Towards a $\Delta$-Gamma Sato multivariate model

Abstract

The increased trading in multi-name financial products has paved the way for the use of multivariate models that are at once computationally tractable and flexible enough to mimic the stylized facts of asset log-returns and of their dependence structure. In this paper we propose a new multivariate Lévy model, the so-called $\varDelta$-Gamma model, where the log-price gains and losses are modeled by separate multivariate Gamma processes, each containing a common and an idiosyncratic component. Furthermore, we extend this multivariate model to the Sato setting, allowing for a moment term structure that is more in line with empirical evidence. We calibrate the two models on single-name option price surfaces and market implied correlations and we show how the $\varDelta$-Gamma Sato model outperforms its Lévy counterpart, especially during periods of market turmoil. The numerical study also reveals the advantages of these new types of multivariate models, compared to a multivariate VG model.

A Maximal attainable correlation: \(\varDelta \)-VG model versus \(\rho \alpha \)-VG model

Let ${\mathbf {B}}(s)$ be a multi-parameter Brownian motion corresponding to a Brownian motion with independent components and Lévy triplet $(\varvec{\mu }, \varvec{\varSigma }, {\mathbf {0}})$ with $\varvec{\varSigma }= diag(\sigma _1^2,\ldots , \sigma _n^2)$, $\varvec{\mu }= (\mu _1,\ldots ,\mu _n)$ and let $\overline{{\mathbf {B}}}_t$ be a multivariate Brownian motion with correlations $\rho ^B_{jk}$, $j,k=1,\ldots ,n$ and Lévy triplet $(\varvec{\mu }^{\rho }, \varvec{\varSigma }^{\rho }, {\mathbf {0}})$ with $\varvec{\mu }^{\rho } = (\mu _1\nu _1,\ldots ,\mu _n\nu _n)$ and $\varvec{\varSigma }^{\rho } = (\rho ^B_{jk}\sigma _j\sigma _k\sqrt{\nu _j\nu _k})_{j,k=1}^n$ with $\rho ^B_{jj}=1$. Under the $\rho \alpha$-VG model of Luciano et al. (2016) the n-dimensional asset log-returns are modeled by the factor-based subordinated Brownian motion

$$\begin{aligned} {\mathbf {Y}}_t = \begin{pmatrix} B^{(1)}\left( X^{(1)}_t\right) +{\overline{B}}^{(1)}(Z_t)\\ \vdots \\ B^{(n)}\left( X^{(n)}_t\right) +{\overline{B}}^{(n)}(Z_t) \end{pmatrix}, \end{aligned}$$

where $X^{(j)}_t$ and $Z_t$ are independent subordinators, independent of ${\mathbf {B}}_t$ and $\overline{{\mathbf {B}}}_t$, with law at unit time $X^{(j)}_1 \sim$ Gamma$(\frac{1}{\nu _j}-\alpha , \frac{1}{\nu _j})$, $j=1,\ldots ,n$ and $Z \sim$ Gamma$(\alpha ,1)$ (see Luciano et al. (2016) for further details).

The linear correlation between two assets j and k, $j \not = k$, is given by

$$\begin{aligned} {\hat{\rho }}^{\rho \alpha VG}_{jk} = \alpha \frac{\nu _j \nu _k \theta _j \theta _k + \rho ^{B}_{jk} \sqrt{\nu _j \nu _k} \sigma _j \sigma _k}{\sqrt{(\sigma _j^2+\nu _j \theta _j^2)(\sigma _k^2+\nu _k \theta _k^2)}}. \end{aligned}$$

(26)

Under the VG model (13) the correlations in the $(\sigma ,\nu ,\theta )$-parametrization become

$$\begin{aligned} {\hat{\rho }}^{\varDelta VG}_{jk} = \alpha _0\frac{\nu _j\nu _k\theta _j\theta _k + \sqrt{\nu _j\nu _k}\sqrt{(2\sigma _j^2+\nu _j\theta _j^2)(2 \sigma _k^2+\nu _k\theta _k^2)}}{2\sqrt{(\sigma _j^2+\nu _j\theta _j^2)( \sigma _k^2+\nu _k\theta _k^2)}} \qquad \forall j\not = k. \end{aligned}$$

The ratio of linear correlation is given by

$$\begin{aligned} \frac{{\hat{\rho }}_{jk}^{\rho \alpha VG}}{{\hat{\rho }}_{jk}^{\varDelta VG}} = \frac{\alpha }{\alpha _0}\frac{2\theta _j\theta _k+2 \rho ^{B}_{jk} \frac{\sigma _j \sigma _k}{\sqrt{\nu _j \nu _k} }}{\theta _j\theta _k + \sqrt{\left( \frac{2\sigma _j^2}{\nu _j}+\theta _j^2\right) \left( \frac{2 \sigma _k^2}{\nu _k}+\theta _k^2\right) }}. \end{aligned}$$

Note that, while the $\varDelta$-VG model correlation can only be positive, the correlation under the $\rho \alpha$-VG model can also take on negative values. Hence, we will compare the maximal attainable positive correlation under both models, where we only consider positive correlations for the $\rho \alpha$-VG model. The correlation coefficient (26) is positive whenever

$$\begin{aligned} \max \left( -1,-\text {sign}(\theta _1\theta _k)\sqrt{\frac{\theta _j^2 \theta _k^2\nu _j\nu _k}{\sigma _j^2\sigma _k^2}}\right) \le \rho ^B_{jk} \le 1. \end{aligned}$$

(27)

The common parameters $\alpha$ and $\alpha _0$ have the same upper bound $0<\alpha <\min _j \left( \frac{1}{\nu _j}\right)$ and $0<\alpha _0<\min _j \left( \frac{1}{\nu _j}\right)$ whenever the marginals are the same. Hence, the ratio of maximal attainable correlation is

$$\begin{aligned} \frac{{\hat{\rho }}_{jk}^{\rho \alpha VG, \max }}{{\hat{\rho }}_{jk}^{\varDelta VG, \max }} =\frac{2+\text {sign}(\theta _j\theta _k) \rho ^{B}_{jk} \sqrt{\frac{2\sigma _j^2 2\sigma _k^2}{\nu _j \theta _j^2\nu _k\theta _k^2}}}{1 + \text {sign}(\theta _j\theta _k)\sqrt{\left( \frac{2\sigma _j^2}{\nu _j\theta _j^2}+1 \right) \left( \frac{2\sigma _k^2}{\nu _k\theta _k^2}+1\right) }}, \end{aligned}$$

which is always smaller than one whenever (27) holds, implying that the $\varDelta$-VG has a wider range of positive correlation than the $\rho \alpha$-VG model. Hence, the upper bound on $\alpha$ for the $\rho \alpha$-VG model is expected to be reached more often than the upper bound on $\alpha _0$ for the $\varDelta$-VG model whenever the market correlation (and hence the model correlation) is positive. Note that this does not indicate that the $\varDelta$-VG model will lead to a better correlation fit. Indeed, due the extra flexibility of the $\rho \alpha$-VG model introduced by the $\rho ^B_{jk}$, the $\rho \alpha$-VG model will result in a better correlation fit whenever the upper bound on $\alpha$ is not too restrictive.