Mathematical Techniques in Finance: Tools for Incomplete Markets

Mathematical Techniques in Finance: Tools for Incomplete Markets

by Ales Cern
     
 

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ISBN-10: 0691088071

ISBN-13: 9780691088075

Pub. Date: 11/03/2003

Publisher: Princeton University Press

Modern finance overlaps with many fields of mathematics, and for students this can represent considerable strain. Mathematical Techniques in Finance is an ideal textbook for Masters finance courses with a significant quantitative element while also being suitable for finance Ph.D. students. Developed for the highly acclaimed Master of Science in Finance program at

Overview

Modern finance overlaps with many fields of mathematics, and for students this can represent considerable strain. Mathematical Techniques in Finance is an ideal textbook for Masters finance courses with a significant quantitative element while also being suitable for finance Ph.D. students. Developed for the highly acclaimed Master of Science in Finance program at Imperial College London, it offers a carefully crafted blend of numerical applications and theoretical grounding in economics, finance, and mathematics. In the best engineering tradition, Ales Cerny mixes tools from calculus, linear algebra, probability theory, numerical mathematics, and programming to analyze in an accessible way some of the most intriguing problems in financial economics. Eighty figures, over 70 worked examples, 25 simple ready-to-run computer programs, and several spreadsheets further enhance the learning experience. Each chapter is followed by a number of classroom-tested exercises with solutions available on the book's web site. Applied mathematics is a craft that requires practice -- this textbook provides plenty of opportunities to practice it and teaches cutting-edge finance into the bargain. Asset pricing is a common theme throughout the book; and readers can follow the development from discrete one-period models to continuous time stochastic processes. This textbook sets itself apart by the comprehensive treatment of pricing and risk measurement in incomplete markets, an area of current research that represents the future in risk management and investment performance evaluation.

Product Details

ISBN-13:
9780691088075
Publisher:
Princeton University Press
Publication date:
11/03/2003
Edition description:
Older Edition
Pages:
352
Product dimensions:
5.90(w) x 9.10(h) x 1.00(d)

