fouque et al to a more complex model with mean reverting stochastic volatility factors. We analyse both regular and singular perturbations to obtain an asymptotic expansion up to second order which can serve as an approximation for the price of non-path-dependent European options. Similar work is done in authors earlier work Canhanga et al where a first-order asymptotic expansion has been developed. Involving the second order terms has the advantage of capturing more accurately the effects of volatility smile and skew on the option pricing. Analytical approximation formula for pricing European Option is presented.

  • 9.
    Canhanga, Betuel
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ni, Ying
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Rancic, Milica
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Numerical Methods on European Options Second Order Asymptotic Expansions for Multiscale Stochastic Volatility2017In: INCPAA 2016 Proceedings: 11th International Conference on Mathematical Problems in Engineering, Aerospace, and Sciences, ICNPAA 2016, La Rochelle, France, 4 - 8 July 2016. / [ed] S. Sivasundaram, 2017, Vol. 1798, p. 020035-1-020035-10, article id 020035Conference paper (Refereed)
    Abstract [en]

    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.

  • 10.
    Carkovs, Jevgenijs
    et al.
    Riga Technical University.
    Malyarenko, AnatoliyMälardalen University, School of Education, Culture and Communication.Pärna, KalevUniversity of Tartu.
    Exploring the world of financial engineering2011Collection (editor) (Other academic)
    Abstract [en]

    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.

  • 11.
    Leonenko, Nikolai
    et al.
    Cardiff University, UK.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Matérn Class Tensor-Valued Random Fields and Beyond2017In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 168, p. 1276-1301Article in journal (Refereed)
    Abstract [en]

    We construct classes of homogeneous random fields on a three-dimensional Euclidean space that take values in linear spaces of tensors of a fixed rank and are isotropic with respect to a fixed orthogonal representation of the group of 3 × 3 orthogonal matrices.The constructed classes depend on finitely many isotropic spectral densities. We say that such a field belongs to either the Matérn or the dual Matérn class if all of the above densities are Matérn or dual Matérn. Several examples are considered.

  • 12.
    Ma, Chunsheng
    et al.
    Wichita State University, USA.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Time-Varying Isotropic Vector Random Fields on Compact Two-Point Homogeneous SpacesIn: Journal of theoretical probability, ISSN 0894-9840, E-ISSN 1572-9230Article in journal (Refereed)
    Abstract [en]

    A general form of the covariance matrix function is derived in this paper for a vector random field that is isotropic and mean square continuous on a compact connected two-point homogeneous space and stationary on a temporal domain. A series representation is presented for such a vector random field which involves Jacobi polynomials and the distance defined on the compact two-point homogeneous space.

  • 13.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication.
    A family of series representations of the multiparameter fractional Brownian motion2011In: SEMINAR ON STOCHASTIC ANALYSIS, RANDOM FIELDS AND APPLICATIONS VI / [ed] Dalang, RC; Dozzi, M; Russo, F, Birkhäuser , 2011, p. 209-226Conference paper (Refereed)
    Abstract [en]

    We derive a family of series representations of the multiparameter fractional Brownian motion in the centred ball of radius R in the N-dimensional space R^N. Some known examples of series representations are shown to be the members of the family under consideration.

  • 14.
    Malyarenko, Anatoliy
    Mälardalen University, Department of Mathematics and Physics.
    A series expansion of a certain class of isotropic Gaussian random fields with homogeneous increments2004Report (Other academic)
  • 15.
    Malyarenko, Anatoliy
    Mälardalen University, Department of Mathematics and Physics.
    Abelian and Tauberian theorems for random fields on two-point homogeneous spaces2003Report (Other scientific)
    Abstract [en]

    We consider centred mean-square continuous random fields for which the incremental variance between two points depends only on the distance between these points. The relations between the asymptotic behaviour of the incremental variance near zero and the asymptotic behaviour of the spectral measure of the field near infinity are investigated. We prove several Abelian and Tauberian theorems in terms of slowly varying functions.

