111,99 €
Mehr erfahren.
- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
A new textbook offering a comprehensive introduction to models and techniques for the emerging field of actuarial Finance Drs. Boudreault and Renaud answer the need for a clear, application-oriented guide to the growing field of actuarial finance with this volume, which focuses on the mathematical models and techniques used in actuarial finance for the pricing and hedging of actuarial liabilities exposed to financial markets and other contingencies. With roots in modern financial mathematics, actuarial finance presents unique challenges due to the long-term nature of insurance liabilities, the presence of mortality or other contingencies and the structure and regulations of the insurance and pension markets. Motivated, designed and written for and by actuaries, this book puts actuarial applications at the forefront in addition to balancing mathematics and finance at an adequate level to actuarial undergraduates. While the classical theory of financial mathematics is discussed, the authors provide a thorough grounding in such crucial topics as recognizing embedded options in actuarial liabilities, adequately quantifying and pricing liabilities, and using derivatives and other assets to manage actuarial and financial risks. Actuarial applications are emphasized and illustrated with about 300 examples and 200 exercises. The book also comprises end-of-chapter point-form summaries to help the reader review the most important concepts. Additional topics and features include: * Compares pricing in insurance and financial markets * Discusses event-triggered derivatives such as weather, catastrophe and longevity derivatives and how they can be used for risk management; * Introduces equity-linked insurance and annuities (EIAs, VAs), relates them to common derivatives and how to manage mortality for these products * Introduces pricing and replication in incomplete markets and analyze the impact of market incompleteness on insurance and risk management; * Presents immunization techniques alongside Greeks-based hedging; * Covers in detail how to delta-gamma/rho/vega hedge a liability and how to rebalance periodically a hedging portfolio. This text will prove itself a firm foundation for undergraduate courses in financial mathematics or economics, actuarial mathematics or derivative markets. It is also highly applicable to current and future actuaries preparing for the exams or actuary professionals looking for a valuable addition to their reference shelf. As of 2019, the book covers significant parts of the Society of Actuaries' Exams FM, IFM and QFI Core, and the Casualty Actuarial Society's Exams 2 and 3F. It is assumed the reader has basic skills in calculus (differentiation and integration of functions), probability (at the level of the Society of Actuaries' Exam P), interest theory (time value of money) and, ideally, a basic understanding of elementary stochastic processes such as random walks.
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 999
Veröffentlichungsjahr: 2019
Ähnliche
Actuarial Finance
Derivatives, Quantitative Models and Risk Management
Mathieu Boudreault and Jean-François Renaud
Université du Québec à Montréal, Canada
This edition first published 2019
© 2019 John Wiley & Sons, Inc.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.
The right of Mathieu Boudreault and Jean-François Renaud to be identified as the authors of this work has been asserted in accordance with law.
Registered Office
John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA
Editorial Office
111 River Street, Hoboken, NJ 07030, USA
For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com.
Wiley also publishes its books in a variety of electronic formats and by print-on-demand. Some content that appears in standard print versions of this book may not be available in other formats.
Limit of Liability/Disclaimer of Warranty While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.
Library of Congress Cataloging-in-Publication data is available for this book.
