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- Muhumuza, Asaph Keikara, 1975-
(author)
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Extreme points of the Vandermonde determinant in numerical approximation, random matrix theory and financial mathematics
- 2020
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Doctoral thesis (other academic/artistic)abstract
- This thesis discusses the extreme points of the Vandermonde determinant on various surfaces, their applications in numerical approximation, random matrix theory and financial mathematics. Some mathematical models that employ these extreme points such as curve fitting, data smoothing, experimental design, electrostatics, risk control in finance and method for finding the extreme points on certain surfaces are demonstrated.The first chapter introduces the theoretical background necessary for later chapters. We review the historical background of the Vandermonde matrix and its determinant, some of its properties that make it more applicable to symmetric polynomials, classical orthogonal polynomials and random matrices.The second chapter discusses the construction of the generalized Vandermonde interpolation polynomial based on divided differences. We explore further, the concept of weighted Fekete points and their connection to zeros of the classical orthogonal polynomials as stable interpolation points.The third chapter discusses some extended results on optimizing the Vandermonde determinant on a few different surfaces defined by univariate polynomials. The coordinates of the extreme points are shown to be given as roots of univariate polynomials.The fourth chapter describes the symmetric group properties of the extreme points of Vandermonde and Schur polynomials as well as application of these extreme points in curve fitting.The fifth chapter discusses the extreme points of Vandermonde determinant to number of mathematical models in random matrix theory where the joint eigenvalue probability density distribution of a Wishart matrix when optimized over surfaces implicitly defined by univariate polynomials.The sixth chapter examines some properties of the extreme points of the joint eigenvalue probability density distribution of the Wishart matrix and application of such in computation of the condition numbers of the Vandermonde and Wishart matrices. The seventh chapter establishes a connection between the extreme points of Vandermonde determinants and minimizing risk measures in financial mathematics. We illustrate this with an application to optimal portfolio selection.The eighth chapter discusses the extension of the Wishart probability distributions in higher dimension based on the symmetric cones in Jordan algebras. The symmetric cones form a basis for the construction of the degenerate and non-degenerate Wishart distributions.The ninth chapter demonstrates the connection between the extreme points of the Vandermonde determinant and Wishart joint eigenvalue probability distributions in higher dimension based on the boundary points of the symmetric cones in Jordan algebras that occur in both the discrete and continuous part of the Gindikin set.
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- Muhumuza, Asaph Keikara, et al.
(author)
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The Wishart Distribution on Symmetric Cones
- 2023
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In: Non-commutative and Non-associative Algebra and Analysis Structures. - : Springer. - 9783031320088 ; , s. 661-684
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Conference paper (peer-reviewed)abstract
- In this paper we discuss the extension of the Wishart probability distributions in higher dimension based on the boundary points of the symmetric cones in Jordan algebras. The symmetric cones form a basis for the construction of the degenerate and non-degenerate Wishart distributions in the field of Herm(m,C), Herm(m,H), Herm(3,O) that denotes respectively the Jordan algebra of all Hermitian matrices of size m× m with complex entries, the skew field H of quaternions, and the algebra O of octonions. This density is characterised by the Vandermonde determinant structure and the exponential weight that is dependent on the trace of the given matrix.
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- Ni, Ying, et al.
(author)
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Exponential asymptotics for nonlinearly perturbed renewal equation with non-polynomial perturbations
- 2008
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In: Journal of Numerical and Applied Mathematics. - Kiev : TBiMC. - 0868-6912. ; 96:1, s. 173-197
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Journal article (peer-reviewed)abstract
- 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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- Silvestrov, Dmitrii, et al.
(author)
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The analytical finance package
- 2007
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In: Theory of Stochastic Processes. - Kiev : TBiMC. - 0321-3900. ; 13 (29):4, s. 201-209
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Journal article (peer-reviewed)abstract
- We describe the Analytical Finance Package, a set of Java applets which is developing at the Mälardalen University.
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- Canhanga, Betuel, 1980-, et al.
(author)
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Analytical and Numerical Studies on the Second Order Asymptotic Expansion Method for European Option Pricing under Two-factor Stochastic Volatilities
- 2018
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In: Communications in Statistics - Theory and Methods. - : Taylor & Francis. - 0361-0926 .- 1532-415X. ; 47:6, s. 1328-1349
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Journal article (peer-reviewed)abstract
- 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.
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- Malyarenko, Anatoliy, 1957-, et al.
(author)
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Sensitivity Analysis of Catastrophe Bond Priceunder the Hull–White Interest Rate Model
- 2016
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In: Engineering Mathematics I. - Cham : Springer. - 9783319420813 - 9783319420820 ; 178, s. 301-314
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Book chapter (peer-reviewed)abstract
- 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.
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- Albuhayri, Mohammed, et al.
(author)
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An Improved Asymptotics of Implied Volatility in the Gatheral Model
- 2022
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In: <em>Springer Proceedings in Mathematics and Statistics</em>. - Cham : Springer Nature. - 9783031178191 - 9783031178207 ; , s. 3-13
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Conference paper (peer-reviewed)abstract
- We study the double-mean-reverting model by Gatheral. Our previous results concerning the asymptotic expansion of the implied volatility of a European call option, are improved up to order 3, that is, the error of the approximation is ultimately smaller that the 1.5th power of time to maturity plus the cube of the absolute value of the difference between the logarithmic security price and the logarithmic strike price.
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- Albuhayri, Mohammed, et al.
(author)
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Asymptotics of Implied Volatility in the Gatheral Double Stochastic Volatility Model
- 2019
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In: Proceedings of 18th Applied Stochastic Models and Data Analysis International Conference with the Demographics 2019 Workshop, Florence, Italy: 11-14 June, 2019. - : ISAST: International Society for the Advancement of Science and Technology. - 9786185180331 ; , s. 81-90
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Conference paper (peer-reviewed)abstract
- The double-mean-reverting model by Gatheral [1] is motivated by empirical dynamics of the variance of the stock price. No closed-form solution for European option exists in the above model. We study the behaviour of the implied volatility with respect to the logarithmic strike price and maturity near expiry and at-the- money. Using the method by Pagliarani and Pascucci [6], we calculate explicitly the first few terms of the asymptotic expansion of the implied volatility within a parabolic region.
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- Albuhayri, Mohammed, et al.
(author)
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Asymptotics of Implied Volatility in the Gatheral Double Stochastic Volatility Model
- 2021
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In: Applied Modeling Techniques and Data Analysis 2. - Hoboken, NJ, USA : John Wiley & Sons. - 9781786306746 ; , s. 27-38
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Book chapter (peer-reviewed)abstract
- The double-mean-reverting model by Gatheral is motivated by empirical dynamics of the variance of the stock price. No closed-form solution for European option exists in the above model. We study the behaviour of the implied volatility with respect to the logarithmic strike price and maturity near expiry and at-the-money. Using the method by Pagliarani and Pascucci, we calculate explicitly the first few terms of the asymptotic expansion of the implied volatility within a parabolic region.
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