The Quadratic Local Variance Gamma Model: an arbitrage-free interpolation of class C3\mathcal{C}^3 for option prices

This paper generalizes the local variance gamma model of Carr and Nadtochiy, to a piecewise quadratic local variance function. The formulation encompasses the piecewise linear Bachelier and piecewise linear Black local variance gamma models. The quadratic local variance function results in an arbitrage-free interpolation of class $\mathcal{C}^3$. The increased smoothness over the piecewise-constant and piecewise-linear representation allows to reduce the number of knots when interpolating raw market quotes, thus providing an interesting alternative to regularization while reducing the computational cost.Comment: arXiv admin note: text overlap with arXiv:2004.0865

Instabilities of Super-Time-Stepping Methods on the Heston Stochastic Volatility Model

This note explores in more details instabilities of explicit super-time-stepping schemes, such as the Runge-Kutta-Chebyshev or Runge-Kutta-Legendre schemes, noticed in the litterature, when applied to the Heston stochastic volatility model. The stability remarks are relevant beyond the scope of super-time-stepping schemes

Instabilities of explicit finite difference schemes with ghost points on the diffusion equation

Ghost, or fictitious points allow to capture boundary conditions that are not located on the finite difference grid discretization. We explore in this paper the impact of ghost points on the stability of the explicit Euler finite difference scheme in the context of the diffusion equation. In particular, we consider the case of a one-touch option under the Black-Scholes model. The observations and results are however valid for a much wider range of financial contracts and models

Stellar black holes: cosmic history and feedback at the dawn of the universe

Significant historic cosmic evolution for the formation rate of stellar black holes is inferred from current theoretical models of the evolution of massive stars, the multiple observations of compact stellar remnants in the near and distant universe, and the cosmic chemical evolution. The mean mass of stellar black holes, the fraction of black holes/neutron stars, and the fraction of black hole high mass X-ray binaries (BH-HMXBs)/solitary black holes increase with redshift. The energetic feedback from large populations of BH-HMXBs form in the first generations of star burst galaxies has been overlooked in most cosmological models of the reionization epoch of the universe. The powerful radiation, jets, and winds from BH-HMXBs heat the intergalactic medium over large volumes of space and keep it ionized until AGN take over. It is concluded that stellar black holes constrained the properties of the faintest galaxies at high redshifts. I present here the theoretical and observational grounds for the historic cosmic evolution of stellar black holes. Detailed calculations on their cosmic impact are presented elsewhere (Mirabel, Dijkstra, Laurent, Loeb, Pritchard, 2011).Comment: 9 pages, 1 color figure. Invited talk at the IAU Symp. 275, Jets at all scales. Held in Buenos Aires on 13-17 September 2010. To be published by Cambridge University Press. Eds. G. Romero, R. Sunyaev and T. Bellon