Measure, Probability, and Mathematical Finance – A Problem–Oriented Approach
A Problem–Oriented Approach
Gebonden Engels 2014 9781118831960Samenvatting
An introduction to the mathematical theory and financial models developed and used on Wall Street
Providing both a theoretical and practical approach to the underlying mathematical theory behind financial models, Measure, Probability, and Mathematical Finance: A Problem–Oriented Approach presents important concepts and results in measure theory, probability theory, stochastic processes, and stochastic calculus. Measure theory is indispensable to the rigorous development of probability theory and is also necessary to properly address martingale measures, the change of numeraire theory, and LIBOR market models. In addition, probability theory is presented to facilitate the development of stochastic processes, including martingales and Brownian motions, while stochastic processes and stochastic calculus are discussed to model asset prices and develop derivative pricing models.
The authors promote a problem–solving approach when applying mathematics in real–world situations, and readers are encouraged to address theorems and problems with mathematical rigor. In addition, Measure, Probability, and Mathematical Finance features:
A comprehensive list of concepts and theorems from measure theory, probability theory, stochastic processes, and stochastic calculus
Over 500 problems with hints and select solutions to reinforce basic concepts and important theorems
Classic derivative pricing models in mathematical finance that have been developed and published since the seminal work of Black and Scholes
Measure, Probability, and Mathematical Finance: A Problem–Oriented Approach is an ideal textbook for introductory quantitative courses in business, economics, and mathematical finance at the upper–undergraduate and graduate levels. The book is also a useful reference for readers who need to build their mathematical skills in order to better understand the mathematical theory of derivative pricing models.
Specificaties
Lezersrecensies
Inhoudsopgave
<p>Financial Glossary xxii</p>
<p>Part I Measure Theory</p>
<p>1 Sets and Sequences 3</p>
<p>2 Measures 15</p>
<p>3 Extension of Measures 29</p>
<p>4 Lebesgue–Stieltjes Measures 37</p>
<p>5 Measurable Functions 47</p>
<p>6 Lebesgue Integration 57</p>
<p>7 The Radon–Nikodym Theorem 77</p>
<p>8 L<sup>P</sup> Spaces 85</p>
<p>9 Convergence 97</p>
<p>10 Product Measures 113</p>
<p>Part II Probability Theory</p>
<p>11 Events and Random Variables 127</p>
<p>12 Independence 141</p>
<p>13 Expectation 161</p>
<p>14 Conditional Expectation 173</p>
<p>15 Inequalities 189</p>
<p>16 Law of Large Numbers 199</p>
<p>17 Characteristic Functions 217</p>
<p>18 Discrete Distributions 227</p>
<p>19 Continuous Distributions 239</p>
<p>20 Central Limit Theorems 257</p>
<p>Part III Stochastic Processes</p>
<p>21 Stochastic Processes 271</p>
<p>22 Martingales 291</p>
<p>23 Stopping Times 301</p>
<p>24 Martingale Inequalities 321</p>
<p>25 Martingale Convergence Theorems 333</p>
<p>26 Random Walks 343</p>
<p>27 Poisson Processes 357</p>
<p>28 Brownian Motion 373</p>
<p>29 Markov Processes 389</p>
<p>30 Lévy Processes 401</p>
<p>Part IV Stochastic Calculus</p>
<p>31 The Wiener Integral 421</p>
<p>32 The Itô Integral 431</p>
<p>33 Extension of the Itô Integral 453</p>
<p>34 Martingale Stochastic Integrals 463</p>
<p>35 The Itô Formula 477</p>
<p>36 Martingale Representation Theorem 495</p>
<p>37 Change of Measure 503</p>
<p>38 Stochastic Differential Equations 515</p>
<p>39 Diffusion 531</p>
<p>40 The Feynman–Kac Formula 547</p>
<p>Part V Stochastic Financial Models</p>
<p>41 Discrete–Time Models 561</p>
<p>42 Black–Scholes Option Pricing Models 579</p>
<p>43 Path–Dependent Options 593</p>
<p>44 American Options 609</p>
<p>45 Short Rate Models 629</p>
<p>46 Instantaneous Forward Rate Models 647</p>
<p>47 LIBOR Market Models 667</p>
<p>References 687</p>
<p>List of Symbols 703</p>
<p>Subject Index 707</p>
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