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An Introduction to Financial Option Valuation :  Mathematics, Stochastics and Computation - Desmond Higham

An Introduction to Financial Option Valuation

Mathematics, Stochastics and Computation

Paperback Published: 14th July 2004
ISBN: 9780521547574
Number Of Pages: 273

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This book is intended for use in a rigorous introductory PhD level course in econometrics, or in a field course in econometric theory. It covers the measure-theoretical foundation of probability theory, the multivariate normal distribution with its application to classical linear regression analysis, various laws of large numbers, central limit theorems and related results for independent random variables as well as for stationary time series, with applications to asymptotic inference of M-estimators, and maximum likelihood theory. Some chapters have their own appendices containing the more advanced topics and/or difficult proofs. Moreover, there are three appendices with material that is supposed to be known. Appendix I contains a comprehensive review of linear algebra, including all the proofs. Appendix II reviews a variety of mathematical topics and concepts that are used throughout the main text, and Appendix III reviews complex analysis. Therefore, this book is uniquely self-contained.

Industry Reviews

'... a well organized and well written text. The book 'does what it says on the cover', is written in plain English and I think is an excellent introductory text. It will be useful to students from a wide range of backgrounds and an essential complement to the standard undergraduate course which embeds mathematical finance into probability theory. Finally, with it being studded with references, it provides an easy entry into deeper material.' Chris Barnett, UK Nonlinear News ' ... this is a very accessible basic introduction to the subject and Des Higham's unique writing style with many quotes and side remarks makes the reading even more enjoyable.' L. Grune, Z. Angew. Math. Mech. 'A colleague and I use Desmond Higham's financial options book in our Computational Finance and Applied Optimal (stochastic) Control courses as a very good computational reference, but some of the motivations are very good too, such as call-put parity and the Black-Scholes derivation. Our students find it very helpful for its MATLAB code and we have cited it in a risk-neutral Monte-Carlo paper.' Floyd B. Hanson, University of Illinois at Chicago 'This book provides a clear introduction to elementary option pricing via Matlab. It is eminently suitable for advanced undergraduates and beginning graduates.' Dr Brad Baxter, Birkbeck College, University of London 'The material is presented in a ... vivid and pedagogical manner. ...It could equally well be ready by people with limited mathematical knowledge wanting to learn the basics of mathematical finance ...' Zentralblatt MATH