  • 16.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication.
    An optimal series expansion of the multiparameter fractional Brownian motion2008In: Journal of theoretical probability, ISSN 0894-9840, E-ISSN 1572-9230, Vol. 21, no 2, p. 459-475Article in journal (Refereed)
    Abstract [en]

    We derive a series expansion for the multiparameter fractional Brownian motion. The derived expansion is proven to be rate optimal.

  • 17.
    Malyarenko, Anatoliy
    Mälardalen University, Department of Mathematics and Physics.
    Functional limit theorems for multiparameter fractional Brownian motion2004Report (Other academic)
    Abstract [en]

    We prove a general functional limit theorem for mul-tiparameter fractional Brownian motion. The functional law ofthe iterated logarithm, functional L ́evy’s modulus of continuityand many other results are its particular cases. Applications toapproximation theory are discussed.

  • 18.
    Malyarenko, Anatoliy
    Mälardalen University, Department of Mathematics and Physics.
    Functional limit theorems for multiparameter fractional Brownian motion2004In: Proceedings of the 6th World Congress of the Bernoulli Society for Mathematical Statistics and Probability, Barcelona, July 26--31 2004, 2004, p. 142-xxxConference paper (Refereed)
  • 19.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication.
    Functional limit theorems for multiparameter fractional Brownian motion2006In: Journal of theoretical probability, ISSN 0894-9840, E-ISSN 1572-9230, Vol. 19, no 2, p. 263-288Article in journal (Refereed)
    Abstract [en]

    We prove a general functional limit theorem for multiparameterfractional Brownian motion. The functional law of the iteratedlogarithm, functional Lévy's modulus of continuity and manyother results are its particular cases. Applications toapproximation theory are discussed.

  • 20.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication.
    Invariant random fields in vector bundles and application to cosmology2011In: Annales de l’Institut Henri Poincaré - Probabilités et Statistiques, ISSN 0246-0203, Vol. 47, no 4, p. 1068-1095Article in journal (Refereed)
    Abstract [en]

    We develop the theory of invariant random fields in vector bundles. The spectral decomposition of an invariant random field in a homogeneous vector bundle generated by an induced representation of a compact connected Lie group G is btained. We discuss an application to the theory of relic radiation, where G = SO(3). A theorem about equivalence of two different groups of assumptions in cosmological theories is proved.

  • 21.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication.
    Invariant random fields in vector bundles and application to cosmologyManuscript (preprint) (Other academic)
    Abstract [en]

    We develop the theory of invariant random fields in vector bundles. The spectral decomposition of an invariant random field in a homogeneous vector bundle generated by an induced representation of a compact connected Lie group $G$ is obtained. We discuss an application to the theory of cosmic microwave background, where $G=SO(3)$. A theorem about equivalence of two different groups of assumptions in cosmological theories is proved.

  • 22.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Invariant random fields on spaces with a group action2013Book (Refereed)
    Abstract [en]

    The author describes the current state of the art in the theory of invariant random fields. This theory is based on several different areas of mathematics, including probability theory, differential geometry, harmonic analysis, and special functions. The present volume unifies many results scattered throughout the mathematical, physical, and engineering literature, as well as it introduces new results from this area first proved by the author. The book also presents many practical applications, in particular in such highly interesting areas as approximation theory, cosmology and earthquake engineering. It is intended for researchers and specialists working in the fields of stochastic processes, statistics, functional analysis, astronomy, and engineering. 

  • 23.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication.
    Lectures on cubature methods in financial engineering2011In: Exploring the world of financíal engineering / [ed] Jeegenijs Carkovs, Anatoliy Malyarenko, Kalev Pärna, Västerås: Mälardalen University , 2011, p. 48-65Chapter in book (Other academic)
    Abstract [en]

    We present a pedagogical introduction into cubature methods on Wiener space and their use in financial engineering. Some important parts of mathematics which are often omitted in study plans, are described in details. These includes the Riemann--Stiltjes integral, tensor products, and elements of Lie theory.

