89 research outputs found
A perturbative approach to acoustic scattering from a vibrating bounded obstacle
No full textIn this paper we study a mathematical model to describe a three dimensional acoustic scattering problem associated to a "vibrating" obstacle that is a bounded simply connected domain contained in the three dimensional real Euclidean space whose shape changes in time. In particular we propose a numerical method based on a perturbation series and the operator expansion method to solve the mathematical model considered. This method makes possible to obtain highly parallelizable algorithms able to compute the solution of the problem considered order by order in perturbation theory, and able to obtain the required solution of the scattering problem summing up the perturbation series. Really impressive speed up factors are observed and reported when the algorithm is executed on the Chiba Cluster, a parallel machine of the Argonne National Laboratory, USA. We validate the mathematical model and the numerical method proposed solving some test problems. The quantitative character of the numerical results obtained is established. The results obtained on the test problems are discussed both from the numerical and the physical point of view. In particular we show that the Doppler spectrum associated to the far field patterns of the scattered acoustic fields depends mainly from the incoming wave and from the excited vibrational modes (Figs. 9–13). The website: http://www.econ.unian.it/recchioni/w7 shows some Applets relative to the numerical examples
An explicitly solvable multi-scale stochastic volatility model: Option pricing and calibration problems
No full textWe introduce an explicitly solvable multiscale stochastic volatility model that generalizes the Heston model. The model describes the dynamics of an asset price and of its two stochastic variances using a system of three Ito stochastic differential equations. The two stochastic variances vary on two distinct time scales and can be regarded as auxiliary variables introduced to model the dynamics of the asset price. Under some assumptions, the transition probability density function of the stochastic process solution of the model is represented as a one-dimensional integral of an explicitly known integrand. In this sense the model is explicitly solvable. We consider the risk-neutral measure associated with the proposed multiscale stochastic volatility model and derive formulae to price European vanilla options (call and put) in the multiscale stochastic volatility model considered. We use the thus-obtained option price formulae to study the calibration problem, that is to study the values of the model parameters, the correlation coefficients of the Wiener processes defining the model, and the initial stochastic variances implied by the “observed” option prices using both synthetic and real data. In the analysis of real data, we use the S&P 500 index and to the prices of the corresponding options in the year 2005. The web site http://www.econ.univpm.it/recchioni/finance/w7 contains some auxiliary material including some animations that helps the understanding of this article. A more general reference to the work of the authors and their coauthors in mathematical finance is the web site http://www.econ.univpm.it/recchioni/finance
The Analysis of Real Data Using a Stochastic Dynamical System Able to Model Spiky Prices
No full textIn this paper we use filtering and maximum likelihood methods to solve a calibration problem for a stochastic dynamical system used to model spiky asset prices. The data used in the calibration problem are the observations at discrete times of the asset price. The model considered has been introduced by V.A. Kholodnyi in [1,2] and describes spiky as-set prices as the product of two independent stochastic processes: the spike process and the process that represents the asset prices in absence of spikes. A Markov chain is used to regulate the transitions between presence and absence of spikes. As suggested in [3] in a different context the calibration problem for this model is translated in a maximum like-lihood problem with the likelihood function defined through the solution of a filtering problem. The estimated values of the model parameters are the coordinates of a constrained maximizer of the likelihood function. Furthermore, given the calibrated model, we develop a sort of tracking procedure able to forecast forward asset prices. Numerical examples using synthetic and real data of the solution of the calibration problem and of the performance of the tracking procedure are presented. The real data studied are electric power price data taken from the UK electricity market in the years 2004-2009. After calibrating the model using the spot prices, the forward prices forecasted with the tracking procedure and the observed forward prices are compared. This comparison can be seen as a way to validate the model, the formu-lation and the solution of the calibration problem and the forecasting procedure. The result of the comparison is satis-factory. In the website: http://www.econ.univpm.it/recchioni/finance/w10 some auxiliary material including animations that helps the understanding of this paper is shown. A more general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/ recchioni/finance
Maximum likelihood estimation of the parameters of a system of stochastic differential equations that models the returns of the index of some classes of hedge funds