9781119137009 (hardback)
Cover Design: Wiley
Cover Image: © precinbe/iStock.com
À Joëlle, Justine, Raphaël et Thomas
À Julie, Étienne et Anne-Sophie
CONTENTS
Cover
Acknowledgments
Preface
Part I Introduction to actuarial finance
1 Actuaries and their environment
1.1 Key concepts
1.2 Insurance and financial markets
1.3 Actuarial and financial risks
1.4 Diversifiable and systematic risks
1.5 Risk management approaches
1.6 Summary
1.7 Exercises
Notes
2 Financial markets and their securities
2.1 Bonds and interest rates
2.2 Stocks
2.3 Derivatives
2.4 Structure of financial markets
2.5 Mispricing and arbitrage opportunities
2.6 Summary
2.7 Exercises
Note
3 Forwards and futures
3.1 Framework
3.2 Equity forwards
3.3 Currency forwards
3.4 Commodity forwards
3.5 Futures contracts
3.6 Summary
3.7 Exercises
Notes
4 Swaps
4.1 Framework
4.2 Interest rate swaps
4.3 Currency swaps
4.4 Credit default swaps
4.5 Commodity swaps
4.6 Summary
4.7 Exercises
Notes
5 Options
5.1 Framework
5.2 Basic options
5.3 Main uses of options
5.4 Investment strategies with basic options
5.5 Summary
5.6 Exercises
Note
6 Engineering basic options
6.1 Simple mathematical functions for financial engineering
6.2 Parity relationships
6.3 Additional payoff design with calls and puts
6.4 More on the put-call parity
6.5 American options
6.6 Summary
6.7 Exercises
Notes
7 Engineering advanced derivatives
7.1 Exotic options
7.2 Event-triggered derivatives
7.3 Summary
7.4 Exercises
Note
8 Equity-linked insurance and annuities
8.1 Definitions and notations
8.2 Equity-indexed annuities
8.3 Variable annuities
8.4 Insurer’s loss
8.5 Mortality risk
8.6 Summary
8.7 Exercises
Notes
Part II Binomial and trinomial tree models
9 One-period binomial tree model
9.1 Model
9.2 Pricing by replication
9.3 Pricing with risk-neutral probabilities
9.4 Summary
9.5 Exercises
Note
10 Two-period binomial tree model
10.1 Model
10.2 Pricing by replication
10.3 Pricing with risk-neutral probabilities
10.4 Advanced actuarial and financial examples
10.5 Summary
10.6 Exercises
11 Multi-period binomial tree model
11.1 Model
11.2 Pricing by replication
11.3 Pricing with risk-neutral probabilities
11.4 Summary
11.5 Exercises
Notes
12 Further topics in the binomial tree model
12.1 American options
12.2 Options on dividend-paying stocks
12.3 Currency options
12.4 Options on futures
12.5 Summary
12.6 Exercises
Note
13 Market incompleteness and one-period trinomial tree models
13.1 Model
13.2 Pricing by replication
13.3 Pricing with risk-neutral probabilities
13.4 Completion of a trinomial tree
13.5 Incompleteness of insurance markets
13.6 Summary
13.7 Exercises
Notes
Part III Black-Scholes-Merton model
14 Brownian motion
14.1 Normal and lognormal distributions
14.2 Symmetric random walks
14.3 Standard Brownian motion
14.4 Linear Brownian motion
14.5 Geometric Brownian motion
14.6 Summary
14.7 Exercises
Notes
15 Introduction to stochastic calculus***
15.1 Stochastic Riemann integrals
15.2 Ito’s stochastic integrals
15.3 Ito’s lemma for Brownian motion
15.4 Diffusion processes
15.5 Summary
15.6 Exercises
Notes
16 Introduction to the Black-Scholes-Merton model
16.1 Model
16.2 Relationship between the binomial and BSM models
16.3 Black-Scholes formula
16.4 Pricing simple derivatives
16.5 Determinants of call and put prices
16.6 Replication and hedging
16.7 Summary
16.8 Exercises
Notes
17 Rigorous derivations of the Black-Scholes formula***
17.1 PDE approach to option pricing and hedging
17.2 Risk-neutral approach to option pricing
17.3 Summary
17.4 Exercises
Notes
18 Applications and extensions of the Black-Scholes formula
18.1 Options on other assets
18.2 Equity-linked insurance and annuities
18.3 Exotic options
18.4 Summary
18.5 Exercises
Notes
19 Simulation methods
19.1 Primer on random numbers
19.2 Monte Carlo simulations for option pricing
19.3 Variance reduction techniques
19.4 Summary
19.5 Exercises
Note
Hedging strategies in practice
20.1 Introduction
20.2 Cash-flow matching and replication
20.3 Hedging strategies
20.4 Interest rate risk management
20.5 Equity risk management
20.6 Rebalancing the hedging portfolio
20.7 Summary
20.8 Exercises
Notes
References
Index
End User License Agreement
List of Tables
Chapter 2
Table 2.1
Table 2.2
Chapter 4
Table 4.1
Chapter 7
Table 7.1
Chapter 8
Table 8.1
Chapter 15
Table 15.1
Chapter 17
Table 17.1
Chapter 19
Table 19.1
Chapter 20
Table 20.1
Guide
Cover
Table of Contents
Part
Pages
iii
iv
v
xvii
xix
xx
xxi
xxii
xxiii
xxiv
1
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
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
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
183
185
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
226
227
228
229
230
231
232
233
234
235
236
237
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
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
325
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
555
556
557
558
559
560
561
562
563
564
565
Acknowledgments
We are deeply grateful to our families for their unconditional support.
