The 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.
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.
About the Authors
Jacques Janssen is now Honorary Professor at the Solvay Business School (ULB) in Brussels, Belgium, having previously taught at EURIA (Euro–Institut d Actuariat, University of West Brittany, Brest, France) and Télécom–Bretagne (Brest, France) as well as being a director of Jacan Insurance and Finance Services, a consultancy and training company.
Oronzio Manca is Professor of thermal sciences at Seconda Università degli Studi di Napoli in Italy. He is currently Associate Editor of ASME Journal of Heat Transfer and Journal of Porous Media and a member of the editorial advisory boards for The Open Thermodynamics Journal, Advances in Mechanical Engineering, The Open Fuels & Energy Science Journal.
Raimondo Manca is Professor of mathematical methods applied to economics, finance and actuarial science at University of Rome La Sapienza in Italy. He is associate editor for the journal Methodology and Computing in Applied Probability. His main research interests are multidimensional linear algebra, computational probability, application of stochastic processes to economics, finance and insurance and simulation models.
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