Verfügbarkeit: Mathematical methods for financial markets

Mathematical methods for financial markets:

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Bibliographische Detailangaben
Beteiligte Personen:	Jeanblanc, Monique 1947- (VerfasserIn), Yor, Marc 1949-2014 (VerfasserIn), Chesney, Marc 1959- (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	London [u.a.] Springer [2009]
Schriftenreihe:	Springer finance textbook
Schlagwörter:	Finance - Mathematical models Finanzmarkt Mathematische Ökonomie Optionspreistheorie Stochastischer Prozess Stochastisches Modell Finanzmathematik Kreditmarkt
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013307502&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Umfang:	xxv, 732 Seiten Illustrationen
ISBN:	9781852333768 9781846287374

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adam_text	Contents Part I Continuous Path Processes Continuous-Path Random Processes: Mathematical Prerequisites .............................................. 3 1.1 Some Definitions ........................................ 3 1.1.1 Measurability ..................................... 3 1.1.2 Monotone Class Theorem .......................... 4 1.1.3 Probability Measures .............................. 5 1.1.4 Filtration ........................................ 5 1.1.5 Law of a Random Variable, Expectation ............. 6 1.1.6 Independence ..................................... 6 1.1.7 Equivalent Probabilities and Radon-Nikodým Densities 7 1.1.8 Construction of Simple Probability Spaces ............ 8 1.1.9 Conditional Expectation ........................... 9 1.1.10 Stochastic Processes ............................... 10 1.1.11 Convergence ...................................... 12 1.1.12 Laplace Transform ................................. 13 1.1.13 Gaussian Processes ................................ 15 1.1.14 Markov Processes ................................. 15 1.1.15 Uniform Integrability .............................. 18 1.2 Martingales ............................................. 19 1.2.1 Definition and Main Properties ...................... 19 1.2.2 Spaces of Martingales .............................. 21 1.2.3 Stopping Times ................................... 21 1.2.4 Local Martingales ................................. 25 1.3 Continuous Semi-martingales ............................. 27 1.3.1 Brackets of Continuous Local Martingales ............ 27 1.3.2 Brackets of Continuous Semi-martingales ............. 29 1.4 Brownian Motion ........................................ 30 1.4.1 One-dimensional Brownian Motion .................. 30 1.4.2 d-dimensional Brownian Motion ..................... 34 Contents 1.4.3 Correlated Brownian Motions ....................... 34 1.5 Stochastic Calculus ...................................... 35 1.5.1 Stochastic Integration .............................. 36 1.5.2 Integration by Parts ............................... 38 1.5.3 Itô s Formula: The Fundamental Formula of Stochastic Calculus ......................................... 38 1.5.4 Stochastic Differential Equations .................... 43 1.5.5 Stochastic Differential Equations: The One- dimensional Case .................................. 47 1.5.6 Partial Differential Equations ....................... 51 1.5.7 Doléans-Dade Exponential .......................... 52 1.6 Predictable Representation Property ....................... 55 1.6.1 Brownian Motion Case ............................. 55 1.6.2 Towards a General Definition of the Predictable Representation Property ........................... 57 1.6.3 Dudley s Theorem ................................. 60 1.6.4 Backward Stochastic Differential Equations ........... 61 1.7 Change of Probability and Girsanov s Theorem .............. 66 1.7.1 Change of Probability .............................. 66 1.7.2 Decomposition of P-Martingales as Q-semi-martingales . 68 1.7.3 Girsanov s Theorem: The One-dimensional Brownian Motion Case ...................................... 69 1.7.4 Multidimensional Case ............................. 72 1.7.5 Absolute Continuity ............................... 73 1.7.6 Condition for Martingale Property of Exponential Local Martingales ................................. 74 1.7.7 Predictable Representation Property under a Change of Probability ..................................... 77 1.7.8 An Example of Invariance of BM under Change of Measure .......................................... 78 Basic Concepts and Examples in Finance .................. 79 2.1 A Semi-martingale Framework ............................ 79 2.1.1 The Financial Market .............................. 80 2.1.2 Arbitrage Opportunities ............................ 83 2.1.3 Equivalent Martingale Measure ..................... 85 2.1.4 Admissible Strategies .............................. 