No full textIn this paper we study an inverse problem for a parabolic partial differential equation. The parabolic partial differential equation considered is the Fokker Planck equation associated to a system of stochastic differential equations and the inverse problem studied consists in finding from suitable data the values of the parameters that appear in the coefficients of this Fokker Planck equation. The data used in the reconstruction of the parameters are observations made at discrete times of the stochastic process solution of the system of stochastic differential equations. That is, the data of the inverse problem are a sample taken at discrete times of some of the components of the random vector solution of the stochastic differential equations and not, as usual, observations made on the solution of the parabolic equation. The choice of the system of stochastic differential equations and of the data used in the inverse problem are motivated by applications in mathematical finance. The stochastic differential equations presented can be used to model the dynamics of the log-returns of the index of some classes of hedge funds, such as, for example, the so called long short equity hedge funds and of some auxiliary variables. The solution of the inverse problem proposed is obtained through the solution of a filtering and of an estimation problem. The solution of these last two problems is based on the knowledge of the joint probability density function of the state variables of the model conditioned to the observations made and to the initial condition. This joint probability density function is solution of an initial value problem for the Kushner equation that in the circumstances considered here can be written as a sequence of initial value problems for the Fokker Planck equation associated to the system of stochastic differential equations with appropriate initial conditions. An integral representation formula for this probability density function is derived and used to develop a numerical procedure to solve the estimation problem using the maximum likelihood method. The Kushner equation provides the relation between the data and the Fokker Planck equation used to solve the inverse problem considered. The computational method proposed has been tested on synthetic data and the results obtained are presented. Some auxiliary material useful to understand this paper including some animations and some numerical experiments can be found in the website http://www.econ.univpm.it/recchioni/finance/w5. A more general reference to the work in mathematical finance of the authors and of their coauthors is the website http://www.econ.univpm.it/recchioni/finance
The SABR Model: Explicit Formulae of the Moments of the Forward Prices/Rates Variable and Series Expansions of the Transition Probability Density and of the Option Prices
No full textThe SABR stochastic volatility model with β-volatility βє(0,1) and an absorbing barrier in zero imposed to the forward prices/rates stochastic process is studied. The presence of (possibly) nonzero correlation between the stochastic differentials that appear on the right hand side of the model equations is considered. A series expansion of the transition probability density function of the model in powers of the correlation coefficient of these stochastic differentials is presented. Explicit formulae for the first three terms of this expansion are derived. These formulae are integrals of known integrands. The zero-th order term of the expansion is a new integral formula containing only elementary functions of the transition probability density function of the SABR model when the correlation coefficient is zero. The expansion is deduced from the final value problem for the backward Kolmogorov equation satisfied by the transition probability density function. Each term of the expansion is defined as the solution of a final value problem for a partial differential equation. The integral formulae that give the solutions of these final value problems are based on the Hankel and on the Kontorovich-Lebedev transforms. From the series expansion of the probability density function we deduce the corresponding expansions of the European call and put option prices. Moreover we deduce closed form formulae for the moments of the forward prices/rates variable. The moment formulae obtained do not involve integrals or series expansions and are expressed using only elementary functions. The option pricing formulae are used to study synthetic and real data. In particular we study a time series (of real data) of futures prices of the EUR/USD currency's exchange rate and of the corresponding option prices. The website:
http://www.econ.univpm.it/recchioni/finance/w18 contains material including animations, an interactive application and an app that helps the understanding of the paper. A more general reference to the work of the authors and of their coauthors in mathematical finance is the website:
http://www.econ.univpm.it/recchioni/finance
A Parallel Code for Time-Dependent acoustic Scattering Involving Passive or Smart Obstacles
No full textA highly parallelizable numerical method to solve three-dimensional time-dependent acoustic obstacle scattering problems involving passive or smart, furtive, realistic obstacles is presented. "Realistic" obstacles have complex geometries, "passive" obstacles do not react by taking an action to pursue a goal when hit by an incoming wave, and "smart furtive" obstacles, when hit by an incoming wave, pursue the goal of being undetectable by circulating a suitable pressure current on their boundaries. Incoming wave packets containing time-harmonic waves of small wavelengths when compared with the characteristic dimension of the obstacles are considered. The features of the computational method proposed to solve these scattering problems that can be exploited in a parallel and/or distributed computing environment are presented. Numerical experiments involving a simplified version of the NASA space shuttle are discussed. The websites: http://www.econ.univpm.it/recchioni/scattering/w12, http://www.econ.univpm.it/recchioni/scattering/w14 contain animations and virtual reality applications showing some numerical experiments relative to the problems studied. A more general reference to the work of some of the authors and of their coworkers in acoustic and electromagnetic scattering is the website: http://www.econ.univpm.it/recchioni/scattering
A perturbative formula to price barrier options with time dependent parameters in the Black and Scholes world
No full textIn this paper, using a perturbative method, a series expansion of the price of a (put up-and-out) barrier option with time-dependent parameters in the Black and Scholes world is obtained. The first three terms of this series are written explicitly as formulae involving some elementary and nonelementary transcendental functions. The formula obtained has been tested on some examples taken from the financial literature and a sufficient condition for the convergence of the perturbation series expansion is given. The numerical experience shows that, in the cases of practical interest considered, the use of the first two or three terms of the series expansion mentioned above guarantees three or four correct significant digits in the prices computed. Similar formulae can be obtained for other types of barrier options such as step options, step barrier options and so on. The website http://www.econ.univpm.it/recchioni/finance/w3 contains some auxiliary material that helps in understanding this paper and the computer programs needed to evaluate the formula obtained
