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ADVANCED DERIVATIVES PRICING AND RISK MANAGEMENT
Advanced Derivatives Pricing and Risk Management
Theory, Tools, and Hands-On Programming Applications
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By
Claudio Albanese, Professor of Mathematical Finance, Imperial College, London, UK
Giuseppe Campolieti, Associate Professor of Mathematics, SHARCNET Chair in Financial Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada

Included in series
Academic Press Advanced Finance ,

Description
Written by leading academics and practitioners in the field of financial mathematics, the purpose of this book is to provide a unique combination of some of the most important and relevant theoretical and practical tools from which any advanced undergraduate and graduate student, professional quant and researcher will benefit. This book stands out from all other existing books in quantitative finance from the sheer impressive range of ready-to-use software and accessible theoretical tools that are provided as a complete package. By proceeding from simple to complex, the authors cover core topics in derivative pricing and risk management in a style that is engaging, accessible and self-instructional. The book contains a wide spectrum of problems, worked-out solutions, detailed methodologies and applied mathematical techniques for which anyone planning to make a serious career in quantitative finance must master. In fact, core portions of the book?s material originated and evolved after years of classroom lectures and computer laboratory courses taught in a world-renowned professional Master?s program in mathematical finance. As a bonus to the reader, the book also gives a detailed exposition on new cutting-edge theoretical techniques with many results in pricing theory that are published here for the first time.

Audience
Students in finance programs, particularly financial engineering.

Contents
I Pricing Theory and Risk Management 11 1 Pricing Theory 13 1.1 Single Period, Finite Financial Models . . . . . . . . . . . . . . . . . 16 1.2 Continuous state spaces . . . . . . . . . . . . . . . . . . 24 1.3 Multivariate Continuous Distributions: Basic Tools . . . . . . . . . . 28 1.4 Brownian Motion, Martingales and Stochastic Integrals . . . . . . . . 35 1.5 Stochastic Differential Equations and Ito?s formula . . . . . . . . . . 46 1.6 Geometric Brownian Motion . . .52 1.7 Forwards and European Calls and Puts . . . . . . . . . . . . . . . . . 61 1.8 Static Hedging and Replication of Exotic Payoffs . . . . . . . . . . . 68 1.9 Continuous Time Financial Models . . . . . . . . . . . . . . . . . . . 77 1.10 Dynamic Hedging and Derivative Asset Pricing in Continuous Time . 84 1.11 Hedging with Forwards and Futures . . . . . . . . . . . . . . . . . . 90 1.12 Pricing formulas of the Black-Scholes type . . . . . . . . . . . . . . 96 1.13 Partial Differential Equations for Pricing Functions and Kernels . . . 108 1.14 American Options . . . . . . . . . . . . . . . . . . . . 114 1.14.1 Arbitrage-Free Pricing and Optimal Stopping Time Formulation 114 1.14.2 Perpetual American Options . . . . . . . . . . . . . . . . . . 125 1.14.3 Properties of the Early-Exercise Boundary . . . . . . . . . . . 127 1.14.4 The PDE and Integral Equation Formulation . . . . . . . . . 129 2 Fixed Income Instruments 135 2.1 Bonds, Futures, Forwards and Swaps . . . . . . . . . . . . . . . . . . 135 2.1.1 Bonds . . . . . . . . . . . . . . . . . . . . . 135 2.1.2 Forward rate agreements . . . . . . . . . . . . . . . . . . . 138 2.1.3 Floating rate notes . . . . . . . . . . . . . . . . . . . . . 