Table of Contents for Pricing and Hedging of Derivative Securities. By Lars Tyge Nielsen. Textbook in continuous-time finance theory. Oxford University Press, 1999.

1  Stochastic Processes  2 

1.1  Basic Notions  2 
1.2  Brownian Motions  5 
1.3  Generalized Brownian Motions  10 
1.4  Information Structures  14 
1.5  Wiener Processes  17 
1.6  Generalized Wiener Processes  21 
1.7  (*) Identification of Processes  26 
1.8  Time Integrals  30 
1.9  Stochastic Integrals  32 
1.10  (*) Predictable Processes  44 
1.11  Summary  46 
1.12  (*) Notes  49 

2  Ito Calculus  52 

2.1  Ito Processes and Ito’s Lemma  52 
2.2  Integrals with Respect to Ito Processes  61 
2.3  Further Manipulations of Ito ‘s Formula  65 
2.4  Stochastic Exponentials  73 
2.5  Girsanov’s Theorem  76 
2.6  Summary  87 
2.7  (*) Notes  89 

3  Gaussian Processes  91 

3.1  Basic Notions  91 
3.2  Deterministic Integrands  93 
3.3  (*) Brownian Bridge Processes  96 
3.4  Conditionally Gaussian One-Factor Processes  103 
3.5  Ornstein-Uhlenbeck Processes  107 
3.6  Summary  114 
3.7  (*) Notes  116 

4  Securities and Trading Strategies  117 

4.1  Elements of the Model  118 
4.2  (*) Almost Simple Trading Strategies  125 
4.3  State Prices  130 
4.4  The Interest Rate and the Prices of Risk  131 
4.5  Existence and Uniqueness of Prices of Risk  133 
4.6  Arbitrage and Admissibility  145 
4.7  (*) The Doubling Strategy  148 
4.8  Changing the Unit of Account  152 
4.9  Summary  156 
4.10  (*) Notes  160 

5  The Martingale Valuation Principle  165 

5.1  Replication of Claims  165 
5.2  Delta Hedging  168 
5.3  Making a Trading Strategy Self-Financing  168 
5.4  Dynamically Complete Markets  172 
5.5  How to Replicate  175 
5.6  Example: Cash-or-Nothing Options  177 
5.7  The State Price Process as a Primitive  182 
5.8  Risk Adjusted Probabilities  185 
5.9  Summary  191 
5.10  (*) Notes  194 

6  The Black-Scholes Model  197 

6.1  Review of the Black-Scholes Economy  197 
6.2  The Value Function  201 
6.3  Cash-or-Nothing Options Revisited  208 
6.4  Asset-Or-Nothing Options  211 
6.5  Standard Call Options  214 
6.6  Standard Put Options  225 
6.7  (*) Black-Scholes and the Heat Equation  228 
6.8  (*) The Black-Scholes PDE: Terminal Data  231 
6.9  (*) The Black-Scholes PDE: Integrability  237 
6.10  (*) The Black-Scholes PDE: Uniqueness  240 
6.11  Summary  242 
6.12  (*) Notes  247 

7  Gaussian Term Structure Models  249 

7.1  Zero-Coupon Bonds and Forward Rates  249 
7.2  The Vasicek Model  253 
7.3  The Risk Adjusted Dynamics as Primitives  261 
7.4  The Vasicek Model: Forward Rates  263 
7.5  The Vasicek Model: Yields  272 
7.6  The Merton Model  280 
7.7  The Extended Vasicek Model  291 
7.8  The Simplified Hull-White Model  301 
7.9  The Continuous-Time Ho-Lee Model  309 
7.10  Summary  309 
7.11  (*) Notes  315 

A  Measure and Probability  317 

A.1  Sigma-Algebras  317 
A.2  Measures and Measure Spaces  323 
A.3  Borel Sigma-Algebras and Lebesgue Measure  328 
A.4  Measurable Mappings  334 
A.5  Convergence in Probability  340 
A.6  Measures and Distribution Functions  341 
A.7  Stochastic Independence  342 

B  Lebesgue Integrals and Expectations  346 

B.1  Lebesgue Integration  346 
B.2  Tonelli’s and Fubini’s Theorems  354 
B.3  Densities and Absolute Continuity  358 
B.4  Locally Integrable Functions  362 
B.5  Conditional Expectations and Probabilities  363 
B.6  Lp-Spaces  367 

C  The Heat Equation  370 

C.1  The Martingale Solution  371 
C.2  The Heat Equation: Initial Data  377 
C.3  The Heat Equation: Integrability  382 
C.4  The Heat Equation: Uniqueness  385 
C.5  Notes  387 

D  Suggested Solutions to Exercises  389 

D.1  Solutions for Chapter 1  389 
D.2  Solutions for Chapter 2  496 
D.3  Solutions for Chapter 3  497 
D.4  Solutions for Chapter 4  497 
D.5  Solutions for Chapter 5  499 
D.6  Solutions for Chapter 6  401 
D.7  Solutions for Chapter 7  413 

E  Suggested Solutions to Exercises  421 

E.1  Solutions for Appendix A  421 
E.2  Solutions for Appendix B  427 

References  434 

Index  439