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More About This Title Actuarial Finance: Derivatives, Quantitative Models and Risk Management
English
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.
English
MATHIEU BOUDREAULT, PHD, is Professor of Actuarial Science in the Département de mathématiques at Université du Québec à Montréal (UQAM), Canada. Fellow of the Society of Actuaries and Associate of the Canadian Institute of Actuaries, his teaching and research interests include actuarial finance, catastrophe modeling and credit risk.
JEAN-FRANÇOIS RENAUD, PHD, is Professor of Actuarial Science in the Département de mathématiques at Université du Québec à Montréal (UQAM), Canada. His teaching and research interests include actuarial finance, actuarial mathematics and applied probability.
English
Part I Introduction to actuarial finance 9
1 The actuary and its environment 11
1.1 Key concepts 12
1.1.1 What is insurance? 12
1.1.2 Actuarial liabilities and financial assets 12
1.1.3 Actuarial functions 13
1.2 Insurance and financial markets 15
1.2.1 Insurance market 15
1.2.2 Financial market 16
1.2.3 Insurance is a derivative 17
1.3 Actuarial and financial risks 18
1.4 Diversifiable and systematic risks 19
1.4.1 Illustrative example 19
1.4.2 Independence 20
1.4.3 Framework 20
1.4.4 Diversifiable risks 21
1.4.5 Systematic risks 23
1.4.6 Partially diversifiable risks 24
1.5 Risk management approaches 26
1.6 Summary 27
1.7 Exercises 29
2 Financial markets and their securities 31
2.1 Bonds and interest rates 31
2.1.1 Characteristics 32
2.1.2 Basics of bond pricing 33
2.1.3 Term structure of interest rates 34
2.2 Stocks 41
2.2.1 Dividends 41
2.2.2 Reinvesting dividends 43
2.3 Derivatives 44
2.3.1 Types of derivatives 44
2.3.2 Uses of derivatives 45
2.4 Structure of financial markets 47
2.4.1 Overview of markets 47
2.4.2 Trading and financial positions 49
2.4.3 Market frictions 50
2.5 Mispricing and arbitrage opportunities 51
2.5.1 Taking advantage of price inconsistencies 52
2.5.2 Arbitrage opportunities and no-arbitrage pricing 53
2.6 Summary 56
2.7 Exercises 59
3 Forwards and futures 63
3.1 Framework 64
3.1.1 Terminology 64
3.1.2 Notation 64
3.1.3 Payoff 65
3.2 Equity forwards 66
3.2.1 Pricing 67
3.2.2 Forward price 70
3.2.3 Discrete and fixed dividends 72
3.2.4 Continuous and proportional dividends 74
3.3 Currency forwards 75
3.3.1 Background 75
3.3.2 Forward exchange rate 77
3.4 Commodity forwards 78
3.5 Futures contracts 79
3.5.1 Futures price 80
3.5.2 Marking-to-market without interest 80
3.5.3 Marking-to-market with interest 82
3.5.4 Equivalence of the futures and forward prices 84
3.5.5 Marking-to-market in practice 86
3.6 Summary 88
3.7 Exercises 90
4 Swaps 93
4.1 Framework 94
4.2 Interest rate swaps 95
4.2.1 Fixed-rate and floating-rate loans 95
4.2.2 Cash flows 96
4.2.3 Valuation 99
4.2.4 Market specifics 104
4.3 Currency swaps 106
4.3.1 Cash flows 107
4.3.2 Valuation 109
4.4 Credit default swaps 110
4.4.1 Cash flows 111
4.4.2 Valuation 113
4.4.3 Comparing a CDS with an insurance policy 114
4.5 Commodity swaps 114
4.6 Summary 116
4.7 Exercises 118
5 Options 121
5.1 Framework 122
5.2 Basic options 125
5.2.1 Call options 125
5.2.2 Put options 128
5.3 Main uses of options 131
5.3.1 Hedging and risk management 131
5.3.2 Speculation 133
5.4 Investment strategies with basic options 134
5.5 Summary 138
5.6 Exercises 140
6 Engineering basic options 143
6.1 Simple mathematical functions for financial engineering 143
6.1.1 Positive part function 144
6.1.2 Maximum function 145
6.1.3 Stop-loss function 145
6.1.4 Indicator function 146
6.2 Parity relationships 147
6.2.1 Simple payoff design 147
6.2.2 Put-call parity 149
6.3 Additional payoff design with calls and puts 151
6.3.1 Decomposing call and put options 151
6.3.2 Binary or digital options 152
6.3.3 Gap options 154
6.4 More on the put-call parity 156
6.4.1 Bounds on European options prices 156
6.4.2 Put-call parity with dividend-paying assets 158
