Mathematical Modeling And Computation In Finance: With Exercises And Python And Matlab Computer Codes

by Cornelis W Oosterlee, Lech A Grzelak

Overview

This book discusses the interplay of stochastics (applied probability theory) and numerical analysis in the field of quantitative finance. The stochastic models, numerical valuation techniques, computational aspects, financial products, and risk management applications presented will enable readers to progress in the challenging field of computational finance.When the behavior of financial market participants changes, the corresponding stochastic mathematical models describing the prices may also change. Financial regulation may play a role in such changes too. The book thus presents several models for stock prices, interest rates as well as foreign-exchange rates, with increasing complexity across the chapters. As is said in the industry, 'do not fall in love with your favorite model.' The book covers equity models before moving to short-rate and other interest rate models. We cast these models for interest rate into the Heath-Jarrow-Morton framework, show relations between the different models, and explain a few interest rate products and their pricing.The chapters are accompanied by exercises. Students can access solutions to selected exercises, while complete solutions are made available to instructors. The MATLAB and Python computer codes used for most tables and figures in the book are made available for both print and e-book users. This book will be useful for people working in the financial industry, for those aiming to work there one day, and for anyone interested in quantitative finance. The topics that are discussed are relevant for MSc and PhD students, academic researchers, and for quants in the financial industry.

Product Details

ISBN-13:	9781786347947
Publisher:	World Scientific Publishing Europe Ltd
Publication date:	11/05/2019
Pages:	576
Product dimensions:	6.69(w) x 9.61(h) x 1.25(d)

