Financial Mathematics: A Comprehensive Treatment (Chapman and Hall/CRC Financial Mathematics Series) Hardcover – March 12, 2014
by Giuseppe Campolieti (Author), Roman N. Makarov (Author)
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ISBN-13: 978-1439892428 ISBN-10: 1439892423 Edition: 1st
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    Series: Chapman and Hall/CRC Financial Mathematics Series
    Hardcover: 829 pages
    Publisher: Chapman and Hall/CRC; 1 edition (March 12, 2014)
    Language: English
    ISBN-10: 1439892423
    ISBN-13: 978-1439892428
    Product Dimensions: 10.1 x 7.2 x 1.8 inches
    Shipping Weight: 3.4 pounds (View shipping rates and policies)
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Contents
List of Figures and Tables xvii
List of Algorithms xxi
Preface xxiii
I Introduction to Pricing and Management of Financial Secu-
rities 1
1 Mathematics of Compounding 3
1.1 Interest and Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Amount Function and Return . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Simple Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.3 Periodic Compound Interest . . . . . . . . . . . . . . . . . . . . . . 7
1.1.4 Continuous Compound Interest . . . . . . . . . . . . . . . . . . . . 10
1.1.5 Equivalent Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.6 Continuously Varying Interest Rates . . . . . . . . . . . . . . . . . 13
1.2 Time Value of Money and Cash Flows . . . . . . . . . . . . . . . . . . . . 15
1.2.1 Equations of Value . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.2 Deterministic Cash Flows . . . . . . . . . . . . . . . . . . . . . . . 17
1.3 Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.1 Simple Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.3.2 Determining the Term of an Annuity . . . . . . . . . . . . . . . . . 24
1.3.3 General Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.3.4 Perpetuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3.5 Continuous Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.4 Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.4.1 Introduction and Terminology . . . . . . . . . . . . . . . . . . . . . 29
1.4.2 Zero-Coupon Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.4.3 Coupon Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.4.4 Serial Bonds, Strip Bonds, and Callable Bonds . . . . . . . . . . . 34
1.5 Yield Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.5.1 Internal Rate of Return and Evaluation Criteria . . . . . . . . . . . 36
1.5.2 Determining Yield Rates for Bonds . . . . . . . . . . . . . . . . . . 37
1.5.3 Approximation Methods . . . . . . . . . . . . . . . . . . . . . . . . 38
1.5.4 The Yield Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2 Primer on Pricing Risky Securities 47
2.1 Stocks and Stock Price Models . . . . . . . . . . . . . . . . . . . . . . . . 47
2.1.1 Underlying Assets and Derivative Securities . . . . . . . . . . . . . 47
2.1.2 Basic Assumptions for Asset Price Models . . . . . . . . . . . . . . 48

2.2 Basic Price Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.2.1 A Single-Period Binomial Model . . . . . . . . . . . . . . . . . . . 50
2.2.2 A Discrete-Time Model with a Finite Number of States . . . . . . 55
2.2.3 Introducing the Binomial Tree Model . . . . . . . . . . . . . . . . . 57
2.2.4 Self-Financing Investment Strategies in the Binomial Model . . . . 61
2.2.5 Log-Normal Pricing Model . . . . . . . . . . . . . . . . . . . . . . . 63
2.3 Arbitrage and Risk-Neutral Pricing . . . . . . . . . . . . . . . . . . . . . . 67
2.3.1 The Law of One Price . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.3.2 A First Look at Arbitrage in the Single-Period Binomial Model . . 69
2.3.3 Arbitrage in the Binomial Tree Model . . . . . . . . . . . . . . . . 71
