Financial mathematics: from discrete to continuous time
"This book is a study of the mathematical ideas and techniques that are important to the two main arms of the area of Financial Mathematics: portfolio optimization and derivative valuation. The text is authored for courses taken by advanced undergraduates, MBA, or other students in quantitative...
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton ; London ; New York
CRC Press
2023
|
| Edition: | First edition |
| Series: | Chapman & Hall/CRC financial mathematics series
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | "This book is a study of the mathematical ideas and techniques that are important to the two main arms of the area of Financial Mathematics: portfolio optimization and derivative valuation. The text is authored for courses taken by advanced undergraduates, MBA, or other students in quantitative finance programs"-- |
| Physical Description: | xvii, 411 Seiten Diagramme |
| ISBN: | 9781498780407 9781032403892 |
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| adam_text | Contents Preface xiii Author xvii 1 Review of Preliminaries Risky Assets ................................................................................. 1.1.1 Single and Multiple Discrete Time Periods................... 1.1.2 Continuous-Time Processes............................................ 1.1.3 Martingales........................................................................ 1.2 Risk Aversion and Portfolios of Assets..................................... 1.2.1 Risk Aversion Constant.................................................. 1.2.2 The Portfolio Problem..................................................... 1-3 Expectation, Variance, and Covariance .................................. 1.3.1 One Variable Expectation............................................... 1.3.2 Expectation for Multiple Random Variables................ 1.3.3 Variance of a Linear Combination ................................ 1.4 Simple Portfolio Optimization .................................................. 1.5 Derivative Assets and Arbitrage .............................................. 1.5.1 Futures.............................................................................. 1.5.2 Arbitrage and Futures..................................................... 1.5.3 Options.............................................................................. 1.6 Valuation of Derivatives in a Single Time Period................... 1.6.1 Replicating Portfolios..................................................... 1.6.2 Risk-Neutral Valuation .................................................. 1.1 2
Portfolio Selection and CAPM Theory 2.1 2.2 2.3 Portfolio Optimization with Multiple Assets ......................... 2.1.1 Lagrange Multipliers........................................................ 2.1.2 Qualitative Behavior......................................................... 2.1.3 Correlated Assets............................................................... 2.1.4 Portfolio Separation and the Market Portfolio............. Capital Market Theory, Part I .................................................. 2.2.1 Linear Algebraic Approach............................................ 2.2.2 Efficient Mean-Standard Deviation Frontier................ Capital Market Theory, Part II........................................ 1 2 4 7 8 11 11 14 18 18 22 26 32 38 38 41 45 51 52 58 65 65 69 71 73 76 82 82 87 99 vii
Contents viii Capital Market Line......................................................... САРМ Formula; Asset ¡3 ............................................... Systematic and Non-Systematic Risk; Pricing Using CAPM..................................................................... 108 2.4 Utility Theory .............................................................................. 2.4.1 Securities and Axioms for InvestorBehavior................ 2.4.2 Indifference Curves, Certainty Equivalent, Risk Aver sion ........................................................................ 118 2.4.3 Examples of Utility Functions........................................ 2.4.4 Absolute and Relative Risk Aversion............................ 2.4.5 Utility Maximization........................................................ 2.5 Multiple Period Portfolio Problems ......................................... 2.5.1 Problem Description and Dynamic Programming Ap proach ..................................................................... 136 2.5.2 Examples........................................................................... 2.5.3 Optimal Portfolios and Martingales ............................ 2.3.1 2.3.2 2.3.3 3 Discrete-Time Derivatives Valuation 3.1 Options Pricing for Multiple Time Periods ............................. 3.1.1 Introduction ...................................................................... 3.1.2 Valuation by Chaining...................................................... 3.1.3 Valuation by Martingales................................................ 3.2 Key Ideas of
