Semi- and nonparametric ARCH processes.
Semi-Markov reliability models with recurrence times and credit rating applications.
Seven Proofs for the Subadditivity of Expected Shortfall
Subadditivity is the key property which distinguishes the popular risk measures Value-at-Risk and Expected Shortfall (ES). In this paper we offer seven proofs of the subadditivity of ES, some found in the literature and some not. One of the main objectives of this paper is to provide a general guideline for instructors to teach the subadditivity of ES in a course. We discuss the merits and suggest appropriate contexts for each proof.With different proofs, different important properties of ES are...
Shape-preserving properties and asymptotic behaviour of the semigroup generated by the Black-Scholes operator
The paper is devoted to a careful analysis of the shape-preserving properties of the strongly continuous semigroup generated by a particular second-order differential operator, with particular emphasis on the preservation of higher order convexity and Lipschitz classes. In addition, the asymptotic behaviour of the semigroup is investigated as well. The operator considered is of interest, since it is a unidimensional Black-Scholes operator so that our results provide qualitative information on the...
Shifts of the term structure of interest rates against which a given portfolio is preimmunized
Small perturbations with large effects on value-at-risk
We show that in the delta-normal model there exist perturbations of the Gaussian multivariate distribution of the returns of a portfolio such that the initial marginal distributions of the returns are statistically undistinguishable from the perturbed ones and such that the perturbed V@R is close to the worst possible V@R which, under some reasonable assumptions, is the sum of the V@Rs of each of the portfolio assets.
Solution of option pricing equations using orthogonal polynomial expansion
We study both analytic and numerical solutions of option pricing equations using systems of orthogonal polynomials. Using a Galerkin-based method, we solve the parabolic partial differential equation for the Black-Scholes model using Hermite polynomials and for the Heston model using Hermite and Laguerre polynomials. We compare the obtained solutions to existing semi-closed pricing formulas. Special attention is paid to the solution of the Heston model at the boundary with vanishing volatility.
Solutions of stochastic differential equations obeying the law of the iterated logarithm, with applications to financial markets.
Some applications of time series models to financial data
Some short elements on hedging credit derivatives
In practice, it is well known that hedging a derivative instrument can never be perfect. In the case of credit derivatives (e.g. synthetic CDO tranche products), a trader will have to face some specific difficulties. The first one is the inconsistence between most of the existing pricing models, where the risk is the occurrence of defaults, and the real hedging strategy, where the trader will protect his portfolio against small CDS spread movements. The second one, which is the main subject of...
Spectral approximation of infinite-dimensional Black-Scholes equations with memory.
Stochastic analysis of Bernoulli processes.
Stock market as a dynamic game with continuum of players
Subprime risk and insurance with regret.
Sugli sviluppi della matematica applicata in un settore interdisciplinare: la finanza matematica
La moderna finanza matematica è un settore interdisciplinare tra economia e matematica che, allo stato attuale, è a forte contenuto matematico, soprattutto probabilistico. Iniziamo questo articolo accennando alle origini di questa disciplina, che non sono molto lontane nel tempo e che erano di natura più economica/econometrica. Successivamente arriveremo a descrivere gli sviluppi più recenti e più tipicamente matematici.
Superconvergence estimates of finite element methods for American options
In this paper we are concerned with finite element approximations to the evaluation of American options. First, following W. Allegretto etc., SIAM J. Numer. Anal. 39 (2001), 834–857, we introduce a novel practical approach to the discussed problem, which involves the exact reformulation of the original problem and the implementation of the numerical solution over a very small region so that this algorithm is very rapid and highly accurate. Secondly by means of a superapproximation and interpolation...
Sur les itérations élémentaires, concernant la formule d'Euler
Symmetry and new solutions of the Black-Scholes equation.