Local Volatility Calibration by Optimal Transport

The calibration of volatility models from observable option prices is a fundamental problem in quantitative finance. The most common approach among industry practitioners is based on the celebrated Dupire's formula [6], which requires the knowledge of vanilla option prices for a continuum of strikes and maturities that can only be obtained via some form of price interpolation. In this paper, we propose a new local volatility calibration technique using the theory of optimal transport. We formulate a time continuous martingale optimal transport problem, which seeks a martingale diffusion process that matches the known densities of an asset price at two different dates, while minimizing a chosen cost function. Inspired by the seminal work of Benamou and Brenier [1], we formulate the problem as a convex optimization problem, derive its dual formulation, and solve it numerically via an augmented Lagrangian method and the alternative direction method of multipliers (ADMM) algorithm. The solution effectively reconstructs the dynamic of the asset price between the two dates by recovering the optimal local volatility function, without requiring any time interpolation of the option prices

On the Optimal Combination of Tensor Optimization Methods

We consider the minimization problem of a sum of a number of functions having Lipshitz $p$-th order derivatives with different Lipschitz constants. In this case, to accelerate optimization, we propose a general framework allowing to obtain near-optimal oracle complexity for each function in the sum separately, meaning, in particular, that the oracle for a function with lower Lipschitz constant is called a smaller number of times. As a building block, we extend the current theory of tensor methods and show how to generalize near-optimal tensor methods to work with inexact tensor step. Further, we investigate the situation when the functions in the sum have Lipschitz derivatives of a different order. For this situation, we propose a generic way to separate the oracle complexity between the parts of the sum. Our method is not optimal, which leads to an open problem of the optimal combination of oracles of a different order

Eigen-transitions in cantilever cylindrical shells subjected to vertical edge loads

A thin cantilever cylindrical shell subjected to a transverse shear force at the free end can experience two distinct modes of buckling, depending on its relative thickness and length. If the former parameter is fixed then a short cylinder buckles in a diffuse manner, while the eigenmodal deformation of a moderately long shell is localised, both axially and circumferentially, near its fixed end. Donnelltype buckling equations for cylindrical shells are here coupled with a non-symmetric membrane basic state to produce a linear boundary-value problem that is shown to capture the transition between the aforementioned instability modes. The main interest lies in exploring the approximate asymptotic separation of the independent variables in the corresponding stability equations, when the eigen-deformation is doubly localised. Comparisons with direct numerical simulations of the full buckling problem provide further insight into the accuracy and limitations of our approximations

Factorization of Operators Through Orlicz Spaces

[EN] We study factorization of operators between quasi-Banach spaces. We prove the equivalence between certain vector norm inequalities and the factorization of operators through Orlicz spaces. As a consequence, we obtain the Maurey-Rosenthal factorization of operators into L-p-spaces. We give several applications. In particular, we prove a variant of Maurey's Extension Theorem.The research of the first author was supported by the National Science Centre (NCN), Poland, Grant No. 2011/01/B/ST1/06243. The research of the second author was supported by Ministerio de Economia y Competitividad, Spain, under project #MTM2012-36740-C02-02Mastylo, M.; Sánchez Pérez, EA. (2017). Factorization of Operators Through Orlicz Spaces. Bulletin of the Malaysian Mathematical Sciences Society. 40(4):1653-1675. https://doi.org/10.1007/s40840-015-0158-5S16531675404Calderón, A.P.: Intermediate spaces and interpolation, the complex method. Stud. Math. 24, 113–190 (1964)Davis, W.J., Garling, D.J.H., Tomczak-Jaegermann, N.: The complex convexity of quasi-normed linear spaces. J. Funct. Anal. 55, 110–150 (1984)Defant, A.: Variants of the Maurey–Rosenthal theorem for quasi Köthe function spaces. Positivity 5, 153–175 (2001)Defant, A., Mastyło, M., Michels, C.: Orlicz norm estimates for eigenvalues of matrices. Isr. J. Math. 132, 45–59 (2002)Defant, A., Sánchez Pérez, E.A.: Maurey–Rosenthal factorization of positive operators and convexity. J. Math. Anal. Appl. 297, 771–790 (2004)Defant, A., Sánchez Pérez, E.A.: Domination of operators on function spaces. Math. Proc. Camb. Phil. Soc. 146, 57–66 (2009)Diestel, J.: Sequences and Series in Banach Spaces. Springer, Berlin (1984)Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge University Press, Cambridge (1995)Dilworth, S.J.: Special Banach lattices and their applications. In: Handbook of the Geometry of Banach Spaces, vol. 1. Elsevier, Amsterdam (2001)Figiel, T., Pisier, G.: Séries alétoires dans les espaces uniformément convexes ou uniformément lisses. Comptes Rendus de l’Académie des Sciences, Paris, Série A 279, 611–614 (1974)Kalton, N.J., Montgomery-Smith, S.J.: Set-functions and factorization. Arch. Math. (Basel) 61(2), 183–200 (1993)Kamińska, A., Mastyło, M.: Abstract duality Sawyer formula and its applications. Monatsh. Math. 151(3), 223–245 (2007)Kantorovich, L.V., Akilov, G.P.: Functional Analysis, 2nd edn. Pergamon Press, Oxford (1982)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1979)Lozanovskii, G.Ya.: On some Banach lattices IV, Sibirsk. Mat. Z. 14, 140–155 (1973) (in Russian); English transl.: Siberian. Math. J. 14, 97–108 (1973)Lozanovskii, G.Ya.:Transformations of ideal Banach spaces by means of concave functions. In: Qualitative and Approximate Methods for the Investigation of Operator Equations, Yaroslavl, vol. 3, pp. 122–147 (1978) (Russian)Mastyło, M., Szwedek, R.: Interpolative constructions and factorization of operators. J. Math. Anal. Appl. 401, 198–208 (2013)Nikišin, E.M.: Resonance theorems and superlinear operators. Usp. Mat. Nauk 25, 129–191 (1970) (Russian)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal Domain and Integral Extension of Operators acting in Function Spaces. Operator Theory: Adv. Appl., vol. 180. Birkhäuser, Basel (2008)Pisier, G.: Factorization of linear operators and geometry of Banach spaces. CBMS Regional Conference Series in Mathematics, vol. 60. Published for the Conference Board of the Mathematical Sciences, Washington, DC (1986)Reisner, S.: On two theorems of Lozanovskii concerning intermediate Banach lattices, geometric aspects of functional analysis (1986/87). Lecture Notes in Math., vol. 1317, pp. 67–83. Springer, Berlin (1988)Wojtaszczyk, P.: Banach Spaces for Analysts. Cambridge University Press, Cambridge (1991

