Option Valuation: A First Course in Financial Mathematics

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Option Valuation: A First Course in Financial Mathematics

Format:  Hardcover,

252 pages

Publisher: Taylor & Francis

Publish Date: Nov 2011

ISBN-13: 9781439889114

ISBN-10: 1439889112

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Option Valuation: A First Course in Financial Mathematics provides a straightforward introduction to the mathematics and models used in the valuation of financial derivatives. It examines the principles of option pricing in detail via standard binomial and stochastic calculus models. Developing the requisite mathematical background as needed, the text presents an introduction to probability theory and stochastic calculus suitable for undergraduate students in mathematics, economics, and finance.

The first nine chapters of the book describe option valuation techniques in discrete time, focusing on the binomial model. The author shows how the binomial model offers a practical method for pricing options using relatively elementary mathematical tools. The binomial model also enables a clear, concrete exposition of fundamental principles of finance, such as arbitrage and hedging, without the distraction of complex mathematical constructs. The remaining chapters illustrate the theory in continuous time, with an emphasis on the more mathematically sophisticated Black-Scholes-Merton model.

Largely self-contained, this classroom-tested text offers a sound introduction to applied probability through a mathematical finance perspective. Numerous examples and exercises help students gain expertise with financial calculus methods and increase their general mathematical sophistication. The exercises range from routine applications to spreadsheet projects to the pricing of a variety of complex financial instruments. Hints and solutions to odd-numbered problems are given in an appendix and a full solutions manual is available for qualifying instructors.

Specifications

Author:
Publisher: Taylor & Francis
Publish Date: Nov 2011
ISBN-13: 9781439889114
ISBN-10: 1439889112
Format: Hardcover
Number of Pages: 252
Shipping Weight (in pounds): 1.15
Product in Inches (L x W x H): 6.1 x 0.8 x 9.3
Walmart No.: 9781439889114

Chapter outline

Prefacep. xi
Interest and Present Valuep. 1
Compound Interestp. 1
Annuitiesp. 3
Bondsp. 6
Rate of Returnp. 7
Exercisesp. 9
Probability Spacesp. 13
Sample Spaces and Eventsp. 13
Discrete Probability Spacesp. 14
General Probability Spacesp. 16
Conditional Probabilityp. 20
Independencep. 22
Exercisesp. 24
Random Variablesp. 27
Definition and General Propertiesp. 27
Discrete Random Variablesp. 29
Continuous Random Variablesp. 32
Joint Distributionsp. 34
Independent Random Variablesp. 35
Sums of Independent Random Variablesp. 38
Exercisesp. 41
Options and Arbitragep. 43
Arbitragep. 44
Classification of Derivativesp. 46
Forwardsp. 46
Currency Forwardsp. 48
Futuresp. 49
Optionsp. 50
Properties of Optionsp. 53
Dividend-Paying Stocksp. 55
Exercisesp. 57
Discrete-Time Portfolio Processesp. 59
Discrete-Time Stochastic Processesp. 59
Self-Financing Portfoliosp. 61
Option Valuation by Portfoliosp. 64
Exercisesp. 66
Expectation of a Random Variablep. 67
Discrete Case: Definition and Examplesp. 67
Continuous Case: Definition and Examplesp. 68
Properties of Expectationp. 69
Variance of a Random Variablep. 71
The Central Limit Theoremp. 73
Exercisesp. 75
The Binomial Modelp. 77
Construction of the Binomial Modelp. 77
Pricing a Claim in the Binomial Modelp. 80
The Cox-Ross-Rubinstien Formulap. 83
Exercisesp. 86
Conditional Expectation and Discrete-Time Martingalesp. 89
Definition of Conditional Expectationp. 89
Examples of Conditional Expectationp. 92
Properties of Conditional Expectationp. 94
Discrete-Time Martingalesp. 96
Exercisesp. 98
The Binomial Model Revisitedp. 101
Martingales in the Binomial Modelp. 101
Change of Probabilityp. 103
American Claims in the Binomial Modelp. 105
Stopping Timesp. 108
Optimal Exercise of an American Claimp. 111
Dividends in the Binomial Modelp. 114
The General Finite Market Modelp. 115
Exercisesp. 117
Stochastic Calculusp. 119
Differential Equationsp. 119
Continuous-Time Stochastic Processesp. 120
Brownian Motionp. 122
Variation of Brownian Pathsp. 123
Riemann-Stieltjes Integralsp. 126
Stochastic Integralsp. 126
The Ito-Doeblin Formulap. 131
Stochastic Differential Equationsp. 136
Exercisesp. 139
The Black-Scholes-Merton Modelp. 141
The Stock Price SDEp. 141
Continuous-Time Portfoliosp. 142
The Black-Scholes-Merton PDEp. 143
Properties of the BSM Call Functionp. 146
Exercisesp. 149
Continuous-Time Martingalesp. 151
Conditional Expectationp. 151
Martingales: Definition and Examplesp. 152
Martinagale Representation Theoremp. 154
Moment Generating Functionsp. 156
Change of Probability and Girsanov's Theoremp. 158
Exercisesp. 161
The BSM Model Revisitedp. 163
Risk-Neutral Valuation of a Derivativep. 163
Proofs of the Valuation Formulasp. 165
Valuation under Pp. 167
The Feynman-Kac Representation Theoremp. 168
Exercisesp. 171
Other Optionsp. 173
Currency Optionsp. 173
Forward Start Optionsp. 175
Chooser Optionsp. 176
Compound Optionsp. 177
Path-Dependent Derivativesp. 178
Barrier Optionsp. 179
Lookback Optionsp. 185
Asian Optionsp. 191
Quantosp. 195
Options on Dividend-Paying Stocksp. 197
Continuous Dividend Streamp. 197
Discrete Dividend Streamp. 198
American Claims in the BSM Modelp. 200
Exercisesp. 203
Sets and Countingp. 209
Solution of the BSM PDEp. 215
Analytical Properties of the BSM Call Functionp. 219
Hints and Solutions to Odd-Numbered Problemsp. 225
Bibliographyp. 247
Indexp. 249

Book description

Option Valuation: A First Course in Financial Mathematics provides a straightforward introduction to the mathematics and models used in the valuation of financial derivatives. It examines the principles of option pricing in detail via standard binomial and stochastic calculus models. Developing the requisite mathematical background as needed, the text presents an introduction to probability theory and stochastic calculus suitable for undergraduate students in mathematics, economics, and finance.

The first nine chapters of the book describe option valuation techniques in discrete time, focusing on the binomial model. The author shows how the binomial model offers a practical method for pricing options using relatively elementary mathematical tools. The binomial model also enables a clear, concrete exposition of fundamental principles of finance, such as arbitrage and hedging, without the distraction of complex mathematical constructs. The remaining chapters illustrate the theory in continuous time, with an emphasis on the more mathematically sophisticated Black-Scholes-Merton model.

Largely self-contained, this classroom-tested text offers a sound introduction to applied probability through a mathematical finance perspective. Numerous examples and exercises help students gain expertise with financial calculus methods and increase their general mathematical sophistication. The exercises range from routine applications to spreadsheet projects to the pricing of a variety of complex financial instruments. Hints and solutions to odd-numbered problems are given in an appendix and a full solutions manual is available for qualifying instructors.

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