  • 24.
    Malyarenko, Anatoliy
    Mälardalen University, Department of Mathematics and Physics.
    Moduli of continuity and fast points for Gaussian isotropic random fields2002Report (Other scientific)
    Abstract [en]

    We consider a class of Gaussian isotropic random fields related to multi-parameter fractional Brownian motion. We calculate both the local and global moduli of continuity as well as the Hausdorff and packing dimensions of the exceptional random sets of fast points for that fields.

  • 25.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Spectral expansions of cosmological fields2015In: Journal of Statistical Science and Application, ISSN 2328-224X, Vol. 3, no 11-12, p. 175-193Article in journal (Refereed)
    Abstract [en]

    We give a review of the theory of random fields defined on the observable part of the Universe that satisfy the cosmological principle, i.e.,invariant with respect to the 6-dimensional group G of theisometries of the time slice of theFriedmann-Lemaitre-Robertson-Walker standard chart. Our new results include proof of spectral expansions of scalar and spin weighted G-invariant cosmological fields in open, flat, and closed cosmological models.

  • 26.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Spectral expansions of random sections of homogeneous vector bundles2017In: Teoriya Imovirnostei ta Matematychna Statystyka (THEORY OF PROBABILITY AND MATHEMATICAL STATISTICS ), ISSN 0868-6904, Vol. 97, p. 142-156Article in journal (Refereed)
    Abstract [en]

    Tiny fluctuations of the Cosmic Microwave Background as well as various observable quantities obtained by spin raising and spin lowering of the effective gravitational lensing potential of distant galaxies and galaxy clusters, are described mathematically as isotropic random sections of homogeneousspin and tensor bundles. We consider the three existing approaches to rigourous constructing of the above objects, emphasising an approach based on the theory of induced group representations. Both orthogonal and unitary representations are treated in a unified manner. Several examples from astrophysics are included.

  • 27.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Spectral expansions of random sections of homogeneous vector bundles2018In: Theory of Probability and Mathematical Statistics, ISSN 0094-9000, Vol. 97, p. 151-165Article in journal (Refereed)
    Abstract [en]

    Tiny fluctuations of the Cosmic Microwave Background as well as various observable quantities obtained by spin raising and spin lowering of the effective gravitational lensing potential of distant galaxies and galaxy clusters are described mathematically as isotropic random sections of homogeneous spin and tensor bundles. We consider the three existing approaches to rigourous construction of the above objects, emphasising an approach based on the theory of induced group representations. Both orthogonal and unitary representations are treated in a unified manner. Several examples from astrophysics are included.

  • 28.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Spectral expansions of tensor-valued random fields2017In: AIP Conference Proceedings, Volume 1798, American Institute of Physics (AIP), 2017, Vol. 1798, p. 1-10, article id 020095Conference paper (Refereed)
    Abstract [en]

    In this paper, we review the theory of random fields that are defined on the space domain ℝ3, take values in a real finite-dimensional linear space V that consists of tensors of a fixed rank, and are homogeneous and isotropic with respect to an orthogonal representation of a closed subgroup G of the group O(3). A historical introduction, the statement of the problem, some current results, and a sketch of proofs are included.

