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Quantization meets Fourier: a new technology for pricing options


In this paper we introduce a novel pricing methodology for a broad class of models for which the characteristic function of the log-asset price can be efficiently computed. The method is based on a new quantization procedure, crucially exploiting for the first time the Fourier transform of the asset process, which fully characterizes the distribution of the log-asset. As opposed to previous quantizations based on Euler (or more sophisticated) discretization schemes, our method reveals to be fast and accurate, to the point that it is possible to calibrate the models on real data. Moreover, our approach allows to price options in multi factor stochastic volatility models including jumps. As a motivating example, we calibrate a Tempered Stable model on market data. This represents the first application of quantization to a pure jump process.

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  1. The quantization problem in infinite dimension, namely when \(\mathbb R^d\) is replaced by a separable Hilbert space or, more generally, by a separable Banach space \((E, \vert \cdot \vert _E)\) (think of E as, e.g., the functional spaces \(L^2_{\mathbb R}([0,T], \lambda )\) or \(\mathcal C([0,T], \mathbb R)\), with \(\lambda \) denoting the Lebesgue measure on \((\mathbb R, \mathcal B(\mathbb R)))\) is known as functional quantization (see e.g. Pagès and Luschgy (2002) for the quantization of stochastic processes viewed as \(L^2([0,1], \lambda )\)-valued random vectors and Pagès and Printems (2005) for an application to finance).

  2. For ease of notation, hereafter we drop N in the notation \(\varGamma ^N\) as we consider a fixed number of points in the grid.

  3. We also mention the optimal quantization website:, where one can download the optimized quadratic quantization grids of the d-dimensional Gaussian distributions \(\mathcal N(0; I_d)\), for \(N = 1\) up to \(10^4\) and for \(d = 1, \dots , 10\). Moreover, at the same link one can also find functional quantization grids of the standard Brownian motion over the interval [0; 1], of the Brownian bridge, as well as a detailed procedure to compute grids for the (normalized) Ornstein-Uhlenbeck process and its bridge.

  4. The function \(\bar{\beta }(x,a,b)\) can be written as

    $$\begin{aligned} \bar{\beta }(x,a,b)&= \beta (a,b) - \beta (x,a,b) = \int _{0}^{1} t^{a-1} (1-t)^{b-1} dt - \int _{0}^{x} t^{a-1} (1-t)^{b-1} dt, \end{aligned}$$

    where \(\beta (a,b)\) (resp. \(\beta (x,a,b)\)) is the complete (resp. incomplete) Euler beta function, see e.g. Abramowitz and Stegun (1970). Therefore, \(\bar{\beta }(x,a,b)\) can be denoted as complementary incomplete Euler beta function. The complete Euler beta function is defined only when \(\mathrm {Re}(a)> 0 , \mathrm {Re}(b)>0\), because otherwise the integral would diverge. Nevertheless, its definition can be extended, by regularization, to negative values of \(\mathrm {Re}(a)\) and \(\mathrm {Re}(b)\), see Ozcag et al. (2008) for a comprehensive study of these special functions. From an implementation perspective, softwares like e.g. Wolfram System already include a regularized version of the Beta and incomplete Beta functions defined for most arguments, by taking into account singular cases.

  5. An important exception is the case when the density is log-concave, for which the Fourier quantization algorithm allows to determine quantizers that are also optimal, see Remark 1.

  6. For sake of brevity we omit the calibration for the other models, for which results are available upon request.

  7. Of course, also the initial variance \(V_0\) should be considered as a parameter to be estimated/calibrated, as it is unobservable.

  8. Note that taking \(a=0\) and \(b\not =0\) corresponds to the 3 / 2 model, for which the mean reversion speed is proportional to the level of the instantaneous variance, contrarily to the Heston (1993) model that has a constant speed. See Grasselli (2016) for further details.

  9. Tempered stable distributions form a six parameter family of infinitely divisible distributions. They include several well-known subclasses like variance gamma distributions of Madan and Milne (1991), bilateral gamma distributions of Kuchler and Tappe (2008) and the CGMY distributions of Carr et al. (2002).

