| 1. |
- Albuhayri, Mohammed
(författare)
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Asymptotics of implied volatility in the Gatheral double stochastic volatility model
- 2022
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- We consider a market model of financial engineering with three factors represented by three correlated Brownian motions. The volatility of the risky asset in this model is the sum of two stochastic volatilities. The dynamic of each volatility is governed by a mean-reverting process. The first stochastic volatility of mean-reversion process reverts to the second volatility at a fast rate, while the second volatility moves slowly to a constant level over time with the state of the economy.The double mean-reverting model by Gatheral (2008) is motivated by empirical dynamics of the variance of the stock price. This model can be consistently calibrated to both the SPX options and the VIX options. However due to the lack of an explicit formula for both the European option price and the implied volatility, the calibration is usually done using time consuming methods like Monte Carlo simulation or the finite difference method.To solve the above issue, we use the method of asymptotic expansion developed by Pagliarani and Pascucci (2017). In paper A, we study the behaviour of the implied volatility with respect to the logarithmic strike price and maturity near expiry and at-the-money. We calculate explicitly the asymptotic expansions of implied volatility within a parabolic region up the second order. In paper B we improve the results obtain in paper A by calculating the asymptotic expansion of implied volatility under the Gatheral model up to order three. In paper C, we perform numerical studies on the asymptotic expansion up to the second order. The Monte-Carlo simulation is used as the benchmark value to check the accuracy of the expansions. We also proposed a partial calibration procedure using the expansions. The calibration procedure is implemented on real market data of daily implied volatility surfaces for an underlying market index and an underlying equity stock for periods both before and during the COVID-19 crisis. Finally, in paper D we check the performance of the third order expansion and compare it with the previous results.
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| 2. |
- Bäck, Per, 1987-
(författare)
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On Hom-associative Ore Extensions
- 2022
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this thesis, we introduce and study hom-associative Ore extensions. These are non-unital, non-associative, non-commutative polynomial rings in which the associativity condition is “twisted” by an additive group homomorphism. In particular, they are examples of hom-associative algebras, and they generalize the classical non-commutative polynomial rings introduced by Ore known as Ore extensions to the non-unital, hom-associative setting. At the same time, when the twisted associativity condition is null, they also generalize to the general non-unital, non-associative setting. We deduce necessary and sufficient conditions for hom-associative Ore extensions to exist, and construct concrete examples thereof. These include hom-associative generalizations of the quantum plane, the universal enveloping algebra of the two-dimensional non-abelian Lie algebra, and the first Weyl algebra, to name a few. The aforementioned algebras turn out to be formal hom-associative deformations of their associative counterparts, the latter two which cannot be formally deformed in the associative setting. Moreover, these are all weakly unital algebras, and we provide a way of embedding any multiplicative, non-unital hom-associative algebra into a multiplicative, weakly unital hom-associative algebra. This generalizes the classical unitalization of non-unital, associative algebras. We then study the hom-associative Weyl algebras in arbitrary characteristic, classify them up to isomorphism, and in the zero characteristic case, we prove that an analogue of the Dixmier conjecture is true. We also study hom-modules over hom-associative rings, and by doing so, we are able to prove a Hilbert's basis theorem for hom-associative Ore extensions. Our theorem includes as special cases both the classical Hilbert's basis theorem for Ore extensions and a Hilbert's basis theorem for unital, non-associative Ore extensions. Last, we construct examples of both hom-associative and non-associative Ore extensions which are all Noetherian by our theorem.
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| 3. |
- Albuhayri, Mohammed, et al.
(författare)
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An Improved Asymptotics of Implied Volatility in the Gatheral Model
- 2022
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Ingår i: <em>Springer Proceedings in Mathematics and Statistics</em>. - Cham : Springer Nature. - 9783031178191 - 9783031178207 ; , s. 3-13
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Konferensbidrag (refereegranskat)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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| 4. |
- Albuhayri, Mohammed, et al.
(författare)
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Numerical Studies of the Implied Volatility Expansions up to Third Order under the Gatheral Model
- 2022
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Konferensbidrag (övrigt vetenskapligt/konstnärligt)abstract
- The Gatheral double stochastic volatility model is a three-factor model with mean-reverting stochastic volatility that reverts to a stochastic long-run mean. Our previous paper investigated the performance of the first and second-order implied volatilities expansions under this model. Moreover, a simple partial calibration method has been proposed. This paper reviews and extends previous results to the third-order implied volatility expansions under the same model. Using Monte-Carlo simulation as the benchmark method, extensive numerical studies are conducted to investigate the accuracy and properties of the third-order expansion.
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| 5. |
- Domingos, Celso Djinja, et al.
(författare)
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Representations of polynomial covariance type commutation relations by linear integral operators on Lp over measure spaces
- 2022
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Ingår i: Stochastic Processes, Statistical Methods, and Engineering Mathematics. - : Springer Nature. - 9783031178191 - 9783031178207 ; , s. 59-95
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Konferensbidrag (refereegranskat)abstract
- Representations of polynomial covariance type commutation relations by linear integral operators on Lp over measures spaces are constructed. Conditions for such representations are described in terms of kernels of the corresponding integral operators. Representation by integral operators are studied both for general polynomial covariance commutation relations and for important classes of polynomial covariance commutation relations associated to arbitrary monomials and to affine functions. Examples of integral operators on Lp spaces representing the covariance commutation relations are constructed. Representations of commutation relations by integral operators with special classes of kernels such as separable kernels and convolution kernels are investigated.