Table of Contents

Preface vii

Acknowledgment ix

Using this Book xi

1 Basics about Stochastic Processes 1

1.1 Stochastic variables 1

1.1.1 Density function, expectation, variance 1

1.1.2 Characteristic function 3

1.1.3 Cumulants and moments 4

1.2 Stochastic processes, martingale property 9

1.2.1 Wiener process 11

1.2.2 Martingales 12

1.2.3 Iterated expectations (Tower property) 13

1.3 Stochastic integration, Itô integral 14

1.3.1 Elementary processes 14

1.3.2 Itô isometry 17

1.3.3 Martingale representation theorem 20

1.4 Exercise set 25

2 Introduction to Financial Asset Dynamics 27

2.1 Geometric Brownian motion asset price process 27

2.1.1 Itô process 29

2.1.2 Itô's lemma 30

2.1.3 Distributions of S(t) and log S(t) 34

2.2 First generalizations 38

2.2.1 Proportional dividend model 38

2.2.2 Volatility variation 39

2.2.3 Time-dependent volatility 39

2.3 Martingales and asset prices 40

2.3.1 P-measure prices 41

2.3.2 Q-measure prices 42

2.3.3 Parameter estimation under real-world measure P 44

2.4 Exercise set 49

3 The Black-Scholes Option Pricing Equation 51

3.1 Option contract definitions 51

3.1.1 Option basics 52

3.1.2 Derivation of the partial differential equation 56

3.1.3 Martingale approach and option pricing 60

3.2 The Feynman-Kac theorem and the Black-Scholes model 61

3.2.1 Closed-form option prices 63

3.2.2 Green's functions and characteristic functions 66

3.2.3 Volatility variations 71

3.3 Delta hedging under the Black-Scholes model 73

3.4 Exercise set 78

4 Local Volatility Models 81

4.1 Black-Scholes implied volatility 81

4.1.1 The concept of implied volatility 82

4.1.2 Implied volatility; implications 86

4.1.3 Discussion on alternative asset price models 86

4.2 Option prices and densities 89

4.2.1 Market implied volatility smile and the payoff 89

4.2.2 Variance swaps 96

4.3 Non-parametric local volatility models 102

4.3.1 Implied volatility representation of local volatility 105

4.3.2 Arbitrage-free conditions for option prices 107

4.3.3 Advanced implied volatility interpolation 111

4.3.4 Simulation of local volatility model 114

4.4 Exercise set 117

5 Jump Processes 121

5.1 Jump diffusion processes 121

5.1.1 Itô's lemma and jumps 124

5.1.2 PIDE derivation for jump diffusion process 127

5.1.3 Special cases for the jump distribution 128

5.2 Feynman-Kac theorem for jump diffusion process 130

5.2.1 Analytic option prices 131

5.2.2 Characteristic function for Merton's model 133

5.2.3 Dynamic hedging of jumps with the Black-Scholes model 137

5.3 Exponential Lévy processes 139

5.3.1 Finite activity exponential Lévy processes 142

5.3.2 PIDE and the Lévy triplet 143

5.3.3 Equivalent martingale measure 145

5.4 Infinite activity exponential Lévy processes 146

5.4.1 Variance Gamma process 146

5.4.2 CGMY process 151

5.4.3 Normal inverse Gaussian process 155

5.5 Discussion on jumps in asset dynamics 156

5.6 Exercise set 159

6 The COS Method for European Option Valuation 163

6.1 Introduction into numerical option valuation 164

6.1.1 Integrals and Fourier cosine series 164

6.1.2 Density approximation via Fourier cosine expansion 165

6.2 Pricing European options by the COS method 169

6.2.1 Payoff coefficients 172

6.2.2 The option Greeks 173

6.2.3 Error analysis COS method 174

6.2.4 Choice of integration range 177

6.3 Numerical COS method results 182

6.3.1 Geometric Brownian Motion 183

6.3.2 CGMY and VG processes 184

6.3.3 Discussion about option pricing 187

6.4 Exercise set 189

7 Multidimensionality, Change of Measure, Affine Processes 193

7.1 Preliminaries for multi-D SDE systems 193

7.1.1 The Cholesky decomposition 194

7.1.2 Multi-D asset price processes 197

7-1.3 Itô's lemma for vector processes 198

7.1.4 Multi-dimensional Feynman-Kac theorem 200

7.2 Changing measures and the Girsanov theorem 201

7.2.1 The Radon-Nikodym derivative 202

7.2.2 Change of numéraire examples 204

7.2.3 From P to Q in the Black-Scholes model 206

7.3 Affine processes 211

7.3.1 Affine diffusion processes 211

7.3.2 Affine jump diffusion processes 216

7.3.3 Affine jump diffusion process and PIDE 217

7.4 Exercise set 219

8 Stochastic Volatility Models 223

8.1 Introduction into stochastic volatility models 224

8.1.1 The Schöbel-Zhu stochastic volatility model 224

8.1.2 The CIR process for the variance 225

8.2 The Heston stochastic volatility model 231

8.2.1 The Heston option pricing partial differential equation 233

8.2.2 Parameter study for implied volatility skew and smile 236

8.2.3 Heston model calibration 238

8.3 The Heston SV discounted characteristic function 242

8.3.1 Stochastic volatility as an affine diffusion process 242

8.3.2 Derivation of Heston SV characteristic function 244

8.4 Numerical solution of Heston PDE 247

8.4.1 The COS method for the Heston model 248

8.4.2 The Heston model with piecewise constant parameters 250

8.4.3 The Bates model 251

8.5 Exercise set 255

9 Monte Carlo Simulation 257

9.1 Monte Carlo basics 257

