Dynamic Portfolio Strategies

by Dokuchaev, Nikolai

• ISBN13: 9780792376484
• ISBN10: 079237648X
• Format: Hardcover
• Copyright: 2002-01-01
• Publisher: Kluwer Academic Pub

Summary

Dynamic Portfolio Strategies: Quantitative Methods and Empirical Rules for Incomplete Information investigates optimal investment problems for stochastic financial market models. It is addressed to academics and students who are interested in the mathematics of finance, stochastic processes, and optimal control, and also to practitioners in risk management and quantitative analysis who are interested in new strategies and methods of stochastic analysis. While there are many works devoted to the solution of optimal investment problems for various models, the focus of this book is on analytical strategies based on "technical analysis" which are model-free. The technical analysis of these strategies has a number of characteristics. Two of the more important characteristics are: (1) they require only historical data, and (2) typically they are more widely used by traders than analysis based on stochastic models. Hence it is the objective of this book to reduce the gap between model-free strategies and strategies that are "optimal" for stochastic models. We hope that researchers, students and practitioners will be interested in some of the new empirically based methods of "technical analysis" strategies suggested in this book and evaluated via stochastic market models.

Table of Contents

List of Figures	xi
List of Tables	xiii
Acknowledgments	xv
Introduction	xvii
Part I Background
Stochastic Market Model	3	(12)
Brief introduction to stochastic market models	3	(3)
Options market	6	(3)
Continuous-time multistock stochastic market model	9	(6)
Part II Model-free empirical strategies and their evaluation
Two Empirical Model-Free ``Winning'' Strategies and Their Statistical Evaluation	15	(18)
A generic discrete-time market model	15	(2)
A bounded risk strategy	17	(6)
The strategy	17	(2)
Estimates of transaction costs	19	(1)
Average performance under probability assumptions	20	(2)
Experiments with historical data	22	(1)
A strategy with a risky numeraire	23	(5)
The strategy	24	(1)
Average performance on a probability space	25	(1)
Experiments	26	(2)
Proofs	28	(5)
Strategies for Investment in Options	33	(10)
Introduction and definitions	33	(2)
The winning strategy	35	(1)
Numerical examples	36	(2)
A consequence for the seller and a paradox	38	(2)
Proofs	40	(3)
Continuous-Time Analogs of ``Winning'' Strategies and Asymptotic Arbitrage	43	(20)
Introduction	43	(1)
Definitions	44	(2)
Unbounded horizon: piecewise constant strategies	46	(2)
Continuous-time strategies for a single stock market with a finite horizon	48	(1)
Strategies for a multi-stock market	49	(1)
Definitions for asymptotic arbitrage	50	(3)
Asymptotic arbitrate for the strategy (4.21)	53	(1)
Proofs	54	(9)
Part III Optimal strategies for the diffusion market model with observable parameters
Optimal Strategies with Direct Observation of Parameters	63	(14)
The market model	63	(2)
Solution via dynamic programming	65	(1)
Solution via optimal claim	66	(5)
Some additional assumptions	67	(1)
Special cases	67	(1)
Replicating special claims	68	(1)
Calculating the optimal strategy	69	(1)
The case of myopic strategies	70	(1)
Proofs	71	(6)
Optimal Portfolio Compression	77	(12)
Problem statement and definitions	77	(3)
Optimal strategy for portfolio compression	80	(2)
A bond market: compression of the bond portfolio	82	(2)
Proofs	84	(5)
Maximin Criterion for Observable But Nonpredictable Parameters	89	(16)
Definitions and problem statement	89	(6)
Optimal solution of the maximin problem	95	(1)
Proofs	96	(9)
A duality theorem	98	(3)
Proof of theorem 7.1	101	(4)
Part IV Optimal strategies based on historical data for markets with nonobservable parameters
Strategies Based on Historical Prices and Volume: Existence Result	105	(20)
The model	105	(4)
A general problem and special cases	109	(2)
The general problem with constraints	109	(1)
Special cases of constraints and costs functions	109	(2)
Solution via dynamic programming	111	(4)
Additional definitions	115	(1)
Existence result for the general case	116	(2)
Auxiliary problem and additional assumptions	116	(1)
Existence result	117	(1)
The optimal strategy as a conditional expectation	118	(1)
Proofs	119	(6)
Solution for Log and Power Utilities with Historical Prices and Volume	125	(16)
Replicating special polynomial claims	125	(2)
Log utility and minimum variance estimation of α	127	(1)
Power utility	127	(1)
Filters (estimators) for the appreciation rate	128	(2)
Portfolio compression for log utility	130	(1)
Some experiments with historical data	130	(5)
Proofs	135	(6)
Solution for General Utilities and Constraints via Parabolic Equations	141	(8)
The model	141	(1)
Problem statement	142	(1)
Additional assumptions	143	(1)
A boundary problem for parabolic equations	144	(1)
The optimal strategy	145	(1)
Proofs	146	(3)
Special Cases and Examples: Replicating with Gap and Goal Achieving	149	(16)
Additional assumptions and problem statement	149	(2)
Explicit formulas for optimal claims for special cases	151	(5)
Goal-achieving problem	151	(2)
Mean-variance criteria	153	(1)
Nonlinear concave utility functions	153	(1)
Nonconnected J (y)	154	(2)
Numerical examples	156	(4)
Solution of the goal-achieving problem	156	(1)
Optimal replication of a put option with a possible gap	157	(1)
Solution with logical constraints	158	(2)
Proofs	160	(5)
Unknown Distribution: Maximin Criterion and Duality Approach	165	(14)
Definitions and problem statement	165	(4)
A duality theorem	169	(1)
Duality approach to the maximin problem	170	(1)
Minimizing with respect to a(.)	171	(1)
An illustrative example	172	(2)
Proofs	174	(5)
On Replication of Claims	179	(16)
Replication of claims using option combinations	179	(4)
Superreplication under uncertainty and transaction costs	183	(7)
Market model and problem setting	184	(4)
Superreplicating strategy	188	(2)
Proofs	190	(5)
References	195	(6)
Index	201