Calibration and advanced simulation schemes for the Wishart stochastic volatility model

Abstract

In this article, we deal with calibration and Monte Carlo simulation of the Wishart stochastic volatility model. Despite the analytical tractability of the considered model, being of affine type, the implementation of Wishart-based stochastic volatility models poses non-trivial challenges from a numerical point of view. The goal of this article is to overcome these problems providing efficient numerical schemes for Monte Carlo simulations. Moreover, a fast and accurate calibration procedure is proposed.

Keywords:

• Wishart process
• Calibration
• Monte Carlo
• Efficient numerical simulation scheme

JEL Code:

• C02
• C63
• G12
• G13

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

D. Marazzina http://orcid.org/0000-0001-6107-9822

Notes

1 We use the notation ${\mathrm{\Sigma }}_{ij}\left(t\right)$ to indicate the element in the ith row and jth column of $\mathrm{\Sigma }\left(t\right)$. If the time dependence is omitted, we assume to refer to time t=0, i.e. ${\mathrm{\Sigma }}_{ij}={\mathrm{\Sigma }}_{ij}\left(0\right)$.

2 Furthermore, in the following we assume it to be nonsingular.

3 Here we exploit the invertibility of $({\mathbb{I}}_d-2i\mathrm{\Theta }(\tau \left)\mathrm{\Lambda }\right)\exp \left(\tau M^{\mathrm{\top }}\right)$ which is given by the invertibility of both $\mathcal{G}$ (proved in the previous Proposition) and $\exp \left(\tau M^{\mathrm{\top }}\right)$ (by definition of matrix exponential).

4 Let $\mathcal{X}$ be a random variable with characteristic function ${\phi }_{\mathcal{X}}\left(\lambda \right)=\mathbb{E}[\exp (\iota \lambda \mathcal{X}\left)\right]$, the computation relies on the standard formula $\mathbb{E}\left[{\mathcal{X}}^n\right]={\iota }^{-n}{\phi }_{\mathcal{X}}^{\left(n\right)}\left(0\right)$. Additionally, we exploit the fact that for a matrix $U=U\left(\lambda \right)$ it holds that $\left(\mathrm{d}/\mathrm{d}\lambda \right)\det \left[U\right]=\det \left[U\right]\mathrm{Tr}\left[U^{-1}\right(\mathrm{d}U/\mathrm{d}\lambda \left)\right]$ and $\left(\mathrm{d}{U}^{-1}/\mathrm{d}\lambda \right)=-U^{-1}\left(\mathrm{d}U/\mathrm{d}\lambda \right)U^{-1}$. We leave the details of the computation to the reader.

5 A square matrix is called stable if all its eigenvalues have strictly negative real part.

6 Specifically, there is no nilpotent matrix (i.e. with all the eigenvalues equal to zero), except the zero matrix, that can be obtained as the product of two non-negative matrices. Moreover, we exploit the fact that if A and B are two non-zero square matrices such that AB=0, then both A and B must be singular. Given that ${\int }_0^{\mathrm{\infty }}\exp \left(uM\right)\exp \left(u{M}^{\mathrm{\top }}\right)\mathrm{d}u$ is positive definite (indeed, it is the integral of a positive definite matrix), we get the result.

7 For example, by considering M, Q and R to be full matrices we have ${\mathit{\pi }}_W=[\beta ,{\mathrm{\Sigma }}_{11},{\mathrm{\Sigma }}_{12},{\mathrm{\Sigma }}_{22},M_{11},M_{12},M_{21},M_{22},Q_{11},Q_{12},Q_{21},Q_{22},R_{11},R_{12},R_{21},R_{22}{]}^{\mathrm{\top }}$ and $N_{\mathrm{W}}=16$.

8 We remark, however, that the computation of function h must be performed for any maturity in the calibration basket.

9 As far as we know, this is the first time that such a result is derived. It provides an efficient tool to calibrate Bi-Heston parameters to market data.

10 The algorithms are implemented via Matlab code on a laptop PC with an Intel Core i7 CPU and 8 GB RAM.

11 Matlab code is available at http://www1.mate.polimi.it/∼marazzina/WMSV.htm

12 Matlab code is available at http://www1.mate.polimi.it/∼marazzina/WMSV.htm

13 As for example exotic options embedded in structured products with daily (or continuous) monitoring of underlying asset price.

14 All tests have been carried out on a laptop PC with an Intel Core i7 CPU and 8 GB RAM. Algorithms are written in Matlab code and then compiled as MEX files to achieve better performances.

15 Matlab code is available at: http://www1.mate.polimi.it/∼marazzina/WMSV.htm