## Journal Articles |

Yann Braouezec; Cyril Grunspan Option Pricing Bounds in a Finite Market Model: A Simple Geometric Approach Using Barycentric Coordinates Journal Article European Journal of Operational Research, 249 (1), pp. 270–280, 2016, ISSN: 0377-2217. @article{Grunspanb, title = {Option Pricing Bounds in a Finite Market Model: A Simple Geometric Approach Using Barycentric Coordinates}, author = { Yann Braouezec and Cyril Grunspan}, editor = {Elsevier}, url = {http://www.sciencedirect.com/science/article/pii/S0377221715007614}, doi = {10.1016/j.ejor.2015.08.024}, issn = {0377-2217}, year = {2016}, date = {2016-02-16}, journal = { European Journal of Operational Research}, volume = {249}, number = {1}, pages = {270–280}, abstract = {The aim of this paper is to provide a new straightforward textitmeasure-free methodology based on a convex hulls to determine the no-arbitrage pricing bounds of an option (European or American). The pedagogical interest of our methodology is also briefly discussed. The central result, which is elementary, is presented for a one period model and is subsequently used for multiperiod models. It shows that a certain point, called the forward point, must lie inside a convex polygon. Multiperiod models are then considered and the pricing bounds of a put option (European and American) are explicitly computed. We then show that the barycentric coordinates of the forward point can be interpreted as a martingale pricing measure. An application is provided for the trinomial model where the pricing measure has a simple geometric interpretation in terms of areas of triangles. Finally, we consider the case of entropic barycentric coordinates in a multi assets framework.}, keywords = {}, pubstate = {published}, tppubtype = {article} } The aim of this paper is to provide a new straightforward textitmeasure-free methodology based on a convex hulls to determine the no-arbitrage pricing bounds of an option (European or American). The pedagogical interest of our methodology is also briefly discussed. The central result, which is elementary, is presented for a one period model and is subsequently used for multiperiod models. It shows that a certain point, called the forward point, must lie inside a convex polygon. Multiperiod models are then considered and the pricing bounds of a put option (European and American) are explicitly computed. We then show that the barycentric coordinates of the forward point can be interpreted as a martingale pricing measure. An application is provided for the trinomial model where the pricing measure has a simple geometric interpretation in terms of areas of triangles. Finally, we consider the case of entropic barycentric coordinates in a multi assets framework. |

Cyril Grunspan Hopf-Galois Systems and Kashiwara Algebras Journal Article Communications in Algebra, 32 (9), pp. 3373-3389, 2008. @article{Grunspanbb, title = {Hopf-Galois Systems and Kashiwara Algebras}, author = { Cyril Grunspan}, url = {http://arxiv.org/pdf/math/0301195.pdf}, doi = {10.1081/AGB-120038639}, year = {2008}, date = {2008-02-01}, journal = {Communications in Algebra}, volume = {32}, number = {9}, pages = {3373-3389}, abstract = {This article is made up with two parts. In the first part, using a recent result of Schauenburg, one generalizes to the case when objects are faithfully flat over the ground ring, the full equivalence between the notions of Hopf-Galois objects and Hopf-Galois systems. In this last description, one gives explicitly an inverse for a Hopf-Galois object T together with its generalized antipode. In the second part of the article, one shows that the Kashiwara algebras introduced by Kashiwara in his study of crystal bases form Hopf-Galois systems under the coaction of a quantized enveloping algebra of a Kac-Moody algebra. Their classical limits are examples of Sridharan algebras.}, keywords = {}, pubstate = {published}, tppubtype = {article} } This article is made up with two parts. In the first part, using a recent result of Schauenburg, one generalizes to the case when objects are faithfully flat over the ground ring, the full equivalence between the notions of Hopf-Galois objects and Hopf-Galois systems. In this last description, one gives explicitly an inverse for a Hopf-Galois object T together with its generalized antipode. In the second part of the article, one shows that the Kashiwara algebras introduced by Kashiwara in his study of crystal bases form Hopf-Galois systems under the coaction of a quantized enveloping algebra of a Kac-Moody algebra. Their classical limits are examples of Sridharan algebras. |

