Search results for ""Author Raimondo Manca""

Book SynopsisThis book provides a detailed study of Financial Mathematics. In addition to the extraordinary depth the book provides, it offers a study of the axiomatic approach that is ideally suited for analyzing financial problems. This book is addressed to MBA's, Financial Engineers, Applied Mathematicians, Banks, Insurance Companies, and Students of Business School, of Economics, of Applied Mathematics, of Financial Engineering, Banks, and more.Table of ContentsPreface xvii Part I. Deterministic Models 1 Chapter 1. Introductory Elements to Financial Mathematics 3 Chapter 2. Theory of Financial Laws 13 Chapter 3. Uniform Regimes in Financial Practice 41 Chapter 4. Financial Operations and their Evaluation: Decisional Criteria 91 Chapter 5. Annuities-Certain and their Value at Fixed Rate 147 Chapter 6. Loan Amortization and Funding Methods 211 Chapter 7. Exchanges and Prices on the Financial Market 289 Chapter 8. Annuities, Amortizations and Funding in the Case of Term Structures 331 Chapter 9. Time and Variability Indicators, Classical Immunization 363 Part II. Stochastic Models 409 Chapter 10. Basic Probabilistic Tools for Finance 411 Chapter 11. Markov Chains 457 Chapter 12. Semi-Markov Processes 481 Chapter 13. Stochastic or Itô Calculus 517 Chapter 14. Option Theory 553 Chapter 15. Markov and Semi-Markov Option Models 607 Chapter 16. Interest Rate Stochastic Models – Application to the Bond Pricing Problem 641 Chapter 17. Portfolio Theory 687 Chapter 18. Value at Risk (VaR) Methods and Simulation 703 Chapter 19. Credit Risk or Default Risk 743 Chapter 20. Markov and Semi-Markov Reward Processes and Stochastic Annuities 791 References 831 Index 839

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Book SynopsisQuantitative finance has become these last years a extraordinary field of research and interest as well from an academic point of view as for practical applications. At the same time, pension issue is clearly a major economical and financial topic for the next decades in the context of the well-known longevity risk. Surprisingly few books are devoted to application of modern stochastic calculus to pension analysis. The aim of this book is to fill this gap and to show how recent methods of stochastic finance can be useful for to the risk management of pension funds. Methods of optimal control will be especially developed and applied to fundamental problems such as the optimal asset allocation of the fund or the cost spreading of a pension scheme. In these various problems, financial as well as demographic risks will be addressed and modelled.Table of ContentsPreface xiii Chapter 1. Introduction: Pensions in Perspective 1 1.1. Pension issues 1 1.2. Pension scheme 7 1.3. Pension and risks 11 1.4. The multi-pillar philosophy 14 Chapter 2. Classical Actuarial Theory of Pension Funding 15 2.1. General equilibrium equation of a pension scheme 15 2.2. General principles of funding mechanisms for DB Schemes 21 2.3. Particular funding methods 22 Chapter 3. Deterministic and Stochastic Optimal Control 31 3.1. Introduction 31 3.2. Deterministic optimal control 31 3.3. Necessary conditions for optimality 33 3.4. The maximum principle 42 3.5. Extension to the one-dimensional stochastic optimal control 45 3.6. Examples 52 Chapter 4. Defined Contribution and Defined Benefit Pension Plans 55 4.1. Introduction 55 4.2. The defined benefit method 56 4.3. The defined contribution method 57 4.4. The notional defined contribution (NDC) method 58 4.5. Conclusions 93 Chapter 5. Fair and Market Values and Interest Rate Stochastic Models 95 5.1. Fair value 95 5.2. Market value of financial flows 96 5.3. Yield curve 97 5.4. Yield