Arithmetic Asian Options under Stochastic Delay Models

Abstract

Motivated by the increasing interest in past-dependent asset pricing models, shown in recent years by market practitioners and prominent authors such as Hobson and Rogers (1998 Hobson, D. and Rogers, L. C. G. 1998. Complete models with stochastic volatility. Mathematical Finance, 8(1): 27–48.  [Google Scholar], Complete models with stochastic volatility, Mathematical Finance, 8(1), pp. 27–48), we explore option pricing techniques for arithmetic Asian options under a stochastic delay differential equation approach. We obtain explicit closed-form expressions for a number of lower and upper bounds and compare their accuracy numerically.     

Comment on: A Note on the Discontinuity Problem in Heston's Stochastic Volatility Model

Abstract

Guo and Hung (2007 Guo, J.-H. and Hung, M.-W. 2007. A note on the discontinuity problem in Heston's stochastic volatility model. Applied Mathematical Finance, 14(4): 339–345. [Taylor & Francis Online] , [Google Scholar]) recently studied the complex logarithm present in the characteristic function of Heston's stochastic volatility model. They proposed an algorithm for the evaluation of the characteristic function that is claimed to preserve its continuity. We show their algorithm is correct, although their proof is not.     

Stochastic Expansion for the Pricing of Call Options with Discrete Dividends

Abstract

In the context of an asset paying affine-type discrete dividends, we present closed analytical approximations for the pricing of European vanilla options in the Black–Scholes model with time-dependent parameters. They are obtained using a stochastic Taylor expansion around a shifted lognormal proxy model. The final formulae are, respectively, first-, second- and third- order approximations w.r.t. the fixed part of the dividends. Using Cameron–Martin transformations, we provide explicit representations of the correction terms as Greeks in the Black–Scholes model. The use of Malliavin calculus enables us to provide tight error estimates for our approximations. Numerical experiments show that this approach yields very accurate results, in particular compared with known approximations of Bos, Gairat and Shepeleva (2003 Bos, R., Gairat, A. and Shepeleva, D. 2003. Dealing with discrete dividends. Risk Magazine, 16: 109–112.  [Google Scholar], Dealing with discrete dividends, Risk Magazine, 16, pp. 109–112) and Veiga and Wystup (2009 Veiga, C. and Wystup, U. 2009. Closed formula for options with discrete dividends and its derivatives. Applied Mathematical Finance, 16(6): 517–531. [Taylor & Francis Online] , [Google Scholar], Closed formula for option with discrete dividends and its derivatives, Applied Mathematical Finance, 16(6), pp. 517–531), and quicker than the iterated integration procedure of Haug, Haug and Lewis (2003 Haug, E. G., Haug, J. and Lewis, A. 2003. Back to basics: a new approach to the discrete dividend problem. Wilmott Magazine, : 37–47.  [Google Scholar], Back to basics: a new approach to the discrete dividend problem, Wilmott Magazine, pp. 37–47) or than the binomial tree method of Vellekoop and Nieuwenhuis (2006 Vellekoop, M. and Nieuwenhuis, J. 2006. Effcient pricing of derivatives on assets with discrete dividends. Applied Mathematical Finance, 13(3): 265–284. [Taylor & Francis Online] , [Google Scholar], Efficient pricing of derivatives on assets with discrete dividends, Applied Mathematical Finance, 13(3), pp. 265–284).     

Combined Optimal Stopping and Singular Stochastic Control

In this article, a simple of combined singular stochastic control and optimal stopping in the jump-diffusion model is formulated and solved. We give sufficient conditions for the existence of an optimal strategy which has the same form as in continuous case given by Davis and Zervos [3 Davis , M.H.A. , and Zervos , M. 1994. A problem of singular stochastic control with discretionary stopping. Annals of Applied Probility 4(1):226–240. [Google Scholar]] and also Karatzas et al. [5 Karatzas , I. , Ocone , D. , Wang , H. , and Zervos , M. 2000 . Finite-fuel singular control with discretionary stopping . Stochastics and Stochastics Reports 71 : 1 – 50 .[Taylor & Francis Online] , [Google Scholar]]. This result is applied to solve explicitly an example of such problem.     

