For portfolio choice problems with proportional transaction costs, we discuss whether or not there exists a "shadow price", i.e., a least favorable frictionless market extension leading to the same optimal strategy and utility. By means of an explicit counter-example, we show that shadow prices...

A well known result in stochastic analysis reads as follows: for an $\mathbb{R}$-valued super-martingale $X = (X_t)_{0\leq t \leq T}$ such that the terminal value $X_T$ is non-negative, we have that the entire process $X$ is non-negative. An analogous result holds true in the no arbitrage theory...

We consider a financial market with one riskless and one risky asset. The super-replication theorem states that there is no duality gap in the problem of super-replicating a contingent claim under transaction costs and the associated dual problem. We give two versions of this theorem. The first...

For portfolio optimisation under proportional transaction costs, we provide a duality theory for general cadlag price processes. In this setting, we prove the existence of a dual optimiser as well as a shadow price process in a generalised sense. This shadow price is defined via a "sandwiched"...

We compare the option pricing formulas of Louis Bachelier and Black-Merton-Scholes and observe -- theoretically as well as for Bachelier's original data -- that the prices coincide very well. We illustrate Louis Bachelier's efforts to obtain applicable formulas for option pricing in pre-computer...

The Mutual Fund Theorem (MFT) is considered in a general semimartingale financial market S with a finite time horizon T, where agents maximize expected utility of terminal wealth. It is established that: 1) Let N be the wealth process of the num\'eraire portfolio (i.e. the optimal portfolio for...

In markets with transaction costs, consistent price systems play the same role as martingale measures in frictionless markets. We prove that if a continuous price process has conditional full support, then it admits consistent price systems for arbitrarily small transaction costs. This result...

We consider the maximization of the long-term growth rate in the Black-Scholes model under proportional transaction costs as in Taksar, Klass and Assaf [Math. Oper. Res. 13, 1988]. Similarly as in Kallsen and Muhle-Karbe [Ann. Appl. Probab., 20, 2010] for optimal consumption over an infinite...

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...

We give an elementary proof of the celebrated Bichteler-Dellacherie Theorem which states that the class of stochastic processes $S$ allowing for a useful integration theory consists precisely of those processes which can be written in the form $S=M+A$, where $M$ is a local martingale and $A$ is...