139 2.1.4 Plain-Vanilla Swaps . . . . . . . . . . . . . . . . . . . . . 140 2.1.5 Constructing the discount curve . . . . . . . . . . . . . . . . 141 2.2 Pricing measures and Black-Scholes formulas . . . . . . . . . . . . . 143 2.2.1 Stock options with stochastic interest rates. . . . . . . . . . . 144 2.2.2 Swaptions. . .. . . . . . . . . . . . . . . . . 145 2.2.3 Caplets. . . . . . . . . . . . . . . . . . . . . 146 2.2.4 Options on Bonds. . . . . . . . . . . . . . . . . . . . . . 147 2.2.5 Futures-forward price spread . . . . . . . . . . . . . . . . . . 147 2.2.6 Bond futures options . . . . . . . . . . .. . . . . . . . . . 149 2.3 One-factor models for the short rate . . . . . . . . . . . . . . . . . . 151 2.3.1 Bond pricing equation . . . . . . . . . . . . . . . . . . . . 151 2.3.2 Hull-White, Ho-Lee and Vasicek Models . . . . . . . . . . . 152 2.3.3 Cox-Ingersoll-Ross model . . . . . . . . . . . . . . . . . . . 158 2.3.4 Flesaker-Hughston model . . . . . . . . . . . . . . . . . . . 163 2.4 Multifactor models . . . . . . . . . . . . . . . . . . . . . 166 2.4.1 HJM with no-arbitrage constraints . . . . . . . . . . . . . . . 167 2.4.2 BGMJ with no-arbitrage constraints . . . . . . . . . . . . . . 169 2.5 Real World Interest Rate Models . . . . . . . . . . . . . . . . . . . . 171 3 Advanced Topics in Pricing Theory: Exotic Options and State Dependent Models 175 3.1 Introduction to Barrier Options . . . . . . . . . . . . . . . . . . . . 177 3.2 Single-Barrier Kernels for the Simplest Model: The Wiener Process . 179 3.2.1 Driftless Case . . . . . . . . . . . . . . . . . . . . . . 179 3.2.2 Brownian Motion with Drift . . . . . . . . . . . . . . . . . . 185 3.3 Pricing Kernels and European Barrier Option Formulas for Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . . . 187 3.4 First Passage Time . . . . . . . . . . . . . . . . . . . . . . 196 3.5 Pricing Kernels and Barrier Option Formulas for Linear and Quadratic Volatility Models . . . . . . . . . . . . . . . . . . . . . 200 3.5.1 Linear Volatility Models Revisited . . . . . . . . . . . . . . 200 3.5.2 Quadratic Volatility Models . . . . . . . . . . . . . . . . . . 208 3.6 Green?s Functions Method for Diffusion Kernels . . . . . . . . . . . 219 3.6.1 Eigenfunction Expansions for the Green?s Function and the Transition Density . . . . . . . . . . . . . . . . . . . . 228 3.7 Kernels for the Bessel Process . . . . . . . . . . . . . . . . . . . . 230 3.7.1 The Barrier-free Kernel: No Absorption . . . . . . . . . . . . 231 3.7.2 The Case of Two Finite Barriers with Absorption . . . . . . . 234 3.7.3 The Case of a Single Upper Finite Barrier with Absorption . . 238 3.7.4 The Case of a Single Lower Finite Barrier with Absorption . . 241 3.8 New Families of Analytical Pricing Formulas: ?From x-Space to FSpace? . . . . .. . . . . . . . . . . . . . . . . . . . 242 3.8.1 Transformation Reduction Methodology . . . . . . . . . . . . 243 3.8.2 Bessel Families of State Dependent Volatility Models . . . . . 249 3.8.3 The 4-Parameter Sub-Family of Bessel Models . . . . . . . . 252 3.8.3.1 Recovering the CEV Model . . . . . . . . . . . . . 256 3.8.3.2 Recovering Quadratic Models . . . . . . . . . . . . 259 3.8.4 Conditions for Absorption or Probability Conservation . . . . 261 3.8.5 Barrier Pricing Formulas for Multi-Parameter Volatility Models 264 3.9 Appendix A: Proof of Lemma 3.1 . . . . . . . . . . . . . . . . . . . 268 3.10 Appendix B: Alternative Proof of Theorem 3.1 . . . . . . . . . . . . 270 3.11 Appendix C: Some Properties of Bessel Functions . . . . . . . . . . . 272 CONTENTS 7 4 Numerical Methods for Value-at-Risk 275 4.1 Risk Factor Models . . . . . . . . . . . . . . . . . . . . . 