The closed-form solution performing a FRFT

Following the last section, we introduce the FRFT technique to derive the price of a zero-coupon bond option. In doing so, we are able to compare the option price coming from the FRFT approach with the appropriate closed-form solution (5.27). 

Given the characteristic function ft,a ) we are able to compute the probabilities by solving the Fourier inversion integral via 

Therefore, the Fast Fourier Transform (FFT) algorithm can be applied, only if we first eliminate the singularity at 0 = 0. Following Carr and Madan [13], we modify the transform 0t z) by multiplying the probability Tl a [ ] with leading to the new transformed probability 

The inner integral can be easily solved and we obtain a mapping between 0t a + (O + i(j)) and the new transform given by 

Finally, we end up with the characteristic function t z) that is well defined at 0 = 0 and standard numerical integration methods are applicable to compute the transformed probability 

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