main-content

## Über dieses Buch

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.

## Inhaltsverzeichnis

### Chapter 1. Stochastic Market Model

Abstract
In this chapter we briefly describe the basic concepts of stochastic market models. Further, we introduce a multystock stochastic continuous-time market model that will be used in the following chapters, and we give some necessary definitions.
Nikolai Dokuchaev

### Chapter 2. Two Empirical Model-Free “Winning” Strategies and Their Statistical Evaluation

Abstract
In this chapter, we consider a generic market model that consists of two assets only: a risky stock and a locally risk-free bond (or bank account). We reduce assumptions on the probability distribution of the price evolution and assume that the price of the stock evolves arbitrarily with interval uncertainty. The dynamics of the bond is exponentially increasing along with interval uncertainty. Under such mild assumptions, the market is incomplete. We further assume that only historical prices are available. Thus, admissible strategies for this model are similar to strategies from “technical analysis” and they are almost model free. We present two original empirical strategies that bound risk closely to a risk-free numéraire and risky numéraire respectively. The important feature of the strategies is that they guarantee a positive average gain for any non-risk-neutral probability measure. Some statistical tests of profitability of these strategies as applied to historical data are provided.
Nikolai Dokuchaev

### Chapter 3. Strategies for Investment in Options

Abstract
We consider strategies for investment in options for the diffusion market model. We show that there exists a correct proportion between put and call options in the portfolio such that the average gain is almost always positive for a generic Black and Scholes model. This gain is zero if and only if the market price of risk is zero. A paradox related to the corresponding loss of option’s seller is also discussed.
Nikolai Dokuchaev

### Chapter 4. Continuous-Time Analogs of “Winning” Strategies and Asymptotic Arbitrage

Abstract
In this chapter, we consider two continuous-time analogue of the model-free winning empirical strategy defined in Theorem 2.2, Chapter 2. These two continuous-time strategies ensure a positive average gain for any non-risk-neutral probability measure; the strategies bound risk and do not require forecasting of the volatility coefficient and appreciation rate estimation. As the number of the traded stocks increases, the strategies converge to arbitrage with a given positive gain that is ensured with probability arbitrarily close to 1.
Nikolai Dokuchaev

### Chapter 5. Optimal Strategies with Direct Observation of Parameters

Abstract
We consider an optimal investment problem for a market consisting of a risk-free bond or bank account and a finite number of risky stocks with correlated stock prices. It is assumed that the stock prices evolve according to an Itô’s stochastic differential equation. The parameters (interest rate, appreciation rate, and volatilities) need not be adapted to the driving Brownian motion, so the market is incomplete. The main assumption here is that all the market parameters, including the appreciation rates, are directly observable. That the condition is restrictive.
Nikolai Dokuchaev

### Chapter 6. Optimal Portfolio Compression

Abstract
In this chapter, we consider the problem of optimal portfolio compression. By this term we mean that admissible strategies may include no more than m different stocks concurrently, where m may be less than the total number n of available stocks.
Nikolai Dokuchaev

### Chapter 7. Maximin Criterion for Observable but Nonpredictable Parameters

Abstract
In this chapter, it is assumed that the risk-free rate, the appreciation rates, and the volatility rates of the stocks are all random; they are not adapted to the driving Brownian motion, and their distributions are unknown, but they are currently observable. Admissible strategies are based on current observations of the stock prices and the aforementioned parameters. The optimal investment problem is stated as a problem with a maximin performance criterion. This criterion is to ensure that a strategy is found such that the minimum of utility over all distributions of parameters is maximal. It is shown that the duality theorem holds for the problem and that the maximin problem is reduced to the minimax problem, with minimization over a single scalar parameter (even for a multistock market). This interesting effect follows from the result of Chapter 6 for the optimal compression problem. Using this effect, the original maximin problem is solved explicitly; the optimal strategy is derived explicitly via solution of a linear parabolic equation.
Nikolai Dokuchaev

### Chapter 8. Strategies Based on Historical Prices and Volume: Problem Statement and Existence Result

Abstract
We consider the investment problem in the class of strategies that do not use direct observations of the appreciation rates of the stocks but rather use historical market data (i.e., stock prices and volume of trade) and prior distributions of the appreciation rates. We formulate the problem statement and prove the existence of optimal strategy for a general case.
Nikolai Dokuchaev

### Chapter 9. Solution for Log and Power Utilities with Historical Prices and Volume

Abstract
We present the explicit solution of an optimal investment problem without additional constraints for log and power utility functions. Results are shown results for numerical experiments with historical data.
Nikolai Dokuchaev

### Chapter 10. Solution for General Utilities and Constraints Via Parabolic Equations

Abstract
We present the solution of an optimal investment problem with additional constraints and utility functions of a very general type, including discontinuous functions. Optimal portfolios are obtained in the class of strategies based on historical prices, when a(t) is random and unobservable, but under some additional restrictions on the prior distributions of market parameters. Optimal investment strategies are expressed via solution of a linear deterministic parabolic backward equation.
Nikolai Dokuchaev

### Chapter 11. Special Cases and Examples: Replicating with Gap and Goal Achieving

Abstract
In this chapter, the optimal portfolio is obtained for the class of strategies based on historical prices under some additional conditions that ensure that the optimal normalized wealth $$\tilde{X}(t)$$ and the optimal strategy π (t) are functions of the current vector $$\tilde{S}(t)$$ of the normalized stock prices. In particular, these conditions are satisfied if σ (t) is deterministic and σ, ã are time independent. A solution of a goal achieving problem and a solution of a problem of optimal replication of a European put option with a possible gap will be given among others. Explicit formulas for optimal claims and numerical examples are provided.
Nikolai Dokuchaev

### Chapter 12. Unknown Distribution: Maximin Criterion and Duality Approach

Abstract
In this chapter, a case is studied in which the appreciation rates, volatilities, and their prior distributions are unknown. The optimal investment problem is stated as a problem with a maximin performance criterion. This criterion is to ensure that a strategy is found such that the utility minimum over all distributions of parameters is maximal. It is shown that the duality theorem holds for the problem. Thus, the maximin problem is reduced to the minimax problem. This minimax problem is computationally a much easier problem.
Nikolai Dokuchaev

### Chapter 13. On Replication of Claims

Abstract
In Chapters 5 and 8, the solution of the optimal investment problem was decomposed on two different problems: calculation of the optimal claim and calculation of a strategy to replicate the optimal claim. In this chapter, we discuss some aspects of replication of given claims. First, some possibilities are considered for replicating the desired claim by purchasing options. Second, an example is considered of an incomplete market with transactions costs and with nonpredictable volatility, when replication is replaced for rational superreplication.
Nikolai Dokuchaev

### Backmatter

Weitere Informationen