279 4.1.1 The lognormal model . . . . . . . . . . . . . . . . . . . . 279 4.1.2 The asymmetric Student?s t model . . . . . . . . . . . . . . . 280 4.1.3 The Parzen model . . . . . . . . . . . . . . . . . . . . . 282 4.1.4 Multivariate models . . . . . . . . . . . . . . . . . . . . . 284 4.2 Portfolio Models . . . . . . . . . . . . . . . . . . . . . 286 4.2.1 _-approximation . . . . . . . .. . . . . . . . . . 287 4.2.2 __-approximation . . . . .. . . . . . . . . . . . . 289 4.3 Statistical estimations for __-portfolios . . . . . . . . . . . . . . . . 291 4.3.1 Portfolio decomposition and portfolio dependent estimation . 291 4.3.2 Testing independence . . . . . . . . . . . . . . . . . . 293 4.3.3 A few implementation issues . . . . . . . . . . . . . . . . . . 295 4.4 Numerical methods for __-portfolios . . . . . . . . . . . . . . . . . 297 4.4.1 Monte Carlo methods and variance reduction . . . . . . . . . 297 4.4.2 Moment methods . . . . . . . . . . . . .. . . . . . . . 300 4.4.3 Fourier Transform of the Moment Generating Function . . . . 303 4.5 The fast convolution method . . . . . . . . . . . . . . . . . . . 305 4.5.1 The pdf of a quadratic random variable . . . . . . . . . . . . 306 4.5.2 Discretization . . . . . . . . . . . . . . . . . 307 4.5.3 Accuracy and convergence . . . . . . . . . . . . . . . . . . 308 4.5.4 The computational details . . . . . . . . . . . . . . . . . . . 308 4.5.5 Convolution with the fast Fourier transform . . . . . . . . . . 308 4.5.6 Computing value-at-risk . . . . . . . . . . . . . . . . . . . . 314 4.5.7 Richardson?s extrapolation improves accuracy . . . . . . . . . 315 4.5.8 Computational complexity . . . . . . . . . . . . . . . . . . . 317 4.6 Examples . . . . . . . . . . . . . . 318 4.6.1 Fat-tails and value-at-risk . . . . . . . . . . . . . . . . . . . . 318 4.6.2 So which result can we trust? . . . . . . . . . . . . . . . . . . 319 4.6.3 Computing the gradient of value-at-risk . . . . . . . . . . . . 319 4.6.4 The value-at-risk gradient and portfolio composition . . . . . 320 4.6.5 Computing the gradient . . . . . . . . . . . . . . . . . . . . 321 4.6.6 Sensitivity analysis and the linear approximation . . . . . . . 323 4.6.7 Hedging with value-at-risk . . . . . . . . . . . . . . . . . . . 324 4.6.8 Adding stochastic volatility . . . . . . . . . . . . . . . . . . 325 4.7 Risk factor aggregation and dimension reduction . . . . . . . . . . . 326 4.7.1 Method 1: reduction with small mean square error . . . . . . 327 4.7.2 Method 2: reduction by low-rank approximation . . . . . . . 329 4.7.3 Absolute versus relative value-at-risk . . . . . . . . . . . . . 332 4.7.4 Example: a comparative experiment . . . . . . . . . . . . . . 332 4.7.5 Example: dimension reduction and optimization . . . . . . . 333 4.8 Perturbation theory . . . . . . . .. . . . . . . . . . 334 4.8.1 When is value-at-risk well-posed? . . . . . . . . . . . . . . . 334 4.8.2 Perturbations of the return model . . . . . . . . . . . . . . . 336 4.8.3 Proof of a first-order perturbation property . . . . . . . . . . 336 4.8.4 Error bounds and the condition number . . . . . . . . . . . . 337 8 CONTENTS 4.8.5 Example: mixture model . . . . . . . . . . . . . . . . . . . . 339 II Numerical Projects in Pricing and Risk Management 353 5 Project: Arbitrage Theory 355 5.1 Basic Terminology and Concepts: Asset Prices, States, Returns and Payoffs . . . . . . . . . . . . . . . . . . . . 355 5.2 Arbitrage Portfolios and The Arbitrage Theorem . . . . . . . . . . . 357 5.3 An example of single period asset pricing: Risk-Neutral Probabilities and Arbitrage . .. . . . . . . . . . . . . . . . . 