Ralf Korn

Optimal Portfolios - Stochastic Models for Optimal Investment and Risk Management in Continuous Time

The focus of the book is the construction of optimal investment strategies in a security market model where the prices follow diffusion processes. It begins by presenting the complete Black-Scholes type model and then moves on to incomplete models and models including constraints and transaction costs. The models and methods presented will include the stochastic control method of Merton, the martingale method of Cox–Huang and Karatzas et al., the log optimal method of Cover and Jamshidian, the value-preserving model of Hellwig etc. Stress is laid on rigorous mathematical presentation and clear economic interpretations while technicalities are kept to the minimum. The underlying mathematical concepts will be provided. No a priori knowledge of stochastic calculus, stochastic control or partial differential equations is necessary (however some knowledge in stochastics and calculus is needed).

Optimal Portfolios with Bounded Capital at Risk

We consider some continuous-time Markowitz type portfolio problems that consist of maximizing expected terminal wealth under the constraint of an upper bound for the Capital-at-Risk. In a Black-Scholes setting we obtain closed form explicit solutions and compare their form and implications to those of the classical continuous-time mean-variance problem. We also consider more general price processes which allow for larger uctuations in the returns.

A Stochastic Control Approach to Portfolio Problems with Stochastic Interest Rates

We consider investment problems where an investor can invest in a savings account, stocks, and bonds and tries to maximize her utility from terminal wealth. In contrast to the classical Merton problem, we assume a stochastic interest rate. To solve the corresponding control problems it is necessary to prove a verification theorem without the usual Lipschitz assumptions.

Portfolio optimisation with strictly positive transaction costs and impulse control

Abstract. 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 nontrivial asymptotically optimal solution for the problem of exponential utility maximisation.

Some applications of impulse control in mathematical finance

Abstract. We consider three applications of impulse control in financial mathematics, a cash management problem, optimal control of an exchange rate, and portfolio optimisation under transaction costs. We sketch the different ways of solving these problems with the help of quasi-variational inequalities. Further, some viscosity solution results are presented.

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.

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.

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.

An Energy Efficient FPGA Accelerator for Monte Carlo Option Pricing with the Heston Model

Today, pricing of derivates (particularly options) in financial institutions is a challenge. Besides the increasing complexity of the products, obtaining fair prices requires more realistic (and therefore complex) models of the underlying asset behavior. Not only due to the increasing costs, energy efficient and accurate pricing of these models becomes more and more important. In this paper we present - to the best of our knowledge - the first FPGA based accelerator for option pricing with the state-of-the-art Heston model. It is based on advanced Monte Carlo simulations. Compared to an 8-core Intel Xeon Server running at 3.07GHz, our hybrid FPGA-CPU-system saves 89% of the energy and provides around twice the speed. The same system reduces the energy consumption per simulation to around 40% of a fully-loaded Nvidia Tesla C2050 GPU. For a three-Virtex-5 chip only accelerator, we expect to achieve the same simulation speed as a Nvidia Tesla C2050 GPU, by consuming less than 3% of the energy at the same time.

Worst-Case Scenario Portfolio Optimization: a New Stochastic Control Approach

We consider the determination of portfolio processes yielding the highest worst-case bound for the expected utility from final wealth if the stock price may have uncertain (down) jumps. The optimal portfolios are derived as solutions of non-linear differential equations which itself are consequences of a Bellman principle for worst-case bounds. A particular application of our setting is to model crash scenarios where both the number and the height of the crash are uncertain but bounded. Also the situation of changing market coefficients after a possible crash is analyzed.