Calibration for Weak Variance-Alpha-Gamma Processes

Abstract

The weak variance-alpha-gamma process is a multivariate Lévy process constructed by weakly subordinating Brownian motion, possibly with correlated components with an alpha-gamma subordinator. It generalises the variance-alpha-gamma process of Semeraro constructed by traditional subordination. We compare three calibration methods for the weak variance-alpha-gamma process, method of moments, maximum likelihood estimation (MLE) and digital moment estimation (DME). We derive a condition for Fourier invertibility needed to apply MLE and show in our simulations that MLE produces a better fit when this condition holds, while DME produces a better fit when it is violated. We also find that the weak variance-alpha-gamma process exhibits a wider range of dependence and produces a significantly better fit than the variance-alpha-gamma process on a S&P500-FTSE100 data set, and that DME produces the best fit in this situation.

Appendix

Lévy Process

The reader is referred to the monographs Applebaum (2009), Bertoin (1996) and Sato (1999) for necessary material on Lévy processes, to Ballotta and Bonfiglioni (2016), Cariboni and Schoutens (2009), Cont and Tankov (2004) for financial applications, while our notation follows Buchmann et al. (2017a) and Buchmann et al. (2017b). For $A\subseteq \mathbb {R}^{n}$, let $A_{*}:=A\backslash \{\textbf {0}\}$ and let $\mathbf {1}_{A}(\omega )$ denote the indicator function. Let $\mathbb {D}:=\{\mathbf {x}\in \mathbb {R}^{n}:\|\mathbf {x}\|\le 1\}$ be the Euclidean unit ball centred at the origin. The law of an n-dimensional Lévy process $X=(X_{1},\dots ,X_{n})=(X(t))_{t\ge 0}$ is determined by its characteristic function ${\Phi }_{X}:={\Phi }_{X(1)}$, with

$${\Phi}_{X(t)}(\boldsymbol{\theta}) := \mathbb{E}\exp(\mathrm{i}\langle\boldsymbol{\theta},{X(t)}\rangle) = \exp(t{\Psi}_{X}(\boldsymbol{\theta})) , \qquad t\!\ge\!0,$$

and Lévy exponent ${\Psi }_{X}:={\Psi }$, where

$$ {\Psi}(\boldsymbol{\theta}) := \mathrm{i} \langle {\boldsymbol{\mu}},\boldsymbol{\theta}\rangle - \frac{1}{2} \|\boldsymbol{\theta}\|^{2}_{\Sigma} +{\int}_{\mathbb{R}^{n}_{*}}\left( e^{\mathrm{i}\langle\boldsymbol{\theta}, \mathbf{x}\rangle} - 1 - \mathrm{i}\langle\boldsymbol{\theta}, \mathbf{x}\rangle \mathbf{1}_{\mathbb{D}}(\mathbf{x})\right) \mathcal{X}(\mathrm{d} \mathbf{x}) , $$

$\boldsymbol {\theta }\in \mathbb {R}^{n}$, $\boldsymbol {\mu }\in \mathbb {R}^{n}$, ${\Sigma }\in \mathbb {R}^{n\times n}$ is a covariance matrix, and $\mathcal {X}$ is a nonnegative Borel measure on $\mathbb {R}^{n}_{*}$ such that ${\int }_{\mathbb {R}^{n}_{*}}(\|\mathbf {x}\|^{2}\wedge 1) \mathcal {X}(\mathrm {d} \mathbf {x})<\infty$. We write $X\sim L^{n}(\boldsymbol {\mu },{\Sigma },\mathcal {X})$ provided X is an n-dimensional Lévy process with canonical triplet $(\boldsymbol {\mu },{\Sigma },\mathcal {X})$.

A subordinator $T\!\sim \!S^{n}(\mathcal {T}) = L^{n}(\boldsymbol {\mu },0,\mathcal {T})$ is drift-less if its drift $\boldsymbol {\mu } - {\int }_{\mathbb {D}_{*}} \mathbf {t} \mathcal {T}(\mathrm {d} \mathbf {t})=\textbf {0}$. All subordinators considered in this paper are drift-less.

An n-dimensional random variable X is self-decomposable if for any $0\!<\!b\!<\!1$, there exists a random variable $Z_{b}$, independent of X, such that $X\stackrel {D}{=} bX+Z_{b}$. A Lévy process X is self-decomposable if $X(1)$ is.

Strongly Subordinated Brownian Motion

Let $B=(B_{1},\dots ,B_{n})\sim BM^{n}(\boldsymbol {\mu },{\Sigma })$ be a Brownian motion and $T = (T_{1},\dots ,T_{n})\!\sim \!S^{n}(\mathcal {T})$ be a drift-less subordinator. A process $B\circ T$ is the traditional or strong subordination ofXand T if $(B\circ T)(t):=(B_{1}(T_{1}(t)),\dots ,B_{n}(T_{n}(t)))$, $t\ge 0$.

Weakly Subordinated Brownian Motion

Let $\mathbf {t} = (t_{1},{\dots } t_{n})\!\in \![0,\infty )^{n}$, $\boldsymbol {\mu } = (\mu _{1},{\dots } \mu _{n})\!\in \!\mathbb {R}^{n}$ and ${\Sigma } = ({\Sigma }_{kl})\!\in \!\mathbb {R}^{n\times n}$ be a covariance matrix. Introduce the outer products $\mathbf {t}\diamond \boldsymbol {\mu }\in \mathbb {R}^{n}$ and $\mathbf {t}\diamond {\Sigma }\!\in \!\mathbb {R}^{n\times n}$ by

$$ \mathbf{t}\diamond\boldsymbol{\mu}:=(t_{1}\mu_{1},\dots,t_{n}\mu_{n})\quad\text{and}\quad(\mathbf{t}\diamond {\Sigma})_{kl}:={\Sigma}_{kl}(t_{k}\wedge t_{l}),\quad 1\le k,l\le\!n. $$

(A.1)

Let $B\!\sim \!BM^{n}(\boldsymbol {\mu },{\Sigma })$ be an n-dimensional Brownian motion and $T\!\sim \!S^{n}(\mathcal {T})$ be an n-dimensional drift-less subordinator. A Lévy process $B\odot T$ is called the weak subordination ofBand T (see Buchmann et al. 2017b, their Proposition 3.1) if it has Lévy exponent

$$ {\Psi}_{B\odot T}(\boldsymbol{\theta})={\int}_{[0,\infty)^{n}_{*}}\left( \exp\left( \mathrm{i}\langle\boldsymbol{\theta},{\mathbf{t}\diamond\boldsymbol{\mu}}\rangle-\frac{1}{2}\|\boldsymbol{\theta}\|^{2}_{\mathbf{t}\diamond{\Sigma}}\right)-1\right) \mathcal{T}(\mathrm{d}\mathbf{t}) , $$

(A.2)

$\boldsymbol {\theta }\in \mathbb {R}^{n}$. Note that a more general definition of weak subordination and a proof of existence is given in Buchmann et al. (2017b).

For independent B and T, $B\circ T\stackrel {D}{=} B\odot T$ if T has indistinguishable components or B has independent components, otherwise $B\circ T$ may not a Lévy process, but $B\odot T$ always is (see Buchmann et al. 2017b, their Proposition 3.3 and 3.9).