358 5.4 Arbitrage detection and the formation of arbitrage portfolios in the Ndimensional case . . . . . . . . . . .. . . . . . . . . . . . . . 360 6 Project: The Black-Scholes (Lognormal) Model 361 6.1 Black-Scholes pricing formula . . . . . . . . . . . . . . . . . . . . 361 6.2 Black-Scholes sensitivity analysis . . . . . . . . . . . . . . . . . . . 365 7 Project: Quantile-quantile plots 367 7.1 Log-returns and standardization . . . . . . . . . . . . . . . . 367 7.2 Quantile-Quantile plots . . . . . . . . . . . . . . . . . . . . . 368 8 Project: Monte Carlo Pricer 371 8.1 Scenario Generation . . . . . . . . . . . . . . . . . . 371 8.2 Calibration . . . . . . . . . . . . . . . . . . 372 8.3 Pricing Equity Basket Options . . . . . . . . . . . . . . . . . . . . 374 9 Project: The Binomial Lattice Model 377 9.1 Building the Lattice . . . . . . . . . . . . . . . . . . . . 377 9.2 Lattice Calibration and Pricing . . . . . . . . . . . . . . . . . . . . 379 10 Project: The Trinomial Lattice Model 383 10.1 Building the Lattice . . . . . . . . . . . . . . . . . . 383 10.2 Pricing procedure . . . . . . . . . . . . . . . . . . . 386 10.3 Calibration . . . . . . . . . . . . . . . 388 10.4 Pricing barrier options . . . . . . . .. . . . . . . . . . . . . 389 10.5 Put-call parity in trinomial lattices . . . . . . . . . . . . . . . . . . . 390 10.6 Computing the sensitivities . . . . . . . . . . . . . . . . . 391 11 Project: Crank-Nicolson option pricer 393 11.1 The Lattice for the Crank-Nicolson pricer . . . . . . . . . . . . . . . 393 11.2 Pricing with Crank-Nicolson . . . . . . . . . . . . . . . . 394 11.3 Calibration . . . . . . . . . . . . . . . . . . 396 11.4 Pricing barrier options . . . . . . . . . . . . . . . . . . 396 CONTENTS 9 12 Project: Static Hedging of Barrier Options 399 12.1 Analytical Pricing Formulas for Barrier Options . . . . . . . . . . . . 399 12.2 Replication of up-and-out barrier options . . . . . . . . . . . . . . . . 402 12.3 Replication of down-and-out barrier options . . . . . . . . . . . . . . 405 13 Project: Variance Swaps 409 13.1 The logarithmic payoff . . . . . . . . . . . . . . . . . . . . 409 13.2 Static Hedging: replication of a logarithmic payoff . . . . . . . . . . 410 14 Project: Monte Carlo VaR for Delta-Gamma Portfolios 415 14.1 Multivariate Normal Distribution . . . . . . . . . . . . . . . 415 14.2 Multivariate Student-t Distributions . . . . . . . . . . . .. . . . . 418 15 Project: Covariance estimation and scenario generation in VaR 421 15.1 Generating covariance matrices of a given spectrum . . . . . . . . . . 421 15.2 Re-estimating the covariance matrix and the spectral shift . . . . . . . 422 16 Project: Interest Rate Trees: Calibration and Pricing 425 16.1 Background Theory . . . . .. . . . . . . . . . . . . . . 425 16.2 Binomial Lattice Calibration for Discount Bonds . . . . . . . . . . . 427 16.3 Binomial pricing of FRAs, Swaps, Caplets, Floorlets, Swaptions and other derivatives . . . . . . . . . . . . . . . . . . 431 16.4 Trinomial Lattice Calibration and Pricing in the Hull-White model . . 437 16.4.1 The First Stage: The Lattice with zero drift . . . . . . . . . . 437 16.4.2 The Second Stage: Lattice calibration with drift and reversion 441 16.4.3 Pricing options . . . . . . . .. . . . . . . . . . . 445 16.5 Calibration and pricing within the Black-Karasinski model . . . . . . 446

Bibliographic details
Hardbound, 426 pages, publication date: SEP-2005
ISBN-13: 978-0-12-047682-4
ISBN-10: 0-12-047682-7
Imprint: ACADEMIC PRESS

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Last update: 5 Sep 2009
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