Hedging, arbitrage and optimality with superlinear frictions

In a continuous-time model with multiple assets described by c\`{a}dl\`{a}g processes, this paper characterizes superhedging prices, absence of arbitrage, and utility maximizing strategies, under general frictions that make execution prices arbitrarily unfavorable for high trading intensity. Such frictions induce a duality between feasible trading strategies and shadow execution prices with a martingale measure. Utility maximizing strategies exist even if arbitrage is present, because it is not scalable at will.Comment: Published at http://dx.doi.org/10.1214/14-AAP1043 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Sticky processes, local and true martingales

We prove that for a so-called sticky process $S$ there exists an equivalent probability $Q$ and a $Q$-martingale $\tilde{S}$ that is arbitrarily close to $S$ in $L^p(Q)$ norm. For continuous $S$, $\tilde{S}$ can be chosen arbitrarily close to $S$ in supremum norm. In the case where $S$ is a local martingale we may choose $Q$ arbitrarily close to the original probability in the total variation norm. We provide examples to illustrate the power of our results and present applications in mathematical finance

Arbitrázs és árazó funkcionálok pénzügyi piacmodellekben = Arbitrage and pricing functionals in financial market models

Kutatásainkban az optimális befektetésekhez, nagy piacokhoz, illetve tranzakciós költséges piacmodellekhez kapcsolódó opcióárazási módszereket tanulmányoztunk. Optimális befektetések létezését bizonyítottuk diszkrét idejű modellekben. Megmutattuk, hogy az optimális stratégiák és a kapcsolódó opcióárak a befektetői magatartás folytonos függvényei. Szintén beláttuk, hogy ha a befektető kockázatkerülése a végtelenhez tart, akkor az általa számított árak a szintetizálási (kockázatmentes) árhoz tartanak. Végtelen sok termékes piac kovariancia-struktúrájára adtunk meg olyan feltételeket, amelyek biztosítják az árazó operátor létezését. Eredményeink kiterjesztik a mikroökonómia jól ismert ""Arbitrázs Árazási Elméletét"". Ilyen ""nagy"" modellek kötvénypiacok leírására alkalmasak. Arányos tranzakciós költségek hatását vizsgáltuk és árazó funkcionálokat konstruáltunk folytonos árfolyamatok esetén, igen általános feltételek mellett. Eredményeink használhatók korábban kezelhetetlen folyamatokra is (pl. frakcionális Brown-mozgás). | In this research programme we studied option pricing methods related to optimal investment, large financial markets and market models with transaction costs. We proved the existence of optimal strategies in discrete-time models. We showed that optimal strategies and option prices related to them are continuous functionals of agents' preferences. We could also establish that the prices computed by agents with risk-aversion tending to infinity will converge to the superreplication (risk-free) price. We found criteria on the covariance structure and the returns of a market with inifinitely many assets which guarantee the existence of a pricing operator. Our results extend the conclusions of the well-known ""Arbitrage Pricing Theory"" of microeconomics. Such ""large"" models are used to describe bond markets. We studied the effects of proportional transaction fees and constructed pricing functionals for continuous asset price processes under general conditions including previously untractable cases (like fractional Brownian motion)

Optimal investment under behavioural criteria -- a dual approach

We consider a discrete-time, generically incomplete market model and a behavioural investor with power-like utility and distortion functions. The existence of optimal strategies in this setting has been shown in a previous paper under certain conditions on the parameters of these power functions. In the present paper we prove the existence of optimal strategies under a different set of conditions on the parameters, identical to the ones which were shown to be necessary and sufficient in the Black-Scholes model. Although there exists no natural dual problem for optimisation under behavioural criteria (due to the lack of concavity), we will rely on techniques based on the usual duality between attainable contingent claims and equivalent martingale measures.Comment: Forthcoming in Banach Center Publications. Some errors have been corrected, in particular in Assumption 2.3

Hiding a drift

In this article we consider a Brownian motion with drift of the form dS_t=\mu_t dt+dB_t\qquadfor t\ge0, with a specific nontrivial $(\mu_t)_{t\geq0}$, predictable with respect to $\mathbb{F}^B$, the natural filtration of the Brownian motion $B=(B_t)_{t\ge0}$. We construct a process $H=(H_t)_{t\ge0}$, also predictable with respect to $\mathbb{F}^B$, such that $((H\cdot S)_t)_{t\ge 0}$ is a Brownian motion in its own filtration. Furthermore, for any $\delta>0$, we refine this construction such that the drift $(\mu_t)_{t\ge0}$ only takes values in $]\mu-\delta,\mu+\delta[$, for fixed $\mu>0$.Comment: Published in at http://dx.doi.org/10.1214/09-AOP469 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org