  • 29.
    Malyarenko, Anatoliy
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Canhanga, Bethuel
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ying, Ni
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Rancic, Milica
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Option pricing and model calibration under multifactor stochastic volatility and stochastic interest rate - an asymptotic expansion approach2017In: Proceedings ASMDA2017 / [ed] Skiadas, Christos H., ISAST: International Society for the Advancement of Science and Technology , 2017, p. 219-231-Conference paper (Refereed)
    Abstract [en]

    Among other limitations, the celebrated Black--Scholes option pricingmodel assumes constant volatility and constant interest rates, which is not supportedby empirical studies on for example implied volatility surfaces. Studiesby many researchers such as Heston in 1993, Christoffersen in 2009, Fouque in2012, Chiarella--Ziveyi in 2013, and the authors' previous work removed the constantvolatility assumption from the Black--Scholes model by introducing one ortwo stochastic volatility factors with constant interest rate. In the present paperwe follow this line but generalize the model by considering also stochasticinterest rate. More specifically, the underlying asset process is governed by amean-reverting interest rate process in addition to two mean-reverting stochasticvolatility processes of fast and slow mean-reverting rates respectively. The focusis to derive an approximating formula for pricing the European option using adouble asymptotic expansion method.

  • 30.
    Malyarenko, Anatoliy
    et al.
    Mälardalen University, School of Education, Culture and Communication.
    Ma, Chunsheng
    Wichita State University.
    Time-Varying Isotropic Vector Random Fields on Compact Two-Point Homogeneous SpacesManuscript (preprint) (Other academic)
    Abstract [en]

    A general form of the covariance matrix function is derived in this paper for a vector random field that is isotropic and mean square continuous on acompact connected two-point homogeneous space and stationary on a temporal domain. A series representation is presented for such a vector random field, which involve Jacobi polynomials and the distance defined on the  compact two-point homogeneous space.

  • 31.
    Malyarenko, Anatoliy
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ni, Ying
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Canhanga, Betuel
    Faculty of Sciences, Department of Mathematics and Computer Sciences,Eduardo Mondlane University, Box 257, Maputo, Mozambique.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Advanced Monte Carlo pricing of European options in a market model with two stochastic volatilities2018In: Proceedings : 5th Stochastic Modeling Techniques and Data Analysis International Conference withDemographics Workshop (SMTDA2018) / [ed] Christos H Skiadas, ISAST: International Society for the Advancement of Science and Technology, 2018, p. 409-422Conference paper (Other academic)
    Abstract [en]

    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 methods 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.

  • 32.
    Malyarenko, Anatoliy
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ostoja-Starzewski, M.
    University of Illinois at Urbana-Champaign, Urbana, United States.
    Random fields related to the symmetry classes of second-order symmetric tensors2018In: Stochastic Processes and Applications: SPAS2017, Västerås and Stockholm, Sweden, October 4-6, 2017 / [ed] Sergei Silvestrov, Anatoliy Malyarenko, Milica Rančić, Springer, 2018, Vol. 271, p. 173-185Chapter in book (Refereed)
    Abstract [en]

    Under the change of basis in the three-dimensional space by means of an orthogonal matrix g, a matrix A of a linear operator is transformed as A → gAg-1 Mathematically, the stationary subgroup of a symmetric matrix under the above action can be either (Formula Presented), when all three eigenvalues of A are different, or (Formula Presented), when two of them are equal, or O(3), when all three eigenvalues are equal. Physically, one typical application relates to dependent quantities like a second-order symmetric stress (or strain) tensor. Another physical setting is that of dependent fields, such as conductivity with such three cases is the conductivity (or, similarly, permittivity, or anti-plane elasticity) second-rank tensor, which can be either orthotropic, transversely isotropic, or isotropic. For each of the above symmetry classes, we consider a homogeneous random field taking values in the fixed point set of the class that is invariant with respect to the natural representation of a certain closed subgroup of the orthogonal group. Such fields may model stochastic heat conduction, electric permittivity, etc. We find the spectral expansions of the introduced random fields.

