Computational methods in finance:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Abingdon
Taylor & Francis
2024
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| Ausgabe: | Second edition |
| Schriftenreihe: | Chapman & Hall/CRC financial mathematics series
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xxix, 621 Seiten Illustrationen |
| ISBN: | 9781498778602 9781032786636 |
Internformat
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Datensatz im Suchindex
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Contents List of Figures xvii List of Tables xxi Preface xxv Acknowledgments xxix I Pricing and Valuation (Traditional) 1 Stochastic Processes and Pricing 1.1 1.2 1.3 1.4 Overview of Options . 1.1.1 Hedging Using Options: A Simple Illustration. 1.1.2 A Visualized Approach to Option Pricing . . 1.1.3 Put-Call Parity . . . . Characteristic Function. 1.2.1 Cumulative Distribution Function via Characteristic Function . 1.2.2 Moments of a Random Variable via the Characteristic Function . . 1.2.3 Characteristic Function of Demeaned Random Variables. 1.2.4 Calculating Jensen’s Inequality Correction. 1.2.5 Calculating the Characteristic Function of the Logarithmic of a Martingale. 1.2.6 Exponential Distribution. 1.2.7 Poisson Distribution. 1.2.8 Gamma Distribution. 1.2.9 Chi-squared Distribution. 1.2.10 Standard Normal Distribution.
1.2.11 Normal Distribution . 1.2.12 Lévy Processes . Stochastic Models for Interest Rates. 1.3.1 Black-Derman-Toy (BDT) Model. 1.3.2 Black-Karasinski (BK) Model. 1.3.3 Ornstein-Uhlenbeck Process. 1.3.4 Cox-Ingersoll-Ross Model. 1.3.5 Heath-Jarrow-Morton (HJM) Models . 1.3.6 Brace-Gatarek-Musiela (BGM) Model . . ; 1.3.7 Affine Term Structure Models (ATSMs) . Stochastic Models of Asset Prices. 1.4.1 Bachelier Model. 1.4.2 Geometric Brownian Motion — Black-Merton-Scholes. 1.4.3 Local Volatility Models — Derman and Kani . 1 3 3 5 6 13 14 14 15 16 16 17 18 19 19 19 19 20 21 22 22 22 22 24 25 26 26 26 27 27 29 ix
X Contents Geometrie Brownian Motion with Stochastic Volatility — Heston Model . 31 1.4.5 Mixing Model — Stochastic Local Volatility (SLV) Model. 1.4.6 Variance Gamma Model. 1.4.7 CGMY Model. 1.4.8 Normal Inverse Gaussian (NIG) Model. 1.4.9 Variance Gamma with Stochastic Arrival (VGSA) Model. 1.4.10 Equity Models and Their Corresponding Interest Rate Models . 1.5 Valuing Derivatives under Various Measures. . 1.5.1 Pricing under the Risk-Neutral Measure. 1.5.2 Change of Probability Measure. 1.5.3 Pricing under Forward Measure. 1.5.4 Pricing under Swap Measure. 1.5.5 Pricing under Share Measure . 1.6 Derivative Pricing Techniques. 1.6.1 Summary of Approaches to Derivatives Pricing . 1.6.2 Classifying PricingModels. 1.6.3 Special
Topics. Problems . « 1.4.4 2 Derivatives Pricing via TransformTechniques Derivatives Pricing via the Fast Fourier Transform . 2.1.1 Call Option Pricing via the Fourier Transform. 2.1.2 Put Option Pricing via the Fourier Transform . 2.1.3 Evaluating the Pricing Integral. 2.1.4 Implementation of Fast Fourier Transform. 2.1.5 Damping Factor a . 2.2 Fractional Fast Fourier Transform. 2.2.1 Formation of Fractional FFT . 2.2.2 Implementation of Fractional FFT . 2.3 Derivatives Pricing via the Fourier-Cosine (COS) Method . 2.3.1 COS Method . 2.3.2 COS Option Pricing for Different Payoffs. 2.3.3 Truncation Range for the COS Method. 2.3.4 Numerical Results for the COS Method. 2.4 Cosine Method for Path-Dependent Options. 2.4.1 Bermudan
