Advanced Mathematical Methods for Finance (eBook)

Giulia Di Nunno and Bernt Øksendal are professors at the University of Oslo. Their work in stochastic analysis, control, and mathematical finance is internationally highly appreciated. They have been chairing the leadership of the large European ESF funded networking program AMaMeF in financial mathematics.

Preface 5
Contents 7
Chapter 1: Dynamic Risk Measures 9
1.1 Introduction 9
1.2 Setup and Notation 11
1.3 Robust Representation 13
1.4 Time Consistency Properties 20
1.4.1 Time Consistency 20
1.4.2 Rejection and Acceptance Consistency 29
1.4.3 Weak Time Consistency 33
1.4.4 A Recursive Construction 35
1.5 The Dynamic Entropic Risk Measure 37
References 41
Chapter 2: Ambit Processes and Stochastic Partial Differential Equations 43
2.1 Introduction 43
2.2 Ambit Processes 45
2.2.1 Background 45
2.2.2 Ambit Fields and Processes 46
2.2.3 Null-Spatial Case: Lévy Semistationary Processes (LSS) 49
2.2.4 Key Example for a BSS Process 49
2.2.5 Generality of BSS 50
2.2.6 Multipower Variations 50
2.2.7 Applications to Turbulence and Finance 53
Tempo-Spatial Settings in Turbulence 53
Modelling Energy Markets by Ambit Fields 54
Spot Price 55
Forward Price 55
2.3 Lévy Bases and the Theory of Walsh 57
2.3.1 Brief Account on the Stochastic Integration Theory of Walsh 57
2.3.2 Lévy Bases and White Noise 58
2.3.3 Lévy Bases and Random Variables in a Hilbert Space 61
2.3.4 Extension of the Stochastic Integration Theory of Walsh 65
2.3.5 Stochastic Partial Differential Equations and Ambit Processes 69
2.4 Lévy Noise Analysis 71
2.4.1 Lévy Bases and Lévy Noise 71
2.4.2 Stochastic Partial Differential Equations and Lévy Noise Analysis 76
2.5 Conclusions 77
Appendix: Lévy Bases and Integration 78
A.1 Introduction 78
A.2 Representation of the Characteristic Function of a Lévy Basis 78
A.3 Integration with Respect to a Lévy Basis 79
A.4 Criteria for Integrability 80
References 81
Chapter 3: Fractional Processes as Models in Stochastic Finance 83
3.1 Introduction 83
3.2 Models and Notions of Arbitrage 84
3.3 Trading with (Almost) Simple Strategies 87
3.4 Trading with Delay-Simple Strategies 91
3.5 Continuous Trading 95
3.6 Trading under Transaction Costs 100
3.7 Approximations 102
3.7.1 Binary Tree Approximations 102
3.7.2 Arbitrage-Free Approximation 105
3.7.3 Microeconomic Approximation 107
3.8 Conclusions 109
3.8.1 Open Problems 109
References 109
Chapter 4: Credit Contagion in a Long Range Dependent Macroeconomic Factor Model 112
4.1 Introduction 112
4.2 The Credit Model 114
4.2.1 The Default Model 114
4.2.2 The Probability Space 115
4.3 A Portfolio with Disjoint Contagion Classes 117
4.3.1 Conditional Infinitesimal Generator of the Default Indicator Process 118
4.3.2 Conditional Infinitesimal Generator of the Default Number Process 120
4.4 The Price of Credit Derivatives as a Function of Psi 121
4.5 Pricing Contingent Claim Depending on the Macroeconomic Process with Credit Risk Contagion 131
4.5.1 Modeling the Macroeconomic Process 131
4.5.2 Pricing Contingent Claims with a Long-Range Dependent Psi 132
4.5.3 Comparison with Markovian Psi 137
References 139
Chapter 5: Modelling Information Flows in Financial Markets 140
5.1 Cash Flow Structures and Market Factors 141
5.2 X-factor Analysis 143
5.3 Information Processes 144
5.4 Brownian-Bridge Information 144
5.5 Assets Paying a Single Dividend 146
5.6 Geometric Brownian Motion Model 147
5.7 Pricing Contingent Claims 148
5.8 Volatility and Correlation 150
5.9 Amount of Information about the Future Cash Flow Contained in the Price Process 152
5.10 Information Disparity and Statistical Arbitrage 153
5.11 Price Formation in Inhomogeneous Markets 156
References 159
Chapter 6: An Overview of Comonotonicity and Its Applications in Finance and Insurance 161
6.1 Comonotonicity 161