2.3.4 Risk-Neutral Probabilities . . . . . . . . . . . . . . . . . . . . . . . 71
2.3.5 Martingale Property . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.3.6 Risk-Neutral Log-Normal Model . . . . . . . . . . . . . . . . . . . 74
2.4 Value at Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.5 Dividend Paying Stock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3 Portfolio Management 83
3.1 Expected Utility Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.1.1 Utility Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.1.2 Mean-Variance Criterion . . . . . . . . . . . . . . . . . . . . . . . . 89
3.2 Portfolio Optimization for Two Assets . . . . . . . . . . . . . . . . . . . . 90
3.2.1 Portfolio of Two Assets . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.2.2 Portfolio Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.2.3 The Minimum Variance Portfolio . . . . . . . . . . . . . . . . . . . 96
3.2.4 Selection of Optimal Portfolios . . . . . . . . . . . . . . . . . . . . 97
3.3 Portfolio Optimization for N Assets . . . . . . . . . . . . . . . . . . . . . 100
3.3.1 Portfolios of Several Assets . . . . . . . . . . . . . . . . . . . . . . 100
3.3.2 The Minimum Variance Portfolio . . . . . . . . . . . . . . . . . . . 102
3.3.3 The Minimum Variance Portfolio Line . . . . . . . . . . . . . . . . 104
3.3.4 Case without Short Selling . . . . . . . . . . . . . . . . . . . . . . . 106
3.3.5 Ecient Frontier and Capital Market Line . . . . . . . . . . . . . . 107
3.4 The Capital Asset Pricing Model . . . . . . . . . . . . . . . . . . . . . . . 110
3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4 Primer on Derivative Securities 115
4.1 Forward Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.1.1 No-Arbitrage Evaluation of Forward Contracts . . . . . . . . . . . 116
4.1.2 Value of a Forward Contract . . . . . . . . . . . . . . . . . . . . . . 119
4.2 Basic Options Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.2.1 Concept of an Option Contract . . . . . . . . . . . . . . . . . . . . 121
4.2.2 Put-Call Parities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.2.3 Properties of European Options . . . . . . . . . . . . . . . . . . . . 125
4.2.4 Early Exercise and American Options . . . . . . . . . . . . . . . . 127
4.2.5 Nonstandard European Options . . . . . . . . . . . . . . . . . . . . 129
4.3 Basics of Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.3.1 Pricing of European-Style Derivatives in the Binomial Tree Model 132
4.3.2 Pricing of American Options in the Binomial Tree Model . . . . . . 138
4.3.3 Option Pricing in the Log-Normal Model: The Black{Scholes{Merton
Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.3.4 Greeks and Hedging of Options . . . . . . . . . . .

4.3.5 Black{Scholes Equation . . . . . . . . . . . . . . . . . . . . . . . . 150
4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
II Discrete-Time Modelling 157
5 Single-Period Arrow{Debreu Models 159
5.1 Specication of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.1.1 Finite-State Economy. Vector Space of Payos. Securities . . . . . 159
5.1.2 Initial Price Vector and Payo Matrix . . . . . . . . . . . . . . . . 162
5.1.3 Portfolios of Base Securities . . . . . . . . . . . . . . . . . . . . . . 163
5.2 Analysis of the Arrow{Debreu Model . . . . . . . . . . . . . . . . . . . . . 164
5.2.1 Redundant Assets and Attainable Securities . . . . . . . . . . . . . 164
5.2.2 Completeness of the Model . . . . . . . . . . . . . . . . . . . . . . 167
5.3 No-Arbitrage Asset Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . 168
5.3.1 The Law of One Price . . . . . . . . . . . . . . . . . . . . . . . . . 168
5.3.2 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
5.3.3 The First Fundamental Theorem of Asset Pricing . . . . . . . . . . 171