Special thanks to Geneviève Gauthier, Carole Bernard and Christiane Lemieux for their impact on our first years of teaching this material. Many thanks to colleagues from both academia and industry who took some of their precious time to review one or several chapters of earlier versions of the book: (in alphabetical order) Maciej Augustyniak, Jean-François Bégin, Bruno Gagnon, Ron Gebhardtsbauer, Jonathan Grégoire, Anne MacKay, Clarence Simard, Tommy Thomassin and Marie-Claude Vachon. Your comments were greatly appreciated and helped improve our book.
We also wish to thank the students who have attended our lectures for their feedback. This book is for you.
Finally, warm thanks to Jean-Mathieu Potvin for turning our handwritten plots into beautiful TikZ figures, to Christel Jackson Book for carefully reviewing each and every example and exercise, and to Adel Benlagra who also contributed to solve and revise exercises in the first half of the book. All remaining errors and typos are ours.
Preface
What is actuarial finance?
The work of actuaries has been strongly impacted by the deregulation of financial markets and the financial innovation that followed. Nowadays, there are embedded options in many life insurance and annuity products. Moreover, the use of weather and catastrophe derivatives in P&C insurance, to manage losses from earthquakes, hurricanes, floods and extreme weather, is constantly increasing, while longevity derivatives have been created to manage longevity risk on pools of pensioners.
As a consequence, in recent years actuarial finance has become an emerging field at the crossing of actuarial science and mathematical finance, i.e. when both actuarial and financial risks have to be taken into account. Despite its common roots with modern financial mathematics, actuarial finance has its own challenges due to:
the very long-term nature (several years or decades) of insurance liabilities;
the presence of mortality risk and other contingencies;
the structure and regulations of the insurance and pension markets.
Therefore, it is now widely recognized that actuaries should have a basic knowledge of:
derivatives and other assets to manage actuarial and financial risks;
embedded options in actuarial liabilities.
A book written by and for actuaries
As we are writing this book for the typical actuarial science (undergraduate) student or practitioner, our main goal is to find an appropriate balance between the level of mathematics and finance in the presentation of classical financial models such as the binomial tree and the Black-Scholes-Merton model. Given the growing complexity of many actuarial liabilities and their ties to financial markets, we also felt the need for a book with actuarial applications at the forefront.
Therefore, this book has been motivated, designed and written for actuaries (and by actuaries). This means that we spend more time on and put more energy into what matters to actuaries, i.e. topics of particular relevance for actuaries are introduced at a more accessible level and analyzed in greater depth. For example, in this book, we:
compare pricing in insurance and financial markets;
present credit default swaps which are largely used by insurance companies to manage credit risk and we compare them with term life insurance;
discuss event-triggered derivatives such as weather, catastrophe and longevity derivatives and how they can be used for risk management;
introduce equity-linked insurance and annuities (EIAs, VAs), relate them to common derivatives and describe how to manage mortality for these products;
put more emphasis on replication in the (two-period and
n
-period) binomial model and in the Black-Scholes-Merton model to better grasp the origins of the so-called actuarial/real-world and risk-neutral probabilities (or the (in)famous
and
measures);
implement and rebalance dynamic hedging strategies over several periods;
introduce pricing and replication in incomplete markets and analyze the impact of market incompleteness on insurance and risk management;
derive the Black-Scholes formula using simple limiting arguments rather than using stochastic calculus;
implement Monte Carlo methods with variance reduction techniques to price simple and exotic derivatives;
present immunization techniques alongside Greeks-based hedging;
cover in details how to delta-gamma/rho/vega hedge a liability and how to rebalance periodically a hedging portfolio.
Moreover, the book contains many actuarial applications and examples to illustrate the relevance of various topics to actuaries.
