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• 1.
Faculty of Sciences, Dept of Mathematics and Computer Sciences, Eduardo Mondlane University, Mozambique.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. 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)

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.

• 2.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. 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)

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).

• 3.
DMI, Eduardo Mondlane University, Maputo, Mozambique.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. 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)

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).

• 4.
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.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. 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)

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.

• 5.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. 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)

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.

• 6.
Faculty of Sciences, Department of Mathematics and Computer Sciences, Eduardo Mondlane University, Maputo, Mozambique.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. 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)

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.

• 7.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. 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)

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.

• 8.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. University of Nairobi, Kenya. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. University of Nairobi, Kenya.
Construction of moment-matching multinomial lattices using Vandermonde matrices and Gröbner bases2017In: AIP Conference Proceedings / [ed] Sivasundaram, S, American Institute of Physics (AIP), 2017, Vol. 1798, p. 020094-1-020094-7, article id 020094Conference paper (Refereed)

In order to describe and analyze the quantitative behavior of stochastic processes, such as the process followed by a financial asset, various discretization methods are used. One such set of methods are lattice models where a time interval is divided into equal time steps and the rate of change for the process is restricted to a particular set of values in each time step. The well-known binomial- and trinomial models are the most commonly used in applications, although several kinds of higher order models have also been examined. Here we will examine various ways of designing higher order lattice schemes with different node placements in order to guarantee moment-matching with the process.

• 9.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
University of Nairobi, Nairobi, Kenya. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. University of Nairobi, Nairobi, Kenya.
Moment-matching multinomial lattices using Vandermonde matrices for option pricing2016In: Stochastic and Data Analysis Methods and Applications in Statistics and Demography: Book 2 / [ed] James R. Bozeman, Teresa Oliveira and Christos H. Skiadas, ISAST , 2016, Vol. 2, p. 15-29Conference paper (Refereed)

Lattice models are discretization methods that divide the life of a financial option into time steps of equal length and model the underlying asset movement at each time step. A financial option of American or European style can be evaluated conveniently via backward induction using a lattice model. The most common lattice models are the well-known binomial- and trinomial lattice models, although severalkinds of higher order models have also been examined in the literature. In the presentpaper we present an explicit scheme for creating a lattice model of arbitrary order and use the Vandermonde matrix to determine suitable parameters. Some selected models created using this scheme are examined with regard to their suitability for option pricing

• 10.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. 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)

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.

• 11.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Faculty of Sciences, Department of Mathematics and Computer Sciences,Eduardo Mondlane University, Box 257, Maputo, Mozambique. 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 (Refereed)

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.

• 12.
Mälardalen University, School of Education, Culture and Communication.
Analytical and Numerical Studies of Perturbed Renewal Equations with Multivariate Non-Polynomial Perturbations2010In: Journal of Applied Quantitative Methods, ISSN 1842-4562, Vol. 5, no 3, p. 411-428Article in journal (Refereed)

The object of study is a model of nonlinearly perturbed continuous-time renewal equation with multivariate non-polynomial perturbations. The characteristics of the distribution generating the renewal equation are assumed to have expansions in a perturbation parameter with respect to a non-polynomial asymptotic. Exponential asymptotics for such a model as well as their applications are given. Numerical studies are performed to gain insights into the asymptotical results.

• 13.
Mälardalen University, School of Education, Culture and Communication.
Asymptotically Improper Perturbed Renewal Equations: Asymptotic Results and Their Applications2011Report (Other academic)

We consider a family of asymptotically improper perturbed renewal equations where the characteristics of the distribution functions generating the perturbed renewal equations are perturbed in a particular way. More specifically, those characteristics are nonlinear functions of the perturbation parameter such that they can be expanded into asymptotic expansions of a non-polynomial type with respect to the perturbation parameter. We give asymptotic results, namely the exponential asymptotic expansions, for the solutions of the perturbed renewal equations. An application to perturbed storage processes is also presented.

• 14.
Mälardalen University, School of Education, Culture and Communication.
Exponential asymptotic expansions and Monte Carlo studies for ruin probabilitiesIn: Journal of Statistical Planning and Inference, ISSN 0378-3758, E-ISSN 1873-1171Article in journal (Refereed)

This paper presents the exponential asymptotic expansions for the ruin probability in a special model of non-linearly perturbed risk processes with non-polynomial perturbations. Monte Carlo studies are performed to investigate the accuracy and other properties of the asymptotic formulas.

• 15.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Exponential asymptotical expansions for ruin probability in a classical risk process with non-polynomial perturbations2014In: Modern Problems in Insurance Mathematics / [ed] Silvestrov, D., Martin-Löf, A. (eds), Springer, 2014, p. 69-94Chapter in book (Refereed)

In this paper we investigate the asymptotical behaviour of ruin probability in a classical compound Poisson risk process associated with perturbations in the claim size distributions and/or other parameters of the risk process. The novelty of this study is that we consider non-polynomial perturbations which include the polynomial perturbations as particular cases. The aim of the study is to develop exponential asymptotical expansions for the ruin probability as the initial capital goesto infinity and the perturbation parameter goes to zero, simultaneously but in a balanced manner. Numerical examples of risk processes with such type of perturbations are also given for illustrative purposes.

• 16.
Mälardalen University, School of Education, Culture and Communication.
Nonlinearly Perturbed Renewal Equation with Perturbations of a Non-polynomial Type2010In: Proceedings of the International Symposium on Stochastic Models in Reliability Engineering, Life Science and Operations Management, Beer Sheva, 2010. / [ed] Frenkel, I., Gertsbakh, I., Khvatskin L., Laslo Z. Lisnianski, A., Beer Sheva: SCE - Shamoon College of Engineering , 2010, p. 754-763Conference paper (Refereed)

The object of study is a model of nonlinearly perturbed continuous-time renewal equation with multivariate non-polynomial perturbations. The characteristics of the distribution generating the renewal equation are assumed to have expansions in the perturbation parameter with respect to a non-polynomial asymptotic scale which can be considered as a generalization of the standard polynomial scale. Exponential asymptotics for such a model are obtained and applications are given.

