arXiv subjects

Stefan Gerhold

Publications and source records attributed to Stefan Gerhold.

At least 37 records · Page 2Linked to original sources

arXiv2019

Consistency of option prices under bid-ask spreads

Given a finite set of European call option prices on a single underlying, we want to know when there is a market model which is consistent with these prices. In contrast to previous studies, we allow models where the underlying trades at a bid-ask spread. The main question then is how large (in terms of a deterministic bound) this spread must be to explain the given prices. We fully solve this problem in the case of a single maturity, and give several partial results for multiple maturities. For the latter, our main mathematical tool is a recent result on approximation by peacocks [S. Gerhold, I.C. Gülüum, arXiv:1512.06640].

q-fin.MF↗

arXiv2019

Asymptotic expansion of Mathieu power series and trigonometric Mathieu series

We consider a generalized Mathieu series where the summands of the classical Mathieu series are multiplied by powers of a complex number. The Mellin transform of this series can be expressed by the polylogarithm or the Hurwitz zeta function. From this we derive a full asymptotic expansion, generalizing known expansions for alternating Mathieu series. Another asymptotic regime for trigonometric Mathieu series is also considered, to first order, by applying known results on the asymptotic behavior of trigonometric series.

math.CA↗

arXiv2019

Large deviations related to the law of the iterated logarithm for Ito diffusions

When a Brownian motion is scaled according to the law of the iterated logarithm, its supremum converges to one as time tends to zero. Upper large deviations of the supremum process can be quantified by writing the problem in terms of hitting times and applying a result of Strassen (1967) on hitting time densities. We extend this to a small-time large deviations principle for the supremum of scaled Ito diffusions, using as our main tool a refinement of Strassen's result due to Lerche (1986).

math.PR↗

arXiv2019

Asymptotics of some generalized Mathieu series

We establish asymptotic estimates of Mathieu-type series defined by sequences with power-logarithmic or factorial behavior. By taking the Mellin transform, the problem is mapped to the singular behavior of certain Dirichlet series, which is then translated into asymptotics for the original series. In the case of power-logarithmic sequences, we obtain precise first order asymptotics. For factorial sequences, a natural boundary of the Mellin transform makes the problem more challenging, but a direct elementary estimate gives reasonably precise asymptotics.

math.CA↗

arXiv2018

Peacocks nearby: approximating sequences of measures

A peacock is a family of probability measures with finite mean that increases in convex order. It is a classical result, in the discrete time case due to Strassen, that any peacock is the family of one-dimensional marginals of a martingale. We study the problem whether a given sequence of probability measures can be approximated by a peacock. In our main results, the approximation quality is measured by the infinity Wasserstein distance. Existence of a peacock within a prescribed distance is reduced to a countable collection of rather explicit conditions. This result has a financial application (developed in a separate paper), as it allows to check European call option quotes for consistency. The distance bound on the peacock than takes the role of a bound on the bid-ask spread of the underlying. We also solve the approximation problem for the stop-loss distance, the Lévy distance, and the Prokhorov distance.

math.PR↗

arXiv2018

Stoichiometry-driven switching between surface reconstructions on SrTiO$_3$(001)

Controlling the surface structure on the atomic scale is a major difficulty for most transition metal oxides; this is especially true for the ternary perovskites. The influence of surface stoichiometry on the atomic structure of the SrTiO$_3$(001) surface was examined with scanning tunneling microscopy, low-energy electron diffraction, low-energy He$^+$ ion scattering (LEIS), and X-ray photoelectron spectroscopy (XPS). Vapor deposition of 0.8 monolayer (ML) strontium and 0.3 ML titanium, with subsequent annealing to 850 °C in 4 $\times$ 10$^{-6}$ mbar O$_2$, reversibly switches the surface between c(4 $\times$ 2) and (2 $\times$ 2) reconstructions, respectively. The combination of LEIS and XPS shows a different stoichiometry that is confined to the top layer. Geometric models for these reconstructions need to take into account these different surface compositions.

