This paper describes the for exact evaluation of higher-order partial derivatives of functions of many variables without explicitly using the analytical expressions for those derivatives. We present and test the table method together with optimization methods which employ first, second and third order derivatives. The actual use of these methods proved to be easy and accurate.

This paper presents a new methodology, quasilinear estimation, for efficiently estimating economic variables reflected in the prices of corporate securities. For example, ex ante bankruptcy costs are not directly observable, however, if these costs are sufficiently large, then current security prices are affected and bankruptcy costs can be indirectly measured. When bankruptcy costs and other relevant parameters are known, there are many numerical solution techniques that can be used to determine security prices. One technique, the method of lines, is compatible with quasilinear estimation, which has been employed extensively in the physical sciences for the estimation of coefficients in differential equation models. We demonstrate that quasilinear estimation is a potentially reliable and efficient technique for the estimation of corporate bankruptcy costs and the asset variance from security prices.

This paper presents a new methodology, quasilinear estimation, for efficiently estimating economic variables reflected in the prices of corporate securities. For example, ex ante bankruptcy costs are not directly observable, however, if these costs are sufficiently large, then current security prices are affected and bankruptcy costs can be indirectly measured. When bankruptcy costs and other relevant parameters are known, there are many numerical solution techniques that can be used to determine security prices. One technique, the method of lines, is compatible with quasilinear estimation, which has been employed extensively in the physical sciences for the estimation of coefficients in differential equation models. We demonstrate that quasilinear estimation is a potentially reliable and efficient technique for the estimation of corporate bankruptcy costs and the asset variance from security prices.