Cristian Pelizzari

A Tabu Search heuristic procedure in Markov chain bootstrapping

Markov chain theory is proving to be a powerful approach to bootstrap finite states processes, especially where time dependence is non linear. In this work we extend such approach to bootstrap discrete time continuous-valued processes. To this purpose we solve a minimization problem to partition the state space of a continuous-valued process into a finite number of intervals or unions of intervals (i.e. its states) and identify the time lags which provide “memory” to the process. A distance is used as objective function to stimulate the clustering of the states having similar transition probabilities. The problem of the exploding number of alternative partitions in the solution space (which grows with the number of states and the order of the Markov chain) is addressed through a Tabu Search algorithm. The method is applied to bootstrap the series of the German and Spanish electricity prices. The analysis of the results confirms the good consistency properties of the method we propose.

Stable classes of technical trading rules

Technical analysis includes a huge variety of trading rules. This fact has always been a serious hindrance to the large number of market efficiency studies implemented either to demonstrate the profitability of market-beating systems or to deny their operational feasibility. For evident reasons it is practically impossible and theoretically weak to systematically analyse the entire body of all trading rules. We therefore propose a novel method to form natural classes of trading rules which are found to be robust to changing market scenarios. In particular, groups are formed adopting a similarity measure based on the investing signals of the trading rules. Our clustering methodology adopts a Markov chain bootstrapping technique to generate differentiated scenarios preserving volume and price joint distributional features. An application is developed on a sample of 674 trading rules. Results show that six groups (here identified as trading styles) are sufficient to explain the large portion of the investing signals variance. We also suggest applications of our results to fund performance measurement and the analysis of financial markets.

Pricing inflation-linked bonds

This paper proposes a pricing model for inflation-linked bonds. Our proposal is developed starting from a Vasicek model of the instantaneous inflation rate process and the Cox, Ingersoll and Ross model for the nominal instantaneous risk-free interest rate process. Instead of adopting the standard approach of a cross-section estimation of the term structure of real interest rates, this work proposes a pricing model based on estimation of the inflation risk premium. The model is applied to Treasury Inflation Protected Securities, which are inflation-linked bonds issued by the U.S. Department of the Treasury. Empirical validation is carried out on data for the period 1999–2005.

Relevant states and memory in Markov chain bootstrapping and simulation

Markov chain theory is proving to be a powerful approach to bootstrap highly nonlinear time series. In this work we provide a method to estimate the memory of a Markov chain (i.e. its order) and to identify its relevant states. In particular the choice of memory lags and the aggregation of irrelevant states are obtained by looking for regularities in the transition probabilities. Our approach is based on an optimization model. More specifically we consider two competing objectives that a researcher will in general pursue when dealing with bootstrapping: preserving the “structural” similarity between the original and the simulated series and assuring a controlled diversification of the latter. A discussion based on information theory is developed to define the desirable properties for such optimal criteria. Two numerical tests are developed to verify the effectiveness of the method proposed here.

Approximating multivariate Markov chains for bootstrapping through contiguous partitions

This paper extends Markov chain bootstrapping to the case of multivariate continuous-valued stochastic processes. To this purpose, we follow the approach of searching an optimal partition of the state space of an observed (multivariate) time series. The optimization problem is based on a distance indicator calculated on the transition probabilities of the Markov chain. Such criterion aims at grouping those states exhibiting similar transition probabilities. A second methodological contribution is represented by the addition of a contiguity constraint, which is introduced to force the states to group only if they have “near” values (in the state space). This requirement meets two important aspects: first, it allows a more intuitive interpretation of the results; second, it contributes to control the complexity of the problem, which explodes with the cardinality of the states. The computational complexity of the optimization problem is also addressed through the introduction of a novel Tabu Search algorithm, which improves both the quality of the solution found and the computing times with respect to a similar heuristic previously advanced in the literature. The bootstrap method is applied to two empirical cases: the bivariate process of prices and volumes of electricity in the Spanish market; the trivariate process composed of prices and volumes of a US company stock (McDonald’s) and prices of the Dow Jones Industrial Average index. In addition, the method is compared with two other well-established bootstrap methods. The results show the good distributional properties of the present proposal, as well as a clear superiority in reproducing the dependence among the data.

A mixed integer linear program to compress transition probability matrices in Markov chain bootstrapping

Bootstrapping time series is one of the most acknowledged tools to study the statistical properties of an evolutive phenomenon. An important class of bootstrapping methods is based on the assumption that the sampled phenomenon evolves according to a Markov chain. This assumption does not apply when the process takes values in a continuous set, as it frequently happens with time series related to economic and financial phenomena. In this paper we apply the Markov chain theory for bootstrapping continuous-valued processes, starting from a suitable discretization of the support that provides the state space of a Markov chain of order

Risk-Averse Equilibrium Modeling and Social Optimality of Cap-and-Trade Mechanisms

k \ge 1

A Mixed Integer Linear Programming Approach to Markov Chain Bootstrapping

k≥1. Even for small k, the number of rows of the transition probability matrix is generally too large and, in many practical cases, it may incorporate much more information than it is really required to replicate the phenomenon satisfactorily. The paper aims to study the problem of compressing the transition probability matrix while preserving the “law” characterising the process that generates the observed time series, in order to obtain bootstrapped series that maintain the typical features of the observed time series. For this purpose, we formulate a partitioning problem of the set of rows of such a matrix and propose a mixed integer linear program specifically tailored for this particular problem. We also provide an empirical analysis by applying our model to the time series of Spanish and German electricity prices, and we show that, in these medium size real-life instances, bootstrapped time series reproduce the typical features of the ones under observation.