Fragility of arbitrage and bubbles in local martingale diffusion models

Guasoni, Paolo; Rásonyi, Miklós

doi:10.1007/s00780-015-0256-0

Cited by 26 publications

(33 citation statements)

References 32 publications

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“…This contrasts with the arguments of e.g. Guasoni and Rasonyi [19], who argue against local-martingale models on the basis that they can always be approximated by martingale models. Short selling bans as a regulatory tool to discourage speculation and stabilise markets have proved to be popular among emerging markets and during times of financial crisis.…”

Section: Introductionmentioning

confidence: 58%

Robust pricing and hedging under trading restrictions and the emergence of local martingale models

2016

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We pursue the robust approach to pricing and hedging in which no probability measure is fixed, but call or put options with different maturities and strikes can be traded initially at their market prices. We allow the inclusion of robust modelling assumptions by specifying a set of feasible paths on which (super)hedging arguments are required to work. In a discrete-time setup with no short selling, we characterise absence of arbitrage and show that if call options are traded, then the usual pricinghedging duality is preserved. In contrast, if only put options are traded, a duality gap may appear. Embedding the results into a continuous-time framework, we show that the duality gap may be interpreted as a financial bubble and link it to strict local martingales. This provides an intrinsic justification of strict local martingales as models for financial bubbles arising from a combination of trading restrictions and current market prices.Keywords Robust pricing and hedging · Financial bubble · Local martingale models · Pricing-hedging duality · Trading restrictions · Short selling constraint · Martingale optimal transport Mathematics Subject Classification (2010) 91G20 · 91B24 · 60G42 · 60G44 JEL Classification C61 · G13 · E32

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Section: Introductionmentioning

confidence: 58%

Robust pricing and hedging under trading restrictions and the emergence of local martingale models

2016

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“…In this context, P is the pricing measure, S is the price process of risky assets. Theorem 5.1 and Corollary 5.2 reiterate the word of caution already pronounced in Guasoni and Rásonyi [11]: an arbitrarily small mis-specification of option and asset prices (that is, mistaking Q for P andS for S) may destroy the "bubble phenomenon" generated by S under P sincẽ S is a martingale under Q, admitting no bubbles.…”

Section: Local Martingalesmentioning

confidence: 64%

“…has a weak solution, unique in law, for all x ∈ R d , then any solution satisfies CFS, a fortiori, stickiness, as shown in Guasoni and Rásonyi [11].…”

Section: Examplesmentioning

confidence: 78%

Sticky processes, local and true martingales

Rásonyi¹,

Sayit²

2018

Bernoulli

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We prove that for a so-called sticky process S there exists an equivalent probability Q and a Q-martingaleS that is arbitrarily close to S in L p (Q) norm. For continuous S,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 an application in mathematical finance. * The first author was supported by the "Lendület" Grant LP2015-6 of the Hungarian Academy of Sciences. Discussions with Martin Keller-Ressel led to formulating the main results of the present paper, we sincerely thank him. We also thank Eberhard Mayerhofer for spotting an error and two anonymous referees for helpful reports which revealed, in particular, another problem in a previous version of this paper. † MTA Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15,

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“…In the frictionless case one typically assumes the existence of an equivalent local martingale measure for the price process having suitable integrability properties to achieve this. In contrast, our sufficient conditions under transaction costs are more robust and hold in a wide variety of models; see [3,14,28,32,31,35,46,45] 1 . They apply in particular to the fractional Black-Scholes model and exponential utility.…”

Section: Introductionmentioning

confidence: 95%

Portfolio optimisation beyond semimartingales: Shadow prices and fractional Brownian motion

Czichowsky¹,

Schachermayer²

2017

Ann. Appl. Probab.

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London School of Economics and Political Science and Universität WienWhile 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 Itô process. As a consequence, we obtain a rather surprising result on the pathwise behaviour of fractional Brownian motion: the trajectories may touch an Itô process in a one-sided manner without reflection. Introduction.Most of the literature in mathematical finance assumes that discounted prices S = (S t ) 0≤t≤T of risky assets are modelled by semimartingales. In frictionless financial markets, where arbitrary amounts of stock can be bought and sold at the same price S t , the semimartingale assumption is necessary. Otherwise, there would exist "arbitrage opportunities" (see [26], Theorem 7.2 for a precise statement) and optimal strategies for utility maximisation problems would fail to exist or yield infinite expected utility (see [2,40,42]).For non-semimartingale models based on fractional Brownian motion (B H t ) t≥0 such as the fractional Black-Scholes model S t = exp(μt + σ B H t ), where μ ∈ R, σ > 0 and Hurst parameter H ∈ (0, 1) \ {

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Fragility of arbitrage and bubbles in local martingale diffusion models

Cited by 26 publications

References 32 publications

Robust pricing and hedging under trading restrictions and the emergence of local martingale models

Robust pricing and hedging under trading restrictions and the emergence of local martingale models

Sticky processes, local and true martingales

Portfolio optimisation beyond semimartingales: Shadow prices and fractional Brownian motion

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