5.3.4 Risk-Neutral Probabilities . . . . . . . . . . . . . . . . . . . . . . . 175
5.3.5 The Second Fundamental Theorem of Asset Pricing . . . . . . . . . 178
5.3.6 Investment Portfolio Optimization . . . . . . . . . . . . . . . . . . 179
5.4 Pricing in an Incomplete Market . . . . . . . . . . . . . . . . . . . . . . . 183
5.4.1 A Trinomial Model of an Incomplete Market . . . . . . . . . . . . . 183
5.4.2 Pricing Nonattainable Payos: The Bid-Ask Spread . . . . . . . . 185
5.5 Change of Numeraire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5.5.1 The Concept of a Numeraire Asset . . . . . . . . . . . . . . . . . . 191
5.5.2 Change of Numeraire in a Binomial Model . . . . . . . . . . . . . . 192
5.5.3 Change of Numeraire in a Multinomial Model . . . . . . . . . . . . 194
5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
6 Introduction to Discrete-Time Stochastic Calculus 207
6.1 A Multi-Period Binomial Probability Model . . . . . . . . . . . . . . . . . 207
6.1.1 The Binomial Probability Space . . . . . . . . . . . . . . . . . . . . 207
6.1.2 Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
6.2 Information Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
6.2.1 Partitions and Their Renements . . . . . . . . . . . . . . . . . . . 216
6.2.2 Sigma-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
6.2.3 Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
6.2.4 Filtered Probability Space . . . . . . . . . . . . . . . . . . . . . . . 227
6.3 Conditional Expectation and Martingales . . . . . . . . . . . . . . . . . . 229
6.3.1 Measurability of Random Variables and Processes . . . . . . . . . . 229
6.3.2 Conditional Expectations . . . . . . . . . . . . . . . . . . . . . . . 230
6.3.3 Properties of Conditional Expectations . . . . . . . . . . . . . . . . 236
6.3.4 Conditioning in the Binomial Model . . . . . . . . . . . . . . . . . 241
6.3.5 Sub-, Super-, and True Martingales . . . . . . . . . . . . . . . . . . 243
6.3.6 Classication of Stochastic Processes . . . . . . . . . . . . . . . . . 245
6.3.7 Stopping Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

7 Replication and Pricing in the Binomial Tree Model 257
7.1 The Standard Binomial Tree Model . . . . . . . . . . . . . . . . . . . . . . 257
7.2 Self-Financing Strategies and Their Value Processes . . . . . . . . . . . . 259
7.2.1 Equivalent Martingale Measures for the Binomial Model . . . . . . 262
7.3 Dynamic Replication in the Binomial Tree Model . . . . . . . . . . . . . . 265
7.3.1 Dynamic Replication of Payos . . . . . . . . . . . . . . . . . . . . 265
7.3.2 Replication and Valuation of Random Cash Flows . . . . . . . . . 273
7.4 Pricing and Hedging Non-Path-Dependent Derivatives . . . . . . . . . . . 274
7.5 Pricing Formulae for Standard European Options . . . . . . . . . . . . . . 278
7.6 Pricing and Hedging Path-Dependent Derivatives . . . . . . . . . . . . . . 281
7.6.1 Average Asset Prices and Asian Options . . . . . . . . . . . . . . . 281
7.6.2 Extreme Asset Prices and Lookback Options . . . . . . . . . . . . 283
7.6.3 Recursive Evaluation of Path-Dependent Options . . . . . . . . . . 283
7.7 American Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
7.7.1 Writer's Perspective: Pricing and Hedging . . . . . . . . . . . . . . 287
7.7.2 Buyer's Perspective: Optimal Exercise . . . . . . . . . . . . . . . . 291
7.7.3 Early-Exercise Boundary . . . . . . . . . . . . . . . . . . . . . . . . 298
7.7.4 Pricing American Options: The Case with Dividends . . . . . . . . 299
7.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
8 General Multi-Asset Multi-Period Model 307
8.1 Main Elements of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . 307
8.2 Assets, Portfolios, and Strategies . . . . . . . . . . . . . . . . . . . . . . . 310
8.2.1 Payos and Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
