Handbooks in Operations Research and Management Science: Financial Engineering (eBook)

Front cover 1
Financial Engineering 4
Copyright page 5
Contents 6
Part I: Introduction 14
Introduction to the Handbook of FinancialEngineering 16
References 24
Chapter 1. An Introduction to Financial Asset Pricing 26
1. Introduction 26
2. Introduction to derivatives and arbitrage 27
3. The core of the theory 34
4. American type derivatives 73
Acknowledgements 80
References 80
Part II: Derivative Securities: Models and Methods 84
Chapter 2. Jump-Diffusion Models for Asset Pricing in Financial Engineering 86
1. Introduction 86
2. Empirical stylized facts 88
3. Motivation for jump-diffusion models 97
4. Equilibrium for general jump-diffusion models 102
5. Basic setting for option pricing 105
6. Pricing call and put option via Laplace transforms 107
7. First passage times 109
8. Barrier and lookback options 113
9. Analytical approximations for American options 116
10. Extension of the jump-diffusion models to multivariate cases 121
References 126
Chapter 3. Modeling Financial Security Returns Using Lévy Processes 130
1. Introduction 130
2. Modeling return innovation distribution using Lévy processes 133
3. Generating stochastic volatility by applying stochastic time changes 140
4. Modeling financial security returns with time-changed Lévy processes 146
5. Option pricing under time-changed Lévy processes 157
6. Estimating Lévy processes with and without time changes 168
7. Concluding remarks 172
Acknowledgements 172
References 173
Chapter 4. Pricing with Wishart Risk Factors 176
1. Introduction 176
2. Wishart process 180
3. Pricing 185
4. Examples 188
5. Concluding remarks 194
References 194
Chapter 5. Volatility 196
1. Introduction 196
2. A model of price formation with microstructure effects 197
3. The variance of the equilibrium price 199
4. Solutions to the inconsistency problem 204
5. Equilibrium price variance estimation: directions for future work 215
6. The variance of microstructure noise: a consistency result 223
7. The benefit of consistency: measuring market quality 223
8. Volatility and asset pricing 229
Acknowledgements 230
References 230
Chapter 6. Spectral Methods in Derivatives Pricing 236
1. Introduction 237
2. Self-adjoint semigroups in Hilbert spaces 243
3. One-dimensional diffusions: general results 250
4. One-dimensional diffusions: a catalog of analytically tractable models 266
5. Symmetric multi-dimensional diffusions 298
6. Introducing jumps and stochastic volatility via time changes 301
7. Conclusion 307
References 307
Chapter 7. Variational Methods in Derivatives Pricing 314
1. Introduction 315
2. European and barrier options in the Black-Scholes-Merton model 318
3. American options in the Black-Scholes-Merton model 328
4. General multi-dimensional jump-diffusion models 333
5. Examples and applications 342
6. Summary 352
References 353
Chapter 8. Discrete Barrier and Lookback Options 356
1. Introduction 356
2. A representation of barrier options via the change of numeraire argument 361
3. Convolution, Broadie-Yamamoto method via the fast Gaussian transform, and Feng-Linetsky method via Hilbert transform 363
4. Continuity corrections 368
5. Perturbation method 374
6. A Laplace transform method via Spitzer's identity 376
7. Which method to use 378
Appendix A. Proof of (1) 379
Appendix B. Calculation of the constant beta 381
References 383
Part III: Interest Rate and Credit Risk Models and Derivatives 388
Chapter 9. Topics in Interest Rate Theory 390
1. Introduction 390
2. Basics 391
3. Forward rate models 394
4. Change of numeraire 400
5. LIBOR market models 403
6. Notes 413
7. Geometric interest rate theory 413
8. Consistency and invariant manifolds 414
9. Existence of nonlinear realizations 424
10. Potentials and positive interest 432
References 447
Chapter 10. Calculating Portfolio Credit Risk 450
1. Introduction 450
2. Problem setting 452
3. Models of dependence 457
4. Conditional loss distributions 464
5. Unconditional loss distributions 470
6. Importance sampling 475
7. Summary 480
References 481
Chapter 11. Valuation of Basket Credit Derivatives in the Credit Migrations Environment 484
1. Introduction 485
2. Notation and preliminary results 489
3. Markovian market model 494
4. Changes of measures and Markovian numeraires 498
5. Valuation of single name credit derivatives 505
6. Valuation of basket credit derivatives 510
7. Model implementation 513
References 520
Part IV: Incomplete Markets 522
Chapter 12. Incomplete Markets 524
1. Introduction 524
2. The over-the-counter market 526
3. Causes of incompleteness 529
4. Pricing and optimization 531
5. Issues in pricing and expected utility examples 541
6. Quadratics 546
7. Entropy and exponential utility 549
8. Loss, quantiles, and prediction 550
