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Mathematical finance: theory, modeling, implementation

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Bibliographic Details
Main Author:	Fries, Christian (Author)
Format:	Book
Language:	English
Published:	Hoboken, NJ Wiley-Interscience 2007
Subjects:	Instruments dérivés (Finances) - Prix - Modèles mathématiques Investissements - Modèles mathématiques Valeurs mobilières - Modèles mathématiques Mathematisches Modell Derivative securities > Prices > Mathematical models Investments > Mathematical models Securities > Mathematical models Finanzmathematik
Links:	http://www.loc.gov/catdir/toc/ecip0713/2007011325.html http://www.loc.gov/catdir/enhancements/fy0739/2007011325-b.html http://www.loc.gov/catdir/enhancements/fy0739/2007011325-d.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016247746&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Physical Description:	XXII, 520 S. Ill., graph. Darst.
ISBN:	9780470047224 0470047224

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adam_text	Contents 1 Introduction 1 1.1 Theory, Modeling, and Implementation ............... 1 1.2 Interest Rate Models and Interest Rate Derivatives ......... 1 1.3 About This Book .......................... 3 1.3.1 How to Read This Book ................. 3 1.3.2 Abridged Versions .................... 3 1.3.3 Special Sections ..................... 4 1.3.4 Notation ......................... 4 1.3.5 Feedback ......................... 5 1.3.6 Resources ......................... 5 I Foundations Foundations 9 2.1 Probability Theory ......................... 9 2.2 Stochastic Processes ......................... 18 2.3 Filtration ............................... 20 2.4 Brownian Motion .......................... 22 2.5 Wiener Measure, Canonical Setup ................. 24 2.6 Ito Calculus ............................. 25 2.6.1 Ito Integral ........................ 28 2.6.2 Ito Process ........................ 30 2.6.3 Ito Lemma and Product Rule ............... 32 2.7 Brownian Motion with Instantaneous Correlation ......... 36 xi 2.8 Martingales ............................. 38 2.8.1 Martingale Representation Theorem........... 38 2.9 Change of Measure ......................... 39 2.10 Stochastic Integration ........................ 44 2.11 Partial Differential Equations (PDEs) ................ 46 2.11.1 Feynman-Kač Theorem ................. 46 2.12 List of Symbols ........................... 48 Replication 49 3.1 Replication Strategies ........................ 49 3.1.1 Introduction ........................ 49 3.1.2 Replication in a Discrete Model ............. 53 3.2 Foundations: Equivalent Martingale Measure ........... 58 3.2.1 Challenge and Solution Outline ............. 58 3.2.2 Steps toward the Universal Pricing Theorem ....... 61 3.3 Excursus: Relative Prices and Risk-Neutral Measures ....... 70 3.3.1 Why relative prices? ................... 70 3.3.2 Risk-Neutral Measure .................. 72 II First Applications 73 4 Pricing of a European Stock Option under the Black-Scholes Model 75 5 Excursus: The Density of the Underlying of a European Call Option 81 6 Excursus: Interpolation of European Option Prices 83 6.1 No-Arbitrage Conditions for Interpolated Prices .......... 83 6.2 Arbitrage Violations through Interpolation ............. 85 6.2.1 Example 1 : Interpolation of Four Prices ......... 85 6.2.2 Example 2: Interpolation of Two Prices ......... 87 6.3 Arbitrage-Free Interpolation of European Option Prices ...... 89 7 Hedging in Continuous and Discrete Time and the Greeks 93 7.1 Introduction ............................. 93 7.2 Deriving the Replications Strategy from Pricing Theory ...... 94 7.2.1 Deriving the Replication Strategy under the Assumption of a Locally Riskless Product .............. 96 хи 7.2.2 Black-Scholes Differential Equation ........... 97 7.2.3 Derivative V(t) as a Function of Its Underlyings S¡(t) . . 97 7.2.4 Example: Replication Portfolio and PDE under a Black- Scholes Model ...................... 