Quantization meets Fourier: a new technology for pricing options

Abstract

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.

Appendices

Appendix

A Proof of Theorem 2.5

Proof

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

Proof

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

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

and

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

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

Proof

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.$\square$