Stefan Gerhold

Transaction Costs, Trading Volume, and the Liquidity Premium

In a market with one safe and one risky asset, an investor with a long horizon, constant investment opportunities and constant relative risk aversion trades with small proportional transaction costs. We derive explicit formulas for the optimal investment policy, its implied welfare, liquidity premium, and trading volume. At the first order, the liquidity premium equals the spread, times share turnover, times a universal constant. The results are robust to consumption and finite horizons. We exploit the equivalence of the transaction cost market to another frictionless market, with a shadow risky asset, in which investment opportunities are stochastic. The shadow price is also found explicitly.

Anisotropic two-dimensional electron gas at SrTiO3(110)

Significance Although still in its infancy, electronics based on all-oxide materials is a rapidly developing field, and strontium titanate is its key player. For this area to thrive, an atomic-scale control and understanding of the materials’ surfaces and interfaces needs to be achieved. A SrTiO3 crystal with (110) orientation automatically forms an overlayer that is more insulating than the bulk and chemically less reactive, akin to the native SiO2 on conventional wafer. With appropriate doping a two-dimensional electron gas (2DEG) forms underneath the SrTiO3(110) surface. This (110) 2DEG is very different from (001): The effective mass here depends on the quantum number, and a completely flat band can be realized. Such a flat band bears good prospects for, among others, magnetism and thermoelectricity. Two-dimensional electron gases (2DEGs) at oxide heterostructures are attracting considerable attention, as these might one day substitute conventional semiconductors at least for some functionalities. Here we present a minimal setup for such a 2DEG––the SrTiO3(110)-(4 × 1) surface, natively terminated with one monolayer of tetrahedrally coordinated titania. Oxygen vacancies induced by synchrotron radiation migrate underneath this overlayer; this leads to a confining potential and electron doping such that a 2DEG develops. Our angle-resolved photoemission spectroscopy and theoretical results show that confinement along (110) is strikingly different from the (001) crystal orientation. In particular, the quantized subbands show a surprising “semiheavy” band, in contrast with the analog 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.

On refined volatility smile expansion in the Heston model

It is known that Hestons stochastic volatility model exhibits moment explosion, and that the critical moment s + can be obtained by solving (numerically) a simple equation. This yields a leading-order expansion for the implied volatility at large strikes: σBS(k, T)2 T ∼ Ψ(s + − 1) × k (Roger Lees moment formula). Motivated by recent ‘tail-wing’ refinements of this moment formula, we first derive a novel tail expansion for the Heston density, sharpening previous work of Drăgulescu and Yakovenko [Quant. Finance, 2002, 2(6), 443–453], and then show the validity of a refined expansion of the type σBS(k, T)2 T = (β1 k 1/2 + β2 + ···)2, where all constants are explicitly known as functions of s +, the Heston model parameters, the spot vol and maturity T. In the case of the ‘zero-correlation’ Heston model, such an expansion was derived by Gulisashvili and Stein [Appl. Math. Optim., 2010, 61(3), 287–315]. Our methods and results may prove useful beyond the Heston model: the entire quantitative analysis is based on affine principles and at no point do we need knowledge of the (explicit, but cumbersome) closed-form expression of the Fourier transform of log S T (equivalently the Mellin transform of S T ). What matters is that these transforms satisfy ordinary differential equations of the Riccati type. Secondly, our analysis reveals a new parameter (the ‘critical slope’), defined in a model-free manner, which drives the second- and higher-order terms in tail and implied volatility expansions.

Asymptotics and duality for the Davis and Norman problem

We revisit the problem of maximizing expected logarithmic utility from consumption over an infinite horizon in the Black–Scholes model with proportional transaction costs, as studied in the seminal paper of Davis and Norman [Math. Operation Research, 15, 1990]. Similar to Kallsen and Muhle-Karbe [Ann. Appl. Probab., 20, 2010], we tackle this problem by determining a shadow price, that is a frictionless price process with values in the bid-ask spread which leads to the same optimization problem. However, we use a different parametrization that facilitates computation and verification. Moreover, for small transaction costs, we determine fractional Taylor expansions of arbitrary order for the boundaries of the no-trade region and the value function. This extends work of Janeček and Shreve [Finance Stoch., 8, 2004], who determined the leading terms of these power series.

A procedure for proving special function inequalities involving a discrete parameter

We define a class of special function inequalities that contains many classical examples, such as the Cauchy-Schwarz inequality, and introduce a proving procedure based on induction and Cylindrical Algebraic Decomposition. We present an array of non-trivial examples that can be done by our method. Most of them have not been proven automatically before. Some difficult well-known inequalities such as the Askey-Gasper inequality and Vietoriss inequality lie in our class as well, but we do not know if our proving procedure terminates for them.

Water Adsorption at the Tetrahedral Titania Surface Layer of SrTiO3(110)-(4 × 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) and photoemission spectroscopy (PES)) study of water interaction with the two-dimensional titania overlayer that terminates the SrTiO3(110)-(4 × 1) surface and consists of TiO4 tetrahedra. STM and core-level and valence band PES show that H2O neither adsorbs nor dissociates on the stoichiometric surface at room temperature, whereas it does dissociate 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.

Asymptotics for a variant of the Mittag–Leffler function

We generalize the Mittag–Leffler function by attaching an exponent to its Taylor coefficients. The main result is an asymptotic formula valid in sectors of the complex plane, which extends the work by Le Roy [Valeurs asymptotiques de certaines séries procédant suivant les puissances entières et positives d’une variable réelle, Bull. des Sciences Math. 24, 1900] and Evgrafov [Asimptoticheskie otsenki i tselye funktsii, 3rd ed., Nauka, Moscow, 1979]. It is established by Plana’s summation formula in conjunction with the saddle point method. As an application, we (re-)prove a non-holonomicity result about powers of the factorial sequence.

Stabilizing Single Ni Adatoms on a Two-Dimensional Porous Titania Overlayer at the SrTiO3(110) Surface.

Nickel vapor-deposited on the SrTiO3(110) surface was studied using scanning tunneling microscopy, photoemission spectroscopy (PES), and density functional theory calculations. This surface forms a (4 × 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.

Point lattices and oscillating recurrence sequences

We consider the following question: which real sequences that satisfy a linear recurrence with constant coefficients are positive for sufficiently large n? We show that the answer is negative for both and if the dominating characteristic roots in the representation of as a generalized power sum comprise either two pairs of conjugate complex roots or one negative real root and one pair of conjugate complex roots. The proof uses results from Diophantine approximation and the geometry of numbers. Concerning the case of a real positive dominating root we show what the answer to the question of positivity is for almost all values of the other dominating roots, provided that all dominating roots are simple.

Nickel-Oxide-Modified SrTiO3(110)-(4 x 1) Surfaces and Their Interaction with Water

Nickel oxide (NiO), deposited onto the strontium titanate (SrTiO3) (110)-(4 × 1) surface, was studied using photoemission spectroscopy (PES), X-ray absorption near edge structure (XANES), and low-energy He+ ion scattering (LEIS), as well as scanning tunneling microscopy (STM). The main motivation for studying this system comes from the prominent role it plays in photocatalysis. The (4 × 1) reconstructed SrTiO3(110) surface was previously found to be remarkably inert toward water adsorption under ultrahigh-vacuum conditions. Nickel oxide grows on this surface as patches without any apparent ordered structure. PES and LEIS reveal an upward band bending, a reduction of the band gap, and reactivity toward water adsorption upon deposition of NiO. Spectroscopic results are discussed with respect to the enhanced reactivity toward water of the NiO-loaded surface.