PART II IT Methods in Finance

Transcription

1 PART II IT Methods in Finance

2 Introduction to Part II This part contains 12 chapters and is devoted to IT methods in finance. There are essentially two ways where IT enters and influences methods used in finance. The first are of course methods which originate in IT and informatics, which are just applied directly or in a suitable adapted form. The second are methods which originate in finance, e. g. mathematical finance, but which need to their efficient implementation a suitable use of IT. This part includes both aspects and it adds as a third the investigation of problems from finance in computer science. Nagurney gives a survey on networks in finance in Chapter 17. The chapter traces the history of networks in finance and overviews a spectrum of relevant methodologies for their formulation, analysis, and solution ranging from optimization techniques to variational inequalities and projected dynamical systems. Numerical examples are provided for illustration purposes. In Chapter 18 van Dinther discusses the role of agent-based simulation for research in economics. Agent-based simulations have produced a variety of interesting contributions to economic research in the past years. Nevertheless, the methodology of economic simulations in general and agent-based simulations in particular is still in question. This is due to the fact that simulations are conducted without considering the underlying theory and links to standard economic approaches. This contribution discusses the use of simulations in economics and reviews the four most common agent-based approaches and their link to economic theory. Agents are also the central theme of the next chapter by Dermietzel. He gives a survey on developments and milestones on the heterogeneous agents approach to financial markets. New models of financial markets have been developed over the last 25 years, which are able to explain empirical phenomena observed in real markets. In contrast to classical financial theory a key feature of these models is the heterogeneity of traders. The models developed from very stylized models in the 1980s to complex dynamical models with realistic trading behaviour. Today these models are used to reproduce and explain the dynamics of financial markets and to investigate improvements and regulations of these sensitive economic mechanisms. This chapter discusses the discrepancies between financial theory

3 380 Introduction to Part II and real market data and provides an overview of important financial market models with heterogeneous traders ranging from 1980 to Chiarella and He develop in Chapter 20 an adaptive model of asset price and wealth dynamics in a financial market with heterogeneous agents and examine the profitability of momentum and contrarian trading strategies. In order to characterise asset prices, wealth dynamics and rational adaptiveness arising from the interaction of heterogeneous agents with constant relative risk aversion (CRRA) utility, an adaptive discrete time equilibrium model in terms of return and wealth proportions (among heterogeneous representative agents) is established. Taking trend followers and contrarians as the main heterogeneous agents in the model, the profitability of momentum and contrarian trading strategies is analysed. The authors show the capability of the model to characterize some of the existing evidence on many of the anomalies observed in financial markets, including the profitability of momentum trading strategies over short time intervals and of contrarian trading strategies over long time intervals, rational adaptiveness of agents, overconfidence and underreaction, overreaction and herd behaviour, excess volatility, and volatility clustering. A survey on simulation methods for stochastic differential equations is given in Chapter 21 by Platen. Stochastic differential equations are becoming the modelling framework in finance, insurance, economics and many other areas of social sciences. Only in rare cases one has explicit solutions. Furthermore, the modelling of market segments or entire markets often involves a large number of factor processes where many quantitative methods fail to work. For these reasons simulation methods have become essential tools in quantitative finance. This chapter provides a brief introduction into the area of simulation methods for stochastic differential equations with focus on finance. The foundations of option pricing are important to many topics in finance. They are discussed by Buchen in Chapter 22. Modern approaches to pricing financial options and derivatives are concerned with the notion of no arbitrage i. e. the concept there are no free lunches in efficient markets. The chapter explores the two principal methods of arbitrage-free pricing and shows how they can be applied to price a wide range of example contracts. The two methods alluded to are: the PDE (partial differential equation) method of Black and Scholes, and the EMM (equivalent martingale measure) method of Harrison and Pliska. Both methods have their origins in the stochastic calculus and ultimately, under certain restrictions, are equivalent, as in-deed they must be if the Law of One Price is to hold. When used together with other analytical tools such as the Gaussian Shift Theorem, Principle of Static Replication and Parity Relations, one is able to price a range of exotic options with both path independent and path dependent payoffs. Sun, Rachev and Fabozzi provide a survey on long-range dependence, fractal processes, and intra-daily data in Chapter 23. With the availability of intra-daily price data, researchers have focused more attention on market microstructure issues to understand and help formulate strategies for the timing of trades. The purpose of this chapter is to provide a brief survey of the research employing intradaily price data. Specifically, the chapter reviews stylized facts of intra-daily data, econometric issues of data analysis, application of intra-daily data in volatility and

