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Mathematics and Financial Economics

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07.11.2023

On intermediate marginals in martingale optimal transportation

verfasst von: Julian Sester

Erschienen in: Mathematics and Financial Economics | Ausgabe 4/2023

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Abstract

We study the influence of additional intermediate marginal distributions on the value of the martingale optimal transport problem. From a financial point of view, this corresponds to taking into account call option prices not only, as usual, for those call options where the respective future maturities coincide with the maturities of some exotic derivative but also additional maturities and then to study the effect on model-independent price bounds for the exotic derivative. We characterize market settings, i.e., combinations of the payoff of exotic derivatives, call option prices and marginal distributions that guarantee improved price bounds as well as those market settings that exclude any improvement. Eventually, we showcase in numerous examples that the consideration of additional price information on vanilla options may have a considerable impact on the resultant model-independent price bounds.

Vorheriger Artikel Investment in two alternative projects with multiple switches and the exit option

Nächster Artikel An elementary proof of the dual representation of Expected Shortfall

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Note that in practice the expiration date of an exchange-traded call option is often the third Friday of a month. Very often, for shorter maturities more expiration dates are available, often on a weekly basis.

A probability measure \(\mu _2\in \mathcal {P}(\mathbb {R})\) is larger in convex order than \(\mu _1\in \mathcal {P}(\mathbb {R})\), abbreviated by \(\mu _1 \preceq \mu _2\), if \(\int _\mathbb {R}f(x) {\mathop {}\!\textrm{d}}\mu _1(x) \le \int _\mathbb {R}f(x) {\mathop {}\!\textrm{d}}\mu _2(x)\) for all convex functions \(f:\mathbb {R}\rightarrow \mathbb {R}\) such that the integrals are finite.

Moreover, note that if the marginals increase in convex order, then the prices of European call options computed as expectations with respect to these marginals also increase, compare [35, Lemma 7.24]. That prices of European call options increase with an increasing maturity is well-known in arbitrage free markets, hence the assumption appears to be natural.

This assumption is mainly imposed to be able to apply a duality result (see [12] or [8]), and could be relaxed to a certain degree, for example by considering semi-continuous payoff functions instead of continuous functions. To avoid the need of distinguishing between lower-semi continuous payoff functions for lower bounds and upper semi-continuous payoff functions for upper bounds, we decided to circumvent this issue by only considering continuous payoff functions. The linear growth condition could be slightly relaxed to functions that do not grow stronger than a sum of \(\mu _i\)-integrable functions, however it is standard to use the linear growth condition as in the definition of \(\mathcal {C}_{{\text {lin}}}(\mathbb {R}^d)\), see e.g. [12, Theorem 1.1].

Compare e.g. [54, Proposition 2.2.] for a proof of the compactness with respect to the Wasserstein distance and the non-emptiness, which both are consequences of Berge’s maximum theorem.

We refer for more details on semi-static trading strategies and its properties to [2, 21, 28] and [56].

A probability kernel is a map where for fixed first component the map is a probability measure, and for fixed second component the map is Borel-measurable.

For any \(\mu ,\nu \in \mathcal {P}(\mathbb {R}^d)\) the Wasserstein distance of order 1 (or Wasserstein 1-distance) is defined as \( W_1(\mu ,\nu ):=\inf _{\mathbb {P}\in \Pi (\mu ,\nu )}\int _{\mathbb {R}^d \times \mathbb {R}^d} \Vert x-y\Vert ~\mathbb {P}({\mathop {}\!\textrm{d}}x, {\mathop {}\!\textrm{d}}y), \) where \(\Vert \cdot \Vert \) denotes the Euclidean norm on \(\mathbb {R}^d\), and where \(\Pi (\mu ,\nu )\) denotes the set of joint distributions of \(\mu \) and \(\nu \), compare also for example [62, Definition 6.1.].

For an extension of the presented setting that enables to formulate model-free super-hedging in a market with frictions, we refer the reader to [20, Section 2] [24, 31, Section 3], [33, Section 6.3] or [53, Appendix A.1].

We denote by \(\delta _x\) the Dirac measure centered on \(x \in \mathbb {R}\), i.e., for any measurable set \(A \subset \mathbb {R}\) we have \(\delta _x(A) = 1\) if \(x\in A\) and 0 else.

As shown in [43], under the unique optimal measure for the maximization problem, the conditional law of \(S_{t_3}\) is supported only on two values, and therefore Proposition 3.7 is applicable to the analogue maximization problem.

We use the mid-prices, i.e., the mean of observed bid and ask prices.

Note that the largest maturities correspond to marginals that are the largest with respect to the convex order.

Here we apply a variant of Lemma 4.1, where we only substitute two of the three \(L^1\)-integrand by functions from \(\mathcal {C}_b(\mathbb {R})\). This is possible due to the argumentation from [12, Appendix] showing that for all \(u\in L^1(\mu _i)\) and for all \(\varepsilon >0\) there exists some \(\widetilde{u} \in \mathcal {C}_b(\mathbb {R})\) such that \(\widetilde{u} \le u\) and \(\mathbb {E}_{\mu _i}[u]-\mathbb {E}_{\mu _i}[\widetilde{u}]<\varepsilon \).

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Titel: On intermediate marginals in martingale optimal transportation
verfasst von: Julian Sester
Publikationsdatum: 07.11.2023
Verlag: Springer Berlin Heidelberg
Erschienen in: Mathematics and Financial Economics / Ausgabe 4/2023
Print ISSN: 1862-9679
Elektronische ISSN: 1862-9660
DOI: https://doi.org/10.1007/s11579-023-00345-9

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