85 2.1.5 Complete Market .................................. 87 2.2 A Diffusion Model ....................................... 89 2.2.1 Absence of Arbitrage .............................. 90 2.2.2 Completeness of the Market ........................ 90 2.2.3 PDE Evaluation of Contingent Claims in a Complete Market ........................................... 92 2.3 The Black and Scholes Model ............................. 93 2.3.1 The Model ....................................... 94 Contents xi 2.3.2 European Call and Put Options ..................... 97 2.3.3 The Greeks .......................................101 2.3.4 General Case .....................................102 2.3.5 Dividend Paying Assets ............................102 2.3.6 Rôle of Information ................................104 2.4 Change of Numéraire ....................................105 2.4.1 Change of Numéraire and Black-Scholes Formula ......106 2.4.2 Self-financing Strategy and Change of Numéraire ......107 2.4.3 Change of Numéraire and Change of Probability ......108 2.4.4 Forward Measure ..................................108 2.4.5 Self-financing Strategies: Constrained Strategies .......109 2.5 Feynman-Kac ...........................................112 2.5.1 Feynman-Kac Formula .............................112 2.5.2 Occupation Time for a Brownian Motion .............113 2.5.3 Occupation Time for a Drifted Brownian Motion ......114 2.5.4 Cumulative Options ...............................116 2.5.5 Quantiles .........................................118 2.6 Ornstein-Uhlenbeck Processes and Related Processes .........119 2.6.1 Definition and Properties ...........................119 2.6.2 Zero-coupon Bond .................................123 2.6.3 Absolute Continuity Relationship for Generalized Vasicek Processes .................................124 2.6.4 Square of a Generalized Vasicek Process ..............127 2.6.5 Powers of ^-Dimensional Radial OU Processes, Alias CIR Processes ....................................128 2.7 Valuation of European Options ............................129 2.7.1 The Garman and Kohlhagen Model for Currency Options ..........................................129 2.7.2 Evaluation of an Exchange Option ...................130 2.7.3 Quanto Options ...................................132 Hitting Times: A Mix of Mathematics and Finance ........135 3.1 Hitting Times and the Law of the Maximum for Brownian Motion .................................................136 3.1.1 The Law of the Pair of Random Variables {Wu Mt) ■ ■ ■ ■ 136 3.1.2 Hitting Times Process .............................138 3.1.3 Law of the Maximum of a Brownian Motion over [0,ű] . 139 3.1.4 Laws of Hitting Times .............................140 3.1.5 Law of the Infimum ................................142 3.1.6 Laplace Transforms of Hitting Times ................143 3.2 Hitting Times for a Drifted Brownian Motion ...............145 3.2.1 Joint Laws of (Mx, X) and (mx, X) at Time t .......145 3.2.2 Laws of Maximum, Minimum, and Hitting Times ......147 3.2.3 Laplace Transforms ................................148 3.2.4 Computation of W(v>(l{Tl,<x)<t} e-XT {x)) ...........149 xii Contents 3.2.5 Normal Inverse Gaussian Law .......................150 3.3 Hitting Times for Geometric Brownian Motion ..............151 3.3.1 Laws of the Pairs (Mf, St) and (mf, St) .............151 3.3.2 Laplace Transforms ................................152 3.3.3 Computation of E(e~XT ^%{Ta{s)<t}) ...............153 3.4 Hitting Times in Other Cases .............................153 3.4.1 Ornstein-Uhlenbeck Processes .......................153 3.4.2 Deterministic Volatility and Nonconstant Barrier ......154 3.5 Hitting Time of a Two-sided Barrier for BM and GBM ......156 3.5.1 Brownian Case ....................................156 3.5.2 Drifted Brownian Motion ...........................159 3.6 Barrier Options .........................................160 3.6.1 Put-Call Symmetry ................................160 3.6.2 Binary Options and A s ............................163 3.6.3 Barrier Options: General Characteristics .............164 3.6.4 Valuation and Hedging of a Regular Down-and-In Call Option When the Underlying is a Martingale .........166 3.6.5 Mathematical Results Deduced from the Previous Approach ........................................169 3.6.6 Valuation and Hedging of Regular Down-and-In Call Options: The General Case .........................172 3.6.7 Valuation and Hedging of Reverse Barrier Options .....175 3.6.8 The Emerging Calls Method ........................177 3.6.9 Closed Form Expressions ...........................178 3.7 Lookback Options .......................................179 3.7.1 Using Binary Options ..............................179 3.7.2 Traditional Approach .............................180 3.8 Double-barrier Options ...................................182 3.9 Other Options ..........................................183 3.9.1 Options Involving a Hitting Time ...................183 3.9.2 Boost Options ....................................184 3.9.3 Exponential Down Barrier Option ...................186 3.10 A Structural Approach to Default Risk .....................188 3.10.1 Merton s Model ...................................188 3.10.2 First Passage Time Models .........................190 3.11 American Options .......................................191 3.11.1 American Stock Options ...........................192 3.11.2 American Currency Options ........................193 3.11.3 Perpetual American Currency Options ...............195 3.12 Real Options ............................................198 3.12.1 Optimal Entry with Stochastic Investment Costs ......198 3.12.2 Optimal Entry in the Presence of Competition ........201 3.12.3 Optimal Entry and Optimal Exit ....................204 3.12.4 Optimal Exit and Optimal Entry in the Presence of Competition ......................................205 Contents xiii 3.12.5 Optimal Entry and Exit Decisions ...................206 Complements on Brownian Motion ........................211 4.1 Local Time .............................................211 4.1.1 A Stochastic Pubini Theorem .......................211 4.1.2 Occupation Time Formula ..........................211 4.1.3 An Approximation of Local Time ....................213 4.1.4 Local Times for Semi-martingales ...................214 4.1.5 Tanaka s Formula .................................214 4.1.6 The Balayage Formula ..............................216 4.1.7 Skorokhod s Reflection Lemma ......................217 4.1.8 Local Time of a Semi-martingale ....................222 4.1.9 Generalized Itô-Tanaka Formula .....................226 4.2 Applications ............................................227 4.2.1 Dupire s Formula ..................................227 4.2.2 Stop-Loss Strategy ................................229 4.2.3 Knock-out BOOST ................................230 4.2.4 Passport Options ..................................232 4.3 Bridges, Excursions, and Meanders ........................232 4.3.1 Brownian Motion Zeros ............................232 4.3.2 Excursions .......................................232 4.3.3 Laws of Tx, dt and gt ..............................233 4.3.4 Laws of (Bt,gt,dt) .................................236 4.3.5 Brownian Bridge ..................................237 4.3.6 Slow Brownian Filtrations ..........................241 4.3.7 Meanders ........................................242 4.3.8 The Azema Martingale .............................243 4.3.9 Drifted Brownian Motion ...........................244 4.4 Parisian Options ........................................246 4.4.1 The Law of (G¿e(W) , WG-,e) ......................249 4.4.2 Valuation of a Down-and-In Parisian Option ..........252 4.4.3 PDE Approach ....................................256 4.4.4 American Parisian Options .........................257 Complements on Continuous Path Processes ...............259 5.1 Time Changes ..........................................259 5.1.1 Inverse of an Increasing Process .....................259 5.1.2 Time Changes and Stopping Times ..................260 5.1.3 Brownian Motion and Time Changes ................261 5.2 Dual Predictable Projections ..............................264 5.2.1 Definitions .......................................264 5.2.2 Examples ........................................266 5.3 Diffusions ..............................................269 5.3.1 (Time-homogeneous) Diffusions .....................270 5.3.2 Scale Function and Speed Measure ..................270 xiv Contents 5.3.3 Boundary Points..................................273 5.3.4 Change of Time or Change of Space Variable .........275 5.3.5 Recurrence .......................................277 5.3.6 Resolvent Kernel and Green Function ................277 5.3.7 Examples ........................................279 5.4 Non-homogeneous Diffusions ..............................281 5.4.1 Kolmogorov s Equations ............................281 5.4.2 Application: Dupire s Formula ......................284 5.4.3 Fokker-Planck Equation ............................286 5.4.4 Valuation of Contingent Claims .....................289 5.5 Local Times for a Diffusion ...............................290 5.5.1 Various Definitions of Local Times ...................290 5.5.2 Some Diffusions Involving Local Time ................291 5.6 Last Passage Times ......................................294 5.6.1 Notation and Basic Results .........................294 5.6.2 Last Passage Time of a Transient Diffusion ...........294 