On optimal arbitrage

Fernholz, Daniel; Karatzas, Ioannis

doi:10.1214/09-aap642

Cited by 60 publications

(114 citation statements)

References 32 publications

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“…Fernholz, Karatzas and Kardaras (2005) derive the existence of arbitrage and relative arbitrage in frictionless markets satisfying a diversity property, which means that no asset can overtake in size the remaining ones. Fernholz and Karatzas (2010) study relative arbitrage strategies that are optimal in the sense of maximal multiplication of wealth, and Ruf (2011) characterizes optimal hedging strategies in the presence of arbitrage.…”

Section: Literature Reviewmentioning

confidence: 99%

Fragility of arbitrage and bubbles in local martingale diffusion models

Guasoni

Rásonyi

2015

Finance Stoch

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For any positive diffusion with minimal regularity, there exists a semimartingale, with uniformly close paths, which is a martingale under an equivalent probability. As a result, in models of asset prices based on such diffusions, arbitrage and bubbles alike disappear under proportional transaction costs, or under small model misspecifications. Thus, local martingale models of arbitrage and bubbles are not robust to small trading and monitoring frictions.Mathematics Subject Classification (2010) Primary 91B28 · Secondary 62P05 JEL Classification G12.

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Section: Literature Reviewmentioning

confidence: 99%

Fragility of arbitrage and bubbles in local martingale diffusion models

Guasoni

Rásonyi

2015

Finance Stoch

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“…[3] and [9] have already observed that the Cauchy problem corresponding to European call options have multiple solutions. (Also see [7] and [1], which consider super hedging prices of call-type options when there are no equivalent local martingale measures.) However, a necessary and sufficient condition under which there is uniqueness/nonuniqueness remains unknown.…”

Section: Proof Of Necessitymentioning

confidence: 99%

On the uniqueness of classical solutions of Cauchy problems

Bayraktar

Xing

2010

Proc. Amer. Math. Soc.

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Abstract. Given that the terminal condition is of at most linear growth, it is well known that a Cauchy problem admits a unique classical solution when the coefficient multiplying the second derivative is also a function of at most linear growth. In this paper, we give a condition on the volatility that is necessary and sufficient for a Cauchy problem to admit a unique solution. Main resultGiven a terminal-boundary condition g :for some constant C > 0, we consider the following Cauchy problem:whereA well-known sufficient condition for (1) to have a unique classical solution among the functions with at most linear growth is that σ itself is of at most linear growth; see, e.g., Chapter 6 of [8] and Theorem 7.6 on page 366 of [10]. On the other hand, consider the SDEThe assumptions on σ we made below (1) ensure that (2) has a unique weak solution which is absorbed at zero. (See [6].) The solution X t,x · is clearly a local martingale.

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“…with the initial condition u(0, x) = 1 , 4) and is in fact the smallest nonnegative (super)solution of the Cauchy problem (1.3), (1.4).…”

Section: Notation and Previous Resultsmentioning

confidence: 99%

“…For the one-dimensional case, the function U of (1. , as well as to [4], [5] and [15]. Proof of Theorem 4.1.…”

Section: Minimalitymentioning

confidence: 87%

Viscosity characterization of the explosion time distribution for diffusions

Wang¹

2017

Bernoulli

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We show that the tail distribution U of the explosion time for a multidimensional diffusion (and more generally, a suitable function U of the Feynman-Kac type involving the explosion time) is a viscosity solution of an associated parabolic partial differential equation (PDE), provided that the dispersion and drift coefficients of the diffusion are continuous. This generalizes a result of Karatzas and Ruf (2015), who characterize U as a classical solution of a Cauchy problem for the PDE in the one-dimensional case, under the stronger condition of local Hölder continuity on the coefficients. We also extend their result to U in the one-dimensional case by establishing the joint continuity of U . Furthermore, we show that U is dominated by any nonnegative classical supersolution of this Cauchy problem. Finally, we consider another notion of weak solvability, that of the distributional (sub/super)solution, and show that U is no greater than any nonnegative distributional supersolution of the relevant PDE.

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On optimal arbitrage

Cited by 60 publications

References 32 publications

Fragility of arbitrage and bubbles in local martingale diffusion models

Fragility of arbitrage and bubbles in local martingale diffusion models

On the uniqueness of classical solutions of Cauchy problems

Viscosity characterization of the explosion time distribution for diffusions

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