The Mathematics of Financial Modeling and Investment Management

by Focardi, Sergio M.; Fabozzi, Frank J.

• ISBN13: 9780471465997
• ISBN10: 0471465992
• Format: Hardcover
• Copyright: 2004-03-29
• Publisher: WILEY

Summary

the mathematics of financial modeling & investment management The Mathematics of Financial Modeling & Investment Management covers a wide range of technical topics in mathematics and finance-enabling the investment management practitioner, researcher, or student to fully understand the process of financial decision-making and its economic foundations. This comprehensive resource will introduce you to key mathematical techniques-matrix algebra, calculus, ordinary differential equations, probability theory, stochastic calculus, time series analysis, optimization-as well as show you how these techniques are successfully implemented in the world of modern finance. Special emphasis is placed on the new mathematical tools that allow a deeper understanding of financial econometrics and financial economics. Recent advances in financial econometrics, such as tools for estimating and representing the tails of the distributions, the analysis of correlation phenomena, and dimensionality reduction through factor analysis and cointegration are discussed in depth. Using a wealth of real-world examples, Focardi and Fabozzi simultaneously show both the mathematical techniques and the areas in finance where these techniques are applied. They also cover a variety of useful financial applications, such as: * Arbitrage pricing * Interest rate modeling * Derivative pricing * Credit risk modeling * Equity and bond portfolio management * Risk management * And much more Filled with in-depth insight and expert advice, The Mathematics of Financial Modeling & Investment Management clearly ties together financial theory and mathematical techniques.

Author Biography

SERGIO FOCARDI is a founding partner of The Intertek Group, a Paris-based firm providing consulting on advanced mathematical methods in banking and finance, and a cofounder of CINEF (Center for Interdisciplinary Research in Economics and Finance) at the University of Genoa, Italy. Focardi’s research interests focus on statistical arbitrage, dynamic factor analysis, and financial modeling in a multiple heterogeneous interacting agents framework. He has published numerous articles and coauthored the books Modeling the Market: New Theories and Techniques and Risk Management: Framework, Methods, and Practice (both published by Wiley). Focardi holds a degree in electronic engineering from the University of Genoa.
FRANK J. FABOZZI, PhD, CFA, is the Frederick Frank Adjunct Professor of Finance at Yale University’s School of Management and Editor of the Journal of Portfolio Management. Fabozzi is a Chartered Financial Analyst and Certified Public Accountant who has authored and edited many acclaimed books in finance. He earned a doctorate in economics from the City University of New York in 1972. He is a Fellow of the International Center for Finance at Yale University.

Table of Contents

Preface	xiv
Acknowledgments	xvi
About the Authors	xviii
Commonly Used Symbols	xix
Abbreviations and Acronyms	xx
From Art to Engineering in Finance	1	(20)
Investment Management Process	2	(8)
Step 1: Setting Investment Objectives	2	(1)
Step 2: Establishing an Investment Policy	2	(4)
