0000000000303089
AUTHOR
Ralf Korn
Contingent claim valuation in a market with different interest rates
The problem of contingent claim valuation in a market with a higher interest rate for borrowing than for lending is discussed. We give results which cover especially the European call and put options. The method used is based on transforming the problem to suitable auxiliary markets with only one interest rate for borrowing and lending and is adapted from a paper of Cvitanic and Karatzas (1992) where the authors study constrained portfolio problems.
Das zeitstetige Marktmodell
Wir betrachten einen Markt, auf dem d+1 Wertpapiere gehandelt werden. Darunter befinden sich d Aktien mit Preisen p 1 , p 2 , ..., p d zur Zeit t=0 und zufalligen Preisen P1(t),..., P d (t) zur Zeit t, sowie ein risikoloses Wertpapier, genannt „Bond“ mit Preis p0 zur Zeit t=0, und deterministischem Preis P0(t) sonst. Das risikolose Wertpapier wird in seiner Modellierung eher einem Sparguthaben als einem Bond entsprechen (siehe unten), es wird aus historischen Grunden weiterhin von uns als Bond bezeichnet. Wir betrachten den endlichen Handlungszeitraum [0, T]. In unserem Modell sei jede beliebige Stuckelung der Wertpapiere zulassig und es gebe keine Transaktionskosten bei Kauf bzw. Verkauf d…
Value preserving portfolio strategies in continuous-time models
We present a new approach for continuous-time portfolio strategies that relies on the principle of value preservation. This principle was developed by Hellwig (1987) for general economic decision and pricing models. The key idea is that an investor should try to consume only so much of his portfolio return that the future ability of the portfolio should be kept constant over time. This ensures that the portfolio will be a long lasting source of income. We define a continuous-time market setting to apply the idea of Hellwig to securities markets with continuous trading and examine existence (and uniqueness) of value-preserving strategies in some widely used market models. Further, we discuss…
Optimal Index Tracking Under Transaction Costs and Impulse Control
We apply impulse control techniques to a cash management problem within a mean-variance framework. We consider the strategy of an investor who is trying to minimise both fixed and proportional transaction costs, whilst minimising the tracking error with respect to an index portfolio. The cash weight is constantly fluctuating due to the stochastic inflow and outflow of dividends and liabilities. We show the existence of an optimal strategy and compute it numerically.
Portfolio optimisation with strictly positive transaction costs and impulse control
One crucial assumption in modern portfolio theory of continuous-time models is the no transaction cost assumption. This assumption normally leads to trading strategies with infinite variation. However, following such a strategy in the presence of transaction costs will lead to immediate ruin. We present an impulse control approach where the investor can change his portfolio only finitely often in finite time intervals. Further, we consider transaction costs including a fixed and a proportional cost component. For the solution of the resulting control problems we present a formal optimal stopping approach and an approach using quasi-variational inequalities. As an application we derive a non…
Optimal control of option portfolios and applications
We present an expected utility maximisation framework for optimally controlling a portfolio of options. By combining the replication approach to option pricing with ideas of the martingale approach to (stock) portfolio optimisation we arrive at an explicit solution of the option portfolio problem. Its characteristics are illustrated by some specific examples. As an application, we calculate an optimal option and consumption strategy for an investor who is obliged to hold a stock position until the time horizon.
Der Erwartungswert-Varianz-Ansatz im Ein-Perioden-Modell
Bevor wir uns mit den zeitstetigen Marktmodellen beschaftigen, wollen wir hier als Einfuhrung ein einfaches Ein-Perioden-Modell betrachten. Der mathematische Startpunkt der Theorie der Portfolio-Optimierung war 1952 die Arbeit von H. Markowitz (1952) uber den Erwartungswert-Varianz-Ansatz zur Beurteilung von Investmentstrategien an Wertpapiermarkten. Aufgrund seiner Einfachheit und Plausibilitat wurde er schnell sehr popular in Theorie und Praxis und wird auch heute noch haufig angewendet. Verdientermasen erhielt Markowitz 1990 zusammen mit zwei anderen Wissenschaftlern den Nobelpreis fur Wirtschaftswissenschaften. Allerdings liegen in der Einfachheit des Erwartungswert-Varianz-Ansatzes auc…
Continuous-time portfolio optimization under terminal wealth constraints
Typically portfolio analysis is based on the expected utility or the mean-variance approach. Although the expected utility approach is the more general one, practitioners still appreciate the mean-variance approach. We give a common framework including both types of selection criteria as special cases by considering portfolio problems with terminal wealth constraints. Moreover, we propose a solution method for such constrained problems.
Value preserving portfolio strategies and the minimal martingale measure
We consider some relations between the minimal martingale measure and the value preserving martingale measure in a continuous-time securities market. Under the assumption of continuous share prices we show that under a structure condition both these martingale measures exist and indeed coincide. This however does not mean that the corresponding concepts of value preserving portfolio strategies and (local) risk minimisation in the area of option hedging in incomplete markets are identical.
Optimal Impulse Control When Control Actions Have Random Consequences
We consider a generalised impulse control model for controlling a process governed by a stochastic differential equation. The controller can only choose a parameter of the probability distribution of the consequence of his control action which is therefore random. We state optimality results relating the value function to quasi-variational inequalities and a formal optimal stopping problem. We also remark that the value function is a viscosity solution of the quasi-variational inequalities which could lead to developments and convergence proofs of numerical schemes. Further, we give some explicit examples and an application in financial mathematics, the optimal control of the exchange rate…