From error bounds to the complexity of first-order descent methods for convex functions

This paper shows that error bounds can be used as effective tools for deriving complexity results for first-order descent methods in convex minimization. In a first stage, this objective led us to revisit the interplay between error bounds and the Kurdyka-\L ojasiewicz (KL) inequality. One can show the equivalence between the two concepts for convex functions having a moderately flat profile near the set of minimizers (as those of functions with H\"olderian growth). A counterexample shows that the equivalence is no longer true for extremely flat functions. This fact reveals the relevance of an approach based on KL inequality. In a second stage, we show how KL inequalities can in turn be employed to compute new complexity bounds for a wealth of descent methods for convex problems. Our approach is completely original and makes use of a one-dimensional worst-case proximal sequence in the spirit of the famous majorant method of Kantorovich. Our result applies to a very simple abstract scheme that covers a wide class of descent methods. As a byproduct of our study, we also provide new results for the globalization of KL inequalities in the convex framework. Our main results inaugurate a simple methodology: derive an error bound, compute the desingularizing function whenever possible, identify essential constants in the descent method and finally compute the complexity using the one-dimensional worst case proximal sequence. Our method is illustrated through projection methods for feasibility problems, and through the famous iterative shrinkage thresholding algorithm (ISTA), for which we show that the complexity bound is of the form $O(q^{k})$ where the constituents of the bound only depend on error bound constants obtained for an arbitrary least squares objective with $\ell^1$ regularization

A user's guide to optimal transport

This text is an expanded version of the lectures given by the first author in the 2009 CIME summer school of Cetraro. It provides a quick and reasonably account of the classical theory of optimal mass transportation and of its more recent developments, including the metric theory of gradient flows, geometric and functional inequalities related to optimal transportation, the first and second order differential calculus in the Wasserstein space and the synthetic theory of metric measure spaces with Ricci curvature bounded from below

