Table Of Contents:

Introduction 1(6)

Audience, Highlights, Agenda 1(3)

Software Installation 4(2)

Acknowledgments 6(1)

Cash Account Evolution 7(18)

Symbolic Solutions of ODEs 7(6)

Numerical Solutions of ODEs 13(12)

Stock Price Evolution 25(58)

What are Stocks? 25(1)

Stock Price Modeling: Stochastic Differential Equations 26(20)

Idea of Stochastic Differential Equations 26(1)

Normal Distribution 27(6)

Markov Random Processes 33(1)

Brownian Motion 34(4)

Stochastic Integral 38(5)

Stochastic Differential Equations 43(3)

Ito Calculus 46(10)

Ito Product Rule 46(1)

Ito Chain Rule 47(1)

Ito Theorem 47(2)

Monte--Carlo Simulation ``Proof'' of the Ito Theorem 49(2)

An Application: Solving the Simplest Stock Price SDE Model 51(5)

Multivariable and Symbolic Ito Calculus 56(23)

Monte--Carlo Solver for m-Dimensional SDEs 56(4)

Symbolic m-Dimensional Ito Chain Rule 60(1)

Some Calculus and Linear Algebra 60(3)

Derivation of the m-Dimensional Ito Chain Rule 63(2)

Implementation of the m-Dimensional Ito Chain Rule 65(3)

Ito Product Rule Revisited 68(1)

The Simplest Price Model: m-Stocks 69(1)

Derivation 69(2)

Example; Using Multinormal Distribution 71(8)

Relationship Between SDEs and PDEs 79(4)

European Style Stock Options 83(58)

What Are Stock Options? 83(4)

Black--Scholes PDE and Hedging 87(20)

Derivation of the Black--Scholes PDE 87(4)

Black--Scholes Hedging Implemented 91(1)

Put Hedging Implemented 91(7)

Call Hedging Implemented 98(4)

Probabilistic Interpretation of the Solution of the Black--Scholes PDE 102(1)

Derivation 102(2)

An Experiment 104(3)

Solving Black--Scholes PDE Symbolically 107(29)

Heat PDE 107(1)

Probabilistic Derivation of the Solution of Heat PDE 107(3)

Solving Heat PDE: Examples of Explicit Solutions 110(2)

Uniqueness for Heat PDE with Exponential Growth at Infinity 112(2)

Monte--Carlo Simulation Solutions of PDEs 114(2)

Reduction of Black--Scholes to Heat PDE 116(4)

Solution of Heat PDE; Back to Black--Scholes PDE 120(8)

Black--Scholes Formulas 128(1)

Options Volatility 128(4)

Put--Call Parity 132(1)

Sensitivity Analysis 132(4)

Generalized Black--Scholes Formulas: Time-Dependent Data 136(5)

Stock Market Statistics 141(56)

Remarks 141(1)

Stock Market Data Import and Manipulation 141(8)

Volatility Estimates: Scalar Case 149(4)

First Method 149(2)

Second Method 151(1)

Experiments: Estimating Volatility 152(1)

Real Data: Estimating Volatility 153(1)

Appreciation Rate Estimates: Scalar Case 153(4)

An Estimate 153(1)

Confidence Interval 154(2)

Monte--Carlo Experiments: Estimating the Appreciation Rate 156(1)

Statistical Experiments: Bayesian and Non-Bayesian 157(11)

Mathematical Framework for Statistical Experiments and Estimation 157(1)

Uniform Prior 158(3)

Non-Uniform Prior 161(2)

Unbounded Parameter Space 163(1)

Non-Uniform Prior 163(2)

``Uniform Prior'' 165(1)

Bayesian and Non-Bayesian Appreciation Rate Estimates 166(2)

Vector Basic Price Model Statistics 168(15)

Volatility in the Vector Base Price Model 168(2)

Non-Bayesian Estimate for the Vector Appreciation Rate 170(1)

Bayesian Statistical Estimate for the Vector Appreciation Rate 171(1)

