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Cover
Cover
Contents
8
Preface
12
Chapter 1. Overview of financial derivatives
14
Chapter 2. Introduction to stochastic processes
18
2.1 Brownian motion
18
2.2 A Brownian model of asset price movements
22
2.3. Ito's formula (or lemma)
23
2.4 Girsanov's theorem
25
2.5 Ito's lemma for multiasset geometric Brownian motion
26
2.6 Ito product and quotient rules in two dimensions
28
2.7 Ito product in n dimensions
31
2.8 The Brownian bridge
32
2.9 Time-transformed Brownian motion
34
2.10 Ornstein-Uhlenbeck process
37
2.11 The Ornstein-Uhlenbeck bridge
40
2.12 Other useful results
44
2.13 Selected problems
46
Chapter 3. Generation of random variates
50
3.1 Introduction
50
3.2 Pseudo-random and quasi-random sequences
51
3.3 Generation of multivariate distributions: independent variates
54
3.4 Generation of multivariate distributions: correlated variates
60
Chapter 4. European options
72
4.1 Introduction
72
4.2 Pricing derivatives using a martingale measure
72
4.3 Put call parity
73
4.4 Vanilla options and the Black-Scholes model
75
4.5 Barrier options
98
Chapter 5. Single asset American options
110
5.1 Introduction
110
5.2 Approximations for vanilla American options
110
5.3 Lattice methods for vanilla options
127
5.4 Grid methods for vanilla options
148
5.5 Pricing American options using a stochastic lattice
185
Chapter 6. Multiasset options
194
6.1 Introduction
194
6.2 The multiasset Black-Scholes equation
194
6.3 Multidimensional Monte Carlo methods
196
6.4 Introduction to multidimensional lattice methods
198
6.5 Two asset options
203
6.6 Three asset options
214
6.7 Four asset options
218
Chapter 7. Other financial derivatives
222
7.1 Introduction
222
7.2 Interest rate derivatives
222
7.3 Foreign exchange derivatives
241
7.4 Credit derivatives
245
7.5 Equity derivatives
250
Chapter 8. C# portfolio pricing application
258
8.1 Introduction
258
8.2 Storing and retrieving the market data
267
8.3 The PricingUtils class and the Analytics_MathLib
275
8.4 Equity deal classes
280
8.5 FX deal classes
293
Appendix A: The Greeks for vanilla European options
302
A.1 Introduction
302
A.2 Gamma
303
A.3 Delta
304
A.4 Theta
305
A.5 Rho
306
A.6 Vega
307
Appendix B: Barrier option integrals
308
B.1 The down and out call
308
B.2 The up and out call
311
Appendix C: Standard statistical results
316
C.1 The law of large numbers
316
C.2 The central limit theorem
316
C.3 The variance and covariance of random variables
318
C.4 Conditional mean and covariance of normal distributions
323
C.5 Moment generating functions
324
Appendix D: Statistical distribution functions
326
D.1 The normal (Gaussian) distribution
326
D.2 The lognormal distribution
328
D.3 The Student's t distribution
330
D.4 The general error distribution
332
Appendix E: Mathematical reference
334
E.1 Standard integrals
334
E.2 Gamma function
334
E.3 The cumulative normal distribution function
335
E.4 Arithmetic and geometric progressions
336
Appendix F: Black-Scholes finite-difference schemes
338
F.1 The general case
338
F.2 The log transformation and a uniform grid
338
Appendix G: The Brownian bridge: alternative derivation
342
Appendix H: Brownian motion: more results
346
H.1 Some results concerning Brownian motion
346
H.2 Proof of Eq. (H.1.2)
347
H.3 Proof of Eq. (H.1.4)
348
H.4 Proof of Eq. (H.1.5)
348
H.5 Proof of Eq. (H.1.6)
348
H.6 Proof of Eq. (H.1.7)
351
H.7 Proof of Eq. (H.1.8)
351
H.8 Proof of Eq. (H.1.9)
351
H.8 Proof of Eq. (H.1.10)
352
Appendix I: The Feynman-Kac formula
354
Appendix J: Answers to problems
356
J.1 Problem 1
356
J.2 Problem 2
357
J.3 Problem 3
358
J.4 Problem 4
359
J.5 Problem 5
359
J.6 Problem 6
360
J.7 Problem 7
361
J.8 Problem 8
363
J.9 Problem 9
363
J.10 Problem 10
365
J.11 Problem 11
367
References
368
Index
374
Glossary
384
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