to maturity for a financial investment and for a bond 99 5.5. Dynamic deterministic continuous time model for an instantaneous interest rate 100 5.6. Stochastic continuous time dynamic model for an instantaneous interest rate 104 5.7. Zero-coupon pricing under the assumption of no arbitrage 114 5.8. Market evaluation of financial flows 130 5.9. Stochastic continuous time dynamic model for asset values 132 5.10. VaR of one asset 136 Chapter 6. Risk Modeling and Solvency for Pension Funds 149 6.1. Introduction 149 6.2. Risks in defined contribution 149 6.3. Solvency modeling for a DC pension scheme 150 6.4. Risks in defined benefit 170 6.5. Solvency modeling for a DB pension scheme 171 Chapter 7. Optimal Control of a Defined Benefit Pension Scheme 181 7.1. Introduction 181 7.2. A first discrete time approach: stochastic amortization strategy 181 7.3. Optimal control of a pension fund in continuous time 194 Chapter 8. Optimal Control of a Defined Contribution Pension Scheme 207 8.1. Introduction 207 8.2. Stochastic optimal control of annuity contracts 208 8.3. Stochastic optimal control of DC schemes with guarantees and under stochastic interest rates 223 Chapter 9. Simulation Models 231 9.1. Introduction231 9.2. The direct method 233 9.3. The Monte Carlo models 250 9.4. Salary lines construction 252 Chapter 10. Discrete Time Semi-Markov Processes (SMP) and Reward SMP 277 10.1. Discrete time semi-Markov processes 277 10.2. DTSMP numerical solutions 280 10.3. Solution of DTHSMP and DTNHSMP in the transient case: a transportation example 284 10.4. Discrete time reward processes 294 10.5. General algorithms for DTSMRWP 304 Chapter 11. Generalized Semi-Markov Non-homogeneous Models for Pension Funds and Manpower Management 307 11.1. Application to pension funds evolution 307 11.2. Generalized non-homogeneous semi-Markov model for manpower management 338 11.3. Algorithms 347 APPENDICES 359 Appendix 1. Basic Probabilistic Tools for Stochastic Modeling 361 Appendix 2. Itô Calculus and Diffusion Processes 397 Bibliography 437 Index 449

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Book SynopsisThis book presents basic stochastic processes, stochastic calculus including Lévy processes on one hand, and Markov and Semi Markov models on the other. From the financial point of view, essential concepts such as the Black and Scholes model, VaR indicators, actuarial evaluation, market values, fair pricing play a central role and will be presented. The authors also present basic concepts so that this series is relatively self-contained for the main audience formed by actuaries and particularly with ERM (enterprise risk management) certificates, insurance risk managers, students in Master in mathematics or economics and people involved in Solvency II for insurance companies and in Basel II and III for banks.Table of ContentsINTRODUCTION xi CHAPTER 1. BASIC PROBABILISTIC TOOLS FOR STOCHASTIC MODELING 1 1.1. Probability space and random variables 1 1.2. Expectation and independence 4 1.3. Main distribution probabilities 7 1.3.1. Binomial distribution 7 1.3.2. Negative exponential distribution 8 1.3.3. Normal (or Laplace–Gauss) distribution 8 1.3.4. Poisson distribution 11 1.3.5. Lognormal distribution 11 1.3.6. Gamma distribution 12 1.3.7. Pareto distribution 13 1.3.8. Uniform distribution 16 1.3.9. Gumbel distribution 16 1.3.10. Weibull distribution 16 1.3.11. Multi-dimensional normal distribution 17 1.3.12. Extreme value distribution 19 1.4. The normal power (NP) approximation 28 1.5. Conditioning 31 1.6. Stochastic processes 39 1.7. Martingales 43 CHAPTER 2. HOMOGENEOUS AND NON-HOMOGENEOUS RENEWAL MODELS 47 2.1. Introduction 47 2.2. Continuous time non-homogeneous convolutions 