99 7.3 Greeks ................................ 102 7.3.1 Greeks of a European Call-Option under the Black- Scholes Model ...................... 103 7.4 Hedging in Discrete Time: Delta and Delta-Gamma Hedging ... 103 7.4.1 Delta Hedging ...................... 105 7.4.2 Error Propagation ..................... 106 7.4.3 Delta-Gamma Hedging .................. 109 7.4.4 Vega Hedging ....................... 113 7.5 Hedging in Discrete Time: Minimizing the Residual Error (Bouchaud-Sornette Method) .................... 113 7.5.) Minimizing the Residual Error at Maturity Τ ...... 115 7.5.2 Minimizing the Residual Error in Each Time Step . ... 117 III Interest Rate Structures, Interest Rate Products, and Analytic Pricing Formulas 119 Motivation and Overview .......................... 12 1 б Interest Rate Structures 123 8.1 Introduction ............................. 123 8.1.1 Fixing Times and Tenor Times .............. 124 8.2 Definitions .............................. 124 8.3 Interest Rate Curve Bootstrapping ................. 130 8.4 Interpolation of Interest Rate Curves ................ 130 8.5 Implementation ........................... 131 9 Simple Interest Rate Products 133 9.1 Interest Rate Products Part 1 : Products without Optionality .... 133 9.1.1 Fix, Floating, and Swap ................. 133 9.1.2 Money Market Account ................. 140 9.2 Interest Rate Products Part 2: Simple Options ........... 142 9.2.1 Cap, Floor, and Swaption ................. 142 9.2.2 Foreign Caplet. Quanto .................. 144 10 The Black Model for a Caplet 147 Xlii 11 Pricing of a Quanto Caplet (Modeling the FFX) 151 11.1 Choice of Numéraire ........................ 151 12 Exotic Derivatives 155 12.1 Prototypical Product Properties ................... 155 12.2 Interest Rate Products Part 3: Exotic Interest Rate Derivatives . . . 157 12.2.1 Structured Bond, Structured Swap, and Zero Structure . 158 12.2.2 Bermudán Option ..................... 162 12.2.3 Bermudán Callable and Bermudán Cancelable ..... 164 12.2.4 Compound Options .................... 166 12.2.5 Trigger Products ..................... 167 12.2.6 Structured Coupons .................... 168 12.2.7 Shout Options ...................... 173 12.3 Product Toolbox ........................... 174 IV Discretization and Numerical Valuation Methods 177 Motivation and Overview .......................... 179 13 Discretization of Time and State Space 181 13.1 Discretization of Time: The Euler and the Milstein Schemes ... 181 13.1.1 Definitions ........................ 183 13.1.2 Time Discretization of a Lognormal Process ....... 185 13.2 Discretization of Paths (Monte Carlo Simulation) ......... 186 13.2.1 Monte Carlo Simulation ................. 187 13.2.2 Weighted Monte Carlo Simulation ............ 187 13.2.3 Implementation ...................... 188 13.2.4 Review .......................... 193 13.3 Discretization of State Space .................... 195 13.3.1 Definitions ........................ 195 13.3.2 Backward Algorithm ................... 197 13.3.3 Review .......................... 197 13.4 Path Simulation through a Lattice: Two Layers .......... 198 14 Numerical Methods for Partial Differential Equations 199 15 Pricing Bermudán Options in a Monte Carlo Simulation 201 15.1 Introduction ............................. 201 15.2 Bermudán Options: Notation .................... 202 15.2.1 Bermudán Callable .................... 203 xiv 15.2.2 Relative Prices ...................... 203 15.3 Bermudán Option as Optimal Exercise Problem.......... 204 15.3.1 Bermudán Option Value as Single (Unconditioned) Ex¬ pectation: The Optimal Exercise Value ......... 204 15.4 Bermudán Option Pricing — The Backward Algorithm ....... 205 15.5 Resimulation ............................ 207 15.6 Perfect Foresight .......................... 207 15.7 Conditional Expectation as Functional Dependence ........ 209 15.8 Binning ............................... 210 15.8.1 Binning as a Least-Square Regression .......... 