Determining a stable relationship between hedge fund index HFRI-Equity and S&P 500 behaviour, using filtering and maximum likelihood
No full textIn this article we test the ability of the stochastic differential model proposed by Fatone et al. [Maximum likelihood estimation of the parameters of a system of stochastic differential equations that models the returns of the index of some classes of hedge funds, J. Inv. Ill-Posed Probl. 15 (2007), pp. 329-362] of forecasting the returns of a long-short equity hedge fund index and of a market index, that is, of the Hedge Fund Research performance Index (HFRI)-Equity index and of the S&P 500 (Standard & Poor 500 New York Stock Exchange) index, respectively. The model is based on the assumptions that the value of the variation of the log-return of the hedge fund index (HFRI-Equity) is proportional up to an additive stochastic error to the value of the variation of the log-return of a market index (S&P 500) and that the log-return of the market index can be satisfactorily modelled using the Heston stochastic volatility model. The model consists of a system of three stochastic differential equations, two of them are the Heston stochastic volatility model and the third one is the equation that models the behaviour of the hedge fund index and its relation with the market index. The model is calibrated on observed data using a method based on filtering and maximum likelihood proposed by Mariani et al. [Maximum likelihood estimation of the Heston stochastic volatility model using asset and option prices: An application of nonlinear filtering theory, Opt. Lett., 2 (2008), pp. 177-222] and further developed in Fatone et al. [Maximum likelihood estimation of the parameters of a system of stochastic differential equations that models the returns of the index of some classes of hedge funds, J. Inv. Ill-Posed Probl. 15 (2007), pp. 329-362; The calibration of the Heston stochastic volatility model using filtering and maximum likelihood methodsin Proceedings of Dynamic Systems and Applications, Vol. 5, G.S. Ladde, N.G. Medhin, C. Peng, and M. Sambandham, eds., Dynamic Publishers, Atlanta, USA, 2008, pp. 170-181]. That is, an inverse problem for the stochastic dynamical system representing the model is solved using the calibration procedure. The data analysed is from January 1990 to June 2007, and are monthly data. For each observation time, they consist of the value at the observation time of the log-returns of the HFRI-Equity and of the S&P 500 indices. The calibration procedure uses appropriate subsets of data, that is the data observed in a 6 months time period. The 6 months data time period used in the calibration is rolled through the time series generating a sequence of calibration problems. The values of the HFRI-Equity and S&P 500 indices forecasted using the calibrated models are compared to the values of the indices observed. The result of the comparison is very satisfactory. The website http://www.econ.univpm.it/recchioni/finance/w8 contains some auxiliary material including some animations that helps the understanding of this article. A more general reference to the work of some of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance
A method to solve an acoustic inverse scattering problem involving smart obstacles
No full textA time-harmonic acoustic inverse scattering problem involving smart obstacles is formulated and a method to solve it is proposed. A smart obstacle is an obstacle that, when hit by an incoming acoustic wave, tries to pursue a given goal circulating a suitable pressure current on its boundary. A pressure current is a quantity whose physical dimension is pressure divided by time. The goals pursued by the smart obstacles that we have considered are the following ones: to be undetectable or to appear with a shape and/or acoustic boundary impedance different from its actual ones eventually in a location in space different from the actual location. The following time-harmonic inverse scattering problem is considered: from the knowledge of several far fields generated by the smart obstacle when hit by known time-harmonic waves, the knowledge of the goal pursued by the smart obstacle and of its acoustic boundary impedance reconstruct the boundary of the obstacle. A method to solve this inverse problem that generalizes the so-called Herglotz function method is proposed. Some numerical experiments that validate the method proposed are presented. The website http://www.econ.univpm.it/recchioni/w13 contains some auxiliary material that helps the understanding of the current paper
High performance algoritms based on a new wavelet expansion for time dependent acoustic obstacle scattering
No full textThis paper presents a highly parallelizable numerical method to solve time dependent acoustic obstacle scattering problems. The method proposed is a generalization of the “operator expansion method” developed by Recchioni and Zirilli [SIAM J. Sci. Comput., 25 (2003), 1158-1186]. The numerical method proposed reduces, via a perturbative approach, the solution of the scattering problem to the solution of a sequence of systems of first kind integral equations. The numerical solution of these systems of integral equations is challenging when scattering problems involving realistic obstacles and small wavelengths are solved. A computational method has been developed to solve these challenging problems with affordable computing resources. To this aim a new way of using the wavelet transform and new bases of wavelets are introduced, and a version of the operator expansion method is developed that constructs directly element by element in a fully parallelizable way. Several numerical experiments involving realistic obstacles and “small” wavelengths are proposed and high dimensional vector spaces are used in the numerical experiments. To evaluate the performance of the proposed algorithm on parallel computing facilities, appropriate speed up factors are introduced and evaluate
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