49 2.2.1. Non-homogeneous convolution product 49 2.3. Homogeneous and non-homogeneous renewal processes 53 2.4. Counting processes and renewal functions 56 2.5. Asymptotical results in the homogeneous case 61 2.6. Recurrence times in the homogeneous case 63 2.7. Particular case: the Poisson process 66 2.7.1. Homogeneous case 66 2.7.2. Non-homogeneous case 68 2.8. Homogeneous alternating renewal processes 69 2.9. Solution of non-homogeneous discrete timevevolution equation 71 2.9.1. General method 71 2.9.2. Some particular formulas 73 2.9.3. Relations between discrete time and continuous time renewal equations 74 CHAPTER 3. MARKOV CHAINS 77 3.1. Definitions 77 3.2. Homogeneous case 78 3.2.1. Basic definitions 78 3.2.2. Markov chain state classification 81 3.2.3. Computation of absorption probabilities 87 3.2.4. Asymptotic behavior 88 3.2.5. Example: a management problem in an insurance company 93 3.3. Non-homogeneous Markov chains 95 3.3.1. Definitions 95 3.3.2. Asymptotical results 98 3.4. Markov reward processes 99 3.4.1. Classification and notation 99 3.5. Discrete time Markov reward processes (DTMRWPs) 102 3.5.1. Undiscounted case 102 3.5.2. Discounted case 105 3.6. General algorithms for the DTMRWP 111 3.6.1. Homogeneous MRWP 112 3.6.2. Non-homogeneous MRWP 112 CHAPTER 4. HOMOGENEOUS AND NON-HOMOGENEOUS SEMI-MARKOV MODELS 113 4.1. Continuous time semi-Markov processes 113 4.2. The embedded Markov chain 117 4.3. The counting processes and the associated semi-Markov process 118 4.4. Initial backward recurrence times 120 4.5. Particular cases of MRP 122 4.5.1. Renewal processes and Markov chains 122 4.5.2. MRP of zero-order (PYKE (1962)) 122 4.5.3. Continuous Markov processes 124 4.6. Examples 124 4.7. Discrete time homogeneous and non-homogeneous semi-Markov processes 127 4.8. Semi-Markov backward processes in discrete time 129 4.8.1. Definition in the homogeneous case 129 4.8.2. Semi-Markov backward processes in discrete time for the non-homogeneous case 130 4.8.3. DTSMP numerical solutions 133 4.9. Discrete time reward processes 137 4.9.1. Undiscounted SMRWP 137 4.9.2. Discounted SMRWP 141 4.9.3. General algorithms for DTSMRWP 144 4.10. Markov renewal functions in the homogeneous case 146 4.10.1. Entrance times 146 4.10.2. The Markov renewal equation 150 4.10.3. Asymptotic behavior of an MRP 151 4.10.4. Asymptotic behavior of SMP 153 4.11. Markov renewal equations for the non-homogeneous case 158 4.11.1. Entrance time 158 4.11.2. The Markov renewal equation 162 CHAPTER 5. STOCHASTIC CALCULUS 165 5.1. Brownian motion 165 5.2. General definition of the stochastic integral 167 5.2.1. Problem of stochastic integration 167 5.2.2. Stochastic integration of simple predictable processes and semi-martingales 168 5.2.3. General definition of the stochastic integral 170 5.3. Itô’s formula 177 5.3.1. Quadratic variation of a semi-martingale 177 5.3.2. Itô’s formula 179 5.4. Stochastic integral with standard Brownian motion as an integrator process 180 5.4.1. Case of simple predictable processes 181 5.4.2. Extension to general integrator processes 183 5.5. Stochastic differentiation 184 5.5.1. Stochastic differential 184 5.5.2. Particular cases 184 5.5.3. Other forms of Itô’s formula 185 5.6. Stochastic differential equations 191 5.6.1. Existence and unicity general theorem 191 5.6.2. Solution of stochastic differential equations 195 5.6.3. Diffusion processes 199 5.7. Multidimensional diffusion processes 202 5.7.1. Definition of multidimensional Itô and diffusion processes 203 5.7.2. Properties of multidimensional diffusion processes 203 5.7.3. Kolmogorov equations 205 5.7.4. The Stroock–Varadhan martingale characterization of diffusion processes 208 5.8. Relation between