ABSTRACT
As is well known, multi-factor stochastic volatility models are necessary to capture the market accurately in pricing financial derivatives. However, the multi-factor models usually require too many parameters to be calibrated efficiently and they do not lead to an analytic pricing formula. The double Heston model is one of them. The approach of this paper for this difficulty is to rescale the double Heston model to reduce the number of the model parameters and obtain a closed form analytic solution formula for variance swaps explicitly. We show that the rescaled double Heston model is as effective as the original double Heston model in terms of fitting to the VIX market data in a stable condition and yet the computing time is much less than that under the double Heston model. However, in a turbulent situation after the start of the COVID-19 pandemic in 2020, we acknowledge that even the double Heston model fails to capture the market accurately.
ABSTRACT
Unlike previous viral diseases, COVID-19 has an "asymptomatic" group that has no symptoms but can still spread the disease to others at the same rate as symptomatic patients who are infected. In the literature, the mass action or standard incidence rates are considered for compartmental models with asymptomatic compartment for studying the transmission dynamics of COVID-19, but the quarantined adjusted incidence rate is not. To bridge this gap, we developed a Susceptible Asymptomatic Infectious Quarantined (SAIQ) model with a Quarantine-Adjusted (QA) incidence to investigate the emergence and containment of COVID-19. COVID-19 models are investigated using various methods, but only a few studies take into account closed-form solutions. The knowledge of closed-form solutions simplifies the construction of the various epidemic indicators that describe the epidemic phenomenon and makes the sensitivity analysis to variations in the data under consideration possible. The closed-form solutions of the systems of four nonlinear first-order ordinary differential equations (ODEs) are established. The Epidemic Peak (EP), Force of Infection (FOI) and Rate of Infection (ROI) are the important indicators for the control and prevention of disease. We examined these indicators using closed-form solutions and particular parameter values. Different disease control scenarios are thoroughly examined. The four scenarios to analyze COVID-19 propagation and containment are (i) lockdown, (ii) quarantine and other preventative measures, (iii) stabilizing the basic reproduction rate to a level where the pandemic can be contained and (iv) containing the epidemic through an appropriate combination of lockdown, quarantine and other preventative measures.