A new general optimal principle of designing explicit finite difference method for valuing derivative securities

Abstract

A new general optimal principle of designing explicit finite difference method was obtained. Several applied cases were put forward to explain the uses of the principle. The validity of the principal was tested by a numeric example. Key words derivative security; explicit finite difference method; implicit finite difference method; numerical method When exact valuation formol~q are unavailable or difficult to get, numerical methods have to be used. There are mainly three methods used for valuing the derivative securities[t'2's] : Monte Carlo simulation, lattices(or "tree") and finite difference methods. Monte Carlo simulation is useful for derivative securities where the payoff is dependent on the history of the underlying variable and where there are several underlying variables Is] . The lattice approach [41 values a derivative security by working backward through a complete price lattice or "tree" of the underlying security. The finite difference methods value a derivative security by solving numerically the differential equation that the derivative security satisfy. Monte Carlo simulation can only be used for European-style options but can cope with a great deal of complexity as far as the payoffs are concerned. Lattice approaches and finite difference methods can accommodate American-style as well as European-style derivative securities. But, it is very difficult to apply when the payoffs depend on the past history of the state variables as well as on their current values. Geske and Shastri c6] provide a careful comparison of finite difference methods and lattice approaches. They found that the researchers computing a smaller number of option values may prefer binomial approximation, while practitioners in the business of computing a large number of option values will generally find that finite difference methods are more efficient.In this paper, the finite difference method is analyzed in detail and an optimal designing principle is put forward. Several cases are listed to explain the use of the principle, and the validity of the principal is tested by a numeric example.