Table of Contents

Prefacexiii
1The Simplest Model of Financial Markets1
1.1One-Period Finite State Model1
1.2Securities and Their Pay-Offs3
1.3Securities as Vectors3
1.4Operations on Securities4
1.5The Matrix as a Collection of Securities6
1.6Transposition6
1.7Matrix Multiplication and Portfolios8
1.8Systems of Equations and Hedging10
1.9Linear Independence and Redundant Securities12
1.10The Structure of the Marketed Subspace14
1.11The Identity Matrix and Arrow-Debreu Securities16
1.12Matrix Inverse17
1.13Inverse Matrix and Replicating Portfolios17
1.14Complete Market Hedging Formula19
1.15Summary20
1.16Notes21
1.17Exercises21
2Arbitrage and Pricing in the One-Period Model25
2.1Hedging with Redundant Securities and Incomplete Market25
2.2Finding the Best Approximate Hedge29
2.3Minimizing the Expected Squared Replication Error32
2.4Numerical Stability of Least Squares34
2.5Asset Prices, Returns and Portfolio Units37
2.6Arbitrage38
2.7No-Arbitrage Pricing40
2.8State Prices and the Arbitrage Theorem42
2.9State Prices and Asset Returns45
2.10Risk-Neutral Probabilities45
2.11State Prices and No-Arbitrage Pricing46
2.12Summary48
2.13Notes49
2.14Appendix: Least Squares with QR Decomposition49
2.15Exercises52
3Risk and Return in the One-Period Model55
3.1Utility Functions55
3.2Expected Utility Maximization59
3.3Reporting Expected Utility in Terms of Money60
3.4Scale-Free Formulation of the Optimal Investment Problem with the HARA Utility62
3.5Quadratic Utility66
3.6Reporting Investment Potential in Terms of Sharpe Ratios70
3.7The Importance of Arbitrage Adjustment77
3.8Portfolio Choice with Near-Arbitrage Opportunities78
3.9Generalization of the Sharpe Ratio82
3.10Summary83
3.11Notes84
3.12Exercises85
4Numerical Techniques for Optimal Portfolio Selection in Incomplete Markets87
4.1Sensitivity Analysis of Portfolio Decisions with the CRRA Utility87
4.2Newton's Algorithm for Optimal Investment with CRRA Utility91
4.3Optimal CRRA Investment Using Empirical Return Distribution93
4.4HARA Portfolio Optimizer99
4.5HARA Portfolio Optimization with Several Risky Assets100
4.6Quadratic Utility Maximization with Multiple Assets104
4.7Summary106
4.8Notes107
4.9Exercises107
5Pricing in Dynamically Complete Markets109
5.1Options and Portfolio Insurance109
5.2Option Pricing110
5.3Dynamic Replicating Trading Strategy113
5.4Risk-Neutral Probabilities in a Multi-Period Model121
5.5The Law of Iterated Expectations124
5.6Summary126
5.7Notes126
5.8Exercises126
6Towards Continuous Time131
6.1IID Returns, and the Term Structure of Volatility131
6.2Towards Brownian Motion133
6.3Towards a Poisson Jump Process142
6.4Central Limit Theorem and Infinitely Divisible Distributions148
6.5Summary149
6.6Notes151
6.7Exercises151
7Fast Fourier Transform153
7.1Introduction to Complex Numbers and the Fourier Transform153
7.2Discrete Fourier Transform (DFT)158
7.3Fourier Transforms in Finance159
7.4Fast Pricing via the Fast Fourier Transform (FFT)164
7.5Further Applications of FFTs in Finance167
7.6Notes171
7.7Appendix172
7.8Exercises174
8Information Management175
8.1Information: Too Much of a Good Thing?175
8.2Model-Independent Properties of Conditional Expectation179
8.3Summary183
8.4Notes184
8.5Appendix: Probability Space184
8.6Exercises188
9Martingales and Change of Measure in Finance193
9.1Discounted Asset Prices Are Martingales193
9.2Dynamic Arbitrage Theorem198
9.3Change of Measure199
9.4Dynamic Optimal Portfolio Selection in a Complete Market204
9.5Summary212
9.6Notes214
9.7Exercises214
10Brownian Motion and Ito Formulae219
10.1Continuous-Time Brownian Motion219
10.2Stochastic Integration and Ito Processes224
10.3Important Ito Processes226
10.4Function of a Stochastic Process: the Ito Formula228
10.5Applications of the Ito Formula229
10.6Multivariate Ito Formula231
10.7Ito Processes as Martingales234
10.8Appendix: Proof of the Ito Formula235
10.9Summary235
10.10Notes236
10.11Exercises237
11Continuous-Time Finance239
11.1Summary of Useful Results239
11.2Risk-Neutral Pricing240
11.3The Girsanov Theorem243
11.4Risk-Neutral Pricing and Absence of Arbitrage247
11.5Automatic Generation of PDEs and the Feynman--Kac Formula252
11.6Overview of Numerical Methods256
11.7Summary257
11.8Notes258
11.9Appendix: Decomposition of Asset Returns into Uncorrelated Components258
11.10Exercises261
12Dynamic Option Hedging and Pricing in Incomplete Markets267
12.1The Risk in Option Hedging Strategies267
12.2Incomplete Market Option Price Bounds283
12.3Towards Continuous Time291
12.4Derivation of Optimal Hedging Strategy297
12.5Summary306
12.6Notes307
12.7Appendix: Expected Squared Hedging Error in the Black--Scholes Model307
12.8Exercises309
Appendix ACalculus313
A.1Notation313
A.2Differentiation316
A.3Real Function of Several Real Variables319
A.4Power Series Approximations321
A.5Optimization324
A.6Integration326
A.7Exercises332
Appendix BProbability337
B.1Probability Space337
B.2Conditional Probability337
B.3Marginal and Joint Distribution340
B.4Stochastic Independence341
B.5Expectation Operator343
B.6Properties of Expectation344
B.7Mean and Variance345
B.8Covariance and Correlation346
B.9Continuous Random Variables349
B.10Normal Distribution354
B.11Quantiles359
B.12Relationships among Standard Statistical Distributions360
B.13Notes361
B.14Exercises361
References369
Index373

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