8.2.2 Static and Dynamic Portfolios . . . . . . . . . . . . . . . . . . . . . 311
8.2.3 Self-Financing Strategies . . . . . . . . . . . . . . . . . . . . . . . . 312
8.2.4 Replication of Payos . . . . . . . . . . . . . . . . . . . . . . . . . 314
8.3 Fundamental Theorems of Asset Pricing . . . . . . . . . . . . . . . . . . . 314
8.3.1 Arbitrage Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . 314
8.3.2 Enhancing the Law of One Price . . . . . . . . . . . . . . . . . . . 315
8.3.3 Equivalent Martingale Measures . . . . . . . . . . . . . . . . . . . . 317
8.3.4 Calculation of Martingale Measures . . . . . . . . . . . . . . . . . . 318
8.3.5 The First and Second FTAPs . . . . . . . . . . . . . . . . . . . . . 321
8.3.6 Pricing and Hedging Derivatives . . . . . . . . . . . . . . . . . . . 323
8.3.7 Radon{Nikodym Derivative Process and Change of Numeraire . . . 324
8.4 Examples of Discrete-Time Models . . . . . . . . . . . . . . . . . . . . . . 327
8.4.1 Binomial Tree Model with Stochastic Volatility . . . . . . . . . . . 327
8.4.2 Binomial Tree Model for Interest Rates . . . . . . . . . . . . . . . . 330
8.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
III Continuous-Time Modelling 335
9 Essentials of General Probability Theory 337
9.1 Random Variables and Lebesgue Integration . . . . . . . . . . . . . . . . . 337
9.2 Multidimensional Lebesgue Integration . . . . . . . . . . . . . . . . . . . . 351
9.3 Multiple Random Variables and Joint Distributions . . . . . . . . . . . . . 353
9.4 Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
9.5 Changing Probability Measures . . . . . . . . . . . . . . . . . . . . . . . . 365

10 One-Dimensional Brownian Motion and Related Processes 369
10.1 Multivariate Normal Distributions . . . . . . . . . . . . . . . . . . . . . . 369
10.1.1 Multivariate Normal Distribution . . . . . . . . . . . . . . . . . . . 369
10.1.2 Conditional Normal Distributions . . . . . . . . . . . . . . . . . . . 370
10.2 Standard Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 371
10.2.1 One-Dimensional Symmetric Random Walk . . . . . . . . . . . . . 371
10.2.2 Formal Denition and Basic Properties of Brownian Motion . . . . 375
10.2.3 Multivariate Distribution of Brownian Motion . . . . . . . . . . . . 378
10.2.4 The Markov Property and the Transition PDF . . . . . . . . . . . 380
10.2.5 Quadratic Variation and Nondierentiability of Paths . . . . . . . . 383
10.3 Some Processes Derived from Brownian Motion . . . . . . . . . . . . . . . 386
10.3.1 Drifted Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . 386
10.3.2 Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . . . . 387
10.3.3 Brownian Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
10.3.4 Gaussian Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
10.4 First Hitting Times and Maximum and Minimum of Brownian Motion . . 391
10.4.1 The Re
ection Principle: Standard Brownian Motion . . . . . . . . 391
10.4.2 Translated and Scaled Driftless Brownian Motion . . . . . . . . . . 398
10.4.3 Brownian Motion with Drift . . . . . . . . . . . . . . . . . . . . . . 400
10.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
11 Introduction to Continuous-Time Stochastic Calculus 411
11.1 The Riemann Integral of Brownian Motion . . . . . . . . . . . . . . . . . 411
11.1.1 The Riemann Integral . . . . . . . . . . . . . . . . . . . . . . . . . 411
11.1.2 The Integral of a Brownian Path . . . . . . . . . . . . . . . . . . . 411
11.2 The Riemann{Stieltjes Integral of Brownian Motion . . . . . . . . . . . . 414
11.2.1 The Riemann{Stieltjes Integral . . . . . . . . . . . . . . . . . . . . 414
11.2.2 Integrals w.r.t. Brownian Motion . . . . . . . . . . . . . . . . . . . 416
11.3 The It^o Integral and Its Basic Properties . . . . . . . . . . . . . . . . . . . 418
