Marginal and dependence uncertainty: bounds, optimal transport, and sharpness

Bartl, Daniel; Kupper, Michael; Lux, Thibaut; Papapantoleon, Antonis; Eckstein, Stephan

doi:10.48550/arxiv.1709.00641

Cited by 4 publications

(7 citation statements)

References 24 publications

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“…The difference between Assumption 2.1 and the no-arbitrage assumption in [15] is that the price of a financial derivative in the present work is not a singleton but can lie anywhere between the corresponding bid and ask prices π, π. Note that there are other notions of no-arbitrage that are weaker than Assumption 2.1, for example the "no uniform strong arbitrage" assumption in Definition 2.1 of Bartl et al [4].…”

Section: Duality In the Presence Of Option-implied Informationmentioning

confidence: 99%

“…The setting of dependence uncertainty is intimately linked with optimal transport theory, and its tools have also been used in order to derive bounds for multi-asset option prices, see e.g. Bartl, Kupper, Lux, Papapantoleon, and Eckstein [4] for a formulation in the presence of additional information on the joint distribution. More recently, Aquino and Bernard [2], Eckstein and Kupper [32] and Eckstein, Guo, Lim, and Oblój [34] have translated the model-free superhedging problem into an optimization problem over classes of functions, by extending results in optimal transport, and used neural networks and the stochastic gradient descent algorithm for the computation of the bounds.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Model-free bounds for multi-asset options using option-implied information and their exact computation

Neufeld¹,

Papapantoleon²,

Xiang³

2020

Preprint

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We consider derivatives written on multiple underlyings in a one-period financial market, and we are interested in the computation of model-free upper and lower bounds for their arbitrage-free prices. We work in a completely realistic setting, in that we only assume the knowledge of traded prices for other single-and multi-asset derivatives, and even allow for the presence of bid-ask spread in these prices. We provide a fundamental theorem of asset pricing for this market model, as well as a superhedging duality result, that allows to transform the abstract maximization problem over probability measures into a more tractable minimization problem over vectors, subject to certain constraints. Then, we recast this problem into a linear semi-infinite optimization problem, and provide two algorithms for its solution. These algorithms provide upper and lower bounds for the prices that are ε-optimal, as well as a characterization of the optimal pricing measures. Moreover, these algorithms are efficient and allow the computation of bounds in high-dimensional scenarios (e.g. when d = 60). Numerical experiments using synthetic data showcase the efficiency of these algorithms, while they also allow to understand the reduction of model-risk by including additional information, in the form of known derivative prices.

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Section: Duality In the Presence Of Option-implied Informationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Model-free bounds for multi-asset options using option-implied information and their exact computation

Neufeld¹,

Papapantoleon²,

Xiang³

2020

Preprint

Self Cite

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show abstract

“…On the contrary, Lux and Papapantoleon [12] showed that for d > 2, the improved Fréchet-Hoeffding bounds are copulas only in trivial cases and proper quasi-copulas otherwise. Moreover, Bartl, Kupper, Lux, Papapantoleon, and Eckstein [1] showed that the improved Fréchet-Hoeffding bounds are not pointwise sharp (or best-possible), even in d = 2, if the aforementioned 'monotonicity' conditions are violated.…”

Section: Copulas Quasi-copulas and Improved Fréchet-hoeffding Boundsmentioning

confidence: 99%

Detection of arbitrage opportunities in multi-asset derivatives markets

Papapantoleon¹,

Sarmiento²

2020

Preprint

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We are interested in the existence of equivalent martingale measures and the detection of arbitrage opportunities in markets where several multi-asset derivatives are traded simultaneously. More specifically, we consider a financial market with multiple traded assets whose marginal risk-neutral distributions are known, and assume that several derivatives written on these assets are traded simultaneously. In this setting, there is a bijection between the existence of an equivalent martingale measure and the existence of a copula that couples these marginals. Using this bijection and recent results on improved Fréchet-Hoeffding bounds in the presence of additional information, we derive sufficient conditions for the absence of arbitrage and formulate an optimization problem for the detection of a possible arbitrage opportunity. This problem can be solved efficiently using numerical optimization routines. The most interesting practical outcome is the following: we can construct a financial market where each multi-asset derivative is traded within its own no-arbitrage interval, and yet when considered together an arbitrage opportunity may arise.

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“…In contrast, Lux and Papapantoleon [24] showed that for d > 2 the bounds Q S,Q * and Q S,Q * are copulas only in degenerate cases, and quasi-copulas otherwise. Moreover, Bartl, Kupper, Lux, and Papapantoleon [2] recently showed that once the constraints of [4,43] are violated then the improved Fréchet-Hoeffding bounds fail to even be pointwise sharp, still in dimension d = 2.…”

Section: Improved Fréchet-hoeffding Bounds Using Subsetsmentioning

confidence: 99%

Model-free bounds on Value-at-Risk using extreme value information and statistical distances

Lux

Papapantoleon

2019

Insurance: Mathematics and Economics

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We derive bounds on the distribution function, therefore also on the Value-at-Risk, of ϕ(X) where ϕ is an aggregation function and X = (X1, . . . , X d ) is a random vector with known marginal distributions and partially known dependence structure. More specifically, we analyze three types of available information on the dependence structure: First, we consider the case where extreme value information, such as the distributions of partial minima and maxima of X, is available. In order to include this information in the computation of Value-at-Risk bounds, we utilize a reduction principle that relates this problem to an optimization problem over a standard Fréchet class, which can then be solved by means of the rearrangement algorithm or using analytical results. Second, we assume that the copula of X is known on a subset of its domain, and finally we consider the case where the copula of X lies in the vicinity of a reference copula as measured by a statistical distance. In order to derive Value-at-Risk bounds in the latter situations, we first improve the Fréchet-Hoeffding bounds on copulas so as to include this additional information on the dependence structure. Then, we translate the improved Fréchet-Hoeffding bounds to bounds on the Value-at-Risk using the so-called improved standard bounds. In numerical examples we illustrate that the additional information typically leads to a significant improvement of the bounds compared to the marginals-only case.

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Marginal and dependence uncertainty: bounds, optimal transport, and sharpness

Cited by 4 publications

References 24 publications

Model-free bounds for multi-asset options using option-implied information and their exact computation

Model-free bounds for multi-asset options using option-implied information and their exact computation

Detection of arbitrage opportunities in multi-asset derivatives markets

Model-free bounds on Value-at-Risk using extreme value information and statistical distances

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