  10. The density \(f_h(g)\) of the gamma increment \(g=\gamma (t+h; \mu , nu) - \gamma (t;\mu ,\nu )\) is given by the gamma density function with mean \(\mu h\) and variance \(\nu h\). See e.g. Madan et al. (1998) and Cont et al. (1997) for background on gamma processes.

  11. As usual, Call (resp. Put) prices are provided when the strike is less (resp. greater) than 100.

  12. We did not perform any optimization in the code. On top of that, we guess that the computational time can be even reduced by migrating to other languages like e.g. \(C{\texttt {++}}\).

  13. The calibration can be easily performed for each of the six models presented, we skip the results (available upon request) for the other models for the sake of brevity. The choice of the dataset is also arbitrary: we tested several dates by getting similar results.


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We are grateful to Alexander Antonov, Mark Joshi, Daniele Marazzina, Gilles Pagès, Andrea Pallavicini, Abass Sagna, Antonino Zanette and the participants to the Quantitative Methods in Finance congress (Sydney, December 2016), SIAM Conference on Financial Mathematics and Engineering (Austin, November 2016) for useful discussions.

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Correspondence to Lucio Fiorin.



A Proof of Theorem 2.5


Let us fix the index j and rewrite the distortion function in Eq. (10) by emphasizing in the summation the four components depending on the variable \(x_{j}\):

$$\begin{aligned} D_p(\varGamma ) =&(\text {terms not depending on} \ x_j) + \int _{x_{j-1}}^{x_{j}^{-}} (z - x_{j-1})^{p} d\mathbb {P}_{S_{T}}(z) + \int _{x_{j}^{-}}^{x_{j}} (x_{j} - z)^{p} d\mathbb {P}_{S_{T}}(z) \\&+ \int _{x_{j}}^{x_{j}^{+}} (z - x_{j})^{p} d\mathbb {P}_{S_{T}}(z) + \int _{x_{j}^{+}}^{x_{j+1}} (x_{j+1} - z )^{p} d\mathbb {P}_{S_{T}}(z). \end{aligned}$$

We then derive with respect to \(x_{j}\), recalling the definition of the density f in (8):

$$\begin{aligned} \frac{\partial D_p(\varGamma )}{\partial x_{j}} =&\frac{1}{2} (x_{j}^{-} - x_{j-1})^{p} f(x_{j}^{-}) - \frac{1}{2} (x_{j} - x_{j}^{-})^{p} f(x_{j}^{-}) + \int _{x_{j}^{-}}^{x_{j}} p (x_{j} - z)^{p-1} f(z) dz\\&+ \frac{1}{2} (x_{j}^{+} - x_{j})^{p} f(x_{j}^{+}) - \int _{x_{j}}^{x_{j}^{+}} p (z - x_{j})^{p-1} f(z) dz - \frac{1}{2}(x_{j+1} - x_{j}^{+} )^{p} f(x_{j}^{+}) \\ =&\frac{1}{2} \left( \frac{x_{j} - x_{j-1}}{2}\right) ^{p} f(x_{j}^{-}) - \frac{1}{2} \left( \frac{x_{j} - x_{j-1}}{2}\right) ^{p} f + \int _{x_{j}^{-}}^{x_{j}} p (x_{j} - z)^{p-1} f(z) dz\\&+ \frac{1}{2} \left( \frac{x_{j+1} - x_{j}}{2}\right) ^{p} f(x_{j}^{+}) - \int _{x_{j}}^{x_{j}^{+}} p (z - x_{j})^{p-1} f(z) dz \ - \frac{1}{2}\left( \frac{x_{j+1} - x_{j}}{2} \right) ^{p} f(x_{j}^{+}) \\ =&\int _{x_{j}^{-}}^{x_{j}} p (x_{j} - z)^{p-1} \left( \frac{1}{\pi } \frac{1}{z} \left( \int _{0}^{+ \infty } \mathrm {Re} \left[ \phi _{T}(u) e^{- i u \log (z)} \right] du \right) \right) dz \\&- \int _{x_{j}}^{x_{j}^{+}} p (z - x_{j})^{p-1} \left( \frac{1}{\pi } \frac{1}{z} \left( \int _{0}^{+ \infty } \mathrm {Re} \left[ \phi _{T}(u) e^{- i u \log (z)} \right] du \right) \right) dz \\ =&\frac{p}{\pi } \int _{0}^{+ \infty } \mathrm {Re} \left[ \phi _{T}(u) \left( \int _{x_{j}^{-}}^{x_{j}} (x_{j} - z)^{p-1} z^{-1 - i u} dz - \int _{x_{j}}^{x_{j}^{+}} (z - x_{j})^{p-1} z^{-1 - iu} dz \right) \right] du, \end{aligned}$$