To meet the above-mentioned objectives, we assume the reader has basic skills in calculus (differentiation and integration of functions), probability (at the level of the Society of Actuaries’ Exam P), interest theory (time value of money) and, ideally, a basic understanding of elementary stochastic processes such as random walks.
Structure of the book
This book is targeted mainly toward undergraduate students in actuarial science and practitioners with an actuarial science background seeking a solid but yet accessible introduction to the quantitative aspects of modern finance. It covers pricing, replication and risk management of derivatives and actuarial liabilities, which are of paramount importance to actuaries in areas such as asset-and-liability management, liability-driven investments, banking, etc.
The book is divided into three parts:
Part 1: Introduction to actuarial finance
Part 2: Binomial and trinomial tree models
Part 3: Black-Scholes-Merton model
In each chapter, on top of the presented material, there is a set of specific objectives, numerical examples, a point-form summary and end-of-chapter exercises. Additional complementary information, such as historic notes or mathematical details, is presented in boxes. Finally, warning icons appear in the margin when a given topic, a concept or a detail deserves extra care or thought.
The content of the book is as follows.
Part 1: Introduction to actuarial finance
After an introductory chapter that puts into perspective the work of actuaries in the financial world, more standard chapters on financial securities, forwards and futures, swaps and options follow. Then, the next two chapters are devoted to the engineering of derivatives payoffs and liabilities, i.e. to the analysis of the structure of payoffs/liabilities, which is at the core of no-arbitrage pricing. Finally, a whole chapter describes insurance products bearing financial risk, namely equity-linked insurance and annuities (ELIAs).
In Chapter 1 – Actuaries and their environment, we put into context the role of the actuary in an insurance company or a pension plan. We explain how to differentiate between (actuarial) liabilities and (financial) assets, and between financial and insurance markets. We describe the various insurance policies and financial securities available and we compare actuarial and financial risks, short- and long-term risks, and diversifiable and systematic risks. Finally, we analyze various risk management methods for systematic risks.
In Chapter 2 – Financial markets and their securities, we provide an introduction to financial markets and financial securities, especially stocks, bonds and derivatives. We present the term structure of interest rates, we calculate the present and future value of cash flows and we explain the impact of dividends on stock prices. We also explain how actuaries can use derivatives and why pricing in the financial market is different from pricing in the insurance market. Finally, we look at price inconsistencies and how to create arbitrage opportunities.
In Chapter 3 – Forwards and futures, we provide an introduction to forwards and futures. We look at situations where forward contracts and futures contracts can be used to manage risks, and we explain the difference between a forward contract and a futures contract. We explain how to replicate the cash flows of forward contracts and calculate the forward price of stocks and of foreign currencies. Finally, we describe the margin balance on long and short positions of futures contracts.
In Chapter 4 – Swaps, we provide an introduction to swaps with an emphasis on those used in the insurance industry, namely interest rate swaps, currency swaps and credit default swaps. We present their characteristics, explain their cash flows and compute their values.
In Chapter 5 – Options, we give an introduction to standard options. We explain the differences between options to buy (call) and options to sell (put), as well as the difference between options and forward contracts. We explain when an option is used for hedging/risk management or for speculating purposes. Finally, we describe various investment strategies using options.
In Chapter 6 – Engineering basic options, we want to understand how to build and relate simple payoffs and then use no-arbitrage arguments to derive parity relationships. We see how to use simple mathematical functions to design simple payoffs and relate basic options and how to create synthetic versions of basic options, including binary options and gap options. Finally, we analyze when American options should be (early-)exercised.
In Chapter 7 – Engineering advanced derivatives, we provide an introduction to exotic/path-dependent options and event-triggered derivatives. We describe the payoff of various derivatives including barrier, Asian, lookback and exchange options, as well as weather, catastrophe and longevity derivatives. We explain why complex derivatives exist and how they can be used. Finally, we show how to use no-arbitrage arguments to identify relationships between the prices of some of these derivatives.
