Quantitative modeling of derivative securities: from theory to practice
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
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Boca Raton
CRC Press
2020
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 322 Seiten |
| ISBN: | 9780367579142 |
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| adam_text | Contents Introduction 1 ix Arbitrage Pricing Theory: The One-Period Model 1 1.1 The Arrow-Debreu Model.................................................................................... 2 1.2 Security-Space Diagram: A Geometric Interpretation of Theorem 1.1............. 8 1.3 Replication.................................................................................................................. 11 1.4 The Binomial Model..................................................................................................13 1.5 Complete and Incomplete Markets........................................................................... 14 1.6 The One-Period Trinomial Model........................................................................... 16 1.7 Exercises......................................................................................................................18 References and Further Reading........................................................................................ 19 2 The Binomial Option Pricing Model 21 2.1 Recursion Relation for Pricing Contingent Claims................................................. 22 2.2 Delta-Hedging and the Replicating Portfolio...........................................................24 2.3 Pricing European Puts and Calls.............................................................................. 26 Portfolio Delta........................................................................................................... 27 Money-Market
Account........................................................................................... 27 Puts ............................................................................................................................28 2.4 Relation Between the Parameters of the Tree and the Stock Price Fluctuations.....................................................................................................28 Calibration of the Volatility Parameter.................................................................... 31 Expected Growth Rate.............................................................................................. 32 Implementation of Binomial Trees........................................................................... 33 2.5 The Limit for dt -*■ 0: Log-Normal Approximation...........................................34 2.6 The Black-Scholes Formula.....................................................................................35 References and Further Reading........................................................................................39 3 Analysis of the Black-Scholes Formula 41 3.1 Delta........................................................................................................................... 42 Option Deltas ...........................................................................................................44 3.2 Practical Delta Hedging...........................................................................................45 3.3 Gamma: The Convexity Factor
..............................................................................48 3.4 Theta: The Time-Decay Factor................................................................................. 51 3.5 The Binomial Model as a Finite-Difference Scheme for the Black-Scholės Equation...........................................................................................54 References and Further Reading....................................................................................... 55 iii
IV CONTENTS 57 57 63 66 70 73 75 76 4 Refinements of the Binomial Model 4.1 Term-Structure of Interest Rates...................................................... 4.2 Constructing a Risk-Neutral Measure with Time-Dependent Volatility 4.3 Deriving a Volatility Term-Structure from Option Market Data . . . 4.4 Underlying Assets That Pay Dividends................................................... 4.5 Futures Contracts as the Underlying Security..................................... 4.6 Valuation of a Stream of Uncertain Cash Flows.................................. References and Further Reading...................................................................... 5 American-Style Options, Early Exercise, and Time-Optionality 5.1 American-Style Options....................................................................................... 5.2 Early-Exercise Premium....................................................................................... 5.3 Pricing American Options Using the Binomial Model: The Dynamic Programming Equation.......................................................................................... 5.4 Hedging................................................................................................................. 5.5 Characterization of the Solution for dt £ 1: Free-Boundary Problem for the Black-Scholes Equation..........................................................................................82 References and Further Reading.......................................................................................88 A A PDE Approach to the
