Klaus Reiner Schenk-Hoppé

Risk Minimization in Stochastic Volatility Models: Model Risk and Empirical Performance

In this paper the performance of locally risk-minimizing delta hedge strategies for European options in stochastic volatility models is studied from an experimental as well as from an empirical perspective. These hedge strategies are derived for a large class of diffusion-type stochastic volatility models, and they are as easy to implement as usual delta hedges. Our simulation results on model risk show that these risk-minimizing hedges are robust with respect to uncertainty and misconceptions about the underlying data generating process. The empirical study, which includes the US sub-prime crisis period, documents that in equity markets risk-minimizing delta hedges consistently outperform usual delta hedges by approximately halving the standard deviation of the profit-and-loss ratio.

The evolution of Walrasian behavior in oligopolies

Abstract This paper studies an explicitly dynamic evolutionary model of Cournot oligopoly in which the behavior of firms is based on imitation of success and experimentation. It is proved that only (evolutionary) stable sets are selected by the unique invariant measure of the induced Markov chain as the probability of experimentation tends to zero. Evolutionary stability refers to a finite population set-up here. We show that stable singletons and symmetric Walrasian equilibria coincide in general. Furthermore, Walrasian equilibria can be approximated by stable proper intervals. These results generalize and extend the approach of Vega-Redondo [Vega-Redondo, F., 1997. The evolution of Walrasian behavior. Econometrica 65, 375–384].

The Stochastic Brusselator: Parametric Noise Destroys Hoft Bifurcation

We perform mainly a numerical study of the bifurcation behavior of the Brusselator under parametric white noise. It was shown before that parametric noise turns the deterministic Hopf bifurcation into a scenario in which the stationary density (unique solution of the Fokker- Planck equation) undergoes a delayed transition from a single-peaked, bellshaped to a crater-type form. We will make this more precise by showing that the stationary density gets a “dent” at the deterministic bifurcation point and develops a local minimum at a later parameter value. In contrast (but not in contradiction) to these findings we will show that, from the view point of random dynamical systems, the deterministic Hopf bifurcation is being “destroyed” by parametric noise in the following sense: For all values of the bifurcation parameter, the system has a unique invariant measure which is, moreover, exponentially stable in the sense that its top Lyapunov exponent is negative. The invariant measure is a random Dirac measure, and its support is the global random attractor of the system.

Exponential Growth of Fixed-Mix Strategies in Stationary Asset Markets

Abstract. The paper analyzes the long-run performance of dynamic investment strategies based on fixed-mix portfolio rules. Such rules prescribe rebalancing the portfolio by transferring funds between its positions according to fixed (time-independent) proportions. The focus is on asset markets where prices fluctuate as stationary stochastic processes. Under very general assumptions, it is shown that any fixed-mix strategy in a stationary market yields exponential growth of the portfolio with probability one.

Volatility-induced financial growth

We show that the volatility of a price process, which is usually regarded as an impediment to financial growth, can serve as an endogenous factor in its acceleration.

On the Micro-Foundations of Money: The Capitol Hill Baby-Sitting Co-Op

This paper contributes to the micro-foundation of money in centralized markets with idiosyncratic uncertainty. It shows existence of stationary monetary equilibria and ensures that there is an optimum quantity of money. The rational solution of our model is compared with actual behavior in a laboratory experiment. The experiment gives support to the theoretical approach.

Financial markets. The joy of volatility

Financial markets. The joy of volatility M. A. H. Dempster a; Igor V. Evstigneev b; Klaus Reiner Schenk-Hoppe c a Centre for Financial Research, Statistical Laboratory, University of Cambridge, Cambridge CB5 8AF, UK b Economic Studies, School of Social Sciences, University of Manchester, Manchester, M13 9PL, UK c School of Mathematics and Leeds University Business School, University of Leeds, Leeds, LS2 9JT, UK

The Great Capitol Hill Baby Sitting Co-Op: Anecdote or Evidence for the Optimum Quantity of Money?

This paper studies a centralized market with idiosyncratic uncertainty and money as a medium of exchange from a theoretical as well as an experimental perspective. In our model, prices are fixed and markets are cleared by rationing. We prove the existence of stationary monetary equilibria and of an optimum quantity of money. The rational solution of our model, which is based on the assumption of individual rationality and rational expectations, is compared with actual behavior in a laboratory experiment. The theoretical results are strongly supported by this experiment.

The von Neumann-Gale growth model and its stochastic generalization

This paper deals with the deterministic and stochastic versions of the von Neumann-Gale model. Von Neumanns (1937) original concern was to determine a balanced path growing at a maximal rate for a linear and stationary technology and a price system supporting that path. Such a pair (a path and a price system) was called a von Neumann equilibrium. Gale (1956) proposed a version of the model assuming a general, not necessarily linear technology. This generalization led to a rich and interesting theory aimed at the analysis of efficient growth paths and based on the concepts introduced by von Neumann. The theory was developed in the 1950s and 1960s by Rockafellar, Radner, McKenzie, Nikaido, Morishima, and others. Its extension to the stochastic case was an open problem for about three decades, and a substantial progress in this direction was made only in the 1990s. The main results obtained are concerned with stochastic generalizations of a von Neumann equilibrium and efficient paths. Fundamental questions about their existence, uniqueness and stability (turnpike properties) are answered. The chapter gives an account of the achievements in the field and outlines new applications of the von Neumann-Gale model in finance related to asset pricing and hedging in securities markets.

From Discrete to Continuous Time Evolutionary Finance Models

This paper aims to open a new avenue for research in continuous-time financial market models with endogenous prices and heterogenous investors. To this end we introduce a discrete-time evolutionary stock market model that accommodates time periods of arbitrary length. The dynamics is time-consistent and allows the comparison of paths with different frequency of trade. The main result in this paper is the derivation of the limit model as the length of the time period tends to zero. The resulting model in continuous time generalizes the workhorse model of mathematical finance by introducing asset prices that are driven by the market interaction of investors following self-financing trading strategies. Our approach also offers a numerical scheme for the simulation of the continuous-time model that satisfies constraints such as market clearing at every time step. An illustration is provided.