Step 3: Selecting a Portfolio Strategy	6	(1)
Step 4: Selecting the Specific Assets	7	(2)
Step 5: Measuring and Evaluating Performance	9	(1)
Financial Engineering in Historical Perspective	10	(1)
The Role of Information Technology	11	(2)
Industry's Evaluation of Modeling Tools	13	(2)
Integrating Qualitative and Quantitative Information	15	(2)
Principles for Engineering a Suite of Models	17	(1)
Summary	18	(3)
Overview of Financial Markets, Financial Assets, and Market Participants	21	(54)
Financial Assets	21	(4)
Financial Markets	25	(9)
Classification of Financial Markets	25	(1)
Economic Functions of Financial Markets	26	(1)
Secondary Markets	27	(7)
Overview of Market Participants	34	(11)
Role of Financial Intermediaries	35	(2)
Institutional Investors	37	(4)
Insurance Companies	41	(1)
Pension Funds	41	(1)
Investment Companies	42	(1)
Depository Institutions	43	(2)
Endowments and Foundations	45	(1)
Common Stock	45	(6)
Trading Locations	45	(1)
Stock Market Indicators	46	(2)
Trading Arrangements	48	(3)
Bonds	51	(6)
Maturity	51	(1)
Par Value	52	(1)
Coupon Rate	52	(3)
Provisions for Paying off Bonds	55	(1)
Options Granted to Bondholders	56	(1)
Futures and Forward Contracts	57	(7)
Futures versus Forward Contracts	58	(1)
Risk and Return Characteristics of Futures Contracts	59	(1)
Pricing of Futures Contracts	59	(4)
The Role of Futures in Financial Markets	63	(1)
Options	64	(5)
Risk-Return for Options	66	(1)
The Option Price	66	(3)
Swaps	69	(1)
Caps and Floors	70	(1)
Summary	71	(4)
Milestones in Financial Modeling and Investment Management	75	(16)
The Precursors: Pareto, Walras, and the Lausanne School	76	(2)
Price Diffusion: Bachelier	78	(2)
The Ruin Problem in Insurance: Lundberg	80	(1)
The Principles of Investment: Markowitz	81	(2)
Understanding Value: Modigliani and Miller	83	(2)
Modigliani-Miller Irrelevance Theorems and the Absence of Arbitrage	84	(1)
Efficient Markets: Fama and Samuelson	85	(1)
Capital Asset Pricing Model: Sharpe, Lintner, and Mossin	86	(1)
The Multifactor CAPM: Merton	87	(1)
Arbitrage Pricing Theory: Ross	88	(1)
Arbitrage, Hedging, and Option Theory: Black, Scholes, and Merton	89	(1)
Summary	90	(1)
Principles of Calculus	91	(50)
Sets and Set Operations	93	(3)
Proper Subsets	93	(2)
Empty Sets	95	(1)
Union of Sets	95	(1)
Intersection of Sets	95	(1)
Elementary Properties of Sets	96	(1)
Distances and Quantities	96	(4)
n-tuples	97	(1)
Distance	98	(1)
Density of Points	99	(1)
Functions	100	(1)
Variables	101	(1)
Limits	102	(1)
Continuity	103	(2)
Total Variation	105	(1)
Differentiation	106	(5)
Commonly Used Rules for Computing Derivatives	107	(4)
Higher Order Derivatives	111	(10)
Application to Bond Analysis	112	(9)
Taylor Series Expansion	121	(6)
Application to Bond Analysis	122	(5)
Integration	127	(4)
Riemann Integrals	127	(2)
Properties of Riemann Integrals	129	(1)
Lebesque-Stieltjes Integrals	130	(1)
Indefinite and Improper Integrals	131	(1)
The Fundamental Theorem of Calculus	132	(2)
Integral Transforms	134	(4)
Laplace Transforms	134	(3)
Fourier Transforms	137	(1)
Calculus in More than One Variable	138	(1)
Summary	139	(2)
Matrix Algebra	141	(24)
Vectors and Matrices Defined	141	(4)
Vectors	141	(3)
Matrices	144	(1)
Square Matrices	145	(3)
Diagonals and Antidiagonals	145	(1)
Identity Matrix	146	(1)
Diagonal Matrix	146	(2)
Upper and Lower Triangular Matrix	148	(1)
Determinants	148	(1)
Systems of Linear Equations	149	(2)
