Luận án tiến sĩ: Mô hình tài chính toán học và phương pháp số - Timothy L. Seaman

Models of Selected Problems in Mathematical Finance and Numerical Methods for Stochastic Differential Equations A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy at George Mason University by Timothy L. Seaman Master of Science California State University, 1972 Director: Harbir Lamba, Associate Professor Department of Mathematical Sciences Fall Semester 2006 George Mason University Fairfax, Virginia UMI Number: 3240839 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted.

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ProQuest Information and Learning Company 300 North Zeeb Road P. Box 1346 Ann Arbor, MI 48106-1346 MODELS OF SELECTED PROBLEMS IN MATHEMATICAL FINANCE AND NUMERICAL METHODS FOR STOCHASTIC DIFFERENTIAL EQUATIONS by Timothy L. Seaman A Dissertation Submitted to the Graduate Faculty of George Mason University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Computational Sciences and Informatics Committee: 26 ⁄ CAL Dr. Harbir Lamba, Director Cn) Fue Dr.

Timothy Sauer 2 = £ /2zZ7 Dr. James Gentle The ce] `———7 Dr. Becker, Associate Dean for men caches Dr. Menas Kafatos, Dean, College of Graduate Programs Science Date: (1/24/06 Ỉ { Fall Semester 2006 George Mason University Fairfax, Virginia il Dedication This is dedicated to my wife Kathy who supported me throughout this entire process.

11 Acknowledgments I would like to thank the members of my committee for their support and patience, and in particular my adviser, Dr. Harbir Lamba, without whose inspiration this endeavor would not have been possible. 1V Table of Contents Page Abstract nh HT. cv 2v TT và 1 1.2 The Efficient Market Hypothesis.3 The Modeling Philosophy .2 Stochastic Differential Equations.

10 Review of the Literature. 2 ng va va va 14 2.1 Agent-Based Simulations of Financial Markets.2 Numerical Solution of Stochastic Differential Equations. 34 Methodology cv ng ng g g kg vi kg kg k ki N v va 45 3.1 A Threshold Model of Investor Behavior .1 Determination of the Asset Price .2 Determination of an Investor’s Position.2 Stochastic Differential Equations .2 The Balanced Milstein Method (Implicit) .3 Adaptivity Using Dual Error Controls .4 Additional Stochastic Error Controls. 68 Results and Discussion.

g gà ànt 70 41 The Financial Market Model.1 Calibration of the Model.2 Simulations without Herding anda=0O.3 Simulations with Herding buta=0.4 The Full Model and Replication of the Stylized Facts .5 Introducing Price Asymmetry into the Model .2 Stochastic Differential Equations .1 Mean-Square Stability .2 Modifications to the Basic Dual-Error-Control Approach .1 Financial Market Model .2 Stochastic Differential Equations .aaA << 112 vi List of Tables Table 3.1 Computed Hurst exponent by each method compared to the Hurst exponent of the fractional Brownian motion used as input.1 For fixed \ = —2, computed values of E(V,’) and the corresponding values Of 66 0ïẼˆ .2 For fixed p? = 4 (u = 2), computed values of E(W;?) and the corre- sponding values of€.3 For a given 7, the average number of steps taken and reasons for halting step size increase, as well as average maximum error per batch and average mean error, using two error controls Fg and ;.4 For a given T, the average number of steps taken and reasons for halting step size increase, as well as average maximum error per batch and average mean error, using four error controls Ey, 1¿, EF, and lạ. Average maximum absolute error and average error at t = 1 of computed solution to Test Problem 2 using Milstein schemes, fixed step sizes hw.6 Non-stiff, Explicit, Adaptive. Average maximum absolute error and average error at t = 1 and average number of steps taken, accepted aud rejected; Test Problem 2.7 Non-stiff, Balanced, Adaptive. Average maximum absolute error and average error at t = 1 and average number of steps taken, accepted and rejected; Test Problem 2, .8 Stiff, Explicit, Adaptive.

Average maximum absolute error and average error at t = 1 and average number of steps taken, accepted and rejected; Test Problem 2.9 Stiff, Balanced, Adaptive. Average maximum absolute error and average error at t = 1 and average number of steps taken, accepted and rejected; Test Problem 2, 2.10 Number of steps for which each error control terminated the increase in step Size. vill List of Figures Figure Page 4.1 Numerical simulation of the model with the herding tension removed (M=100), 2.2 Numerical simulation of the model with the herding tension removed 89).3 Numerical simulation of the model with herding but with a = 0 over a 40-year period. See text for details.4 Numerical simulation of the full model (with herding and with a # 0) over a 40-year period.

See text for details.5 The cumulative distributions of the log-price returns (positive only, negative only and absolute value) along with the line of best fit.6 Decay of correlation of volatility and approximation with power law.7 Distribution of normalized logarithmic returns separated into increases (positive) and decreases (negative).8 Cumulative distribution of bull market normed absolute value of rela- tive price changes.9 Cumulative distribution of bear market normed absolute value of rel- ative price changes. 2 IH Abstract MODELS OF SELECTED PROBLEMS IN MATHEMATICAL FINANCE AND NUMERICAL METHODS FOR STOCHASTIC DIFFERENTIAL EQUATIONS Timothy L. George Mason University, 2006 Dissertation Director: Dr. Harbir Lamba The research conducted for this dissertation addresses two different problems from computational science and mathematical modeling.

