A numerical method for inversion of Laplace transform F(s) of probability densities, given for re... more A numerical method for inversion of Laplace transform F(s) of probability densities, given for real s⩾0 only, is proposed. Fractional calculus allows us to have expected values from F(s) or F′(s). Then the maximum entropy technique provides the analytical form of the approximant. Fractional moments are mainly investigated. Entropy and cross-entropy convergence are proved. Some numerical tests are illustrated.
Bulletin of the Seismological Society of America, Jun 1, 2003
In seismic hazard analysis there are quantities that are dominated by the distribution of strong ... more In seismic hazard analysis there are quantities that are dominated by the distribution of strong earthquakes. A typical example is the peak ground acceleration with a 500-year return period at a specific site. This quantity, a (500), is particularly important from the engineering point of view. Due to the scanty number of strong earthquakes in seismic catalogs, the available data are generally not sufficient for a statistical validation of a magnitude distribution model. Different models can explain available data with the same level of statistical likelihood, even if they lead, ceteris paribus , to values of a (500) at the site that are significantly different. Which one is more credible and should be favored for seismic hazard analysis at the considered site? In most cases the traditional fitting tests do not give a meaningful answer. In this article we introduce a particular definition of credibility of a magnitude model, based on the distribution of expected errors in the evaluation of a (500) at a specific site. We show that this concept of credibility is a useful tool for selecting the more appropriate of two competing models.
Sometimes it is not possible to obtain a single parametric density with the desired tail behavior... more Sometimes it is not possible to obtain a single parametric density with the desired tail behavior to fit a given data set. Splicing two different parametric densities is a useful process in such cases. Since the two parts depend on local data, a question arises over how best to assemble the two parts so that the properties of the whole data set are taken into account. We propose an application of the method of maximum entropy in the mean to splice the two parts together in such a way that the resulting global density has the first two moments of the full data set.
The Hausdorff, Hamburger and Stieltjes moment problem and the sequence of maximum entropy (MaxEnt... more The Hausdorff, Hamburger and Stieltjes moment problem and the sequence of maximum entropy (MaxEnt) approx- imates with given moments are considered. Thanks to MaxEnt formalism it is proved, through a unified procedure, MaxEnt approximate converges in entropy to the unknown distribution when the Hausdorff, Hamburger or Stieltjes moment problem are determinate and the underlying distribution has finite or (minus) infinite entropy.
The paper is devoted to pricing options characterized by discontinuities in the terminal conditio... more The paper is devoted to pricing options characterized by discontinuities in the terminal condition. Finite difference schemes are examined to highlight how discontinuities can generate numerical drawbacks such as spurious oscillations. We propose a finite difference scheme that is free of spurious oscillations and satisfies both the positivity requirement and maximum principle, as it is demanded for the financial and diffusive solution of the original Black-Scholes partial differential equation. We explore examples of discrete double barrier knock-out call options and the results are in very good agreement with those in the literature.
In this paper we investigate the use of finite difference and finite element schemes when applied... more In this paper we investigate the use of finite difference and finite element schemes when applied to the valuation of exotic options characterized by discontinuities in the payoff function. In particular, we will conduct a numerical analysis of several common schemes in order to give a better understanding of the numerical problems associated with the valuation of non-standard options.
In this paper we explore the numerical diffusion introduced by two nonstandard finite difference ... more In this paper we explore the numerical diffusion introduced by two nonstandard finite difference schemes applied to the Black-Scholes partial differential equation for pricing discontinuous payoff and low volatility options. Discontinuities in the initial conditions require applying nonstandard non-oscillating finite difference schemes such as the exponentially fitted finite difference schemes suggested by D. Duffy and the Crank-Nicolson variant scheme of Milev-Tagliani. We present a short survey of these two schemes, investigate the origin of the respective artificial numerical diffusion and demonstrate how it could be diminished.
For a given set of moments whose predetermined values represent the available information, we con... more For a given set of moments whose predetermined values represent the available information, we consider the case where the Maximum Entropy (MaxEnt) solutions for Stieltjes and Hamburger reduced moment problems do not exist. Genuinely relying upon MaxEnt rationale we find the distribution with largest entropy and we prove that this distribution gives the best approximation of the true but unknown underlying distribution. Despite the nice properties just listed, the suggested approximation suffers from some numerical drawbacks and we will discuss this aspect in detail in the paper.