5.6.3 Last Passage Time Before Hitting a Level ............297 5.6.4 Last Passage Time Before Maturity ..................298 5.6.5 Absolutely Continuous Compensator ................301 5.6.6 Time When the Supremum is Reached ...............302 5.6.7 Last Passage Times for Particular Martingales ........303 5.7 Pitman s Theorem about (2Mt -Wt) ......................306 5.7.1 Time Reversal of Brownian Motion ..................306 5.7.2 Pitman s Theorem .................................307 5.8 Filtrations ..............................................309 5.8.1 Strong and Weak Brownian Filtrations ...............310 5.8.2 Some Examples ...................................312 5.9 Enlargements of Filtrations ...............................315 5.9.1 Immersion of Filtrations ...........................315 5.9.2 The Brownian Bridge as an Example of Initial Enlargement .....................................318 5.9.3 Initial Enlargement: General Results .................319 5.9.4 Progressive Enlargement ...........................323 5.10 Filtering the Information .................................329 5.10.1 Independent Drift .................................329 5.10.2 Other Examples of Canonical Decomposition .........330 5.10.3 Innovation Process ................................331 6 A Special Family of Diffusions: Bessel Processes ...........333 6.1 Definitions and First Properties ...........................333 6.1.1 The Euclidean Norm of the n-Dimensional Brownian Motion ...........................................333 6.1.2 General Definitions ................................334 6.1.3 Path Properties ...................................337 6.1.4 Infinitesimal Generator .............................337 Contents xv 6.1.5 Absolute Continuity............................... 339 6.2 Properties ..............................................342 6.2.1 Additivity of BESQ s ..............................342 6.2.2 Transition Densities ...............................343 6.2.3 Hitting Times for Bessel Processes ...................345 6.2.4 Lamperti s Theorem ...............................347 6.2.5 Laplace Transforms ................................349 6.2.6 BESQ Processes with Negative Dimensions ...........353 6.2.7 Squared Radial Ornstein-Uhlenbeck ..................356 6.3 Cox-Ingersoll-Ross Processes ..............................356 6.3.1 GIR Processes and BESQ ..........................357 6.3.2 Transition Probabilities for a CIR Process ............358 6.3.3 CIR Processes as Spot Rate Models .................359 6.3.4 Zero-coupon Bond .................................361 6.3.5 Inhomogeneous CIR Process ........................364 6.4 Constant Elasticity of Variance Process ....................365 6.4.1 Particular Case μ = 0..............................366 6.4.2 CEV Processes and CIR Processes ...................368 6.4.3 CEV Processes and BESQ Processes .................368 6.4.4 Properties ........................................370 6.4.5 Scale Functions for CEV Processes ..................371 6.4.6 Option Pricing in a CEV Model .....................372 6.5 Some Computations on Bessel Bridges .....................373 6.5.1 Bessel Bridges ....................................373 6.5.2 Bessel Bridges and Ornstein-Uhlenbeck Processes ......374 6.5.3 European Bond Option ............................376 6.5.4 American Bond Options and the CIR Model ..........378 6.6 Asian Options ..........................................381 6.6.1 Parity and Symmetry Formulae .....................382 6.6.2 Laws of Ä%] and Ąv) .............................383 6.6.3 The Moments of At ................................388 6.6.4 Laplace Transform Approach .......................389 6.6.5 PDE Approach ....................................391 6.7 Stochastic Volatility .....................................392 6.7.1 Black and Scholes Implied Volatility .................392 6.7.2 A General Stochastic Volatility Model ...............392 6.7.3 Option Pricing in Presence of Non-normality of Returns: The Martingale Approach ..................393 6.7.4 Hull and White Model .............................396 6.7.5 Closed-form Solutions in Some Correlated Cases .......398 6.7.6 PDE Approach .................................... 401 6.7.7 Heston s Model ................................... 401 6.7.8 Mellin Transform ................................. 403 Contents Part II Jump Processes Default Risk: An Enlargement of Filtration Approach .....407 7.1 A Toy Model ...........................................407 7.1.1 Defaultable Zero-coupon with Payment at Maturity ... 