Financial Market Model Research Articles

Dynamic Financial Market Model (introduction)

Dynamic Financial Market Model (introduction)

On the Pathwise Large Deviations of Stochastic Differential and Functional Differential Equations with Applications to Finance

The thesis deals with the asymptotic behaviour of highly nonlinear stochastic differential equations, as well as linear and nonlinear functional differential equations. Both ordinary functional and neutral equations are analysed. In the first chapter, a class of nonlinear stochastic differential equations which satisfy the Law of the Iterated Logarithm is studied, and the results applied to a �financial market model. Mainly scalar equations are considered in the �first chapter. The second chapter deals with a more general class of finite-dimensional nonlinear SDEs and SFDEs, employing comparison and time change methods, as well as martingale inequalities, to determine the almost sure rate of growth of the running maximum of functionals of the solution. The third chapter examines the exact almost sure rate of growth of the large deviations for affine stochastic functional differential equations, and for equations with additive noise which are subject to relatively weak nonlinearities at infinity. The fourth chapter extends conventional conditons for existence and uniqueness of neutral functional differential equations to the stochastic case. The �final chapter deals with large fluctuations of stochastic neutral functional differential equations.

Illiquidity and Derivative Valuation

In illiquid markets, option traders may have an incentive to increase their portfolio value by using their impact on the dynamics of the underlying. We provide a mathematical framework within which to value derivatives under market impact in a multi-player framework by introducing strategic interactions into the Almgren & Chriss (2001) model. Specifically, we consider a financial market model with several strategically interacting players that hold European contingent claims and whose trading decisions have an impact on the price evolution of the underlying. We establish existence and uniqueness of equilibrium results and show that the equilibrium dynamics can be characterized in terms of a coupled system of possibly non-linear PDEs. For the linear cost function used in Almgren & Chriss (2001), we obtain (semi) closed form solutions for risk neutral or CARA investors. Finally, we indicate how spread crossing costs discourage market manipulation.

Multiplicative Approximation of Wealth Processes Involving No-Short-Sale Strategies

A financial market model with general semimartingale asset-price processes and where agents can only trade using no-short-sale strategies is considered. We show that wealth processes using continuous trading can be approximated very closely by wealth processes using simple combinations of buy-and-hold trading. This approximation is based on controlling the proportions of wealth invested in the assets. As an application, the utility maximization problem is considered and it is shown that optimal utilities and wealth processes resulting from continuous trading can be approximated arbitrarily well by the use of simple combinations of buy-and-hold strategies.

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. The main result is the derivation of the limit of a discretetime evolutionary stock market model as the length of the time period tends to zero. The resulting explicit 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.

On “Acquisition of Information in Financial Markets”

In their paper Information Acquisition in Financial Markets (this journal, 2000), Barlevy and Veronesi present a model of a one-period financial market, and claim that for an open set of parameter values, the value of information increases with the mass of informed agents. That claim is shown here to be false. The property of strategic substitution is robust in their model.

Dynamic effects of increasing heterogeneity in financial markets

Despite canonical behavioural financial market models [Day R, Huang W. Bulls, bears and market sheep. J Econ Behav Org 1990;14:299–329], that use different types of agents (i.e., fundamentalist vs. chartists), we develop a model in which the source of instability is the interaction of groups that are homogeneous in the strategy they use, but have heterogeneous beliefs about the fundamental value of the asset. Specifically, heterogeneity arises among two groups of fundamentalists that follow gurus. We show that an increasing distance between beliefs (the degree of heterogeneity), leads first (i) to a pitchfork bifurcation to arise secondly (ii) it generates, together with a larger reaction to misalignment of both market maker and agents, the appearance of a periodic, or even, chaotic, price fluctuation; (iii) finally a homoclinic bifurcation [Dieci R, Bischi GI, Gardini L. From bi-stability to chaotic oscillations in a macroeconomic model. Chaos, Solitons & Fractals 2001;12:805–22] transforms a two piece chaotic set into a one piece chaotic set that generates bull and bear markets.

A Threshold Model of Financial Markets

We proposed a model of interacting market agents based on the generalized Ising spin model. The agents can take three actions: \buy, \sell, or \stay inactive. We deflned a price evolution in terms of the system magnetization. The model reproduces main stylized facts of real markets such as: fat-tailed distribution of returns and volatility clustering.