Linear Independence and Rank	151	(1)
Hankel Matrix	152	(1)
Vector and Matrix Operations	153	(7)
Vector Operations	153	(3)
Matrix Operations	156	(4)
Eigenvalues and Eigenvectors	160	(1)
Diagonalization and Similarity	161	(1)
Singular Value Decomposition	162	(1)
Summary	163	(2)
Concepts of Probability	165	(36)
Representing Uncertainty with Mathematics	165	(2)
Probability in a Nutshell	167	(2)
Outcomes and Events	169	(1)
Probability	170	(1)
Measure	171	(1)
Random Variables	172	(1)
Integrals	172	(2)
Distributions and Distribution Functions	174	(1)
Random Vectors	175	(3)
Stochastic Processes	178	(2)
Probabilistic Representation of Financial Markets	180	(1)
Information Structures	181	(1)
Filtration	182	(2)
Conditional Probability and Conditional Expectation	184	(2)
Moments and Correlation	186	(2)
Copula Functions	188	(1)
Sequences of Random Variables	189	(2)
Independent and Identically Distributed Sequences	191	(1)
Sum of Variables	191	(3)
Gaussian Variables	194	(3)
The Regression Function	197	(2)
Linear Regression	197	(2)
Summary	199	(2)
Optimization	201	(16)
Maxima and Minima	202	(2)
Lagrange Multipliers	204	(2)
Numerical Algorithms	206	(6)
Linear Programming	206	(5)
Quadratic Programming	211	(1)
Calculus of Variations and Optimal Control Theory	212	(2)
Stochastic Programming	214	(2)
Summary	216	(1)
Stochastic Integrals	217	(22)
The Intuition Behind Stochastic Integrals	219	(6)
Brownian Motion Defined	225	(5)
Properties of Brownian Motion	230	(2)
Stochastic Integrals Defined	232	(4)
Some Properties of Ito Stochastic Integrals	236	(1)
Summary	237	(2)
Differential Equations and Difference Equations	239	(28)
Differential Equations Defined	240	(1)
Ordinary Differential Equations	240	(3)
Order and Degree of an ODE	241	(1)
Solution to an ODE	241	(2)
Systems of Ordinary Differential Equations	243	(3)
Closed-Form Solutions of Ordinary Differential Equations	246	(3)
Linear Differential Equations	247	(2)
Numerical Solutions of Ordinary Differential Equations	249	(7)
The Finite Difference Method	249	(7)
Nonlinear Dynamics and Chaos	256	(3)
Fractals	258	(1)
Partial Differential Equations	259	(6)
Diffusion Equation	259	(2)
Solution of the Diffusion Equation	261	(2)
Numerical Solution of PDEs	263	(2)
Summary	265	(2)
Stochastic Differential Equations	267	(16)
The Intuition Behind Stochastic Differential Equations	268	(3)
Ito Processes	271	(1)
The 1-Dimensional Ito Formula	272	(2)
Stochastic Differential Equations	274	(2)
Generalization to Several Dimensions	276	(2)
Solution of Stochastic Differential Equations	278	(4)
The Arithmetic Brownian Motion	280	(1)
The Ornstein-Uhlenbeck Process	280	(1)
The Geometric Brownian Motion	281	(1)
Summary	282	(1)
Financial Econometrics: Time Series Concepts, Representations, and Models	283	(32)
Concepts of Time Series	284	(2)
Stylized Facts of Financial Time Series	286	(2)
Infinite Moving-Average and Autoregressive Representation of Time Series	288	(9)
Univariate Stationary Series	288	(1)
The Lag Operator L	289	(3)
Stationary Univariate Moving Average	292	(1)
Multivariate Stationary Series	293	(2)
Nonstationary Series	295	(2)
ARMA Representations	297	(8)
Stationary Univariate ARMA Models	297	(3)
Nonstationary Univariate ARMA Models	300	(1)
Stationary Multivariate ARMA Models	301	(3)
Nonstationary Multivariate ARMA Models	304	(1)
Markov Coefficients and ARMA Models	304	(1)
Hankel Matrices and ARMA Models	305	(1)
