Financial Friction and Multiplicative Markov Market Game

We study optimal investment with multiple assets in the presence of small proportional transaction costs. Rather than computing an asymptotically optimal no-trade region, we optimize over suitable trading frequencies. We derive explicit formulas for these and the associated welfare losses due to small transaction costs in a general, multidimensional diffusion setting, and compare their performance to a number of alternatives using Monte Carlo simulations.

We propose a multi-agent model of an asset market and study conditions that guarantee that the strategy of an individual agent cannot outperform the market. The model assumes a mean-field approximation of the market by considering an infinite number of infinitesimal agents who use the same strategy and another infinitesimal agent with a different strategy who tries to outperform the market. We show that the optimal strategy for the market agents is to split their investment b...

In this paper we consider a discrete-time risk sensitive portfolio optimization over a long time horizon with proportional transaction costs. We show that within the log-return i.i.d. framework the solution to a suitable Bellman equation exists under minimal assumptions and can be used to characterize the optimal strategies for both risk-averse and risk-seeking cases. Moreover, using numerical examples, we show how a Bellman equation analysis can be used to construct or refin...

The paper studies problem of continuous time optimal portfolio selection for a incom- plete market diffusion model. It is shown that, under some mild conditions, near optimal strategies for investors with different performance criteria can be constructed using a limited number of fixed processes (mutual funds), for a market with a larger number of available risky stocks. In other words, a dimension reduction is achieved via a relaxed version of the Mutual Fund Theorem.

This survey is an introduction to asymptotic methods for portfolio-choice problems with small transaction costs. We outline how to derive the corresponding dynamic programming equations and simplify them in the small-cost limit. This allows to obtain explicit solutions in a wide range of settings, which we illustrate for a model with mean-reverting expected returns and proportional transaction costs. For even more complex models, we present a policy iteration scheme that allo...

We consider the problem of optimizing the expected logarithmic utility of the value of a portfolio in a binomial model with proportional transaction costs with a long time horizon. By duality methods, we can find expressions for the boundaries of the no-trade-region and the asymptotic optimal growth rate, which can be made explicit for small transaction costs. Here we find that, contrary to the classical results in continuous time, the size of the no-trade-region as well as t...

In this article we study an optimal stopping/optimal control problem which models the decision facing a risk-averse agent over when to sell an asset. The market is incomplete so that the asset exposure cannot be hedged. In addition to the decision over when to sell, the agent has to choose a control strategy which corresponds to a feasible wealth process. We formulate this problem as one involving the choice of a stopping time and a martingale. We conjecture the form of the s...

We study the origins of the $\sqrt{dt}$ effect in finance and SDE. In particular, we show, in the game-theoretic framework, that market volatility is a consequence of the absence of riskless opportunities for making money and that too high volatility is also incompatible with such opportunities. More precisely, riskless opportunities for making money arise whenever a traded security has fractal dimension below or above that of the Brownian motion and its price is not almost c...

In a market with one safe and one risky asset, an investor with a long horizon, constant investment opportunities, and constant relative risk aversion trades with small proportional transaction costs. We derive explicit formulas for the optimal investment policy, its implied welfare, liquidity premium, and trading volume. At the first order, the liquidity premium equals the spread, times share turnover, times a universal constant. Results are robust to consumption and finite ...

We consider the problem of finding optimal strategies that maximize the average growth-rate of multiplicative stochastic processes. For a geometric Brownian motion the problem is solved through the so-called Kelly criterion, according to which the optimal growth rate is achieved by investing a constant given fraction of resources at any step of the dynamics. We generalize these finding to the case of dynamical equations with finite carrying capacity, which can find applicatio...