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Risk-neutral valuation: pricing and hedging of financial derivatives

Author: Bingham, N. H. ; Kiesel, R. Series: Springer finance Publisher: Springer, 2004.Edition: 2nd ed.Language: EnglishDescription: 437 p. ; 24 cm.ISBN: 9781852334581Type of document: BookBibliography/Index: Includes bibliographical references and index
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Book Europe Campus
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Print HG106 .B56 2004
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001227846
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Book Middle East Campus
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Print HG106 .B56 2004
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Includes bibliographical references and index

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Risk-Neutral Valuation Pricing and Hedging of Financial Derivatives Contents Preface to the Second Edition .................................................................................. vii Preface to the First Edition ........................................................................................ ix 1. Derivative Background ............................................................................................. 1 1.1 Financial Markets and Instruments .......................................................... 2 1.1.1 Derivative Instruments...................................................................... 2 1.1.2 Underlying Securities ........................................................................ 4 1.1.3 Markets ................................................................................................... 5 1.1.4 Types of Traders .................................................................................. 6 1.1.5 Modeling Assumptions ...................................................................... 6 1.2 Arbitrage............................................................................................................... 8 1.3 Arbitrage Relationships ............................................................................... 11 1.3.1 Fundamental Determinants of Option Values........................ 11 1.3.2 Arbitrage Bounds .............................................................................. 13 1.4 Single-period Market Models...................................................................... 15 1.4.1 A Fundamental Example ............................................................... 15 1.4.2 A Single-period Model ..................................................................... 18 1.4.3 A Few Financial-economic Considerations .............................. 25 Exercises .................................................................................................................. 26 2. Probability Background ......................................................................................... 2.1 Measure.............................................................................................................. 2.2 Integral .............................................................................................................. 2.3 Probability ........................................................................................................ 2.4 Equivalent Measures and Radon-Nikodym Derivatives..................... 2.5 Conditional Expectation............................................................................... 2.6 Modes of Convergence .................................................................................. 2.7 Convolution and Characteristic Functions ........................................... 2.8 The Central Limit Theorem ......................................................................... 2.9 Asset Return Distributions ......................................................................... 2.10 Infinite Divisibility and the Lévy-Khintchine Formula ................... 2.11 Elliptically Contoured Distributions...................................................... 2.12 Hyberbolic Distributions ........................................................................... 29 30 34 37 42 44 51 53 57 61 63 65 67 Exercises ................................................................................................................... 71 3. Stochastic Processes in Discrete Time ........................................................... 3.1 Information and Filtrations ........................................................................ 3.2 Discrete-parameter Stochastic Processes .............................................. 3.3 Definition and Basic Properties of Martingales .................................... 3.4 Martingale Transforms ................................................................................. 3.5 Stopping Times and Optional Stopping................................................... 3.6 The Snell Envelope and Optimal Stopping............................................. 3.7 Spaces of Martingales ................................................................................... 3.8 Markov Chains................................................................................................. Exercises ................................................................................................................... 4. Mathematical Finance in Discrete Time ...................................................... 4.1 The Model ....................................................................................................... 4.2 Existence of Equivalent Martingale Measures.................................... 4.2.1 The No-arbitrage Condition ........................................................ 4.2.2 Risk-Neutral Pricing ...................................................................... 4.3 Complete Markets: Uniqueness of EMMs ............................................ 4.4 The Fundamental Theorem of Asset Pricing: Risk-Neutral Valuation....................................................................................................... 4.5 The Cox-Ross-Rubinstein Model ............................................................ 4.5.1 Model Structure............................................................................... 4.5.2 Risk-neutral Pricing ...................................................................... 4.5.3 Hedging............................................................................................... 4.6 Binomial Approximations.......................................................................... 4.6.1 Model Structure............................................................................... 4.6.2 The Black-Scholes Option Pricing Formula ........................... 4.6.3 Further Limiting Models .............................................................. 