Free-Boundary Condition............................................ 89 A.l A Proof of the Free Boundary Condition................................................................ 90 6 Trinomial Model and Finite-Difference Schemes 93 6.1 Trinomial Model........................................................................................................... 93 6.2 Stability Analysis ........................................................................................................ 95 6.3 Calibration of the Model...............................................................................................96 6.4 “Tree-Trimming” and Far-Field Boundary Conditions............................................100 6.5 Implicit Schemes.......................................................................................................... 103 References and Further Reading........................................................................................ 106 7 Brownian Motion and Ito Calculus 107 7.1 Brownian Motion........................................................................................................107 7.2 Elementary Properties of Brownian Paths................................................................ 109 7.3 Stochastic Integrals .................................................................................................. լ լ լ 7.4 Ito’s Lemma.................................................................................................................. η 7.5 Ito Processes and Ito
Calculus.................................................................................... 120 References and Further Reading.................................................................................. շշշ A Properties of the Ito Integral .................................................................................. լ 23 8 Introduction to Exotic Options: Digital and Barrier Options 8.1 Digital Options...................................................................... European Digitals............................................................ American Digitals...................................................... 8.2 Barrier Options.................................................. Pricing Barrier Options Using Trees or Lattices Closed-Form Solutions........................................ Hedging Barrier Options........................................ 8.3 Double Barrier Options Range Discount Note ........................... Range Accruals ................................. Double Knock-out Options.................... References and Further Reading 127 128 128 . 135 . 139 . 141 . 142 . 145 . 146 . 147 . 148 . 150 . 150
CONTENTS A A.l A.2 В B.l B.2 v Proofs of Lemmas 8.1 and 8.2 151 A Consequence of the Invariance of Brownian Motion Under Reflections . . .151 The Case џ փ 0.....................................................................................................153 Closed-Form Solutions for Double-Barrier Options............................................155 Exit Probabilities of a Brownian Trajectory from a Strip — B Z A . . . .155 Applications to Pricing Barrier Options..............................................................158 9 Ito Processes, Continuous-Time Martingales, and Girsanov’s Theorem 161 9.1 Martingales and Doob-Meyer Decomposition..................................................... 161 9.2 Exponential Martingales......................................................................................... 163 9.3 Girsanov’s Theorem................................................................................................165 References and Further Reading......................................................................................168 A Proof of Equation (9.11) 169 10 Continuous-Time Finance: An Introduction 171 10.1 The Basic Model...................................................................................................... 171 10.2 Trading Strategies...................................................................................................173 10.3 Arbitrage Pricing Theory............................................................. 176 References and Further
Reading...................................................................................... 181 11 Valuation of Derivative Securities 183 11.1 The General Principle............................................................................................ 183 11.2 Black-Scholes Model............................................................................................ 185 11.3 Dynamic Hedging and Dynamic Completeness.................................................. 189 11.4 Fokker-Planck Theory: Computing Expectations Using PDEs..........................193 References and Further Reading...................................................................................... 196 A Proof of Proposition 11.5..................................................................................... 197 12 Fixed-Income Securities and the Term-Structure of Interest Rates 199 12.1 Bonds...................................................................................................................... 199 12.2 Duration...................................................................................................................206 12.3 Term Rates, Forward Rates, and Futures-Implied Rates.................................... 209 12.4 Interest-Rate Swaps............................................................................................... 212 12.5 Caps and Floors......................... 217 12.6 Swaptions and Bond Options.................................................................................. 218 12.7 Instantaneous Forward Rates:
Definition...............................................................221 12.8 Building an Instantaneous Forward-Rate Curve..................................................224 References and Further Reading......................................................................................227 13 The Heath-Jarrow-Morton Theorem and Multidimensional Term-Structure Models 229 13.1 The Heath-Jarrow-Morton Theorem..................................................................... 230 13.2 The Но-Lee Model ............................................................................................... 234 13.3 Mean Reversion: The Modified Vasicek or Hull-White Model ........................ 237 13.4 Factor Analysis of the Term-Structure..................................................................239 13.5 Example: Construction of a Two-Factor Model with Parametric Components.........................................................................................245 13.6 More General Volatility Specifications in the HJM Equation.............................. 248 References and Further Reading..................................................................................... 251