11.3.1 The It^o Integral for Simple Processes . . . . . . . . . . . . . . . . . 418
11.3.2 Properties of the It^o Integral . . . . . . . . . . . . . . . . . . . . . 420
11.4 It^o Processes and Their Properties . . . . . . . . . . . . . . . . . . . . . . 424
11.4.1 Gaussian Processes Generated by It^o Integrals . . . . . . . . . . . . 424
11.4.2 It^o Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
11.4.3 Quadratic (Co-)Variation . . . . . . . . . . . . . . . . . . . . . . . 427
11.5 It^o's Formula for Functions of BM and It^o Processes . . . . . . . . . . . . 429
11.5.1 It^o's Formula for Functions of BM . . . . . . . . . . . . . . . . . . 429
11.5.2 It^o's Formula for It^o Processes . . . . . . . . . . . . . . . . . . . . 432
11.6 Stochastic Dierential Equations . . . . . . . . . . . . . . . . . . . . . . . 435
11.6.1 Solutions to Linear SDEs . . . . . . . . . . . . . . . . . . . . . . . 435
11.6.2 Existence and Uniqueness of a Strong Solution of an SDE . . . . . 439
11.7 The Markov Property, Feynman{Kac Formulae, and Transition CDFs and
PDFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
11.7.1 Forward Kolmogorov PDE . . . . . . . . . . . . . . . . . . . . . . . 452
11.7.2 Transition CDF/PDF for Time-Homogeneous Diusions . . . . . . 453
11.8 Radon{Nikodym Derivative Process and Girsanov's Theorem . . . . . . . 455
11.8.1 Some Applications of Girsanov's Theorem . . . . . . . . . . . . . . 460
11.9 Brownian Martingale Representation Theorem . . . . . . . . . . . . . . . 464
11.10 Stochastic Calculus for Multidimensional BM . . . . . . . . . . . . . . . . 466
11.10.1 The It^o Integral and It^o's Formula for Multiple Processes on Multidimensional
BM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466

11.10.2 Multidimensional SDEs, Feynman{Kac Formulae, and Transition
CDFs and PDFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
11.10.3 Girsanov's Theorem for Multidimensional BM . . . . . . . . . . . . 487
11.10.4 Martingale Representation Theorem for Multidimensional BM . . . 489
11.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
12 Risk-Neutral Pricing in the (B; S) Economy: One Underlying Stock 497
12.1 Replication (Hedging) and Derivative Pricing in the Simplest Black{Scholes
Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
12.1.1 Pricing Standard European Calls and Puts . . . . . . . . . . . . . . 504
12.1.2 Hedging Standard European Calls and Puts . . . . . . . . . . . . . 507
12.1.3 Europeans with Piecewise Linear Payos . . . . . . . . . . . . . . . 510
12.1.4 Power Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
12.1.5 Dividend Paying Stock . . . . . . . . . . . . . . . . . . . . . . . . . 515
12.2 Forward Starting and Compound Options . . . . . . . . . . . . . . . . . . 520
12.3 Some European-Style Path-Dependent Derivatives . . . . . . . . . . . . . 526
12.3.1 Risk-Neutral Pricing under GBM . . . . . . . . . . . . . . . . . . . 529
12.3.2 Pricing Single Barrier Options . . . . . . . . . . . . . . . . . . . . . 532
12.3.3 Pricing Lookback Options . . . . . . . . . . . . . . . . . . . . . . . 539
12.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
13 Risk-Neutral Pricing in a Multi-Asset Economy 553
13.1 General Multi-Asset Market Model: Replication and Risk-Neutral Pricing 554
13.2 Black{Scholes PDE and Delta Hedging for Standard Multi-Asset Derivatives
within a General Diusion Model . . . . . . . . . . . . . . . . . . . . . . . 561
13.2.1 Standard European Option Pricing for Multi-Stock GBM . . . . . 564
13.2.2 Explicit Pricing Formulae for the GBM Model . . . . . . . . . . . . 566
13.2.3 Cross-Currency Option Valuation . . . . . . . . . . . . . . . . . . 575
13.3 Equivalent Martingale Measures: Derivative Pricing with General Numeraire
Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580
13.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
