Verfügbarkeit: Stochastic calculus for finance | Technische Universität München

Stochastic calculus for finance: 2 Continuous-time models

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Bibliographische Detailangaben
Beteilige Person:	Shreve, Steven E. (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	Berlin [u.a.] Springer 2008
Ausgabe:	corr. 8. print.
Schriftenreihe:	Springer finance : Textbook
Schlagwörter:	Análisis estocástico Modelos matemáticos Sistema financiero
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016697265&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Umfang:	XIX, 550 S. graph. Darst.
ISBN:	0387401016 9780387401010

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Datensatz im Suchindex

DE-BY-TUM_call_number	0102 MAT 639 2001 A 33757-2(1,2008)
DE-BY-TUM_katkey	2445543
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adam_text	Contents 1 General Probability Theory ................................ 1 1.1 Infinite Probability Spaces................................ 1 1.2 Random Variables and Distributions ....................... 7 1.3 Expectations ............................................ 13 1.4 Convergence of Integrals .................................. 23 1.5 Computation of Expectations ............................. 27 1.6 Change of Measure ...................................... 32 1.7 Summary ............................................... 39 1.8 Notes .................................................. 41 1.9 Exercises ............................................... 41 2 Information and Conditioning ............................. 49 2.1 Information and σ -algebras ............................... 49 2.2 Independence ........................................... 53 2.3 General Conditional Expectations ......................... 66 2.4 Summary ............................................... 75 2.5 Notes .................................................. 77 2.6 Exercises ............................................... 77 3 Brownian Motion .......................................... 83 3.1 Introduction ............................................ 83 3.2 Scaled Random Walks ................................... 83 3.2.1 Symmetric Random Walk .......................... 83 3.2.2 Increments of the Symmetric Random Walk .......... 84 3.2.3 Martingale Property for the Symmetric Random Walk . 85 3.2.4 Quadratic Variation of the Symmetric Random Walk .. 85 3.2.5 Scaled Symmetric Random Walk .................... 86 3.2.6 Limiting Distribution of the Scaled Random Walk ..... 88 3.2.7 Log-Normal Distribution as the Limit of the Binomial Model ........................................... 91 3.3 Brownian Motion ........................................ 93 Contents 3.3.1 Definition of Brownian Motion...................... 93 3.3.2 Distribution of Brownian Motion.................... 95 3.3.3 Filtration for Brownian Motion ..................... 97 3.3.4 Martingale Property for Brownian Motion............ 98 3.4 Quadratic Variation ..................................... 98 3.4.1 First-Order Variation.............................. 99 3.4.2 Quadratic Variation ...............................101 3.4.3 Volatility of Geometrie Brownian Motion.............106 3.5 Markov Property........................................ 107 3.6 First Passage Time Distribution...........................108 3.7 Reflection Principle......................................Ill 3.7.1 Reflection Equality ................................ Ill 3.7.2 First Passage Time Distribution .....................112 3.7.3 Distribution of Brownian Motion and Its Maximum .... 113 3.8 Summary ...............................................115 3.9 Notes ..................................................116 3.10 Exercises ...............................................117 Stochastic Calculus ........................................125 4.1 Introduction ............................................125 4.2 Itô s Integral for Simple Integrands ........................125 4.2.1 Construction of the Integral ........................126 4.2.2 Properties of the Integral ...........................128 4.3 Itô s Integral for General Integrands .......................132 4.4 Itô-Doeblin Formula .....................................137 4.4.1 Formula for Brownian Motion .......................137 4.4.2 Formula for Ito Processes ...........................143 4.4.3 Examples ........................................147 4.5 Black-Scholes-Merton Equation ...........................153 4.5.1 Evolution of Portfolio Value ........................154 4.5.2 Evolution of Option Value ..........................155 4.5.3 Equating the Evolutions ............................156 4.5.4 Solution to the Black-Scholes-Merton Equation ........158 4.5.5 The Greeks .......................................159 4.5.6 Put-Call Parity ...................................162 4.6 Multivariable Stochastic Calculus ..........................164 4.6.1 Multiple Brownian Motions .........................164 4.6.2 Itô-Doeblin Formula for Multiple Processes ...........165 4.6.3 Recognizing a Brownian Motion .....................168 4.7 Brownian Bridge ........................................172 4.7.1 Gaussian Processes ................................172 4.7.2 Brownian Bridge as a Gaussian Process ..............175 4.7.3 Brownian Bridge as a Scaled Stochastic Integral .......176 4.7.4 Multidimensional Distribution of the Brownian Bridge . 178 4.7.5 Brownian Bridge as a Conditioned Brownian Motion . . . 