Numerical Computation of Model Prices

Abstract

In this chapter we develop a new pricing algorithm to compute model prices for the derivatives contracts previously discussed. Here, we distinguish, as before, between contracts with unconditional and conditional exercise rights. The distinction is made because of the separate fundamental calculation procedure for these prices. Whereas derivatives with unconditional exercise rights can be calculated in terms of the general characteristic function ψ(x _t , z, w₀,w, g0, g, τ) and in terms of the relevant moment-generating function¹²⁸, respectively, without evaluating any integral at all if the characteristic function is known in closed form, we need for option-type contracts to apply a numerical integration scheme in order to calculate their model prices. Carr and Madan (1999) showed in their prominent article a very convenient method to compute option prices for a given strike range, using the FFT. The advantage in applying the FFT to option-pricing problems, is its considerable computational speed improvement compared to other numerical integration schemes. Due to the payoff transform methodology, we use another pricing algorithm, which shares the same desirable, numerical properties of the FFT. Unfortunately, implementing the pricing approach according to Lewis (2001), it is necessary to impose the transform with respect to the strike. Therefore, one cannot use the FFT any longer to obtain option prices in one pass for a strike range¹²⁹. In order to circumvent this problem within the payoff-transform pricing approach, we need an another numerical algorithm.

128 See Section 5.2.