Mathematical Methods for Finance – Tools for Asset and Risk Management

<p>Preface xi</p>
<p>About the Authors xvii</p>
<p>CHAPTER 1 Basic Concepts: Sets, Functions, and Variables 1</p>
<p>Introduction 2</p>
<p>Sets and Set Operations 2</p>
<p>Distances and Quantities 6</p>
<p>Functions 10</p>
<p>Variables 10</p>
<p>Key Points 11</p>
<p>CHAPTER 2 Differential Calculus 13</p>
<p>Introduction 14</p>
<p>Limits 15</p>
<p>Continuity 17</p>
<p>Total Variation 19</p>
<p>The Notion of Differentiation 19</p>
<p>Commonly Used Rules for Computing Derivatives 21</p>
<p>Higher–Order Derivatives 26</p>
<p>Taylor Series Expansion 34</p>
<p>Calculus in More Than One Variable 40</p>
<p>Key Points 41</p>
<p>CHAPTER 3 Integral Calculus 43</p>
<p>Introduction 44</p>
<p>Riemann Integrals 44</p>
<p>Lebesgue–Stieltjes Integrals 47</p>
<p>Indefinite and Improper Integrals 48</p>
<p>The Fundamental Theorem of Calculus 51</p>
<p>Integral Transforms 52</p>
<p>Calculus in More Than One Variable 57</p>
<p>Key Points 57</p>
<p>CHAPTER 4 Matrix Algebra 59</p>
<p>Introduction 60</p>
<p>Vectors and Matrices Defined 61</p>
<p>Square Matrices 63</p>
<p>Determinants 66</p>
<p>Systems of Linear Equations 68</p>
<p>Linear Independence and Rank 69</p>
<p>Hankel Matrix 70</p>
<p>Vector and Matrix Operations 72</p>
<p>Finance Application 78</p>
<p>Eigenvalues and Eigenvectors 81</p>
<p>Diagonalization and Similarity 82</p>
<p>Singular Value Decomposition 83</p>
<p>Key Points 83</p>
<p>CHAPTER 5 Probability: Basic Concepts 85</p>
<p>Introduction 86</p>
<p>Representing Uncertainty with Mathematics 87</p>
<p>Probability in a Nutshell 89</p>
<p>Outcomes and Events 91</p>
<p>Probability 92</p>
<p>Measure 93</p>
<p>Random Variables 93</p>
<p>Integrals 94</p>
<p>Distributions and Distribution Functions 96</p>
<p>Random Vectors 97</p>
<p>Stochastic Processes 100</p>
<p>Probabilistic Representation of Financial Markets 102</p>
<p>Information Structures 103</p>
<p>Filtration 104</p>
<p>Key Points 106</p>
<p>CHAPTER 6 Probability: Random Variables and Expectations 107</p>
<p>Introduction 109</p>
<p>Conditional Probability and Conditional Expectation 110</p>
<p>Moments and Correlation 112</p>
<p>Copula Functions 114</p>
<p>Sequences of Random Variables 116</p>
<p>Independent and Identically Distributed Sequences 117</p>
<p>Sum of Variables 118</p>
<p>Gaussian Variables 120</p>
<p>Appproximating the Tails of a Probability Distribution: Cornish–Fisher Expansion and Hermite Polynomials 123</p>
<p>The Regression Function 129</p>
<p>Fat Tails and Stable Laws 131</p>
<p>Key Points 144</p>
<p>CHAPTER 7 Optimization 147</p>
<p>Introduction 148</p>
<p>Maxima and Minima 149</p>
<p>Lagrange Multipliers 151</p>
<p>Numerical Algorithms 156</p>
<p>Calculus of Variations and Optimal Control Theory 161</p>
<p>Stochastic Programming 163</p>
<p>Application to Bond Portfolio: Liability–Funding Strategies 164</p>
<p>Key Points 178</p>
<p>CHAPTER 8 Difference Equations 181</p>
<p>Introduction 182</p>
<p>The Lag Operator L 183</p>
<p>Homogeneous Difference Equations 183</p>
<p>Recursive Calculation of Values of Difference Equations 192</p>
<p>Nonhomogeneous Difference Equations 195</p>
<p>Systems of Linear Difference Equations 201</p>
<p>Systems of Homogeneous Linear Difference Equations 202</p>
<p>Key Points 209</p>
<p>CHAPTER 9 Differential Equations 211</p>
<p>Introduction 212</p>
<p>Differential Equations Defined 213</p>
<p>Ordinary Differential Equations 213</p>
<p>Systems of Ordinary Differential Equations 216</p>
<p>Closed–Form Solutions of Ordinary Differential Equations 218</p>
<p>Numerical Solutions of Ordinary Differential Equations 222</p>
<p>Nonlinear Dynamics and Chaos 228</p>
<p>Partial Differential Equations 231</p>
<p>Key Points 237</p>
<p>CHAPTER 10 Stochastic Integrals 239</p>
<p>Introduction 240</p>
<p>The Intuition behind Stochastic Integrals 243</p>
<p>Brownian Motion Defined 248</p>
<p>Properties of Brownian Motion 254</p>
<p>Stochastic Integrals Defined 255</p>
<p>Some Properties of Ito Stochastic Integrals 259</p>
<p>Martingale Measures and the Girsanov Theorem 260</p>
<p>Key Points 266</p>
<p>CHAPTER 11 Stochastic Differential Equations 267</p>
<p>Introduction 268</p>
<p>The Intuition behind Stochastic Differential Equations 269</p>
<p>Ito Processes 272</p>
<p>Stochastic Differential Equations 273</p>
<p>Generalization to Several Dimensions 276</p>
<p>Solution of Stochastic Differential Equations 278</p>
<p>Derivation of Ito s Lemma 282</p>
<p>Derivation of the Black–Scholes Option Pricing Formula 284</p>
<p>Key Points 291</p>
<p>Index 293</p>