American options with gradual exercise under proportional transaction costs

American options in a multi-asset market model with proportional transaction costs are studied in the case when the holder of an option is able to exercise it gradually at a so-called mixed (randomized) stopping time. The introduction of gradual exercise leads to tighter bounds on the option price when compared to the case studied in the existing literature, where the standard assumption is that the option can only be exercised instantly at an ordinary stopping time. Algorithmic constructions for the bid and ask prices and the associated superhedging strategies and optimal mixed stopping times for an American option with gradual exercise are developed and implemented, and dual representations are established

The Schroedinger Problem, Levy Processes Noise in Relativistic Quantum Mechanics

The main purpose of the paper is an essentially probabilistic analysis of relativistic quantum mechanics. It is based on the assumption that whenever probability distributions arise, there exists a stochastic process that is either responsible for temporal evolution of a given measure or preserves the measure in the stationary case. Our departure point is the so-called Schr\"{o}dinger problem of probabilistic evolution, which provides for a unique Markov stochastic interpolation between any given pair of boundary probability densities for a process covering a fixed, finite duration of time, provided we have decided a priori what kind of primordial dynamical semigroup transition mechanism is involved. In the nonrelativistic theory, including quantum mechanics, Feyman-Kac-like kernels are the building blocks for suitable transition probability densities of the process. In the standard "free" case (Feynman-Kac potential equal to zero) the familiar Wiener noise is recovered. In the framework of the Schr\"{o}dinger problem, the "free noise" can also be extended to any infinitely divisible probability law, as covered by the L\'{e}vy-Khintchine formula. Since the relativistic Hamiltonians $|\nabla |$ and $\sqrt {-\triangle +m^2}-m$ are known to generate such laws, we focus on them for the analysis of probabilistic phenomena, which are shown to be associated with the relativistic wave (D'Alembert) and matter-wave (Klein-Gordon) equations, respectively. We show that such stochastic processes exist and are spatial jump processes. In general, in the presence of external potentials, they do not share the Markov property, except for stationary situations. A concrete example of the pseudodifferential Cauchy-Schr\"{o}dinger evolution is analyzed in detail. The relativistic covariance of related waveComment: Latex fil

A counter-example to an option pricing formula under transaction costs

In the paper by Melnikov and Petrachenko (Finance Stoch. 9: 141–149, 2005), a procedure is put forward for pricing and replicating an arbitrary European contingent claim in the binomial model with bid-ask spreads. We present a counter-example to show that the option pricing formula stated in that paper can in fact lead to arbitrage. This is related to the fact that under transaction costs a superreplicating strategy may be less expensive to set up than a strictly replicating one

Utility maximizing entropy and the second law of thermodynamics

Expected utility maximization problems in mathematical finance lead to a generalization of the classical definition of entropy. It is demonstrated that a necessary and sufficient condition for the second law of thermodynamics to operate is that any one of the generalized entropies should tend to its minimum value of zero