408 7.1.2 Defaultable Zero-coupon with Payment at Hit .........410 7.2 Toy Model and Martingales ...............................412 7.2.1 Key Lemma ......................................412 7.2.2 The Fundamental Martingale .......................412 7.2.3 Hazard Function ..................................413 7.2.4 Incompleteness of the Toy Model, non Arbitrage Prices 415 7.2.5 Predictable Representation Theorem .................415 7.2.6 Risk-neutral Probability Measures ...................416 7.2.7 Partial Information: Duffie and Lando s Model ........418 7.3 Default Times with a Given Stochastic Intensity .............418 7.3.1 Construction of Default Time with a Given Stochastic Intensity .........................................418 7.3.2 Conditional Expectation with Respect to Tt .........419 7.3.3 Enlargements of Filtrations .........................420 7.3.4 Conditional Expectations with Respect to Qt .........420 7.3.5 Conditional Expectations of ^oo-Measurable Random Variables .........................................422 7.3.6 Correlated Defaults: Copula Approach ...............423 7.3.7 Correlated Defaults: Jarrow and Yu s Model ..........425 7.4 Conditional Survival Probability Approach .................426 7.4.1 Conditional Expectations ...........................427 7.5 Conditional Survival Probability Approach and Immersion ___428 7.5.1 (ft)-Hypothesis and Arbitrages ......................429 7.5.2 Pricing Contingent Claims ..........................430 7.5.3 Correlated Defaults: Kusuoka s Example .............431 7.5.4 Stochastic Barrier .................................432 7.5.5 Predictable Representation Theorems ................432 7.5.6 Hedging Contingent Claims with DZC ...............434 7.6 General Case: Without the (Tť)-Hypothesis .................437 7.6.1 An Example of Partial Observation ..................437 7.6.2 Two Defaults, Trivial Reference Filtration ............440 7.6.3 Initial Times ......................................442 7.6.4 Explosive Defaults .................................444 7.7 Intensity Approach ......................................445 7.7.1 Definition ........................................445 7.7.2 Valuation Formula .................................446 7.8 Credit Default Swaps ....................................446 7.8.1 Dynamics of the CDS s Price in a single name setting .. 447 7.8.2 Dynamics of the CDS s Price in a multi-name setting .. 448 Contents xvii 7.9 PDE Approach for Hedging Defaultable Claims .............449 7.9.1 Defaultable Asset with Total Default ................449 7.9.2 PDE for Valuation ................................450 7.9.3 General Case .....................................454 Poisson Processes and Ruin Theory .......................457 8.1 Counting Processes and Stochastic Integrals ................457 8.2 Standard Poisson Process .................................459 8.2.1 Definition and First Properties ......................459 8.2.2 Martingale Properties ..............................461 8.2.3 Infinitesimal Generator .............................464 8.2.4 Change of Probability Measure: An Example .........465 8.2.5 Hitting Times .....................................466 8.3 Inhomogeneous Poisson Processes .........................467 8.3.1 Definition ........................................467 8.3.2 Martingale Properties ..............................467 8.3.3 Watanabe s Characterization of Inhomogeneous Poisson Processes .................................468 8.3.4 Stochastic Calculus ................................469 8.3.5 Predictable Representation Property .................473 8.3.6 Multidimensional Poisson Processes ..................474 8.4 Stochastic Intensity Processes .............................475 8.4.1 Doubly Stochastic Poisson Processes .................475 8.4.2 Inhomogeneous Poisson Processes with Stochastic Intensity .........................................476 8.4.3 ItÔ s Formula .....................................476 8.4.4 Exponential Martingales ...........................477 8.4.5 Change of Probability Measure ......................478 8.4.6 An Elementary Model of Prices Involving Jumps ......479 8.5 Poisson Bridges .........................................480 8.5.1 Definition of the Poisson Bridge .....................480 8.5.2 Harness Property .................................. 481 8.6 Compound Poisson Processes .............................483 8.6.1 Definition and Properties ...........................483 8.6.2 Integration Formula ...............................484 8.6.3 Martingales .......................................485 8.6.4 Itô s Formula .....................................492 8.6.5 Hitting Times ..................................... 492 8.6.6 Change of Probability Measure ...................... 494 8.6.7 Price Process ..................................... 495 8.6.8 Martingale Representation Theorem .................496 8.6.9 Option Pricing .................................... 497 8.7 Ruin Process ............................................ 