Options. 2.4.2 Discretely Monitored Barrier Options. 2.5 Saddlepoint Method. 2.5.1 Generalized Lugannani-Rice Approximation. 2.5.2 Option Prices as Tail Probabilities . 2.5.3 Lugannani-Rice Approximation for Option Pricing. 2.5.4 Implementation of the Saddlepoint Approximation . 2.5.5 Numerical Results for Saddlepoint Methods . :. 2.6 Power or Leverged Option Pricing via the Fourier Transform. Problems . Case Study. 2.1 3 Introduction to Finite Differences 3.1 3.2 Taylor Expansion. Finite Difference Method. 31 32 35 36 37 39 39 39 41 42 44 45 46 47 47 47 48 49 49 50 55 57 59 60 64 67 69 70 71 74 75 76 79 80 81 82 83 85 86 88 89 93 95 97 102 102 105
Contents xi 3.2.1 Explicit Discretization. 3.2.2 Implicit Discretization. 3.2.3 Crank-Nicolson Discretization. 3.2.4 Multi-Step Scheme. . 3.3 Stability Analysis. 3.3.1 Stability of the Explicit Scheme. 3.3.2 Stability of the Implicit Scheme. 3.3.3 Stability of the Crank-Nicolson Scheme . 3.3.4 Stability of the Multi-Step Scheme. 3.4 Derivative Approximation by Finite Differences: Generic Approach . 3.5 Matrix Equations Solver. 3.5.1 Tridiagonal Matrix Solver. 3.5.2 Pentadiagonal Matrix Solver. Problems. . Case Study. ■. 106 109 112 115 118 122 122 123 123 123 125 126 127 129 133 4 Derivative Pricing via Numerical
Solutions of PDEs Option Pricing under the Generalized Black-Merton-Scholes PDE. 4.1.1 Explicit Discretization. 4.1.2 Implicit Discretization. 4.1.3 Crank-Nicolson Discretization. 4.2 Boundary Conditions and Critical Points. 4.2.1 Implementing Boundary Conditions. 4.2.2 Implementing Deterministic Jump Conditions. . 4.3 Nonuniform Grid Points. 4.3.1 Unequal Sub-intervals . 4.3.2 Coordinate Transformation. 4.4 Dimension Reduction. 4.5 Pricing Path-Dependent Options in a Diffusion Framework. 4.5.1 Bermudan Options . . 4.5.2 American Options. 4.5.3 Barrier Options. 4.6 Forward
PDEs. 4.6.1 Vanilla Calls. 4.6.2 Down-and-Out Calls. 4.6.3 Up-and-Out Calls. 4.7 Finite Differences in Higher Dimensions. 4.7.1 Heston Stochastic Volatility Model. 4.7.2 Options Pricing under the Heston PDE. 4.7.3 Alternative Direction Implicit (ADI) Scheme. 4.7.4 Heston PDE. 4.7.5 Numerical Results and Conclusion . Problems. Case Studies . 4.1 5 Derivative Pricing viaNumerical Solutions of PIDEs 5.1 5.2 Option Pricing. Numerical Solution of PIDEs (a Generic Example) . 5.2.1 Derivation of the PIDE. 5.2.2
Discretization. 5.2.3 Evaluation of theIntegral Term. 135 137 137 139 140 141 142 146 147 147 148 151 152 152 154 158 162 163 163 164 165 167 169 177 181 182 186 191 194 194 194 195 200 201
xii Contents 5.2.4 Difference Equation. ,. American Options. 5.3.1 Heaviside Term — Synthetic Dividend Process. 5.3.2 Numerical Experiments. 5.4 PIDE Solutions for Lévy Processes. . 5.5 Forward PIDEs. 5.5.1 American Options. 5.5.2 Down-and-Out and Up-and-Out Calls. 5.6 Calculation of gj and #2 . Problems. Case Studies . 5.3 6 Simulation Methods for Pricing and valuation 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 Random Number Generation. 6.1.1 Linear Congruential Generators. 6.1.2 Tests on Randomness. 6.1.3 Standard Uniform Distribution . Sampling from Various Different