State-Space Representation	305	(4)
Equivalence of State-Space and ARMA Representations	308	(1)
Integrated Series and Trends	309	(4)
Summary	313	(2)
Financial Econometrics: Model Selection, Estimation, and Testing	315	(36)
Model Selection	315	(2)
Learning and Model Complexity	317	(2)
Maximum Likelihood Estimate	319	(5)
Linear Models of Financial Time Series	324	(1)
Random Walk Models	324	(3)
Correlation	327	(2)
Random Matrices	329	(3)
Multifactor Models	332	(6)
CAPM	334	(1)
Asset Pricing Theory (APT) Models	335	(1)
PCA and Factor Models	335	(3)
Vector Autoregressive Models	338	(1)
Cointegration	339	(6)
State-Space Modeling and Cointegration	342	(1)
Empirical Evidence of Cointegration in Equity Prices	343	(2)
Nonstationary Models of Financial Time Series	345	(4)
The ARCH/GARCH Family of Models	346	(1)
Markov Switching Models	347	(2)
Summary	349	(2)
Fat Tails, Scaling, and Stable Laws	351	(42)
Scaling, Stable Laws, and Fat Tails	352	(10)
Fat Tails	352	(1)
The Class £ of Fat-Tailed Distributions	353	(5)
The Law of Large Numbers and the Central Limit Theorem	358	(2)
Stable Distributions	360	(2)
Extreme Value Theory for IID Processes	362	(16)
Maxima	362	(6)
Max-Stable Distributions	368	(1)
Generalized Extreme Value Distributions	368	(1)
Order Statistics	369	(2)
Point Process of Exceedances or Peaks over Threshold	371	(2)
Estimation	373	(5)
Eliminating the Assumption of IID Sequences	378	(10)
Heavy-Tailed ARMA Processes	381	(1)
ARCH/GARCH Processes	382	(1)
Subordinated Processes	383	(1)
Markov Switching Models	384	(1)
Estimation	384	(1)
Scaling and Self-Similarity	385	(3)
Evidence of Fat Tails in Financial Variables	388	(3)
On the Applicability of Extreme Value Theory in Finance	391	(1)
Summary	392	(1)
Arbitrage Pricing: Finite-State Models	393	(48)
The Arbitrage Principle	393	(2)
Arbitrage Pricing in a One-Period Setting	395	(7)
State Prices	397	(1)
Risk-Neutral Probabilities	398	(1)
Complete Markets	399	(3)
Arbitrage Pricing in a Multiperiod Finite-State Setting	402	(21)
Propagation of Information	402	(1)
Trading Strategies	403	(1)
State-Price Deflator	404	(1)
Pricing Relationships	405	(9)
Equivalent Martingale Measures	414	(2)
Risk-Neutral Probabilities	416	(7)
Path Dependence and Markov Models	423	(1)
The Binomial Model	423	(4)
Risk-Neutral Probabilities for the Binomial Model	426	(1)
Valuation of European Simple Derivatives	427	(2)
Valuation of American Options	429	(1)
Arbitrage Pricing in a Discrete-Time, Continuous-State Setting	430	(5)
APT Models	435	(4)
Testing APT	436	(3)
Summary	439	(2)
Arbitrage Pricing: Continuous-State, Continuous-Time Models	441	(30)
The Arbitrage Principle in Continuous Time	441	(4)
Trading Strategies and Trading Gains	443	(2)
Arbitrage Pricing in Continuous-State, Continuous-Time	445	(2)
Option Pricing	447	(7)
Stock Price Processes	447	(1)
Hedging	448	(1)
The Black-Scholes Option Pricing Formula	449	(3)
Generalizing the Pricing of European Options	452	(2)
State-Price Deflators	454	(3)
Equivalent Martingale Measures	457	(2)
Equivalent Martingale Measures and Girsanov's Theorem	459	(4)
The Diffusion Invariance Principle	461	(1)
Application of Girsanov's Theorem to Black-Scholes Option Pricing Formula	462	(1)
Equivalent Martingale Measures and Complete Markets	463	(1)
Equivalent Martingale Measures and State Prices	464	(2)
Arbitrage Pricing with a Payoff Rate	466	(1)
Implications of the Absence of Arbitrage	467	(1)
