139,99 €
Mehr erfahren.
- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
This 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.
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 279
Veröffentlichungsjahr: 2015
Ähnliche
Table of Contents
Cover
Title
Copyright
Introduction
1: Basic Probabilistic Tools for Stochastic Modeling
1.1. Probability space and random variables
1.2. Expectation and independence
1.3. Main distribution probabilities
1.4. The normal power (NP) approximation
1.5. Conditioning
1.6. Stochastic processes
1.7. Martingales
2: Homogeneous and Non-homogeneous Renewal Models
2.1. Introduction
2.2. Continuous time non-homogeneous convolutions
2.3. Homogeneous and non-homogeneous renewal processes
2.4. Counting processes and renewal functions
2.5. Asymptotical results in the homogeneous case
2.6. Recurrence times in the homogeneous case
2.7. Particular case: the Poisson process
2.8. Homogeneous alternating renewal processes
2.9. Solution of non-homogeneous discrete time evolution equation
3: Markov Chains
3.1. Definitions
3.2. Homogeneous case
3.3. Non-homogeneous Markov chains
3.4. Markov reward processes
3.5. Discrete time Markov reward processes (DTMRWPs)
3.6. General algorithms for the DTMRWP
4: Homogeneous and Non-homogeneous Semi-Markov Models
4.1. Continuous time semi-Markov processes
4.2. The embedded Markov chain
4.3. The counting processes and the associated semi-Markov process
4.4. Initial backward recurrence times
4.5. Particular cases of MRP
4.6. Examples
4.7. Discrete time homogeneous and non-homogeneous semi-Markov processes
4.8. Semi-Markov backward processes in discrete time
4.9. Discrete time reward processes
4.10. Markov renewal functions in the homogeneous case
4.11. Markov renewal equations for the non-homogeneous case
5: Stochastic Calculus
5.1. Brownian motion
5.2. General definition of the stochastic integral
5.3. Itô’s formula
5.4. Stochastic integral with standard Brownian motion as an integrator process
5.5. Stochastic differentiation
5.6. Stochastic differential equations
5.7. Multidimensional diffusion processes
5.8. Relation between the resolution of PDE and SDE problems. The Feynman–Kac formula [PLA 06]
5.9. Application to option theory
6: Lévy Processes
6.1. Notion of characteristic functions
6.2. Lévy processes
6.3. Lévy–Khintchine formula
6.4. Subordinators
6.5. Poisson measure for jumps
6.6. Markov and martingale properties of Lévy processes
6.7. Examples of Lévy processes
6.8. Variance gamma (VG) process
6.9. Hyperbolic Lévy processes
6.10. The Esscher transformation
6.11. The Brownian–Poisson model with jumps
6.12. Complete and incomplete markets
6.13. Conclusion
7: Actuarial Evaluation, VaR and Stochastic Interest Rate Models
7.1. VaR technique
7.2. Conditional VaR value
7.3. Solvency II
7.4. Fair value
7.5. Dynamic stochastic time continuous time model for instantaneous interest rate
7.6. Zero-coupon pricing under the assumption of no arbitrage
7.7. Market evaluation of financial flows
Bibliography
Index
End User License Agreement
List of Tables
1: Basic Probabilistic Tools for Stochastic Modeling
Table 1.1. Impact of dissymmetry between z
α
and λ
α
5: Stochastic Calculus
Table 5.1. Example of application of the BS formula
6: Lévy Processes
Table 6.1. Characteristic functions of some distributions
7: Actuarial Evaluation, VaR and Stochastic Interest Rate Models
Table 7.1. VaR values for one asset with the normal distribution
Table 7.2. VaR values for one asset with the truncated normal distribution
Table 7.3. VaR values for a protection business portfolio with the normal distribution
Table 7.4. VaR values, for example
Table 7.5. Corresponding VaR values with a Gaussian model
Table 7.6. Ratio VaR – mean for α=0.995
Table 7.7. Some CVaR values
Table 7.8. Zero-coupons values with OUV
Table 7.9. Zero-coupon values with CIR
Table 7.10. Spread OUV and CIR for the four scenarios
Table 7.11. Present values with the four CIR scenarios
Table 7.12. PV with flat yield curves
Table 7.13. Evaluation comparison
List of Illustrations
1: Basic Probabilistic Tools for Stochastic Modeling
Figure 1.1. Pareto distribution function with Θ=1,β=1
Figure 1.2. Pareto distribution with β=3,θ=1
Figure 1.3. Graph of the Gumbel distribution
Figure 1.4. Graph of the Fréchet distribution (β=2,1)
Figure 1.5. Graph of the Weibull distribution (β=2,1)
2: Homogeneous and Non-homogeneous Renewal Models
Figure 2.1. Sample path of a counting process
Figure 2.2. Backward (age) and forward (excess) recurrence times and spread
Figure 2.3. Age events analysis
Figure 2.4. Spread events analysis
3: Markov Chains
Figure 3.1. Graph associated to matrix [3.21]
Figure 3.2. Graph associated to matrix [3.23]
Figure 3.3. Graph of the states
4: Homogeneous and Non-homogeneous Semi-Markov Models
Figure 4.1. A trajectory of a SMP
Figure 4.2. Homogeneous SMP backward time
Figure 4.3. Non-homogeneous SMP backward time
Figure 4.4. Non-homogeneous semi-Markov backward trajectory
Figure 4.5. HSMP with backward time trajectory
Figure 4.6. Backward time axis
5: Stochastic Calculus
Figure 5.1. Example 1 of a trajectory of a standard Brownian motion
Figure 5.2. Example 2 of a trajectory of a standard Brownian motion
Figure 5.3. Holder’s gain at maturity for call option
Figure 5.4. Holder’s gain at maturity for put option
Figure 5.5. Net gains at maturity for the holder of call and put options ns
6: Lévy Processes
Figure 6.1. A possible sample path of the process N
Figure 6.2. A sample path of the α process with occurrence of the ruin event
Figure 6.3. Evolution (end of 2004) of the title of Mercer, stock (source: http://www.yahoo.fr.)