Cyril Grunspan Quantizations of the Witt Algebra and of Simple Lie algebras in characteristic p Journal Article Journal of Algebra, 280 (1), pp. 145-161, 2004. @article{Grunspanbb, title = {Quantizations of the Witt Algebra and of Simple Lie algebras in characteristic p}, author = { Cyril Grunspan}, url = {http://www.sciencedirect.com/science/article/pii/S002186930400273X}, doi = {doi:10.1016/j.jalgebra.2004.04.016}, year = {2004}, date = {2004-08-01}, journal = {Journal of Algebra}, volume = {280}, number = {1}, pages = {145-161}, abstract = {We explicitly quantize the Witt algebra in characteristic 0 equipped with its Lie bialgebra structures discovered by Taft. Then, we study the reduction modulo p of our formulas. This gives p−1p−1 families of polynomial noncocommutative deformations of a restricted enveloping algebra of a simple Lie algebra in characteristic p (of Cartan type). In particular, this yields new families of noncommutative and noncocommutative Hopf algebras of dimension pppp in char p.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We explicitly quantize the Witt algebra in characteristic 0 equipped with its Lie bialgebra structures discovered by Taft. Then, we study the reduction modulo p of our formulas. This gives p−1p−1 families of polynomial noncocommutative deformations of a restricted enveloping algebra of a simple Lie algebra in characteristic p (of Cartan type). In particular, this yields new families of noncommutative and noncocommutative Hopf algebras of dimension pppp in char p. |

Cyril Grunspan Quantum Torsors Journal Article Journal of Pure and Applied Algebra, (184), pp. 229-255, 2003. @article{Grunspanbb, title = {Quantum Torsors}, author = { Cyril Grunspan}, url = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.234.8852}, year = {2003}, date = {2003-01-01}, journal = {Journal of Pure and Applied Algebra}, number = {184}, pages = {229-255}, abstract = {The following text is a short version of a forthcoming preprint about torsors. The adopted viewpoint is an old reformulation of torsors recalled recently by Kontsevich [Kon]. We propose a unification of the definitions of torsors in algebraic geometry and in Poisson manifolds (Example 2 and section 2.2). We introduce the notion of a quantum torsor (Definition 2.1). Any quantum torsor is equipped with two comodule-algebra structures over Hopf algebras and these structures commute with each other (Theorem 3.1.) In the finite dimensional case, these two Hopf algebras share the same finite dimension (Proposition 3.1). We show that any Galois extension of a field is a torsor (Example 4) and that any torsor is a Hopf-Galois extension (section 3.2). We give also examples of non-commutative torsors without character (Example 5). Torsors can be composed (Theorem 3.2). This leads us to define for any Hopf algebra, a new group-invariant, its torsors invariant (Theorem 3.3). We show how Parmentier’s quantization formalism of “affine Poisson groups ” is part of our theory of torsors (Theorem 3.4). 1.}, keywords = {}, pubstate = {published}, tppubtype = {article} } The following text is a short version of a forthcoming preprint about torsors. The adopted viewpoint is an old reformulation of torsors recalled recently by Kontsevich [Kon]. We propose a unification of the definitions of torsors in algebraic geometry and in Poisson manifolds (Example 2 and section 2.2). We introduce the notion of a quantum torsor (Definition 2.1). Any quantum torsor is equipped with two comodule-algebra structures over Hopf algebras and these structures commute with each other (Theorem 3.1.) In the finite dimensional case, these two Hopf algebras share the same finite dimension (Proposition 3.1). We show that any Galois extension of a field is a torsor (Example 4) and that any torsor is a Hopf-Galois extension (section 3.2). We give also examples of non-commutative torsors without character (Example 5). Torsors can be composed (Theorem 3.2). This leads us to define for any Hopf algebra, a new group-invariant, its torsors invariant (Theorem 3.3). We show how Parmentier’s quantization formalism of “affine Poisson groups ” is part of our theory of torsors (Theorem 3.4). 1. |

Cyril Grunspan Discrete Quantum Drinfeld-Sokolov Correspondence Journal Article Communications in Mathematical Physics, 226 (3), pp. 627-662, 2002. @article{Grunspanbb, title = {Discrete Quantum Drinfeld-Sokolov Correspondence}, author = { Cyril Grunspan}, url = {http://www.researchgate.net/publication/225384888_Discrete_Quantum_DrinfeldSokolov_Correspondence}, doi = {10.1007/s002200200626}, year = {2002}, date = {2002-01-01}, journal = {Communications in Mathematical Physics}, volume = {226}, number = {3}, pages = {627-662}, abstract = {We construct a discrete quantum version of the Drinfeld–Sokolov correspondence for the sine-Gordon system. The classical version of this correspondence is a birational Poisson morphism between the phase space of the discrete sine-Gordon system and a Poisson homogeneous space. Under this correspondence, the commuting higher mKdV vector fields correspond to the action of an Abelian Lie algebra. We quantize this picture (1) by quantizing this Poisson homogeneous space, together with the action of the Abelian Lie algebra, (2) by quantizing the sine-Gordon phase space, (3) by computing the quantum analogues of the integrals of motion generating the mKdV vector fields, and (4) by constructing an algebra morphism taking one commuting family of derivations to the other one.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We construct a discrete quantum version of the Drinfeld–Sokolov correspondence for the sine-Gordon system. The classical version of this correspondence is a birational Poisson morphism between the phase space of the discrete sine-Gordon system and a Poisson homogeneous space. Under this correspondence, the commuting higher mKdV vector fields correspond to the action of an Abelian Lie algebra. We quantize this picture (1) by quantizing this Poisson homogeneous space, together with the action of the Abelian Lie algebra, (2) by quantizing the sine-Gordon phase space, (3) by computing the quantum analogues of the integrals of motion generating the mKdV vector fields, and (4) by constructing an algebra morphism taking one commuting family of derivations to the other one. |