where we used the Eq. (8). Now we study separately the two last integrals. For the first one we use the change of variable \(t = z/x_{j}\) to get

$$\begin{aligned} \int _{x_{j}^{-}}^{x_{j}} (x_{j} - z)^{p-1} z^{-1 - i u} dz =&\,\, x_{j}^{p-1 - i u } \int _{\frac{x_{j}^{-}}{x_{j}}}^{1} \left( 1 -t \right) ^{p-1} t ^{-1 - i u} dt \\ =&\, x_{j}^{p-1 - i u } \bar{\beta }\left( \frac{x_{j}^{-}}{x_{j}}, - i u , p \right) , \end{aligned}$$

where the function \(\bar{\beta }\) is defined in (12), while for the second integral we use \(1/t=z/x_j\) and we find

$$\begin{aligned} \int _{x_{j}}^{x_{j}^{+}} (z - x_{j})^{p-1} z^{-1 - iu} dz =&\, x_{j}^{p-1 - i u } \int _{1}^{\frac{x_{j}}{x_{j}^{+}}} \left( \frac{1}{t} - 1 \right) ^{p-1} \left( \frac{1}{t} \right) ^{-1 - i u} \left( - \frac{1}{t^{2}} \right) dt \\ =&\,x_{j}^{p-1 - i u } \bar{\beta } \left( \frac{x_{j}}{x_{j}^{+}},1 - p + i u, p \right) . \end{aligned}$$

Summing up the two expressions we get

$$\begin{aligned} \frac{\partial D_p(\varGamma )}{\partial x_{j}}=\frac{p}{\pi } x_{j}^{p-1} \int _{0}^{+ \infty } \mathrm {Re} \left[ \phi _{T}(u) e^{- i u \log (x_{j})} \left( \bar{\beta }\left( \frac{x_{j}^{-}}{x_{j}}, -i u, p \right) - \bar{\beta }\left( \frac{x_{j}}{x_{j}^{+}},1-p+ i u,p \right) \right) \right] du , \end{aligned}$$

which gives the result.\(\square \)

B Proof of Theorem 2.6


First of all, notice that we can rewrite \(\frac{\partial D_p(\varGamma )}{\partial x_{i}}\) as seen in the proof of Theorem 1:

$$\begin{aligned} \frac{\partial D_p(\varGamma )}{\partial x_{j}}&=\int _{x_{j}^{-}}^{x_{j}} p (x_{j} - z)^{p-1} f(z) dz - \int _{x_{j}}^{x_{j}^{+}} p (z - x_{j})^{p-1} f(z) dz. \end{aligned}$$

We now compute the upper and lower diagonal components. Noting that \(x_{j+1}\) and \(x_{j-1}\) appear only in the endpoints of the interval of integration, we easily have that

$$\begin{aligned} \frac{\partial ^{2} D_p(\varGamma ) }{\partial x_{j} \partial x_{j+1}}&= - \frac{1}{2} p (x_{j}^{+} - x_{j})^{p-1} f(x_{j}^{+}) \\&= - \frac{p}{2 \pi } \frac{1}{x_{j}^{+}} \left( \frac{x_{j+1} - x_{j}}{2} \right) ^{p-1} \int _{0}^{+ \infty } \mathrm {Re} \left[ \phi _{T}(u) e^{- i u \log (x_{j}^{+})} \right] du, \end{aligned}$$


$$\begin{aligned} \frac{\partial ^{2} D_p(\varGamma ) }{\partial x_{j} \partial x_{j-1}}&= - \frac{1}{2} p (x_{j} - x_{j}^{-})^{p-1} f(x_{j}^{-}) \\&= - \frac{p}{2 \pi } \frac{1}{x_{j}^{-}} \left( \frac{x_{j} - x_{j-1}}{2} \right) ^{p-1} \int _{0}^{+ \infty } \mathrm {Re} \left[ \phi _{T}(u) e^{- i u \log (x_{j}^{-})} \right] du. \end{aligned}$$

