The SINC way: A fast and accurate approach to Fourier pricing
Fabio Baschetti, Giacomo Bormetti, Silvia Romagnoli, Pietro Rossi
aa r X i v : . [ q -f i n . P R ] S e p Rough Heston: the
SINC way
Fabio Baschetti ∗ Giacomo Bormetti † Silvia Romagnoli ‡ Pietro Rossi § ¶ September 2, 2020
Abstract
The goal of this paper is to investigate the method outlined by one of us (PR) inCherubini et al. (2009) to compute option prices. We named it the SINC approach.While the COS method by Fang and Osterlee (2009) leverages the Fourier-cosine ex-pansion of truncated densities, the SINC approach builds on the Shannon SamplingTheorem revisited for functions with bounded support. We provide several importantresults which were missing in the early derivation: i) a rigorous proof of the convergeof the SINC formula to the correct option price when the support growths and thenumber 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 COSformula in several settings; iv) a numerical challenge against alternative Fast Fourierspecifications, such as Carr and Madan (1999) and Lewis (2000); v) an extensivepricing exercise under the rough Heston model of Jaisson and Rosenbaum (2015); vi)formulas to evaluate numerically the moments of a truncated density. The advantagesof the SINC approach are numerous. When compared to benchmark methodologies,
SINC provides the most accurate and fast pricing computation. The method natu-rally lends itself to price all options in a smile concurrently by means of Fast Fourier ∗ Department of Statistics, University of Bologna, Italy E-mail: [email protected] † Department of Mathematics, University of Bologna, Italy E-mail: [email protected] ‡ Department of Statistics, University of Bologna, Italy E-mail: [email protected] § Prometeia S.p.A., Bologna, Italy E-mail: [email protected] ¶ We wish to thank Jim Gatheral and Giulia Livieri for several comments they provided us. echniques, boosting fast calibration. Pricing requires to resort only to odd momentsin the Fourier space. Keywords : option pricing; rough Heston model; Fourier expansion; COS method;Fast Fourier methods
The search of numerically efficient approaches to price options is the subject of intensiveresearch. This fact comes with no surprise, since the ubiquitous presence and crucial roleplayed by contingent claims in modern finance. Without fear of contradiction, it can be af-firmed that, when the characteristic function (CF for short) of the log-price process is knownin analytic or semi-analytic form, the current standard solution to the pricing problem isthe COS method by Fang and Oosterlee (2009). COS – a short-name for Fourier-cosineexpansion – builds on the idea that it is computationally convenient to transform the ex-pectation of the payoff with respect to the risk-neutral probability density function (PDFfor short) into a linear combination of products of Fourier-cosine coefficients of the payoffand the density. To achieve this goal, one initially pays some price – the key step in theCOS development is the approximation of true PDF by a truncated version with boundedsupport – but this trick eventually reveals to be the crucial step to obtain the best per-forming pricing formula so far.Our paper leverages the same idea of truncating the PDF, due to one of us (PR) andoutlined in Cherubini et al. (2009), but from a different perspective. It exploits a well-known result which applies to periodic functions with limited bandwidth, i.e. the ShannonSampling Theorem. The formal symmetry between the forward and backward Fouriertransform readily provides the intuition that Shannon’s result can be adapted to functionswith limited support in the direct space. As an interesting outcome of the applicationof the Sampling Theorem, one can express Plain Vanilla put and call prices, and digitaloption constituents, as a Fourier-sinc expansion. Given that the sinc function is the Fourier2ransform of the rectangular function, it is not surprising that it may play a crucial rolein representing expectations with respect to truncated densities. The convolution betweenthe sinc function – which conveys the information related to the bounded support – andthe Fourier transform of the Heaviside step function – which characterizes the point ofdiscontinuity of the digital options – lends itself to analytic simplification by means ofHilbert transforms. As a result, the option price can be represented as a series expansionwhich only requires the CF computation of the log-price process for odd moments. Werefer to this method as
SINC approach . As an important contribution, in this paper weprove in a rigorous way that the numerical error induced by the PDF truncation and byapproximating a double infinite Fourier series by a finite sum can be made arbitrary small.The need to know the CF in order to apply both COS and
SINC is in principle alimitation of the approach, which however turns out to be a quite mild drawback. Theliterature on stochastic models where it is natural to work in the Fourier space is hugeand ever growing (see Cherubini et al. (2009) for an overview of the topic). The suc-cessful application of Fourier analysis to price options was pioneered by Chen and Scott(1992); Heston (1993); Bates (1996); Bakshi and Chen (1997); Scott (1997). The pub-lication of Duffie et al. (2000) definitely celebrated the role of the transform analysis indynamic asset pricing models when the state vector follows an affine jump-diffusion. Thepapers by Carr and Madan (1999) and Lewis (2000, 2001) contributed in a significant wayto this stream of research in quantitative finance. In the former, the authors introduceda simple analytic expression for the Fourier transform of the option value, which allowsto exploit the considerable computational power of the Fast Fourier Transform (FFT) inthe inversion stage. The introduction of FFT techniques boosted the way to real-time cal-ibration, pricing, and hedging. In the latter contributions, Lewis (2000, 2001) detailed arepresentation of the option price in terms of the CF which is rooted on a clever extensionof the Fourier transform in the complex domain. His approach is naturally prone to the ap-plication of FFT, too. It is often preferred to the Carr and Madan (1999) approach, whichrequires the introduction of an auxiliary damping parameter.
