Stochastic Calculus and Financial Applications

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Springer Science & Business Media, 2001 - Business & Economics - 300 pages
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The Wharton School course on which the book is based is designed for energetic students who have had some experience with probability and statistics, but who have not had advanced courses in stochastic processes. Even though the course assumes only a modest background, it moves quickly and - in the end - students can expect to have the tools that are deep enough and rich enough to be relied upon throughout their professional careers. The course begins with simple random walk and the analysis of gambling games. This material is used to motivate the theory of martingales, and, after reaching a decent level of confidence with discrete processes, the course takes up the more demanding development of continuous time stochastic process, especially Brownian motion. The construction of Brownian motion is given in detail, and enough material on the subtle properties of Brownian paths is developed so that the student should sense of when intuition can be trusted and when it cannot. The course then takes up the It┐ integral and aims to provide a development that is honest and complete without being pedantic. With the It┐ integral in hand, the course focuses more on models. Stochastic processes of importance in Finance and Economics are developed in concert with the tools of stochastic calculus that are needed in order to solve problems of practical importance. The financial notion of replication is developed, and the Black-Scholes PDE is derived by three different methods. The course then introduces enough of the theory of the diffusion equation to be able to solve the Black-Scholes PDE and prove the uniqueness of the solution.
 

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Contents

Random Walk and First Step Analysis
ix
13 Tossing an Unfair Coin
3
14 Numerical Calculation and Intuition
5
16 Exercises
7
First Martingale Steps
9
22 New Martingales from Old
11
23 Revisiting the Old Ruins
13
24 Submartingales
15
93 Matching Product Process Coefficients
137
94 Existence and Uniqueness Theorems
140
95 Systems of SDEs
146
96 Exercises
147
Arbitrage and SDEs
151
102 The BlackScholes Model
154
103 The BlackScholes Formula
156
104 Two Original Derivations
158

25 Doobs Inequalities
17
26 Martingale Convergence
20
27 Exercises
24
Brownian Motion
27
31 Covariances and Characteristic Functions
28
32 Visions of a Series Approximation
31
33 Two Wavelets
33
34 Wavelet Representation of Brownian Motion
34
35 Scaling and Inverting Brownian Motion
38
36 Exercises
39
Martingales The Next Steps
41
42 Conditional Expectations
42
43 Uniform Integrability
45
44 Martingales in Continuous Time
48
45 Classic Brownian Motion Martingales
53
46 Exercises
56
Richness of Paths
59
52 Not Too Smooth
61
53 Two Reflection Principles
64
54 The Invariance Principle and Donskers Theorem
68
55 Random Walks Inside Brownian Motion
70
56 Exercises
75
It˘ Integration
77
It˘s Integral as a Process
80
Benefits and Costs
83
65 Pathwise Interpretation of Ito Integrals
85
66 Approximation in H▓
88
67 Exercises
91
Localization and It˘s Integral
93
72 An Intuitive Representation
97
73 Why Just L▓LOC ?
100
74 Local Martingales and Honest Ones
101
75 Alternative Fields and Changes of Time
104
76 Exercises
107
It˘s Formula
109
82 First Consequences and Enhancements
113
83 Vector Extension and Harmonic Functions
118
84 Functions of Processes
121
85 The General It˘ Formula
124
86 Quadratic Variation
126
87 Exercises
132
Stochastic Differential Equations
135
105 The Perplexing Power of a Formula
163
106 Exercises
165
The Diffusion Equation
167
112 Solutions of the Diffusion Equation
170
113 Uniqueness of Solutions
176
114 How to Solve the BlackScholes PDE
180
115 Uniqueness and the BlackScholes PDE
185
116 Exercises
187
Representation Theorems
189
122 The Martingale Representation Theorem
194
123 Continuity of Conditional Expectations
199
124 Representation via Time Change
201
125 Levys Characterization of Brownian Motion
202
126 Bedrock Approximation Techniques
204
127 Exercises
209
Girsanov Theory
211
132 Tilting a Process
213
133 Simplest Girsanov Theorem
216
134 Creation of Martingales
219
135 Shifting the General Drift
220
136 Exponential Martingales and Novikovs Condition
223
137 Exercises
227
Arbitrage and Martingales
231
142 The Valuation Formula in Continuous Time
233
143 The BlackScholes Formula via Martingales
239
144 American Options
242
145 SelfFinancing and SelfDoubt
244
146 Admissible Strategies and Completeness
250
147 Perspective on Theory and Practice
255
148 Exercises
257
The FeynmanKac Connection
261
152 The FeynmanKac Connection for Brownian Motion
263
153 LÚvys Arcsin Law
265
154 The FeynmanKac Connection for Diffusions
268
155 FeynmanKac and the BlackScholes PDEs
269
156 Exercises
272
Mathematical Tools
275
Comments and Credits
283
Bibliography
291
Index
295
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Page iii - The course begins with simple random walk and the analysis of gambling games. This material is used to motivate the theory of martingales, and, after reaching a decent level of confidence with discrete processes, the course takes up the more demanding development of continuous-time stochastic processes, especially Brownian motion.
Page iii - Stochastic processes of importance in finance and economics are developed in concert with the tools of stochastic calculus that are needed to solve problems of practical importance.

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About the author (2001)

J. Michael Steele is C.F. Koo Professor of Statistics at the Wharton School of the University of Pennsylvania.

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