Let us now consider the main diagonal:

$$\begin{aligned} \frac{\partial ^{2} D_p(\varGamma ) }{\partial x_{j} \partial x_{j}}=&- \frac{1}{2} p (x_{j}^{+} - x_{j})^{p-1} f(x_{j}^{+}) + \int _{x_{j}^{-}}^{x_{j}} p (p - 1) (x_{j} - z)^{p-2} f(z) dz \\&+ \int _{x_{j}}^{x_{j}^{+}} p (p - 1) (z - x_{j})^{p-2} f(z) dz - \frac{1}{2} p (x_{j} - x_{j}^{-})^{p-1} f(x_{j}^{-}) \\ =&\,\, p (p - 1) \left( \int _{x_{j}^{-}}^{x_{j}} (x_{i} - z)^{p-2} f(z) dz + \int _{x_{j}}^{x_{j}^{+}} (z - x_{j})^{p-2} f(z) dz\right) \\&+ \frac{\partial ^{2} D_p(\varGamma ) }{\partial x_{j} \partial x_{j+1}} + \frac{\partial ^{2} D_p(\varGamma ) }{\partial x_{j} \partial x_{j-1}}. \end{aligned}$$

Note that the integrals in the last expression are similar to the ones encountered in the proof of Theorem 1 when replacing the exponent \(p-2\) with \(p-1\), so the same computations lead to the result.\(\square \)

C Proof of Theorem 2.11


The proof adapts to our context the arguments of Graf and Luschgy (2000) and Cambanis and Gerr (1983) and will be developed in several steps.

Step 0. As a preliminary step, we observe that from

$$\begin{aligned} f(z) \sim z^{-\alpha } \quad \text {when } z \rightarrow +\infty \end{aligned}$$

and the fact that \(S_T\) has finite \(p-th\) moment, it follows that \(\alpha > p+1\). Similarly, the polynomial behavior at 0

$$\begin{aligned} f(z) \sim z^{\gamma } \quad \text {when } z \rightarrow 0 \end{aligned}$$

and \(\mathbb {E}\left[ S_T^{p} \right] < + \infty \) implies that \(\gamma > -(1 + p)\). These conditions together guarantee that \( || f ||_{\frac{1}{p+1}} <+ \infty \).

Step 1 Now we develop the expression of the distortion function in the Voronoï partition as follows:

$$\begin{aligned} D_{N}\left( x_{1}, \dots , x_{N} \right) =&\sum _{i=1}^{N} \int _{C_{i}} |z - x_{i}|^{p} f(z) dz \\ =&\displaystyle \int _{0}^{ x_{1} } ( x_{1} - z)^{p} f(z) dz + \int _{x_{1} }^{\frac{x_{1} + x_{2}}{2} } (z - x_{1})^{p} f(z) dz \\&+ \int _{\frac{x_{1} + x_{2}}{2}}^{x_{2} } (x_{2} - z)^{p} f(z) dz + \int _{x_{2}}^{\frac{x_{2} + x_{3}}{2} } (z - x_{2})^{p} f(z) dz + \dots \\&+ \int _{\frac{x_{N-2} + x_{N-1}}{2}}^{x_{N-1} } (x_{N-1} - z)^{p} f(z) dz + \int _{ x_{N-1}}^{\frac{x_{N-1} + x_{N}}{2} } (z - x_{N-1})^{p} f(z) dz \\&+ \int _{\frac{x_{N-1} + x_{N}}{2}}^{x_{N} } (x_{N} - z)^{p} f(z) dz + \int _{x_{N}}^{+ \infty } (z - x_{N})^{p} f(z) dz. \end{aligned}$$