212 15.9 Foresight Bias ............................ 214 15.10 Regression Methods — Least-Square Monte Carlo ......... 215 15.10.1 Least-Square Approximation of the Conditional Expectation215 15.10.2 Example: Evaluation of a Bermudán Option on a Stock (Backward Algorithm with Conditional Expectation Esti¬ mator) ........................... 216 15.10.3 Example: Evaluation of a Bermudán Callable ...... 217 15.10.4 Implementation ...................... 222 15.10.5 Binning as Linear Least-Square Regression ....... 223 15.11 Optimization Methods ....................... 224 15.11.1 Andersen Algorithm for Bermudán Swaptions ...... 224 15.11.2 Review of the Threshold Optimization Method ..... 225 15.11.3 Optimization of Exercise Strategy: A More General For¬ mulation ......................... 228 15.11.4 Comparison of Optimization Method and Regression Method .......................... 228 15.12 Duality Method: Upper Bound for Bermudán Option Prices . . . . 230 15.12.1 Foundations ........................ 230 15.12.2 American Option Evaluation as Optimal Stopping Problem232 15.13 Primal-Dual Method: Upper and Lower Bound .......... 235 16 Pricing Path-Dependent Options in a Backward Algorithm 237 16.1 State Space Extension ........................ 237 16.2 Implementation ........................... 238 16.3 Path-Dependent Bermudán Options ................ 239 16.4 Examples .............................. 240 16.4.1 Evaluation of a Snowball in a Backward Algorithm . . . 240 16.4.2 Evaluation of a Autocap in a Backward Algorithm . . . . 240 17 Sensitivities (Partial Derivatives) of Monte Carlo Prices 243 17.1 Introduction ............................. 243 xv 17.2 Problem Description ........................ 244 17.2.1 Pricing using Monte-Carlo Simulation.......... 244 17.2.2 Sensitivities from Monte Carlo Pricing ......... 245 17.2.3 Example: The Linear and the Discontinuous Payout . . 245 17.2.4 Example: Trigger Products ................ 247 17.3 Generic Sensitivities: Bumping the Model ............. 249 17.4 Sensitivities by Finite Differences ................. 251 17.4.1 Example: Finite Differences Applied to Smooth and Dis¬ continuous Payout .................... 252 17.5 Sensitivities by Pathwise Differentiation .............. 254 17.5.1 Example: Delta of a European Option under a Black- Scholes Model ...................... 254 17.5.2 Pathwise Differentiation for Discontinuous Payouts . . . 255 17.6 Sensitivities by Likelihood Ratio Weighting ............ 256 17.6.1 Example: Delta of a European Option under a Black- Scholes Model Using Pathwise Derivative ........ 257 17.6.2 Example: Variance Increase of the Sensitivity when using Likelihood Ratio Method for Smooth Payouts ...... 257 17.7 Sensitivities by Malliavin Weighting ................ 258 17.8 Proxy Simulation Scheme ...................... 259 18 Proxy Simulation Schemes for Monte Carlo Sensitivities and Importance Sampling 261 18.1 Full Proxy Simulation Scheme ................... 261 18.1.1 Pricing under a Proxy Simulation Scheme ........ 262 18.1.2 Calculation of Monte Carlo Weights ........... 262 18.1.3 Sensitivities by Finite Differences on a Proxy Simulation Scheme .......................... 263 18.1.4 Localization ........................ 264 18.1.5 Object-Oriented Design ................. 265 18.1.6 Importance Sampling ................... 265 18.2 Partial Proxy Simulation Schemes ................. 268 18.2.1 Linear Proxy Constraint ................. 268 18.2.2 Comparison to Full Proxy Scheme Method ....... 269 18.2.3 Nonlinear Proxy Constraint ............... 269 18.2.4 Transition Probability from a Nonlinear Proxy Constraint 271 18.2.5 Sensitivity with Respect to the Diffusion Coefficients —Vega274 18.2.6 Example: LIBOR Target Redemption Note ....... 274 18.2.7 Example: CMS Target Redemption Note ........ 276 18.3 Localized Proxy Simulation Schemes ............... 279 18.3.1 Problem Description ................... 279 xvi 18.3.2 Solution .......................... 282 18.3.3 Partial Proxy Simulation Scheme (revisited) ....... 