Standard Koszul standardly stratified algebras

In this paper, we prove that every standard Koszul (not necessarily graded) standardly stratified algebra is also Koszul. This generalizes a similar result of [3 Ágoston, I., Dlab, V., Lukács, E. (2003). Quasi-hereditary extension algebras. Algebras Represent. Theory 6:97–117.[Crossref], [Web of Science ®] , [Google Scholar]] on quasi-hereditary algebras.     

Splitting of operations for alternative and Malcev structures

In this paper, we define pre-Malcev algebras and alternative quadri-algebras and prove that they generalize pre-Lie algebras and quadri-algebras, respectively, to the alternative setting. We use the results and techniques from [4 Bai, C., Bellier, O., Guo, L., Ni, X. (2013). Splitting of operations, Manin products, and Rota-Baxter operators. Int. Math. Res. Not. 2013(3):485–524. [Google Scholar], 14 Gubarev, V. Y., Kolesnikov, P. S. (2013). Embedding of dendriform algebras into Rota-Baxter algebras. Cent. Eur. J. Math. 11(2):226–245.[Crossref], [Web of Science ®] , [Google Scholar]] to discuss and give explicit computations of different constructions in terms of bimodules, splitting of operations, and Rota–Baxter operators.     

Superconvergence of Solution Derivatives for the Shortley–Weller Difference Approximation of Poisson's Equation. II. Singularity Problems

Abstract

This is a continued analysis on superconvergence of solution derivatives for the Shortley–Weller approximation in Li (Li, Z. C., Yamamoto, T., Fang, Q. ([2003] Li, Z. C., Yamamoto, T. and Fang, Q. 2003. Superconvergence of solution derivatives for the Shortley–Weller difference approximation of Poisson's equation, Part I. Smoothness problems. J. Comp. and Appl. Math., 152(2): 307–333.  [Google Scholar]): Superconvergence of solution derivatives for the Shortley–Weller difference approximation of Poisson's equation, Part I. Smoothness problems. J. Comp. and Appl. Math. 152(2):307–333), which is to explore superconvergence for unbounded derivatives near the boundary. By using the stretching function proposed in Yamamoto (Yamamoto, T. ([2002] Yamamoto, T. 2002. Convergence of consistant and inconsistent finite difference schemes and an acceleration technique. J. Comp. Appl. Math., 140: 849–866. [Crossref], [Web of Science ®] , [Google Scholar]): Convergence of consistant and inconsistent finite difference schemes and an acceleration technique. J. Comp. Appl. Math. 140:849–866), the second order superconvergence for the solution derivatives can be established. Moreover, numerical experiments are provided to support the error analysis made. The analytical approaches in this article are non-trivial, intriguing, and different from Li, Z. C., Yamamoto, T., Fang, Q. ([2003] Li, Z. C., Yamamoto, T. and Fang, Q. 2003. Superconvergence of solution derivatives for the Shortley–Weller difference approximation of Poisson's equation, Part I. Smoothness problems. J. Comp. and Appl. Math., 152(2): 307–333.  [Google Scholar]). This article also provides the superconvergence analysis for the bilinear finite element method and the finite difference method with nine nodes.     

Calderón problem for connections

In this paper, we consider the problem of identifying a connection ? on a vector bundle up to gauge equivalence from the Dirichlet-to-Neumann map of the connection Laplacian ?*? over conformally transversally anisotropic (CTA) manifolds. This was proved in [9 Dos Santos Ferreira, D., Kenig, C., Salo, M., Uhlmann, G. (2009). Limiting Carleman weights and anisotropic inverse problems. Invent. Math. 178:119–171.[Crossref], [Web of Science ®] , [Google Scholar]] for line bundles in the case of the transversal manifold being simple—we generalize this result to the case where the transversal manifold only has an injective ray transform. Moreover, the construction of suitable Gaussian beam solutions on vector bundles is given for the case of the connection Laplacian and a potential, following the works of [11 Dos Santos Ferreira, D., Kurylev, Y., Lassas, M., Salo, M. (2016). The Calderón problem in transversally anisotropic geometries. J. Eur. Math. Soc., 18:2579–2626.[Crossref], [Web of Science ®] , [Google Scholar]]. This in turn enables us to construct the Complex Geometrical Optics (CGO) solutions and prove our main uniqueness result. We also reduce the problem to a new non-abelian X-ray transform for the case of simple transversal manifolds and higher rank vector bundles. Finally, we prove the recovery of a flat connection in general from the DN map, up to gauge equivalence, using an argument relating the Cauchy data of the connection Laplacian and the holonomy.     