  • 33.
    Malyarenko, Anatoliy
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ostoja-Starzewski, Martin
    University of Illinois at Urbana-Champaign, US.
    A Random Field Formulation of Hooke’s Law in All Elasticity Classes2017In: Journal of elasticity, ISSN 0374-3535, E-ISSN 1573-2681, Vol. 127, no 2, p. 269-302Article in journal (Refereed)
    Abstract [en]

    For each of the 8 symmetry classes of elastic materials, we consider a homogeneousrandom field taking values in the fixed point set V of the corresponding class, that is isotropic with respect to the natural orthogonal representation of a group lying between the isotropy group of the class and its normaliser. We find the general form of the correlation tensors of orders 1 and 2 of such a field, and the field’s spectral expansion.

  • 34.
    Malyarenko, Anatoliy
    et al.
    Mälardalen University, School of Education, Culture and Communication.
    Ostoja-Starzewski, Martin
    University of Illinois at Urbana-Champaign, USA.
    A random field formulation of Hooke’s law in all elasticity classesManuscript (preprint) (Other academic)
    Abstract [en]

    For each of the 8 isotropy classes of elastic materials, we consider a homogeneous random field taking values in the fixed point set V of the corresponding class, that is isotropic with respect to the natural orthogonal representation of a group lying between the isotropy group of the class and its normaliser. We find the general form of the correlation tensors of orders 1 and 2 of such a field, and the field's spectral expansion.

  • 35.
    Malyarenko, Anatoliy
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ostoja-Starzewski, Martin
    University of Illinois at Urbana-Champaign, USA.
    Fractal planetary rings: energy inequalities and random field model2017In: International Journal of Modern Physics B, ISSN 0217-9792, Vol. 31, no 30, article id 1750236Article in journal (Refereed)
    Abstract [en]

    This study is motivated by a recent observation, based on photographs from the Cassini mission, that Saturn’s rings have a fractal structure in radial direction. Accordingly, two questions are considered: (1) What Newtonian mechanics argument in support of such a fractal structure of planetary rings is possible? (2) What kinematics model of such fractal rings can be formulated? Both challenges are based on taking planetary rings’ spatial structure as being statistically stationary in time and statistically isotropic in space, but statistically nonstationary in space. An answer to the first challenge is given through an energy analysis of circular rings having a self-generated, noninteger-dimensional mass distribution [V. E. Tarasov, Int. J. Mod Phys. B 19, 4103 (2005)]. The second issue is approached by taking the random field of angular velocity vector of a rotating particle of the ring as a random section of a special vector bundle. Using the theory of group representations, we prove that such a field is completely determined by a sequence of continuous positive-definite matrix-valued functions defined on the Cartesian square F^2 of the radial cross-section F, where F is a fat fractal.

  • 36.
    Malyarenko, Anatoliy
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ostoja-Starzewski, Martin
    University of Illinois at Urbana-Champaign, Champaign, United States .
    Spectral Expansion of Three-Dimensional Elasticity Tensor Random Fields2016In: Engineering Mathematics I: Electromagnetics, Fluid Mechanics, Material Physics and Financial Engineering / [ed] Sergei Silvestrov; Milica Rancic, Springer, 2016, p. 281-300Chapter in book (Refereed)
    Abstract [en]

    We consider a random field model of the 21-dimensional elasticity tensor. Representation theory is used to obtain the spectral expansion of the model in terms of stochastic integrals with respect to random measures.

  • 37.
    Malyarenko, Anatoliy
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ostoja-Starzewski, Martin
    University of Illinois at Urbana-Champaign.
    Spectral expansions of homogeneous and isotropic tensor-valued random fields2016In: Zeitschrift für angewandte Matematik und Physik ZAMP, ISSN 1420-9039, Vol. 67, no 3, article id 59Article in journal (Refereed)
    Abstract [en]

    We establish spectral expansions of tensor-valued homogeneous and isotropic random fields in terms of stochastic integrals with respect to orthogonal scattered random measures previously known only for the case of tensor rank 0. The fields under consideration take values in the 3-dimensional Euclidean space E3 and in the space S2(E3) of symmetric rank 2 tensors over E3. We find a link between the theory of random fields and the theory of finite-dimensional convex compact sets. These random fields furnish stepping-stone for models of rank 1 and rank 2 tensor-valued fields in continuum physics, such as displacement, velocity, stress, strain, providing appropriate conditions (such as the governing equation or positive-definiteness) are imposed.

