SPX Calibration of Option Approximations under Rough Heston Model

The volatility of stock return does not follow the classical Brownian motion, but instead it follows a form that is closely related to fractional Brownian motion. Taking advantage of this information, the rough version of classical Heston model also known as rough Heston model has been derived as the macroscopic level of microscopic Hawkes process where it acts as a high-frequency price process. Unlike the pricing of options under the classical Heston model, it is significantly harder to price options under rough Heston model due to the large computational cost needed. Previously, some studies have proposed a few approximation methods to speed up the option computation. In this study, we calibrate five different approximation methods for pricing options under rough Heston model to SPX options, namely a third-order Padé approximant, three variants of fourth-order Padé approximant, and an approximation formula made from decomposing the option price. The main purpose of this study is to fill in the gap on lack of numerical study on real market options. The numerical experiment includes calibration of the mentioned methods to SPX options before and after the Lehman Brothers collapse.

Analysis of Transient Thermal Distribution in a Convective–Radiative Moving Rod Using Two-Dimensional Differential Transform Method with Multivariate Pade Approximant

The transient temperature distribution through a convective-radiative moving rod with temperature-dependent internal heat generation and non-linearly varying temperature-dependent thermal conductivity is elaborated in this investigation. Symmetries are intrinsic and fundamental features of the differential equations of mathematical physics. The governing energy equation subjected to corresponding initial and boundary conditions is non-dimensionalized into a non-linear partial differential equation (PDE) with the assistance of relevant non-dimensional terms. Then the resultant non-dimensionalized PDE is solved analytically using the two-dimensional differential transform method (2D DTM) and multivariate Pade approximant. The consequential impact of non-dimensional parameters such as heat generation, radiative, temperature ratio, and conductive parameters on dimensionless transient temperature profiles has been scrutinized through graphical elucidation. Furthermore, these graphs indicate the deviations in transient thermal profile for both finite difference method (FDM) and 2D DTM-multivariate Pade approximant by considering the forced convective and nucleate boiling heat transfer mode. The results reveal that the transient temperature profile of the moving rod upsurges with the change in time, and it improves for heat generation parameter. It enriches for the rise in the magnitude of Peclet number but drops significantly for greater values of the convective-radiative and convective-conductive parameters.

EXACT SOLUTION OF VAN DER POL NONLINEAR OSCILLATORS ON FINITE DOMAIN BY PADE APPROXIMANT AND ADOMIAN DECOMPOSITION METHODS

This paper is concerned with a thorough investigation in achieving exact analytical solution for the Van der Pol (VDP) nonlinear oscillators models via Adomian decomposition method (ADM). The models are nonlinear time dependent second order ordinary differential equations. ADM has already been applied, in existing literatures, to obtain approximate results. But, we adapt the method by adjusting the source term; a procedure that is base on the asymptotic Taylor's series expansion on the term that would have resulted to proliferation of terms during the invertible process. Then, the rational Pade Approximant is applied to clarify and get a better understanding of the uniqueness and convergence of our findings. Two models were used as illustrations and their result pictured to indicate their behaviour in the given domains. And, we found that the adaptation on the models yielded exact results which were further displayed in constructed tables.