cond-mat.mtrl-sci↗

arXiv2018

Stabilizing Single Ni adatoms on a Two-dimensional Porous Titania Overlayer at the SrTiO$_3$(110) Surface

Nickel vapor-deposited on the SrTiO$_3$(110) surface was studied using scanning tunneling microscopy, photoemission spectroscopy (PES), and density functional theory calculations. This surface forms a (4 $\times$ 1) reconstruction, composed of a 2-D titania structure with periodic six- and ten-membered nanopores. Anchored at these nanopores, Ni single adatoms are stabilized at room temperature. PES measurements show that the Ni adatoms create an in-gap state located at 1.9 eV below the conduction band minimum and induce an upward band bending. Both experimental and theoretical results suggest that Ni adatoms are positively charged. Our study produces well-dispersed single-adatom arrays on a well-characterized oxide support, providing a model system to investigate single-adatom catalytic and magnetic properties.

cond-mat.mtrl-sci↗

arXiv2018

Vacancy clusters at domain boundaries and band bending at the SrTiO$_3$(110) surface

Antiphase domain boundaries (APDBs) in the ($n \times 1$)($n=4,5$) reconstructions of the SrTiO$_3$(110) surface were studied with scanning tunneling microscopy, x-ray photoemission spectroscopy, and density functional theory (DFT) calculations. Two types of APDBs form on each reconstruction; they consist of Ti$_x$O$_y$ vacancy clusters with a specific stoichiometry. The presence of these clusters is controlled by the oxygen pressure during annealing. The structural models of the vacancy clusters are resolved with DFT, which also shows that their relative stability depends on the chemical potential of oxygen. The surface band bending can be tuned by controlling the vacancy clusters at the domain boundaries.

cond-mat.mtrl-sci↗

arXiv2018

Well-Ordered In Adatoms at the In$_2$O$_3$(111) Surface Created by Fe Deposition

Metal deposition on oxide surfaces usually results in adatoms, clusters, or islands of the deposited material, where defects in the surface often act as nucleation centers. Here an alternate configuration is reported. After the vapor deposition of Fe on the In$_2$O$_3$(111) surface at room temperature, ordered adatoms are observed with scanning tunneling microscopy. These are identical to the In adatoms that form when the sample is reduced by heating in ultrahigh vacuum. Density functional theory (DFT) calculations confirm that Fe interchanges with In in the topmost layer, pushing the excess In atoms to the surface where they arrange as a well-ordered adatom array.

cond-mat.mtrl-sci↗

arXiv2018

Moment Explosions in the Rough Heston Model

We show that the moment explosion time in the rough Heston model [El Euch, Rosenbaum 2016, arxiv:1609.02108] is finite if and only if it is finite for the classical Heston model. Upper and lower bounds for the explosion time are established, as well as an algorithm to compute the explosion time (under some restrictions). We show that the critical moments are finite for all maturities. For negative correlation, we apply our algorithm for the moment explosion time to compute the lower critical moment.

q-fin.MF↗

arXiv2017

Dynamic trading under integer constraints

In this paper we investigate discrete time trading under integer constraints, that is, we assume that the offered goods or shares are traded in integer quantities instead of the usual real quantity assumption. For finite probability spaces and rational asset prices this has little effect on the core of the theory of no-arbitrage pricing. For price processes not restricted to the rational numbers, a novel theory of integer arbitrage free pricing and hedging emerges. We establish an FTAP, involving a set of absolutely continuous martingale measures satisfying an additional property. The set of prices of a contingent claim is no longer an interval, but is either empty or dense in an interval. We also discuss superhedging with integral portfolios.

q-fin.MF↗

arXiv2016

Small-maturity asymptotics for the at-the-money implied volatility slope in Lévy models

We consider the at-the-money strike derivative of implied volatility as the maturity tends to zero. Our main results quantify the behavior of the slope for infinite activity exponential Lévy models including a Brownian component. As auxiliary results, we obtain asymptotic expansions of short maturity at-the-money digital call options, using Mellin transform asymptotics. Finally, we discuss when the at-the-money slope is consistent with the steepness of the smile wings, as given by Lee's moment formula.