Distributions. 6.2.1 Inverse .Transform Method. . ■ 6.2.2 Acceptance-Rejection Method. 6.2.3 Univariate Standard Normal Random Variables. 6.2.4 Multivariate Normal RandomVariables. 6.2.5 Cholesky Factorization. Models of Dependence. 6.3.1 Full Rank Gaussian Copula Model . 6.3.2 Correlating Gaussian Components in a Variance Gamma Representation . . . 6.3.3 Linear Mixtures of Independent Lévy Processes. Stochastic Interpolation . 6.4.1 Brownian Bridge. 6.4.2 Variance Gamma Bridge. Monte Carlo Integration. 6.5.1 Quasi-Monte Carlo Methods. 6.5.2 Latin Hypercube Sampling Methods . Numerical Integration of
Stochastic Differential Equations. 6.6.1 Euler Scheme. 6.6.2 Milstein Scheme. 6.6.3 Runge-Kutta Scheme. Simulating SDEs under Different Stochastic Models. 6.7.1 Geometric Brownian Motion. 6.7.2 Ornstein-Uhlenbeck Process. 6.7.3 CIR Process. 6.7.4 Heston Stochastic Volatility Model. 6.7.5 Variance Gamma Process . 6.7.6 Variance Gamma with StochasticArrival (VGSA) Process. 6.7.7 Merton-Jump Diffusion Model. 6.7.8 Discrete Time Double GammaSV Model (Hirsa-Madan). American Option Pricing Example . 6.8.1 Problem Formulation. 6.8.2 Longstaff Schwartz Method. Output/Simulation Analysis. 204 208 210 211 213 214 214 217 221 222 223 229 231 232 233 233 234 234 238 245 249 250 252 253 253 253 254
254 255 256 260 262 263 264 265 265 266 266 267 267 267 269 271 275 278 278 279 280 281
II Contents xiii 6.9.1 Chebyshev’s Inequality. 6.9.2 Central Limit Theorem (CLT). 6.10 Variance Reduction Techniques. 6.10.1 Control Variate Method . . . 6.10.2 Antithetic Variates Method . 6.10.3 Conditional Monte Carlo Methods . 6.10.4 Importance Sampling Methods . 6.10.5 Stratified Sampling Methods. 6.10.6 Common Random Numbers. 6.10.7 Path-wise Estimator. 6.10.8 The Likelihood Ratio Method . 6.11 Concluding Remarks. Problems. 283 283 289 289 291 293 295 298 302 303 308 312 313 Pricing and Valuation (ML/DL-based) 7 Supervised Deep Neural Networks for Pricing Traditional Pricing. Labels and Parameter
Selection. 7.2.1 European Options. 7.2.2 Barrier Options. 7.2.3 American Options. 7.3 Brief Introduction to Neural Networks . . 7.3.1 Activation Functions g. 7.3.2 Convolutional Neural Networks. 7.3.3 More Hidden Layers or More Neurons Per Layer?. 7.3.4 Dropout. 7.3.5 Choice of Optimizer and Convergence. 7.4 Training. 7.4.1 Structure/Architecture. 7.4.2 Diagnostics . 7.5 Validation. 7.5.1 European Options. 7.5.2 Barrier Options. 7.5.3 American
Options. 7.6 Deep Neural Network versus Recurrent Neural Network. 7.7 Findings and Observations. Problems. Case Studies . 7.1 7.2 8 An Unsupervised Deep Learning Approach to Solving Partial Integro-Differential Equations 8.1 8.2 8.3 Introduction. 8.1.1 PIDE for Option Pricing. Neural Network as the Solution to the PIDE.; 8.2.1 Traditional Multi-layer Perceptron . 8.2.2 Neural Network Solution. 8.2.3 Singular Terms in the Network . 8.2.4 Full Structure of the Neural Network. Loss Function. 325 327 327 333 334 335 336 336 339 348 348 349 350 351 351 353 354 354 356 357 359 359 361 362 364 365 367 368 368 369 370 372 374
Contents xiv 8.3.1 Initial and Boundary Conditions. ,. 8.3.2 Derivation of Loss Function. 8.3.3 Summarized Algorithms. 8.4 Calculation. 8.4.1 Derivatives and Integral. 8.4.2 Extrapolation of the Price Function in theIntegral. 8.5 Numerical Experiments. 8.5.1 Range of Parameters and Distribution of Samples. . 8.5.2 Scope of Application of the Method . . 8.5.3 Hyper-parameters and Training Results. 8.5.4 Short Maturity Fitting. 8.5.5 Calculation Speed. 8.5.6 Greeks . 8.6 Conclusion. Case Studies . Model Calibrationand Parameter Estimation III 9 374 375 376 376 376 377 377 378 379 379 381 381 382 383 384 385 Model Calibration 387 Calibration
