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.
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References
See Lee (2004), p. 61. However, comparing the structure in equation (4.21) it is possible to obtain model prices with the help of a FFT procedure for different levels of g (xt).
We find it natural to use the FFT and the IFFT algorithm to obtain the desired Fourier Transformation. Other numerical integration schemes are also possible, like for example the numerical integration via Laguerre polynomials as used in Tahani (2004).
This topic is covered comprehensively in Nagel (2001), Appendix 4.
In case of the Fong and Vasicek (1991a) model, we made the same experience as mentioned in Tahani (2004), Footnote 4, and compute values of the characteristic function with help of an explicit Runge-Kutta algorithm in the first place. Thus, besides the prevention of discontinuities, the Runge-Kutta algorithm can be more efficient than the explicit computation of the confluent hypergeometric function.
We use the same discretization scheme for α(K) as used in Lee (2004). The advantage, in contrast to the discretization schemes applied in Carr and Madan (1999) and Raible (2000), is the possibility to adjust the numerical scheme for the lower bound of the strike rates. Thus, one does not necessarily have to compute option prices for negligible strike rates, which is a more efficient procedure.
See Cooley and Tukey (1965).
The fractional Fourier Transformation parameter ζ in this thesis corresponds to α in the original article of Bailey and Swarztrauber (1994).
The detailed derivation of the FRFT algorithm is given in Bailey and Swarztrauber (1994).
See Lee (2004) Table 2 and 3. The same observation is made in Lord and Kahl (2007), Figure 1.
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(2008). Numerical Computation of Model Prices. In: Pricing Interest-Rate Derivatives. Lecture Notes in Economics and Mathematical Systems, vol 607. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77066-4_6
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DOI: https://doi.org/10.1007/978-3-540-77066-4_6
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