4.7 American Options ........................................................................................ 4.7.1 Theory.................................................................................................. 4.7.2 American Options in the CRR Model ....................................... 4.8 Further Contingent Claim Valuation in Discrete Time.................... 4.8.1 Barrier Options ............................................................................... 4.8.2 Lookback Options .......................................................................... 4.8.3 A Three-period Example ............................................................... 4.9 Multifactor Models ........................................................................................ 4.9.1 Extended Binomial Model............................................................ 4.9.2 Multinomial Models ....................................................................... Exercises .................................................................................................................. 75 75 77 78 80 82 88 94 96 98 101 101 105 105 112 116 118 121 122 124 126 130 130 131 136 138 138 141 143 143 144 145 147 147 148 150 5. Stochastic Processes in Continuous Time .................................................... 153 5.1 Filtrations; Finite-dimensional Distributions ....................................... 153 5.2 Classes of Processes...................................................................................... 155 5.2.1 Martingales......................................................................................... 155 5.2.2 Gaussian Processes ........................................................................ 158 5.2.3 Markov Processes ............................................................................ 158 5.2.4 Diffusions ........................................................................................... 159 5.3 Brownian Motion............................................................................................ 160 5.3.1 Definition and Existence ............................................................... 160 5.3.2 Quadratic Variation of Brownian Motion ................................. 167 5.3.3 Properties of Brownian Motion..................................................... 171 5.3.4 Brownian Motion in Stochastic Modeling ................................. 173 5.4 Point Processes .............................................................................................. 175 5.4.1 Exponential Distribution ............................................................... 175 5.4.2 The Poisson Process ........................................................................ 176 5.4.3 Compound Poisson Processes ..................................................... 176 5.4.4 Renewal Processes............................................................................ 177 5.5 Levy Processes ............................................................................................... 179 5.5.1 Distributions ..................................................................................... 179 5.5.2 Levy Processes................................................................................... 181 5.5.3 Levy Processes and the Levy-Khintchine Formula................. 183 5.6 Stochastic Integrals; Ito Calculus ............................................................ 187 5.6.1 Stochastic Integration..................................................................... 187 5.6.2 Itô's Lemma........................................................................................ 193 5.6.3 Geometric Brownian Motion.......................................................... 196 5.7 Stochastic Calculus for Black-Scholes Models..................................... 198 5.8 Stochastic Differential Equations ............................................................. 202 5.9 Likelihood Estimation for Diffusions ...................................................... 206 5.10 Martingales, Local Martingales and Semi-martingales ................... 209 5.10.1 Definitions ....................................................................................... 209 5.10.2 Semi-martingale Calculus........................................................... 211 5.10.3 Stochastic Exponentials .............................................................. 215 5.10.4 Semi-martingale Characteristics .............................................. 217 5.11 Weak Convergence of Stochastic Processes......................................... 219 5.11.1 The Spaces Cd and Dd................................................................... 219 5.11.2 Definition and Motivation ........................................................... 220 5.11.3 Basic Theorems of Weak Convergence..................................... 222 5.11.4 Weak Convergence Results for Stochastic Integrals............ 223 Exercises ................................................................................................................. 225 6. Mathematical Finance in Continuous Time .................................................. 6.1 Continuous-time Financial Market Models .......................................... 6.1.1 The Financial Market Model.......................................................... 6.1.2 Equivalent Martingale Measures.................................................. 6.1.3 Risk-neutral Pricing ........................................................................ 229 229 229 232 235 6.1.4 Changes of Numéraire ........................................................ 6.2 The Generalized Black-Scholes Model ........................................... 6.2.1 The Model ............................................................................. 6.2.2 Pricing and Hedging Contingent Claims............................. 6.2.3 The Greeks........................................................................... 6.2.4 Volatility ................................................................................ 6.3 Further Contingent Claim Valuation ............................................ 6.3.1 American Options ............................................................... 6.3.2 Asian Options ..................................................................... 6.3.3 Barrier Options .................................................................. 6.3.4 Lookback Options ............................................................... 6.3.5 Binary Options ................................................................... 6.4 Discrete- versus Continuous-time Market Models ........................ 6.4.1 Discrete- to Continuous-time Convergence Reconsidered................................................. 6.4.2 Finite Market Approximations ............................................ 6.4.3 Examples of Finite Market Approximations........................ 6.4.4 Contiguity............................................................................ 