arXiv subjects

Christoph Kühn

Publications and source records attributed to Christoph Kühn.

12 recordsLinked to original sources

arXiv2026

Optimal investment under capital gains taxes

We generalize classical existence results for expected utility maximization in discrete time frictionless market models to models with capital gains taxes. We consider the realistic but mathematically challenging rule that losses do not trigger negative taxes but can only be offset against potential gains in the future. Central to the analysis is a well-known phenomenon from arbitrage-free markets with proportional transaction costs that does not exist in arbitrage-free frictionless markets: an investment in specific quantities of stocks that is completely riskless but may provide an advantage over holding money in the bank account. As a result of this phenomenon, on an infinite probability space, no-arbitrage does not imply that the set of attainable terminal wealth is closed in probability. We provide simple sufficient conditions for closedness. Then, we characterize the closure of the set of attainable terminal wealth, thereby identifying precisely the source of non-closedness. As a by-product, we obtain a new construction for an integrable majorant that dominates the utilities of all nonnegative terminal wealth attainable from a given initial capital in a frictionless market and that works directly in multiperiod models.

q-fin.MF↗

arXiv2024

The fundamental theorem of asset pricing with and without transaction costs

We prove a version of the fundamental theorem of asset pricing (FTAP) in continuous time that is based on the strict no-arbitrage condition and that is applicable to both frictionless markets and markets with proportional transaction costs. We consider a market with a single risky asset whose ask price process is higher than or equal to its bid price process. Neither the concatenation property of the set of wealth processes, that is used in the proof of the frictionless FTAP, nor some boundedness property of the trading volume of admissible strategies usually argued with in models with a nonvanishing bid-ask spread need to be satisfied in our model.

q-fin.MF↗

arXiv2024

Insider trading in discrete time Kyle games

We present a new discrete time version of Kyle's (1985) classic model of insider trading, formulated as a generalised extensive form game. The model has three kinds of traders: an insider, random noise traders, and a market maker. The insider aims to exploit her informational advantage and maximise expected profits while the market maker observes the total order flow and sets prices accordingly. First, we show how the multi-period model with finitely many pure strategies can be reduced to a (static) social system in the sense of Debreu (1952) and prove the existence of a sequential Kyle equilibrium, following Kreps and Wilson (1982). This works for any probability distribution with finite support of the noise trader's demand and the true value, and for any finite information flow of the insider. In contrast to Kyle (1985) with normal distributions, equilibria exist in general only in mixed strategies and not in pure strategies. In the single-period model we establish bounds for the insider's strategy in equilibrium. Finally, we prove the existence of an equilibrium for the game with a continuum of actions, by considering an approximating sequence of games with finitely many actions. Because of the lack of compactness of the set of measurable price functions, standard infinite-dimensional fixed point theorems are not applicable.

q-fin.TR↗

arXiv2021

Semimartingale price systems in models with transaction costs beyond efficient friction

A standing assumption in the literature on proportional transaction costs is efficient friction. Together with robust no free lunch with vanishing risk, it rules out strategies of infinite variation, as they usually appear in frictionless markets. In this paper, we show how the models with and without transaction costs can be unified. The bid and the ask price of a risky asset are given by cádlág processes which are locally bounded from below and may coincide at some points. In a first step, we show that if the bid-ask model satisfies "no unbounded profit with bounded risk" for simple strategies, then there exists a semimartingale lying between the bid and the ask price process. In a second step, under the additional assumption that the zeros of the bid-ask spread are either starting points of an excursion away from zero or inner points from the right, we show that for every bounded predictable strategy specifying the amount of risky assets, the semimartingale can be used to construct the corresponding self-financing risk-free position in a consistent way. Finally, the set of most general strategies is introduced, which also provides a new view on the frictionless case.

q-fin.MF↗

arXiv2020

picFoam: An OpenFOAM based electrostatic Particle-in-Cell solver

picFoam is a fully kinetic electrostatic Particle-in-Cell(PIC) solver, including Monte Carlo Collisions(MCC), for non-equilibrium plasma research in the open-source framework of OpenFOAM. The solver's modular design, based on the same principles used in OpenFOAM, makes it highly flexible, by allowing the user to choose different methods at run time, and extendable, by building upon templated modular classes. The implementation of the PIC method employing the finite volume method, allows it to simulate on arbitrary geometries in one to three dimensions. OpenFOAM's barycentric particle tracking is used effectively to performe charge and field weighting from the Lagrangian particle based description to the Eulerian field description and backwards without computational expensive particle searching algorithm. picFoam also includes open and general circuit boundary models for the description of real plasma devices.

physics.plasm-ph↗

arXiv2019

Prospective strict no-arbitrage and the fundamental theorem of asset pricing under transaction costs