Discrete Probability, Part I ............................... 3.2.1 Algebras and Measurability............................................. 3.2.2 Independence...................................................................... 3.3 Key Ideas of Discrete Probability, Part II............................... 3.3.1 Conditional Expectation................................................... 3.3.2 Application to Pricing Models......................................... 3.4 Fundamental Theorems of Options Pricing ............................ 3.4.1 The Market Model............................................................ 3.4.2 Gain, Arbitrage, and Attainability................................ 3.4.3 Martingale Measures and the Fundamental Theorems 3.5 Valuation of Non-Vanilla Options ............................................ 3.5.1 American and Bermudan Options................................... 3.5.2 Barrier Options.................................................................. 3.5.3 Asian Options .................................................................. 3.5.4 Two-Asset Derivatives...................................................... 3.6 Derivatives Pricing by Simulation ............................................ 3.6.1 Setup and Algorithm......................................................... 3.6.2 Examples............................................................................ 3.7 From Discrete to Continuous Time (A Preview) ................... 99 104 114 114 123 127 129 136 138 148 153 153 153 156 163 171 172 181 185 185 196 202 203 208 212 223 223 232 237
238 243 243 246 253
Contents 4 Continuous Probability Models ix 261 4.1 Continuous Distributions and Expectation ............................. 261 4.1.1 Densities and Cumulative DistributionFunctions . . . 261 4.1.2 Expectation....................................................................... 265 4.1.3 Normal and Lognormal Distributions ......................... 268 4.2 Joint Distributions........................................................................ 274 4.2.1 Basic Ideas....................................................................... 274 4.2.2 Marginal and Conditional Distributions...................... 278 4.2.3 Independence..................................................................... 282 4.2.4 Covariance and Correlation........................................... 284 4.2.5 Bivariate Normal Distribution........................................ 288 4.3 Measurability and Conditional Expectation............................ 295 4.3.1 Sigma Algebras................................................................. 295 4.3.2 Random Variables and Measurability............................. 298 4.3.3 Continuous Conditional Expectation............................. 300 4.4 Brownian Motion and Geometric Brownian Motion ............. 309 4.4.1 Random Walk Processes.................................................. 310 4.4.2 Standard Brownian Motion............................................ 312 4.4.3 Non-Standard Brownian Motion................................... 314 4.4.4 Geometric Brownian Motion ......................................... 317 4.4.5 Brownian Motion and