Formulation. 9.1.1 General Formulation. 9.1.2 Weighted Least-Squares Formulation. 9.1.3 Regularized Calibration Formulations. 9.2 Calibration of a Single Underlier Model. 9.2.1 Black-Merton-Scholes Model . 9.2.2 Local Volatility Model. 9.2.3 Constant Elasticity of Variance (CEV) Model. 9.2.4 Heston Stochastic Volatility Model. .’ . . 9.2.5 Mixing Model — Stochastic Local Volatility (SLV) Model. 9.2.6 Variance Gamma Model. 9.2.7 CGMY Model. 9.2.8 Variance Gamma with Stochastic Arrival Model. 9.2.9 Lévy Models. 9.3 Interest Rate Models. 9.3.1 Short Rate Models. 9.3.2 Multi-Factor Short Rate
Models. 9.3.3 Affine Term Structure Models (ATSMs) . 9.3.4 Forward Rate (HJM) Models . 9.3.5 LIBOR Market Models. 9.4 Credit Derivative Models. 9.5 Model Risk. 9.6 Construction of the Discount Curve. 9.6.1 LIBOR Yield Instruments. 9.6.2 Constructing the Yield Curve. 9.6.3 Polynomial Splines for Constructing DiscountCurves. Problems. Case Studies . 390 391 391 391 392 392 393 398 399 402 403 404 405 408 409 411 424 429 430 434 434 435 438 438 441 444 450 452 9.1 10 Filtering and Parameter Estimation 10.1 Filtering. 464 466
Contents XV 10.1.1 Construction of p(xfc|zi:fc). 10.2 Likelihood Function. . 10.3 Kalman Filter. 10.3.1 Underlying Model. 10.3.2 Posterior Estimate Covariance under Optimal Kalman Gain and Interpretation of the Optimal Kalman Gain. 10.4 Non-Linear Filters. 10.5 Extended Kalman Filter. 10.6 Unscented Kalman Filter. 10.6.1 Predict. 10.6.2 Update. 10.6.3 Implementation of Unscented Kalman Filter (UKF). 10.7 Square Root Unscented Kalman Filter (SR UKF). 10.8 Particle Filter. . 10.8.1 Sequential Importance Sampling (SIS) Particle Filtering. 10.8.2 Sampling Importance Resampling (SIR) Particle Filtering. 10.8.3 Problem of Resampling in Particle Filter and Possible Panaceas . . 10.9 Markov Chain Monte Carlo
(MCMC). 10.9.1 History of MCMC. 10.9.2 The Metropolis-Hastings Algorithm. 10.9.3 Gibbs Sampling. 10.9.4 Convergence Diagnostics. Problems. Case Study. · 467 468 473 474 IV Appendices A Interest Rate Definitions A.l A.2 Borrowing and Lending Rates. Forward Rate Agreement (FRA). A.2.1 Simple (Simply Compounded) Forward Rate. A.2.2 Simple Spot Rate. A.2.3 Continuously Compounded ForwardRate . A.2.4 Continuously CompoundedSpot Rate . A.2.5 Instantaneous Forward Rate. A.2.6 Instantaneous Short Rate . A.3 Zero-Coupon Bond Price. A.4 Futures Contracts. A.5 Swaps
. A.5.1 Terms and Payments. В Arbitrage Restrictions on Option Premiums B.l 478 481 482 484 485 486 487 498 501 503 503 514 515 515 516 520 540 541 543 545 547 547 548 549 549 549 550 550 550 550 550 551 551 553 Simple No-arbitrage Arguments. 553 C Derivative Approximation by Finite Differences in Higher Dimensional Case 555 Derivative Approximation byFinite Differences: Generic Approach in k-dimensional. C.l.l Calculating CoefficientsСг^,.,.^. C.2 Examples. C.l 555 556 558