6.5 American options 160
6.5.1 Lower bounds on American options prices 161
6.5.2 Early exercise of American calls 162
6.5.3 Early exercise of American puts 163
6.6 Summary 164
6.7 Exercises 165
7 Engineering advanced derivatives 169
7.1 Exotic options 169
7.1.1 Barrier options 170
7.1.2 Lookback options 175
7.1.3 Asian options 177
7.1.4 Exchange options 178
7.2 Event-triggered derivatives 180
7.2.1 Weather derivatives 180
7.2.2 Catastrophe derivatives 182
7.2.3 Longevity derivatives 183
7.3 Summary 185
7.4 Exercises 187
8 Equity-linked insurance and annuities 191
8.1 Definitions and notations 193
8.2 Equity-indexed annuities 194
8.2.1 Additional notation 194
8.2.2 Indexing methods 195
8.3 Variable annuities 198
8.3.1 Sub-account dynamics 199
8.3.2 Typical guarantees 200
8.4 Insurer’s loss 205
8.4.1 Equity-indexed annuities 205
8.4.2 Variable annuities 206
8.5 Mortality risk 207
8.6 Summary 212
8.7 Exercises 214
Part II Binomial and trinomial tree models 219
9 One-period binomial tree model 221
9.1 Model 221
9.1.1 Risk-free asset 222
9.1.2 Risky asset 222
9.1.3 Derivatives 225
9.2 Pricing by replication 227
9.3 Pricing with risk-neutral probabilities 233
9.4 Summary 237
9.5 Exercises 238
10 Two-period binomial tree model 241
10.1 Model 241
10.1.1 Risk-free asset 242
10.1.2 Risky asset 242
10.1.3 Derivatives 251
10.2 Pricing by replication 255
10.2.1 Trading strategies/portfolios 255
10.2.2 Backward recursive algorithm 259
10.3 Pricing with risk-neutral probabilities 266
10.3.1 Simplified tree 270
10.4 Advanced actuarial and financial examples 271
10.4.1 Stochastic interest rates 271
10.4.2 Discrete dividends 275
10.4.3 Guaranteed minimum withdrawal benefits 279
10.5 Summary 281
10.6 Exercises 284
11 Multi-period binomial tree model 287
11.1 Model 287
11.1.1 Risk-free asset 288
11.1.2 Risky asset 288
11.1.3 Derivatives 294
11.1.4 Labelling the nodes 294
11.1.5 Path-dependent payoffs 298
11.2 Pricing by replication 301
11.2.1 Trading strategies, portfolios 301
11.2.2 Portfolio value process 302
11.2.3 Self-financing strategies 303
11.2.4 Replicating strategy 304
11.2.5 Backward recursive procedure 305
11.2.6 Algorithm 307
11.3 Pricing with risk-neutral probabilities 312
11.4 Summary 317
11.5 Exercises 319
12 Further topics in the binomial tree model 323
12.1 American options 323
12.1.1 American put options 323
12.1.2 American call options 330
12.2 Options on dividend-paying stocks 332
12.3 Currency options 335
12.4 Options on futures 339
12.4.1 Futures price in a binomial environment 340
12.4.2 Replication and risk-neutral pricing 340
12.5 Summary 344
12.6 Exercises 347
13 Market incompleteness and one-period trinomial tree models 349
13.1 Model 351
13.1.1 Risk-free asset 351
13.1.2 Risky asset 351
13.1.3 Derivatives 354
13.2 Pricing by replication 355
13.2.1 (In)complete markets 355
13.2.2 Intervals of no-arbitrage prices 358
13.2.3 Super- and sub-replicating strategies 362
13.3 Pricing with risk-neutral probabilities 367
13.3.1 Maximal and minimal probabilities 367
13.3.2 Fundamental Theorem of Asset Pricing 369
13.3.3 Finding risk-neutral probabilities 371
13.3.4 Risk-neutral pricing 372
13.4 Completion of a trinomial tree 374
13.4.1 Trinomial tree with three basic assets 375
13.5 Incompleteness of insurance markets 378
13.6 Summary 382
13.7 Exercises 384
Part III Black-Scholes-Merton model 389
14 Brownian motion 391
14.1 Normal and lognormal distributions 392
14.1.1 Normal distribution 392
14.1.2 Lognormal distribution 394
14.2 Symmetric random walks 398
14.2.1 Markovian property 400
14.2.2 Martingale property 400
14.3 Standard Brownian motion 401
14.3.1 Construction as the limit of symmetric random walks 401
14.3.2 Definition 406
14.3.3 Distributional properties 407
14.3.4 Markovian property 409
14.3.5 Martingale property 411
14.3.6 Simulation 412
14.4 Linear Brownian motion 414
14.4.1 Distributional properties 415
14.4.2 Markovian property 418
14.4.3 Martingale property 419
