Dan Stefanica

A scalable FETI-DP algorithm with non-penetration mortar conditions on contact interface

By combining FETI algorithms of dual-primal type with recent results for bound constrained quadratic programming problems, we develop an optimal algorithm for the numerical solution of coercive variational inequalities. The model problem is discretized using non-penetration conditions of mortar type across the potential contact interface, and a FETI-DP algorithm is formulated. The resulting quadratic programming problem with bound constraints is solved by a scalable algorithm with a known rate of convergence given in terms of the spectral condition number of the quadratic problem. Numerical experiments for non-matching meshes across the contact interface confirm the theoretical scalability of the algorithm.

FETI and FETI-DP Methods for Spectral and Mortar Spectral Elements: A Performance Comparison

We investigate the numerical performance of several FETI and FETI-DP algorithms, for both spectral and mortar spectral elements on geometrically conforming discretizations of the computational domain.

Tighter Bounds for Implied Volatility

We establish bounds on Black–Scholes implied volatility that improve on the uniform bounds previously derived by Tehranchi. Our upper bound is uniform, while the lower bound holds for most options likely to be encountered in practical applications. We further demonstrate the practical effectiveness of our new bounds by showing how the efficiency of the bisection algorithm is improved for a snapshot of SPX option quotes.

Pólya-Based Approximation for the ATM-Forward Implied Volatility

We introduce a closed form approximation for the implied volatility of ATM-forward options. The relative error of this approximation is uniformly bounded for all option maturities and implied volatilities. The approximation is extremely precise, having relative error less than 10−6 for all options with integrated volatility less than 1.9, such as options with maturity less than three years and implied volatility less than 100%. Moreover, the approximate implied volatilities fall within the implied volatility bid-ask spread for all the liquid options, such as options with volatility less than 200% and maturity less than nine years.

An Overview of Scalable FETI—DP Algorithms for Variational Inequalities

We review our recent results concerning optimal algorithms for numerical solution of both coercive and semi-coercive variational inequalities by combining dual-primal FETI algorithms with recent results for bound and equality constrained quadratic programming problems. The convergence bounds that guarantee the scalability of the algorithms are presented. These results are confirmed by numerical experiments.

Choosing Nonmortars: Does it Influence the Performance of FETI-DP Algorithms?

We investigate whether different choices of nonmortar sides for the geometrically conforming partitions inherent to FETI-DP influence the convergence of the algorithms for four different preconditioners. We conclude experimentally that they do not, although better condition number estimates exist for a Neumann-Dirichlet choice of nonmortars.

An Explicit Implied Volatility Formula

We show that an explicit approximate implied volatility formula can be obtained from a Black–Scholes formula approximation that is 2% accurate. The relative error of the approximate implied volatility is uniformly bounded for options with any moneyness and with arbitrary large or small option maturities and volatilities, including for long dated options and options on highly volatile underlying assets. For options within a large trading range, such as options with maturity less than five years and implied volatility less than 150%, the error of the approximate implied volatility relative to the Black–Scholes implied volatility is less than 10% points.

A Sharp Polya-Based Approximation to the Normal CDF

We introduce a closed form approximation to the cumulative distribution function of the standard normal variable involving only five explicit constants with an approximation error of 5.79 x 10^{-6} across the entire range of real numbers. With its simple form and applicability for all real numbers, our approximation surpasses either in computational efficiency or in accuracy, and most often in both, other approximation formulas based on numerical algorithms or ad-hoc approximations. An extensive overview and classification of the existing approximations from the literature is included.

A Sharp Approximation for ATM-Forward Option Prices and Implied Volatilites

In this paper, we provide an approximation formula for at-the-money forward options based on a Polya approximation of the cumulative density function of the standard normal distribution, and prove that the relative error of this approximation is uniformly bounded for options with arbitrarily large (or small) maturities and implied volatilities. This approximation is viable in practice: for options with implied volatility less than 95% and maturity less than three years, which includes the large majority of traded options, the values given by the approximation formula fall within the tightest typical implied vol bid–ask spreads. The relative errors of the corresponding approximate option values are also uniformly bounded for all maturities and implied volatilities. The error bounds established here are the first results in the literature holding for all integrated volatilities, and are vastly superior to those of two other approximation formulas analyzed in this paper, including the Brenner–Subrahmanyam formula.

A Balancing Algorithm for Mortar Methods

The balancing methods are hybrid nonoverlapping Schwarz domain decomposition methods from the Neumann-Neumann family. They are efficient and easy to implement. We present a new balancing algorithm for mortar finite element methods. We prove a condition number estimate which depends polylogarithmically on the number of nodes on each subregion edge and does not depend on the number of subregions of the partition of the computational domain, just as in the conforming finite element case.