Let \(\xi _{1}=\underset{z \in \left[ x_{1}, \frac{ x_{1} + x_{2}}{2} \right] }{{\text {argmax}}} f(z)\), \(\xi _{i}=\underset{z \in C_{i}}{{\text {argmax}}} f(z)\), for \(i=2, \dots , N-1\), and \(\xi _{N}=\underset{z \in \left[ \frac{x_{N-1} + x_{N}}{2}, x_{N} \right] }{{\text {argmax}}} f(z)\), then

$$\begin{aligned} D_{N}\left( x_{1}, \dots , x_{N} \right) \le&\displaystyle \int _{0}^{ x_{1} } (x_{1}- z )^{p} f(z) dz + f(\xi _{1}) \int _{x_{1}}^{\frac{x_{1} + x_{2}}{2} } (z - x_{1})^{p} dz + f(\xi _{2}) \int _{\frac{x_{1} + x_{2}}{2}}^{x_{2}} ( x_{2} - z )^{p} dz \\&+ f(\xi _{2}) \int _{x_{2}}^{\frac{x_{2} + x_{3}}{2}} ( z - x_{2})^{p} dz + \dots +f(\xi _{N-1}) \int _{\frac{x_{N-2} + x_{N-1}}{2}}^{x_{N-1} } (x_{N-1} - z)^{p} dz\\&+ f(\xi _{N-1}) \int _{x_{N-1}}^{\frac{x_{N-1} + x_{N}}{2} } (z - x_{N-1})^{p} dz + f(\xi _{N}) \int _{\frac{x_{N-1} + x_{N}}{2}}^{x_{N} } (z - x_{N})^{p} dz \\&+ \int _{x_{N}}^{+ \infty } (z - x_{N})^{p} f(z) dz\\ =&\int _{0}^{ x_{1} } (x_{1} - z)^{p} f(z) dz + \sum _{i=1}^{N-1} \frac{ f(\xi _{i})+f(\xi _{i+1})}{p+1} \left( \frac{x_{i+1} - x_{i }}{2}\right) ^{p+1} \\&+ \int _{x_{N}}^{+ \infty } (z - x_{N})^{p} f(z) dz. \end{aligned}$$

Step 2 Let us define a quantization grid \((\bar{x}_{1}, \dots , \bar{x}_{N})\) such that

$$\begin{aligned} \frac{ \displaystyle \int _{0}^{\bar{x}_{i}} f^{\frac{1}{p+1}} (z) dz }{ \displaystyle || f ||^{\frac{1}{p+1}}_{\frac{1}{p+1}}} = \frac{ \displaystyle \int _{0}^{\bar{x}_{i}} f^{\frac{1}{p+1}} (z) dz }{\displaystyle \int _{0}^{+ \infty } f^{\frac{1}{p+1}} (z) dz} = \frac{2i -1 }{2 N}, \quad \text {for } i=1, \dots , N. \end{aligned}$$

This definition is well posed from Step 0. We have that, for \(i=1, \dots , N-1\)

$$\begin{aligned} \displaystyle \int _{\bar{x}_{i}}^{\bar{x}_{i+1}} f^{\frac{1}{p+1}} (z) dz = \frac{ || f ||^{\frac{1}{p+1}}_{\frac{1}{p+1}}}{N}, \end{aligned}$$

and, using the mean value theorem (notice that f is continuous), we obtain that for \(i = 1, \dots , N-1\) there exists \(\zeta _{i} \in \left[ \bar{x}_{i} , \bar{x}_{i+1} \right] \) such that

$$\begin{aligned} \left( \bar{x}_{i+1} - \bar{x}_{i} \right) ^{p} = \frac{ || f ||^{\frac{p}{p+1}}_{\frac{1}{p+1}}}{ f^{\frac{p}{p+1}} (\zeta _{i} ) N^p}, \end{aligned}$$

where the denominator is always bounded away from zero.