4 Introduction to Part II 381 liquidity research, and the applications to market microstructure theory. Longrange dependence is observed in intra-daily data. Because fractal processes or fractional integrated models are usually used to model long-range dependence, we also provide a review of fractal processes and long-range dependence in order to consider them in future research using intra-daily data. Bayesian applications to the investment management process are the central theme of Chapter 24 by Bagasheva, Rachev, Hsu and Fabozzi. The usual investment management practice is to assume that the parameter values estimated from the available data sample are the population parameter values. Failing to take into account the errors intrinsic in sample estimates, however, could lead to suboptimal asset allocations. Bayesian methods provide the theoretical framework and tools to account for estimation risk. Moreover, they allow a researcher or a practitioner to incorporate their subjective views about the model parameters into the asset allocation process. In this chapter, an overview of some aspects of the investment management process within a Bayesian setting is presented. The chapter highlights the advantages that setting provides over the classical (frequentist) estimation framework. Racheva-Iotova and Stoyanov discuss a unified framework for constructing optimal portfolio strategies in Chapter 25. It is based on an optimization problem involving the CVaR risk measure and an appropriately selected multivariate model for stock returns. The authors provide an empirical example using the Russell 2000 universe. In Chapter 26 Gilli, Maringer and Winker give a survey on applications of heuristics in finance. Optimisation is crucial in many areas of finance, but often quite demanding because the type of objective functions and the constraints that have to be considered cannot be handled with traditional optimization techniques. A rather novel group of methods are heuristics. These methods are less restricted in their applicability, and they perform search and optimisation in a non-deterministic fashion by repeatedly updating one ore a whole set of candidate solutions and by incorporating stochastic elements. Frequently, these methods are inspired by principles found in nature, e. g., evolution, they are typically not tailored for a certain type of problem, and they are flexible enough to be adapted to complex optimization problems. This chapter presents some of the most popular of these methods and demonstrates how they can be applied to financial problems including portfolio optimization and model selection. These methods are found not only to work reliably, but also to allow a better analysis of complex problems that could not be addressed with traditional methods. Another class of heuristic approaches to solve several complex problems in finance stem from the area of machine learning. They are surveyed by Chalup and Mitschele in Chapter 27 on kernel methods. Kernel methods are a class of powerful machine learning algorithms which are able to solve non-linear tasks. The chapter presents a concise overview of a selection of relevant machine learning methods and a survey of applications to show how kernel methods have been applied in finance. The overview of learning concepts addresses methods for dimensionality reduction, regression, and classification. The concept of kernelisation which can be used in order to transform classical linear machine learning methods into non-linear kernel methods is emphasised. The survey of applications of kernel

5 382 Introduction to Part II methods in finance covers the areas of credit risk management, market risk management, and discusses possible future application fields. It concludes with a brief overview of relevant software toolboxes. In the last years there were several related results developed, showing that special problems form finance are provably complex, i. e. their complexity is so high that one is in need of heuristic methods to approach solutions. Some of these are results on the complexity of decision problems in foreign exchange markets. These problems are discussed in Chapter 28 by Cai and Deng. They study the computational complexity problem of finding arbitrage in foreign exchange markets, taking market frictions into consideration. The problem is approached through several steps, refining the computational complexity issue with respect to arbitrage. At first, it is investigated the computational complexity to detect the existence of an arbitrage opportunity in a general market model. This problem has a negative solution, since it turns out to be NP-complete to identify arbitrage. Secondly, market models are described for which polynomial time algorithms exist to locate an arbitrage opportunity. It is shown that two important market structure models belong to this category. Thirdly, the impact of futures on the complexity of finding arbitrage is explored. Finally, the minimal number of necessary transactions to bring the exchange system back to some non-arbitrage states is studied. The results show that different exchange systems exhibit different computational complexities for finding arbitrage. The complexity understandings help to shed new light on understanding how monetary system models are adopted and evolved in reality.