Breadth and characteristic sequence of nilpotent Lie algebras

The notion of breadth of a nilpotent Lie algebra was introduced and used to approach problems of classification up to isomorphism in [5 Khuhirun, B., Misra, K. C., Stitzinger, E. (2015). On nilpotent Lie algebras of small breadth. J. Algebra 444:328–338.[Crossref], [Web of Science ®] , [Google Scholar]]. In the present paper, we study this invariant in terms of characteristic sequence, another invariant, introduced by Goze and Ancochea in [1 Ancochea-Bermúdez, J. M., Goze, M. (1986). Sur la classification des algèbres de Lie nilpotentes de dimension 7. C. R. Acad. Sci. Paris 302:611–613. [Google Scholar]]. This permits to complete the determination of Lie algebras of breadth 2 studied in [5 Khuhirun, B., Misra, K. C., Stitzinger, E. (2015). On nilpotent Lie algebras of small breadth. J. Algebra 444:328–338.[Crossref], [Web of Science ®] , [Google Scholar]] and to begin the work for Lie algebras with breadth greater than 2.     

Poisson deleting derivations algorithm and Poisson spectrum

Cauchon [5 Cauchon, G. (2003). Effacement des dérivations et spectres premiers des algèbres quantiques. J. Algebra 260(2):476–518.[Crossref], [Web of Science ®] , [Google Scholar]] introduced the so-called deleting derivations algorithm. This algorithm was first used in noncommutative algebra to prove catenarity in generic quantum matrices, and then to show that torus-invariant primes in these algebras are generated by quantum minors. Since then this algorithm has been used in various contexts. In particular, the matrix version makes a bridge between torus-invariant primes in generic quantum matrices, torus orbits of symplectic leaves in matrix Poisson varieties and totally non-negative cells in totally non-negative matrix varieties [12 Goodearl, K. R., Launois, S., Lenagan, T. (2011). Torus invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves. Math. Z. 269(1):29–45.[Crossref], [Web of Science ®] , [Google Scholar]]. This led to recent progress in the study of totally non-negative matrices such as new recognition tests [18 Launois, S., Lenagan, T. (2014). E?cient recognition of totally non-negative matrix cells. Found. Comput. Math. 14:371–387.[Crossref], [Web of Science ®] , [Google Scholar]]. The aim of this article is to develop a Poisson version of the deleting derivations algorithm to study the Poisson spectra of the members of a class 𝒫 of polynomial Poisson algebras. It has recently been shown that the Poisson Dixmier–Moeglin equivalence does not hold for all polynomial Poisson algebras [2 Bell, J., Launois, S., Sanchez, O. L., Moosa, R. Poisson algebras via model theory and differential-algebraic geometry. J. Eur. Math. Soc. (to appear). [Google Scholar]]. Our algorithm allows us to prove this equivalence for a significant class of Poisson algebras, when the base field is of characteristic zero. Finally, using our deleting derivations algorithm, we compare topologically spectra of quantum matrices with Poisson spectra of matrix Poisson varieties.     

Assessing the Performance of Different Volatility Estimators: A Monte Carlo Analysis

Abstract

We test the performance of different volatility estimators that have recently been proposed in the literature and have been designed to deal with problems arising when ultra high-frequency data are employed: microstructure noise and price discontinuities. Our goal is to provide an extensive simulation analysis for different levels of noise and frequency of jumps to compare the performance of the proposed volatility estimators. We conclude that the maximum likelihood estimator filter (MLE-F), a two-step parametric volatility estimator proposed by Cartea and Karyampas (2011a Cartea, Á. and Karyampas, D. 2011a. The relationship between the volatility of returns and the number of jumps in financial markets, SSRN eLibrary, Working Paper Series, SSRN.  [Google Scholar]; The relationship between the volatility returns and the number of jumps in financial markets, SSRN eLibrary, Working Paper Series, SSRN), outperforms most of the well-known high-frequency volatility estimators when different assumptions about the path properties of stock dynamics are used.     