Experiments: Statistics for Vector Basic Price Model 172(1)

Experiment Set Up 172(2)

Estimate of σ.σT 174(1)

Non-Bayesian Estimate for a 175(1)

Bayesian Estimate for a 176(2)

Volatility and Appreciation Rate Estimates for Real Vector Data 178(1)

Data Import and Formatting 178(1)

Estimating and Using σ.σT 179(1)

Non-Bayesian Appreciation Rate Estimates 180(1)

Bayesian Appreciation Rate Estimates 181(2)

Dynamic Statistics: Filtering of Conditional Gaussian Processes 183(14)

Conditional Gaussian Filtering 183(1)

Kalman--Bucy Filtering Implemented 184(1)

General Equations 184(2)

Recursive Estimates for the Appreciation Rate 186(2)

An Experiment 188(4)

Conditional Gaussian Filtering Implemented: An Example 192(1)

Filtering Equations 192(1)

Monte--Carlo Simulation 193(4)

Implied Volatility for European Options 197(70)

Remarks 197(1)

Option Market Data 198(6)

Black--Scholes Theory vs. Market Data: Implied Volatility 204(16)

Black--Scholes Theory and Market Data 204(4)

Constant Implied Volatility 208(1)

Single Option Implied Volatility 208(4)

Average Implied Volatility 212(1)

The Least ``Square'' Constant Implied Volatility 212(3)

Time-Dependent Implied Volatility 215(1)

Time-Dependent Implied Volatility: ``Symbolic'' Solutions 215(3)

Market Timing? 218(2)

Numerical PDEs, Optimal Control, and Implied Volatility 220(47)

Remarks 220(1)

Tridiagonal Implicit Finite Difference Solution of Parabolic PDEs 220(10)

Pricing European Options with Stock Price Dependent Data 230(3)

Dupire Partial Differential Equation 233(1)

Formulation and Verification 233(3)

Numerical Solution of the Strike Price Dependent Dupire PDE 236(1)

A Numerical Implied Volatility Problem 236(1)

Optimal Control of Differential Equations 236(1)

Optimal Control of ODEs with Quadratic Cost: Explicit Solutions 236(3)

An Example 239(4)

An Extension 243(2)

Yet Another Extension 245(2)

General Implied Volatility: Optimal Control of Dupire PDEs 247(6)

Computational Example: Call Implied Volatility for QQQ 253(1)

Data Import 253(1)

Single Iterative Step of the Steepest Descent Method 254(7)

Iteration 261(6)

American Style Stock Options 267(68)

Remarks 267(1)

American Options and Obstacle Problems 268(41)

American Options, Optimal Stopping, and Obstacle Problem 268(5)

Equivalent Formulations of Obstacle Problems 273(1)

Variational Inequality Problem 273(1)

Calculus of Variations Problem 274(3)

Complementarity Problem 277(1)

Fully Non-Linear PDE Problem 278(1)

Semilinear PDE Problem 1 278(2)

Semilinear PDE Problem 2 280(1)

Free Boundary Value Problem 281(1)

Computing Solution: Free Boundary Value Problem 281(2)

Computing Solution: Maximal Boundary Value Problem 283(1)

Free Boundary Value Problem: Non-Uniqueness 284(3)

Maximal Boundary Value Problem: Uniqueness 287(2)

``Perpetual American Options'' 289(1)

Steady State Obstacle Problem for Black--Scholes PDE 289(4)

Steady State Obstacle Problem for Dupire PDE 293(3)

Fast Numerical Solution of Obstacle Problems for Black--Scholes PDE 296(7)

Fast Numerical Solution of Obstacle Problems for the Dupire PDE 303(6)

General Implied Volatility for American Options 309(26)

Implied Volatility via Optimal Control of Obstacle Problems 309(4)

Tridiagonal Solver for Parabolic PDEs in Non-Cylindrical Domains 313(5)

Computational Example: Put Implied Volatility for QQQ 318(1)