14 American Options 593
14.1 Basic Properties of Early-Exercise Options . . . . . . . . . . . . . . . . . . 593
14.2 Arbitrage-Free Pricing of American Options . . . . . . . . . . . . . . . . 596
14.2.1 Optimal Stopping Formulation and Early-Exercise Boundary . . . 596
14.2.2 The Smooth Pasting Condition . . . . . . . . . . . . . . . . . . . . 598
14.2.3 Put-Call Symmetry Relation . . . . . . . . . . . . . . . . . . . . . . 601
14.2.4 Dynamic Programming Approach for Bermudan Options . . . . . . 602
14.3 Perpetual American Options . . . . . . . . . . . . . . . . . . . . . . . . . . 603
14.3.1 Pricing a Perpetual Put Option . . . . . . . . . . . . . . . . . . . . 603
14.3.2 Pricing a Perpetual Call Option . . . . . . . . . . . . . . . . . . . . 606
14.4 Finite-Expiration American Options . . . . . . . . . . . . . . . . . . . . . 606
14.4.1 The PDE Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 606
14.4.2 The Integral Equation Formulation . . . . . . . . . . . . . . . . . . 609
14.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612

15 Interest-Rate Modelling and Derivative Pricing 615
15.1 Basic Fixed Income Instruments . . . . . . . . . . . . . . . . . . . . . . . 615
15.1.1 Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
15.1.2 Forward Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616
15.1.3 Arbitrage-Free Pricing . . . . . . . . . . . . . . . . . . . . . . . . . 617
15.1.4 Fixed Income Derivatives . . . . . . . . . . . . . . . . . . . . . . . 619
15.2 Single-Factor Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620
15.2.1 Diusion Models for the Short Rate Process . . . . . . . . . . . . . 621
15.2.2 PDE for the Zero-Coupon Bond Value . . . . . . . . . . . . . . . . 622
15.2.3 Ane Term Structure Models . . . . . . . . . . . . . . . . . . . . . 624
15.2.4 The Ho{Lee Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 625
15.2.5 The Vasicek Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 626
15.2.6 The Cox{Ingersoll{Ross Model . . . . . . . . . . . . . . . . . . . . 628
15.3 Heath{Jarrow{Morton Formulation . . . . . . . . . . . . . . . . . . . . . . 630
15.3.1 HJM under Risk-Neutral Measure . . . . . . . . . . . . . . . . . . . 631
15.3.2 Relationship between HJM and Ane Yield Models . . . . . . . . 634
15.4 Multifactor Ane Term Structure Models . . . . . . . . . . . . . . . . . . 637
15.4.1 Gaussian Multifactor Models . . . . . . . . . . . . . . . . . . . . . 638
15.4.2 Equivalent Classes of Ane Models . . . . . . . . . . . . . . . . . . 639
15.5 Pricing Derivatives under Forward Measures . . . . . . . . . . . . . . . . . 640
15.5.1 Forward Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 640
15.5.2 Pricing Stock Options under Stochastic Interest Rates . . . . . . . 642
15.5.3 Pricing Options on Zero-Coupon Bonds . . . . . . . . . . . . . . . 644
15.6 LIBOR Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
15.6.1 LIBOR Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
15.6.2 Brace{Gatarek{Musiela Model of LIBOR Rates . . . . . . . . . . . 646
15.6.3 Pricing Caplets, Caps, and Swaps . . . . . . . . . . . . . . . . . . . 648
15.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650
16 Alternative Models of Asset Price Dynamics 653
16.1 Stochastic Volatility Diusion Models . . . . . . . . . . . . . . . . . . . . 653
16.1.1 Local Volatility Models . . . . . . . . . . . . . . . . . . . . . . . . . 653
16.1.2 Constant Elasticity of Variance Model . . . . . . . . . . . . . . . . 656
16.1.3 The Heston Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 661
16.2 Models with Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662
16.2.1 The Poisson Process . . . . . . . . . . . . . . . . . . . . . . . . . . 662
16.2.2 Jump-Diusion Models with a Compound Poisson Component . . . 665
