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In the first part of this thesis, we introduce the concept of prospective strict no-arbitrage for discrete-time financial market models with proportional transaction. The prospective strict no-arbitrage condition, which is a variant of strict no-arbitrage, is slightly weaker than the robust no-arbitrage condition. It still implies that the set of portfolios attainable from zero initial endowment is closed in probability. Consequently, prospective strict no-arbitrage implies the existence of consistent prices, which may lie on the boundary of the bid-ask spread. A weak version of prospective strict no-arbitrage turns out to be equivalent to the existence of a consistent price system.
In continuous-time financial market models with proportional transaction costs, efficient friction, i.e., nonvanishing transaction costs, is a standing assumption. Together with robust no free lunch with vanishing risk, it rules out strategies of infinite variation which usually appear in frictionless financial markets. In the second part of this thesis, we show how models with and without transaction costs can be unified. The bid and the ask price of a risky asset are given by cadlag 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 long-only 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.
Die vorliegende Dissertation analysiert Großinvestorhandelsstrategien in illiquiden Finanzmärkten. Ein Großinvestor beeinflusst die Preise der Wertpapiere, die er handelt, so dass der daraus resultierende Feedbackeffekt berücksichtigt werden muss. Der Preisprozess wird durch eine Familie von cadlag Semimartingalen modelliert, die in dem zusätzlichen Parameter stetig differenzierbar ist. Ziel ist es, eine möglichst allgemeine Strategiemenge zu bestimmen, für die eine Vermögensdynamik definiert werden kann. Es sind dies vorhersehbare Prozesse von wohldefinierter quadratischer Variation entlang Stoppzeiten. Sie erweisen sich als laglad. Die Vermögensdynamikzerlegung zeigt, dass bei stetigen adaptierten Strategien von endlicher Variation (zahme Strategien) die quadratischen Transaktionskostenterme verschwinden und der Gewinnprozess nur noch aus einem nichtlinearen stochastischen Integral besteht. Es wird gezeigt, unter welchen Bedingungen gewisse Approximationen vorhersehbarer laglad Strategien durch adaptierte stetige Strategien von endlicher Variation möglich sind. Im Fall, dass der Approximationsfehler für die Risikoeinstellung des Großinvestors erträglich ist, kann er seine Investmentziele durch Verwendung dieser zahmen Strategien, Liquiditätskosten vermeidend, erreichen. In diesem Fall ist der Gewinnprozess durch das nicht-lineare stochastische Integral gegeben.
We provide a mathematical framework to model continuous time trading in limit order markets of a small investor whose transactions have no impact on order book dynamics. The investor can continuously place market and limit orders. A market order is executed immediately at the best currently available price, whereas a limit order is stored until it is executed at its limit price or canceled. The limit orders can be chosen from a continuum of limit prices.
In this framework we show how elementary strategies (hold limit orders with only finitely many different limit prices and rebalance at most finitely often) can be extended in a suitable
way to general continuous time strategies containing orders with infinitely many different limit prices. The general limit buy order strategies are predictable processes with values in the set of nonincreasing demand functions (not necessarily left- or right-continuous in the price variable). It turns out that this family of strategies is closed and any element can be approximated by a sequence of elementary strategies.
Furthermore, we study Merton’s portfolio optimization problem in a specific instance of this framework. Assuming that the risky asset evolves according to a geometric Brownian
motion, a proportional bid-ask spread, and Poisson execution times for the limit orders of the small investor, we show that the optimal strategy consists in using market orders to keep the
proportion of wealth invested in the risky asset within certain boundaries, similar to the result for proportional transaction costs, while within these boundaries limit orders are used to profit from the bid-ask spread.
In the first part of the thesis, 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. In addition, we give an example for tax-efficient strategies for which the tax payment flow can be computed explicitly.
In the second part of the thesis, 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.
We study the price-setting problem of market makers under perfect competition in continuous time. Thereby we follow the classic Glosten-Milgrom model that defines bid and ask prices as the expectation 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 by solving a non-standard 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. In a different setting we construct a counterexample to uniqueness. Further, we discuss the behavior of the spread by a convergence result and simulations.