Data Import 318(2)

Single Iterative Steepest Descent Step 320(7)

Iteration 327(3)

Final Result 330(5)

Optimal Portfolio Rules 335(68)

Remarks 335(1)

Utility of Wealth 336(2)

Merton's Optimal Portfolio Rule Derived and Implemented 338(29)

Derivation of Equations: Cash Transactions, Wealth Evolution 338(3)

A Non-Optimal Hedging Strategy Implemented 341(1)

Data 341(1)

Market Evolution 342(1)

Monte--Carlo Simulation of Trading 343(6)

Stochastic Control Problem 349(1)

Derivation of the Hamilton--Jacobi--Bellman PDE 349(2)

Monge--Ampere PDEs 351(2)

Derivation of Merton's Monge--Ampere PDE 353(1)

Symbolic Solution of Merton's Monge--Ampere PDE 354(3)

Optimal Portfolio Hedging Strategy Implemented 357(5)

Fringe Issues 362(1)

An Alternative Approach 362(2)

Optimal Portfolio Balance Evolution 364(1)

Time-Dependent Market Dynamics 365(2)

Portfolio Rules under Appreciation Rate Uncertainty 367(6)

Statement of the Problem and Solution of the Easy Case 367(2)

A New Portfolio Rule via Calculus of Variations 369(4)

Portfolio Optimization under Equality Constraints 373(22)

Portfolio Optimization under General Affine Constraints 373(1)

Affine Constraints: Introduction 373(2)

Wealth Volatility Minimization 375(3)

Stochastic Control under Affine Constraints on the Portfolio 378(6)

Examples 384(2)

Time-Dependent Market Dynamics 386(1)

Portfolios Under Affine Constraints and Appreciation Uncertainty 387(3)

A Quadratic Constraint: Constraint on Wealth Volatility 390(1)

Solution under the Constraint on Wealth Volatility 390(2)

Affine Constraint together with Constraint on Wealth Volatility 392(3)

Portfolio Optimization under Inequality Constraints 395(8)

Advanced Trading Strategies 403(70)

Remarks 403(1)

Reduced Monge--Ampere PDEs of Advanced Portfolio Hedging 404(41)

Advanced Optimal Portfolio Hedging Problems 404(1)

The Fundamental Trichotomy 404(1)

Market Dynamics 405(1)

Market Dynamics Example: Appreciation-Rate Reversing Model 406(4)

Stochastic Control Problems 410(1)

Derivation of the Monge--Ampere PDEs 411(1)

No Constraints 411(3)

Affine Constraints 414(4)

Reduced Monge--Ampere PDEs of Advanced Optimal Portfolio Hedging 418(1)

No Constraints 418(4)

General Affine Constraints 422(5)

Computational Examples of Advanced Portfolio Hedging 427(1)

Advanced Portfolios of Stocks with and without Affine Constraints 427(8)

Advanced Hedging of Options 435(10)

Hypoelliptic Obstacle Problems in Optimal Momentum Trading 445(28)

Problems 445(1)

SDE Model: Price/Trend Process 446(1)

ODE Motivation 446(2)

Price/Trend Process 448(5)

Hypoellipticity of the Infinitesimal Generator of the Price/Trend Process 453(1)

L: Infinitesimal Generator of the Price/Trend Process 453(1)

Hypoellipticity and Probability: Simple Examples 454(5)

Hypoellipticity of the Infinitesimal Generator L 459(1)

Dirichlet Problem for L 460(1)

Optimal Momentum Trading of Stocks 461(1)

Obstacle Problem for L: When is it optimal to sell a stock? 461(3)

``Implicit'' Obstacle Problems for L 464(3)

Optimal Momentum Trading of Stock Options 467(1)

Obstacle Problem for ∂/∂t + L: When to sell a call/put option? 467(4)

Implicit Obstacle Problem for ∂/∂t + L: When to buy a call/put option? 471(2)
Bibliography 473(4)
Index 477