SINC is naturally suitedfor the computation by means of FFT. Then, not only
SINC is rooted on a parsimonious3epresentation of the option payoff, which requires to sample the CF at optimal points, butit expresses the payoff as a transform where the log-moneyness is the conjugate variable inthe direct space. As a consequence, all options in a smile can be computed concurrentlywith O ( N log N ) complexity, where N is the number of sample points in Fourier space,enhancing the computational advantage of SINC with respect to COS.The stream of research inspired by the general framework introduced in Duffie et al.(2000) is vast. It ranges from models to equity and exchange rate option pricing, to in-terest rate derivative pricing, credit risk, and systemic risk modeling. We do not try areview of the literature, which will surely result in a deficient list. Instead, we focus on arestricted but stimulating and flourishing field, the modeling of financial volatility for pric-ing purposes. Nonetheless, the scope of applicability of the
SINC methodology is wider,as it will become soon clear. The main reason of the interest in volatility modeling is that,recently, the celebrated Heston model has been revisited in several respects. Jaisson et al.(2015) showed that the Hawkes-based (Hawkes, 1971a,b) market microstructure modelof Bacry et al. (2013) under nearly-unstable conditions converges in law to the Hestonmodel. A modification of the microstructure model more aligned with financial data consid-ers the case of nearly-unstable heavy tailed Hawkes processes, that is Hawkes processes withan hyperbolic decaying kernel (Hardiman et al., 2013; Bacry et al., 2016). Under this morerealistic setting, Jaisson et al. (2016) proved that the process converges to an integratedfractional diffusion. The most surprising fact of this result is that the persistence observedat high frequency is washed out by the aggregation at longer time scale. The limitingprocess is very irregular, with a derivative behaving as a fractional Brownian motion withHurst exponent smaller than 0.5 and close to zero. For this reason, the limiting process isdubbed rough Heston (rHeston for short). Gatheral et al. (2018) demonstrated for a widerange of assets that the historical volatility is rougher than a Brownian motion, and thatthe empirical moment of order q of the log-volatility increments are consistent with a scalingwith Hurst exponent of order 0.1. Similar findings are reported in Bennedsen et al. (2016)under the historical measure, while Livieri et al. (2018) investigated the rough behavior of A Hawkes process is nearly-unstable when the L norm of the regression kernel is almost one. H . In empirical investigations, both under the pricing and the historicalmeasures, H is found to be of order 0.05-0.1, thus motivating the use of the rational ap-proximation. An alternative approach is provided by the Adams scheme (Diethelm et al.,2004), possibly combined with a power series expansion and Richardson-Romberg extrap-olation (Callegaro et al., 2020). We are currently testing these alternatives but postponethe analysis of their numerical viability in a companion paper. For the moment, we remarkthat in a calibration exercise it is hard to dispute the computational gain of the rationalapproximation.As a second main contribution of our paper, we challenge SINC against COS and
FFT-SINC against Carr and Madan (1999) and Lewis (2000, 2001) approaches computed viaFFT. Through extensive pricing under the forward variance specification, we assess thesuperiority in pricing accuracy of
SINC with respect to competitors. The comparison isperformed keeping the same number N F of points sampled in the Fourier space equal for allmethodologies. We believe this is the most fair way to claim the relative performance of thedifferent algorithms, since the number of times the CF needs to be computed in rHestonrepresents the most time consuming step in pricing. Under this specification, when SINC is5hallenged against COS, the superiority of the former is apparent. When the full power ofFFT is exploited, the reduction of the numerical complexity of
SINC vs COS is sizable anddramatic, making
SINC our preferred choice. As a matter of fact when dealing with therHeston model, the main computational burden comes from the solution of the fractionalRiccati equation needed to get the CF. This part greatly outweights the cost of pricingeven a highly populated smile and the burden of using FT is twice as big as that of FFTin our exercise. Very much different is the situation where the CF is known analitically; inthat case the advantage of having a natural FFT formulation would be very large.Last, but not least, as a side result of
SINC approach, we detail in the Appendix anovel analytical methodology to approximate the moments of a random variable startingfrom the CF.The remainder of the paper is organized as follows. In Section 2 we discuss the
SINC formula and in Section 3 we characterize the numerical error. Then, in Section 4 wereview the rHeston model by El Euch and Rosenbaum (2019). Sections 5 and 6 presentthe numerical results from the pricing exercise by means of the
SINC and FFT-
SINC specifications, respectively. Section 7 draws the most relevant conclusions. The Appendixprovides technical details. SINC at a glance
The
SINC approach to price options is rooted on the following definition of a Fourier pair g ( x ) = ¯ F [ˆ g ( ω )] = Z R e − i πxω ˆ g ( ω ) dω, ˆ g ( ω ) = F [ g ( x )] = Z R e + i πxω g ( x ) dx, where ¯ F and F stand for the forward Fourier operator and the inverse Fourier operator,respectively. Under the assumption of null interest rate and dividend yield, i.e. r = 0 and q = 0 , it exploits the following decomposition of a European put into Cash or Nothing6CoN hereafter) plus Asset or Nothing (AoN) options, i.e. E [( K − S T ) + ] = K E [ { s T