Working with Equivalent Martingale Measures	468	(1)
Summary	468	(3)
Portfolio Selection Using Mean-Variance Analysis	471	(40)
Diversification as a Central Theme in Finance	472	(2)
Markowitz's Mean-Variance Analysis	474	(3)
Capital Market Line	477	(5)
Deriving the Capital Market Line	478	(3)
What is Portfolio M?	481	(1)
Risk Premium in the CML	482	(1)
The CML and the Optimal Portfolio	482	(3)
Utility Functions and Indifference Curves	482	(2)
Selection of the Optimal Portfolio	484	(1)
Extension of the Markowitz Mean-Variance Model to Inequality Constraints	485	(2)
A Second Look at Portfolio Choice	487	(4)
The Return Forecast	487	(1)
The Utility Function	488	(2)
Optimizers	490	(1)
A Global Probabilistic Framework for Portfolio Selection	490	(1)
Relaxing the Assumption of Normality	491	(1)
Multiperiod Stochastic Optimization	492	(2)
Application to the Asset Allocation Decision	494	(13)
The Inputs	495	(5)
Portfolio Selection: An Example	500	(3)
Inclusion of More Asset Classes	503	(4)
Extensions of the Basic Asset Allocation Model	507	(2)
Summary	509	(2)
Capital Asset Pricing Model	511	(18)
CAPM Assumptions	512	(1)
Systematic and Nonsystematic Risk	513	(3)
Security Market Line	516	(2)
Estimating the Characteristic Line	518	(1)
Testing The CAPM	518	(5)
Deriving the Empirical Analogue of the CML	518	(1)
Empricial Implications	519	(1)
General Findings of Empirical Tests of the CAPM	520	(1)
A Critique of Tests of the CAPM	520	(1)
Merton and Black Modifications of the CAPM	521	(1)
CAPM and Random Matrices	522	(1)
The Conditional CAPM	523	(1)
Beta, Beta Everywhere	524	(1)
The Role of the CAPM in Investment Management Applications	525	(1)
Summary	526	(3)
Multifactor Models and Common Trends for Common Stocks	529	(22)
Multifactor Models	530	(7)
Determination of Factors	532	(5)
Dynamic Market Models of Returns	537	(1)
Estimation of State-Space Models	538	(1)
Dynamic Models for Prices	538	(8)
Estimation and Testing of Cointegrated Systems	543	(1)
Cointegration and Financial Time Series	544	(2)
Nonlinear Dynamic Models for Prices and Returns	546	(3)
Summary	549	(2)
Equity Portfolio Management	551	(42)
Integrating the Equity Portfolio Management Process	551	(1)
Active versus Passive Portfolio Management	552	(1)
Tracking Error	553	(7)
Backward-Looking versus Forward-Looking Tracking Error	555	(1)
The Impact of Portfolio Size, Benchmark Volatility, and Portfolio Beta on Tracking Error	556	(4)
Equity Style Management	560	(4)
Types of Equity Styles	560	(2)
Style Classification Systems	562	(2)
Passive Strategies	564	(2)
Constructing an Indexed Portfolio	564	(1)
Index Tracking and Cointegration	565	(1)
Active Investing	566	(11)
Top-Down Approaches to Active Investing	566	(1)
Bottom-Up Approaches to Active Investing	567	(1)
Fundamental Law of Active Management	568	(3)
Strategies Based on Technical Analysis	571	(2)
Nonlinear Dynamic Models and Chaos	573	(1)
Technical Analysis and Statistical Nonlinear Pattern Recognition	574	(1)
Market-Neutral Strategies and Statistical Arbitrage	575	(2)
Application of Multifactor Risk Models	577	(12)
Risk Decomposition	577	(5)
Portfolio Construction and Risk Control	582	(1)
Assessing the Exposure of a Portfolio	583	(4)
Risk Control Against a Stock Market Index	587	(1)
Tilting a Portfolio	587	(2)
Summary	589	(4)
Term Structure Modeling and Valuation of Bonds and Bond Options	593	(56)
