Dynamic portfolio optimization with transaction costs and state-dependent drift

The problem of dynamic portfolio choice with transaction costs is often addressed by constructing a Markov Chain approximation of the continuous time price processes. Using this approximation, we present an efficient numerical method to determine optimal portfolio strategies under time- and state-dependent drift and proportional transaction costs. This scenario arises when investors have behavioral biases or the actual drift is unknown and needs to be estimated. Our numerical method solves dynamic optimal portfolio problems with an exponential utility function for time-horizons of up to 40 years. It is applied to measure the value of information and the loss from transaction costs using the indifference principle

Optimal hedging of Derivatives with transaction costs

We investigate the optimal strategy over a finite time horizon for a portfolio of stock and bond and a derivative in an multiplicative Markovian market model with transaction costs (friction). The optimization problem is solved by a Hamilton-Bellman-Jacobi equation, which by the verification theorem has well-behaved solutions if certain conditions on a potential are satisfied. In the case at hand, these conditions simply imply arbitrage-free ("Black-Scholes") pricing of the derivative. While pricing is hence not changed by friction allow a portfolio to fluctuate around a delta hedge. In the limit of weak friction, we determine the optimal control to essentially be of two parts: a strong control, which tries to bring the stock-and-derivative portfolio towards a Black-Scholes delta hedge; and a weak control, which moves the portfolio by adding or subtracting a Black-Scholes hedge. For simplicity we assume growth-optimal investment criteria and quadratic friction.Comment: Revised version, expanded introduction and references 17 pages, submitted to International Journal of Theoretical and Applied Finance (IJTAF

Option Pricing and Hedging with Small Transaction Costs

An investor with constant absolute risk aversion trades a risky asset with general It\^o-dynamics, in the presence of small proportional transaction costs. In this setting, we formally derive a leading-order optimal trading policy and the associated welfare, expressed in terms of the local dynamics of the frictionless optimizer. By applying these results in the presence of a random endowment, we obtain asymptotic formulas for utility indifference prices and hedging strategies in the presence of small transaction costs.Comment: 20 pages, to appear in "Mathematical Finance

A recursive algorithm for multivariate risk measures and a set-valued Bellman's principle

A method for calculating multi-portfolio time consistent multivariate risk measures in discrete time is presented. Market models for $d$ assets with transaction costs or illiquidity and possible trading constraints are considered on a finite probability space. The set of capital requirements at each time and state is calculated recursively backwards in time along the event tree. We motivate why the proposed procedure can be seen as a set-valued Bellman's principle, that might be of independent interest within the growing field of set optimization. We give conditions under which the backwards calculation of the sets reduces to solving a sequence of linear, respectively convex vector optimization problems. Numerical examples are given and include superhedging under illiquidity, the set-valued entropic risk measure, and the multi-portfolio time consistent version of the relaxed worst case risk measure and of the set-valued average value at risk.Comment: 25 pages, 5 figure

Components of the Czech Koruna Risk Premium in a Multiple-Dealer FX Market

The paper proposes a continuous time model of an FX market organized as a multiple dealership. The model reflects a number of salient features of the Czech koruna spot market. The dealers have costly access to the best available quotes. They interpret signals from the joint dealer-customer order flow and decide upon their own quotes and trades in the inter-dealer market. Each dealer uses the observed order flow to improve the subjective estimates of the relevant aggregate variables, which are the sources of uncertainty. One of the risk factors is the size of the cross-border dealer transactions in the FX market. These uncertainties have diffusion form and are dealt with according to the principles of portfolio optimization in continuous time. The model is used to explain the country, or risk, premium in the uncovered national return parity equation for the koruna/euro exchange rate. The two country premium terms that I identify in excess of the usual covariance term (a consequence of the 'Jensen inequality effect') are the dealer heterogeneity-induced inter-dealer market order flow component and the dealer Bayesian learning component. As a result, a 'dealer-based total return parity' formula links the exchange rate to both the 'fundamental' factors represented by the differential of the national asset returns, and the microstructural factors represented by heterogeneous dealer knowledge of the aggregate order flow and the fundamentals. Evidence on the cross-border order flow dependence of the Czech koruna risk premium, in accordance with the model prediction, is documented.Bayesian learning, FX microstructure, optimizing dealer, uncovered parity.