Binomial Branch Processes . . . 320 4.5 Introduction to Stochastic Differential Equations................... 324 4.5.1 Meaning of the General SDE......................................... 325 4.5.2 Ito’s Formula..................................................................... 327 4.5.3 Geometric Brownian Motion as Solution....................... 330 5 Derivative Valuation in Continuous Time 5.1 Black-Scholes Via Limits ........................................................... 5.1.1 Black-Scholes Formula..................................................... 5.1.2 Limiting Approach............................................................ 5.1.3 Put Options and Put-Call Parity................................... 5.2 Black-Scholes Via Martingales .................................................. 5.2.1 Trading Strategies and Martingale Valuation............. 5.2.2 Martingale Measures......................................................... 5.2.3 Asset and Bond Binaries.................................................. 5.2.4 Other Binary Derivatives............................................... 5.3 Black-Scholes Via Differential Equations ................................ 5.3.1 Deriving the PDE............................................................ 5.3.2 Boundary Conditions; Solving the PDE....................... 5.4 Checking Black-Scholes Assumptions ...................................... 5.4.1 Normality of Rates of Return......................................... 5.4.2 Stability of Parameters .................................................. 5.4.3 Independence of
Rates of Return................................... 335 335 335 336 341 345 345 347 351 355 360 360 363 367 368 376 378
x Contents A Multivariate Normal Distribution A.l Review of Matrix Concepts......................................................... A.2 Multivariate Normal Distribution ............................................ 385 385 387 B Answers to Selected Exercises 395 Bibliography 405 Index 407
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| adam_txt |
Contents Preface xiii Author xvii 1 Review of Preliminaries Risky Assets . 1.1.1 Single and Multiple Discrete Time Periods. 1.1.2 Continuous-Time Processes. 1.1.3 Martingales. 1.2 Risk Aversion and Portfolios of Assets. 1.2.1 Risk Aversion Constant. 1.2.2 The Portfolio Problem. 1-3 Expectation, Variance, and Covariance . 1.3.1 One Variable Expectation. 1.3.2 Expectation for Multiple Random Variables. 1.3.3 Variance of a Linear Combination . 1.4 Simple Portfolio Optimization . 1.5 Derivative Assets and Arbitrage . 1.5.1 Futures. 1.5.2 Arbitrage and Futures. 1.5.3 Options. 1.6 Valuation of Derivatives in a Single Time Period. 1.6.1 Replicating Portfolios. 1.6.2 Risk-Neutral Valuation . 1.1 2
Portfolio Selection and CAPM Theory 2.1 2.2 2.3 Portfolio Optimization with Multiple Assets . 2.1.1 Lagrange Multipliers. 2.1.2 Qualitative Behavior. 2.1.3 Correlated Assets. 2.1.4 Portfolio Separation and the Market Portfolio. Capital Market Theory, Part I . 2.2.1 Linear Algebraic Approach. 2.2.2 Efficient Mean-Standard Deviation Frontier. Capital Market Theory, Part II. 1 2 4 7 8 11 11 14 18 18 22 26 32 38 38 41 45 51 52 58 65 65 69 71 73 76 82 82 87 99 vii
Contents viii Capital Market Line. САРМ Formula; Asset ¡3 . Systematic and Non-Systematic Risk; Pricing Using CAPM. 108 2.4 Utility Theory . 2.4.1 Securities and Axioms for InvestorBehavior. 2.4.2 Indifference Curves, Certainty Equivalent, Risk Aver sion . 118 2.4.3 Examples of Utility Functions. 2.4.4 Absolute and Relative Risk Aversion. 2.4.5 Utility Maximization. 2.5 Multiple Period Portfolio Problems . 2.5.1 Problem Description and Dynamic Programming Ap proach . 136 2.5.2 Examples. 2.5.3 Optimal Portfolios and Martingales . 2.3.1 2.3.2 2.3.3 3 Discrete-Time Derivatives Valuation 3.1 Options Pricing for Multiple Time Periods . 3.1.1 Introduction . 3.1.2 Valuation by Chaining. 3.1.3 Valuation by Martingales. 3.2 Key Ideas of
Discrete Probability, Part I . 3.2.1 Algebras and Measurability. 3.2.2 Independence. 3.3 Key Ideas of Discrete Probability, Part II. 3.3.1 Conditional Expectation. 3.3.2 Application to Pricing Models. 3.4 Fundamental Theorems of Options Pricing . 3.4.1 The Market Model. 3.4.2 Gain, Arbitrage, and Attainability. 3.4.3 Martingale Measures and the Fundamental Theorems 3.5 Valuation of Non-Vanilla Options . 3.5.1 American and Bermudan Options. 3.5.2 Barrier Options. 3.5.3 Asian Options . 3.5.4 Two-Asset Derivatives. 3.6 Derivatives Pricing by Simulation . 3.6.1 Setup and Algorithm. 3.6.2 Examples. 3.7 From Discrete to Continuous Time (A Preview) . 99 104 114 114 123 127 129 136 138 148 153 153 153 156 163 171 172 181 185 185 196 202 203 208 212 223 223 232 237
238 243 243 246 253