CONTENTS 14 Exponential-Affine Models 14.1 A Characterization of EA Models............................................................................ 255 14.2 Gaussian State-Variables: General Formulas........................................................... 258 14.3 Gaussian Models: Explicit Formulas.........................................................................261 14.4 Square-Root Processes and the Non-Central Chi-Squared Distribution .... 264 14.5 One-Factor Square-Root Model: Discount Factors and Forward Rates..............268 References and Further Reading......................................................................................... 222 A Behavior of Square-Root Processes for Large Times...........................................273 В Characterization of the Probability Density Function of Square-Root Processes................................................................................................275 C The Square-Root Diffusion with υ = 1.................................................................. 277 15 Interest-Rate Options 279 15.1 Forward Measures......................................................................................................279 Definition and Examples............................................................................................279 15.2 Commodity Options with Stochastic Interest Rate................................................. 282 15.3 Options on Zero-Coupon Bonds............................................................................... 283 15.4 Money-Market Deposits with Yield
Protection........................................................285 Forward Rates and Forward Measures..................................................................... 286 15.5 Pricing Caps................................................................................................................289 General Considerations ............................................................................................ 289 Cap Pricing with Gaussian Models............................................................................ 292 Cap Pricing with Square-Root Models......................................................................293 Cap Pricing and Implied Volatilities......................................................................... 297 15.6 Bond Options and Swaptions...................................................................................... 299 General Pricing Relations......................................................................................... 299 Jamshidian’s Theorem................................................................................................ 301 Volatility Analysis....................................................................................................... ЗОЗ 15.7 Epilogue: The Brace-Gatarek-Musiela model......................................................... 308 References and Further Reading.......................................................................................... Յլշ Index 313
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| adam_txt |
Contents Introduction 1 ix Arbitrage Pricing Theory: The One-Period Model 1 1.1 The Arrow-Debreu Model. 2 1.2 Security-Space Diagram: A Geometric Interpretation of Theorem 1.1. 8 1.3 Replication. 11 1.4 The Binomial Model.13 1.5 Complete and Incomplete Markets. 14 1.6 The One-Period Trinomial Model. 16 1.7 Exercises.18 References and Further Reading. 19 2 The Binomial Option Pricing Model 21 2.1 Recursion Relation for Pricing Contingent Claims. 22 2.2 Delta-Hedging and the Replicating Portfolio.24 2.3 Pricing European Puts and Calls. 26 Portfolio Delta. 27 Money-Market
Account. 27 Puts .28 2.4 Relation Between the Parameters of the Tree and the Stock Price Fluctuations.28 Calibration of the Volatility Parameter. 31 Expected Growth Rate. 32 Implementation of Binomial Trees. 33 2.5 The Limit for dt -*■ 0: Log-Normal Approximation.34 2.6 The Black-Scholes Formula.35 References and Further Reading.39 3 Analysis of the Black-Scholes Formula 41 3.1 Delta. 42 Option Deltas .44 3.2 Practical Delta Hedging.45 3.3 Gamma: The Convexity Factor
.48 3.4 Theta: The Time-Decay Factor. 51 3.5 The Binomial Model as a Finite-Difference Scheme for the Black-Scholės Equation.54 References and Further Reading. 55 iii
IV CONTENTS 57 57 63 66 70 73 75 76 4 Refinements of the Binomial Model 4.1 Term-Structure of Interest Rates. 4.2 Constructing a Risk-Neutral Measure with Time-Dependent Volatility 4.3 Deriving a Volatility Term-Structure from Option Market Data . . . 4.4 Underlying Assets That Pay Dividends. 4.5 Futures Contracts as the Underlying Security. 4.6 Valuation of a Stream of Uncertain Cash Flows. References and Further Reading. 5 American-Style Options, Early Exercise, and Time-Optionality 5.1 American-Style Options. 5.2 Early-Exercise Premium. 5.3 Pricing American Options Using the Binomial Model: The Dynamic Programming Equation. 5.4 Hedging. 5.5 Characterization of the Solution for dt £ 1: Free-Boundary Problem for the Black-Scholes Equation.82 References and Further Reading.88 A A PDE Approach to the
Free-Boundary Condition. 89 A.l A Proof of the Free Boundary Condition. 90 6 Trinomial Model and Finite-Difference Schemes 93 6.1 Trinomial Model. 93 6.2 Stability Analysis . 95 6.3 Calibration of the Model.96 6.4 “Tree-Trimming” and Far-Field Boundary Conditions.100 6.5 Implicit Schemes. 103 References and Further Reading. 106 7 Brownian Motion and Ito Calculus 107 7.1 Brownian Motion.107 7.2 Elementary Properties of Brownian Paths. 109 7.3 Stochastic Integrals . լ լ լ 7.4 Ito’s Lemma.\\η 7.5 Ito Processes and Ito