Step 3 In this step we provide a bound for the quantization error. We define

$$\begin{aligned} E_{N}(x_{1}, \dots , x_{N}): = \sum _{i=1}^{N-1} \frac{ f(\xi _{i})+f(\xi _{i+1})}{p+1} \left( \frac{x_{i+1} - x_{i }}{2}\right) ^{p+1} , \end{aligned}$$

so that

$$\begin{aligned} D_{N}(x_{1}, \dots , x_{N}) \le E_{N}(x_{1}, \dots , x_{N}) + \int _{0}^{x_{1}} (x_{1} - z)^{p} f(z) dz + \int _{x_{N}}^{+ \infty } (z - x_{N})^{p} f(z) dz. \end{aligned}$$

We now use the quantization grid \((\bar{x}_{1}, \dots , \bar{x}_{N})\) defined in Step 2. Let us define, in a similar way as in Step 1, \(\bar{\xi }_{1}=\underset{z \in \left[ \bar{x}_{1}, \frac{ \bar{x}_{1} + \bar{x}_{2}}{2} \right] }{{\text {argmax}}} f(z)\), \(\bar{\xi }_{i}=\underset{z \in \left[ \frac{\bar{x}_{i-1} + \bar{x}_{i}}{2}, \frac{\bar{x}_{i} + \bar{x}_{i+1}}{2} \right] }{{\text {argmax}}} f(z)\), for \(i=2, \dots , N-1\), and \(\bar{\xi }_{N}=\underset{z \in \left[ \frac{\bar{x}_{N-1} + \bar{x}_{N}}{2}, x_{N} \right] }{{\text {argmax}}} f(z)\), then:

$$\begin{aligned} E_{N}(\bar{x}_{1}, \dots , \bar{x}_{N})&= \sum _{i=1}^{N-1} \frac{ f(\bar{\xi }_{i})+f(\bar{\xi }_{i+1})}{2^{p+1}(p+1)} \left( \bar{x}_{i+1} - \bar{x}_{i }\right) ^{p} \left( \bar{x}_{i+1} - \bar{x}_{i }\right) + \\&= \frac{ || f ||^{\frac{p}{p+1}}_{\frac{1}{p+1}}}{ 2^{p+1}(p+1) N^p} \sum _{i=1}^{N-1} \frac{ f(\bar{\xi }_{i})+f(\bar{\xi }_{i+1})}{ f^{\frac{p}{p+1}} (\zeta _{i} ) } \left( \bar{x}_{i+1} - \bar{x}_{i }\right) . \end{aligned}$$

By definition of the quantization error \(e_{p,N}(S_T,\varGamma )\), we have that

$$\begin{aligned} \left( e_{p,N}(S_T,\varGamma ) \right) ^{p}\le D_{N}(\bar{x}_{1}, \dots , \bar{x}_{N}) \le&E_{N}(\bar{x}_{1}, \dots , \bar{x}_{N}) \\&+ \int _{0}^{\bar{x}_{1}} ( \bar{x}_{1} - z)^{p} f(z) dz + \int _{\bar{x}_{N}}^{+ \infty } (z - \bar{x}_{N})^{p} f(z) dz. \end{aligned}$$

In the next step we study the asymptotic behavior of \(e_{p,N}\) when \(N \rightarrow + \infty \).

Step 4 Note that when \(N \rightarrow + \infty \), \(\bar{x}_{1} \rightarrow 0\) and \(\bar{x}_{N} \rightarrow +\infty \), so that

$$\begin{aligned} \sum _{i=1}^{N-1} \frac{ f(\bar{\xi }_{i})+f(\bar{\xi }_{i+1})}{ f^{\frac{p}{p+1}} (\zeta _{i} ) } \left( \bar{x}_{i+1} - \bar{x}_{i }\right)&\rightarrow 2 \int _{0}^{+ \infty } f^{\frac{1}{p+1}}(z) dz = 2 || f ||_{\frac{1}{p+1}}^{\frac{1}{p+1}}. \end{aligned}$$

Now we show that the integrals \(\displaystyle \int _{\bar{x}_{N}}^{+ \infty } (z - \bar{x}_{N})^{p} f(z) dz\) and \(\displaystyle \int _{\bar{x}_{N}}^{+ \infty } (z - \bar{x}_{N})^{p} f(z) dz\) are of order \(\displaystyle o\left( \frac{1}{N^p} \right) \). For the first one we recall that, from Step 2,

$$\begin{aligned} \displaystyle \int _{\bar{x}_{N}}^{+ \infty } f^{\frac{1}{p+1}} (z) dz = \frac{ || f ||^{\frac{1}{p+1}}_{\frac{1}{p+1}}}{2 N}, \end{aligned}$$