The Larson–Sweedler theorem for weak multiplier Hopf algebras

The Larson–Sweedler theorem says that a finite-dimensional bialgebra with a faithful integral is a Hopf algebra [15 Larson, R. G., Sweedler, M. E. (1969). An associative orthogonal bilinear form for Hopf algebras. Amer. J. Math. 91:75–93.[Crossref], [Web of Science ®] , [Google Scholar]]. The result has been generalized to finite-dimensional weak Hopf algebras by Vecsernyés [44 Vecsernyés, P. (2003). Larson–Sweedler theorem and the role of grouplike elements in weak Hopf algebras. J. Algebra 270:471–520. See also arXiv: 0111045v3 [math.QA] for an extended version.[Crossref], [Web of Science ®] , [Google Scholar]]. In this paper, we show that the result is still true for weak multiplier Hopf algebras. The notion of a weak multiplier bialgebra was introduced by Böhm et al. in [4 Böhm, G., Gómez-Torecillas, J., López-Centella, E. (2015). Weak multiplier bialgebras. Weak multiplier bialgebras. 367(12):8681–8872. See also arXiv: 1306.1466 [math.QA]. [Google Scholar]]. In this note it is shown that a weak multiplier bialgebra with a regular and full coproduct is a regular weak multiplier Hopf algebra if there is a faithful set of integrals. Weak multiplier Hopf algebras are introduced and studied in [40 Van Daele, A., Wang, S. (2015). Weak multiplier Hopf algebras I. The main theory. J. Ange. Math. (Crelles J.) 705:155–209, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/crelle-2013-0053, July 2013. See also arXiv:1210.4395v1 [math.RA].[Web of Science ®] , [Google Scholar]]. Integrals on (regular) weak multiplier Hopf algebras are treated in [43 Van Daele, A., Wang, S. (2016). Weak multiplier Hopf algebras III. Integrals and duality. Preprint University of Leuven (Belgium) and Southeast University of Nanjing (China), See arXiv: 1701.04951.v3 [math.RA]. [Google Scholar]]. This result is important for the development of the theory of locally compact quantum groupoids in the operator algebra setting, see [13 Kahng, B.-J., Van Daele, A. A class of C*-algebraic locally compact quantum groupoids I. Preprint Canisius College Buffalo (USA) and University of Leuven (Belgium). [Google Scholar]] and [14 Kahng, B.-J., Van Daele, A. A class of C*-algebraic locally compact quantum groupoids II. Preprint Canisius College Buffalo (USA) and University of Leuven (Belgium). [Google Scholar]]. Our treatment of this material is motivated by the prospect of such a theory.     

Existence and Uniqueness of Solutions to Fokker–Planck Type Equations with Irregular Coefficients

We study the existence and the uniqueness of the solution to a class of Fokker–Planck type equations with irregular coefficients, more precisely with coefficients in Sobolev spaces W ^{1, p} . Our arguments are based upon the DiPerna–Lions theory of renormalized solutions to linear transport equations and related equations [5 Di Perna , R. J. , Lions , P. L. ( 1989 ). Ordinary differential equations, transport theory and Sobolev spaces . Invent. Math. 98 ( 3 ): 511 – 547 . [Google Scholar]]. The present work extends the results of our previous article [14 Le Bris , C. , Lions , P. L. ( 2004 ). Renormalized solutions of some transport equations with partially W ^1,1 velocities and applications . Annali di Matematica Pura ed Applicata 183 : 97 – 130 . [Google Scholar]], where only the simpler case of a Fokker–Planck equation with constant diffusion matrix was addressed. The consequences of the present results on the well-posedness of the associated stochastic differential equations are only outlined here. They will be more thoroughly examined in a forthcoming work [15 Le Bris , C. , Lions , P. L. Generalized flows for stochastic differential equations with irregular coefficients. In preparation . [Google Scholar]].     

