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Abstract:
The main addition is the result that the validity of Novikov condition property is not robust for the mean-reverting model, i.e., this model is the edge of the area when Novikov condition is satisfied for all time interval. In addition, the proofs in this version have been simplified.

diffusion market, mean-reverting model, arbitrage, technical analysis, self-financing strategies, universal portfolio

Abstract:
We study arbitrage opportunities and possible speculative opportunities for diffusion mean-reverting market models. We prove that the Novikov condition is satisfied for any time interval and for any set of parameters. It is non-trivial because the appreciation rate has conditionally Gaussian distribution converging to a stationary limit. It follows that the mean-reverting model is arbitrage free for any finite time interval. Further, we found that this model still has some speculative opportunities: a growth for a wide enough set of expected utilities may be achieved for a strategy that does not require any hypothesis on market parameters and does not use estimation of these parameters. In other words, this strategy is similar to strategies from technical analysis and Cover's universal portfolio.

diffusion market, mean-reverting model, arbitrage, technical analysis, self-financing strategies, universal portfolio

Abstract:
The paper studies option price models for diffusion market with random volatility and with pricing rules based on risk-neutral valuation. It is found that there are some properties of implied volatility that are presented for all possible risk neutral measures for the case when random historical volatility does not depend on the driving Brownian motion. Properties of the pairs consisting of implied volatility and implied risk-free interest rate are also studied.

diffusion market model, stochastic volatility, volatility smile, implied volatiliy, implied parameters

Abstract:
The standard implied volatility definition presents its value as an implicit function of several parameters, including the risk-free interest rate. This approach ignores the fact that, in reality, the risk free interest rate is unknown and need to be forecasted, because the option price depends on its future curve. Therefore, the standard implied volatility is a conditional: it depends on the future values of the risk free rate. Instead, we suggest to calculate two implied parameters: the implied volatility and the implied average cumulative risk free interest rate. They can be found unconditionally from a system of two equations. We found that very simple models with random volatilities (for instance, with two point distributions) allow to generate various volatility smiles and skews with this approach.

volatility smile, volatility skew, market models, Black-Scholes, diffusion market, stochastic volatility

Abstract:
We consider optimal investment problems for a diffusion market model with non-observable random drifts that evolve as an Ito's process. Admissible strategies do not use direct observations of the market parameters, but rather use historical stock prices. For a non-linear problem with a general performance criterion, the optimal portfolio strategy is expressed via the solution of a scalar minimization problem and a linear parabolic equation with coefficients generated by the Kalman filter.

Optimal portfolio, non-observable parameters, Kalman filter

Abstract:
We study a bond pricing problem for a case when the short term interest rate is a random process with unknown prior distribution and evolution law. We assume that there is a stock and options on this stock with observable prices; the stock volatility can be random. We assume that the option prices are generated by a risk-neutral valuation method and that they are correlated with the short term interest rate that generates the bond price. We suggest to price bonds via volatility and cumulative risk free interest rate inferred from stock and option prices. These parameters can be found unconditionally from a system of two equations. We found that the implied forward risk free rate inferred from system of put and call options does not depend of the option strike price, and, therefore, it can be effectively used for bond pricing.

Bond price, diffusion market models, Black-Scholes, stochastic volatility, implied volatility, implied forward risk-free rate

Abstract:
The paper studies trading strategies for currency exchange market which do not employ any distribution assumptions on exchange rate evolution. It is found that there are strategies that ensure some speculative opportunities for currency exchange under condition of currency corridor or soft peg.

portfolio strategies, currency exchange, soft peg, currency corridor, currency trading, speculations

Abstract:
The paper presents a pricing rule for market models with random volatility with an uncertainty in its evolution law. It is shown that the most popular existing models allow a possibility that the option price calculated for random volatility with an error in volatility forecasts is lower that the price for the market with zero error of volatility forecast. We suggest and study a pricing rule that eliminates this possibility and is consistent with the volatility smile. The rule is based on maximization of the price via a class of possible equivalent risk-neutral measures. In Markovian setting, it requires to solve a parabolic Bellman equation. For this equation, some existence results and a prior estimates are obtained.

Diffusion market model, stochastic volatility, volatility smile, Hamilton-Jacobi-Bellman equation

Abstract:
The paper studies discrete time mean-reverting market models. It is shown that a correct choice of initial conditions ensures existence of an equivalent martingale measure for any finite time horizon. This leads to a pricing rule for options and absence of arbitrage. Further, it is shown that this model still allows some speculative opportunities: a gain for a wide enough set of expected utilities can be achieved for a strategy that does not require any hypothesis on market parameters and does not use estimation of these parameters.

discrete time market, mean-reverting model, arbitrage, technical analysis, self-financing strategies, universal portfolio

Abstract:
The paper investigates a problem of bounded risk portfolio selection for a multi-period market in the case when only historical prices are available, and all market parameters are not observable. We present a strategy which bounds risk closely to a risk-free investment and guarantees at the same time a positive average gain for any non-risk-neutral probability measure.

bounded risk, portfolio selection, non-observable parameters, stochastic market models

Abstract:
The paper investigates a problem of bounded risk portfolio selection for a multi-period market in the case when only historical prices are available, and all market parameters are not observable. We present a strategy which bounds risk closely to a risk-free investment and guarantees at the same time a positive average gain for any non-risk-neutral probability measure.

bounded risk, portfolio selection, non-observable parameters, stochastic market models

Abstract:
Optimal investment problem is studied for discrete time market models with serial correlations. It is found sufficient conditions that ensure that the optimal strategy is myopic for the case of power or log utility function. The possibility of discrete time approximation of optimal continuous time strategies for diffusion market is analyzed.