6.2 Convex Bounds for Sums of Random Variables 164
6.2.1 Sums of Comonotonic Random Variables 164
6.2.2 Convex Bounds for Sums of Random Variables 166
6.3 Further Developments of the Theory 168
6.4 Applications of the Theory of Comonotonicity 171
6.4.1 Derivatives Pricing and Hedging 171
6.4.2 Risk Management: Risk Sharing, Optimal Investment, Capital Allocation 175
Risk Measures and Risk Sharing 175
Optimal Investment Strategies 176
Capital Allocation 177
6.4.3 Life Insurance and Pensions 178
6.5 Conclusion 180
References 180
Chapter 7: A General Maximum Principle for Anticipative Stochastic Control and Applications to Insider Trading 186
7.1 Introduction 187
7.2 Framework 189
7.2.1 Malliavin Calculus for Lévy Processes 190
7.2.2 Malliavin Calculus and Forward Integral 192
Forward Integral and Malliavin Calculus for B(·) 192
Forward Integral and Malliavin Calculus for Ñ(·,·) 194
7.3 A Stochastic Maximum Principle for Insider 196
7.4 Controlled Itô-Lévy Processes 200
7.5 Applications to Some Special Cases of Filtrations 202
7.5.1 D-commutable Filtrations 202
7.5.2 Smoothly Anticipative Filtrations 205
7.6 Application to Optimal Insider Portfolio 209
7.6.1 Case Gt=FGt, Gt[0,t]. See (7.45) 210
7.6.2 Case Gt= Ftvsigma(B(T)). See (7.51) 211
7.7 Application to Optimal Insider Consumption 212
Appendix: Proof of Theorem 7.13 214
References 225
Chapter 8: Analyticity of the Wiener-Hopf Factors and Valuation of Exotic Options in Lévy Models 227
8.1 Introduction 227
Important Remark 229
8.2 Lévy Processes 229
8.2.1 Notation 230
8.2.2 Analytic Extension, Fixed-Time Case 231
8.2.3 Analytic Extension, Exponential Time Case 234
8.3 The Wiener-Hopf Factorization 235
8.3.1 Analyticity 236
8.3.2 Inversion 240
8.4 Lévy Processes: Examples and Properties 241
8.4.1 Continuity Properties 242
8.4.2 Examples 242
8.5 Applications in Finance 244
8.5.1 Lookback Options 245
8.5.2 One-Touch Options 246
8.5.3 Equity Default Swaps 247
References 248
Chapter 9: Optimal Liquidation of a Pairs Trade 250
9.1 Introduction 250
9.2 Solving the Optimal Stopping Problem 251
9.3 Dependence on Parameters 253
9.4 Including a Discount Factor 255
References 257
Chapter 10: A PDE-Based Approach for Pricing Mortgage-Backed Securities 259
10.1 Introduction 259
10.2 MBSs Modeling 262
10.2.1 MBS Cash-Flows 263
10.2.2 The MBS Market 265
10.2.3 Pricing MBSs: A PDE Approach 271
10.3 Viscosity Solutions 276
10.4 Financial Motivations for the Regularity of U 277
10.4.1 Stochastic Representation 279
10.5 The Numerical Solution 283
10.5.1 The Numerical Approximation 284
10.5.2 The Numerical Tests 287
References 292
Chapter 11: Nonparametric Methods for Volatility Density Estimation 294
11.1 Introduction 294
11.2 The Continuous-Time Model 296
11.3 Kernel Deconvolution 298
11.3.1 Construction of the Estimator 298
11.3.2 An Application to the Amsterdam AEX Index 301
11.4 Wavelet Deconvolution 302
11.5 Penalized Projection Estimators 306
11.6 Estimation for Discrete-Time Models 308
11.6.1 Discrete-Time Models 308
11.6.2 Density Estimation 309
11.6.3 Regression Function Estimation 310
11.7 Concluding Remarks 312
References 312
Chapter 12: Fractional Smoothness and Applications in Finance 314
12.1 Introduction 314
The Discrete-Time Hedging Error as an Important Application 315
Organization of the Paper 316
Assumptions 316
12.2 Definition of Fractional Smoothness and Basic Properties 317
12.3 Connection to Real Interpolation 319
Multidimensional Black-Scholes-Samuelson Model 321
12.4 Examples 322
12.5 Applications 324
12.5.1 Weak Limits of Error Processes 324
12.5.2 L2-estimates of the Tracking Error 326
Equidistant Time Nets 327
Nonequidistant Time Nets 328
Concluding Remarks 329
12.6 Further Developments 330
12.6.1 Backward Stochastic Differential Equations 330
12.6.2 Lévy Processes 331
12.6.3 Multigrid Monte Carlo Methods 331
References 331