7: Actuarial Evaluation, VaR and Stochastic Interest Rate Models
Figure 7.1. Risk classification (source: QIS5 technical specifications)
Figure 7.2. The two levels of capital requirements in the Solvency II framework
Figure 7.3. Yield curve in November 2014 in year maturities up to May 2020 (source: IA, France)
Figure 7.4. A sample path for an OUV mode
Figure 7.5. A sample path for a CIR model
Figure 7.6. Yield curve with the scenario CIR IV
Guide
Cover
Table of Contents
Begin Reading
Pages
cover
Contents
iii
iv
xi
xii
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
301
302
303
304
305
306
307
309
310
311
312
313
Basic Stochastic Processes
Pierre Devolder
Jacques Janssen
Raimondo Manca
Jacques Janssen
First published 2015 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd
27-37 St George’s Road
London SW19 4EU
UK
www.iste.co.uk
John Wiley & Sons, Inc.
111 River Street
Hoboken, NJ 07030
USA
www.wiley.com
© ISTE Ltd 2015
The rights of Pierre Devolder, Jacques Janssen and Raimondo Manca to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Control Number: 2015942727
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-84821-882-6
Introduction
This book will present basic stochastic processes for building models in insurance, especially in life and non-life insurance as well as credit risk for insurance companies. Of course, stochastic methods are quite numerous; so we have deliberately chosen to consider to use those induced by two big families of stochastic processes: stochastic calculus including Lévy processes and Markov and semi-Markov models. From the financial point of view, essential concepts such as the Black and Scholes model, VaR indicators, actuarial evaluation, market values and fair pricing play a key role, and they will be presented in this volume.
This book is organized into seven chapters. Chapter 1 presents the essential probability tools for the understanding of stochastic models in insurance. The next three chapters are, respectively, devoted to renewal processes (Chapter 2), Markov chains (Chapter 3) and semi-Markov processes both homogeneous and non-time homogeneous (Chapter 4) in time. This fact is important as new non-homogeneous time models are now becoming more and more used to build realistic models for insurance problems.
Chapter 5 gives the bases of stochastic calculus including stochastic differential equations, diffusion processes and changes of probability measures, therefore giving results that will be used in Chapter 6 devoted to Lévy processes. Chapter 6 is devoted to Lévy processes. This chapter also presents an alternative to basic stochastic models using Brownian motion as Lévy processes keep the properties of independent and stationary increments but without the normality assumption.
Finally, Chapter 7 presents a summary of Solvency II rules, actuarial evaluation, using stochastic instantaneous interest rate models, and VaR methodology in risk management.
Our main audience is formed by actuaries and particularly those specialized in entreprise risk management, insurance risk managers, Master’s degree students in mathematics or economics, and people involved in Solvency II for insurance companies and in Basel II and III for banks. Let us finally add that this book can also be used as a standard reference for the basic information in stochastic processes for students in actuarial science.
From the odd value of this exponent, it follows that:
−γ1<0 gives a right dissymmetry giving a maximum of the density function situated to the right and a distribution with a left heavy queue.
b) The kurtosis coefficient also due to Fisher is defined as follows:
Its interpretation refers to the normal distribution for which its value is 3. Also some authors refer to the excess of kurtosis given by γ1-3 of course null in the normal case.
For γ2<3, distributions are called leptokurtic, being more plated around the mean than in the normal case and with heavy queues.
For γ2>3, distributions are less plated around the mean than in the normal case and with heavy queues.