182 Contents XI 4.8 Summary ...............................................183 4.9 Notes..................................................187 4.10 Exercises ...............................................189 Risk-Neutral Pricing .......................................209 5.1 Introduction ............................................209 5.2 Risk-Neutral Measure ....................................210 5.2.1 Girsanov s Theorem for a Single Brownian Motion .....210 5.2.2 Stock Under the Risk-Neutral Measure ...............214 5.2.3 Value of Portfolio Process Under the Risk-Neutral Measure ..........................................217 5.2.4 Pricing Under the Risk-Neutral Measure .............218 5.2.5 Deriving the Black-Scholes-Merton Formula ...........218 5.3 Martingale Representation Theorem .......................221 5.3.1 Martingale Representation with One Brownian Motion . 221 5.3.2 Hedging with One Stock ...........................222 5.4 Fundamental Theorems of Asset Pricing ....................224 5.4.1 Girsanov and Martingale Representation Theorems .... 224 5.4.2 Multidimensional Market Model .....................226 5.4.3 Existence of the Risk-Neutral Measure ...............228 5.4.4 Uniqueness of the Risk-Neutral Measure ..............231 5.5 Dividend-Paying Stocks ..................................234 5.5.1 Continuously Paying Dividend ......................235 5.5.2 Continuously Paying Dividend with Constant Coefficients .......................................237 5.5.3 Lump Payments of Dividends .......................238 5.5.4 Lump Payments of Dividends with Constant Coefficients239 5.6 Forwards and Futures ....................................240 5.6.1 Forward Contracts .................................240 5.6.2 Futures Contracts .................................241 5.6.3 Forward-Futures Spread ...........................247 5.7 Summary ...............................................248 5.8 Notes ..................................................250 5.9 Exercises ...............................................251 Connections with Partial Differential Equations ...........263 6.1 Introduction ............................................263 6.2 Stochastic Differential Equations ..........................263 6.3 The Markov Property ....................................266 6.4 Partial Differential Equations .............................268 6.5 Interest Rate Models .....................................272 6.6 Multidimensional Feynman-Kac Theorems ..................277 6.7 Summary ...............................................280 6.8 Notes ..................................................281 6.9 Exercises ...............................................282 XII Contents 7 Exotic Options ............................................295 7.1 Introduction ............................................295 7.2 Maximum of Brownian Motion with Drift .................. 295 7.3 Knock-out Barrier Options ...............................299 7.3.1 Up-and-Out Call ..................................300 7.3.2 Black-Scholes-Merton Equation .....................300 7.3.3 Computation of the Price of the Up-and-Out Call .....304 7.4 Lookback Options .......................................308 7.4.1 Floating Strike Lookback Option ....................308 7.4.2 Black-Scholes-Merton Equation .....................309 7.4.3 Reduction of Dimension ............................312 7.4.4 Computation of the Price of the Lookback Option .....314 7.5 Asian Options ..........................................320 7.5.1 Fixed-Strike Asian Call ............................320 7.5.2 Augmentation of the State ..........................321 7.5.3 Change of Numéraire ..............................323 7.6 Summary ...............................................331 7.7 Notes ..................................................331 7.8 Exercises ...............................................332 8 American Derivative Securities ............................339 8.1 Introduction ............................................339 8.2 Stopping Times .........................................340 8.3 Perpetual American Put ..................................345 8.3.1 Price Under Arbitrary Exercise .....................346 8.3.2 Price Under Optimal Exercise .......................349 8.3.3 Analytical Characterization of the Put Price ..........351 8.3.4 Probabilistic Characterization of the Put Price ........353 8.4 Finite-Expiration American Put ...........................356 8.4.1 Analytical Characterization of the Put Price ..........357 8.4.2 Probabilistic Characterization of the Put Price ........359 8.5 American Call ..........................................361 8.5.1 Underlying Asset Pays No Dividends .................361 8.5.2 Underlying Asset Pays Dividends ....................363 8.6 Summary ...............................................368 8.7 Notes ..................................................369 8.8 Exercises ...............................................370 9 Change of Numéraire ......................................375 9.1 Introduction ............................................375 9.2 Numéraire ..............................................376 9.3 Foreign and Domestic Risk-Neutral Measures ...............381 9.3.1 The Basic Processes ...............................381 9.3.2 Domestic Risk-Neutral Measure .....................383 9.3.3 Foreign Risk-Neutral Measure .......................385 Contents XIII 9.3.4 Siegeľs Exchange Rate Paradox.....................387 9.3.5 Forward Exchange Rates...........................388 9.3.6 Garman-Kohlhagen Formula ........................390 9.3.7 Exchange Rate Put-Call Duality ....................390 9.4 Forward Measures .......................................392 9.4.1 Forward Price .....................................392 9.4.2 