A numerical method for inversion of Laplace transform F(s) of probability densities, given for re... more A numerical method for inversion of Laplace transform F(s) of probability densities, given for real s⩾0 only, is proposed. Fractional calculus allows us to have expected values from F(s) or F′(s). Then the maximum entropy technique provides the analytical form of the approximant. Fractional moments are mainly investigated. Entropy and cross-entropy convergence are proved. Some numerical tests are illustrated.
Bulletin of the Seismological Society of America, Jun 1, 2003
In seismic hazard analysis there are quantities that are dominated by the distribution of strong ... more In seismic hazard analysis there are quantities that are dominated by the distribution of strong earthquakes. A typical example is the peak ground acceleration with a 500-year return period at a specific site. This quantity, a (500), is particularly important from the engineering point of view. Due to the scanty number of strong earthquakes in seismic catalogs, the available data are generally not sufficient for a statistical validation of a magnitude distribution model. Different models can explain available data with the same level of statistical likelihood, even if they lead, ceteris paribus , to values of a (500) at the site that are significantly different. Which one is more credible and should be favored for seismic hazard analysis at the considered site? In most cases the traditional fitting tests do not give a meaningful answer. In this article we introduce a particular definition of credibility of a magnitude model, based on the distribution of expected errors in the evaluation of a (500) at a specific site. We show that this concept of credibility is a useful tool for selecting the more appropriate of two competing models.
Sometimes it is not possible to obtain a single parametric density with the desired tail behavior... more Sometimes it is not possible to obtain a single parametric density with the desired tail behavior to fit a given data set. Splicing two different parametric densities is a useful process in such cases. Since the two parts depend on local data, a question arises over how best to assemble the two parts so that the properties of the whole data set are taken into account. We propose an application of the method of maximum entropy in the mean to splice the two parts together in such a way that the resulting global density has the first two moments of the full data set.
The Hausdorff, Hamburger and Stieltjes moment problem and the sequence of maximum entropy (MaxEnt... more The Hausdorff, Hamburger and Stieltjes moment problem and the sequence of maximum entropy (MaxEnt) approx- imates with given moments are considered. Thanks to MaxEnt formalism it is proved, through a unified procedure, MaxEnt approximate converges in entropy to the unknown distribution when the Hausdorff, Hamburger or Stieltjes moment problem are determinate and the underlying distribution has finite or (minus) infinite entropy.
The paper is devoted to pricing options characterized by discontinuities in the terminal conditio... more The paper is devoted to pricing options characterized by discontinuities in the terminal condition. Finite difference schemes are examined to highlight how discontinuities can generate numerical drawbacks such as spurious oscillations. We propose a finite difference scheme that is free of spurious oscillations and satisfies both the positivity requirement and maximum principle, as it is demanded for the financial and diffusive solution of the original Black-Scholes partial differential equation. We explore examples of discrete double barrier knock-out call options and the results are in very good agreement with those in the literature.
In this paper we investigate the use of finite difference and finite element schemes when applied... more In this paper we investigate the use of finite difference and finite element schemes when applied to the valuation of exotic options characterized by discontinuities in the payoff function. In particular, we will conduct a numerical analysis of several common schemes in order to give a better understanding of the numerical problems associated with the valuation of non-standard options.
In this paper we explore the numerical diffusion introduced by two nonstandard finite difference ... more In this paper we explore the numerical diffusion introduced by two nonstandard finite difference schemes applied to the Black-Scholes partial differential equation for pricing discontinuous payoff and low volatility options. Discontinuities in the initial conditions require applying nonstandard non-oscillating finite difference schemes such as the exponentially fitted finite difference schemes suggested by D. Duffy and the Crank-Nicolson variant scheme of Milev-Tagliani. We present a short survey of these two schemes, investigate the origin of the respective artificial numerical diffusion and demonstrate how it could be diminished.
For a given set of moments whose predetermined values represent the available information, we con... more For a given set of moments whose predetermined values represent the available information, we consider the case where the Maximum Entropy (MaxEnt) solutions for Stieltjes and Hamburger reduced moment problems do not exist. Genuinely relying upon MaxEnt rationale we find the distribution with largest entropy and we prove that this distribution gives the best approximation of the true but unknown underlying distribution. Despite the nice properties just listed, the suggested approximation suffers from some numerical drawbacks and we will discuss this aspect in detail in the paper.
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Papers by Aldo Tagliani