Computational Financial Mathematics Using Mathematica

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ISBN: 9780817641979 | 0817641971
Cover: Hardcover
Copyright: 11/1/2002

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CO-PUBLISHED WITH TELOSThis book provides a beautiful overview of what mathematics and MATHEMATICA¿ can do for finance. Sophisticated theories are presented in a rigorous but user-friendly, practical style, which, with the programming capabilities of MATHEMATICA, help the reader develop good intuition in real trading.Key features: Entire book is on cross-platform CD written in MATHEMATICA * quick introduction to MATHEMATICA provided * minimal prerequisites: good understanding of calculus and some differential equations * a highly original presentation of optimal portfolio diversification.The book is designed for instructors and students, and most importantly, will meet the everyday trading needs of the professional¿the analytically inclined individual investor who wants to solve various problems encountered when investing and trading in stocks and stock options.

Introduction

(6)

Audience, Highlights, Agenda

(3)

Software Installation

(2)

Acknowledgments

(1)

Cash Account Evolution

(18)

Symbolic Solutions of ODEs

(6)

Numerical Solutions of ODEs

(12)

Stock Price Evolution

(58)

What are Stocks?

(1)

Stock Price Modeling: Stochastic Differential Equations

(20)

Idea of Stochastic Differential Equations

(1)

Normal Distribution

(6)

Markov Random Processes

(1)

Brownian Motion

(4)

Stochastic Integral

(5)

Stochastic Differential Equations

(3)

Ito Calculus

(10)

Ito Product Rule

(1)

Ito Chain Rule

(1)

Ito Theorem

(2)

Monte--Carlo Simulation ``Proof'' of the Ito Theorem

(2)

An Application: Solving the Simplest Stock Price SDE Model

(5)

Multivariable and Symbolic Ito Calculus

(23)

Monte--Carlo Solver for m-Dimensional SDEs

(4)

Symbolic m-Dimensional Ito Chain Rule

(1)

Some Calculus and Linear Algebra

(3)

Derivation of the m-Dimensional Ito Chain Rule

(2)

Implementation of the m-Dimensional Ito Chain Rule

(3)

Ito Product Rule Revisited

(1)

The Simplest Price Model: m-Stocks

(1)

Derivation

(2)

Example; Using Multinormal Distribution

(8)

Relationship Between SDEs and PDEs

(4)

European Style Stock Options

(58)

What Are Stock Options?

(4)

Black--Scholes PDE and Hedging

(20)

Derivation of the Black--Scholes PDE

(4)

Black--Scholes Hedging Implemented

(1)

Put Hedging Implemented

(7)

Call Hedging Implemented

(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

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Computational Financial Mathematics Using Mathematica

Summary

Table of Contents

Supplemental Materials