Options Prices in Incomplete Markets

Jacod, Jean; Protter, Philip; Crépey, Stéphane; Jeanblanc, Monique; Nikeghbali, Ashkan

doi:10.1051/proc/201756072

Cited by 5 publications

(4 citation statements)

References 14 publications

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“…3.4] for a discussion of this weaker notion of completeness. Below, we provide an example where the market is complete for the large filtration G, but not even quasi-complete for the smaller filtration F; this example answers negatively a conjecture put forth in Jacod and Protter [18].…”

Section: A Further Example Where Incompleteness Arises Though Filtratsupporting

confidence: 64%

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Filtration shrinkage, the structure of deflators, and failure of market completeness

Kardaras

Ruf

2020

Finance Stoch

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We analyse the structure of local martingale deflators projected on smaller filtrations. In a general continuous-path setting, we show that the local martingale parts in the multiplicative Doob–Meyer decomposition of projected local martingale deflators are themselves local martingale deflators in the smaller information market. Via use of a Bayesian filtering approach, we demonstrate the exact mechanism of how updates on the possible class of models under less information result in the strict supermartingale property of projections of such deflators. Finally, we demonstrate that these projections are unable to span all possible local martingale deflators in the smaller information market, by investigating a situation where market completeness is not retained under filtration shrinkage.

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Section: A Further Example Where Incompleteness Arises Though Filtratsupporting

confidence: 64%

“…The material in Sect. 6.4 also yields a counterexample to a conjecture put forth in Jacod and Protter [18].…”

Section: Introductionmentioning

confidence: 56%

Filtration shrinkage, the structure of deflators, and failure of market completeness

Kardaras

Ruf

2020

Finance Stoch

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“…As mentioned in the introduction, the answer to the motivating problem depends on the kind of option, on the specific model for the price fluctuations of the underlying asset and on the option pricing method. While leaving the comfortable world of complete markets [13], one has the freedom of choosing from a plethora of option pricing methods (see the references in [13] for a short survey). Therefore, a first natural extension of the present work would imply the analysis of different option pricing methods.…”

Section: Discussionmentioning

confidence: 99%

Limit theorems for prices of options written on semi-Markov processes

Scalas

Toaldo

2021

Theor. Probability and Math. Statist.

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We consider plain vanilla European options written on an underlying asset that follows a continuous time semi-Markov multiplicative process. We derive a formula and a renewal type equation for the martingale option price. In the case in which intertrade times follow the Mittag-Leffler distribution, under appropriate scaling, we prove that these option prices converge to the price of an option written on geometric Brownian motion time-changed with the inverse stable subordinator. For geometric Brownian motion time changed with an inverse subordinator, in the more general case when the subordinator’s Laplace exponent is a special Bernstein function, we derive a time-fractional generalization of the equation of Black and Scholes.

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“…While leaving the comfortable world of complete markets [13], one has the freedom of choosing from a plethora of option pricing methods (see the references in [13] for a short survey).…”

Section: Discussionmentioning

confidence: 99%

Limit theorems for prices of options written on semi-Markov processes

Scalas¹,

Toaldo²

2021

Preprint

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We consider plain vanilla European options written on an underlying asset that follows a continuous time semi-Markov multiplicative process. We derive a formula and a renewal type equation for the martingale option price. In the case in which intertrade times follow the Mittag-Leffler distribution, under appropriate scaling, we prove that these option prices converge to the price of an option written on geometric Brownian motion time-changed with the inverse stable subordinator. For geometric Brownian motion time changed with an inverse subordinator, in the more general case when the subordinator's Laplace exponent is a special Bernstein function, we derive a time-fractional generalization of the equation of Black and Scholes.

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Options Prices in Incomplete Markets

Cited by 5 publications

References 14 publications

Filtration shrinkage, the structure of deflators, and failure of market completeness

Filtration shrinkage, the structure of deflators, and failure of market completeness

Limit theorems for prices of options written on semi-Markov processes

Limit theorems for prices of options written on semi-Markov processes

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