Backward stochastic partial differential equations related to utility maximization and hedging

We study the utility maximization problem, the problem of minimization of the hedging error and the corresponding dual problems using dynamic programming approach. We consider an incomplete financial market model, where the dynamics of asset prices are described by an ℝd-valued continuous semimartingale. Under some regularity assumptions, we derive the backward stochastic PDEs for the value functions related to these problems, and for the primal problem, we show that the strategy is optimal if and only if the corresponding wealth process satisfies a certain forward SDE. As examples we consider the mean-variance hedging problem and the cases of power, exponential, logarithmic utilities, and corresponding dual problems.

Uncertainty Modeling of Improved Fuzzy Functions With Evolutionary Systems

This paper introduce a type-2 fuzzy function system for uncertainty modeling using evolutionary algorithms (ET2FF). The type-1 fuzzy inference systems (FISs) with fuzzy functions, which do not entail if ... then rule bases, have demonstrated better performance compared to traditional FIS. Nonetheless, the performance of these approaches is usually affected by their uncertain parameters. The proposed method implements a three-phase learning strategy to capture the uncertainties in fuzzy function systems induced by learning parameters, as well as fuzzy function structures. The improved fuzzy clustering initially finds hidden structures, and the genetic learning algorithm optimizes interval type-2 fuzzy sets to capture their optimum uncertainty interval. The proposed ET2FF architecture is evaluated using an extensive suite of real-life applications such as manufacturing process and financial market modeling. The results show that the proposed ET2FF method is comparable--if not superior--to earlier FIS in terms of generalization performance and robustness.

BSDEs driven by Lévy process with enlarged filtration and applications in finance

In this paper, we study the solution of a one-dimensional backward stochastic differential equation driven by Teugels martingales with enlarged filtration. As an application, we will try to compare the strategies of an insider trader and a non-insider one on a financial market modeling by a Lévy process.

An Optimal Control Problem Associated with SDEs Driven by Lévy-Type Processes

In this article, we consider an optimal control problem associated with jump type stochastic differential equations driven by Lévy-type processes. The problem arises from portfolio optimization for the pair of the wealth process and the cumulative consumption process in (incomplete) financial market models. We establish the existence and the uniqueness of (constrained) viscosity solutions to the associated the integro-differential Hamilton–Jacobi–Bellman equation.

On financial markets based on telegraph processes

The paper develops a new class of financial market models. These models are based on generalised telegraph processes: Markov random flows with alternating velocities and jumps occurring when the velocities are switching. While such markets may admit an arbitrage opportunity, the model under consideration is arbitrage-free and complete if directions of jumps in stock prices are in a certain correspondence with their velocity and interest rate behaviour. An analog of the Black–Scholes fundamental differential equation is derived, but, in contrast with the Black–Scholes model, this equation is hyperbolic. Explicit formulas for prices of European options are obtained using perfect and quantile hedging.

The Use of Agent-Based Financial Market Models to Test the Effectiveness of Regulatory Policies

Summary Models with heterogeneous interacting agents have proven to be quite successful in the past. For instance, such models are able to mimic the dynamics of financial markets quite well. The goal of our paper is to explore whether this approach may offer new insights into the working of certain regulatory policies such as transaction taxes, central bank interventions and trading halts. Although this strand of research is rather novel, we argue that agent-based models may be used as artificial laboratories to improve our understanding of how regulatory policy tools function.

Utility maximization under a shortfall risk constraint

The article analyzes optimal portfolio choice of utility maximizing agents in a general continuous-time financial market model under a joint budget and downside risk constraint. The risk constraint is given in terms of a class of convex risk measures. We do not impose any specific assumptions on the price processes of the underlying assets. We analyze under which circumstances the risk constraint is binding. We provide a closed-form solution to the optimization problem in a general semimartingale framework. For a complete market, the wealth maximization problem is equivalent to a dynamic portfolio optimization problem.

The Impact of Competition and Information on Intraday Trading

In a dynamic model of financial market trading multiple heterogeneously informed traders choose when to place orders. Better informed traders trade immediately, worse informed delay even though they expect the public expectation to move against them. This behavior causes distinct patterns with decreasing spreads and probability of informed trading (PIN) and increasing volume. Competition increases market participation and volume, and it causes more pronounced spread and less pronounced volume patterns. Systematic improvements in information increase spreads, volume, and market participation. Very short-lived private information inverts the volume pattern.

