We 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.

### 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

#### Abstract

We 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.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11581/200447`
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