1,721,011 research outputs found
Mathematical models of 'active' obstacles in acoustic scattering in Control and Boundary Analysis
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An explicitly solvable multi-scale stochastic volatility model: option pricing and calibration
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 onedimensional 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. © 2009 Wiley Periodicals, Inc
The Use of Ordinary Differential Equations in Quadratic Maximization with Integer Constraints
No full textWe consider the problem of maximizing a quadratic function on the set {-1,1}^n. This problem is related to some graph partitioning problems. We propose a path following method to compute an upper bound to the previous maximization problem. Numerical implementation of the proposed method and related numerical experience are presented
A method to compute the transition probability density associated to a multifactor Cox-Ingersoll-Ross model of the term structure of interest rates with no drift term
No full textWe consider an n-dimensional square root process and we obtain a formula involving series expansions for the associated transition probability density. The process mentioned previously can be used to model forward rates, future prices, forward prices and, as a consequence, can be used to price derivatives on these underlyings. The formula that we propose for the transition probability density has been obtained using appropriately a perturbative expansion in the correlation coefficients of the square root process, the Fourier transform and the method of characteristics to solve first-order hyperbolic partial differential equations. The computational effort needed to evaluate this formula is polynomial with respect to the dimension n of the space spanned by the square root process when the order where all the series involved in the transition probability density formula are truncated is fixed. This strategy gives an accuracy that some numerical tests show approximately constant for a wide range of values of n. Some examples of prices of financial derivatives whose evaluation involves integrals in two, twenty and one hundred dimensions (i.e. n = 2, 20, 100), that is derivatives on two, twenty and one hundred assets, where accurate results can be obtained are shown. An experiment shows that the formula derived here for the transition probability density is well suited for parallel computing. This feature makes the formula computationally very attractive to price derivatives of the LIBOR market such as caplets or swaptions since the use of parallel computing and the formula makes it possible to evaluate derivatives on several tens of underlyings in negligible times. The website http://www.econ.univpm.it/recchioni/finance/w1 contains an interactive tool that helps with the understanding of this paper and a portable software library that makes it possible to the user to exploit the formula derived in this paper to evaluate the transition probability densities of its own models and the prices of the associated financial derivatives. (c) 2007 Elsevier Ltd. All rights reserved
Some mathematical models of furtivity and masking problems in time dependent acoustic obstacle scattering
No full textA quadratically convergent method for linear programming
No full textAbstractA new method to solve linear programming problems is introduced. This method follows a path defined by a system of o.d.e., and for nondegenerate problem is quadratically convergent
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