The SINC way: a fast and accurate approach to Fourier pricing

Abstract

The goal of this paper is to investigate the method outlined by one of us (P. R.) in Cherubini, U., Della Lunga, G., Mulinacci, S. and Rossi, P. [Fourier Transform Methods in Finance, 2009 (John Wiley & Sons Inc.).] to compute option prices. We name it the SINC approach. While the COS method by Fang, F. and Oosterlee, C.W. [A novel pricing method for european options based on Fourier-cosine series expansions. SIAM. J. Sci. Comput., 2009, 31(2), 826–848.] leverages the Fourier-cosine expansion of truncated densities, the SINC approach builds on the Shannon Sampling Theorem revisited for functions with bounded support. We provide several results which were missing in the early derivation: (i) a rigorous proof of the convergence of the SINC formula to the correct option price when the support grows and the number of Fourier frequencies increases; (ii) ready to implement formulas for put, Cash-or-Nothing, and Asset-or-Nothing options; (iii) a systematic comparison with the COS formula for several log-price models; iv) a numerical challenge against alternative Fast Fourier specifications, such as Carr, P. and Madan, D. [Option valuation using the fast Fourier transform. J. Comput. Finance, 1999, 2(4), 61–73.] and Lewis, A.L. [Option Valuation Under Stochastic Volatility with Mathematica Code, 2000 (Newport Beach: Finance Press).]; (v) an extensive pricing exercise under the rough Heston model of Jaisson, T. and Rosenbaum, M. [Limit theorems for nearly unstable Hawkes processes. Ann. Appl. Probab., 2015, 25(2), 600–631.]; (vi) formulas to evaluate numerically the moments of a truncated density. The advantages of the SINC approach are numerous. When compared to benchmark methodologies, SINC provides the most accurate and fast pricing computation. The method naturally lends itself to price all options in a smile concurrently by means of Fast Fourier techniques, boosting fast calibration. Pricing requires resorting only to odd moments in the Fourier space.

Keywords:

• Option pricing
• Rough Heston model
• Fourier expansion
• COS method
• Fast Fourier methods

JEL Codes:

• G12
• G13
• C6—Mathematical Methods and Programming < C—Mathematical and Quantitative Methods

Acknowledgments

We thank the Managing Editor and three anonymous Referees for several comments which helped improving the manuscript a lot.

We also warmly thank Fulvio Corsi, Nicola Fusari, Jim Gatheral, Giulia Livieri, and Andrea Pallavicini for the many insightful comments they provided us.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Monte Carlo methods represent an alternative approach to pricing under rough volatility. We do not consider them here, because it is quite aside from our main message. We refer the interested reader to Bennedsen et al. 2017, McCrickerd and Pakkanen 2018, Bayer et al. 2020 for recent developments.

2 The general formula for non zero interest rate and dividend yield is readily recovered by setting $s_T=\log \left(S_T/S_0\right)-(r-q)T$ and $k=\log \left(K/S_0\right)-(r-q)T$ and reads $\mathrm{P}\mathrm{u}\mathrm{t}(t=0,S_0)={\mathrm{e}}^{-qT}{S}_0\mathbb{E}\left[\right({\mathrm{e}}^k-{\mathrm{e}}^{s_T}{)}^{+}].$

3 The derivation of the AoN put price is perfectly equivalent to the CoN one. We decide to skip it because going through each steps would not add anything new.

4 As done before for the pricing formula, we detail the case of CoN put options. Similar results for the AoN puts can be readily derived.

5 The Riemann–Liouville fractional derivative of a function f is defined as $\begin{array}{r}{D}^{\alpha }f\left(t\right)=\frac{1}{\mathrm{\Gamma }(1-\alpha )}\frac{\mathrm{d}}{\mathrm{d}t}{\int }_0^t(t-s{)}^{-\alpha }f(s)\mathrm{d}s\alpha \in [0,1),\end{array}$ provided that it exists. Similarly the fractional integral, provided that it exists, is given by $\begin{array}{r}{I}^{\alpha }f\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^t(t-s{)}^{\alpha -1}f(s)\mathrm{d}s\alpha \in (0,1].\end{array}$

6 We only focus on options which are struck at K = 0.80 but stress that the pattern does not depend on the moneyness (as one can guess from table 10).

7 In a concrete situation the standard strategy is to estimate the forward variance curve as a difference on the variance swap curve. The fair value of a variance swap is computed using the methodologies explained in Fukasawa 2012 and an iteration procedure is subsequently performed to match model and market at-the-money volatilities through shifting and scaling.

8 https://github.com/fabioBaschetti/SINC-method. This also contains the surfaces that we use and all the codes one needs to reproduce our results.

9 It is possible to derive an analytic bound for ${ϵ}_2$, assuming some mild regularity for the PDF. The reasoning is similar to that in Fang and Oosterlee (2009).

10 We thank an anonymous referee for pointing this out.