Option pricing formulas under a change of numeraire

We present some formulations of the Cox-Ross-Rubinstein and Black-Scholes formulas for European options obtained through a suitable change of measure, which corresponds to a change of numèraire for the underlying price process. Among other consequences, a closed formula for the price of an European call option at each node of the multi-period binomial tree is achieved, too. Some of the results contained herein, though comparable with analogous ones appearing elsewhere in the financial literature, provide however a supplementary widening and deepening in view of useful applications in the more challenging framework of incomplete markets. This last issue, having the present paper as a preparatory material, will be treated extensively in a forthcoming paper

Generalized Bernstein Durrmeyer operators and the associated limit semigroup

AbstractThe aim of this paper is the study of a new sequence of positive linear approximation operators Mn, λ on C([0, 1]) which generalize the classical Bernstein–Durrmeyer operators. After proving a Voronovskaja-type result, we show that there exists a strongly continuous positive contraction semigroup on C([0, 1]) which may be expressed in terms of powers of these operators. As a direct consequence, we are able to represent explicitly the solutions of the Cauchy problems associated with a particular class of second order differential operators

Semigroups generated by ordinary differential operators in L^{1}(I)

In this paper we give necessary and sufficient conditions for the existence of a C_0 semigroup in L_1(I) (I real interval) generated by a second-order differential operator when suitable boundary conditions at the endpoints are imposed. AMS Mathematical Classification: 47D05, 34B05 Key words: generators of semigroups, second-order differential operators, boundary condition

Overiterated Linear Operator and Asymptotic Behaviour of Semigroups

The results provided hereby are related to the asymptotic behaviour of certain strongly continuous semigroups, which may be expressed in terms of iterates of positive linear operators, in the sense of Altomare’s theory. We present some applications to concrete cases involving continuous and discrete type operators, namely the Beta and the Stancu operators. Mathematics Subject Classification (2000). Primary 41A35; Secondary 47D06. Keywords. Bounded linear operators, strongly continuous semigroups, asymptotic behaviour, iterates, overiterates

Gamma-type operators and the Black-Scholes semigroup

AbstractWe study Gamma-type operators from the analytic and probabilistic viewpoint in the setting of weighted continuous function spaces and estimate the rate of convergence of their iterates towards their limiting semigroup, providing, in this way, a quantitative version of the classical Trotter approximation theorem. The semigroup itself has some interest, since it is generated by the Black–Scholes operator, frequently occurring in the theory of option pricing in mathematical finance