Lucia Caramellino

Pricing and hedging American options by Monte Carlo methods using a Malliavin calculus approach

Following the pioneering papers of Fournié, Lasry, Lebouchoux, Lions and Touzi, an important work concerning the applications of the Malliavin calculus in numerical methods for mathematical finance has come after. One is concerned with two problems: computation of a large number of conditional expectations on one hand and computation of Greeks (sensitivities) on the other hand. A significant test of the power of this approach is given by its application to pricing and hedging American options. The paper gives a global and simplified presentation of this topic including the reduction of variance techniques based on localization and control variables. A special interest is given to practical implementation, number of numerical tests are presented and their performances are carefully discussed.

Dependence and Aging Properties of Lifetimes with Schur-Constant Survival Functions

For n-dimensional survival functions, we study some probabilistic aspects of the Schur-constant property. The latter is of interest in that it extends the “lack-of-memory” property in a Bayesian context. Some general facts are studied in detail, and related results about interdependence, aging, and extendibility are presented.

Convergence and regularity of probability laws by using an interpolation method

Fournier and Printems [Bernoulli 16 (2010) 343–360] have recently established a methodology which allows to prove the absolute continuity of the law of the solution of some stochastic equations with Holder continuous coefficients. This is of course out of reach by using already classical probabilistic methods based on Malliavin calculus. By employing some Besov space techniques, Debussche and Romito [Probab. Theory Related Fields 158 (2014) 575–596] have substantially improved the result of Fournier and Printems. In our paper, we show that this kind of problem naturally fits in the framework of interpolation spaces: we prove an interpolation inequality (see Proposition 2.5) which allows to state (and even to slightly improve) the above absolute continuity result. Moreover, it turns out that the above interpolation inequality has applications in a completely different framework: we use it in order to estimate the error in total variance distance in some convergence theorems.

Monte Carlo methods for pricing and hedging American options in high dimension

We numerically compare some recent Monte Carlo algorithms devoted to the pricing and hedging American options in high dimension. In particular, the comparison concerns the quantization method of Barraquand-Martineau and an algorithm based on Malliavin calculus. The (pure) Malliavin calculus algorithm improves the precision of the computation of the delta but, merely for pricing purposes, is uncompetitive with respect to other Monte Carlo methods in terms of computing time. Here, we propose to suitably combine the Malliavin calculus approach with the Barraquand-Martineau algorithm, using a variance reduction technique based on control variables. Numerical tests for pricing and hedging American options in high dimension are given in order to compare the different methodologies.

Regularity of probability laws by using an interpolation method

One of the outstanding applications of Malliavin calculus is the criterion of regularity of the law of a functional on the Wiener space (presented in Section 2.3). The functional involved in such a criterion has to be regular in Malliavin sense, i.e., it has to belong to the domain of the differential operators in this calculus. As long as solutions of stochastic equations are concerned, this amounts to regularity properties of the coefficients of the equation.

Asymptotic development for the CLT in total variation distance

The aim of this paper is to study the asymptotic expansion in total variation in the central limit theorem when the law of the basic random variable is locally lower-bounded by the Lebesgue measure (or equivalently, has an absolutely continuous component): we develop the error in powers of n^{−1/2} and give an explicit formula for the approximating measure.

Strassen’s law of the iterated logarithm for diffusion processes for small time

We study the Strassens law of the iterated logarithm for diffusion processes for small values of the parameter. For the Brownian Motion this result can be obtained by time reversal, a technique which is not easy to reproduce for diffusion processes. A number of examples and applications are discussed.

Stochastic integration by parts and functional Itô calculus.

This volume contains lecture notes from the courses given by Vlad Bally and Rama Cont at the Barcelona Summer School on Stochastic Analysis (July 2012). The notes of the course by Vlad Bally, co-authored with Lucia Caramellino, develop integration by parts formulas in an abstract setting, extending Malliavins work on abstract Wiener spaces. The results are applied to prove absolute continuity and regularity results of the density for a broad class of random processes. Rama Conts notes provide an introduction to the Functional Ito Calculus, a non-anticipative functional calculus that extends the classical Ito calculus to path-dependent functionals of stochastic processes. This calculus leads to a new class of path-dependent partial differential equations, termed Functional Kolmogorov Equations, which arise in the study of martingales and forward-backward stochastic differential equations. This book will appeal to both young and senior researchers in probability and stochastic processes, as well as to practitioners in mathematical finance.

Large deviation estimates of the crossing probability for pinned Gaussian processes

The paper deals with the asymptotic behavior of the bridge of a Gaussian process conditioned to stay in n fixed points at n fixed past instants. In particular, functional large deviation results are stated for small time. Several examples are considered: integrated or not fractional Brownian motions and m-fold integrated Brownian motion. As an application, the asymptotic behavior of the exit probability is studied and used for the practical purpose of the numerical computation, via Monte Carlo methods, of the hitting probability up to a given time of the unpinned process.

Large and Moderate Deviations for Random Walks on Nilpotent Groups

We prove large and moderate deviation estimates for products of i.i.d. r.v.s taking values on simply connected nilpotent Lie groups as a consequence of large and moderate deviation results for stochastic processes which are solutions of O.D.E. with random coefficients.