16.2.3 The Variance Gamma Model . . . . . . . . . . . . . . . . . . . . . 668
16.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669
IV Computational Techniques 673
17 Introduction to Monte Carlo and Simulation Methods 675
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
17.1.1 The \Hit-or-Miss" Method . . . . . . . . . . . . . . . . . . . . . . . 676
17.1.2 The Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . 676
17.1.3 Approximation Error and Condence Interval . . . . . . . . . . . . 677
17.1.4 Parallel Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . 678
17.1.5 One Monte Carlo Application: Numerical Integration . . . . . . . . 679
17.2 Generation of Uniformly Distributed Random Numbers . . . . . . . . . . 680
17.2.1 Uniform Probability Distributions . . . . . . . . . . . . . . . . . . . 680

17.2.2 Linear Congruential Generator . . . . . . . . . . . . . . . . . . . . 681
17.3 Generation of Nonuniformly Distributed Random Numbers . . . . . . . . 685
17.3.1 Transformations of Random Variables . . . . . . . . . . . . . . . . 686
17.3.2 Inversion Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687
17.3.3 Composition Methods . . . . . . . . . . . . . . . . . . . . . . . . . 692
17.3.4 Acceptance-Rejection Methods . . . . . . . . . . . . . . . . . . . . 697
17.3.5 Multivariate Sampling . . . . . . . . . . . . . . . . . . . . . . . . . 702
17.4 Simulation of Random Processes . . . . . . . . . . . . . . . . . . . . . . . 705
17.4.1 Simulation of Brownian Processes . . . . . . . . . . . . . . . . . . . 706
17.4.2 Simulation of Gaussian Processes . . . . . . . . . . . . . . . . . . . 708
17.4.3 Diusion Processes: Exact Simulation Methods . . . . . . . . . . . 709
17.4.4 Diusion Processes: Approximation Schemes . . . . . . . . . . . . 712
17.4.5 Simulation of Processes with Jumps . . . . . . . . . . . . . . . . . 716
17.5 Variance Reduction Methods . . . . . . . . . . . . . . . . . . . . . . . . . 719
17.5.1 Numerical Integration by a Direct Monte Carlo Method . . . . . . 720
17.5.2 Importance Sampling Method . . . . . . . . . . . . . . . . . . . . . 721
17.5.3 Change of Probability Measure . . . . . . . . . . . . . . . . . . . . 723
17.5.4 Control Variate Method . . . . . . . . . . . . . . . . . . . . . . . . 724
17.5.5 Antithetic Variate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727
17.5.6 Conditional Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . 728
17.5.7 Stratied Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . 729
17.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738
18 Numerical Applications to Derivative Pricing 739
18.1 Overview of Deterministic Numerical Methods . . . . . . . . . . . . . . . 739
18.1.1 Quadrature Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . 739
18.1.2 Finite-Dierence Methods . . . . . . . . . . . . . . . . . . . . . . . 744
18.2 Pricing European Options . . . . . . . . . . . . . . . . . . . . . . . . . . . 754
18.2.1 Pricing European Options by Quadrature Rules . . . . . . . . . . . 754
18.2.2 Pricing European Options by the Monte Carlo Method . . . . . . . 759
18.2.3 Pricing European Options by Tree Methods . . . . . . . . . . . . . 762
18.2.4 Pricing European Options by PDEs . . . . . . . . . . . . . . . . . . 769
18.2.5 Calibration of Asset Price Models to Empirical Data . . . . . . . . 774
18.3 Pricing Early-Exercise and Path-Dependent Options . . . . . . . . . . . . 776
18.3.1 Pricing American and Bermudan Options . . . . . . . . . . . . . . 776
18.3.2 Pricing Asian Options . . . . . . . . . . . . . . . . . . . . . . . . . 783
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 788
Appendix: Some Useful Integral Identities and Symmetry Properties of
Normal Random Variables 789
Glossary of Symbols and Abbreviations 791
References 795
Index 799

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