Speed and biases of Fourier-based pricing choices: a numerical analysis

ABSTRACT

We compare the CPU effort and pricing biases of seven Fourier-based implementations. Our analyses show that truncation and discretization errors significantly increase as we move away from the Black–Scholes–Merton framework. We rank the speed and accuracy of the competing choices, showing which methods require smaller truncation ranges and which are the most efficient in terms of sampling densities. While all implementations converge well in the Bates jump-diffusion model, Attari's formula is the only Fourier-based method that does not blow up for any Variance Gamma parameter values. In terms of speed, the use of strike vector computations significantly improves the computational burden, rendering both fast Fourier transforms (FFT) and plain delta-probability decompositions inefficient. We conclude that the multi-strike version of the COS method is notably faster than any other implementation, whereas the strike-optimized Carr Madan's formula is simultaneously faster and more accurate than the FFT, thus questioning its use.

KEYWORDS:

• Option pricing
• Fourier transforms
• jump processes
• pricing errors
• speed comparisons

2010 AMS SUBJECT CLASSIFICATIONS:

• 30E10
• 60H35
• 65C20
• 65T50

Acknowledgments

We thank the anonymous referees and the editor for their quick review and useful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Ricardo Crisóstomo  http://orcid.org/0000-0001-8482-3247.

Notes

1 We define the strike grid as $k_m=-k_{max}+(m-1)\mathrm{\Delta }k+\ln \left(S_0\right)$ with $m=1,\dots ,N$ and $k=\ln K$. This choice entails setting the log FFT strikes symmetrically centered around $K=S_0$.

2 For example, out of the 4096 FFT prices calculated by Carr and Madan [7], only about 67 fall within the $\pm 20\mathrm{\%}$ log-strike interval [9].

3 We employ the truncation range $[a,b]=[c_1-L\sqrt{c_2+\sqrt{c_4}},c_1+L\sqrt{c_2+\sqrt{c_4}}]$ where $c_n$ denotes the n-th cumulant of $\ln \left(S_T/K\right)$.

4 The CM-OPT and FFT-SA rely on the same pricing approach and can evaluate any specific strike. Therefore, when the same truncation range and integration grid is used, they are equivalent in terms of accuracy

5 Following Fang and Oosterlee [14], we compute the truncation range in the Bates model through the first two cumulants of $\ln \left(S_T/K\right)$. This choice, however, leaves the 4th-cumulant $c_4$ out of the calculation, thus resulting in larger values of L.

6 Except for the ITM and OTM strikes in the FFT, where interpolation errors result in biases of $O\left({10}^{-3}\right)$ or lower

7 For instance, when $\theta =0$ the AVG distribution is symmetric, and v alone determines the excess kurtosis, which is equal to $3(1+v)$.

8 This methodology replicates the numerical results in [32] and the related example in [19], for which [25] reports a high precision value of 11.3700278104.