6.5 Further Applications of the Risk-neutral Valuation Principle ....................................................................... 6.5.1 Futures Markets................................................................ 6.5.2 Currency Markets .............................................................. Exercises ............................................................................................... 239 242 242 250 254 255 258 258 260 263 266 269 270 270 271 274 280 281 281 285 287 7. Incomplete Markets .................................................................................. 289 7.1 Pricing in Incomplete Markets ....................................................... 289 7.1.1 A General Option-Pricing Formula ..................................... 289 7.1.2 The Esscher Measure ........................................................ 292 7.2 Hedging in Incomplete Markets ..................................................... 295 7.2.1 Quadratic Principles .......................................................... 296 7.2.2 The Financial Market Model................................................ 297 7.2.3 Equivalent Martingale Measures........................................ 299 7.2.4 Hedging Contingent Claims................................................. 300 7.2.5 Mean-variance Hedging and the Minimal ELMM ............... 305 7.2.6 Explicit Example.................................................................. 307 7.2.7 Quadratic Principles in Insurance .................................... 312 7.3 Stochastic Volatility Models ........................................................... 314 7.4 Models Driven by Levy Processes.................................................... 318 7.4.1 Introduction ...................................................................... 318 7.4.2 General Levy-process Based Financial Market Model........................................................................ 319 7.4.3 Existence of Equivalent Martingale Measures ................... 321 7.4.4 Hyperbolic Models: The Hyperbolic Levy Process ............... 323 8. Interest Rate Theory ................................................................................ 327 8.1 The Bond Market ........................................................................... 328 8.1.1 The Term Structure of Interest Rates ............................... 328 8.1.2 Mathematical Modelling...................................................... 330 8.1.3 Bond Pricing, ..................................................................... 334 8.2 Short-rate Models ........................................................................ 336 8.2.1 The Term-structure Equation ........................................... 337 8.2.2 Martingale Modelling .......................................................... 338 8.2.3 Extensions: Multi-Factor Models ....................................... 342 8.3 Heath-Jarrow-Morton Methodology............................................... 343 8.3.1 The Heath-Jarrow-Morton Model Class ............................. 343 8.3.2 Forward Risk-neutral Martingale Measures ..................... 346 8.3.3 Completeness .................................................................... 348 8.4 Pricing and Hedging Contingent Claims ...................................... 350 8.4.1 Short-rate Models ............................................................. 350 8.4.2 Gaussian HJM Framework................................................ 351 8.4.3 Swaps ............................................................................... 353 8.4.4 Caps.................................................................................. 354 8.5 Market Models of LIBOR- and Swap-rates.................................... 356 8.5.1 Description of the Economy ............................................... 356 8.5.2 LIBOR Dynamics Under the Forward LIBOR Measure................................................................... 357 8.5.3 The Spot LIBOR Measure..................................................... 361 8.5.4 Valuation of Caplets and Floorlets in the LMM ................ 362 8.5.5 The Swap Market Model ..................................................... 363 8.5.6 The Relation Between LIBOR- and Swap-market Models........................................................ 367 8.6 Potential Models and the Flesaker-Hughston Framework........... 368 8.6.1 Pricing Kernels and Potentials .......................................... 368 8.6.2 The Flesaker-Hughston Framework ................................. 370 Exercises .............................................................................................. 372 9. Credit Risk................................................................................................... 9.1 Aspects of Credit Risk .................................................................... 9.1.1 The Market.......................................................................... 9.1.2 What Is Credit Risk? .......................................................... 9.1.3 Portfolio Risk Models........................................................... 9.2 Basic Credit Risk Modeling............................................................ 9.3 Structural Models ........................................................................ 9.3.1 Merton's Model .................................................................... 9.3.2 A Jump-diffusion Model..................................................... 9.3.3 Structural Model with Premature Default ........................ 9.3.4 Structural Model with Stochastic Interest Rates............... 9.3.5 Optimal Capital Structure ­ Leland's Approach ............... 9.4 Reduced Form Models ................................................................... 375 376 376 376 377 378 379 379 382 384 388 389 390 9.5 Credit Derivatives .......................................................................... 9.6 Portfolio Credit Risk Models........................................................... 9.7 Collateralized Debt Obligations (CDOs)......................................... 9.7.1 Introduction ..................................................................... 9.7.2 Review of Modelling Methods............................................... 399 400 404 404 405 A. Hilbert Space................................................................................................ 409 B. Projections and Conditional Expectations ............................................ 411 C. The Separating Hyperplane Theorem...................................................... 415 Bibliography........................................................................................................ 417 Index.................................................................................................................. 433

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