Contents 4 Continuous Probability Models ix 261 4.1 Continuous Distributions and Expectation . 261 4.1.1 Densities and Cumulative DistributionFunctions . . . 261 4.1.2 Expectation. 265 4.1.3 Normal and Lognormal Distributions . 268 4.2 Joint Distributions. 274 4.2.1 Basic Ideas. 274 4.2.2 Marginal and Conditional Distributions. 278 4.2.3 Independence. 282 4.2.4 Covariance and Correlation. 284 4.2.5 Bivariate Normal Distribution. 288 4.3 Measurability and Conditional Expectation. 295 4.3.1 Sigma Algebras. 295 4.3.2 Random Variables and Measurability. 298 4.3.3 Continuous Conditional Expectation. 300 4.4 Brownian Motion and Geometric Brownian Motion . 309 4.4.1 Random Walk Processes. 310 4.4.2 Standard Brownian Motion. 312 4.4.3 Non-Standard Brownian Motion. 314 4.4.4 Geometric Brownian Motion . 317 4.4.5 Brownian Motion and
Binomial Branch Processes . . . 320 4.5 Introduction to Stochastic Differential Equations. 324 4.5.1 Meaning of the General SDE. 325 4.5.2 Ito’s Formula. 327 4.5.3 Geometric Brownian Motion as Solution. 330 5 Derivative Valuation in Continuous Time 5.1 Black-Scholes Via Limits . 5.1.1 Black-Scholes Formula. 5.1.2 Limiting Approach. 5.1.3 Put Options and Put-Call Parity. 5.2 Black-Scholes Via Martingales . 5.2.1 Trading Strategies and Martingale Valuation. 5.2.2 Martingale Measures. 5.2.3 Asset and Bond Binaries. 5.2.4 Other Binary Derivatives. 5.3 Black-Scholes Via Differential Equations . 5.3.1 Deriving the PDE. 5.3.2 Boundary Conditions; Solving the PDE. 5.4 Checking Black-Scholes Assumptions . 5.4.1 Normality of Rates of Return. 5.4.2 Stability of Parameters . 5.4.3 Independence of
Rates of Return. 335 335 335 336 341 345 345 347 351 355 360 360 363 367 368 376 378
x Contents A Multivariate Normal Distribution A.l Review of Matrix Concepts. A.2 Multivariate Normal Distribution . 385 385 387 B Answers to Selected Exercises 395 Bibliography 405 Index 407 |
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| spelling | Hastings, Kevin J. 1955- Verfasser (DE-588)172128277 aut Financial mathematics from discrete to continuous time Kevin J. Hastings First edition Boca Raton ; London ; New York CRC Press 2023 xvii, 411 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Chapman & Hall/CRC financial mathematics series "This book is a study of the mathematical ideas and techniques that are important to the two main arms of the area of Financial Mathematics: portfolio optimization and derivative valuation. The text is authored for courses taken by advanced undergraduates, MBA, or other students in quantitative finance programs"-- Bewertung (DE-588)4006340-9 gnd rswk-swf Finanzmathematik (DE-588)4017195-4 gnd rswk-swf Portfolio Selection (DE-588)4046834-3 gnd rswk-swf Derivat Wertpapier (DE-588)4381572-8 gnd rswk-swf Business mathematics (DE-588)4123623-3 Lehrbuch gnd-content Finanzmathematik (DE-588)4017195-4 s Portfolio Selection (DE-588)4046834-3 s Derivat Wertpapier (DE-588)4381572-8 s Bewertung (DE-588)4006340-9 s DE-604 Erscheint auch als Online-Ausgabe 978-0-429-11365-9 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034069574&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hastings, Kevin J. 1955- Financial mathematics from discrete to continuous time Bewertung (DE-588)4006340-9 gnd Finanzmathematik (DE-588)4017195-4 gnd Portfolio Selection (DE-588)4046834-3 gnd Derivat Wertpapier (DE-588)4381572-8 gnd |
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| title | Financial mathematics from discrete to continuous time |
| title_auth | Financial mathematics from discrete to continuous time |
| title_exact_search | Financial mathematics from discrete to continuous time |
| title_exact_search_txtP | Financial mathematics from discrete to continuous time |
| title_full | Financial mathematics from discrete to continuous time Kevin J. Hastings |
| title_fullStr | Financial mathematics from discrete to continuous time Kevin J. Hastings |
| title_full_unstemmed | Financial mathematics from discrete to continuous time Kevin J. Hastings |
| title_short | Financial mathematics |
| title_sort | financial mathematics from discrete to continuous time |
| title_sub | from discrete to continuous time |
| topic | Bewertung (DE-588)4006340-9 gnd Finanzmathematik (DE-588)4017195-4 gnd Portfolio Selection (DE-588)4046834-3 gnd Derivat Wertpapier (DE-588)4381572-8 gnd |
| topic_facet | Bewertung Finanzmathematik Portfolio Selection Derivat Wertpapier Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034069574&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT hastingskevinj financialmathematicsfromdiscretetocontinuoustime |