282 18.3.4 Localized Proxy Simulation Scheme ........... 285 18.3.5 Example: Euler Schemes ................. 286 18.3.6 Implementation ...................... 286 18.3.7 Examples and Numerical Results ............ 287 V Pricing Models for Interest Rate Derivatives 293 Motivation and Overview .......................... 295 19 LIBOR Market Model 297 19.1 Derivation of the Drift Term .................... 299 19.1.1 Derivation of the Drift Term under the Terminal Measure 299 19.1.2 Derivation of the Drift Term under the Spot LIBOR Measure301 19.1.3 Derivation of the Drift Term under the Гд -Forward МеаѕигеЗОЗ 19.2 The Short Period Bond P(Tm{t)+ t) ................. 304 19.2.1 Role of the Short Bond in a LIBOR Market Model . . . 304 19.2.2 Link to Continuous Time Tenors ............. 304 19.2.3 Drift of the Short Bond in a LIBOR Market Model . . . 304 19.3 Discretization and (Monte Carlo) Simulation ............ 305 19.3.1 Generation of the (Time-Discrete) Forward Rate Process 305 19.3.2 Generation of the Sample Paths ............. 306 19.3.3 Generation of the Numéraire ............... 306 19.4 Calibration — Choice of the Free Parameters ............ 307 19.4.1 Choice of the Initial Conditions ............. 308 19.4.2 Choice of the Volatilities ................. 308 19.4.3 Choice of the Correlations ................ 311 19.4.4 Covariance Structure ................... 313 19.4.5 Analytic Evaluation of Caplets. S waptions and Swap Rate Covariance ........................ 314 19.5 Interpolation of Forward Rates in the LIBOR Market Model ... 319 19.5.1 Interpolation of the Tenor Structure {T¡ ......... 319 19.6 Object-Oriented Design ....................... 323 19.6.1 Reuse of Implementation ................. 324 19.6.2 Separation of Product and Model ............ 324 19.6.3 Abstraction of Model Parameters ............ 324 19.6.4 Abstraction of Calibration ................ 325 20 Swap Rate Market Models 329 XVII 20.1 The Swap Measure ......................... 330 20.2 Derivation of the Drift Term .................... 331 20.3 Calibration — Choice of the Free Parameters ............ 332 20.3.1 Choice of the Initial Conditions ............. 332 20.3.2 Choice of the Volatilities ................. 332 21 Excursus: Instantaneous Correlation and Terminal Correlation 335 21.1 Definitions .............................. 335 21.2 Terminal Correlation Examined in a LIBOR Market Model Example 336 21.2.1 Decorrelation in a One-Factor Model .......... 337 21.2.2 Impact of the Time Structure of the Instantaneous Volatil¬ ity on Caplet and Swaption Prices ............ 339 21.2.3 Swaption Value as a Function of Forward Rates ..... 340 21.3 Terminal Correlation Is Dependent on the Equivalent Martingale Measure ............................... 342 21.3.1 Dependence of the Terminal Density on the Martingale Measure .......................... 342 22 Heath-Jarrow-Morton Framework: Foundations 345 22.1 Short-Rate Process in the HJM Framework ............. 346 22.2 The HJM Drift Condition ...................... 347 23 Short-Rate Models 351 23.1 Introduction ............................. 351 23.2 The Market Price of Risk ...................... 352 23.3 Overview: Some Common Models ................. 354 23.4 Implementations ........................... 355 23.4.1 Monte Carlo Implementation of Short-Rate Models . . . 355 23.4.2 Lattice Implementation of Short-Rate Models ...... 355 24 Heath-Jarrow-Morton Framework: Immersion of Short-Rate Models and LIBOR Market Model 357 24.1 Short-Rate Models in the HJM Framework ............. 357 24.1.1 Example: The Но -Lee Model in the HJM Framework . . 358 24.1.2 Example: The Hull-White Model in the HJM Framework 359 24.2 LIBOR Market Model in the HJM Framework ........... 360 24.2.1 HJM Volatility Structure of the LIBOR Market Model . 360 24.2.2 LIBOR Market Model Drift under the QB Measure ... 362 24.2.3 LIBOR Market Model as a Short Rate Model ...... 364 xviii 25 Excursus: Shape of the Interest Rate Curve under Mean Rever¬ sion and a Multifactor Model 365 25.1 Model ................................ 