  • 38.
    Malyarenko, Anatoliy
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ostoja-Starzewski, Martin
    University of Illinois at Urbana-Champaign.
    Statistically isotropic tensor random fields: correlation structures2014In: Mathematics and Mechanics of Complex Systems, ISSN 2325-3444, Vol. 2, no 2, p. 209-231Article in journal (Refereed)
    Abstract [en]

    Let V be a real finite-dimensional vector space. We introduce some physical problems that may be described by V-valued homogeneous and isotropic random fields on R 3 . We propose a general method for calculation of expectations and two-point correlation functions of such fields. Our results are equivalent to classical results by Robertson, when V = R 3 , and those by Lomakin, when V is the space of symmetric second-rank tensors over R 3 . Our solution involves an analogue of the classical Clebsch–Gordan coefficients.

  • 39.
    Malyarenko, Anatoliy
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ostoja-Starzewski, Martin
    University of Illinois at Urbana-Champaign.
    Tensor Random Fields in Continuum Mechanics2018In: Encyclopedia of Continuum Mechanics / [ed] Altenbach, Holm and Öchsner, Andreas, Berlin, Heidelberg: Springer Berlin/Heidelberg, 2018, p. 1-9Chapter in book (Refereed)
  • 40.
    Malyarenko, Anatoliy
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ostoja-Starzewski, Martin
    University of Illinois at Urbana-Champaign, USA.
    Tensor-Valued Random Fields for Continuum Physics2018Book (Refereed)
  • 41.
    Malyarenko, Anatoliy
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ostoja-Starzewski, Martin
    University of Illinois at Urbana-Champaign, USA.
    Tensor-Valued Random Fields in Continuum Physics2016In: Materials with internal structure: Multiscale and Multifield Modeling and Simulation / [ed] P. Trovalusci, Berlin/Heidelberg: Springer Science+Business Media B.V., 2016, p. 75-88Chapter in book (Refereed)
    Abstract [en]

    This article reports progress on homogeneous isotropic tensor random fields (TRFs) for continuum mechanics. The basic thrust is on determinin most general representations of the correlation functions as well as their spectral expansions. Once this is accomplished, the second step is finding the restrictionsdictated by a particular physical application. Thus, in the case of fields of material properties (like conductivity and stiffness), the restriction resides in the positive-definiteness, whereby a connection to experiments and/or computational micromechanics can be established. On the other hand, in the case of fields of dependent properties (e.g., stress, strain and displacement), restrictions are due to the respective field equations.

  • 42.
    Malyarenko, Anatoliy
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ostoja-Starzewski, Martin
    University of Illinois in Urbana-Champaign, USA.
    The spectral expansion of the elasticity random field2014In: AIP Conference Proceedings 1637 / [ed] S. Sivasundaram, 2014, p. 647-655Conference paper (Refereed)
    Abstract [en]

    We consider a deformable body that occupies a region D in the plane. In our model, the body's elasticity tensor H (x) is the restriction to D of a second-order mean-square continuous random field. Under translation, the expected value and the correlation tensor of the field H (x) do not change. Under action of an arbitrary element k of the orthogonal group O (2), they transform according to the reducible orthogonal representation k bar right arrow S-2 (S-2 (k)) of the above group. We find the spectral expansion of the correlation tensor R (x) of the elasticity field as well as the expansion of the field itself in terms of stochastic integrals with respect to a family of orthogonal scattered random measures.

