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Advanced Derivatives Pricing and Risk Management
 
 

Advanced Derivatives Pricing and Risk Management, 1st Edition

Theory, Tools, and Hands-On Programming Applications

 
Advanced Derivatives Pricing and Risk Management, 1st Edition,Claudio Albanese,Giuseppe Campolieti,ISBN9780080488097
 
 
 

  &      

Academic Press

9780080488097

426

Explores important topics and cutting-edge research about financial derivatives and risk management supported by numerical projects on the accompanying CD-ROM

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Key Features

*Includes easy-to-implement VB/VBA numerical software libraries
*Proceeds from simple to complex in approaching pricing and risk management problems
*Provides analytical methods to derive cutting-edge pricing formulas for equity derivatives

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.

Readership

Students in finance programs, particularly financial engineering.

Claudio Albanese

Affiliations and Expertise

Professor of Mathematical Finance, Imperial College, London, UK

Giuseppe Campolieti

Affiliations and Expertise

Associate Professor of Mathematics, SHARCNET Chair in Financial Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada

Advanced Derivatives Pricing and Risk Management, 1st Edition

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

Quotes and reviews

“Albanese and Campolieti carefully select the most important and relevant topics in financial derivatives pricing and risk management. Their work strikes a fine balance between theory and financial practice. A dozen carefully designed numerical projects are included that serve to introduce students to actual implementation issues in pricing and risk management. The book is succinctly written, with clear and insightful descriptions of state-of-the-art financial models. The style of presentation demonstrates the authors' unique pedagogical exposition of the quantitative and financial concepts in derivative pricing and risk management. Advanced Derivatives Pricing and Risk Management is destined to be a valuable text and reference for students and practitioners in the field of financial engineering.”
— Yue Kuen Kwok, Associate Professor, Department of Mathematics, Hong Kong University of Science and Technology

“The set of projects on the accompanying CDROM give students and professors the opportunity to work in a simulated environment and can be used, as is the goal here, to train students in building software modules for pricing, hedging, etc. The projects enhance the understanding of the material and extend the book's usefulness by enabling students to tackle other situations not explicitly addressed in the modules provided. VBA is easy to learn and can facilitate rapid developments of real applications. In addition, the choice of Excel as the Graphic User Interface (GUI) is very appropriate. Furthermore, the existence of a built-in visual basic editor allows users to see the code, modify it to suit different needs and to experiment with it. Hence these features facilitate student learning to produce software themselves.”
— Eliezer Prisman, Nigel Martin Chair in Finance, Director of the Financial Engineering Collaborative Diploma, Schulich School of Business, York University, Toronto

"...provides a combination of theoretical and practical tools from which any advanced undergraduate and graduate student, professional quant and researcher will benefit. It differs from existing books in quantitative finance from the ready-to-use software and accessible theoretical tools provided as a complete package. As a bonus, the book also gives a detailed exposition on cutting-edge theoretical techniques published here for the first time."
- Technical Analysis of Stocks & Commodities
 
 
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