Optimal portfolio, discrete time market, diffusion market, myopic strategies

Abstract:
The paper studies multi-period discrete time market models with serial correlations. We found the optimal strategy in mean-variance and goal achieving setting for the case when there are serial correlations and when the parameter process that causes correlations is currently observable.

discrete time market, multi-period market, serial correlation, optimal portfolio, mean variance portfolio, goal achieving

Abstract:
The paper studies multi-stock discrete time market models with serial correlations and with some management costs. We found a market structure that ensures that the optimal strategy is myopic for the case of either power or log utility function.

discrete time market, serial correlation, optimal portfolio, myopic strategies

Abstract:
We study an optimal investment problem for a diffusion market model such that the risk-free rate, the appreciation rates and the volatility of the stocks are all random; they are not necessarily adapted to the driving Brownian motion, and their distributions are unknown, but they are supposed to be currently observable. The optimal investment problem is stated as a problem with a maximin performance criterion which leads to maximization of the minimum of expected utility over all distributions of parameters. We found that including the non-discounted wealth to the utility (in addition to the discounted wealth) implies an additional condition for the saddle point of the maximin problem: it must be a solution of an additional minimization problem for the risk-free return. This is different from the case when the utility depends on the discounted wealth only. Using these results, the maximin problem is reduced to a linear parabolic equation and minimization (over two scalar parameters). It is an important addition to the result obtained in the author's paper (2006).

minimax problems, optimal portfolio, stochastic control

Abstract:
We study optimal investment problem for a market model where the evolution of risky assets is described by Ito's equations. The risk-free rate, the appreciation rates, and the volatility of the stocks are all random; they are not necessary adapted to the driving Brownian motion, their distributions are unknown, and they are supposed to be currently observable. The optimal investment problem is stated as a problem with a maximin performance criterion to ensure that a strategy is found such that the minimum of expected utility over all possible parameters is maximal. We show that a saddle point exists and can be found via minimization over a single scalar parameter. The {\it maximin} problem is solved for a very general case via solution of a linear parabolic equation with explicit fundamental solution.

continuous market, uncertainty, optimal portfolio, minimax problems, saddle point

Abstract:
The paper suggest a modification of an American option such that the option holder can exercise the option early before the expiration, and he or she can retract later this decision to exercise. This feature gives additional flexibility and risk protection for the option holder. The contract conditions have clear economical sense; they are very straightforward and easy to follow. On the other hand, pricing of these options is challenging and requires significant computational efforts. Some rules and theorems for the prices are given.

American options, exotic options, Irish options, pricing rules

Abstract:
We consider optimal gradual liquidation of equity from a risky asset for continuous time stochastic market model. The owner of the risky asset uses this equity as a source of steady cash flow by borrowing money permanently against this equity. At the terminal time, there is no equity for him in this asset, and the bank gains ownership of this asset. Optimal strategy is obtained explicitly.

equity liquidation, optimal strategy, contingent claim replication

Abstract:
We study a modification of an American option such that the option holder can exercise the option early before the expiration, and he or she can revert later this decision to exercise a number of times. This feature gives additional flexibility and risk protection for the option holder. We found that, for the Black-Scholes market model, the price of call options with this feature is the same as for European call, i.e., the additional flexibility costs nothing, similarly to the situation with American and European call options. For the market model with zero interest rate, the price of put options with this feature is also the same as for the standard European put options. Therefore, these options can be more competitive than standard American options.

American options, compound options, exotic options, multiple exercise, multiple rescissions, multiple stopping, pricing rules, Irish options

Abstract:
We found conditions that ensure that the optimal strategy is myopic for the case of exponential utility function for multi-stock discrete time market model with serial correlations.

stochastic discrete time market, optimal portfolio, myopic strategies, exponential utility

Abstract:
This short note describes some statistical tests and experiments for serial correlations of historical stock prices. More precisely, some parameters calculated via empirical characteristics functions are compared with the same parameters for time series with known degree of correlation.

financial time series, serial correlations, non-parametric methods, technical analysis, statistical validation

Abstract:
This short note discusses some examples of implementation of discrete time myopic strategies to continuous time stochastic market models.

optimal portfolio, myopic strategies, discrete time market, diffusion market, discretization, first exit times

Abstract:
We study the optimal investment problem for a continuous time incomplete market model such that the risk-free rate, the appreciation rates and the volatility of the stocks are all random; they are assumed to be independent from the driving Brownian motion, and they are supposed to be currently observable. It is shown that some weakened version of Mutual Fund Theorem holds for this market for general class of utilities; more precisely, it is shown that the supremum of expected utilities can be achieved on a sequence of strategies with a certain distribution of risky assets that does not depend on risk preferences described by different utilities.

optimal portfolio, Mutual Fund Theorem, continuous time market models

Abstract:
The paper presents sufficient conditions of predictability for continuous time processes in deterministic setting. We found that processes with exponential decay on energy for higher frequencies are predictable in some weak sense on some finite time horizon defined by the rate of decay. Moreover, this predictability can be achieved uniformly over classes of processes. Some explicit formulas for predictors are suggested.

nonparametric methods, spectral analysis, forecasting

Abstract:
This is the final version of the paper published on-line in September 23, 2004 at SSRN: http://ssrn.com/abstract=594869. The main addition is the example illustrating the choice of the stocks and the options to match the required risk neutral measure.

Bond price, diffusion market models, Black-Scholes, stochastic volatility, implied volatility, implied forward risk-free rate