  • 43.
    Malyarenko, Anatoliy
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ostoja-Starzewski, Martin
    University of Illinois at Urbana-Champaign.
    Towards stochastic continuum damage mechanics2019In: International Journal of Solids and Structures, ISSN 0020-7683, E-ISSN 1879-2146Article in journal (Refereed)
    Abstract [en]

    In classical continuum damage mechanics, the distribution of cracks over differently oriented planes is an even deterministic function defined on the unit sphere. The coefficients of its Fourier expansion are completely symmetric and completely traceless tensors of even rank, the so-called fabric or damage tensors. We propose a stochastic generalisation of the above described mathematical model, where damage tensors are mean-square continuous wide-sense homogeneous and isotropic random fields.

  • 44.
    Malyarenko, Anatoliy
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Röman, J.
    Swedbank, Stockholm, Sweden.
    Schyberg, Oskar
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Sensitivity analysis of catastrophe bond price under the hull-white interest rate model2016In: Springer Proceedings in Mathematics & statistics, ISSN 2194-1017, E-ISSN 2194-1009, Vol. 178, p. 301-314Article in journal (Refereed)
    Abstract [en]

    We consider a model, where the natural risk index is described by the Merton jump-diffusion while the risk-free interest rate is governed by theHull-White stochastic differential equation. We price a catastrophe bond with payoff depending on finitely many values of the underlying index. The sensitivities of the bond price with respect to the initial condition, volatility of the diffusion component, and jump amplitude, are calculated using the Malliavin calculus approach.

  • 45.
    Malyarenko, Anatoliy
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Röman, Jan
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Swedbank, Sweden.
    Schyberg, Oskar
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Sensitivity Analysis of Catastrophe Bond Priceunder the Hull–White Interest Rate Model2016In: Engineering Mathematics I: Electromagnetics, Fluid Mechanics, Material Physics and Financial Engineering / [ed] Sergei Silvestrov; Milica Rancic, Springer, 2016Chapter in book (Refereed)
    Abstract [en]

    We consider a model, where the natural risk index is described by the Merton jump-diffusion while the risk-free interest rate is governed by the Hull–White stochastic differential equation. We price a catastrophe bond with payoff depending on finitely many values of the underlying index. The sensitivities of the bond price with respect to the initial condition, volatility of the diffusion component, and jump amplitude, are calculated using the Malliavin calculus approach.

  • 46.
    Malyarenko, Anatoliy
    et al.
    Mälardalen University, Department of Mathematics and Physics.
    Silvestrov, Dmitrii S.
    Mälardalen University, Department of Mathematics and Physics.
    Silvestrova, Evelina
    Mälardalen University, Department of Mathematics and Physics.
    Stochastic modelling of insurance business with dynamical control of investments2004In: 6th World Congress of Bernoulli Society for Mathematical Statistics and Probability, Barcelona, July 26--31 2004, 2004, p. page 181-Conference paper (Other academic)
  • 47.
    Mishchenko, Kateryna
    et al.
    Mälardalen University, Department of Mathematics and Physics.
    Mishchenko, Volodymyr
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication.
    Adapted Downhill Simplex Method for Pricing Convertible Bonds2007In: Theory of Stochastic Processes, ISSN 0321-3900, Vol. 13, no 4, p. 130-147Article in journal (Refereed)
    Abstract [en]

    The paper is devoted to modeling optimal exercise strategies of thebehavior of investors and issuers working with convertible bonds.This implies solution of the problems of stock price modeling, payoffcomputation and minimax optimization.Stock prices (underlying asset) were modeled under the assumptionof the geometric Brownian motion of their values. The Monte Carlomethod was used for calculating the real payoff which is the objectivefunction. The minimax optimization problem was solved using thederivative-free Downhill Simplex method.The performed numerical experiments allowed to formulate recommendationsfor the choice of appropriate size of the initial simplex inthe Downhill Simplex Method, the number of generated trajectoriesof underlying asset, the size of the problem and initial trajectories ofthe behavior of investors and issuers.

