Implicit transaction costs and the fundamental theorems of asset pricing

This paper studies arbitrage pricing theory in financial markets with implicit transaction costs. We extend the existing theory to include the more realistic possibility that the price at which the investors trade is dependent on the traded volume. The investors in the market always buy at the ask and sell at the bid price. Implicit transaction costs are composed of two terms, one is able to capture the bid-ask spread, and the second the price impact. Moreover, a new definition of a self-financing portfolio is obtained. The self-financing condition suggests that continuous trading is possible, but is restricted to predictable trading strategies having c\'adl\'ag (right-continuous with left limits) and c\'agl\'ad (left-continuous with right limits) paths of bounded quadratic variation and of finitely many jumps. That is, c\'adl\'ag and c\'agl\'ad predictable trading strategies of infinite variation, with finitely many jumps and of finite quadratic variation are allowed in our setting. Restricting ourselves to c\'agl\'ad predictable trading strategies, we show that the existence of an equivalent probability measure is equivalent to the absence of arbitrage opportunities, so that the first fundamental theorem of asset pricing (FFTAP) holds. It is also shown that the use of continuous and bounded variation trading strategies can improve the efficiency of hedging in a market with implicit transaction costs. To better understand how to apply the theory proposed we provide an example of an implicit transaction cost economy that is linear and non-linear in the order size.Comment: International Journal of Theoretical and Applied Finance, 20(04) 201

A fundamental theorem of asset pricing for continuous time large financial markets in a two filtration setting

We present a version of the fundamental theorem of asset pricing (FTAP) for continuous time large financial markets with two filtrations in an $L^p$-setting for $1 \leq p < \infty$. This extends the results of Yuri Kabanov and Christophe Stricker \cite{KS:06} to continuous time and to a large financial market setting, however, still preserving the simplicity of the discrete time setting. On the other hand it generalizes Stricker's $L^p$-version of FTAP \cite{S:90} towards a setting with two filtrations. We do neither assume that price processes are semi-martigales, (and it does not follow due to trading with respect to the \emph{smaller} filtration) nor that price processes have any path properties, neither any other particular property of the two filtrations in question, nor admissibility of portfolio wealth processes, but we rather go for a completely general (and realistic) result, where trading strategies are just predictable with respect to a smaller filtration than the one generated by the price processes. Applications range from modeling trading with delayed information, trading on different time grids, dealing with inaccurate price information, and randomization approaches to uncertainty

Superreplication under Model Uncertainty in Discrete Time

We study the superreplication of contingent claims under model uncertainty in discrete time. We show that optimal superreplicating strategies exist in a general measure-theoretic setting; moreover, we characterize the minimal superreplication price as the supremum over all continuous linear pricing functionals on a suitable Banach space. The main ingredient is a closedness result for the set of claims which can be superreplicated from zero capital; its proof relies on medial limits.Comment: 14 pages; forthcoming in 'Finance and Stochastics

Portfolio optimisation beyond semimartingales: shadow prices and fractional Brownian motion

While absence of arbitrage in frictionless financial markets requires price processes to be semimartingales, non-semimartingales can be used to model prices in an arbitrage-free way, if proportional transaction costs are taken into account. In this paper, we show, for a class of price processes which are not necessarily semimartingales, the existence of an optimal trading strategy for utility maximisation under transaction costs by establishing the existence of a so-called shadow price. This is a semimartingale price process, taking values in the bid ask spread, such that frictionless trading for that price process leads to the same optimal strategy and utility as the original problem under transaction costs. Our results combine arguments from convex duality with the stickiness condition introduced by P. Guasoni. They apply in particular to exponential utility and geometric fractional Brownian motion. In this case, the shadow price is an Ito process. As a consequence we obtain a rather surprising result on the pathwise behaviour of fractional Brownian motion: the trajectories may touch an Ito process in a one-sided manner without reflection.Comment: To appear in Annals of Applied Probability. We would like to thank Junjian Yang for careful reading of the manuscript and pointing out a mistake in an earlier versio

Shadow prices for continuous processes

In a financial market with a continuous price process and proportional transaction costs we investigate the problem of utility maximization of terminal wealth. We give sufficient conditions for the existence of a shadow price process, i.e.~a least favorable frictionless market leading to the same optimal strategy and utility as in the original market under transaction costs. The crucial ingredients are the continuity of the price process and the hypothesis of "no unbounded profit with bounded risk". A counter-example reveals that these hypotheses cannot be relaxed