q-fin.PR↗

arXiv2016

Fe3O4(110)-(1x3) Revisited: Periodic (111) Nano-Facets

The structure of the Fe3O4(110)-(1x3) surface was studied with scanning tunneling microscopy (STM), low-energy electron diffraction (LEED), and reflection high energy electron diffraction (RHEED). The so-called one-dimensional reconstruction is characterised by bright rows that extend hundreds of nanometers in the [1-10] direction and have a periodicity of 2.52 nm in [001] in STM. It is concluded that this reconstruction is the result of a periodic faceting to expose {111}-type planes with a lower surface energy.

cond-mat.mtrl-sci↗

arXiv2016

Option Pricing in the Moderate Deviations Regime

We consider call option prices in diffusion models close to expiry, in an asymptotic regime ("moderately out of the money") that interpolates between the well-studied cases of at-the-money options and out-of-the-money fixed-strike options. First and higher order small-time moderate deviation estimates of call prices and implied volatility are obtained. The expansions involve only simple expressions of the model parameters, and we show in detail how to calculate them for generic local and stochastic volatility models. Some numerical examples for the Heston model illustrate the accuracy of our results.

q-fin.PR↗

arXiv2014

Refined wing asymptotics for the Merton and Kou jump diffusion models

Refining previously known estimates, we give large-strike asymptotics for the implied volatility of Merton's and Kou's jump diffusion models. They are deduced from call price approximations by transfer results of Gao and Lee. For the Merton model, we also analyse the density of the underlying and show that it features an interesting "almost power law" tail.

q-fin.PR↗

arXiv2013

Disproof of a conjecture by Rademacher on partial fractions

In his book Topics in Analytic Number Theory, Rademacher considered the generating function of partitions into at most $N$ parts, and conjectured certain limits for the coefficients of its partial fraction decomposition. We carry out an asymptotic analysis that disproves this conjecture, thus confirming recent observations of Sills and Zeilberger (Journal of Difference Equations and Applications 19, 2013), who gave strong numerical evidence against the conjecture.

math.NT↗

arXiv2013

Water Adsorption at the Tetrahedral Titania Surface Layer of SrTiO$_3$(110)-(4$\times$1)

The interaction of water with oxide surfaces is of great interest for both fundamental science and applications. We present a combined theoretical [density functional theory (DFT)] and experimental [Scanning Tunneling Microscopy (STM), photoemission spectroscopy (PES)] study of water interaction with the two-dimensional titania overlayer that terminates the SrTiO$_3$(110)-(4$\times$1) surface and consists of TiO$_4$ tetrahedra. STM, core-level and valence band PES show that H$_2$O neither adsorbs nor dissociates on the stoichiometric surface at room temperature, while it dissociates at oxygen vacancies. This is in agreement with DFT calculations, which show that the energy barriers for water dissociation on the stoichiometric and reduced surfaces are 1.7 and 0.9 eV, respectively. We propose that water weakly adsorbs on two-dimensional, tetrahedrally coordinated overlayers.

cond-mat.mtrl-sci↗

arXiv2013

Anisotropic two-dimensional electron gas at SrTiO3(110) protected by its native overlayer

Two dimensional electron gases (2DEGs) at a oxide heterostructures are attracting considerable attention, as these might substitute conventional semiconductors for novel electronic devices [1]. Here we present a minimal set-up for such a 2DEG -the SrTiO3(110)-(4 x 1) surface, natively terminated with one monolayer of chemically-inert titania. Oxygen vacancies induced by synchrotron radiation migrate under- neath this overlayer, this leads to a confining potential and electron doping such that a 2DEG develops. Our angular resolved photoemission spectroscopy (ARPES) and theoretical results show that confinement along (110) is strikingly different from a (001) crystal orientation. In particular the quantized subbands show a surprising "semi-heavy" band, in contrast to the analogue in the bulk, and a high electronic anisotropy. This anisotropy and even the effective mass of the (110) 2DEG is tunable by doping, offering a high flexibility to engineer the properties of this system.

cond-mat.mtrl-sci↗