  • 48.
    Murara, Jean-Paul
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Modelling electricity price series using regime-switching GARCH model2015In: ASMDA 2015 Proceedings: 16th Applied Stochastic Models and Data Analysis International Conference with 4th Demographics 2015 Workshop / [ed] Christos H. Skiadas, ISAST: International Society for the Advancement of Science and Technology , 2015, p. 713-725Conference paper (Refereed)
    Abstract [en]

    This paper implements and analyzes the Regime-Switching GARCH modelusing real NordPool Electricity spot data. We allow the model parameters to switch between a regular regime and a non-regular regime, which is justied by a so-called structural break behaviour of electricity price series. In splitting the two regimes we consider three criteria, namely the intercountry price difference criterion, the capacity/flow difference criterion and the spikes-in-Finland criterion. We study the correlation relationships among these criteria using the mean-square contingency coefficient and the co-occurrence measure. We also estimate our model parameters and present empirical validity of the model.

  • 49.
    Ni, Ying
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Canhanga, Betuel
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Approximation Methods of European Option Pricing in Multiscale Stochastic Volatility Model2017In: INCPAA 2016 Proceedings: 11th International Conference on Mathematical Problems in Engineering, Aerospace, and Sciences, ICNPAA 2016, La Rochelle, France, 4 - 8 July 2016. / [ed] S. Sivasundaram, American Institute of Physics (AIP), 2017, Vol. 1798, p. 020112-1-020112-10, article id 020112Conference paper (Refereed)
    Abstract [en]

    In the classical Black-Scholes model for financial option pricing, the asset price follows a geometric Brownian motion with constant volatility. Empirical findings such as volatility smile/skew, fat-tailed asset return distributions have suggested that the constant volatility assumption might not be realistic. A general stochastic volatility model, e.g. Heston model, GARCH model  and SABR volatility model , in which the variance/volatility itself follows typically a mean-reverting stochastic process, has shown to be superior in terms of capturing the empirical facts. However in order to capture more features of the volatility smile a two-factor, of double Heston type, stochastic volatility model is more useful as shown by Christoffersen, Heston and Jacobs.  We consider one specific type of such two-factor volatility models in which the volatility has multiscale mean-reversion rates. Our model contains two mean-reverting volatility processes with a fast and a slow reverting rate respectively. We consider the European option pricing problem under one type of the multiscale stochastic volatility model where the two volatility processes act as independent factors in the asset price process.  The novelty in this chapter is an approximating analytical solution using asymptotic expansion method which extends the authors earlier research in Canhanga et al. In addition we propose a numerical approximating solution using Monte-Carlo simulation. For completeness and for comparison we also implement the semi-analytical solution by Chiarella and Ziveyi using method of characteristics, Fourier and bivariate Laplace transforms.

  • 50.
    Ni, Ying
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Engström, Christopher
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Wallin, Fredrik
    Mälardalen University, School of Business, Society and Engineering, Future Energy Center.
    Building-type classification based on measurements of energy consumption data2015In: New Trends in Stochastic Modeling and Data Analysis / [ed] Raimondo Manca, Sally McClean, Christos H SkiadasISAST 2015, ISAST: International Society for the Advancement of Science and Technology , 2015, p. 287-298Chapter in book (Refereed)
    Abstract [en]

    In this paper we apply data-mining techniques to a classication problemon actual electricity consumption data from 350 Swedish households. Morespecically we use measurements of hourly electricity consumption during one monthand t classication models to the given data. The goal is to classify and later predict whether the building type of a specic household is an apartmentor a detached house. This classication/prediction problem becomes important ifone has a consumption time series for a household with unknown building type. Tocharacterise each household, we compute from the data some selected statistical attributesand also the load prole throughout the day for that household. The most important task here is to select a good representative set of feature variables, whichis solved by ranking the variable importance using technique of random forest. Wethen classify the data using classication tree method and linear discriminant analysis.The predictive power of the chosen classication models is plausible.

12 1 - 50 of 67
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