Basic Principles of Valuation of Debt Instruments	594	(2)
Yield-to-Maturity Measure	596	(3)
Premium Par Yield	598	(1)
Reinvestment of Cash Flow and Yield	598	(1)
The Term Structure of the Interest Rates and the Yield Curve	599	(13)
Limitations of Using the Yield to Value a Bond	602	(1)
Valuing a Bond as a Package of Cash Flows	603	(1)
Obtaining Spot Rates from the Treasury Yield Curve	603	(3)
Using Spot Rates to the Arbitrage-Free Value of a Bond	606	(1)
The Discount Function	606	(1)
Forward Rates	607	(1)
Swap Curve	608	(4)
Classical Economic Theories About the Determinants of the Shape of the Term Structure	612	(6)
Expectations Theories	613	(5)
Market Segmentation Theory	618	(1)
Bond Valuation Formulas in Continuous Time	618	(5)
The Term Structure of Interest Rates in Continuous Time	623	(15)
Spot Rates: Continuous Case	624	(1)
Forward Rates: Continuous Case	625	(1)
Relationships for Bond and Option Valuation	626	(1)
The Feynman-Kac Formula	627	(5)
Multifactor Term Structure Model	632	(2)
Arbitrage-Free Models versus Equilibrium Models	634	(1)
Examples of One-Factor Term Structure Models	635	(3)
Two-Factor Models	638	(1)
Pricing of Interest-Rate Derivatives	638	(2)
The Heath-Jarrow-Morton Model of the Term Structure	640	(3)
The Brace-Gatarek-Musiela Model	643	(1)
Discretization of Ito Processes	644	(2)
Summary	646	(3)
Bond Portfolio Management	649	(30)
Management versus a Bond Market Index	649	(12)
Tracking Error and Bond Portfolio Strategies	651	(1)
Risk Factors and Portfolio Management Strategies	652	(2)
Determinants of Tracking Error	654	(1)
Illustration of the Multifactor Risk Model	654	(7)
Liability-Funding Strategies	661	(16)
Cash Flow Matching	664	(3)
Portfolio Immunization	667	(5)
Scenario Optimization	672	(1)
Stochastic Programming	673	(4)
Summary	677	(2)
Credit Risk Modeling and Credit Default Swaps	679	(58)
Credit Default Swaps	679	(4)
Single-Name Credit Default Swaps	680	(1)
Basket Default Swaps	681	(2)
Legal Documentation	683	(1)
Credit Risk Modeling: Structural Models	683	(13)
The Black-Scholes-Merton Model	685	(5)
Geske Compound Option Model	690	(4)
Barrier Structural Models	694	(2)
Advantages and Drawbacks of Structural Models	696	(1)
Credit Risk Modeling: Reduced Form Models	696	(14)
The Poisson Process	697	(1)
The Jarrow-Turnbull Model	698	(5)
Transition Matrix	703	(3)
The Duffie-Singleton Model	706	(4)
General Observations on Reduced Form Models	710	(1)
Pricing Single-Name Credit Default Swaps	710	(8)
General Framework	711	(1)
Survival Probability and Forward Default Probability: A Recap	712	(1)
Credit Default Swap Value	713	(3)
No Need For Stochastic Hazard Rate or Interest Rate	716	(1)
Delivery Option in Default Swaps	716	(1)
Default Swaps with Counterparty Risk	717	(1)
Valuing Basket Default Swaps	718	(16)
The Pricing Model	718	(4)
How to Model Correlated Default Processes	722	(12)
Summary	734	(3)
Risk Management	737	(20)
Market Completeness	738	(6)
The Mathematics of Market Completeness	739	(3)
The Economics of Market Completeness	742	(2)
Why Manage Risk?	744	(1)
Risk Models	745	(2)
Market Risk	745	(1)
Credit Risk	746	(1)
Operational Risk	746	(1)
Risk Measures	747	(4)
Risk Management in Asset and Portfolio Management	751	(4)
Factors Driving Risk Management	752	(1)
Risk Measurement in Practice	752	(1)
Getting Down to the Lowest Level	753	(1)
Regulatory Implications of Risk Measurement	754	(1)
Summary	755	(2)
Index	757