In Chapter 8 – Equity-linked insurance and annuities, we give an introduction to a large class of insurance products known as equity-linked insurance and annuities. First, we present relationships and differences between ELIAs and other derivatives. Then, after defining three indexing methods, we show how to compute the benefit(s) of typical guarantees included in ELIAs. We explain how equity-indexed annuities and variable annuities are funded and analyze the losses tied to these products. Finally, we explain how mortality is accounted for when risk managing ELIAs.
Part 2: Binomial and trinomial tree models
In the second part, we focus on the binomial tree model and the trinomial tree model, two discrete-time market models for the replication, hedging and pricing of financial derivatives and equity-linked products. By keeping the level of mathematics low, the binomial model allows greater emphasis on replication (as opposed to pricing), a concept of paramount importance for actuaries in asset and liability management. The intuition gained from the binomial model will be used repeatedly in the Black-Scholes-Merton model. Finally, as market incompleteness is a crucial concept in insurance markets, the trinomial tree model is treated in a chapter of its own. This model is simple and yet powerful enough to illustrate the idea of market incompleteness and its consequences for hedging and pricing.
In Chapter 9 – One-period binomial tree model, we first describe the basic assets available and we identify the assumptions on which this model is based. Then, we explain how to build a one-period binomial tree. Most of the chapter is devoted to the pricing of derivatives by replication of their payoff, from which we obtain risk-neutral pricing formulas.
In Chapter 10 – Two-period binomial tree model, we consider again the replication and the pricing of options and other derivatives, but now in a two-step tree. First, we explain how to build a two-period binomial tree using three one-period binomial trees. Then, we build dynamic replicating strategies to price options, from which we obtain risk-neutral pricing formulas. Finally, we determine how to price options in more complex situations: path-dependent options, options on assets that pay dollar dividends, variable annuities or stochastic interest rates.
In Chapter 11 – Multi-period binomial tree model, we see how to build a general binomial tree. We relate the asset price observed at a given time step to the binomial distribution and we highlight the differences between simple options and path-dependent options. We explain how to set up the dynamic replicating strategy to price an option and then obtain risk-neutral pricing formulas in a multi-period setup.
In Chapter 12 – Further topics in the binomial tree model, we extend the binomial tree to more realistic situations. We determine replicating portfolios and derive risk-neutral formulas for American-style options, options on stocks paying continuous dividends, currency options and futures options.
In Chapter 13 – Market incompleteness and one-period trinomial tree models, we define market incompleteness and we present the one-period trinomial tree model. We build sub-replicating and super-replicating portfolios for derivatives and explain that, in incomplete markets, there is a range of prices that prevent arbitrage opportunities. Then, we derive bounds on the admissible risk-neutral probabilities and relate the resulting prices to sub- and super-replicating portfolios in a one-period trinomial tree model. Second, we show how to replicate derivatives if the model has three traded assets. Also, we examine the risk management implications of ignoring possible outcomes when replicating a derivative. Finally, we analyze how actuaries cope with the incompleteness of insurance markets.
Part 3: Black-Scholes-Merton model
The third and last part of the book is devoted to the Black-Scholes-Merton model, the famous Black-Scholes formula and its applications in insurance. Both the model and its main formula are presented without the use of stochastic calculus; justifications are provided mainly by using the detailed work done previously in the binomial model and taking the appropriate limits. For the sake of completeness, a more classical treatment with stochastic calculus is also presented in two starred chapters, which can be skipped. In the last chapters, we apply generalizations of the Black-Scholes formula to price more advanced derivatives and equity-linked products, we provide an introduction to simulation methods and, finally, we present several sensitivity-based hedging strategies for equity risk, interest rate risk and volatility risk.
In Chapter 14 – Brownian motion, we provide the necessary background on Brownian motion to understand the Black-Scholes-Merton model and how to price and manage (hedge) options in that model. We also focus on simulation and estimation of this process, which are very important in practice. First, we provide an introduction to the lognormal distribution and compute truncated expectations and the stop-loss transform of a lognormally distributed random variable. Then, we define standard Brownian motion as the limit of random walks and present its basic properties. Linear and geometric Brownian motions are defined as transformations of standard Brownian motion. Finally, we show how to simulate standard, linear and geometric Brownian motions to generate scenarios, and how to estimate a geometric Brownian motion from a given data set.
