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Review of Derivatives Research

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27-02-2019

Towards a $\Delta $-Gamma Sato multivariate model

Authors: Lynn Boen, Florence Guillaume

Published in: Review of Derivatives Research | Issue 1/2020

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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.

next article Conditional risk-neutral density from option prices by local polynomial kernel smoothing with no-arbitrage constraints

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The quoted paper by Ballotta and Bonfiglioli is mainly theoretical. An estimation implementation can be found in Loregian et al. (2015). Applications of this model to credit risk are presented in Ballotta and Fusai (2015) and Ballotta et al. (2019).

Note that the joint distribution of ${\mathbf {G}}$ in Definition 4 is not necessarily self-decomposable in the multivariate sense, but Q-selfdecomposable (see Barndorff-Nielsen et al. 2001).

$X_t$ follows a DG Sato distribution if (3) holds with $X_1$ DG distributed.

X follows a VG Sato distribution if (3) holds with $X_1$ VG distributed.

Empirical data shows that many stocks included in the major stock indices are in fact positively correlated, see f.i. the correlation indices issued by the Chicago Board Options (2009).

To check the robustness of the calibration results to a change of the objective function, we also performed the calibration of the different models using the multivariate Average Relative Percentage Error

$$\begin{aligned} \text{ MARPE } = \sum _{j=1}^n \frac{\text{ ARPE }^{(j)}}{n} = \sum _{j=1}^n \left( \frac{1}{n}\sqrt{\frac{1}{N^{(j)}}\sum _{k=1}^{N^{(j)}} \frac{|P_k^{(j)}-{\hat{P}}_k^{(j)}|}{P_k^{(j)}}}\right) \end{aligned}$$

instead of the objective function (15) to fit the single-name option prices. The use of the functional MARPE instead of MRMSE leads to similar results for both the decoupled and the joint calibration as the ones in Sect. 5, which are available to the interested reader upon request.

Whenever possible, the calibration of the correlation structure should be done on pair-wise market implied correlations, which requires a deep and active option trading market for any basket composed of any pair of the n underlyings. For each pair (j, k) of assets, we then have

$$\begin{aligned} \rho ^{(jk)}(t) = \frac{\text{ Var }\left( \frac{S_t^{(jk)}-S_0^{(jk)}}{S_0^{(jk)}} \right) -\omega _i^{(jk)}\text{ Var }\left( \frac{S_t^{(j)}-S_0^{(j)}}{S_0^{(j)}} \right) -\omega _j^{(jk)}\text{ Var }\left( \frac{S_t^{(k)}-S_0^{(k)}}{S_0^{(k)}} \right) }{2\left( \omega _j^{(jk)}\right) ^*\left( \omega _k^{(jk)}\right) ^*\sqrt{\text{ Var } \left( \frac{S_t^{(j)}-S_0^{(j)}}{S_0^{(j)}}\right) \text{ Var }\left( \frac{S_t^{(k)}-S_0^{(k)}}{S_0^{(k)}}\right) }}, \end{aligned}$$

where $S_t^{(jk)}=\omega _j^{(jk)}S_t^{(j)}+\omega _k^{(jk)}S_t^{(k)}$ and $\left( \omega _j^{(jk)}\right) ^*=\omega _j^{(jk)}S_0^{(jk)}/S_0^{(jk)}$.

mean [($\hbox {MRMSE}^{L \acute{e}{} vy }$- $\hbox {MRMSE}^{Sato }$)/($\hbox {MRMSE}^{L \acute{e}{} vy }$)]

$\text {mean}\left[ {(\text{ MRMSE }^{{VG }}-\text{ MRMSE }^{\varDelta {Gamma }})/(\text{ MRMSE }^{{VG }})}\right] $

$\text {mean}{\left[ (\text{ MRMSE }^{{Joint }}-\text{ MRMSE }^{{Decoup }})/(\text{ MRMSE }^{{Joint }})\right] }$.

Note that the parameters of the positive jump process are less stable than those of the negative jump process for both the $\varDelta $-Gamma Lévy and Sato models, hence the variability in magnitude order for the upper bound on $\alpha _0$.

A decoupled calibration can also be performed by taking $\alpha _j^* = \alpha _j+\alpha _{0,1}+\alpha _{0,2}$ and $\eta _j^* = \eta _j + \eta _{0,1} + \eta _{0,2}$. The marginal structure under the decoupled setting will then be the same as under the decoupled one-factor $\varDelta $-Gamma models. Besides, the constraints $\alpha _{0,1}+\alpha _{0,2} \in (0, \min (\alpha _j^{*}, j=1, \ldots , n))$ and $\eta _{0,1}+\eta _{0,2} \in (0, \min (\eta _j^{*}, j=1, \ldots , n))$ would have to hold such that the maximal attainable pairwise correlation would be the same as under the one-factor model. Hence, such extension is not recommended under the decoupled setting since the model parsimony will decrease while not improving the model flexibility.

$\text {mean}\left( \frac{\text{ RMSE }^{1factor}-\text{ RMSE }^{2factor}}{\text{ RMSE }^{1factor}}\right) $

When a negative market correlation is observed, the $\rho \alpha $-VG model will always outperform the $\varDelta $-VG model, since the latter can only model positive correlations. One of the extensions proposed in Remark 3 should then be applied to the $\varDelta $-VG model in order to incorporate negative correlations.

The results are available to the interested reader.

Note that we do not use the MLE to calibrate the different models under the ${\mathbb {P}}$-measure because the marginal densities are expressed in terms of special functions (Whittaker W functions for the $\varDelta $-Gamma model and modified Bessel functions of the third kind for the VG model). To evaluate such functions, we would have to resort to numerical approximations that often lead to significant numerical errors and hence, to unreliable parameter estimates for the models under consideration. On the other hand, the method of moments leads to a straightforward calibration, since the parameter estimates are given analytically as a function of the empirical moments.

Note that a calibration under the ${\mathbb {P}}$-measure only makes sense for the Lévy models since Sato processes are by construction processes with non-stationary increments, such that an historical calibration of Sato models would then lead to parameter estimates that strongly depend on the origin of the time series.

A similar moment matching system can also be used under the ${\mathbb {Q}}$-measure to calibrate the marginal parameters under the decoupled setting, provided the calibration is performed on one single maturity. When considering the whole set of quoted maturities, the moment matching system becomes overdetermined, such that one has to resort to some optimization problem and one loses the main advantage of the method (see Guillaume and Schoutens 2013).

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Title: Towards a -Gamma Sato multivariate model
Authors: Lynn Boen
Florence Guillaume
Publication date: 27-02-2019
Publisher: Springer US
Published in: Review of Derivatives Research / Issue 1/2020
Print ISSN: 1380-6645
Electronic ISSN: 1573-7144
DOI: https://doi.org/10.1007/s11147-019-09155-y