The first problem concerns the relaxation of the Efficient Market Hypothesis and the development of agent-based models that can help explain the consistent, but poorly understood, non-Gaussian statistical properties of real financial markets. A simple yet robust class of models is introduced in which the agents are driven both by rational considerations and also by less rational, but psychologically plausible, factors. It is shown that these models are capable of reproducing many of the im- portant features of real markets. Furthermore, the results suggest that the two most significant deviations from Gaussian behavior in the price returns, namely ‘fat tails’ and volatility clustering, may have different causes allowing for future models that can more accurately reproduce them.

The second area of research is the analysis and development of efficient variable timestepping algorithms for the numerical solution of stochastic differential equations. The starting point is a previously introduced algorithm that uses a dual-error-control strategy. One local error estimate corresponds to the error in the drift and the other to the diffusion. It is proved that under certain modes of operation the algorithm is mean-square stable for a class of test problems with multiplicative noise.

Additional error controls are then introduced to determine whether they result in improved performance on different test problems. The research demonstrates that algorithms based upon such error controls are feasible and can result in large efficiency and stability improvements over their fixed timestepping counterparts. Chapter 1: Introduction In this dissertation, I conduct research in two areas: one concerns the development of mathematical models of financial markets that do not satisfy the usual efficiency and rationality assumptions; the other relates to the efficient solution of stochastic differential equation (SDEs) using adaptive timestepping algorithms.1 Financial Markets The economic records of nations, industries and even individual companies are full of irregular cycles, bubbles and crashes. The tulip mania of seventeenth century Netherlands and the South Sea Bubble of the eighteenth century are well-known examples of booming, speculative markets that eventually crashed [95].

By the same token, the stock market crashes of 1929, 1987 and 2000 sent economic reverberations to every corner of society, and each crash was preceded by a booming financial market. To this day, none of these crashes are thoroughly understood. Consequently, we are still at risk, that is to say, our economy is still susceptible to severe market crashes.1 A Historical Perspective The study of the economy, or ‘political economy’ as it is sometimes known, goes back only a few hundred years, to the time of Adam Smith. Much of the work of the classical economists - Smith, Ricardo, Malthus and Marx — focused on growth and change, such as the problems caused by economic fluctuations and unemployment [80].

Most of these eminent figures worked in Britain, and the fact that Great Britain was a leader in the Industrial Revolution is hardly coincidental. Adam Smith published his 1 well-known Wealth of Nations in the latter half of the eighteenth century. Malthus, nowadays best remembered for his doomsday predictions about population growth, published his work towards the end of the same century, while Ricardo published in the first half of the nineteenth century. The famed German Karl Marx spent the latter part of his life in Great Britain, where he wrote Das Kapital.

Unfortunately, Marx’s economic theories have been widely ignored by the general reader, because they were championed so dogmatically by the failed communist regimes of the twentieth century. These classical economists are known today for their theories about economics. Their theoretical models were surely based on empirical evidence, but how well were these theories and models tested? Around 1870, more quantitative analysis began to enter the economic picture. Among others, Léon Walras, a trained physicist in Lausanne, Switzerland, introduced mathematical systems of analysis to the study of economics.

The contributions Walras made had a great impact on economic theory. Nowadays, typical undergraduate courses in microeconomics begin with demand curves and supply curves and build to “Walrasian General Equilibrium” [67,80]. Despite the usefulness of Walrasian General Equilibrium in understanding the general theory of markets, some economists object to the fact that money plays no role in the Walrasian microeconomic explanation of markets. Further, this lack creates a dichotomy in the transition to macroeconomics [67].

Other economists complain that, although the Walrasian mathematical approach applies more rigorous arguments to much economic reasoning, this approach can be carried too far [80]. Economics is not a physical science; there are no rigid, universal laws of human behavior. The Walrasian view, that of economics as a smoothly running machine that can be explained by mathematical arguments, tends to separate economics from its original, people-oriented foundations. In the same spirit of using mathematics to explain economic phenomena, Louis Bachelier in 1900 wrote his Ph.

Théorie de la Speculation, about the theory of financial markets. Although his ideas were not widely disseminated at the time, Bachelier is now recognized as one of the first modern researchers of financial markets. He proposed that the price of a stock moved in a random fashion and, consequently, that this seemingly erratic motion of the stock market price was equivalent to a random walk. If these assertions were true, then changes in stock price would be distributed in the well-known bell-shaped, or Gaussian, curve.

Bachelier’s assertion of unpredictability also lent credence to the fact that it is very difficult for even the best investors to beat the market in the long run. Bachelier, and later others, argued that in a liquid market, such as stock or foreign exchange, any significant correlation in returns would be quickly noticed and acted upon to generate a profit. Even small opportunities for arbitrage would not linger long, but instead would be rapidly turned into profit. Ultimately, the liquidity and efficiency of the market effectively cancel meaningful correlations in the market.

This leads one to conclude that the lack of arbitrage opportunity is in fact an integral feature of the market. In time, this fundamental concept was expanded and came to be known as the ‘Efficient Market Hypothesis.’ For over 50 years, Bachelier’s Gaussian distribution of the random changes in stock prices stood the test of time, drawing few detractors.