the resolution of PDE and SDE problems. The Feynman–Kac formula 209 5.8.1. Terminal payoff 209 5.8.2. Discounted payoff function 210 5.8.3. Discounted payoff function and payoff rate 210 5.9. Application to option theory 213 5.9.1. Options 213 5.9.2. Black and Scholes model 216 5.9.3. The Black and Scholes partial differential equation (BSPDE) and the BS formula 216 5.9.4. Girsanov theorem 219 5.9.5. The risk-neutral measure and the martingale property 221 5.9.6. The risk-neutral measure and the evaluation of derivative products 224 CHAPTER 6. LÉVY PROCESSES 227 6.1. Notion of characteristic functions 227 6.2. Lévy processes 228 6.3. Lévy–Khintchine formula 230 6.4. Subordinators 234 6.5. Poisson measure for jumps 234 6.5.1. The Poisson random measure 234 6.5.2. The compensated Poisson process 235 6.5.3. Jump measure of a Lévy process 236 6.5.4. The Itô–Lévy decomposition 236 6.6. Markov and martingale properties of Lévy processes 237 6.6.1. Markov property 237 6.6.2. Martingale properties 239 6.6.3. Itô formula 240 6.7. Examples of Lévy processes 240 6.7.1. The lognormal process: Black and Scholes process 240 6.7.2. The Poisson process 241 6.7.3. Compensated Poisson process 242 6.7.4. The compound Poisson process 242 6.8. Variance gamma (VG) process 244 6.8.1. The gamma distribution 244 6.8.2. The VG distribution 245 6.8.3. The VG process 246 6.8.4. The Esscher transformation 247 6.8.5. The Carr–Madan formula for the European call 249 6.9. Hyperbolic Lévy processes 250 6.10. The Esscher transformation 252 6.10.1. Definition 252 6.10.2. Option theory with hyperbolic Lévy processes 253 6.10.3. Value of the European option call 255 6.11. The Brownian–Poisson model with jumps 256 6.11.1. Mixed arithmetic Brownian–Poisson and geometric Brownian–Poisson processes 256 6.11.2. Merton model with jumps 258 6.11.3. Stochastic differential equation (SDE) for mixed arithmetic Brownian–Poisson and geometric Brownian–Poisson processes 261 6.11.4. Value of a European call for the lognormal Merton model 264 6.12. Complete and incomplete markets 264 6.13. Conclusion 265 CHAPTER 7. ACTUARIAL EVALUATION, VAR AND STOCHASTIC INTEREST RATE MODELS 267 7.1. VaR technique 267 7.2. Conditional VaR value 271 7.3. Solvency II 276 7.3.1. The SCR indicator 276 7.3.2. Calculation of MCR 278 7.3.3. ORSA approach 279 7.4. Fair value 280 7.4.1. Definition 280 7.4.2. Market value of financial flows 281 7.4.3. Yield curve 281 7.4.4. Yield to maturity for a financial investment and a bond 283 7.5. Dynamic stochastic time continuous time model for instantaneous interest rate 284 7.5.1. Instantaneous deterministic interest rate 284 7.5.2. Yield curve associated with a deterministic instantaneous interest rate 285 7.5.3. Dynamic stochastic continuous time model for instantaneous interest rate 286 7.5.4. The OUV stochastic model 287 7.5.5. The CIR model 289 7.6. Zero-coupon pricing under the assumption of no arbitrage 292 7.6.1. Stochastic dynamics of zero-coupons 292 7.6.2. The CIR process as rate dynamic 295 7.7. Market evaluation of financial flows 298 BIBLIOGRAPHY 301 INDEX 309

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Book SynopsisThe aim of this book is to promote interaction between engineering, finance and insurance, as these three domains have many models and methods of solution in common for solving real-life problems. The authors point out the strict inter-relations that exist among the diffusion models used in engineering, finance and insurance. In each of the three fields, the basic diffusion models are presented and their strong similarities are discussed. Analytical, numerical and Monte Carlo simulation methods are explained with a view to applying them to obtain the solutions