This paper derives a closed-form solutin for the price of the European and semi-Amirican callable bond for two popular one-factor models of the term structure of interest rates which have been proposed by Vasicek as well as Cox, Ingersoll, and Ross. the price is derived by means of repeated use of Green's function, which, in turn, is derived from a series solution of the partial differential equation to value a discount bond. the boundary conditions which lead to the well-known formulae for the price of a discount bond are also identified. the algorithm to implement the explicit solution relies on numerical quadrature involving Green's function. It offers both higher accuracy and higher speed of computation than finite difference methods, which suffer from numerical instabilites due to discontinuous boundary values. For suitably small time steps, the proposed algorithm can also be applied to American callable bonds or to any American-type option with Green's function being explicitly known. Copyright 1996 Blackwell Publishers.

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This paper investigates the impact of land-use zoning such as building-height and building-density restrictions on housing rent, land rent, the design parameters of a building, and population density in a residential city. A theoretical land-use model with several new features is used. Building capital, the demand for, and the supply of housing are explicitly defined. Housing is treated as a multi-dimensional good, the elements of which are floor area, finishing cost of a dwelling unit, garden space, and building height. Building costs are the sum of structural frame costs, which are a function of building height and building area, and finishing costs, which are independent of design parameters. Zoning restrictions are explicitly considered. The utility and profit maximisations are simultaneous through the housing attributes, which are implicitly priced. There are six major conclusions: First, the relative gradients of the endogeneous variables, all with regard to distance, depend not only on demand specifications but also on structural frame cost elasticities, i.e. supply specifications. Secondly, in empirical tests the free market and zoning restrictions can be discriminated with regard to housing rent if, and only if, housing attributes are taken into account. Thirdly, both the relative housing-rent gradient and land-rent gradient are the same with zoning restrictions. Fourthly, the finishing cost of a dwelling unit depends on the household's taste and income and is the same in all cases. Fifthly, while in the case of the free market a residential city with one income class could consist of single-family homes only, this is impossible in the case of binding zoning restrictions. Sixthly, the cost of zoning restrictions to tenants, or the loss of utility in terms of money, can be obtained from the indirect utility function evaluated at the equilibrium.