Klaus Pötzelberger

Stochastic Equilibrium: Learning by Exponential Smoothing

This article considers three standard asset pricing models with adaptive agents and stochastic dividends. The models only differ in the parameters to be estimated. We assume that only limited information is used to construct estimators. Therefore, parameters are not estimated consistently. More precisely, we assume that the parameters are estimated by exponential smoothing, where past parameters are down-weighted and the weight of recent observations does not decrease with time. This situation is familiar for applications in finance. Even if time series of volatile stocks or bonds are available for a long time, only recent data is used in the analysis. In this situation the prices do not converge and remain a random variable. This raises the question how to describe equilibrium behavior with stochastic prices. However, prices can reveal properties such as ergodicity, such that the law of the price process converges to a stationary law, which provides a natural and useful extension of the idea of equilibrium behavior of an economic system for a stochastic setup. It is this implied law of the price process that we investigate in this paper. We provide conditions for the ergodicity and analyze the stationary distribution. (authors abstract)

Consistent pricing of warrants and traded options

Abstract Warrants issuance affects the stock price process of the issuing company. This change in the stock price process leads to subsequent changes in the prices of options written on the issuing companys stocks. In this paper, we investigate the effects of warrants issuance on the prices of traded options (bought and sold by third parties) already outstanding at the time of warrants issuance. We show how these options can be valued as portfolios of standard and compound options written on the stock of an otherwise similar company without warrants and derive a closed-form solution for the Black-Scholes model. An application of these results to empirical data shows that warrants issuance can have very large effects on the prices of outstanding traded options.

On the Fisher information of discretized data

In this paper we study the loss of Fisher information in approximating a continuous distribution by a multinomial distribution coming from a partition of the sample space into a finite number of intervals. We describe and characterize the Fisher information as a function of the partition chosen especially for location parameters. For a small number of intervals theconsequences of the choice is demonstrated by instructive examples. For increasing numbers of intervals we give the asymptotically optimal partition.

Consistency of the empirical quantization error

The asymptotic behavior of the quantization error allows the definition of a dimension for a probability distribution P, the quantization dimension. This concept fits into standard geometric measure theory, as the quantization dimension is always between the Hausdorff and the box-counting dimension. It is operational in the sense that the empirical quantization error may be computed and used as an estimator of the quantization error of P. We study the empirical quantization error in case the number of prototypes increases with the size of the sample. Results of the consistency of the empirical quantization error and estimators of the quantization dimensions of distributions are given. The results depend on geometrical properties of optimal partitions, such as the eccentricity of the cells and the number of neighbors.

Improving the Monte Carlo estimation of boundary crossing probabilities by control variables

Abstract. We propose an efficient Monte Carlo approach to compute boundary crossing probabilities (BCP) for Brownian motion and a large class of diffusion processes, the method of adaptive control variables. For the Brownian motion the boundary b (or the boundaries in case of two-sided boundary crossing probabilities) is approximated by a piecewise linear boundary , which is linear on m intervals. Monte Carlo estimators of the corresponding BCP are based on an m-dimensional Gaussian distribution. Let N denote the number of (univariate) Gaussian variables used. The mean squared error for the boundary is of order , leading to a mean squared error for the boundary b of order with , if the difference of the (exact) BCPs for b and is . Typically, for infinite-dimensional Monte Carlo methods, the convergence rate is less than the finite-dimensional . Let be a further approximating boundary which is linear on k intervals. If k is small compared to m, the corresponding BCP may be estimated with high accuracy. The BCP for as control variable improves the convergence rate of the Monte Carlo estimator to with . The constant depends on the correlation of the estimators for and . We show that this method of adaptive control variable improves the convergence rate considerably. Iterating control variables leads to a rate of convergence (of the mean squared error) of order , reducing the problem of estimating the BCP to an essentially finite-dimensional problem.

Dilution, anti-dilution and corporate positions in options on the company's own stocks

Abstract In this paper, we analyse options that are bought or sold by the company on whose stocks these options are written, leading to dilution and anti-dilution effects. We provide valuation equations for the European versions of such options, and discuss conditions for existence and uniqueness of their prices. Option prices to be paid or received for these options by the company are shown to be different from those that apply for standard options (which are bought and sold by outside investors). Since the options become part of the companys assets/liabilities, the stochastic process followed by the stock price changes. We demonstrate how the new stock price process can be derived, and discuss economic implications of our results. Numerical examples illustrate our findings.

Asymptotic quantization of probability distributions

We give a brief introduction to results on the asymptotics of quantization errors. The topics discussed include the quantization dimension, asymptotic distributions of sets of prototypes, asymptotically optimal quantizations, approximations and random quantizations.

Equilibrium and Learning in a Non-Stationary Environment

This article considers three standard asset pricing models with adaptive agents and stochastic non-stationary dividends. We assume that the parameters are estimated by exponential smoothing, such that prices and returns remain random variables. This paper provides sufficient conditions for the ergodicity of the return process and checks whether the perceived law assumed by the bounded rational agents can be considered to be sound with the returns observed. (authors abstract)

Estimating the quantization dimension: Diffusion processes

We present estimators of the dimension of the support of a probability distribution. These estimators are derived from the concept of quantization dimension. For the general case consistency results are discussed. Versions of the estimators may be applied for instance to estimate the dimension of the driving Brownian motion of Ito processes.

Chapter 32 – Equilibrium and Learning in a Non-Stationary Environment

Publisher Summary This chapter considers three standard asset-pricing models with adaptive agents and stochastic non-stationary dividends. It is assumed that the parameters are estimated by exponential smoothing, such that prices and returns remain random variables. Sufficient conditions are provided for the ergodicity of the return process and checks, whether the perceived law assumed by the bounded rational agents can be considered sound with the returns observed. To discuss equilibrium behavior the concepts of stationarity and ergodicity provide a familiar extension of equilibrium behavior in a stochastic system. It is found that the forecast model is called weak consistent if the implied asset prices and the price process perceived by the agents is integrated of the same order. Given a benchmark forecast model and a performance measure for the quality of the predictions, a forecast rule is strong consistent if it does not deviate too much from the benchmark considered. It is shown that that neither complicated assumptions about the perceived laws nor about the learning rules is necessary to derive returns exhibiting autoregressive effects and non-linearities.