so that

$$\begin{aligned} \lim _{N \rightarrow +\infty } N^p \int _{\bar{x}_{N}}^{+ \infty } (z - \bar{x}_{N})^{p} f(z) dz&= \frac{2^{p} }{|| f ||_{\frac{1}{p+1}}^{\frac{p}{p+1}}} \lim _{N \rightarrow +\infty } \frac{\displaystyle \int _{\bar{x}_{N}}^{+ \infty } (z - \bar{x}_{N})^{p} f(z) dz}{\left( \displaystyle \int _{\bar{x}_{N}}^{+ \infty } f^{\frac{1}{p+1}}(z) dz \right) ^{p} }, \end{aligned}$$

therefore we are reduced to the study of

$$\begin{aligned} \lim _{y \rightarrow +\infty } \frac{\displaystyle \int _{y}^{+ \infty } (z - y)^{p} f(z) dz}{\left( \displaystyle \int _{y}^{+ \infty } f^{\frac{1}{p+1}}(z) dz \right) ^{p} } . \end{aligned}$$

From Step 0 we have that \(f(z) \sim z^{- \alpha }\) when \(z \rightarrow +\infty \) for \(\alpha >p+1\), then

$$\begin{aligned} \lim _{y \rightarrow +\infty } \frac{\displaystyle \int _{y}^{+ \infty } (z - y)^{p} f(z) dz}{ \left( \displaystyle \int _{y}^{+ \infty } f^{\frac{1}{p+1}}(z) dz \right) ^{p} }&= \lim _{y \rightarrow +\infty } \frac{\displaystyle \int _{y}^{+ \infty } (z - y)^{p} z^{-\alpha } dz}{\left( \displaystyle \frac{y^{- \frac{\alpha }{p+1} + 1} }{ \frac{\alpha }{p+1} - 1}\right) ^{p} }. \end{aligned}$$

Let us focus on the numerator:

$$\begin{aligned} \displaystyle \int _{y}^{+ \infty } (z - y)^{p} z^{-\alpha } dz&\overset{ z = \frac{y}{t} }{=} y^{p+1-\alpha } \int _{0}^{1 } \left( 1 - t\right) ^{p} t ^{\alpha - 2 - p} dt \\&= y^{p+1-\alpha } \beta (\alpha - 1 - p,p+1), \end{aligned}$$

where the Beta function is well defined here as both \(\alpha - 1 - p\) and \(p+1 \) are positive. In conclusion we get

$$\begin{aligned} \lim _{N \rightarrow +\infty } N^p \int _{\bar{x}_{N}}^{+ \infty } (z - \bar{x}_{N})^{p} f(z) dz&=0, \end{aligned}$$

that is \(\displaystyle \int _{\bar{x}_{N}}^{+ \infty } (z - \bar{x}_{N})^{p} f(z) dz\) is \(\displaystyle o\left( \frac{1}{N^p} \right) \). The second integral \(\displaystyle \int _{\bar{x}_{N}}^{+ \infty } (z - \bar{x}_{N})^{p} f(z) dz\) can be handled similarly using the fact that f(z) behaves as \(z^{\gamma }\) at 0 with \(\gamma > - 1\).

Step 5 Using the results in Step 4, we have that, when \(N \rightarrow + \infty \),

$$\begin{aligned} \left( e_{p,N}(S_T,\varGamma ) \right) ^{p} \le D_{N}(\bar{x}_{1}, \dots , \bar{x}_{N})&\le \frac{|| f ||^{\frac{p}{p+1}}_{\frac{1}{p+1}}}{2^{p+1}(p+1)} \frac{2 || f ||_{\frac{1}{p+1}}^{\frac{1}{p+1}} }{N^p} + o\left( \frac{1}{N^p} \right) + o\left( \frac{1}{N^p} \right) \\&\sim \frac{|| f ||_{\frac{1}{p+1}}}{2^{p} (p+1)} \frac{1}{N^p}, \end{aligned}$$

and the theorem is proved.\(\square \)

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Callegaro, G., Fiorin, L. & Grasselli, M. Quantization meets Fourier: a new technology for pricing options. Ann Oper Res 282, 59–86 (2019).

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