Zero-Coupon Bond as Numéraire ....................392 9.4.3 Option Pricing with a Random Interest Rate .........394 9.5 Summary ...............................................397 9.6 Notes ..................................................398 9.7 Exercises ...............................................398 10 Term-Structure Models ....................................403 10.1 Introduction ............................................403 10.2 Affine- Yield Models ......................................405 10.2.1 Two-Factor Vasicek Model ..........................406 10.2.2 Two-Factor CIR Model ............................420 10.2.3 Mixed Model .....................................422 10.3 Heath-Jarrow-Morton Model ..............................423 10.3.1 Forward Rates ....................................423 10.3.2 Dynamics of Forward Rates and Bond Prices .........425 10.3.3 No-Arbitrage Condition ............................426 10.3.4 HJM Under Risk-Neutral Measure ...................429 10.3.5 Relation to Affine- Yield Models .....................430 10.3.6 Implementation of HJM ............................432 10.4 Forward LIBOR Model ...................................435 10.4.1 The Problem with Forward Rates ...................435 10.4.2 LIBOR and Forward LIBOR .........................436 10.4.3 Pricing a Backset LIBOR Contract ..................437 10.4.4 Black Caplet Formula ..............................438 10.4.5 Forward LIBOR and Zero-Coupon Bond Volatilities . . . 440 10.4.6 A Forward LIBOR Term-Structure Model ............442 10.5 Summary ...............................................447 10.6 Notes ..................................................450 10.7 Exercises ...............................................451 11 Introduction to Jump Processes ...........................461 11.1 Introduction ............................................461 11.2 Poisson Process .........................................462 11.2.1 Exponential Random Variables ......................462 11.2.2 Construction of a Poisson Process ...................463 11.2.3 Distribution of Poisson Process Increments ...........463 11.2.4 Mean and Variance of Poisson Increments ............466 11.2.5 Martingale Property ...............................467 11.3 Compound Poisson Process ...............................468 XIV Contents 11.3.1 Construction of a Compound Poisson Process .........468 11.3.2 Moment-Generating Function .......................470 11.4 Jump Processes and Their Integrals ........................473 11.4.1 Jump Processes ...................................474 11.4.2 Quadratic Variation ...............................479 11.5 Stochastic Calculus for Jump Processes ....................483 11.5.1 Itò-Doeblin Formula for One Jump Process ...........483 11.5.2 Itô-Doeblin Formula for Multiple Jump Processes .....489 11.6 Change of Measure ......................................492 11.6.1 Change of Measure for a Poisson Process .............493 11.6.2 Change of Measure for a Compound Poisson Process . . . 495 11.6.3 Change of Measure for a Compound Poisson Process and a Brownian Motion ............................502 11.7 Pricing a European Call in a Jump Model ..................505 11.7.1 Asset Driven by a Poisson Process ...................505 11.7.2 Asset Driven by a Brownian Motion and a Compound Poisson Process ...................................512 11.8 Summary ...............................................523 11.9 Notes ..................................................525 ll.lOExercises ...............................................525 A Advanced Topics in Probability Theory ....................527 A.I Countable Additivity ....................................527 A. 2 Generating σ -algebras .................................... 530 A.3 Random Variable with Neither Density nor Probability Mass Function ...............................................531 В Existence of Conditional Expectations .....................533 С Completion of the Proof of the Second Fundamental Theorem of Asset Pricing ..................................535 References .....................................................537 Index ..........................................................545
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author	Shreve, Steven E.
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indexdate	2024-12-20T13:18:23Z
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isbn	0387401016 9780387401010
language	English
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publisher	Springer
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series2	Springer finance : Textbook
spellingShingle	Shreve, Steven E. Stochastic calculus for finance Análisis estocástico Modelos matemáticos Sistema financiero
title	Stochastic calculus for finance
title_auth	Stochastic calculus for finance
title_exact_search	Stochastic calculus for finance
title_full	Stochastic calculus for finance 2 Continuous-time models Steven E. Shreve
title_fullStr	Stochastic calculus for finance 2 Continuous-time models Steven E. Shreve
title_full_unstemmed	Stochastic calculus for finance 2 Continuous-time models Steven E. Shreve
title_short	Stochastic calculus for finance
title_sort	stochastic calculus for finance continuous time models
topic	Análisis estocástico Modelos matemáticos Sistema financiero
topic_facet	Análisis estocástico Modelos matemáticos Sistema financiero
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016697265&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV019352359
work_keys_str_mv	AT shrevestevene stochasticcalculusforfinance2

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