In discrete time markets with proportional transaction costs, Schachermayer (2004) shows that robust no-arbitrage is equivalent to the existence of a strictly consistent price system. In this paper, we introduce the concept of prospective strict no-arbitrage that is a variant of the strict no-arbitrage property from Kabanov, Rásonyi, and Stricker (2002). The prospective strict no-arbitrage condition is slightly weaker than robust no-arbitrage, and it implies that the set of portfolios attainable from zero initial endowment is closed in probability. A weak version of prospective strict no-arbitrage turns out to be equivalent to the existence of a consistent price system. In contrast to the fundamental theorem of asset pricing of Schachermayer (2004), the consistent frictionless prices may lie on the boundary of the bid-ask spread. On the technical level, a crucial difference to Schachermayer (2004) and Kabanov-Rásonyi-Stricker (2003) is that we prove closedness without having at hand that the null-strategies form a linear space.

q-fin.MF↗

arXiv2018

How local in time is the no-arbitrage property under capital gains taxes ?

In frictionless financial markets, no-arbitrage is a local property in time. This means that a discrete time model is arbitrage-free if and only if there does not exist a one-period-arbitrage. With capital gains taxes, this equivalence fails. For a model with a linear tax and one non-shortable risky stock, we introduce the concept of robust local no-arbitrage (RLNA) as the weakest local condition which guarantees dynamic no-arbitrage. Under a sharp dichotomy condition, we prove (RLNA). Since no-one-period-arbitrage is necessary for no-arbitrage, the latter is sandwiched between two local conditions, which allows us to estimate its non-locality. Furthermore, we construct a stock price process such that two long positions in the same stock hedge each other. This puzzling phenomenon that cannot occur in arbitrage-free frictionless markets (or markets with proportional transaction costs) is used to show that no-arbitrage alone does not imply the existence of an equivalent separating measure if the probability space is infinite. Finally, we show that the model with a linear tax on capital gains can be written as a model with proportional transaction costs by introducing several fictitious securities.

q-fin.PM↗

arXiv2018

Nash equilibria for game contingent claims with utility-based hedging

Game contingent claims (GCCs) generalize American contingent claims by allowing the writer to recall the option as long as it is not exercised, at the price of paying some penalty. In incomplete markets, an appealing approach is to analyze GCCs like their European and American counterparts by solving option holder's and writer's optimal investment problems in the underlying securities. By this, partial hedging opportunities are taken into account. We extend results in the literature by solving the stochastic game corresponding to GCCs with both continuous time stopping and trading. Namely, we construct Nash equilibria by rewriting the game as a non-zero-sum stopping game in which players compare payoffs in terms of their exponential utility indifference values. As a by-product, we also obtain an existence result for the optimal exercise time of an American claim under utility indifference valuation by relating it to the corresponding nonlinear Snell envelope.

math.PR↗

arXiv2015

Modeling capital gains taxes for trading strategies of infinite variation

In this article we show that the payment flow of a linear tax on trading gains from a security with a semimartingale price process can be constructed for all càglàd and adapted trading strategies. It is characterized as the unique continuous extension of the tax payments for elementary strategies w.r.t. the convergence "uniformly in probability". In this framework we prove that under quite mild assumptions dividend payoffs have almost surely a negative effect on investor's after-tax wealth if the riskless interest rate is always positive.

q-fin.PM↗

arXiv2015

Optimal Liquidity Provision

A small investor provides liquidity at the best bid and ask prices of a limit order market. For small spreads and frequent orders of other market participants, we explicitly determine the investor's optimal policy and welfare. In doing so, we allow for general dynamics of the mid price, the spread, and the order flow, as well as for arbitrary preferences of the liquidity provider under consideration.

q-fin.TR↗

arXiv2014

Optimal Selling Time of a Stock under Capital Gains Taxes

We investigate the impact of capital gains taxes on optimal investment decisions in a quite simple model. Namely, we consider a risk neutral investor who owns one risky stock from which she assumes that it has a lower expected return than the riskless bank account and determine the optimal stopping time at which she sells the stock to invest the proceeds in the bank account up to the maturity date. In the case of linear taxes and a positive riskless interest rate, the problem is nontrivial because at the selling time the investor has to realize book profits which triggers tax payments. We derive a boundary that is continuous and increasing in time and decreasing in the volatility of the stock such that the investor sells the stock at the first time its price is smaller or equal to this boundary.

q-fin.PM↗

arXiv2012

Price-Setting of Market Makers: A Filtering Problem with an Endogenous Filtration

We study the price-setting problem of market makers under risk neutrality and perfect competition in continuous time. Thereby we follow the classic Glosten-Milgrom model that defines bid and ask prices as expectations of a true value of the asset given the market makers' partial information that includes the customers trading decisions. The true value is modeled as a Markov process that can be observed by the customers with some noise at Poisson times. We analyze the price-setting problem in a mathematically rigorous way by solving a filtering problem with an endogenous filtration that depends on the bid and ask price process quoted by the market maker. Under some conditions we show existence and uniqueness of the price processes.

q-fin.TR↗