Calculus. 120 References and Further Reading. շշշ A Properties of the Ito Integral . լ 23 8 Introduction to Exotic Options: Digital and Barrier Options 8.1 Digital Options. European Digitals. American Digitals. 8.2 Barrier Options. Pricing Barrier Options Using Trees or Lattices Closed-Form Solutions. Hedging Barrier Options. 8.3 Double Barrier Options Range Discount Note . Range Accruals . Double Knock-out Options. References and Further Reading 127 128 128 . 135 . 139 . 141 . 142 . 145 . 146 . 147 . 148 . 150 . 150
CONTENTS A A.l A.2 В B.l B.2 v Proofs of Lemmas 8.1 and 8.2 151 A Consequence of the Invariance of Brownian Motion Under Reflections . . .151 The Case џ փ 0.153 Closed-Form Solutions for Double-Barrier Options.155 Exit Probabilities of a Brownian Trajectory from a Strip — B Z A . . . .155 Applications to Pricing Barrier Options.158 9 Ito Processes, Continuous-Time Martingales, and Girsanov’s Theorem 161 9.1 Martingales and Doob-Meyer Decomposition. 161 9.2 Exponential Martingales. 163 9.3 Girsanov’s Theorem.165 References and Further Reading.168 A Proof of Equation (9.11) 169 10 Continuous-Time Finance: An Introduction 171 10.1 The Basic Model. 171 10.2 Trading Strategies.173 10.3 Arbitrage Pricing Theory. 176 References and Further
Reading. 181 11 Valuation of Derivative Securities 183 11.1 The General Principle. 183 11.2 Black-Scholes Model. 185 11.3 Dynamic Hedging and Dynamic Completeness. 189 11.4 Fokker-Planck Theory: Computing Expectations Using PDEs.193 References and Further Reading. 196 A Proof of Proposition 11.5. 197 12 Fixed-Income Securities and the Term-Structure of Interest Rates 199 12.1 Bonds. 199 12.2 Duration.206 12.3 Term Rates, Forward Rates, and Futures-Implied Rates. 209 12.4 Interest-Rate Swaps. 212 12.5 Caps and Floors. 217 12.6 Swaptions and Bond Options. 218 12.7 Instantaneous Forward Rates:
Definition.221 12.8 Building an Instantaneous Forward-Rate Curve.224 References and Further Reading.227 13 The Heath-Jarrow-Morton Theorem and Multidimensional Term-Structure Models 229 13.1 The Heath-Jarrow-Morton Theorem. 230 13.2 The Но-Lee Model . 234 13.3 Mean Reversion: The Modified Vasicek or Hull-White Model . 237 13.4 Factor Analysis of the Term-Structure.239 13.5 Example: Construction of a Two-Factor Model with Parametric Components.245 13.6 More General Volatility Specifications in the HJM Equation. 248 References and Further Reading. 251
CONTENTS 14 Exponential-Affine Models 14.1 A Characterization of EA Models. 255 14.2 Gaussian State-Variables: General Formulas. 258 14.3 Gaussian Models: Explicit Formulas.261 14.4 Square-Root Processes and the Non-Central Chi-Squared Distribution . 264 14.5 One-Factor Square-Root Model: Discount Factors and Forward Rates.268 References and Further Reading. 222 A Behavior of Square-Root Processes for Large Times.273 В Characterization of the Probability Density Function of Square-Root Processes.275 C The Square-Root Diffusion with υ = 1. 277 15 Interest-Rate Options 279 15.1 Forward Measures.279 Definition and Examples.279 15.2 Commodity Options with Stochastic Interest Rate. 282 15.3 Options on Zero-Coupon Bonds. 283 15.4 Money-Market Deposits with Yield
Protection.285 Forward Rates and Forward Measures. 286 15.5 Pricing Caps.289 General Considerations . 289 Cap Pricing with Gaussian Models. 292 Cap Pricing with Square-Root Models.293 Cap Pricing and Implied Volatilities. 297 15.6 Bond Options and Swaptions. 299 General Pricing Relations. 299 Jamshidian’s Theorem. 301 Volatility Analysis. ЗОЗ 15.7 Epilogue: The Brace-Gatarek-Musiela model. 308 References and Further Reading. Յլշ Index 313 |
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| spelling | Avellaneda, Marco 1955-2022 Verfasser (DE-588)171583396 aut Quantitative modeling of derivative securities from theory to practice Marco Avellaneda in collaboration with Peter Laurence Boca Raton CRC Press 2020 © 2000 XII, 322 Seiten txt rdacontent n rdamedia nc rdacarrier Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Derivat Wertpapier (DE-588)4381572-8 gnd rswk-swf Derivat Wertpapier (DE-588)4381572-8 s Mathematisches Modell (DE-588)4114528-8 s DE-604 Laurence, Peter Verfasser aut Äquivalent Avellaneda, Marco Quantitative modeling of derivative securities 2000 978-1-58488-031-8 Digitalisierung UB Bamberg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032533728&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Avellaneda, Marco 1955-2022 Laurence, Peter Quantitative modeling of derivative securities from theory to practice Mathematisches Modell (DE-588)4114528-8 gnd Derivat Wertpapier (DE-588)4381572-8 gnd |
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| title | Quantitative modeling of derivative securities from theory to practice |
| title_auth | Quantitative modeling of derivative securities from theory to practice |
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| title_full | Quantitative modeling of derivative securities from theory to practice Marco Avellaneda in collaboration with Peter Laurence |
| title_fullStr | Quantitative modeling of derivative securities from theory to practice Marco Avellaneda in collaboration with Peter Laurence |
| title_full_unstemmed | Quantitative modeling of derivative securities from theory to practice Marco Avellaneda in collaboration with Peter Laurence |
| title_short | Quantitative modeling of derivative securities |
| title_sort | quantitative modeling of derivative securities from theory to practice |
| title_sub | from theory to practice |
| topic | Mathematisches Modell (DE-588)4114528-8 gnd Derivat Wertpapier (DE-588)4381572-8 gnd |
| topic_facet | Mathematisches Modell Derivat Wertpapier |
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