We prove a version of the Fundamental Theorem of Asset Pricing, which applies to Kabanov's approach to foreign exchange markets under transaction costs. The financial market is modelled by a d x d matrix-valued stochastic process Sigma_t_t=0^T specifying the mutual bid and ask prices between d assets. We introduce the notion of ``robust no arbitrage", which is a version of the no arbitrage concept, robust with respect to small changes of the bid ask spreads of Sigma_t_t=0^T. Dually, we interpret a concept used by Kabanov and his co-authors as "strictly consistent price systems". We show that this concept extends the notion of equivalent martingale measures, playing a well-known role in the frictionless case, to the present setting of bid-ask processes Sigma_t_t=0^T. The main theorem states that the bid-ask process Sigma_t_t=0^T satisfies the robust no arbitrage condition if it admits a strictly consistent pricing system. This result extends the theorems of Harrison-Pliska and Dalang-Morton-Willinger to the present setting, and also generalizes previous results obtained by Kabanov, Rasonyi and Stricker. An example of a 5-times-5-dimensional process Sigma_t_t=0^2 shows that, in this theorem, the robust no arbitrage condition cannot be replaced by the so-called strict no arbitrage condition, thus answering negatively a question raised by Kabanov, Rasonyi and Stricker. (author's abstract) / Series: Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

Title: Exponential control of homogeneous Markov processes Author: Pavol Stanek Department: Department of Probability and Mathematical Statistics, MFF UK Supervisor: Mgr. Peter Dostál Ph.D., Department of Probability and Mathematical Statistics, MFF UK Abstract: This master thesis concerns exponential control of Markov decision chains. An iterative alghorithm for finding a control, that maximizes a long term growth rate of expected utility is developed. The utility is measured by exponential utility function. The algorithm is derived for both discrete time and continuous time chain. Subsequently, the results are applied on the problem of optimally managing port- folio with proportional transaction costs. The dynamics of the investor's position is derived and the consequent process is approximated by Markov chain. Using the iterative alghorithm, the optimal trading strategy is numerically found. Keywords: exponential control, Markov chain, portfolio optimization, proportional transaction costs 1

In this thesis we study the utility maximization problem for assets whose prices are cointegrated, which arises from the investment practice of convergence trading and its special forms, pairs trading and spread trading. The major theme in the first two chapters of the thesis, is to investigate the assumption of market-neutrality of the optimal convergence trading strategies, which is a ubiquitous assumption taken by practitioners and academics alike. This assumption lacks a theoretical justification and, to the best of our knowledge, the only relevant study is Liu and Timmermann (2013) which implies that the optimal convergence strategies are, in general, not market-neutral. We start by considering a minimalistic pairs-trading scenario with two cointegrated stocks and solve the Merton investment problem with power and logarithmic utilities. We pay special attention to when/if the stochastic control problem is well-posed, which is overlooked in the study done by Liu and Timmermann (2013). In particular, we show that the problem is ill-posed if and only if the agent’s risk-aversion is less than a constant which is an explicit function of the market parameters. This condition, in turn, yields the necessary and sufficient condition for well-posedness of the Merton problem for all possible values of agent’s risk-aversion. The resulting well-posedness condition is surprisingly strict and, in particular, is equivalent to assuming the optimal investment strategy in the stocks to be market-neutral. Furthermore, it is shown that the well-posedness condition is equivalent to applying Novikov’s condition to the market-price of risk, which is a ubiquitous sufficient condition for imposing absence of arbitrage. To the best of our knowledge, these are the only theoretical results for supporting the assumption of market-neutrality of convergence trading strategies. We then generalise the results to the more realistic setting of multiple cointegrated assets, assuming risk factors that effects the asset returns, and general utility functions for investor’s preference. In the process of generalising the bivariate results, we also obtained some well-posedness conditions for matrix Riccati differential equations which are, to the best of our knowledge, new. In the last chapter, we set up and justify a Merton problem that is related to spread-trading with two futures assets and assuming proportional transaction costs. The model possesses three characteristics whose combination makes it different from the existing literature on proportional transaction costs: 1) finite time horizon, 2) Multiple risky assets 3) stochastic opportunity set. We introduce the HJB equation and provide rigorous arguments showing that the corresponding value function is the viscosity solution of the HJB equation. We end the chapter by devising a numerical scheme, based on the penalty method of Forsyth and Vetzal (2002), to approximate the viscosity solution of the HJB equation.