xvi Contents C.2.1 One-dimensional Examples. ,. C.2.2 Two-dimensional Examples . C.2.3 Higher-dimensional Examples. D Derivation of Characteristic Function D.l D.2 Exponential Distribution — Characteristic Function of Exponential Distribution. 564 Heston Model — Characteristic Function of the Log Asset Price . . E Evaluation of the PIDE E.l E.2 E.3 Split of the Integral in the PIDE. Pre-calculations. Numerical Integral . F Optimization and OptimizationMethodology F.l F.2 F.3 F.4 F.5 F.6 F.7 Converting ML into Optimization. The Major Issue in Learning Problems. Zero-order, or Gradient-free Routines. . F.3.1 Grid Search. . . . . F.3.2 Gradient-free Routines. Gradient-based
Routines. F.4.1 Gradient Descent Method. Second-Ordered Methods. F.5.1 Using Unconstrained Optimization for Linear Constrained Input. . Trust Region Methods for ConstrainedProblems . Expectation-Maximization (EM)Algorithm. 558 559 562 564 566 572 572 573 574 575 576 577 577 577 578 582 583 590 591 593 594 Bibliography 595 Index 615 |
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| author | Hirsa, Ali |
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| dewey-hundreds | 300 - Social sciences |
| dewey-ones | 332 - Financial economics |
| dewey-raw | 332.6322201518 |
| dewey-search | 332.6322201518 |
| dewey-sort | 3332.6322201518 |
| dewey-tens | 330 - Economics |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | Second edition |
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| id | DE-604.BV049928490 |
| illustrated | Illustrated |
| indexdate | 2025-01-28T09:07:13Z |
| institution | BVB |
| isbn | 9781498778602 9781032786636 |
| language | English |
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| physical | xxix, 621 Seiten Illustrationen |
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| record_format | marc |
| series2 | Chapman & Hall/CRC financial mathematics series |
| spelling | Hirsa, Ali Verfasser (DE-588)1350769827 aut Computational methods in finance Ali Hirsa Second edition Abingdon Taylor & Francis 2024 xxix, 621 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Chapman & Hall/CRC financial mathematics series Derivat Wertpapier (DE-588)4381572-8 gnd rswk-swf Preisbildung (DE-588)4047103-2 gnd rswk-swf Finanzmathematik (DE-588)4017195-4 gnd rswk-swf Finanzmathematik (DE-588)4017195-4 s Derivat Wertpapier (DE-588)4381572-8 s Preisbildung (DE-588)4047103-2 s DE-604 Erscheint auch als Online-Ausgabe 978-0-4290-9474-3 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=035266899&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hirsa, Ali Computational methods in finance Derivat Wertpapier (DE-588)4381572-8 gnd Preisbildung (DE-588)4047103-2 gnd Finanzmathematik (DE-588)4017195-4 gnd |
| subject_GND | (DE-588)4381572-8 (DE-588)4047103-2 (DE-588)4017195-4 |
| title | Computational methods in finance |
| title_auth | Computational methods in finance |
| title_exact_search | Computational methods in finance |
| title_full | Computational methods in finance Ali Hirsa |
| title_fullStr | Computational methods in finance Ali Hirsa |
| title_full_unstemmed | Computational methods in finance Ali Hirsa |
| title_short | Computational methods in finance |
| title_sort | computational methods in finance |
| topic | Derivat Wertpapier (DE-588)4381572-8 gnd Preisbildung (DE-588)4047103-2 gnd Finanzmathematik (DE-588)4017195-4 gnd |
| topic_facet | Derivat Wertpapier Preisbildung Finanzmathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=035266899&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT hirsaali computationalmethodsinfinance |