Appearances of pseudo-bosons from Black-Scholes equation

Abstract

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,

One-directional quantum mechanical dynamics and an application to decision making

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A description of pseudo-bosons in terms of nilpotent Lie algebras

“…The reader may refer to [20,22,24], in order to see that we are dealing with a well known notion in geometry and algebra. Similar constructions involve in fact not only Lie algebras, but even the structure of compact groups (see [22,Theorems 9.24,9.39,9.42] for details). We conclude with a surprising fact in our context of study: Proof.…”

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