Elisa Alòs

Stochastic integration with respect to the fractional Brownian motion

We develop a stochastic calculus for the fractional Brownian motion with Hurst parameter using the techniques of the Malliavin calculus. We establish estimates in L p , maximal inequalities and a continuity criterion for the stochastic integral. Finally, we derive an Itôs formula for integral processes.

On the Short-Time Behavior of the Implied Volatility for Jump-Diffusion Models With Stochastic Volatility

In this paper we use Malliavin calculus techniques to obtain an expression for the short-time behavior of the at-the-money implied volatility skew for a generalization of the Bates model, where the volatility does not need to be neither a difussion, nor a Markov process as the examples in section 7 show. This expression depends on the derivative of the volatility in the sense of Malliavin calculus.

An Extension of Ito's Formula for Anticipating Processes

In this paper we introduce a class of square integrable processes, denoted by LF, defined in the canonical probability space of the Brownian motion, which contains both the adapted processes and the processes in the Sobolev space L2,2. The processes in the class LF satisfy that for any time t, they are twice weakly differentiable in the sense of the stochastic calculus of variations in points (r, s) such that r ∨ s ≥ t. On the other hand, processes belonging to the class LF are Skorohod integrable, and the indefinite Skorohod integral has properties similar to those of the Ito integral. In particular we prove a change-of-variable formula that extends the classical Itô formula. Those results are generalization of similar properties proved by Nualart and Pardoux(7) for processes in L2,2.

Stochastic partial differential equations with Dirichlet white-noise boundary conditions

Abstract The paper is devoted to one-dimensional nonlinear stochastic partial differential equations of parabolic type with non homogeneous Dirichlet boundary conditions of white-noise type. We formulate a set of conditions that a random field must satisfy to solve the equation. We show that a unique solution exists and that we can write it in terms of the stochastic kernel related to the problem. This formulation allows us to study the basic properties of the solution, as the continuity and the boundary-layer behavior, by means of Malliavin calculus.

A decomposition formula for option prices in the Heston model and applications to option pricing approximation

By means of classical Itô calculus, we decompose option prices as the sum of the classical Black–Scholes formula, with volatility parameter equal to the root-mean-square future average volatility, plus a term due to correlation and a term due to the volatility of the volatility. This decomposition allows us to develop first- and second-order approximation formulas for option prices and implied volatilities in the Heston volatility framework, as well as to study their accuracy for short maturities. Numerical examples are given.

A Hull and White formula for a general stochastic volatility jump-diffusion model with applications to the study of the short-time behavior of the implied volatility

In this paper, generalizing results in Alòs, León and Vives (2007b), we see that the dependence of jumps in the volatility under a jump-diffusion stochastic volatility model, has no effect on the short-time behaviour of the at-the-money implied volatility skew, although the corresponding Hull and White formula depends on the jumps. Towards this end, we use Malliavin calculus techniques for Lévy processes based on Løkka (2004), Petrou (2006), and Solé, Utzet and Vives (2007).

Anticipating stochastic Volterra equations

In this paper we establish the existence and uniqueness of a solution for stochastic Volterra equations assuming that the coefficients F(t,s,x) and Gi(t,s,x) are Ft-measurable, for s[less-than-or-equals, slant]t, where {Ft} denotes the filtration generated by the driving Brownian motion. We impose some differentiability assumptions on the coefficients, in the sense of the Malliavin calculus, in the time interval [s,t]. Some properties of the solution are discussed.

STABILITY FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH DIRICHLET WHITE-NOISE BOUNDARY CONDITIONS

In this paper we study the long-time behaviour a stochastic parabolic equation perturbed through the boundary; the perturbation is represented by a nonhomogeneous Dirichlet boundary condition of white-noise type. The existence of the solution for an equation of this kind was the object of a previous paper of the same authors. The estimates proved therein allow to study the asymptotic properties of the solution; we show that there exists a unique invariant measure that is exponentially mean square stable.

Stochastic heat equation with white-noise drift

Abstract We study the existence and uniqueness of the solution for a one-dimensional anticipative stochastic evolution equation driven by a two-parameter Wiener process Wt,x and based on a stochastic semigroup p(s,t,y,x). The kernel p(s,t,y,x) is supposed to be measurable with respect to the increments of the Wiener process on [s,t]× R . The results are based on Lp-estimates for the Skorohod integral. As a application we deduce the existence of a weak solution for the stochastic partial differential equation ∂u ∂t = ∂ 2 u ∂x 2 + v (t,x) ∂u ∂x +F(t,x,u) ∂ 2 W ∂t∂x , where v (t,x) is a white-noise in time.

VALUATION OF BARRIER OPTIONS VIA A GENERAL SELF‐DUALITY

Classical put-call symmetry relates the price of puts and calls under a suitable dual market transform. One well‐known application is the semistatic hedging of path-dependent barrier options with European options. This, however, in its classical form requires the price process to observe rather stringent and unrealistic symmetry properties. In this paper, we develop a general self-duality theorem to develop valuation schemes for barrier options in stochastic volatility models with correlation.