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.
Introduction xiii
Chapter 1 Diffusion Phenomena and Models 1 (16)
1.1 General presentation of diffusion 1 (5)
process
1.2 General balance equations 6 (4)
1.3 Heat conduction equation 10 (2)
1.4 Initial and boundary conditions 12 (5)
Chapter 2 Probabilistic Models of Diffusion 17 (30)
Processes
2.1 Stochastic differentiation 17 (2)
2.1.1 Definition 17 (1)
2.1.2 Examples 18 (1)
2.2 Ito's formula 19 (5)
2.2.1 Stochastic differential of a 19 (1)
product
2.2.2 Ito's formula with time dependence 19 (2)
2.2.3 Interpretation of Ito's formula 21 (1)
2.2.4 Other extensions of Ito's formula 21 (3)
2.3 Stochastic differential equations 24 (4)
(SDE)
2.3.1 Existence and unicity general 24 (4)
theorem (Gikhman and Skorokhod)
2.3.2 Solution of SDE under the 28 (1)
canonical form
2.4 Ito and diffusion processes 28 (4)
2.4.1 Ito processes 28 (1)
2.4.2 Diffusion processes 29 (2)
2.4.3 Kolmogorov equations 31 (1)
2.5 Some particular cases of diffusion 32 (4)
processes
2.5.1 Reduced form 32 (1)
2.5.2 The OUV 32 (2)
(Ornstein-Uhlenbeck-Vasicek) SDE
2.5.3 Solution of the SDE of 34 (2)
Black-Scholes-Samuelson
2.6 Multidimensional diffusion processes 36 (5)
2.6.1 Multidimensional SDE 36 (1)
2.6.2 Multidimensional Ito and 36 (1)
diffusion processes
2.6.3 Properties of multidimensional 37 (1)
diffusion processes
2.6.4 Kolmogorov equations 38 (3)
2.7 The Stroock-Varadhan martingale 41 (1)
characterization of diffusions (Karlin
and Taylor)
2.8 The Feynman-Kac formula (Platen and 42 (5)
Heath)
2.8.1 Terminal condition 42 (1)
2.8.2 Discounted payoff function 43 (1)
2.8.3 Discounted payoff function and 44 (3)
payoff rate
Chapter 3 Solving Partial Differential 47 (38)
Equations of Second Order
3.1 Basic definitions on PDE of second 47 (4)
order
3.1.1 Notation 47 (1)
3.1.2 Characteristics 48 (2)
3.1.3 Canonical form of PDE 50 (1)
3.2 Solving the heat equation 51 (14)
3.2.1 Separation of variables 54 (1)
3.2.2 Separation of variables in the 55 (2)
rectangular Cartesian coordinates
3.2.3 Orthogonality of functions 57 (1)
3.2.4 Fourier series 58 (1)
3.2.5 Sturm-Liouville problem 59 (2)
3.2.6 One-dimensional homogeneous 61 (4)
problem in a finite medium
3.3 Solution by the method of Laplace 65 (10)
transform
3.3.1 Definition of the Laplace 65 (2)
transform
3.3.2 Properties of the Laplace 67 (8)
transform
3.4 Green's functions 75 (10)
3.4.1 Green's function as auxiliary 76 (2)
problem to solve diffusive problems
3.4.2 Analysis for determination of 78 (7)
Green's function
Chapter 4 Problems in Finance 85 (26)
4.1 Basic stochastic models for stock 85 (5)
prices
4.1.1 The Black, Scholes and Samuelson 85 (4)
model
4.1.2 BSS model with deterministic 89 (1)
variation of μ and σ
4.2 The bond investments 90 (3)
4.2.1 Introduction 90 (1)
4.2.2 Yield curve 91 (1)
4.2.3 Yield to maturity for a financial 92 (1)
investment and for a bond
4.3 Dynamic deterministic continuous time 93 (5)
model for instantaneous interest rate
4.3.1 Instantaneous interest rate 93 (1)
4.3.2 Particular cases 94 (1)
4.3.3 Yield curve associated with 94 (1)
instantaneous interest rate
4.3.4 Examples of theoretical models 95 (3)
4.4 Stochastic continuous time dynamic 98 (12)
model for instantaneous interest rate
4.4.1 The OUV stochastic model 99 (6)
4.4.2 The CIR model (1985) 105 (5)
4.5 Multidimensional Black and Scholes 110 (1)
model
Chapter 5 Basic PDE in Finance 111 (34)
5.1 Introduction to option theory 111 (4)