9.1.1 Monte Carlo integration 260

9.1.2 Path simulation of stochastic differential equations 265

9.2 Stochastic Euler and Milstein schemes 266

9.2.1 Euler scheme 266

9.2.2 Milstein scheme: detailed derivation 269

9.3 Simulation of the CIR process 274

9.3.1 Challenges with standard discretization schemes 274

9.3.2 Taylor-based simulation of the CIR process 276

9.3.3 Exact simulation of the CIR model 278

9.3.4 The Quadratic Exponential scheme 279

9.4 Monte Carlo scheme for the Heston model 283

9.4.1 Example of conditional sampling and integrated variance 283

9.4.2 The integrated CIR process and conditional sampling 285

9.4.3 Almost exact simulation of the Heston model 288

9.4.4 Improvements of Monte Carlo simulation 292

9.5 Computation of Monte Carlo Greeks 294

9.5.1 Finite differences 295

9.5.2 Pathwise sensitivities 297

9.5.3 Likelihood ratio method 301

9.6 Exercise set 306

10 Forward Start Options; Stochastic Local Volatility Model 309

10.1 Forward start options 309

10.1.1 Introduction into forward start options 310

10.1.2 Pricing under the Black-Scholes model 311

10.1.3 Pricing under the Heston model 314

10.1.4 Local versus stochastic volatility model 316

10.2 Introduction into stochastic-local volatility model 319

10.2.1 Specifying the local volatility 320

10.2.2 Monte Carlo approximation of SLV expectation 327

10.2.3 Monte Carlo AES scheme for SLV model 330

10.3 Exercise set 336

11 Short-Rate Models 339

11.1 Introduction to interest rates 339

11.1.1 Bond securities, notional 340

11.1.2 Fixed-rate bond 341

11.2 Interest rates in the Heath-J arrow-Morton framework 343

11.2.1 The HJM framework 343

11.2.2 Short-rate dynamics under the HJM framework 347

11.2.3 The Hull-White dynamics in the HJM framework 349

11.3 The Hull-White model 352

11.3.1 The solution of the Hull-White SDE 352

11.3.2 The HW model characteristic function 353

11.3.3 The CIR model under the HJM framework 356

11.4 The HJM model under the T-forward measure 359

11.4.1 The Hull-White dynamics under the T-forward measure 360

11.4.2 Options on zero-coupon bonds under Hull-White model 362

11.5 Exercise set 365

12 Interest Rate Derivatives and Valuation Adjustments 367

12.1 Basic interest rate derivatives and the Libor rate 368

12.1.1 Libor rate 368

12.1.2 Forward rate agreement 370

12.1.3 Floating rate note 371

12.1.4 Swaps 371

12.1.5 How to construct a yield curve 375

12.2 More interest rate derivatives 378

12.2.1 Caps and floors 378

12.2.2 European swaptions 383

12.3 Credit Valuation Adjustment and Risk Management 386

12.3.1 Unilateral Credit Value Adjustment 392

12.3.2 Approximations in the calculation of CVA 395

12.3.3 Bilateral Credit Value Adjustment (BCVA) 396

12.3.4 Exposure reduction by netting 397

12.4 Exercise set 400

13 Hybrid Asset Models, Credit Valuation Adjustment 405

13.1 Introduction to affine hybrid asset models 406

13.1.1 Black-Scholes Hull-White (BSHW) model 406

13.1.2 BSHW model and change of measure 408

13.1.3 Schöbel-Zhu Hull-White (SZHW) model 413

13.1.4 Hybrid derivative product 416

13.2 Hybrid Heston model 417

13.2.1 Details of Heston Hull-White hybrid model 418

13.2.2 Approximation for Heston hybrid models 420

13.2.3 Monte Carlo simulation of hybrid Heston SDEs 428

13.2.4 Numerical experiment, HHW versus SZHW model 431

13.3 CVA exposure profiles and hybrid models 433

13.3.1 CVA and exposure 434

13.3.2 European and Bermudan options example 434

13.4 Exercise set 439

14 Advanced Interest Rate Models and Generalizations 445

14.1 Libor market model 446

14.1.1 General Libor market model specifications 446

14.1.2 Libor market model under the HJM framework 449

14.2 Lognormal Libor market model 451

14.2.1 Change of measure in the LMM 452

14.2.2 The LMM under the terminal measure 453

14.2.3 The LMM under the spot measure 454

14.2.4 Convexity correction 457

14.3 Parametric local volatility models 460

14.3.1 Background, motivation 460

14.3.2 Constant Elasticity of Variance model (CEV) 461

14.3.3 Displaced diffusion model 467

14.3.4 Stochastic volatility LMM 470

14.4 Risk management: The impact of a financial crisis 475

14.4.1 Valuation in a negative interest rates environment 476

14.4.2 Multiple curves and the Libor rate 479

14.4.3 Valuation in a multiple curves setting 484

14.5 Exercise set 486

15 Cross-Currency Models 489

15.1 Introduction into the FX world and trading 490

15.1.1 FX markets 490

15.1.2 Forward FX contract 491

15.1.3 Pricing of FX options, the Black-Scholes case 493

15.2 Multi-currency FX model with short-rate interest rates 495

15.2.1 The model with correlated, Gaussian interest rates 496

15.2.2 Pricing of FX options 498

15.2.3 Numerical experiment for the FX-HHW model 505

15.2.4 CVA for FX swaps 508

15.3 Multi-currency FX model with interest rate smile 510

15.3.1 Linearization and forward characteristic function 514

15.3.2 Numerical experiments with the FX-HLMM model 517

15.4 Exercise set 522

References 525

Index 541