84) into the CoN Equation (2), straightforwardly writing E [ { s T The Modified Hilbert transform H − of a given function g is the result of aconvolution of the distribution δ − ( x ) with the function itself. This formally translates as: H − [ g ( y )] = Z g ( x ) δ − ( y − x ) dx = i π Z g ( x ) y − x + iε dx. In particular, the Appendix (Section C) proves that Z sinc [2 πX c ( ω − ω n )] ω + iε dω = 2 πi H − [ sinc (2 πX c ω n )] = 12 X c ω n (1 − e i πX c ω n ) , (7)which is sufficient to specialize the CoN put as E [ { s T 12 + 2 π N/ X n =1 n − (cid:20) sin(2 πkω n − ) ℜ (cid:2) ˆ f ( ω n − ) (cid:3) − cos(2 πkω n − ) ℑ (cid:2) ˆ f ( ω n − ) (cid:3)(cid:21) , (8)9here ℜ and ℑ denote the real and imaginary parts, respectively. We show the validity ofthis final formula in Appendix (Section D), and claim that the AoN option is priced in avery similar way, except that the CF needs to be evaluated for a complex argument, i.e. E [ e s T { s T Out of the N + 1 terms that we included in the expansions, only N/ survive.They correspond to the positive odd frequencies. Equations (8) and (9) finally guarantee that the put option price in Equation (1) accom-modates the following form: E [( K − S T ) + ] ≃ 12 ( K − S )+ 2 π N/ X n =1 n − (cid:20) sin(2 πkω n − ) ℜ (cid:2) K ˆ f ( ω n − ) − S ˆ f ( ω n − − i π ) (cid:3) − cos(2 πkω n − ) ℑ (cid:2) K ˆ f ( ω n − ) − S ˆ f ( ω n − − i π ) (cid:3)(cid:21) . (10)Equations (8) - (10) represent the main formulas of this paper. They express CoN, AoN,and Plain Vanilla put options in an extremely simple and compact form.To ease the interpretation of the results in the numerical sections and the comparisonamong different benchmark methodologies, we introduce the notation N F to refer to thenumber of times the CF needs to be evaluated to compute the option price. For instance,to price a CoN, it is sufficient to sample the CF N F = N/ times at points ω n − ( N/ times at shifted points ω n − − i/ (2 π ) for the AoN) and to weight them with a suitableimaginary phase and the inverse of the integer odd numbers. The price of the Plain Vanillaput is readily recovered from AoN and CoN, thus by means of N F = N/ valuations ofthe CF. In the next sections, we are going to support the computational effectiveness of10he SINC formulas, by challenging them against the COS ones and showing how the SINC approach can be readily adapted to the FFT framework. SINC One merit of SINC is that it is readily adapted to the stiff structure of the FFT algorithm;the computational speed of the Fast Fourier Transform is crucial for any concrete applica-tion within the calibration process and the extension comes with almost no effort in oursetting.Our assumption is to price a discrete grid of strikes k m = m X c N , − N/ ≤ m < N/ and tofit the remaining points, when needed, by linear interpolation from bucket to bucket.Digital put prices at the aforementioned vector of strikes are now calculated as follows E [ e as T { s T In spite of the fact that the index n runs from to N − , a closer inspection eveals that the computation of q n only requires the evaluation of the CF at N/ differentfrequencies. Indeed, all q n for even n are identically zero. The described procedure generates prices for CoN and AoN digitals indexed by the strikes n (2 X C /N ) . To recover the price for different strikes (not belonging to the grid) we per-form a linear interpolation. The interpolation error can be reduced by increasing thenumber of terms in the expansion or resorting to the fractional FFT framework (see forinstance Chourdakis (2005)). An analysis similar to that performed in Fang and Oosterlee (2009) shows that there arethree sources of error affecting the SINC formula: the approximation of the true PDF witha truncated density, the replacement of a double infinite sum with a finite sum, and thesubstitution of the Fourier coefficients for the truncated density with the Fourier transformof the true PDF valued at discrete points. To characterize in a quantitative way the threeerror components, we proceed as follows.The error associated to our approach can be written as ǫ = Z f ( s T ) θ ( k − s T ) ds T − − i πX c + N/ X n = − N/ e − i πkω n − ˆ f ( ω n − ) ω n − = Z f ( s T ) θ ( k − s T ) ds T − Z X c − X c f ( s T ) θ ( k − s T ) ds T + Z X c − X c f ( s T ) θ ( k − s T ) ds T − − i πX c + N/ X n = − N/ e − i πkω n − ˆ f ( ω n − ) ω n − . Exploiting the fact that Z X c − X c f ( s T ) θ ( k − s T ) ds T = 12 + i πX c + ∞ X −∞ e − i πkω n − f {− X c ≤ s T ≤ X c } V ( ω n − ) ω n − , we can write ǫ = Z f ( s T ) θ ( k − s T ) ds T − Z X c − X c f ( s T ) θ ( k − s T ) ds T As done before for the pricing formula, we detail the case for the AoN put options. Similar results forthe CoN puts can be readily derived. i πX c + ∞ X −∞ e − i πkω n − f {− X c ≤ s T ≤ X c } V ( ω n − ) ω n − − i πX c + N/ X n = − N/ e − i πkω n − ˆ f ( ω n − ) ω n − = Z f ( s T ) θ ( k − s T ) ds T − Z X c − X c f ( s T ) θ ( k − s T ) ds T + i πX c X | n | >N/ e − i πkω n − f {− X c ≤ s T ≤ X c } V ( ω n − ) ω n − + i πX c + N/ X − N/ e − i πkω n − f {− X c ≤ s T ≤ X c } V ( ω n − ) − ˆ f ( ω n − ) ω n − . (13)The PDF truncation error reads ǫ . = Z f ( s T ) θ ( k − s T ) ds T − Z X c − X c f ( s T ) θ ( k − s T ) ds T , where we introduce the same notation, ǫ , used in Fang and Oosterlee (2009). The secondand last components of the error in Equation (13), that we refer to with ǫ and ǫ to con-form with the notation in Fang and Oosterlee (2009), are the error contributions due to thetruncation of a double infinite Fourier series and the replacement of the Fourier coefficientsof the truncated PDF with the Fourier transform of the true PDF, respectively.Such a decomposition of the overall error is the starting point when proving that the SINC price converges to the true option price: technical reasons and assumptions essential forthe proof are given in the Appendix (Section F), where we bound the magnitude for eachof the components in Equation (13) and conclude that the error can be made arbitrarilysmall by increasing the number of Fourier modes N and the truncation range [ − X c , X c ] . Ease of transposition to the FFT form makes the SINC approach very well suited for cal-ibration, and the present paper wants to show that this is an efficient solution for roughvolatility models as well. In particular, we will take the rough Heston model as a reference.We recall it in the following for the readers’ convenience.13he (generalized) rough Heston model emerging from El Euch and Rosenbaum (2018) isdescribed by the following equations: dS t = S t p V t { ρdB t + p − ρ dB ⊥ t } ,V t = V + λ Γ( H + ) Z t θ ( s ) − V s ( t − s ) − H ds + ν Γ( H + ) Z t √ V s ( t − s ) − H dB s , where V , λ , and ν are positive real numbers, ρ ∈ [ − , . The deterministic function θ ( t ) is positive and satisfies few constraints specified in El Euch and Rosenbaum (2018).The coefficient H ∈ (0 , / is shown to govern the smoothness of the volatility, whosetrajectories enjoy Hölder continuity H − ǫ for any ǫ > . It is therefore clear that thechoice H < / allows for a rough behavior of the volatility process and the case H = 1 / amounts to the classical Heston model with time-dependent mean reversion level.El Euch and Rosenbaum (2018) proved also that the product λθ ( · ) is directly inferredfrom the time- forward variance curve ξ ( t ) = E [ V t |F ] = E [ V t ] , leading to the followingspecification of the model for λ → : dS t = S t p V t { ρdB t + p − ρ dB ⊥ t } ,V t = ξ ( t ) + ν Γ( H + ) Z t √ V s ( t − s ) − H dB s . Remark 3. The forward variance curve is easily obtained from the variance swap curve bydifferentiation (see El Euch et al., 2019) and variance swaps valued as in Fukasawa (2012). This is extremely convenient for calibration purposes thanks to the reduced dimensionalityof the problem. We will consequently work under this last specification throughout the restof the paper, thus placing ourselves in the same setting of El Euch et al. (2019).The forward variance curve is a state variable in the model and it also enters the CF of theasset log-price (see El Euch and Rosenbaum (2018) for further details): ϕ ( a, t ) = E (cid:20) exp (cid:26) ia log (cid:18) S t S (cid:19)(cid:27)(cid:21) = exp (cid:18) Z t D α h ( a, t − s ) ξ ( s ) ds (cid:19) , α = H + , h ( a, t ) is the unique continuous solution of the fractional Riccati equation D α h ( a, t ) = − a ( a + i ) + iaρνh ( a, t ) + ν h ( a, t ) , I − α h ( a, 0) = 0 , (16)and D α , I − α denote the Reimann-Liouville fractional derivative and fractional integral oforder α and − α , respectively .Now, Equation (16) is a rough version of the Riccati ODE which emerges in the classicalHeston model with zero mean reversion. Here, the standard derivative is replaced by afractional one. However, such a small change is not painless: the rHeston Riccati equationhas no explicit solution and needs to be approximated using numerical methods whichare not really plain. In this paper we are not discussing the general issue of an efficientcomputation of the CF. More precisely, given any approximation to the CF we want toshow that the SINC is a very effective method to perform pricing. We will stick with therational approximation to the CF of Gatheral and Radoicic (2019) and discuss our resultswithin that contest. A second paper, in preparation, will center around the tricky aspectsencountered when trying to compute the CF. SINC at work In this section, we perform numerical tests to assess the accuracy of the SINC approach.We price European puts and their digital components separately and span over variousmaturities and moneynesses. The idea is to compare the SINC method with the COSmethod. This second one is known to be very robust and more accurate than any other The Reimann-Liouville fractional derivative of a function f is defined as D α f ( t ) = 1Γ(1 − α ) ddt Z t ( t − s ) − α f ( s ) ds α ∈ [0 , , provided that it exists. Similarly the fractional integral, provided that it exists, is given by I α f ( t ) = 1Γ( α ) Z t ( t − s ) α − f ( s ) ds α ∈ (0 , . SINC , for the chosen strikes, maturities, andparameter sets, is almost always better than COS when computing call and put options.Always orders of magnitude better when dealing with digital options. Furthermore, SINC enjoys the non negligible advantage to be tailor made for the FFT, while the COS, as weknow, does not have a painless transition.We said our experiments are under the rHeston model, with the specific prescription thatthe forward variance form of El Euch et al. (2019) is used. Parameters are as follows H = 0 . ν = 0 . ρ = − . , and the forward variance curve supposed to be flat at ξ ( · ) = 0 . . We consider strikes forall the regions of moneyness, i.e. K = { . , . , . } , at both short and long maturities,i.e. T = { . , } .As for truncation of the PDF, we resort to Equation (5), where the cumulants are given by c = m c = m − m c = m − m m + 6 m m − m , and the moments have been computed by the techniques explained in the Appendix (Sec-tion E) . The object of our study is the accuracy of the methods, at this stage, and weconsequently take L = 100 in spite of different indications in Fang and Oosterlee (2009).Benchmarks are built by pushing SINC and COS at very high precision ( N F = 2 ) andtaking all the digits they have in common - or at most ten if they happen to have more.Table 1 aggregates CoN and AoN puts and reports maximum absolute errors with respectto the benchmark for both SINC and COS at different values of N F , for T = 1 . Thesuperiority of the SINC method w.r.t. the COS, for digital option is strikingly evident. The numerical results presented in this Section are computed assuming X h = − X l = X c . This isequivalent to assume m = 0 . F K 256 512 1024 2048 40960.80 4.9272e − 03 2.6647e − 04 5.5175e − ⋆ ⋆ SINC 1.00 1.3679e − 02 3.5341e − 04 4.5323e − ⋆ ⋆ − 03 4.6770e − 04 9.2800e − ⋆ ⋆ − 02 1.1897e − 02 4.6170e − 03 2.5080e − 04 5.1600e − − 01 3.3614e − 02 1.3558e − 02 2.8836e − 04 3.0859e − − 01 2.2281e − 02 2.5574e − 03 5.4213e − 04 1.0379e − SINC and COS at dif-ferent values of N F . Benchmarks are as follows: CoN = {0.0746857077, 0.3677803881,0.9746184153}, AoN = {0.0477997904, 0.3222614106, 0.9673515242} for strikes K = { . , . , . } respectively; T = 1 . Stars ( ⋆ ) mean that the price fully conforms withthe benchmark (up to the number of digits of the benchmark itself).When dealing with put options, the COS method partially catches up the SINC . Put, aswell as call options, are the difference between two digital options, and this introducescancellations mildly benefiting the SINC but greatly benefiting the COS method. As aconsequence, for put or call options, the performance of the COS is still inferior to theSINC but not in such a striking way as it is for digital options.We stress that the numbers we will see in the following tables do not depend on the mon-eyness and only focus on options which are struck at K = 0 . . Table 1 confirms thatthe