to the different problems presented in the book. Advanced topics such as nonlinear problems, Lévy processes and semi-Markov models in interactions with the diffusion models are discussed, as well as possible future interactions among engineering, finance and insurance. Contents 1. Diffusion Phenomena and Models.2. Probabilistic Models of Diffusion Processes.3. Solving Partial Differential Equations of Second Order.4. Problems in Finance.5. Basic PDE in Finance.6. Exotic and American Options Pricing Theory.7. Hitting Times for Diffusion Processes and Stochastic Models in Insurance.8. Numerical Methods.9. Advanced Topics in Engineering: Nonlinear Models.10. Lévy Processes.11. Advanced Topics in Insurance: Copula Models and VaR Techniques.12. Advanced Topics in Finance: Semi-Markov Models.13. Monte Carlo Semi-Markov Simulation Methods.Table of ContentsIntroduction xiii Chapter 1 Diffusion Phenomena and Models 1 1.1 General presentation of diffusion process 1 1.2 General balance equations 6 1.3 Heat conduction equation 10 1.4 Initial and boundary conditions 12 Chapter 2 Probabilistic Models of Diffusion Processes 17 2.1 Stochastic differentiation 17 2.2 Itô’s formula 19 2.3 Stochastic differential equations (SDE) 24 2.4 Itô and diffusion processes 28 2.5 Some particular cases of diffusion processes 32 2.6 Multidimensional diffusion processes 36 2.7 The Stroock–Varadhan martingale characterization of diffusions (Karlin and Taylor) 41 2.8 The Feynman–Kac formula (Platen and Heath) 42 Chapter 3 Solving Partial Differential Equations of Second Order 47 3.1 Basic definitions on PDE of second order 47 3.2 Solving the heat equation 51 3.3 Solution by the method of Laplace transform 65 3.4 Green’s functions 75 Chapter 4 Problems in Finance 85 4.1 Basic stochastic models for stock prices 85 4.2 The bond investments 90 4.3 Dynamic deterministic continuous time model for instantaneous interest rate 93 4.4 Stochastic continuous time dynamic model for instantaneous interest rate 98 4.5 Multidimensional Black and Scholes model 110 Chapter 5 Basic PDE in Finance 111 5.1 Introduction to option theory 111 5.2 Pricing the plain vanilla call with the Black–Scholes–Samuelson model 115 5.3 Pricing no plain vanilla calls with the Black-Scholes-Samuelson model 120 5.4 Zero-coupon pricing under the assumption of no arbitrage 127 Chapter 6 Exotic and American Options Pricing Theory 145 6.1 Introduction 145 6.2 The Garman–Kohlhagen formula 146 6.3 Binary or digital options 149 6.4 “Asset or nothing” options 150 6.5 Numerical examples 152 6.6 Path-dependent options 153 6.7 Multi-asset options 157 6.8 American options 165 Chapter 7 Hitting Times for Diffusion Processes and Stochastic Models in Insurance 177 7.1 Hitting or first passage times for some diffusion processes 177 7.2 Merton’s model for default risk 193 7.3 Risk diffusion models for insurance 201 Chapter 8 Numerical Methods 219 8.1 Introduction 219 8.2 Discretization and numerical differentiation 220 8.3 Finite difference methods 222 9.1 Nonlinear model in heat conduction 232 Chapter 9 Advanced Topics in Engineering: Nonlinear Models 231 9.2 Integral method applied to diffusive problems 233 9.3 Integral method applied to nonlinear problems 239 9.4 Use of transformations in nonlinear problems 243 Chapter 10 Lévy Processes 255 10.1 Motivation 255 10.2 Notion of characteristic functions 257 10.3 Lévy processes 257 10.4 Lévy–Khintchine formula 259 10.5 Examples of Lévy processes 261 10.6 Variance gamma (VG) process 264 10.7 The Brownian–Poisson model with jumps 266 10.8 Risk neutral measures for Lévy models in finance 275 10.9 Conclusion 276 Chapter 11 Advanced Topics in Insurance: Copula Models and VaR Techniques 277 11.1 Introduction 