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This article deals with the computation of the probability, for a GBM (geometric Brownian motion) process, to hit sequences of one-sided stochastic boundaries defined as GBM processes, over a closed time interval. Explicit formulae are obtained, allowing the analytical valuation of all the main kinds of barrier options in a much more general setting than the usual one assuming constant or time-dependent, deterministic barriers. The numerical implementation of all stated formulae is shown to be easy, fast and accurate. The practical applications are potentially substantial, since barrier options play a major role in quantitative finance, not only as intensively traded contracts on their own, but also as the building blocks of a large variety of structured products. Barrier options are also an important tool in financial modelling, used to measure default risk in the so-called "structural" models. [ABSTRACT FROM AUTHOR]

In this article, two general results are provided about the multidimensional Black–Scholes partial differential equation: its fundamental solution is derived, and it is shown how to turn it into the standard heat equation in whatever dimension. A fundamental connection is established between the multivariate normal distribution and the linear second order partial differential operator of parabolic type. These results allow to compute new closed form formulae for the valuation of multiasset options, with possible boundary crossing conditions, thus partially alleviating the « curse of dimensionality », at least in moderate dimension. [ABSTRACT FROM AUTHOR]

This article provides explicit integration rules for the quadrivariate and the pentavariate normal distribution. By analytically reducing the dimension of the problem and simplifying the functions to be integrated, these rules form the basis for a numerical evaluation scheme yielding an observed maximum error in the order of 10− 7 and a computational time of less than 10− 6 s. The implementation is very straightforward as it is based on a classical Gauss-Legendre quadrature. Order statistics are also dealt with. [ABSTRACT FROM AUTHOR]

This paper shows how to value multiasset options analytically in a modeling framework that combines both continuous and discontinuous variations in the underlying equity or foreign exchange processes and a stochastic, two-factor yield curve. All correlations are taken into account, between the factors driving the yield curve, between fixed income and equity as asset classes, and between the individual equity assets themselves. The valuation method is applied to three of the most popular two-asset options. [ABSTRACT FROM AUTHOR]

This paper provides explicit formulae for the probability that an arithmetic or a geometric Brownian motion will not cross an absorbing boundary defined as a step function during a finite time interval. Various combinations of downward and upward steps are handled. Numerical computation of the survival probability is done quasi-instantaneously and with utmost precision. The sensitivity of the survival probability to the number and the ordering of the steps in the boundary is analyzed. [ABSTRACT FROM AUTHOR]

This paper provides new explicit results for some boundary crossing distributions in a multidimensional geometric Brownian motion framework when the boundary is a piecewise constant function of time. Among their various possible applications, they enable accurate and efficient analytical valuation of a large number of option contracts traded in the financial markets belonging to the classes of barrier and look-back options. [ABSTRACT FROM AUTHOR]