14.4.4 Simulation 419
14.5 Geometric Brownian motion 421
14.5.1 Distributional properties 422
14.5.2 Markovian property 424
14.5.3 Martingale property 426
14.5.4 Simulation 426
14.5.5 Estimation 427
14.6 Summary 430
14.7 Exercises 433
15 Introduction to stochastic calculus 437
15.1 Stochastic Riemann integrals 438
15.2 Ito’s stochastic integrals 441
15.2.1 Riemann sums 443
15.2.2 Elementary stochastic processes 444
15.2.3 Ito-integrable stochastic processes 449
15.2.4 Properties 451
15.3 Ito’s Lemma for Brownian motion 455
15.4 Diffusion processes 458
15.4.1 Stochastic differential equations 458
15.4.2 Ito’s lemma for diffusion processes 460
15.4.3 Geometric Brownian motion 462
15.4.4 Ornstein-Uhlenbeck process 463
15.4.5 Square-root diffusion process 465
15.5 Summary 466
15.6 Exercises 468
16 Introduction to the Black-Scholes-Merton model 469
16.1 Model 470
16.1.1 Risk-free asset 470
16.1.2 Risky asset 470
16.1.3 Derivatives 472
16.2 Relationship between the binomial and BSM models 473
16.2.1 Second look at the binomial model 474
16.2.2 Convergence of the binomial model 475
16.2.3 Formal proof 477
16.2.4 Risk-neutral probabilities 480
16.3 Black-Scholes formula 481
16.3.1 Limit of binomial models 482
16.3.2 Stop-loss transforms 486
16.3.3 Dynamic formula 487
16.4 Pricing simple derivatives 489
16.4.1 Forward contracts 490
16.4.2 Binary options 490
16.4.3 Gap options 492
16.5 Determinants of call and put prices 494
16.5.1 Stock price 495
16.5.2 Strike price 495
16.5.3 Risk-free rate 496
16.5.4 Volatility 496
16.5.5 Time to maturity 496
16.6 Replication and hedging 497
16.6.1 Trading strategies/portfolios 497
16.6.2 Replication for call and put options 500
16.6.3 Replication for simple derivatives 503
16.6.4 Delta-hedging strategy 506
16.7 Summary 511
16.8 Exercises 513
17 Rigorous derivations of the Black-Scholes formula 517
17.1 PDE approach to option pricing and hedging 518
17.1.1 Partial differential equations 518
17.1.2 Feynman-Kac formula 519
17.1.3 Deriving the Black-Scholes PDE 521
17.1.4 Solving the Black-Scholes PDE 524
17.1.5 Black-Scholes formula 525
17.2 Risk-neutral approach to option pricing 526
17.2.1 Probability measure 527
17.2.2 Changes of probability measure 529
17.2.3 Girsanov theorem 533
17.2.4 Risk-neutral probability measures 534
17.2.5 Risk-neutral dynamics 535
17.2.6 Risk-neutral pricing formulas 536
17.3 Summary 539
17.4 Exercises 540
18 Applications and extensions of the Black-Scholes formula 543
18.1 Options on other assets 543
18.1.1 Options on dividend-paying stocks 544
18.1.2 Currency options 548
18.1.3 Futures options and Black’s formula 548
18.1.4 Exchange options 550
18.2 Equity-linked insurance and annuities 553
18.2.1 Investment guarantees 553
18.2.2 Equity-indexed annuities 555
18.2.3 Variable annuities 560
18.3 Exotic options 564
18.3.1 Asian options 565
18.3.2 Lookback options 571
18.3.3 Barrier options 574
18.4 Summary 576
18.5 Exercises 578
19 Simulation methods 581
19.1 Primer on random numbers 582
19.1.1 Uniform random numbers 582
19.1.2 Inverse transform method 582
19.1.3 Normal random numbers 584
19.2 Monte Carlo simulations for option pricing 585
19.2.1 Notation 586
19.2.2 Application to option pricing 587
19.3 Variance reduction techniques 593
19.3.1 Stratified sampling 593
19.3.2 Antithetic variates 599
19.3.3 Control variates 606
19.4 Summary 611
19.5 Exercises 614
20 Hedging strategies in practice 617
20.1 Introduction 618
20.2 Cash-flow matching and replication 620
20.3 Hedging strategies 622
20.3.1 Taylor series expansions 623
20.3.2 Matching sensitivities 624
20.4 Interest rate risk management 626
20.4.1 Sensitivities 626
20.4.2 Duration matching 629
20.4.3 Duration-convexity matching 631
20.4.4 Immunization 632
20.5 Equity risk management 633
20.5.1 Greeks 633
20.5.2 Delta hedging 636
20.5.3 Delta-gamma hedging 639
20.5.4 Hedging with additional Greeks 643
20.6 Rebalancing the hedging portfolio 647
20.7 Summary 650
20.8 Exercises 653
Index 659