convergence to the true option price is much faster for SINC than it is for COS, whendealing with digital options, and this is markedly evident in Figure 1. The idea behindthese charts is that we take our benchmarks as a reference, compute prices with the twomethods by increasing log N F of one unit per time and stop when we have reached accuracy The reason for this choice is that the traditional method of Carr and Madan (1999) is known to exhibitpathological behaviors when the option is deep OTM and the maturity very short. On the other hand, theCOS is insensible to this and we wish to show that so also is the SINC. 17f five significant digits on both the CoN and AoN. Not surprisingly SINC meets the target S I NC p r i c e CoNAoNPUT C O S p r i c e CoNAoNPUT Figure 1: Convergence of the SINC (lhs) and the COS (rhs) method. Red (dashed), blue(dot-dashed), and black (bold) lines are the CoN, AoN, and put options, respectively. Lightblue horizontal lines denote the benchmarks. T = 1 and K = 0 . .for much lower N F , but we still should recognize that the COS price for the put come closerto the SINC performance than its digital components. Note that oscillations last longeron the digital components than the put option itself on the rhs of Figure 1, but they allare less stable than the corresponding SINC prices on the lhs of the same Figure.Before commenting Table 2, it is important to stress a major difference between the SINC and COS methods. While COS uses the same frequencies for building the CoN and theAoN, thus computing the CF N F times for deriving both digital and Plain Vanilla prices, SINC uses different frequencies for CoN and AoN. Consequently, it computes put pricesby evaluating the CF N F times, half of them to compute the CoN price and half for theAoN. This is a huge advantage to COS. Nonetheless, the results in Table 2 confirm thesuperior performance of the SINC , where the put option prices for three different strikes, K = { . , . , . } , and maturity T = 1 are reported for different values of N F .Moreover, we also consider the complex situation where T = 0 . . If this seems too short,it is still something one may encounter during the calibration process. It is consequently18 F K 256 512 1024 2048 40960.80 1.5951e − 03 1.2334e − 03 6.5822e − 05 1.3886e − ⋆ SINC 1.00 4.6365e − 03 2.1574e − 04 6.5050e − 05 1.4452e − ⋆ − 03 1.6262e − 04 7.3393e − 05 1.1195e − ⋆ − 03 8.3628e − 04 4.9324e − 05 1.0182e − 06 1.1195e − − 02 1.6422e − 03 9.4302e − 05 9.3428e − 06 9.7198e − − 03 2.4196e − 03 2.4696e − 04 1.0429e − 06 2.0835e − SINC and COS at different valuesof N F . Benchmarks are as follows: put = {0.0119487757, 0.0455189774, 0.2021905741}for strikes K = { . , . , . } , respectively; T = 1 . Stars ( ⋆ ) mean that the price fullyconform with the benchmark (up to the number of digits of the benchmark itself).useful to understand whether the SINC method is robust with respect to pricing optionswhose expiration is within a couple of days. We only look at the case K = 0 . , as usual,and repeat the same analysis as before. The pattern we deduce from Table 3 is very S I NC p r i c e CoNAoNPUT C O S p r i c e CoNAoNPUT Figure 2: Convergence of SINC (lhs) and COS (rhs) method. Red (dashed), blue (dot-dashed), and black (bold) lines are the CoN, AoN, and put options, respectively. Lightblue horizontal lines denote the benchmarks T = 0 . and K = 0 . .similar to what we have seen for longer maturities. To ensure fairness, the prices reported19INC COS N F abs.err. N F abs.err.CoN CoN256 AoN 256 AoNPUT PUTCoN 6.9486e − 07 CoN512 AoN 7.2486e − 07 512 AoNPUT PUT 6.6597e − − 10 CoN1024 AoN 3.4810e − 10 1024 AoNPUT 1.6908e − 07 PUT 2.3578e − ⋆ CoN 6.9047e − ⋆ − − 10 PUT 2.8114e − ⋆ CoN 3.6300e − ⋆ − ⋆ PUT ⋆ Table 3: Absolute errors for SINC and COS at different values of N F for T = 0 . and K = 0 . . Benchmarks are as follows: CoN = 2.42220e − 05, AoN = 1.88150e − 05, put =5.625e − 07. Stars ( ⋆ ) mean that the price fully conform with the benchmark (up to thenumber of digits of the benchmark itself). Horizontal lines mean that the number returnedby the algorithm is not meaningful (negative or indistinguishable from zero.)on each line are computed by means of the same number of sampled frequencies (speci-fied in the N F columns), independently on the approach. It is worth to notice that for N F = 256 neither SINC nor COS provide meaningful values. We need larger values of N F to reach satisfactory accuracy, but this is an obvious consequence of the much morepeaked feature of the PDF implying more difficulties to approximate it in a series expan-sion (see Figure 3). For N F = 512 , apparently only COS is capable to provide a sensible20umber. However, it is important to stress that the error (6.6597e − 06) is ten times largerthan the benchmark values (5.625e − 07) rendering the COS value useless. Starting from N F = 1024 , both methodologies provide sensible values but the accuracy of the SINC issuperior. Figure 2 confirms that SINC prices converge well before the COS counterpart,for digital options, while for Plain Vanilla the gap between the two methods is less marked. -2 -1.5 -1 -0.5 0 0.5 1 1.5 201234567 P D F -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.80102030405060 P D F Figure 3: PDF of the asset log-price in the rough Heston model for T = 1 (lhs) and T = 0 . (rhs). H = 0 . , ν = 0 . , and ρ = − . .So if the SINC method is at least as accurate and more efficient than COS - in the sensethat convergence to the true option price is generally faster - it is also evident from ourexplorations that it is very robust with respect to the option’s specifics.We finally want to give one reason to not strictly follow the prescriptions in Fang and Oosterlee(2009) and set L = 100 in Equation 5. In fact, it turns out that strict bounds as they sug-gest (they set L = 10 ) end up with cutting regions of the PDF in the rHeston model whosecontribution is not negligible. In Table 4, we quote the absolute differences from the bench-mark for a couple of options whose price is computed using a smaller support ( L = 10 ) forthe PDF of the asset log-return. Even though, apparently, COS is more accurate, indeedthe benchmark price is never reproduced. This is going to result in larger errors where oneprices indexes whose forward