277 11.2 Sklar theorem (1959) 279 11.3 Particular cases and Fréchet bounds 280 11.4 Dependence 288 11.5 Applications in finance: pricing of the bivariate digital put option 293 11.6 VaR application in insurance 296 Chapter 12 Advanced Topics in Finance: Semi-Markov Models 307 12.1 Introduction 307 12.2 Homogeneous semi-Markov process 308 12.3 Semi-Markov option model 328 12.4 Semi-Markov VaR models 332 12.5 Conclusion 339 Chapter 13 Monte Carlo Semi-Markov Simulation Methods 341 13.1 Presentation of our simulation model 341 13.2 The semi-Markov Monte Carlo model in a homogeneous environment 345 13.3 A credit risk example 350 13.4 Semi-Markov Monte Carlo with initial recurrence backward time in homogeneous case 362 13.5 The SMMC applied to claim reserving problem 363 13.6 An example of claim reserving calculation 366 Conclusion 379 Bibliography 381 Index 393

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Book SynopsisWith the impact of the recent financial crises, more attention must be given to new models in finance rejecting “Black-Scholes-Samuelson” assumptions leading to what is called non-Gaussian finance. With the growing importance of Solvency II, Basel II and III regulatory rules for insurance companies and banks, value at risk (VaR) – one of the most popular risk indicator techniques plays a fundamental role in defining appropriate levels of equities. The aim of this book is to show how new VaR techniques can be built more appropriately for a crisis situation.VaR methodology for non-Gaussian finance looks at the importance of VaR in standard international rules for banks and insurance companies; gives the first non-Gaussian extensions of VaR and applies several basic statistical theories to extend classical results of VaR techniques such as the NP approximation, the Cornish-Fisher approximation, extreme and a Pareto distribution. Several non-Gaussian models using Copula methodology, Lévy processes along with particular attention to models with jumps such as the Merton model are presented; as are the consideration of time homogeneous and non-homogeneous Markov and semi-Markov processes and for each of these models. Contents 1. Use of Value-at-Risk (VaR) Techniques for Solvency II, Basel II and III.2. Classical Value-at-Risk (VaR) Methods.3. VaR Extensions from Gaussian Finance to Non-Gaussian Finance.4. New VaR Methods of Non-Gaussian Finance.5. Non-Gaussian Finance: Semi-Markov Models.Table of ContentsINTRODUCTION ix CHAPTER 1. USE OF VALUE-AT-RISK (VAR) TECHNIQUES FOR SOLVENCY II, BASEL II AND III 1 1.1. Basic notions of VaR 1 1.2. The use of VaR for insurance companies 6 1.3. The use of VaR for banks 13 1.4. Conclusion 16 CHAPTER 2. CLASSICAL VALUE-AT-RISK (VAR) METHODS 17 2.1. Introduction 17 2.2. Risk measures 18 2.3. General form of the VaR 19 2.4. VaR extensions: tail VaR and conditional VaR 25 2.5. VaR of an asset portfolio 28 2.6. A simulation example: the rates of investment of assets 32 CHAPTER 3. VAR EXTENSIONS FROM GAUSSIAN FINANCE TO NON-GAUSSIAN FINANCE 35 3.1. Motivation 35 3.2. The normal power approximation 37 3.3. VaR computation with extreme values 40 3.4. VaR value for a risk with Pareto distribution 56 3.5. Conclusion 62 CHAPTER 4. NEW VAR METHODS OF NON-GAUSSIAN FINANCE 63 4.1. Lévy processes 63 model with jumps 76 4.2. Copula models and VaR techniques 90 4.3. VaR for insurance 109 CHAPTER 5. NON-GAUSSIAN FINANCE: SEMI-MARKOV MODELS 115 5.1. Introduction 115 5.2. Homogeneous semi-Markov process 116 5.3. Semi-Markov option model 139 5.4. Semi-Markov VaR models 143 5.5. The Semi-Markov Monte Carlo Model in a homogeneous environment 147 CONCLUSION 159 BIBLIOGRAPHY 161 INDEX 165

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