497 8.7.1 Ruin Probability .................................. 497 8.7.2 Integral Equation ................................. 498 xviii Contents 8.7.3 An Example...................................... 498 8.8 Marked Point Processes ..................................501 8.8.1 Random Measure ..................................501 8.8.2 Definition ........................................501 8.8.3 An Integration Formula ............................503 8.8.4 Marked Point Processes with Intensity and Associated Martingales .......................................503 8.8.5 Girsanov s Theorem ...............................504 8.8.6 Predictable Representation Theorem .................504 8.9 Poisson Point Processes ..................................505 8.9.1 Poisson Measures ..................................505 8.9.2 Point Processes ...................................506 8.9.3 Poisson Point Processes ............................506 8.9.4 The Ito Measure of Brownian Excursions .............507 9 General Processes: Mathematical Facts ....................509 9.1 Some Basic Facts about càdlàg Processes ...................509 9.1.1 An Illustrative Lemma .............................509 9.1.2 Finite Variation Processes, Pure Jump Processes ......510 9.1.3 Some σ -algebras ................................... 512 9.2 Stochastic Integration for Square Integrable Martingales ......513 9.2.1 Square Integrable Martingales .......................513 9.2.2 Stochastic Integral .................................516 9.3 Stochastic Integration for Semi-martingales .................517 9.3.1 Local Martingales .................................517 9.3.2 Quadratic Covariation and Predictable Bracket of Two Local Martingales .............................519 9.3.3 Orthogonality .....................................521 9.3.4 Semi-martingales ..................................522 9.3.5 Stochastic Integration for Semi-martingales ...........524 9.3.6 Quadratic Covariation of Two Semi-martingales .......525 9.3.7 Particular Cases ...................................525 9.3.8 Predictable Bracket of Two Semi-martingales .........527 9.4 Itô s Formula and Girsanov s Theorem .....................528 9.4.1 Itô s Formula: Optional and Predictable Forms ........528 9.4.2 Semi-martingale Local Times .......................531 9.4.3 Exponential Semi-martingales ......................532 9.4.4 Change of Probability, Girsanov s Theorem ...........534 9.5 Existence and Uniqueness of the e.m.m .....................537 9.5.1 Predictable Representation Property .................537 9.5.2 Necessary Conditions for Existence .................538 9.5.3 Uniqueness Property ...............................542 9.5.4 Examples ........................................543 9.6 Self-financing Strategies and Integration by Parts ............544 Contents xix 9.6.1 The Model.......................................545 9.6.2 Self-financing Strategies and Change of Numéraire .....545 9.7 Valuation in an Incomplete Market ........................547 9.7.1 Replication Criteria ................................548 9.7.2 Choice of an Equivalent Martingale Measure ..........549 9.7.3 Indifference Prices .................................550 10 Mixed Processes ...........................................551 10.1 Definition ..............................................551 10.2 Itô s Formula ...........................................552 10.2.1 Integration by Parts ...............................552 10.2.2 Itô s Formula: One-dimensional Case .................553 10.2.3 Multidimensional Case ............................555 10.2.4 Stochastic Differential Equations ....................556 10.2.5 Feynman-Kac Formula .............................557 10.2.6 Predictable Representation Theorem .................558 10.3 Change of Probability ....................................559 10.3.1 Exponential Local Martingales ......................559 10.3.2 Girsanov s Theorem ...............................560 10.4 Mixed Processes in Finance ...............................561 10.4.1 Computation of the Moments .......................561 10.4.2 Symmetry ........................................562 10.4.3 Hitting Times ..................................... 563 10.4.4 Affine Jump-Diffusion Model ........................565 10.4.5 General Jump-Diffusion Processes ...................569 10.5 Incompleteness ..........................................569 10.5.1 The Set of Risk-neutral Probability Measures ........570 10.5.2 The Range of Prices for European Call Options .......572 10.5.3 General Contingent Claims .........................575 10.6 Complete Markets with Jumps ............................578 10.6.1 A Three Assets Model ............................. 578 10.6.2 Structure Equations ............................... 