price is thousands of times greater than what we have here.21oN AoN PUT T = 1 K = 0 . SINC 4.5248e − 07 2.1203e − 08 3.4072e − T = 1 K = 0 . COS 2.4376e − 09 5.1893e − 09 3.1792e − T = 0 . K = 1 SINC 2.0305e − 07 1.3157e − 07 7.1488e − T = 0 . K = 1 COS 1.4036e − 10 1.1111e − 08 1.1252e − L = 10 inEquation 5.We actually do that with entire surfaces in the next section: for some of the smiles understudy, the value of L = 10 , suggested by Fang and Oosterlee (2009) is clearly inadequate,while 100 seems to be a good choice everywhere we tested both methods. There mightbe ’optimal’ values somewhere in between 10 and 100. We did not look into that detailedoptimization. SINC One strong merit of the SINC approach is that it is easily adapted to the FFT form, thisfact is crucial for calibration where several strikes have to be computed simultaneously forany given maturity. While the extension is not immediate with COS, one still can counton other methods that are based on a naive discretization of Lewis integral or Carr-Madantraditional technique. In any case, if the former is known to be very slow when it comes toconvergence to the true option price, the latter is not really accurate and very sensitive tothe options specifics and its tuning parameters. We therefore wish to show that FFT- SINC requires much lower N F to reach some specified accuracy on the implied volatilities, and itis to be consequently regarded as a benchmark method for calibration.For these purposes, we now price the same volatility surfaces as in El Euch et al. (2019)using model parameters resulting from their calibration, and report average errors on theimplied volatility for each smile. The numbers we quote refer to the lowest N F needed22o make such an average error on the implied volatility smaller than − , and observethat this is achieved with far more effort on the aforementioned alternatives than it isfor FFT- SINC . Our benchmark prices are computed as the usual intersection between thehigh-precision SINC and COS candidates. Implementation details for each of the FFTmethods are given in the following: • FFT- SINC is given with a truncation range for the PDF which complies to Equa-tion (5), with L = 100 ; • for Carr-Madan method we fix the dumping parameter at α CM = 0 . and the upperlimit of integration as a CM = 1500 ; • FFT-Lewis extends the integration range according to the usual rule, with L = 5000 .As for the model itself, then, we maintain the forward variance form that we have reportedin Section 4. The forward variance curve is estimated as a difference on the variance swapcurve, and the fair value of a variance swap computed using the methodologies explained inFukasawa (2012). An iteration procedure is subsequently performed to match model andmarket at-the-money volatilities through shifting and scaling.Calibrated parameters for August 14, 2013 are reported in El Euch et al. (2019): H = 0 . ν = 0 . ρ = − . . With these numbers we compute put option prices for the entire surface based on theFFT methods above. Table 5 reports average errors on the implied volatility for each ofthe quoted smiles at the lowest value of N F that satisfies our prior condition. We read-ily observe that reaching the desired accuracy is much faster for the FFT- SINC than forCarr-Madan method and, even more, for a naive discretization of Lewis integral.Maturity FFT- SINC FFT-Lewis Carr-Madan N F = 2048 N F = 262144 N F = 8192 − 05 2.1e − 04 8.0e − − 05 3.4e − 04 2.6e − − 05 1.3e − 04 1.6e − − 05 7.6e − 05 1.2e − − 05 8.3e − 05 9.2e − − 05 1.2e − 04 6.1e − − 05 1.0e − 04 5.8e − − 05 7.0e − 05 3.5e − − 05 8.2e − 05 2.7e − − 04 2.4e − 04 2.6e − − 05 8.0e − 05 1.8e − − 04 1.6e − 04 1.8e − − 04 1.4e − 04 1.1e − − 04 7.9e − 05 9.0e − − 04 2.8e − 04 9.0e − − 04 9.0e − 05 6.0e − − 04 4.4e − 04 3.7e − − 04 9.5e − 05 9.0e − − 04 2.4e − 04 2.0e − N F satisfying the condition that the average error per surface is smaller than one basispoint. Remark 4. Increasing the accuracy of the FFT- SINC to any other level is immediate bysimply using an higher N F , but the same thing is not trivial with Carr-Madan method inview of the documented problems we listed at the beginning of the section. We repeat the same analysis for a second date in El Euch et al. (2019). For May 19, 2017calibrated parameters are: H = 0 . ν = 0 . ρ = − . . Table 6 confirms what we have seen for the first surface, thus corroborating our claims fora superior performance of the FFT- SINC . 24aturity FFT- SINC FFT-Lewis Carr-Madan N = 2048 N = 262144 N = 8192 − 05 9.4e − 05 7.3e − − 04 1.2e − 04 4.0e − − 05 2.5e − 04 2.7e − − 04 1.5e − 04 1.9e − − 05 1.7e − 04 1.4e − − 05 1.4e − 04 1.3e − − 04 1.7e − 04 1.5e − − 04 1.7e − 04 1.3e − − 04 1.4e − 04 9.6e − − 04 1.5e − 04 1.1e − − 04 1.4e − 04 1.0e − − 05 1.9e − 04 6.5e − − 04 1.1e − 04 8.4e − − 04 1.2e − 04 8.7e − − 04 1.0e − 04 6.0e − − 04 1.0e − 04 6.5e − − 04 1.2e − 04 7.3e − − 04 1.8e − 04 5.9e − − 04 1.1e − 04 5.3e − − 04 1.1e − 04 4.7e − − 04 1.2e − 04 3.3e − − 04 1.3e − 04 2.8e − − 04 1.1e − 04 2.2e − − 04 1.1e − 04 2.3e − − 04 1.7e − 04 2.2e − − 04 2.6e − 04 2.6e − − 04 9.7e − 05 1.5e − − 04 2.6e − 04 1.3e − − 04 1.2e − 04 1.0e − − 04 1.7e − 04 7.0e − − 04 1.1e − 04 7.0e − − 04 1.2e − 04 8.6e − − 04 1.2e − 04 8.9e − − 04 1.6e − 04 7.5e − − 04 1.9e − 04 4.0e − N F satisfying the condition that the average error per surface is smaller than one basis point. The paper investigates the SINC approach when pricing European options. SINC is shownto be superior to well-known benchmark methodologies. At variance with COS, it allowsfor an immediate extension to the FFT form. This fact is essential in any calibration exer-cise. We therefore claim that SINC is a promising approach, regarding both the precisionit achieves and its numerical efficiency. The numbers we produce in Sections 5 and 6 leavefew space for interpretation. They can be obviously reproduced for any other model whoseCF is known in closed or semi-closed form. In this respect, our focus on the rHeston modelis motivated by the spurring interest on rough volatility models and it is by no meansdictated by any limitation of the SINC approach.The idea behind SINC is that one first writes put options as a linear combination of digitalAsset-or-Nothing and Cash-or-Nothing options. The expectation