Kothe dual of Banach lattices generated by vector measures

We study the Kothe dual spaces of Banach function lattices generated by abstract methods having roots in the theory of interpolation spaces. We apply these results to Banach spaces of integrable functions with respect to Banach space valued countably additive vector measures. As an application we derive a description of the Banach dual of a large class of these spaces, including Orlicz spaces of integrable functions with respect to vector measuresThe first author was supported by the Foundation for Polish Science (FNP). The second author was supported by the Ministerio de Economia y Competitividad (Spain) under Grant #MTM2012-36740-C02-02.Mastylo, M.; Sánchez Pérez, EA. (2014). Kothe dual of Banach lattices generated by vector measures. Monatshefte fur Mathematik. 173(4):541-557. https://doi.org/10.1007/s00605-013-0560-8S5415571734Aronszajn, N., Gagliardo, E.: Interpolation spaces and interpolation methods. Ann. Mat. Pura. Appl. 68, 51–118 (1965)Bartle, R.G., Dunford, N., Schwartz, J.: Weak compactness and vector measures. Canad. J. Math. 7, 289–305 (1955)Brudnyi, Yu.A., Krugljak, N.Ya.: Interpolation functors and interpolation spaces $$I$$ I . North-Holland, Amsterdam (1991)Curbera, G.P.: Operators into $$L^1$$ L 1 of a vector measure and applications to Banach lattices. Math. Ann. 293, 317–330 (1992)Curbera, G.P., Ricker, W.J.: The Fatou property in $$p$$ p -convex Banach lattices. J. Math. Anal. Appl. 328, 287–294 (2007)Delgado, O.: Banach function subspaces of $$L^1$$ L 1 of a vector measure and related Orlicz spaces. Indag. Math. 15(4), 485–495 (2004)Diestel, J., Jr., Uhl, J.J.: Vector measures, Amer. Math. Soc. Surveys 15, Providence, R.I. (1977)Fernández, A., Mayoral, F., Naranjo, F., Sánchez-Pérez, E.A.: Spaces of $$p$$ p -integrable functions with respect to a vector measure. Positivity 10, 1–16 (2006)Ferrando, I., Rodríguez, J.: The weak topology on $$L_p$$ L p of a vector measure. Topol. Appl. 155, 1439–1444 (2008)Ferrando, I., Sánchez Pérez, E.A.: Tensor product representation of the (pre)dual of the $$L_p$$ L p -space of a vector measure. J. Aust. Math. Soc. 87, 211–225 (2009)Galaz-Fontes, F.: The dual space of $$L^p$$ L p of a vector measure. Positivity 14(4), 715–729 (2010)Kamińska, A.: Indices, convexity and concavity in Musielak-Orlicz spaces, dedicated to Julian Musielak. Funct. Approx. Comment. Math. 26, 67–84 (1998)Kantorovich, L.V., Akilov, G.P.: Functional analysis, 2nd edn. Pergamon Press, New York (1982)Krein, S.G., Petunin, Yu.I., Semenov, E.M.: Interpolation of linear operators. In: Translations of mathematical monographs, 54. American Mathematical Society, Providence, R.I., (1982)Lewis, D.R.: Integration with respect to vector measures. Pacific. J. Math. 33, 157–165 (1970)Lewis, D.R.: On integrability and summability in vector spaces. Ill. J. Math. 16, 583–599 (1973)Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces II. Springer, Berlin (1979)Lozanovskii, G.Ya.: On some Banach lattices, (Russian). Sibirsk. Mat. Z. 10, 419–430 (1969)Musielak, J.: Orlicz spaces and modular spaces. In: Lecture Notes in Math. 1034, Springer-Verlag, Berlin (1983)Okada, S.: The dual space of $$L^1(\mu )$$ L 1 ( μ ) of a vector measure $$\mu$$ μ . J. Math. Anal. Appl. 177, 583–599 (1993)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal domain and integral extension of operators acting in function spaces, operator theory. Adv. Appl., vol. 180, Birkhäuser, Basel (2008)Rao, M.M., Zen, Z.D.: Applications of Orlicz spaces. Marcel Dekker, Inc., New York (2002)Rivera, M.J.: Orlicz spaces of integrable functions with respect to vector-valued measures. Rocky Mt. J. Math. 38(2), 619–637 (2008)Sánchez Pérez, E.A.: Compactness arguments for spaces of $$p$$ p -integrable functions with respect to a vector measure and factorization of operators through Lebesgue-Bochner spaces. Ill. J. Math. 45(3), 907–923 (2001)Sánchez Pérez, E.A.: Vector measure duality and tensor product representation of $$L_p$$ L p spaces of vector measures. Proc. Amer. Math. Soc. 132, 3319–3326 (2004)Zaanen, A.C.: Integration. North Holland, Amsterdam (1967

Relation between the Kantorovich-Wasserstein metric and the Kullback-Leibler divergence

We discuss a relation between the Kantorovich-Wasserstein (KW) metric and the Kullback-Leibler (KL) divergence. The former is defined using the optimal transport problem (OTP) in the Kantorovich formulation. The latter is used to define entropy and mutual information, which appear in variational problems to find optimal channel (OCP) from the rate distortion and the value of information theories. We show that OTP is equivalent to OCP with one additional constraint fixing the output measure, and therefore OCP with constraints on the KL-divergence gives a lower bound on the KW-metric. The dual formulation of OTP allows us to explore the relation between the KL-divergence and the KW-metric using decomposition of the former based on the law of cosines. This way we show the link between two divergences using the variational and geometric principles