• 17.
Mälardalen University, School of Education, Culture and Communication.
Nonlinearly Perturbed Renewal Equations: asymptotic Results and Applications2011Doctoral thesis, comprehensive summary (Other academic)

In this thesis we investigate a model of nonlinearly perturbed continuous-time renewal equation. Some characteristics of the renewal equation are assumed to have non-polynomial perturbations, more specifically they can be expanded with respect to a non-polynomial asymptotic scale. The main result of the present study is exponential asymptotic expansions for the solution of the perturbed renewal equation. These asymptotic results are also applied to various applied probability models like perturbed risk processes, perturbed M/G/1 queues and perturbed dam/storage processes.

The thesis is based on five papers where the model described above is successively studied.

• 18.
Mälardalen University, School of Education, Culture and Communication.
NONLINEARLY PERTURBED RENEWAL EQUATIONS: THE NON-POLYNOMIAL CASE2012In: Theory of Probability and Mathematical Statistics, ISSN 0094-9000, Vol. 84, p. 117-129Article in journal (Refereed)

Models of nonlinearly perturbed renewal equations with non-polynomial perturbations are studied. Exponential asymptotic expansions are given for the solutions to the perturbed renewal equations under consideration. An application to perturbed M/G/1/ queues is presented.

• 19.
Mälardalen University, School of Education, Culture and Communication.
Perturbed Renewal Equations with Non-Polynomial Perturbations2010Licentiate thesis, comprehensive summary (Other academic)

This thesis deals with a model of nonlinearly perturbed continuous-time renewal equation with nonpolynomial perturbations. The characteristics, namely the defect and moments, of the distribution function generating the renewal equation are assumed to have expansions with respect to a non-polynomial asymptotic scale: $\{\varphi_{\nn} (\varepsilon) =\varepsilon^{\nn \cdot \w}, \nn \in \mathbf{N}_0^k\}$  as $\varepsilon \to 0$, where $\mathbf{N}_0$ is the set of non-negative integers, $\mathbf{N}_0^k \equiv \mathbf{N}_0 \times \cdots \times \mathbf{N}_0, 1\leq k <\infty$ with the product being taken $k$ times and $\w$ is a $k$ dimensional parameter vector that satisfies certain properties. For the one-dimensional case, i.e., $k=1$, this model reduces to the model of nonlinearly perturbed renewal equation with polynomial perturbations which is well studied in the literature.  The goal of the present study is to obtain the exponential asymptotics for the solution to the perturbed renewal equation in the form of exponential asymptotic expansions and present possible applications.

The thesis is based on three papers which study successively the model stated above. Paper A investigates the two-dimensional case, i.e. where $k=2$. The corresponding asymptotic exponential expansion for the solution to the perturbed renewal equation is given. The asymptotic results are applied to an example of the perturbed risk process, which leads to diffusion approximation type asymptotics for the ruin probability.  Numerical experimental studies on this example of perturbed risk process are conducted in paper B, where Monte Carlo simulation are used to study the accuracy and properties of the asymptotic formulas. Paper C presents the asymptotic results for the more general case where the dimension $k$ satisfies $1\leq k <\infty$, which are applied to the asymptotic analysis of the ruin probability in an example of perturbed risk processes with this general type of non-polynomial perturbations.  All the proofs of the theorems stated in paper C are collected in its supplement: paper D.

• 20.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. 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)

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.

• 21.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. 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)

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.

• 22.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Business, Society and Engineering, Future Energy Center.
Investigating the added values of high frequency energy consumption data using data mining techniques2014In: AIP Conference Proceedings 1637 (2014): Volume number: 1637; Published: 10 december 2014 / [ed] Seenith Sivasundaram, AIP Publishing , 2014, p. 734-743Conference paper (Refereed)

In this paper we apply data-mining techniques to customer classification and clustering tasks on actual electricity consumption data from 350 Swedish households. For the classification task we classify households into different categories based on some statistical attributes of their energy consumption measurements. For the clustering task, we use average daily load diagrams to partition electricity-consuming households into distinct groups. The data contains electricity consumption measurements on each 10-minute time interval for each light source and electrical appliance. We perform the classification and clustering tasks using four variants of processeddata sets corresponding to the 10-minute total electricity consumption aggregated from all electrical sources, the hourly total consumption aggregated over all 10-minute intervals during that clock hour, the total consumption over each four-hour intervals and finally the daily total consumption. The goal is to see if there are any differences in using data sets of various frequency levels. We present the comparison results and investigate the added value of the high-frequency measurements, for example 10-minute measurements, in terms of its influence on customer clustering and classification.

• 23.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Exponential asymptotics for nonlinearly perturbed renewal equation with non-polynomial perturbations2008In: Journal of Numerical and Applied Mathematics, ISSN 0868-6912, Vol. 96, no 1, p. 173-197Article in journal (Refereed)

The model of nonlinearly perturbedcontinuous-time renewal equation is studied in this paper.The perturbation conditions considered involve asymptoticalexpansions with respect to asymptotic scale$\{\varphi_{n,m}(\varepsilon) = \varepsilon^{n +m\omega}\}$,with $n, m$ being non-negative integers and $\omega >1$ beingirrational number. Such asymptotical scale results in non-polynomialtype of asymptotic expansions for solutions for perturbed renewalequations. An example of risk processes with perturbations describedabove and asymptotic expansions in diffusion approximation for ruinprobabilities in this model are given.

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