Dynamic portfolio optimization with liquidity cost and market impact: a simulation-and-regression approach

We present a simulation-and-regression method for solving dynamic portfolio optimization problems in the presence of general transaction costs, liquidity costs and market impact. This method extends the classical least squares Monte Carlo algorithm to incorporate switching costs, corresponding to transaction costs and transient liquidity costs, as well as multiple endogenous state variables, namely the portfolio value and the asset prices subject to permanent market impact. To handle endogenous state variables, we adapt a control randomization approach to portfolio optimization problems and further improve the numerical accuracy of this technique for the case of discrete controls. We validate our modified numerical method by solving a realistic cash-and-stock portfolio with a power-law liquidity model. We identify the certainty equivalent losses associated with ignoring liquidity effects, and illustrate how our dynamic optimization method protects the investor's capital under illiquid market conditions. Lastly, we analyze, under different liquidity conditions, the sensitivities of certainty equivalent returns and optimal allocations with respect to trading volume, stock price volatility, initial investment amount, risk aversion level and investment horizon.     

Estimation and Prediction of a Non-Constant Volatility

In this paper we study volatility functions. Our main assumption is that the volatility is a function of time and is either deterministic, or stochastic but driven by a Brownian motion independent of the stock. Our approach is based on estimation of an unknown function when it is observed in the presence of additive noise. The set up is that the prices are observed over a time interval [0, t], with no observations over (t, T), however there is a value for volatility at T. This value is may be inferred from options, or provided by an expert opinion. We propose a forecasting/interpolating method for such a situation. One of the main technical assumptions is that the volatility is a continuous function, with derivative satisfying some smoothness conditions. Depending on the degree of smoothness there are two estimates, called filters, the first one tracks the unknown volatility function and the second one tracks the volatility function and its derivative. Further, in the proposed model the price of option is given by the Black–Scholes formula with the averaged future volatility. This enables us to compare the implied volatility with the averaged estimated historical volatility. This comparison is done for three companies and has shown that the two estimates of volatility have a weak statistical relation.     

Environmental priorities of Russia

This paper analyzes Russia’s high priority environmental problems requiring effective and urgent solution based on large-scale costs.     

Solutions and Simulations of Some One-Dimensional Stochastic Differential Equations

We consider a one dimensional SDE dX _t  = μ(X _t )dt + σ(X _t )dB _t . We give a new general formula for solutions that involves solving an associated ordinary differential equation. Explicit solutions are obtained in cases where the ODE has such. This recovers linear case but also some non-linear cases. In any case our approach leads to a new simulation scheme that returns positive values for processes on I R⁺{{\hskip 0.02in \hbox{\rm I}\hskip -.02in \hbox{\rm R}{^+}}}, which is advantageous when modelling prices or rates.