Paris Dauphine University

About: Paris Dauphine University is a education organization based out in Paris, France. It is known for research contribution in the topics: Context (language use) & Population. The organization has 1766 authors who have published 6909 publications receiving 162747 citations. The organization is also known as: Paris Dauphine & Dauphine.

Computing the implied volatility in stochastic volatility models

Abstract: The Black-Scholes model [6, 23] has gained wide recognition on financial markets. One of its shortcomings, however, is that it is inconsistent with most observed option prices. Although the model can still be used very efficiently, it has been proposed to relax its assumptions, and, for instance, to consider that the volatility of the underlying asset S is no longer a constant but rather a stochastic process. There are two well-known approaches to achieve this goal. In the first class of models, the volatility is assumed to depend on the variables t (time) and S, giving rise to the so-called local volatility models. The second one, conceptually more ambitious, considers that the volatility has a stochastic component of its own. In the latter, the number of factors is increased by the amount of stochastic factors entering the volatility modeling. Both models are of practical interest. In these contexts, it is relevant to express the resulting prices in terms of implied volatilities. Given a price, the Black-Scholes implied volatility is determined, for each given product (that is for each given strike and expiry date defining, say, the call option) as the unique value of the volatility parameter for which the BlackScholes pricing formula agrees with that given price. Actually, it is common practice on trading floors to quote and to observe prices in this way. A great advantage of having prices expressed in such dimensionless units is to provide easy comparison between products with different characteristics. In principle, the implied volatility can be inferred from computed options prices by inverting the Black-Scholes formula. It is more convenient, however, to directly analyze the implied volatility. Indeed, this approach allows us to shed light on qualitative properties that would otherwise be more difficult to establish. In particular, we derive here several asymptotic formulae that are of practical interest, for example, in the calibration problem. The latter—an inverse problem that consists

Contingent Claims and Market Completeness in a Stochastic Volatility Model

Abstract: In an incomplete market framework, contingent claims are of particular interest since they improve the market efficiency. This paper addresses the problem of market completeness when trading in contingent claims is allowed. We extend recent results by Bajeux and Rochet (1996) in a stochastic volatility model to the case where the asset price and its volatility variations are correlated. We also relate the ability of a given contingent claim to complete the market to the convexity of its price function in the current asset price. This allows us to state our results for general contingent claims by examining the convexity of their “admissible arbitrage prices.”

Existence of Weak Solutions for the Motion of Rigid Bodies in a Viscous Fluid

Abstract: . We study the evolution of a finite number of rigid bodies within a viscous incompressible fluid in a bounded domain of \(\R^d$ $(d=2$ or $3)\) with Dirichlet boundary conditions. By introducing an appropriate weak formulation for the complete problem, we prove existence of solutions for initial velocities in \(H^1_0(\Omega)\). In the absence of collisions, solutions exist for all time in dimension 2, whereas in dimension 3 the lifespan of solutions is infinite only for small enough data.