Cyril Grunspan On Integrals of Motion of The Discrete Sine-Gordon System Journal Article Letters in Mathematical Physics, 54 (2), pp. 101-121, 2000, ISSN: 1573-0530. @article{Grunspanbb, title = {On Integrals of Motion of The Discrete Sine-Gordon System}, author = { Cyril Grunspan}, editor = {Springer Kluwer Academic Publishers}, url = {http://link.springer.com/article/10.1023%2FA%3A1011078706129}, doi = {10.1023/A:1011078706129}, issn = {1573-0530}, year = {2000}, date = {2000-10-01}, journal = {Letters in Mathematical Physics}, volume = {54}, number = {2}, pages = {101-121}, abstract = {We give explicit formulas for some densities of integrals of motion for the discrete sine-Gordon system (quantum or not). The generating function for the densities of integrals of motion may be seen as the expansion of the logarithm of a certain continued fraction (possibly quantum). In the case of q root of the unity, we show that these integrals of motion can be identified to the classical integrals of motion.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We give explicit formulas for some densities of integrals of motion for the discrete sine-Gordon system (quantum or not). The generating function for the densities of integrals of motion may be seen as the expansion of the logarithm of a certain continued fraction (possibly quantum). In the case of q root of the unity, we show that these integrals of motion can be identified to the classical integrals of motion. |

## Unpublished |

Cyril Grunspan A Note on the Equivalence between the Normal and the Lognormal Implied Volatility : A Model Free Approach Unpublished 2011. @unpublished{Grunspan, title = {A Note on the Equivalence between the Normal and the Lognormal Implied Volatility : A Model Free Approach}, author = { Cyril Grunspan}, url = {http://arxiv.org/pdf/1112.1782.pdf}, year = {2011}, date = {2011-12-09}, abstract = {First, we show that implied normal volatility is intimately linked with the incomplete Gamma function. Then, we deduce an expansion on implied normal volatility in terms of the time - value of a European call option. Then, we formulate an equivalence between the implied normal volatility and the lognormal implied volatility with any strike and any model. This generalizes a known result for the SABR model. Finally, we address the issue of the 'breakeven move' of a delta - hedged portfolio.}, keywords = {}, pubstate = {published}, tppubtype = {unpublished} } First, we show that implied normal volatility is intimately linked with the incomplete Gamma function. Then, we deduce an expansion on implied normal volatility in terms of the time - value of a European call option. Then, we formulate an equivalence between the implied normal volatility and the lognormal implied volatility with any strike and any model. This generalizes a known result for the SABR model. Finally, we address the issue of the 'breakeven move' of a delta - hedged portfolio. |

Cyril Grunspan Asymptotic Expansions of the Lognormal Implied Volatility Unpublished 2011. @unpublished{Grunspanbb, title = {Asymptotic Expansions of the Lognormal Implied Volatility}, author = { Cyril Grunspan}, url = {http://papers.ssrn.com/sol3/papers.cfm?abstract-id=1965977}, year = {2011}, date = {2011-11-29}, abstract = {We invert the Black-Scholes formula. We consider the cases low strike, large strike, short maturity and large maturity. We give explicitly the first 5 terms of the expansions. A method to compute all the terms by induction is also given. At the money, we have a closed form formula for implied lognormal volatility in terms of a power series in call price.}, keywords = {}, pubstate = {published}, tppubtype = {unpublished} } We invert the Black-Scholes formula. We consider the cases low strike, large strike, short maturity and large maturity. We give explicitly the first 5 terms of the expansions. A method to compute all the terms by induction is also given. At the money, we have a closed form formula for implied lognormal volatility in terms of a power series in call price. |

Cyril Grunspan On Some Forms of Lie bialgebras and Quantum Groups Unpublished 2003. @unpublished{Grunspanbb, title = {On Some Forms of Lie bialgebras and Quantum Groups}, author = { Cyril Grunspan}, year = {2003}, date = {2003-01-01}, keywords = {}, pubstate = {published}, tppubtype = {unpublished} } |

Cyril Grunspan Quantum Homogeneous Spaces and Toda Theory on Lattices Unpublished 1999. @unpublished{Grunspanbb, title = {Quantum Homogeneous Spaces and Toda Theory on Lattices}, author = { Cyril Grunspan}, year = {1999}, date = {1999-01-01}, keywords = {}, pubstate = {published}, tppubtype = {unpublished} } |

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