9. Pricing kernel restrictions 553
10. Ambiguity and robustness 557
11. Calibration 563
12. Conclusion 564
Acknowledgements 567
Appendix A. Definition of incompleteness and fundamental theorems 567
Appendix B. Financial perspectives on incompleteness 569
References 571
Chapter 13. Option Pricing: Real and Risk-Neutral Distributions 578
1. Introduction 579
2. Implications of the absence of arbitrage 580
3. Additional restrictions implied by utility maximization 583
4. Special case: one period without transaction costs 587
5. Special case: one period with transaction costs and general payoffs 591
6. Special case: two periods without transaction costs and general payoffs 592
7. Special case: two periods with transaction costs and general payoffs 593
8. Multiple periods without transaction costs and with convex payoffs 594
9. Multiple periods with transaction costs and with convex payoffs 596
10. Empirical results 598
11. Concluding remarks 601
Acknowledgements 602
References 602
Chapter 14. Total Risk Minimization Using Monte Carlo Simulations 606
1. Introduction 606
2. Discrete hedging criteria 612
3. Total risk minimization in the Black-Scholes framework 616
4. Total risk minimization in a stochastic volatility framework 631
5. Shortfall risk minimization 638
6. Conclusions 645
References 647
Chapter 15. Queuing Theoretic Approaches to Financial Price Fluctuations 650
1. Introduction 651
2. Agent-based models of financial markets 652
3. Microstructure models with inert investors 662
4. Outlook and conclusion 684
Acknowledgements 687
References 687
Part V: Risk Management 692
Chapter 16. Economic Credit Capital Allocation and Risk Contributions 694
1. Introduction 695
2. Credit portfolio models and general framework 697
3. Capital allocation and risk contributions 701
4. Credit risk contributions in analytical models 706
5. Numerical methods to compute risk contributions 714
6. Case studies 719
7. Summary and further research 730
Appendix A 734
References 737
Chapter 17. Liquidity Risk and Option Pricing Theory 740
1. Introduction 740
2. The model 742
3. The extended first fundamental theorem 746
4. The extended second fundamental theorem 748
5. Example (extended Black-Scholes economy) 754
6. Economies with supply curves for derivatives 756
7. Transaction costs 758
8. Examples of supply curves 760
9. Conclusion 764
Acknowledgement 764
Appendix A 764
References 774
Chapter 18. Financial Engineering: Applications in Insurance 776
1. Introduction 776
2. Insurance products and markets 778
3. Premium principles and risk measures 781
4. Risk management for life insurance 783
5. Variable annuities 788
6. Guaranteed annuity options 794
7. Conclusions 797
Acknowledgements 797
References 798
Part VI: Portfolio Optimization 800
Chapter 19. Dynamic Portfolio Choice and Risk Aversion 802
1. Introduction 802
2. Optimality and state pricing 806
3. Recursive utility 817
4. Modeling risk aversion 827
5. Scale-invariant solutions 834
6. Extensions 846
Acknowledgements 852
References 852
Chapter 20. Optimization Methods in Dynamic Portfolio Management 858
1. Introduction 858
2. Formulation 859
3. Approximation methods 862
4. Solution methods 870
5. Extensions and conclusions 873
Acknowledgements 874
References 874
Chapter 21. Simulation Methods for Optimal Portfolios 880
1. Introduction 880
2. The consumption-portfolio choice problem 882
3. Simulation methods for portfolio computation 891
4. Asymptotic properties of portfolio estimators 900
5. Performance evaluation: a numerical study 916
6. Conclusion 920
Acknowledgement 922
Appendix A. An introduction to Malliavin calculus 922
Appendix B. Proofs 928
References 935
Chapter 22. Duality Theory and Approximate Dynamic Programming for Pricing American Options and Portfolio Optimization 938
1. Introduction 938
2. Pricing American options 940
3. Portfolio optimization 950
References 960
Chapter 23. Asset Allocation with Multivariate Non-Gaussian Returns 962
1. Introduction 962
2. Non-Gaussian investment 964
3. Modeling distributions 966
4. Exponential utility and investment in zero cost VG cash flows 968
5. Identifying the joint distribution of returns 971
6. Non-Gaussian and Gaussian investment compared 973
7. Conclusion 975
Appendix A. Formal analysis of skewness preference and kurtosis aversion 976
Appendix B. Proof of Theorem 4.1 977
Appendix C. Proof of Theorem 4.2 979
References 981
Chapter 24. Large Deviation Techniques and Financial Applications 984
1. Introduction 984
2. Large deviation techniques 985
3. Applications to portfolio management 992
4. Tail risk of portfolios 999
5. Application to simulation 1000
6. Incomplete markets 1005
7. Conclusions and potential topics for future research 1010
Acknowledgements 1011
References 1011
Subject Index 1014