arXiv subjects

Frederik S. Herzberg

Publications and source records attributed to Frederik S. Herzberg.

3 recordsLinked to original sources

arXiv2017

Minimal axiomatic frameworks for definable hyperreals with transfer

We modify the definable ultrapower construction of Kanovei and Shelah (2004) to develop a ZF-definable extension of the continuum with transfer provable using countable choice only, with an additional mild hypothesis on well-ordering implying properness. Under the same assumptions, we also prove the existence of a definable, proper elementary extension of the standard superstructure over the reals. Keywords: definability; hyperreal; superstructure; elementary embedding.

math.LO↗

arXiv2006

Approximate Bermudan option pricing based on the réduite or cubature: soundness and characterisation of perpetual prices as fixed points

In this paper, it is shown that Bermudan option pricing based on either the réduite (in a one-dimensional setting: piecewise harmonic interpolation) or cubature -- is sensible from an economic vantage point: Any sequence of thus-computed prices for Bermudan options with increasing sets of exercise times is increasing. Furthermore, under certain regularity assumptions on the payoff function and provided the exercise times are equidistant of exercise mesh size $h$, it has a supremum which coincides with the least fixed point of the approximate pricing algorithm -- this algorithm being perceived as a map that assigns to any real-valued function $f$ (on the basket of underlyings) the approximate value of the European option of maturity $h$ and payoff function $f$.

math.PR↗

arXiv2005

A brief note on the soundness of Bermudan option pricing via cubature

The subject of this study is an iterative Bermudan option pricing algorithm based on (high-dimensional) cubature. We show that the sequence of Bermudan prices (as functions of the underlying assets' logarithmic start prices) resulting from the iteration is bounded and increases monotonely to the approximate perpetual Bermudan option price; the convergence is linear in the supremum norm with the discount factor being the convergence factor. Furthermore, we prove a characterisation of this approximated perpetual Bermudan price as the smallest fixed point of the iteration procedure.

math.PR↗