The Cauchy Problem for the Klein–Gordon–Schrödinger System in Low Dimensions Below the Energy Space

ABSTRACT

We show that the Klein–Gordon–Schrödinger system in one, two, and three dimensions has a global solution below the energy space. The proof uses the I-method recently introduced by Colliander et al. (2001 Colliander , J. , Keel , M. , Staffilani , G. , Takaoka , H. , Tao , T. ( 2001 ). Global well-posedness for Schrödinger equations with derivative . SIAM J. Math. Anal. 33 ( 3 ): 649 – 669 . [CROSSREF]  [Google Scholar]) and mixed type Strichartz estimates for the solutions of Schrödinger and Klein–Gordon equations, respectively.     

ON FREE PRODUCTS OF ABSOLUTE GALOIS GROUPS. II

In [1] Ershov, Yu. L. 1997. On Free Products of Absolute Galois Groups. Dokl. Math., 56(3): 915–917.  [Google Scholar], the author gave a positive solution to the problem in the survey of Jarden [2] Jarden, M. 1996. “Infinite Galois Theory”. In Handbook of Algebra I Amsterdam: Elsevier Sci.. [Crossref] , [Google Scholar] on the closedness of the class of profinite groups that are isomorphic to absolute Galois groups of fields with respect to finite free products. In [3] Mel'nikov, O. V. 1999. On Free Products of Absolute Galois Groups. Sib. Mat. J., 40(1): 95–99. [Crossref], [Web of Science ®] , [Google Scholar], O. V. Mel'nikov solved this problem for separable profinite groups ([3] Mel'nikov, O. V. 1999. On Free Products of Absolute Galois Groups. Sib. Mat. J., 40(1): 95–99. [Crossref], [Web of Science ®] , [Google Scholar] was done earlier than [1] Ershov, Yu. L. 1997. On Free Products of Absolute Galois Groups. Dokl. Math., 56(3): 915–917.  [Google Scholar]). In the same case, a more exact result on the absolute Galois groups of fields of fixed characteristic was obtained there. The proof proposed in 4-5 Koenigsmann, J. in press. Relatively Projective Groups as Absolute Galois Groups. Haran, D., Jarden, M. and Koenigsmann, J. in press. Free Products of Absolute Galois Groups.   is simpler than that in [1] Ershov, Yu. L. 1997. On Free Products of Absolute Galois Groups. Dokl. Math., 56(3): 915–917.  [Google Scholar] and, in addition, provides the results of Mel'nikov.

On February, 2000, the author (knowing nothing about 4-5 Koenigsmann, J. in press. Relatively Projective Groups as Absolute Galois Groups. Haran, D., Jarden, M. and Koenigsmann, J. in press. Free Products of Absolute Galois Groups.  ) found one more proof of these results. In the author opinion, this proof is the simplest and the construction used in the proof, as well as its properties (cf. Propositio n 1) can have other applications.     

Backward Uniqueness for the Heat Equation in Cones

It was shown in [4 Escauriaza , L. , Seregin , G. , ?verák , V. ( 2003 ). L _{3, ∞}-solutions of the Navier–Stokes equations and backward uniqueness . Uspekhi Mat. Nauk 58:3–44 (in Russian); translation in Russian Math. Surveys 58 : 211 – 250 .[Crossref], [Web of Science ®] , [Google Scholar], 14 Seregin , G. , ?verák , V. ( 2002 ). The Navier–Stokes equations and backward uniqueness. Nonlinear problems in mathematical physics and related topics, II . In : Int. Math. Ser. (N.Y.) . Vol. 2 . New York : Kluwer/Plenum , pp. 353 – 366 . [Google Scholar]] that a bounded solution of the heat equation in a half-space which becomes zero at some time must be identically zero, even though no assumptions are made on the boundary values of the solutions. In a recent example, Luis Escauriaza showed that this statement fails if the half-space is replaced by cones with opening angle smaller than 90°. Here we show that the result remains true for cones with opening angle larger than 110°.