List of illustrationsp. xiii
Prefacep. xvii
Optionsp. 1
What are options?p. 1
Why do we study options?p. 2
How are options traded?p. 4
Typical option pricesp. 6
Other financial derivativesp. 7
Notes and referencesp. 7
Program of Chapter 1 and walkthroughp. 8
Option valuation preliminariesp. 11
Motivationp. 11
Interest ratesp. 11
Short sellingp. 12
Arbitragep. 13
Put-call parityp. 13
Upper and lower bounds on option valuesp. 14
Notes and referencesp. 16
Program of Chapter 2 and walkthroughp. 17
Random variablesp. 21
Motivationp. 21
Random variables, probability and meanp. 21
Independencep. 23
Variancep. 24
Normal distributionp. 25
Central Limit Theoremp. 27
Notes and referencesp. 28
Program of Chapter 3 and walkthroughp. 29
Computer simulationp. 33
Motivationp. 33
Pseudo-random numbersp. 33
Statistical testsp. 34
Notes and referencesp. 40
Program of Chapter 4 and walkthroughp. 41
Asset price movementp. 45
Motivationp. 45
Efficient market hypothesisp. 45
Asset price datap. 46
Assumptionsp. 48
Notes and referencesp. 49
Program of Chapter 5 and walkthroughp. 50
Asset price model: Part Ip. 53
Motivationp. 53
Discrete asset modelp. 53
Continuous asset modelp. 55
Lognormal distributionp. 56
Features of the asset modelp. 57
Notes and referencesp. 59
Program of Chapter 6 and walkthroughp. 60
Asset price model: Part IIp. 63
Computing asset pathsp. 63
Timescale invariancep. 66
Sum-of-square returnsp. 68
Notes and referencesp. 69
Program of Chapter 7 and walkthroughp. 71
Black-Scholes PDE and formulasp. 73
Motivationp. 73
Sum-of-square increments for asset pricep. 74
Hedgingp. 76
Black-Scholes PDEp. 78
Black-Scholes formulasp. 80
Notes and referencesp. 82
Program of Chapter 8 and walkthroughp. 83
More on hedgingp. 87
Motivationp. 87
Discrete hedgingp. 87
Delta at expiryp. 89
Large-scale testp. 92
Long-Term Capital Managementp. 93
Notesp. 94
Program of Chapter 9 and walkthroughp. 96
The Greeksp. 99
Motivationp. 99
The Greeksp. 99
Interpreting the Greeksp. 101
Black-Scholes PDE solutionp. 101
Notes and referencesp. 102
Program of Chapter 10 and walkthroughp. 104
More on the Black-Scholes formulasp. 105
Motivationp. 105
Where is [mu]?p. 105
Time dependencyp. 106
The big picturep. 106
Change of variablesp. 108
Notes and referencesp. 111
Program of Chapter 11 and walkthroughp. 111
Risk neutralityp. 115
Motivationp. 115
Expected payoffp. 115
Risk neutralityp. 116
Notes and referencesp. 118
Program of Chapter 12 and walkthroughp. 120
Solving a nonlinear equationp. 123
Motivationp. 123
General problemp. 123
Bisectionp. 123
Newtonp. 124
Further practical issuesp. 127
Notes and referencesp. 127
Program of Chapter 13 and walkthroughp. 128
Implied volatilityp. 131
Motivationp. 131
Implied volatilityp. 131
Option value as a function of volatilityp. 131
Bisection and Newtonp. 133
Implied volatility with real datap. 135
Notes and referencesp. 137
Program of Chapter 14 and walkthroughp. 137
Monte Carlo methodp. 141
Motivationp. 141
Monte Carlop. 141
Monte Carlo for option valuationp. 144
Monte Carlo for Greeksp. 145
Notes and referencesp. 148
Program of Chapter 15 and walkthroughp. 149
Binomial methodp. 151
Motivationp. 151
Methodp. 151
Deriving the parametersp. 153
Binomial method in practicep. 154
Notes and referencesp. 156
Program of Chapter 16 and walkthroughp. 159
Cash-or-nothing optionsp. 163
Motivationp. 163
Cash-or-nothing optionsp. 163
Black-Scholes for cash-or-nothing optionsp. 164
Delta behaviourp. 166
Risk neutrality for cash-or-nothing optionsp. 167
Notes and referencesp. 168
Program of Chapter 17 and walkthroughp. 170
American optionsp. 173
Motivationp. 173
American call and putp. 173
Black-Scholes for American optionsp. 174
Binomial method for an American putp. 176
Optimal exercise boundaryp. 177
Monte Carlo for an American putp. 180
Notes and referencesp. 182
Program of Chapter 18 and walkthroughp. 183
Exotic optionsp. 187
Motivationp. 187
Barrier optionsp. 187
Lookback optionsp. 191
Asian optionsp. 192
Bermudan and shout optionsp. 193
Monte Carlo and binomial for exoticsp. 194
Notes and referencesp. 196
Program of Chapter 19 and walkthroughp. 199
Historical volatilityp. 203
Motivationp. 203
Monte Carlo-type estimatesp. 203
Accuracy of the sample variance estimatep. 204
Maximum likelihood estimatep. 206
Other volatility estimatesp. 207
Example with real datap. 208
Notes and referencesp. 209
Program of Chapter 20 and walkthroughp. 210
Monte Carlo Part II: variance reduction by antithetic variatesp. 215
Motivationp. 215
The big picturep. 215
Dependencep. 216
Antithetic variates: uniform examplep. 217
Analysis of the uniform casep. 219
Normal casep. 221
Multivariate casep. 222
Antithetic variates in option valuationp. 222
Notes and referencesp. 225
Program of Chapter 21 and walkthroughp. 225
Monte Carlo Part III: variance reduction by control variatesp. 229
Motivationp. 229
Control variatesp. 229
Control variates in option valuationp. 231
Notes and referencesp. 232
Program of Chapter 22 and walkthroughp. 234
Finite difference methodsp. 237
Motivationp. 237
Finite difference operatorsp. 237
Heat equationp. 238
Discretizationp. 239
FTCS and BTCSp. 240
Local accuracyp. 246
Von Neumann stability and convergencep. 247
Crank-Nicolsonp. 249
Notes and referencesp. 251
Program of Chapter 23 and walkthroughp. 252
Finite difference methods for the Black-Scholes PDEp. 257
Motivationp. 257
FTCS, BTCS and Crank-Nicolson for Black-Scholesp. 257
Down-and-out call examplep. 260
Binomial method as finite differencesp. 261
Notes and referencesp. 262
Program of Chapter 24 and walkthroughp. 265
Referencesp. 267
Indexp. 271
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780521547574
ISBN-10: 0521547571
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 273
Published: 14th July 2004
Country of Publication: GB
Dimensions (cm): 24.13 x 17.02  x 1.78
Weight (kg): 0.48

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