579 10.7 Valuation of Options ..................................... 582 10.7.1 The Valuation of European Options .................584 10.7.2 American Option .................................. 586 11 Levy Processes ............................................ 11.1 Infinitely Divisible Random Variables ..................... 592 11.1.1 Definition ........................................ J92 11.1.2 Self-decomposable Random Variables ................ 11.1.3 Stable Random Variables ........................... 59° 11.2 Levy Processes .......................................... 11.2.1 Definition and Main Properties ...................... 11.2.2 Poisson Point Processes, Levy Measures .............. 60J. 11.2.3 Lévy-Khintchine Formula for a Levy Process .......... 60b Contents 11.2.4 Itô s Formulae for a One-dimensional Levy Process .... 612 11.2.5 Itô s Formula for Lévy-Itô Processes .................613 11.2.6 Martingales .......................................615 11.2.7 Harness Property ..................................620 11.2.8 Representation Theorem of Martingales in a Levy Setting ......................................621 11.3 Absolutely Continuous Changes of Measures ................623 11.3.1 Esscher Transform .................................623 11.3.2 Preserving the Levy Property with Absolute Continuity 625 11.3.3 General Case .....................................627 11.4 Fluctuation Theory ......................................628 11.4.1 Maximum and Minimum ...........................628 11.4.2 Pecherskii-Rogozin Identity .........................631 11.5 Spectrally Negative Levy Processes ........................632 11.5.1 Two-sided Exit Times ..............................632 11.5.2 Laplace Exponent of the Ladder Process .............633 11.5.3 D. Kendall s Identity ..............................633 11.6 Subordinators ...........................................634 11.6.1 Definition and Examples ...........................634 11.6.2 Levy Characteristics of a Subordinated Process .......636 11.7 Exponential Levy Processes as Stock Price Processes .........636 11.7.1 Option Pricing with Esscher Transform ..............636 11.7.2 A Differential Equation for Option Pricing ............637 11.7.3 Put-call Symmetry ................................638 11.7.4 Arbitrage and Completeness ........................639 11.8 Variance-Gamma Model ..................................639 11.9 Valuation of Contingent Claims ...........................641 11.9.1 Perpetual American Options ........................641 List of Special Features, Probability Laws, and Functions .. 647 A.I Main Formulae ..........................................647 A. 1.1 Absolute Continuity Relationships ...................647 A. 1.2 Bessel Processes ...................................648 A.1.3 Brownian Motion ..................................649 A.1.4 Diffusions ........................................650 A.1.5 Finance ..........................................650 A. 1.6 Girsanov s Theorem ...............................651 A.I. 7 Hitting Times .....................................651 A.1.8 Itô s Formulae ....................................651 A.1.9 Levy Processes ....................................653 A.I. 10 Semi-martingales ..................................654 A.2 Processes ...............................................655 A.3 Some Main Models ......................................655 A.4 Some Important Probability Distributions ..................656 A.4.1 Laws with Density .................................656 Contents xxi A.4.2 Some Algebraic Properties for Special r.v. s ...........656 A.4.3 Poisson Law ......................................657 A.4.4 Gamma and Inverse Gaussian Law ..................658 A.4.5 Generalized Inverse Gaussian and Normal Inverse Gaussian .........................................659 A.4.6 Variance Gamma VG(ct, ν,θ) .......................661 A.4.7 Tempered Stable ТЅ(У±, C±,M±) ..................661 A.5 Special Functions ........................................662 A.5.1 Gamma and Beta Functions ........................662 A.5.2 Bessel Functions ..................................662 A.5.3 Hermite Functions .................................663 A.5.4 Parabolic Cylinder Functions .......................663 A.5.5 Airy Function .....................................663 A.5.6 Kummer Functions ...............................664 A.5.7 Whittaker Functions ...............................664 A. 5.8 Some Laplace Transforms ..........................665 References .....................................................667 В Some Papers and Books on Specific Subjects ..............709 B.I Theory of Continuous Processes ...........................709 B.I.I Books ............................................709 B.1.2 Stochastic Differential Equations ....................709 B.1.3 Backward SDE ....................................709 B.1.4 Martingale Representation Theorems ................710 B.I. 5 Enlargement of Filtrations ..........................710 B.I. 6 Exponential Functionals ............................710 В.1.7 Uniform Integrability of Martingales .................710 B.2 Particular Processes .....................................710 B.2.1 Ornstein-Uhlenbeck Processes ......................710 B.2.2 CIR Processes .................................... 