Chapter 13: Liquidity Models in Continuous and Discrete Time 333
13.1 What Is Illiquidity? 333
13.2 Optimal Execution Problem 335
13.2.1 The First Approach 335
13.2.2 Continuous-Time Models 339
13.2.3 Models of Limit Order Books 342
13.3 Option Hedging for Large Traders 346
13.4 Supply Curve Models 355
13.5 Expected Utility Maximization in Illiquid Markets 361
13.6 Price Manipulation strategies in Price Impact Models 362
References 364
Chapter 14: Some New BSDE Results for an Infinite-Horizon Stochastic Control Problem 366
14.1 Introduction 366
14.2 Preliminaries and Overview 368
14.3 The BSDE on a Finite Horizon 370
14.4 The BSDE on an Infinite Horizon 375
14.5 The Stochastic Control Problem on an Infinite Horizon 379
14.6 Solving the Stochastic Control Problem via the BSDE 386
14.6.1 The Bounded Case 387
14.6.2 The Positive Case 389
14.6.3 The General Case 391
14.6.4 Consequences for the Stochastic Control Problem 393
References 394
Chapter 15: Functionals Associated with Gradient Stochastic Flows and Nonlinear SPDEs 395
15.1 Introduction 395
15.2 Preliminaries 399
15.3 Gradient Representation of Stochastic Flow and Construction of a Solution of Nonlinear SPDE 402
15.4 Applications 408
15.4.1 Pathwise Solutions of Burgers Equations with Stochastic Perturbations 408
15.4.2 A Filtering Problem for SDEs Associated with Parameterized Backward Parabolic Equations 410
References 412
Chapter 16: Pricing and Hedging of Rating-Sensitive Claims Modeled by F-doubly Stochastic Markov Chains 414
16.1 Introduction 414
16.2 General Notion of Hedging a Payment Stream 416
16.3 Doubly Stochastic Markov Chains 423
16.4 Valuation of Defaultable Rating-Sensitive Claims with Ratings Given by a Doubly Stochastic Markov Chain 426
16.4.1 Description of Claims 426
16.4.2 Pricing of Rating-Sensitive Claims 428
16.4.3 Examples of Pricing of Selected Instruments 432
Defaultable Bond with Fractional Recovery of Par Value 432
Credit Sensitive Note (CNS)-Resetting at Coupon Payment Date 433
Credit Sensitive Note-Continuous Coupon Payments 433
Credit Default Swap 433
16.5 Hedging of Rating-Sensitive Claims 435
16.5.1 Hedging with Rating Digital Options 436
16.5.2 Hedging with General Rating-Sensitive Claims 446
References 449
Chapter 17: Exotic Derivatives under Stochastic Volatility Models with Jumps 451
17.1 Introduction 451
17.2 Markov Chains and Phase-Type Distributions 454
17.2.1 Finite-State Markov Chains 454
17.2.2 (Double) Phase-Type Distributions 456
17.3 Stochastic Volatility Models with Jumps 460
17.3.1 A Class of Regime-Switching Models 462
17.3.2 Two-Step Approximation Procedure 466
Markov Chain Approximation of the Variance Process 466
Approximation of a Lévy Process 468
17.3.3 Convergence of the Approximation Procedure 469
17.4 European and Volatility Derivatives 474
17.4.1 Call and Put Options 474
17.4.2 Implied Volatility at Extreme Strikes 475
17.4.3 Forward Starting Options and the Forward Smile 476
17.4.4 Volatility Derivatives 478
17.5 First Passage Times for Regime-Switching Processes 482
17.5.1 Three Key Matrices 482
17.5.2 Matrix Wiener-Hopf Factorisation 483
17.5.3 First-Passage into a Half-Line 484
17.5.4 Joint Distribution of the Maximum and Minimum 486
17.5.5 Valuing a Double-Barrier Rebate Option 489
17.6 Double-No-Touch and Other Barrier Options 490
17.7 Embedding of the Process (X,Z) 492
17.8 Ladder Processes 494
17.8.1 Proof of Lemma 17.28 498
17.8.2 Proof of Theorem 17.21 500
17.8.3 Proof of Theorem 17.24 502
References 503
Chapter 18: Asymptotics of HARA Utility from Terminal Wealth under Proportional Transaction Costs with Decision Lag or Execution Delay and Obligatory Diversification 505
18.1 Introduction 505
18.2 Discounted GOP 510
18.3 Long-Run GOP 512
18.4 Discounted Discrete-Time PUOP 517
18.5 Long-Run PUOP 524
18.6 General Form of the Long-Run Bellman Equations 530
References 531