Dans cette thèse, nous nous intéressons à des sujets différents en mathématiques financières, tous liés aux imperfections de marché et à la technique fondamentale de la maximisation d'utilité. Elle comporte trois parties. Dans la première, qui se base sur deux papiers, nous considérons le problème d'investissement optimal sur un marché financier avec coûts de transaction proportionnels. On commence par étudier le problème d'investissement dans le cas où la fonction d'utilité est multivariée (ce qui s'adapte particulièrement bien aux marchés des devises) et l'agent a une dotation initiale aléatoire, qui peut s'interpréter comme une option ou un autre contrat dérivé. Après avoir analysé les propriétés du problème et de son dual, nous utilisons ces résultats pour examiner, dans ce contexte, certains aspects d'une technique de pricing devenue populaire dans le cadre des marchés incomplets, l'évaluation par indifférence d'utilité. Dans le deuxième chapitre, nous étudions le problème d'existence d'un ensemble de prix (appelés "prix fictifs" ou "shadow prices") qui offrirait la même utilité maximale à l'agent si le marché n'avait pas de frictions. Ces résultats sont utiles pour clarifier le lien entre la théorie classique des marchés sans frictions et la littérature en croissance rapide sur les coûts de transaction. Dans la deuxième partie de cette thèse, nous considérons le problème d'évaluation de produits dérivés par indifférence d'utilité dans des marchés incomplets, où la source d'incomplétude provient du fait que certains actifs ne peuvent pas être échangés sur le marché, ce qui est le cas par exemple dans le cadre des modèles structurels pour le prix de l'électricité. Sous certaines hypothèses, nous dérivons une caractérisation en terme d'équations différentielles stochastiques rétrogrades (EDSR) pour le prix, et nous nous concentrons ensuite sur les options européennes en établissant en particulier l'existence d'une stratégie de couverture optimale, même lorsque le payoff présente des discontinuités et est éventuellement non borné. Dans la dernière partie, nous analysons un simple problème de principal-agent à horizon fini, où le principal est essentiellement interprété comme un régulateur et l'agent comme une entreprise qui produit certaines émissions polluantes. Nous traitons séparément les problèmes du principal et de l'agent et nous utilisons la théorie des EDSR pour fournir des conditions nécessaires et suffisantes d'optimalité. Nous effectuons également des analyses de sensibilité et nous montrons des résultats numériques dans le but de fournir une meilleure compréhension du comportement des agents. / In this thesis we deal with different topics in financial mathematics, that are all related to market imperfections and to the fundamental technique of utility maximization. The work consists of three parts. In the first one, which is based on two papers, we consider the problem of optimal investment on a financial market with proportional transaction costs. We initially study the investment problem in the case where the utility function is multivariate (which is particularly suitable on currency markets) and the agent is endowed with a random claim, which can be interpreted as an option or another derivative contract. After analyzing the properties of the primal and dual problems, we apply those results to investigate, in this context, some aspects of a popular pricing technique in incomplete markets, i.e. utility indifference evaluation. In the second contribution to the transaction costs literature, we investigate the existence problem for a set of prices (called shadow prices) that would provide the same maximal utility to the agent if the market did not have frictions. These results shed some light on the link between the classical theory of frictionless markets and the quickly growing literature on transaction costs. In the second part of this thesis we consider the utility indifference pricing problem in incomplete markets, where the source of incompleteness comes from the fact that some assets in the market cannot be actively traded, which is the case for example in the framework of structural models for electricity prices. We provide a BSDE characterization for the price under mild assumptions, and then focus on the case of European claims by establishing in particular the existence of an optimal hedging strategy even when the claim presents discontinuities and is possibly unbounded. In the last contribution we analyze a simple principal-agent problem in finite time horizon, where the principal is mainly interpreted as a regulator and the agent as a firm producing some kind of polluting emissions. We separately treat both the agent's and the principal's problems and use the BSDE theory for providing necessary and sufficient conditions for optimality. We also perform some sensitivity analyses and give numerical results in order to provide a better understanding of the agents' behavior.