Positive Expectations Feedback Experiments and Number Guessing Games as Models of Financial Markets

In repeated number guessing games choices typically converge quickly to the Nash equilibrium. In positive expectations feedback experiments, however, convergence to the equilibrium price tends to be very slow, if it occurs at all. Both types of experimental designs have been suggested as modeling essential aspects of financial markets. In order to isolate the source of the differences in outcomes we present several new treatments in this paper. We conclude that the feedback strength (i.e. the ‘p-value’ in standard number guessing games) is essential for the results. Furthermore, positive expectations feedback experiments may provide good representations of highly speculative markets while standard number guessing games model financial markets with more emphasis on dividend yield and value stocks.

Competitive Rational Expectations Equilibria without Apology

In a standard financial market model with asymmetric information with a finite number N of risk-averse informed traders, competitive rational expectations equilibria provide a good approximation to strategic equilibria as long as N is not too small: equilibrium prices in each situation converge to each other at a rate of 1/N as the market becomes large. The approximation is particularly good when the informationally adjusted risk bearing capacity of traders is not very large. This is not the case if informed traders are close to risk neutral. Both equilibria converge to the competitive equilibrium of an idealized limit continuum economy as the market becomes large at a slower rate of 1/(square root of N) and, therefore, the limit equilibrium need not be a good approximation of the strategic equilibrium in moderately large markets.

PARALLELIZATION OF THE Α‐STABLE MODELLING ALGORITHMS

Stable distributions have a wide sphere of application: probability theory, physics, electronics, economics, sociology. Particularly important role they play in financial mathematics, since the classical models of financial market, which are based on the hypothesis of the normality, often become inadequate. However, the practical implementation of stable models is a nontrivial task, because the probability density functions of α‐stable distributions have no analytical representations (with a few exceptions). In this work we exploit the parallel computing technologies for acceleration of numerical solution of stable modelling problems. Specifically, we are solving the stable law parameters estimation problem by the maximum likelihood method. If we need to deal with a big number of long financial series, only the means of parallel technologies can allow us to get results in a adequate time. We have distinguished and defined several hierarchical levels of parallelism. We show that coarse‐grained Multi‐Sets parallelization is very efficient on computer clusters. Fine‐grained Maximum Likelihood level is very efficient on shared memory machines with Symmetric multiprocessing and Hyper‐threading technologies. Hybrid application, which is utilizing both of those levels, has shown superior performance compared to single level (MS) parallel application on cluster of Pentium 4 HT nodes.

Financial market models with Lévy processes and time-varying volatility

Asset management and pricing models require the proper modeling of the return distribution of financial assets. While the return distribution used in the traditional theories of asset pricing and portfolio selection is the normal distribution, numerous studies that have investigated the empirical behavior of asset returns in financial markets throughout the world reject the hypothesis that asset return distributions are normally distribution. Alternative models for describing return distributions have been proposed since the 1960s, with the strongest empirical and theoretical support being provided for the family of stable distributions (with the normal distribution being a special case of this distribution). Since the turn of the century, specific forms of the stable distribution have been proposed and tested that better fit the observed behavior of historical return distributions. More specifically, subclasses of the tempered stable distribution have been proposed. In this paper, we propose one such subclass of the tempered stable distribution which we refer to as the “KR distribution”. We empirically test this distribution as well as two other recently proposed subclasses of the tempered stable distribution: the Carr–Geman–Madan–Yor (CGMY) distribution and the modified tempered stable (MTS) distribution. The advantage of the KR distribution over the other two distributions is that it has more flexible tail parameters. For these three subclasses of the tempered stable distribution, which are infinitely divisible and have exponential moments for some neighborhood of zero, we generate the exponential Lévy market models induced from them. We then construct a new GARCH model with the infinitely divisible distributed innovation and three subclasses of that GARCH model that incorporates three observed properties of asset returns: volatility clustering, fat tails, and skewness. We formulate the algorithm to find the risk-neutral return processes for those GARCH models using the “change of measure” for the tempered stable distributions. To compare the performance of those exponential Lévy models and the GARCH models, we report the results of the parameters estimated for the S&P 500 index and investigate the out-of-sample forecasting performance for those GARCH models for the S&P 500 option prices.