5.2 Pricing the plain vanilla call with 115 (5)
the Black-Scholes-Samuelson model
5.2.1 The PDE of Black and Scholes 115 (3)
5.2.2 The resolution of the PDE of 118 (2)
Black and Scholes without dividend
repartition
5.3 Pricing no plain vanilla calls with 120 (7)
the Black-Scholes-Samuelson model
5.3.1 The Girsanov theorem 121 (1)
5.3.2 Application to the 122 (5)
Black-Scholes-Samuelson model
5.4 Zero-coupon pricing under the 127 (18)
assumption of no arbitrage
5.4.1 Stochastic dynamics of zero 127 (3)
coupons
5.4.2 Application of the no arbitrage 130 (1)
principle and risk premium
5.4.3 Partial differential equation for 131 (3)
the structure of zero coupons
5.4.4 Values of zero coupons without 134 (7)
arbitrage opportunity for particular
cases
5.4.5 Value of a call on zero coupon 141 (1)
5.4.6 Option on bond with coupons 142 (2)
5.4.7 A numerical example 144 (1)
Chapter 6 Exotic and American Options 145 (32)
Pricing Theory
6.1 Introduction 145 (1)
6.2 The Garman-Kohlhagen formula 146 (3)
6.3 Binary or digital options 149 (1)
6.3.1 Definition 149 (1)
6.3.2 Pricing of a call cash or nothing 149 (1)
6.3.3 Case of the put cash or nothing 150 (1)
6.4 "Asset or nothing" options 150 (2)
6.4.1 Definition 150 (1)
6.4.2 Pricing a call asset or nothing 151 (1)
6.4.3 Premium of the put asset or 151 (1)
nothing
6.5 Numerical examples 152 (1)
6.6 Path-dependent options 153 (4)
6.6.1 The barrier options 153 (3)
6.6.2 Lookback options 156 (1)
6.6.3 Asiatic (or average) options 157 (1)
6.7 Multi-asset options 157 (8)
6.7.1 Definitions 157 (1)
6.7.2 The multi-dimensional Black and 158 (1)
Scholes equation
6.7.3 Outperformance or Margrabe option 159 (2)
6.7.4 Other related type options 161 (2)
6.7.5 General case 163 (2)
6.8 American options 165 (12)
6.8.1 Early exercise in case of no 165 (1)
dividend repartition
6.8.2 Early exercise in case of 165 (2)
dividend repartition
6.8.3 The formula of Barone-Adesi and 167 (7)
Whaley (BAW): approximated formula for
American options
6.8.4 Discretization and simulation 174 (3)
Chapter 7 Hitting Times for Diffusion 177 (42)
Processes and Stochastic Models in Insurance
7.1 Hitting or first passage times for 177 (16)
some diffusion processes
7.1.1 First definitions 177 (2)
7.1.2 Distribution of hitting times for 179 (2)
the non-standard Brownian motion
7.1.3 The Gaussian inverse (or normal 181 (2)
inverse) distribution
7.1.4 Other absorption problems for 183 (3)
Brownian motion
7.1.5 Other absorption problems for 186 (1)
non-standard Brownian processes
7.1.6 Results with probabilistic 187 (6)
reasoning
7.2 Merton's model for default risk 193 (8)
7.2.1 Introduction 193 (1)
7.2.2 Merton's model 194 (5)
7.2.3 The Longstaff and Schwartz model 199 (2)
7.3 Risk diffusion models for insurance 201 (18)
7.3.1 Introduction 201 (1)
7.3.2 The diffusion process (Cox and 202 (3)
Miller, Gerber)
7.3.3 First ALM model (ALM I) (Janssen) 205 (7)
7.3.4 Second ALM model (ALM II) 212 (7)
(Janssen)
Chapter 8 Numerical Methods 219 (12)
8.1 Introduction 219 (1)
8.2 Discretization and numerical 220 (2)
differentiation
8.3 Finite difference methods 222 (9)
Chapter 9 Advanced Topics in Engineering: 231 (24)
Nonlinear Models
9.1 Nonlinear model in heat conduction 232 (1)
9.2 Integral method applied to diffusive 233 (6)
problems
9.3 Integral method applied to nonlinear 239 (4)
problems
9.4 Use of transformations in nonlinear 243 (12)
problems
9.4.1 Kirchhoff transformation 243 (3)
