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Actuarial Finance: Derivatives, Quantitative Models and Risk Management

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Actuarial Finance: Derivatives, Quantitative Models and Risk Management

Mathieu Boudreault, Jean-François Renaud

ISBN: 978-1-119-13702-3 March 2019 592 Pages

Description

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.

Acknowledgments xvii

Preface xix

Part I Introduction to actuarial finance 1

1 Actuaries and their environment 3

1.1 Key concepts 3

1.2 Insurance and financial markets 6

1.3 Actuarial and financial risks 8

1.4 Diversifiable and systematic risks 9

1.5 Risk management approaches 15

1.6 Summary 16

1.7 Exercises 17

2 Financial markets and their securities 21

2.1 Bonds and interest rates 21

2.2 Stocks 29

2.3 Derivatives 32

2.4 Structure of financial markets 35

2.5 Mispricing and arbitrage opportunities 38

2.6 Summary 42

2.7 Exercises 44

3 Forwards and futures 49

3.1 Framework 49

3.2 Equity forwards 52

3.3 Currency forwards 59

3.4 Commodity forwards 61

3.5 Futures contracts 62

3.6 Summary 70

3.7 Exercises 72

4 Swaps 75

4.1 Framework 76

4.2 Interest rate swaps 77

4.3 Currency swaps 87

4.4 Credit default swaps 90

4.5 Commodity swaps 93

4.6 Summary 95

4.7 Exercises 96

5 Options 99

5.1 Framework 100

5.2 Basic options 102

5.3 Main uses of options 107

5.4 Investment strategies with basic options 110

5.5 Summary 114

5.6 Exercises 116

6 Engineering basic options 119

6.1 Simple mathematical functions for financial engineering 119

6.2 Parity relationships 122

6.3 Additional payoff design with calls and puts 126

6.4 More on the put-call parity 129

6.5 American options 133

6.6 Summary 136

6.7 Exercises 137

7 Engineering advanced derivatives 141

7.1 Exotic options 141

7.2 Event-triggered derivatives 150

7.3 Summary 154

7.4 Exercises 156

8 Equity-linked insurance and annuities 159

8.1 Definitions and notations 160

8.2 Equity-indexed annuities 161

8.3 Variable annuities 165

8.4 Insurer’s loss 171

8.5 Mortality risk 173

8.6 Summary 177

8.7 Exercises 179

Part II Binomial and trinomial tree models 183

9 One-period binomial tree model 185

9.1 Model 185

9.2 Pricing by replication 190

9.3 Pricing with risk-neutral probabilities 195

9.4 Summary 198

9.5 Exercises 199

10 Two-period binomial tree model 201

10.1 Model 201

10.2 Pricing by replication 212

10.3 Pricing with risk-neutral probabilities 220

10.4 Advanced actuarial and financial examples 225

10.5 Summary 233

10.6 Exercises 236

11 Multi-period binomial tree model 239

11.1 Model 239

11.2 Pricing by replication 250

11.3 Pricing with risk-neutral probabilities 259

11.4 Summary 263

11.5 Exercises 265

12 Further topics in the binomial tree model 269

12.1 American options 269

12.2 Options on dividend-paying stocks 276

12.3 Currency options 279

12.4 Options on futures 282

12.5 Summary 287

12.6 Exercises 289

13 Market incompleteness and one-period trinomial tree models 291

13.1 Model 292

13.2 Pricing by replication 296

13.3 Pricing with risk-neutral probabilities 306

13.4 Completion of a trinomial tree 313

13.5 Incompleteness of insurance markets 316

13.6 Summary 319

13.7 Exercises 321

Part III Black-Scholes-Mertonmodel 325

14 Brownian motion 327

14.1 Normal and lognormal distributions 327

14.2 Symmetric random walks 333

14.3 Standard Brownian motion 336

14.4 Linear Brownian motion 347

14.5 Geometric Brownian motion 351

14.6 Summary 359

14.7 Exercises 362

15 Introduction to stochastic calculus*** 365

15.1 Stochastic Riemann integrals 366

15.2 Ito’s stochastic integrals 368

15.3 Ito’s lemma for Brownian motion 380

15.4 Diffusion processes 382

15.5 Summary 389

15.6 Exercises 391

16 Introduction to the Black-Scholes-Mertonmodel 393

16.1 Model 394

16.2 Relationship between the binomial and BSM models 397

16.3 Black-Scholes formula 403

16.4 Pricing simple derivatives 410

16.5 Determinants of call and put prices 414

16.6 Replication and hedging 417

16.7 Summary 428

16.8 Exercises 430

17 Rigorous derivations of the Black-Scholes formula*** 433

17.1 PDE approach to option pricing and hedging 433

17.2 Risk-neutral approach to option pricing 440

17.3 Summary 451

17.4 Exercises 452

18 Applications and extensions of the Black-Scholes formula 455

18.1 Options on other assets 455

18.2 Equity-linked insurance and annuities 463

18.3 Exotic options 473

18.4 Summary 484

18.5 Exercises 485

19 Simulation methods 487

19.1 Primer on random numbers 488

19.2 Monte Carlo simulations for option pricing 490

19.3 Variance reduction techniques 497

19.4 Summary 513

19.5 Exercises 516

20 Hedging strategies in practice 519

20.1 Introduction 520

20.2 Cash-flow matching and replication 521

20.3 Hedging strategies 523

20.4 Interest rate risk management 527

20.5 Equity risk management 533

20.6 Rebalancing the hedging portfolio 546

20.7 Summary 548

20.8 Exercises 551

References 555

Index 557