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| 6. |
- Elhamdadi, Mohamed, et al.
(författare)
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Derivation problem for quandle algebras
- 2022
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Ingår i: International Journal of Algebra and Computation. - 0218-1967 .- 1793-6500. ; 32:05, s. 985-1007
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Tidskriftsartikel (refereegranskat)abstract
- The purpose of this paper is to introduce and investigate the notion of derivation for quandle algebras. More precisely, we describe the symmetries on structure constants providing a characterization for a linear map to be a derivation. We obtain a complete characterization of derivations in the case of quandle algebras of dihedral quandles over fields of characteristic zero, and provide the dimensionality of the Lie algebra of derivations. Many explicit examples and computations are given over both zero and positive characteristic. Furthermore, we investigate inner derivations, in the sense of Schafer for non-associative structures. We obtain necessary conditions for the Lie transformation algebra of quandle algebras of Alexander quandles, with explicit computations in low dimensions.
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| 7. |
- Harrathi, F., et al.
(författare)
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Kupershmidt operators on Hom-Malcev algebras and their deformation
- 2022
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Ingår i: International Journal of Geometric Methods in Modern Physics (IJGMMP). - : World Scientific. - 0219-8878.
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Tidskriftsartikel (refereegranskat)abstract
- The main feature of Hom-algebras is that the identities defining the structures are twisted by linear maps. The purpose of this paper is to introduce and study a Hom-type generalization of pre-Malcev algebras, called Hom-pre-Malcev algebras. We also introduce the notion of Kupershmidt operators of Hom-Malcev and Hom-pre-Malcev algebras and show the connections between Hom-Malcev and Hom-pre-Malcev algebras using Kupershmidt operators. Hom-pre-Malcev algebras generalize Hom-pre-Lie algebras to the Hom-alternative setting and fit into a bigger framework with a close relationship with Hom-pre-alternative algebras. Finally, we establish a deformation theory of Kupershmidt operators on a Hom-Malcev algebra in consistence with the general principles of deformation theories and introduce the notion of Nijenhuis elements.
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| 8. |
- Laraiedh, Ismail, et al.
(författare)
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Constructions of BiHom-X algebras and bimodules of some BiHom-dialgebras
- 2022
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Ingår i: Algebra and Discrete Mathematics. - : Lugansk Taras Shevchenko National University. - 1726-3255 .- 2415-721X. ; 34:2, s. 273-316
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Tidskriftsartikel (refereegranskat)abstract
- The aim of this paper is to introduce and to develop several methods for constructions of BiHom-X algebras by extending composition methods, and by using Rota-Baxter operators and some elements of centroids. The bimodules of BiHom-left symmetric dialgebras, BiHom-associative dialgebras and BiHom-tridendriform algebra are deőned, and it is shown that a sequence of this kind of bimodules can be constructed. Their matched pairs of BiHom-left symmetric, BiHom-associative dialgebras BiHom-tridendriform algebra are introduced and methods for their constructions and properties are investigated.
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| 9. |
- Muhumuza, Asaph Keikara, et al.
(författare)
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Connections between the extreme points for Vandermonde determinants and minimizing risk measure in financial mathematics
- 2022
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Ingår i: <em>Springer Proceedings in Mathematics and Statistics</em>. - : Springer Nature. - 9783031178191 ; , s. 587-623
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Konferensbidrag (refereegranskat)abstract
- The extreme points of Vandermonde determinants when optimized on surfaces like spheres and cubes have various applications in random matrix theory, electrostatics and financial mathematics. In this study, we apply the extreme points of Vandermonde determinant when optimized on various surfaces to risk minimization in financial mathematics. We illustrate this by constructing the efficient frontiers represented by spheres, cubes and other general surfaces as applies to portfolio theory. The extreme points of Vandermonde determinant lying on such surfaces as efficient frontier would be used to determine the set of assets with minimum risk and maximum returns. This technique can also applied in optimal portfolio selection and asset pricing.
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| 10. |
- Muhumuza, Asaph Keikara, et al.
(författare)
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Extreme points of the Vandermonde determinant and Wishart ensemble on symmetric cones
- 2022
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Ingår i: <em>Springer Proceedings in Mathematics and Statistics</em>. - Cham : Springer Nature. - 9783031178191 ; , s. 625-649
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Konferensbidrag (refereegranskat)abstract
- In this paper we demonstrate the extreme points of the Wishart joint eigenvalue probability distributions in higher dimension based on the boundary points of the symmetric cones in Jordan algebras. The extreme of points of theVandermonde4 determinant are defined to be a set of boundary points of the symmetric cones that occur in both the discrete and continuous part of the Gindikin set. The symmetric cones form a basis for the construction of the degenerate and non-degenerate Wishart ensembles in Herm(m,C), Herm(m,H), Herm(3,O) denoting 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.
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