710 B.2.3 CEV Processes ....................................711 B.2.4 Bessel Processes ...................................711 B.3 Processes with Discontinuous Paths ........................711 B.3.1 Some Books ...................................... 7n B.3.2 Survey Papers .................................... 7n B.4 Hitting Times ........................................... 711 B.5 Levy Processes .......................................... 7 B.5.1 Books ............................................ 712 B.5.2 Some Papers ...................................... 712 B.6 Some Books on Finance .................................. ^ B.6.1 Discrete Time ..................................... 712 B.6.2 Continuous Time .................................. 712 B.6.3 Collective Books .................................. 712 B.6.4 History .......................................... 713 xxii Contents B.7 Arbitrage...............................................713 B.8 Exotic Options ..........................................713 8.8.1 Books ............................................713 8.8.2 Articles ..........................................713 Index of Authors ..............................................715 Index of Symbols ..............................................723 Subject Index .................................................725
any_adam_object	1
author	Jeanblanc, Monique 1947- Yor, Marc 1949-2014 Chesney, Marc 1959-
author_GND	(DE-588)171430689 (DE-588)120628635 (DE-588)170190595
author_facet	Jeanblanc, Monique 1947- Yor, Marc 1949-2014 Chesney, Marc 1959-
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author_sort	Jeanblanc, Monique 1947-
author_variant	m j mj m y my m c mc
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dewey-search	332.0151
dewey-sort	3332.0151
dewey-tens	330 - Economics
discipline	Mathematik Wirtschaftswissenschaften
format	Book
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illustrated	Illustrated
indexdate	2024-12-20T12:09:35Z
institution	BVB
isbn	9781852333768 9781846287374
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-013307502
oclc_num	475350194
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owner	DE-824 DE-91G DE-BY-TUM DE-384 DE-19 DE-BY-UBM DE-83 DE-1051 DE-355 DE-BY-UBR DE-945 DE-11 DE-188 DE-898 DE-BY-UBR DE-20 DE-M382
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physical	xxv, 732 Seiten Illustrationen
publishDate	2009
publishDateSearch	2009
publishDateSort	2009
publisher	Springer
record_format	marc
series2	Springer finance textbook
spellingShingle	Jeanblanc, Monique 1947- Yor, Marc 1949-2014 Chesney, Marc 1959- Mathematical methods for financial markets Finance - Mathematical models Finanzmarkt stw Mathematische Ökonomie stw Optionspreistheorie stw Stochastischer Prozess stw Stochastisches Modell (DE-588)4057633-4 gnd Finanzmathematik (DE-588)4017195-4 gnd Kreditmarkt (DE-588)4073788-3 gnd
subject_GND	(DE-588)4057633-4 (DE-588)4017195-4 (DE-588)4073788-3
title	Mathematical methods for financial markets
title_auth	Mathematical methods for financial markets
title_exact_search	Mathematical methods for financial markets
title_full	Mathematical methods for financial markets Monique Jeanblanc ; Marc Yor ; Marc Chesney
title_fullStr	Mathematical methods for financial markets Monique Jeanblanc ; Marc Yor ; Marc Chesney
title_full_unstemmed	Mathematical methods for financial markets Monique Jeanblanc ; Marc Yor ; Marc Chesney
title_short	Mathematical methods for financial markets
title_sort	mathematical methods for financial markets
topic	Finance - Mathematical models Finanzmarkt stw Mathematische Ökonomie stw Optionspreistheorie stw Stochastischer Prozess stw Stochastisches Modell (DE-588)4057633-4 gnd Finanzmathematik (DE-588)4017195-4 gnd Kreditmarkt (DE-588)4073788-3 gnd
topic_facet	Finance - Mathematical models Finanzmarkt Mathematische Ökonomie Optionspreistheorie Stochastischer Prozess Stochastisches Modell Finanzmathematik Kreditmarkt
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013307502&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT jeanblancmonique mathematicalmethodsforfinancialmarkets AT yormarc mathematicalmethodsforfinancialmarkets AT chesneymarc mathematicalmethodsforfinancialmarkets

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