defining their values is aconvolution between the density of the asset log-return and the payoff function. Then, theconvolution theorem for Fourier transforms guarantees that each price can be expressed asthe integral over a shifted CF. By approximating the CF of the true density with the CFof a truncated PDF, one can fully exploit the potential of the Shannon Sampling Theorem.26t allows to represent the CF at any point by means of a discrete set of frequencies andexpress it as a Fourier-sinc expansion. Making use of the closed-form representation ofthe Modified Hilbert transform of the sinc function one can achieve simple and compactformulas for digital and Plain Vanilla put option prices. Moreover, these formulas lendthemself to fast computation by means of FFT. The paper provides a rigorous proof ofthe converge of the SINC formula to the correct option price when the support growthsand the number of Fourier frequencies increases. It also investigates several technicalprescriptions, such as the computation of truncation bounds by means of a novel techniqueto compute the cumulants from the CF or the sensitivity of the option prices to the number N F of frequencies sampled in the Fourier-space. Through an extensive pricing exercise, itassesses the superior performance of the SINC approach with respect to the competitorCOS methodology. As far as the FFT specification is concerned, the paper challenges SINC against the FFT specification of the Lewis formula and the Carr-Madan approach. In bothcases, SINC proves to be accurate and robust to option’s specification.27 eferences Bacry, E., S. Delattre, M. Hoffmann, and J.-F. Muzy (2013). Modelling microstructurenoise with mutually exciting point processes. Quantitative Finance 13 (1), 65–77.Bacry, E., T. Jaisson, and J.-F. Muzy (2016). Estimation of slowly decreasing Hawkes ker-nels: Application to high-frequency order book dynamics. 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The integral (17) can be computedremaining on the real axis but moving the singularity on the negative imaginary axis asillustrated in Figure (4). When x < we can close the integration contour on the upper Re( ω )Im( ω ) o ε Figure 4: Possible integration path when integrand is exp( i πωx ) / ( ω + iε ) .half plane as in Figure (5) and since there is no pole inside the integration path the resultis zero. On the other hand, when x > we can close the contour in the lower half plane asin Figure (6). Since we are running clockwise the result will be: Z dω e − i πωx δ − ( ω ) = i π Z Γ dω e − i πωx ω + iε = i π (cid:2) − πie − i πω ( − iε ) (cid:3) = 1 x > . e( ω )Im( ω ) o ε Figure 5: The integration path when integrand is exp( − i πωx ) / ( ω + iε ) and x < . Re( ω )Im( ω ) o ε Γ Figure 6: The integration path when integrand is exp( − i πωx ) / ( ω + iε ) and x < . B The Shannon Sampling Theorem Let us consider a function c ( x ) whose domain is centered around the origin, i.e. c ( x ) : [ − X c , X c ] → R . Its Fourier transform is defined as ˆ c ( ω ) = Z X c − X c e i πωx c ( x ) dx, and the Fourier Inversion Theorem guarantees that the original function can be written c ( x ) = 12 X c ∞ X n = −∞ ˆ c ( ω n ) e − i πω n x . An immediate consequence is that ˆ c ( ω ) = 12 X c ∞ X n = −∞ ˆ c ( ω n ) Z X c − X c e i π ( ω − ω n ) x dx = 12 X c ∞ X n = −∞ ˆ c ( ω n ) e i π ( ω − ω n ) X c − e − i π ( ω − ω n ) X c i π ( ω − ω n )= ∞ X n = −∞ ˆ c ( ω n ) sin[2 π ( ω − ω n ) X c ]2 π ( ω − ω n ) X c ∞ X n = −∞ ˆ c ( ω n ) sinc [2 π ( ω − ω n ) X c ] . Similarly, for a function z ( x ) defined over a bounded interval I z = { x : X l ≤ x ≤ X h } , we get back to the same case as above by properly shifting the function z , i.e. c ( x ) . = z ( x + X m ) , X m = X h + X l . Hence, knowledge of this next fact ˆ c ( ω n ) = Z X c − X c e i πω n x c ( x ) dx = e − i πω n X m Z X h X l e i πω n x c ( x − X m ) dx = e − i πω n X m ˆ z ( ω n ) makes it not difficult to show that ˆ z ( ω ) = Z X h X l e i πωx z ( x ) dx = Z X c − X c e i πω ( x + X m ) z ( x + X m ) dx = e i πωX m Z X c − X c e i πωx c ( x ) dx = e i πωX m ˆ c ( ω )= ∞ X n = −∞ e i πωX m ˆ c ( ω n ) sinc [2 π ( ω − ω n ) X c ]= ∞ X n = −∞ ˆ z ( ω n ) sinc [2 π ( ω − ω n ) X c ] . C The Modified Hilbert Transform The object of our interest are integrals which take the following form Z sinc [ a ( ω − y )] ω + iε dω = 2 πi H − [ sinc ( ay )] and their solution based on an application of the Modified Hilbert transform of Definition1. Then H − [ sinc ( ay )] = Z sinc ( ax ) δ − ( y − x ) dx = Z Z e − i πωx F [ sinc ( ax )] dωδ − ( y − x ) dx = Z (cid:18) π | a | Z e − i πωx [ − | a | π <ω< | a | π ] dω (cid:19) δ − ( y − x ) dx π | a | Z e − i πωy [ − | a | π <ω< | a | π ] dω Z e + i πω ( y − x ) δ − ( y − x ) dx = π | a | Z e − i πωy θ ( − ω ) [ − | a | π <ω< | a | π ] dω = π | a | Z − | a | π e − i πωy dω = 1 − iy | a | (1 − e iy | a | ) , where we make use of the fact that the Fourier transform of the sinc function complies to F [ sinc ( ax )] = Z e i πωx sin( ax ) ax dx = Z e i πωx sin( ax ) ax − aiε dx = 1 a Z e i πωx e iax − e − iax i ( x − iε ) dx = πa Z e i πωx e iax − e − iax πi ( x − iε ) dx = πa (cid:20) Z e i π ( ω + a π ) x πi ( x − iε ) dx − Z e i π ( ω − a π ) x πi ( x − iε ) dx (cid:21) = πa (cid:20) Z ie − i π ( ω + a π ) x π ( x + iε ) dx − Z ie − i π ( ω − a π ) x π ( x + iε ) dx (cid:21) = πa (cid:20) θ (cid:18) ω + a π (cid:19) − θ (cid:18) ω − a π (cid:19)(cid:21) = π | a | [ − | a | π <ω< | a | π ] . We consequently conclude that our target integral admits solutions of an exponential type Z sinc [ a ( ω − y )] ω + iε dω = πy | a | (1 − e iy | a | ) , and aptly choosing a = 2 πX c and y = ω n finally proves the desired result of Equation (7). D An Explicit Formulation for the CoN Put Price This section derives an explicit formulation of the CoN put price, in terms of sin and cosfunctions multiplying real and imaginary parts of the Fourier transform ˆ f . We have E [ { s T 12 + iπ N/ X n =1 n − (cid:20) e − i πkω n − ˆ f ( ω n − ) − e i πkω n − ˆ f † ( ω n − ) (cid:21) . Properly rearranging terms based on Euler’s formula, we obtain − π N/ X n =1 n − (cid:20) cos(2 πkω n − ) ℑ (cid:2) ˆ f ( ω n − ) (cid:3) − sin(2 πkω n − ) ℜ (cid:2) ˆ f ( ω n − ) (cid:3)(cid:21) . Numerical Moments of q -th Order The computation