365 25.2 Interpretation of the Figures ..................... 366 25.3 Mean Reversion ........................... 367 25.4 Factors ................................ 368 25.5 Exponential Volatility Function ................... 369 25.6 Instantaneous Correlation ...................... 371 26 Ritchken-Sakarasubramanian Framework: HJM with Low Markov Dimension 373 26.1 Introduction ............................. 373 26.2 Cheyette Model ........................... 374 26.3 Implementation: PDE ........................ 375 27 Markov Functional Models 377 27.1 Introduction ............................. 377 27.1.1 The Markov Functional Assumption (Independent of the Model Considered) .................... 378 27.1.2 Outline of This Chapter ................. 379 27.2 Equity Markov Functional Model .................. 379 27.2.1 Markov Functional Assumption ............. 379 27.2.2 Example: The Black-Scholes Model ........... 380 27.2.3 Numerical Calibration to a Full Two-Dimensional Euro¬ pean Option Smile Surface ................ 381 27.2.4 Interest Rates ....................... 383 27.2.5 Model Dynamics ..................... 384 27.2.6 Implementation ...................... 390 27.3 LIBOR Markov Functional Model ................. 390 27.3.1 LIBOR Markov Functional Model in Terminal Measure 390 27.3.2 LIBOR Markov Functional Model in Spot Measure ... 396 27.3.3 Remark on Implementation ................ 400 27.3.4 Change of Numéraire in a Markov Functional Model . . 401 27.4 Implementation: Lattice ....................... 403 27.4.1 Convolution with the Normal Probability Density .... 404 27.4.2 State Space Discretization ................ 407 VI Extended Models 409 XIX 28 Credit Spreads 411 28.1 Introduction — Different Types of Spreads ............. 411 28.1.1 Spread on a Coupon ................... 411 28.1.2 Credit Spread ....................... 411 28.2 Defaultable Bonds .......................... 412 28.3 Integrating Deterministic Credit Spread into a Pricing Model ... 414 28.3.1 Deterministic Credit Spread ............... 415 28.3.2 Implementation ...................... 416 28.4 Receiver s and Payer s Credit Spreads ............... 418 28.4.1 Example: Defaultable Forward Starting Coupon Bond . 419 28.4.2 Example: Option on a Defaultable Coupon Bond .... 420 29 Hybrid Models 421 29.1 Cross-Currency LIBOR Market Model ............... 421 29.1.1 Derivation of the Drift Term under Spot Measure .... 422 29.1.2 Implementation ...................... 426 29.2 Equity Hybrid LIBOR Market Model ................ 426 29.2.1 Derivation of the Drift Term under Spot-Measure .... 426 29.2.2 Implementation ...................... 428 29.3 Equity Hybrid Cross-Currency LIBOR Market Model ....... 428 29.3.1 Dynamic of the Foreign Stock under Spot Measure . . . 429 29.3.2 Summary ......................... 430 29.3.3 Implementation ...................... 431 VII Implementation 433 30 Object-Oriented Implementation in Java™ 435 30.1 Elements of Object-Oriented Programming: Class and Objects . . 435 30.1.1 Example: Class of a Binomial Distributed Random Variable436 30.1.2 Constructor ........................ 438 30.1.3 Methods: Getter, Setter, and Static Methods ....... 438 30.2 Principles of Object Oriented Programming ............ 440 30.2.1 Encapsulation and Interfaces ............... 440 30.2.2 Abstraction and Inheritance ............... 444 30.2.3 Polymorphism ...................... 447 30.3 Example: A Class Structure for One-Dimensional Root Finders . . 449 30.3.1 Root Finder for General Functions ............ 449 30.3.2 Root Finder for Functions with Analytic Derivative: New¬ ton s Method ....................... 451 xx 30.3.3 Root Finder for Functions with Derivative Estimation: Secant Method ...................... 452 30.4 Anatomy of a Java™ Class ..................... 458 30.5 Libraries ............................... 460 30.5.1 Java™ 2 Platform, Standard Edition (j2se) ........ 