9.4.2 Similarity methods 246 (9)
Chapter 10 Levy Processes 255 (22)
10.1 Motivation 255 (2)
10.2 Notion of characteristic functions 257 (1)
10.3 Levy processes 257 (2)
10.4 Levy-Khintchine formula 259 (2)
10.5 Examples of Levy processes 261 (3)
10.6 Variance gamma (VG) process 264 (2)
10.7 The Brownian-Poisson model with jumps 266 (9)
10.7.1 Mixed arithmetic 266 (3)
Brownian-Poisson and geometric
Brownian-Poisson processes
10.7.2 Merton model with jumps 269 (2)
10.7.3 Stochastic differential equation 271 (3)
(SDE) for mixed arithmetic
Brownian-Poisson and geometric
Brownian-Poisson processes
10.7.4 Value of a European call for the 274 (1)
lognormal Merton model
10.8 Risk neutral measures for Levy 275 (1)
models in finance
10.9 Conclusion 276 (1)
Chapter 11 Advanced Topics in Insurance: 277 (30)
Copula Models and VaR Techniques
11.1 Introduction 277 (2)
11.2 Sklar theorem (1959) 279 (1)
11.3 Particular cases and Frechet bounds 280 (8)
11.3.1 Particular cases 280 (1)
11.3.2 Frechet bounds 281 (1)
11.3.3 Examples of copula 281 (4)
11.3.4 The normal copula 285 (2)
11.3.5 Estimation of copula 287 (1)
11.4 Dependence 288 (5)
11.4.1 Conditional probabilities 288 (1)
11.4.2 The correlation coefficient 289 (4)
τ of Kendall
11.5 Applications in finance: pricing of 293 (3)
the bivariate digital put option
11.6 VaR application in insurance 296 (11)
11.6.1 VaR of one risky asset 296 (7)
11.6.2 The VaR concept in relation with 303 (4)
Solvency II
Chapter 12 Advanced Topics in Finance: 307 (34)
Semi-Markov Models
12.1 Introduction 307 (1)
12.2 Homogeneous semi-Markov process 308 (20)
12.2.1 Basic definitions 308 (2)
12.2.2 Basic properties 310 (4)
12.2.3 Particular cases of MRP 314 (3)
12.2.4 Asymptotic behavior of SMP 317 (1)
12.2.5 Non-homogeneous semi-Markov 318 (2)
process
12.2.6 Discrete time homogeneous and 320 (3)
non-homogeneous semi-Markov processes
12.2.7 Homogeneous semi-Markov backward 323 (2)
processes in discrete time
12.2.8 Discrete time non-homogeneous 325 (3)
backward semi-Markov processes
12.3 Semi-Markov option model 328 (4)
12.3.1 General model 328 (2)
12.3.2 Particular case: semi-Markov 330 (1)
Black-Scholes model
12.3.3 Numerical application for the 330 (2)
semi-Markov Black-Scholes model
12.4 Semi-Markov VaR models 332 (7)
12.4.1 The normal power (NP) 332 (1)
approximation
12.4.2 The Cornish-Fisher approximation 333 (1)
12.4.3 VaR computation with a Pareto 333 (2)
distribution
12.4.4 VaR semi-Markov models 335 (1)
12.4.5 Numerical applications for the 336 (2)
semi-Markov VaR model
12.4.6 Semi-Markov extension of the 338 (1)
Merton's model
12.5 Conclusion 339 (2)
Chapter 13 Monte Carlo Semi-Markov 341 (38)
Simulation Methods
13.1 Presentation of our simulation model 341 (4)
13.2 The semi-Markov Monte Carlo model in 345 (5)
a homogeneous environment
13.3 A credit risk example 350 (12)
13.3.1 Discrete time homogeneous 350 (2)
semi-Markov reliability model
13.3.2 A classical example of 352 (3)
reliability for a mechanical system
13.3.3 The semi-Markov reliability 355 (3)
credit risk models
13.3.4 A simplified example 358 (4)
13.4 Semi-Markov Monte Carlo with initial 362 (1)
recurrence backward time in homogeneous
case
13.5 The SMMC applied to claim reserving 363 (3)
problem
13.6 An example of claim reserving 366 (13)
calculation
13.6.1 Example of the merging process 368 (11)
Conclusion 379 (2)
Bibliography 381 (12)
Index 393
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