of the moments of a distribution requires to manage integrals which arenot always ensured to admit a closed form solution. Nevertheless, the knowledge of the CFallows to evaluate them numerically. This fact is of crucial importance when truncatingthe PDF within the SINC method but should be clearly recognized to have a much widerscope. That is why we suppress dependence on s T and talk about a random variable X defined over the support [ − X c , X c ] , in this section.Let us first recall the next fundamental relation between the q -th order moment of X andits CF φ X : E [ X q ] = ( i π ) − q d q d ω q φ X ( ω ) (cid:12)(cid:12)(cid:12)(cid:12) ω =0 then, if we apply the Shannon Sampling Theorem = ( i π ) − q ∞ X n = −∞ φ X ( ω n ) d q d ω q sinc (2 π ( ω − ω n ) X c ) (cid:12)(cid:12)(cid:12)(cid:12) ω =0 and perform a simple change of variable, we have = ( iX c ) q ∞ X n = −∞ φ X ( ω n ) d q d t q sinc ( t ) (cid:12)(cid:12)(cid:12)(cid:12) t = nπ . (18)Furthermore, a power series expansion of the sinc function, i.e.sinc ( t ) = ∞ X n =0 ( − n t n (2 n + 1)! is readily obtained given the corresponding expansion for the sin function, and this clearlyjustifies a number of properties. Among them we have the following:odd derivatives are such thatd q +1 d t q +1 sinc ( t ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = 0 by parity of the sinc function terms of the following typed q +1 d t q +1 sinc ( t ) (cid:12)(cid:12)(cid:12)(cid:12) t = nπ are odd with respect to n q d t q sinc ( t ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = ( − q q + 1 by the theory of Taylor series terms of the following typed q d t q sinc ( t ) (cid:12)(cid:12)(cid:12)(cid:12) t = nπ are even with respect to n These properties play a fundamental role when specifying Equation (18) for some given q .We report the explicit formulation of the first few moments next: m = E [ X ] = − X c ∞ X n =1 ℑ (cid:2) φ X ( ω n ) (cid:3) ( − n nπ ,m = E [ X ] = X c X c ∞ X n =1 ℜ (cid:2) φ X ( ω n ) (cid:3) ( − n ( nπ ) ,m = E [ X ] = − X c ∞ X n =1 ℑ (cid:2) φ X ( ω n ) (cid:3)(cid:20) ( − n nπ (cid:18) − nπ ) (cid:19)(cid:21) ,m = E [ X ] = X c X c ∞ X n =1 ℜ (cid:2) φ X ( ω n ) (cid:3)(cid:20) ( − n ( nπ ) (cid:18) − nπ ) (cid:19)(cid:21) . F Error Analysis (proof ) The overall error ǫ is equal to the sum ǫ + ǫ + ǫ and its norm can be bounded as | ǫ | ≤ ǫ + | ǫ | + | ǫ | . Arguing in the same way as in the COS paper, ǫ can be made arbitrarily small by choosinga sufficiently high value for X c . As far as ǫ is concerned, it is clear from Equation (6) thatit corresponds to the remainder of a series converging to E [ { s T 36o bound the last quantity, we can proceed following two strategies, which are based upondifferent assumptions. We first recall that ˆ f ( ω n − ) − f {− X c ≤ s T ≤ X c } V ( ω n − ) = Z R \ [ − X c ,X c ] f ( s T ) e i πω n − s T ds T . To ensure converge of AoN and Plain Vanilla call prices, for s T >> the PDF f ( s T ) hasto satisfy f ( s T ) ≤ Ce − βs T , with C > and β > . For s T << − , we assume the following condition – typicallysatisfied by commonly used stochastic models for log-returns f ( s T ) ≤ Ce γs T , with γ > . Then, | ǫ | ≤ π N/ X n = − N/ | n − | (cid:12)(cid:12)(cid:12)(cid:12)Z R \ [ − X c ,X c ] f ( s T ) e i πω n − s T ds T (cid:12)(cid:12)(cid:12)(cid:12) ≤ π N/ X n = − N/ | n − | Z R \ [ − X c ,X c ] f ( s T ) ds T ≤ π N/ X n =0 n + 1 Z R \ [ − X c ,X c ] f ( s T ) ds T ≤ π (2 + log( N/ Z R \ [ − X c ,X c ] f ( s T ) ds T ≤ Cπ (2 + log( N/ (cid:18) γ e − γX c + 1 β e − βX c (cid:19) . Naming δ = min( β, γ ) > , we obtain | ǫ | ≤ Cπ (2 + log( N/ e − δX c . To conclude, it is sufficient to choose X c proportional to log( N/ . Practically, thisassumption amounts to choose L proportional to log( N/ in (5). Then, ǫ can be madearbitrarily small by increasing N .An alternative strategy allows to reach the same conclusion, without assuming the depen-dence of X c on N , but under a different hypothesis about the asymptotic behavior of thedensity f ( s T ) . We can split the integral R R \ [ − X c ,X c ] f ( s T ) e i πω n − s T ds T in two terms, I and I , with I ( ω n − ) = Z − X c −∞ f ( s T ) e i πω n − s T ds T and I ( ω n − ) = Z + ∞ X c f ( s T ) e i πω n − s T ds T , 37o that | ǫ | ≤ π N/ X n =1 n − | I ( ω n − ) + I ( ω n − ) | + 1 π N/ X n =1 n + 1 (cid:12)(cid:12)(cid:12) I † ( ω n +1 ) + I † ( ω n +1 ) (cid:12)(cid:12)(cid:12) . (19)Let us consider I ( ω n − ) and define the variable y via the relation s T = y + X c n − . Then, I ( ω n − ) = − Z + ∞ X c − X c / (2 n − e i πω n − y f (cid:18) y + X c n − (cid:19) dy = − Z + ∞ X c e i πω n − y f (cid:18) y + X c n − (cid:19) dy − Z X c X c − X c / (2 n − e i πω n − y f (cid:18) y + X c n − (cid:19) dy . It follows that I ( ω n − ) = 12 Z + ∞ X c e i πω n − y (cid:18) f ( y ) − f (cid:18) y + X c n − (cid:19)(cid:19) dy − Z X c X c − X c / (2 n − e i πω n − y f (cid:18) y + X c n − (cid:19) dy so | I ( ω n − ) | ≤ Z + ∞ X c (cid:12)(cid:12)(cid:12)(cid:12) f ( y ) − f (cid:18) y + X c n − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dy + 12 Z X c X c − X c / (2 n − f (cid:18) y + X c n − (cid:19) dy . We now assume that f ( s T ) is monotonically converging to zero for sufficiently large | s T | .The argument of the modulus is positive, so | I ( ω n − ) | ≤ Z X c ( n − ) X c f ( s T ) ds T + 12 Z X c ( n − ) X c f ( s T ) ds T ≤ X c n − f ( X c ) . Defining s T = y − X c / (2 n − , it readily follows that | I ( ω n − ) | ≤ X c n − f ( − X c ) . Similar results hold for I † ( ω n +1 ) and I † ( ω n +1 ) . From Equation (19), we obtain | ǫ | ≤ X c π ( f ( X c ) + f ( − X c )) N/ X n =1 (cid:18) n − + 1(2 n + 1) (cid:19) X c π ( f ( X c ) + f ( − X c )) N/ − N/ X n =1 n − . The partial sum in the last term converges to a positive constant for N → + ∞ . So, byexpressing P N/ n =1 as P + ∞ n =1 − P n>N/ , we can bound | ǫ | as follows | ǫ | ≤ X c π ( f ( X c ) + f ( − X c )) ( η − QN/ O (cid:18) N/ (cid:19) ) , for suitable constants η and Q . By increasing N , the last two terms converge to zero. Toconclude, it is sufficient to assume the existence of the first moment of s T . Indeed, thisimplies that f ( s T ) = o (1 /s T ) for | s T | → + ∞ . Then, X c f ( X c ) and X c f ( − X c ) can be madearbitrarily small by choosing X cc