460 30.5.2 Java™ 2 Platform, Enterprise Edition (j2ee) ....... 460 30.5.3 Colt ............................ 460 30.5.4 Commons-Math: The Jakarta Mathematics Library ... 461 30.6 Some Final Remarks ........................ 461 30.6.1 Object-Oriented Design (OOD)/Unined Modeling Lan¬ guage (UML) ....................... 461 VIII Appendices 463 A A Small Collection of Common Misconceptions 465 В Tools (Selection) 467 B.I Generation of Random Numbers .................. 467 B.I.I Uniform Distributed Random Variables ......... 467 B. 1.2 Transformation of the Random Number Distribution via the Inverse Distribution Function ............. 468 B.1.3 Normal Distributed Random Variables .......... 468 B.1.4 Poisson Distributed Random Variables .......... 468 B. 1.5 Generation of Paths of an «-Dimensional Brownian Motion469 B.2 Factor Decomposition — Generation of Correlated Brownian Motion 471 B.3 Factor Reduction .......................... 472 B.4 Optimization (One-Dimensional): Golden Section Search ..... 475 B.5 Linear Regression .......................... 476 B.6 Convolution with Normal Density ................. 477 С Exercises 479 D Java™ Source Code (Selection) 487 D.I Java™ Classes for Chapter 30 ................... 487 List of Symbols 493 List of Figures 495 List of Tables 499 XXI List of Listings 500 Bibliography 503 Index 511 XXII
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illustrated	Illustrated
indexdate	2024-12-20T13:07:20Z
institution	BVB
isbn	9780470047224 0470047224
language	English
lccn	2007011325
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-016247746
oclc_num	263711966
open_access_boolean
owner	DE-473 DE-BY-UBG DE-355 DE-BY-UBR DE-92 DE-29 DE-945 DE-20 DE-521 DE-634 DE-91G DE-BY-TUM DE-11 DE-19 DE-BY-UBM DE-706
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physical	XXII, 520 S. Ill., graph. Darst.
publishDate	2007
publishDateSearch	2007
publishDateSort	2007
publisher	Wiley-Interscience
record_format	marc
spellingShingle	Fries, Christian Mathematical finance theory, modeling, implementation Instruments dérivés (Finances) - Prix - Modèles mathématiques Investissements - Modèles mathématiques Valeurs mobilières - Modèles mathématiques Mathematisches Modell Derivative securities Prices Mathematical models Investments Mathematical models Securities Mathematical models Finanzmathematik (DE-588)4017195-4 gnd
subject_GND	(DE-588)4017195-4
title	Mathematical finance theory, modeling, implementation
title_auth	Mathematical finance theory, modeling, implementation
title_exact_search	Mathematical finance theory, modeling, implementation
title_full	Mathematical finance theory, modeling, implementation Christian Fries
title_fullStr	Mathematical finance theory, modeling, implementation Christian Fries
title_full_unstemmed	Mathematical finance theory, modeling, implementation Christian Fries
title_short	Mathematical finance
title_sort	mathematical finance theory modeling implementation
title_sub	theory, modeling, implementation
topic	Instruments dérivés (Finances) - Prix - Modèles mathématiques Investissements - Modèles mathématiques Valeurs mobilières - Modèles mathématiques Mathematisches Modell Derivative securities Prices Mathematical models Investments Mathematical models Securities Mathematical models Finanzmathematik (DE-588)4017195-4 gnd
topic_facet	Instruments dérivés (Finances) - Prix - Modèles mathématiques Investissements - Modèles mathématiques Valeurs mobilières - Modèles mathématiques Mathematisches Modell Derivative securities Prices Mathematical models Investments Mathematical models Securities Mathematical models Finanzmathematik
url	http://www.loc.gov/catdir/toc/ecip0713/2007011325.html http://www.loc.gov/catdir/enhancements/fy0739/2007011325-b.html http://www.loc.gov/catdir/enhancements/fy0739/2007011325-d.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016247746&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT frieschristian mathematicalfinancetheorymodelingimplementation

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