In Chapter 15 – Introduction to stochastic calculus***, we provide a heuristic introduction to stochastic calculus based on Brownian motion by defining Ito’s stochastic integral and stochastic differential equations (SDEs). First, we define stochastic integrals and look at their basic properties, including the computations of the mean and variance of a given stochastic integral. Then, we show how to apply Ito’s lemma in simple situations. Next, we explain how a stochastic process can be the solution to a stochastic differential equation. Finally, we study the SDEs for linear and geometric Brownian motions, the Ornstein-Uhlenbeck process and the square-root process, and understand the role played by their coefficients.
In Chapter 16 – Introduction to the Black-Scholes-Merton model, we lay the foundations of the famous Black-Scholes-Merton (BSM) market model and we provide a heuristic approach to the Black-Scholes formula. More specifically, we present the main assumptions of the Black-Scholes-Merton model, including the dynamics of the risk-free and risky assets, and connect the Black-Scholes-Merton model to the binomial model. We explain the difference between real-world (actuarial) and risk-neutral probabilities. Then, we compute call and put options prices with the Black-Scholes formula and price simple derivatives using risk-neutral probabilities. Also, we analyze the impact of various determinants of the call or put option price. Finally, we derive replicating portfolios for simple derivatives and show how to implement a delta-hedging strategy over several periods.
In Chapter 17 – Rigorous derivations of the Black-Scholes formula***, we provide a more advanced treatment of the BSM model. More precisely, we provide two rigorous derivations of the Black-Scholes formula using either partial differential equations (PDEs) or changes of probability measures. In the first part, we define PDEs and show a link with diffusion processes as given by the Feynman-Kač formula. Then, we derive and solve the Black-Scholes PDE for simple payoffs and we show how to price and replicate simple derivatives. In the second part, we explain the effect of changing the probability measure on random variables and on Brownian motions (Girsanov theorem). Then, we compute the price of simple and exotic derivatives using the risk-neutral probability measure.
In Chapter 18 – Applications and extensions of the Black-Scholes formula, we analyze the pricing of options and other derivatives such as options on dividend-paying assets, currency options and futures options, but also insurance products such as investment guarantees, equity-indexed annuities and variable annuities, as well as exotic options (Asian, lookback and barrier options). Also, we explain how to compute the break-even participation rate or annual fee for common equity-linked insurance and annuities.
In Chapter 19 – Simulation methods, we apply simulation techniques to compute approximations of the no-arbitrage price of derivatives under the BSM model. As the price of most complex derivatives does not have a closed-form expression, we illustrate the techniques by pricing simple and path-dependent derivatives with crude Monte Carlo methods. Then, we describe three variance reduction techniques, namely stratified sampling, antithetic and control variates, to accelerate convergence of the price estimator.
In Chapter 20 – Hedging strategies in practice, we analyze various risk management practices, mostly hedging strategies used for interest rate risk and equity risk management. First, we apply cash-flow matching or replication to manage interest rate risk and equity risk. Then, we define the so-called Greeks. We explain how Taylor series expansions can be used for risk management purposes and highlight the similarities between duration-(convexity) matching and delta(-gamma) hedging. We show how to implement delta(-gamma) hedging, delta-rho hedging and delta-vega hedging to assets and liabilities sensitive to changes in both the underlying asset price and the other corresponding financial quantity. Finally, we compute the new hedging portfolio (rebalancing) as conditions in the market evolve.
Further reading
This book strives to find a balance between actuarial science, finance and mathematics. As such, the reader looking for additional information should find the following references useful.
For readers seeking a more advanced treatment of stochastic calculus and/or mathematical finance, important references include (in alphabetical order): Baxter & Rennie [1], Björk [2], Cvitanic & Zapatero [3], Lamberton & Lapeyre [4], Mikosch [5], Musiela & Rutkowski [6] and both volumes of Shreve, [7] and [8].
From a finance and/or business perspective: Boyle & Boyle [9], Hull [10], McDonald [11] and Wilmott [12].
On simulation methods: Devroye [13] and Glasserman [14].
Finally, for more details on ELIAs, there are two key references: Hardy [15] and Kalberer & Ravindran [16].
