Pseudo Hermitian formulation of the quantum Black–Scholes Hamiltonian

Jana, Tapas Kumar; Roy, Provas Kumar

doi:10.1016/j.physa.2011.12.012

Cited by 11 publications

(17 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…With these definitions, in [12] was also shown that B = ρ −2 A † ρ 2 . Hence B = A † if ρ = 1 1, which is true only if β = 0, see (3.3).…”

Section: D-pbs From the Black-scholes Equationmentioning

confidence: 98%

“…In [10,11,12] some real potential V (x) is added to the operator H BS , defining in this way a new operator, H ef f = H BS + V (x). The economical interpretation of V (x), see [10], is that it can be used to represent a certain class of options; we will not insist on that here.…”

Section: D-pbs From the Black-scholes Equationmentioning

confidence: 99%

“…These equalities could be made rigorous acting on functions of D. In [12] the authors' interest was focused on the factorization of H ef f , using standard results in supersymmetric quantum mechanics, [13]. H ef f can indeed be written as…”

Section: D-pbs From the Black-scholes Equationmentioning

confidence: 99%

“…all well defined on D. Going back to the economic aspect of the model, following [10,11,12] the next step consists in the computation of the price kernel, x, e −τ H x ′ . Here |x is the generalized eigenstate of the position operator written, in agreement with [10,11,12], as a Dirac bra.…”

Section: θ Is a Positive Operatormentioning

confidence: 99%

“…Here |x is the generalized eigenstate of the position operator written, in agreement with [10,11,12], as a Dirac bra.…”

Section: θ Is a Positive Operatormentioning

confidence: 99%

See 4 more Smart Citations

Appearances of pseudo-bosons from Black-Scholes equation

Багарелло

2016

Journal of Mathematical Physics

View full text Add to dashboard Cite

It is a well known fact that the Black-Scholes equation admits an alternative representation as a Schrödinger equation expressed in terms of a non self-adjoint hamiltonian. We show how pseudo-bosons, linear or not, naturally arise in this context, and how they can be used in the computation of the pricing kernel. IntroductionPseudo-Hermitian quantum mechanics, together with its many relatives, received an increasing interest by the physicists community and, more recently, by the mathematicians too, because of its possible applications in concrete systems and also in connection with several interesting mathematical properties appearing in systems driven by non-hermitian Hamiltonians. We refer to [1]-[4] for some recent and not so recent reviews or volumes dedicated to this subject.Most examples of non hermitian Hamiltonians come from physics, see [5]-[8] just to cite a few, or from mathematics, see [9], where the interest is more focused to a rigorous treatment of the models under consideration, and to their main constituents.But, as shown in [10] and, recently, in [11, 12], more examples of this kind of operators come from an unexpected field of research, i.e. from economics. In fact, with a suitable change of variable, the Black-Scholes equation can be rewritten as a Schrödinger equation, but with a non self-adjoint hamiltonian. This fact was used in [10]-[12] to shown how quantum mechanical techniques can be useful to compute the pricing kernel for various situations, and how in some cases the Black-Scholes Hamiltonian can be factorized and used also as an interesting example in supersymmetric quantum mechanics, [13]. In this paper we show that similar (or identical) models also provide examples of D-pseudo bosons (D-PBs) and non linear D-PBs (D-NLPBs), which are excitations recently introduced by us, considering suitable deformations of the canonical commutation relations (CCRs). We also discuss how pricing kernels can be defined and computed in these cases.The paper is organized as follows: in the next section we give a brief mathematical introduction to D-PBs and D-NLPBs, useful to keep the paper self-contained. In Section 3 we show how D-PBs arise naturally out of the Black-Scholes equation, while in Section 4 we discuss the appearance of D-NLPBs in the same context. Section 5 contains our conclusions. Mathematical preliminariesIn view of their use in Sections 3 and 4, we devote this section to some preliminary definitions and results on D-PBs and D-NLPBs. We refer to [5] and to [14]-[16] for more details. Something about D-PBsLet H be a given Hilbert space with scalar product ., . and related norm . . Let further a and b be two operators on H, with domains D(a) and D(b) respectively, a † and b † their adjoints,

show abstract

“…With these definitions, in [12] was also shown that B = ρ −2 A † ρ 2 . Hence B = A † if ρ = 1 1, which is true only if β = 0, see (3.3).…”

Section: D-pbs From the Black-scholes Equationmentioning

confidence: 98%

Section: D-pbs From the Black-scholes Equationmentioning

confidence: 99%

Section: D-pbs From the Black-scholes Equationmentioning

confidence: 99%

Section: θ Is a Positive Operatormentioning

confidence: 99%

“…Here |x is the generalized eigenstate of the position operator written, in agreement with [10,11,12], as a Dirac bra.…”

Section: θ Is a Positive Operatormentioning

confidence: 99%

See 3 more Smart Citations

Appearances of pseudo-bosons from Black-Scholes equation

Багарелло

2016

Journal of Mathematical Physics

View full text Add to dashboard Cite

show abstract

Susy for Non-Hermitian Hamiltonians, with a View to Coherent States

Багарелло

2020

Math Phys Anal Geom

View full text Add to dashboard Cite

We propose an extended version of supersymmetric quantum mechanics which can be useful if the Hamiltonian of the physical system under investigation is not Hermitian. The method is based on the use of two, in general different, superpotentials. Bi-coherent states of the Gazeau-Klauder type are constructed and their properties are analyzed. Some examples are also discussed, including an application to the Black-Scholes equation, one of the most important equations in Finance.

show abstract

Solving the Deformed Woods–Saxon Potential with $$\eta $$-Pseudo-hermetic Generator

Hafezghoran

Bakhshi

2021

Arab J Sci Eng

View full text Add to dashboard Cite

Pseudo Hermitian formulation of the quantum Black–Scholes Hamiltonian

Cited by 11 publications

References 17 publications

Appearances of pseudo-bosons from Black-Scholes equation

Appearances of pseudo-bosons from Black-Scholes equation

Susy for Non-Hermitian Hamiltonians, with a View to Coherent States

Solving the Deformed Woods–Saxon Potential with $$\eta $$-Pseudo-hermetic Generator

Contact Info

Product

Resources

About