The application of optimization methods to prediction issues is a continually exploring field. In line with this, this paper investigates the connectedness between the infected cases of COVID-19 and US fear index from a forecasting perspective. The complex characteristics of implied volatility risk index such as non-linearity structure, time-varying and non-stationarity motivate us to apply a nonlinear polynomial Hammerstein model with known structure and unknown parameters. We use the Hybrid Particle Swarm Optimization (HPSO) tool to identify the model parameters of nonlinear polynomial Hammerstein model. Findings indicate that, following a nonlinear polynomial behaviour cascaded to an autoregressive with exogenous input (ARX) behaviour, the fear index in US financial market is significantly affected by COVID-19-infected cases in the US, COVID-19-infected cases in the world and COVID-19-infected cases in China, respectively. Statistical performance indicators provided by the developed models show that COVID-19-infected cases in the US are particularly powerful in predicting the Cboe volatility index compared to COVID-19-infected cases in the world and China (MAPE (2.1013%); R2 (91.78%) and RMSE (0.6363 percentage points)). The proposed approaches have also shown good convergence characteristics and accurate fits of the data.

From the Chinese city of Wuhan, the world is going through difficult conditions these days caused by the rapidly spreading COVID-19 pandemic, which has strangled the worldwide economy, the financial markets, the oil market, the banking and insurance industries [

In this paper, we focus on the financial market in the United States of America (USA), which has suffered historic losses in the first three months of 2020. Daily FT [

This leads investors into a vicious cycle of random selling and further declines in stock prices. The team in [

Given these circumstances that are caused by COVID-19, traders find themselves in a state of blurred vision about the future and more specifically regarding the expected cash flows that can affect the value of a company. Indeed, forecasting the fear index in the US market has become a necessity for traders and other market participants. Thus, many studies in financial economics have examined the forecasting of the implied volatility risk index. The authors in [

The objective of this paper is to study the predictive power of COVID-19 cases worldwide, in the United States of America and in China for the US financial market fear index. This is done using the Hammerstein-ARX approach estimated by a hybrid particle swarm optimization (HPSO). Many reasons have been put forward in the literature to justify the use of HPSO to estimate the Hammerstein-ARX approach. The authors in [

The author in [

Our research is based on the literature dealing with the response of financial markets to several disasters and crises. First, we refer to [

This study contributes to the previous studies in several ways. First, to the best of our knowledge, no studies have investigated the effect of COVID-19 outbreak on the fear index of U.S. Financial market. This study is the first to explore the forecasting power of the infected cases of COVID-19 in USA, in World and in China for the Cboe’s volatility index. Second, the complex pricing of Cboe’s volatility index caused by the spillover effect of different disasters and pandemics on financial markets prompts us to apply the Artificial Intelligence (AI) Techniques. Driven by their minor complexity and costs, greater accuracy, and fast processing times, this study is the pioneer to use the Hammerstein models tuned by a particle swarm optimization (PSO) algorithm to forecast Cboe’s volatility index during the COVID-19 outbreak. Third, our estimated findings provide important implications for policymakers and investors when considering the forecasting power of useful information’s about infected cases COVID-19 for the uncertainty in US financial market to better respectively tacking measures to ensure satisfactory level of stability and confidence in US financial market and formulating optimal investment strategies and efficient use of hedging instruments.

The rest of this article is organized as follows. Section 2 describes the research methodology. The data and preliminary results is presented and analyzed in Section 3. Section 4 summarizes the empirical findings, while Section 5 provides conclusion and implications.

The main objective of this paper is to predict the fear index in the US financial market using the COVID-19 infected cases in the world, in the US and in China. This is achieved by developing a Hammerstein-ARX model. The parameters of the model were identified and estimated using hybrid particle swarm optimization (HPSO). The Hammerstein nonlinear part will be modelled using a polynomial function.

In this study, we first estimate the parameters of a relevant model reflecting the effect of the COVID-19 infected cases on the Cboe’s volatility index from a forecasting perspective.

From a dynamical systems perspective, the input of the present system is the infected cases of Covid-19,

Let

In the proposed HPSO algorithm [

Each particle represents an element in D-dimensional vector. The ith particle position is defined as:

We consider the prior best position of any particle in the swarm as follows:

The velocity of any particle in the swarm i as follows:

The position visited on previous occasion by the ith particle is defined as

The principle of the HPSO search for the optimal solution is based on the combination of three trends: (i) Following the current search direction, (ii) following the local search during which the particle records the best previous position; and (iii) the global search ability in which each particle in the swarm try to follow the leader current realization.

Accordingly, the particles evolve following the move equations:

Since the Hammerstein-ARX model parameters include integers, the corresponding dimensions (d, p, q and r) are updated according to:

We consider

In addition, the optimization of the fitness function will be achieved through the minimization of the total of the squared residuals between the observed values and the predicted values of the Cboe’s volatility index defined as follows:

To achieve this, the PSO identification algorithm is run in three steps [

For robustness reasons, we compare our proposed Hammerstein ARX model with the two classical time series models such as the heterogeneous autoregressive model (HARX) and the fractionally integrated autoregressive moving average models (ARFIMAX (p, d, q)). This is very essential to show the advantage of the complex machine learning models over the classical and traditional approaches. First, the ARFIMAX (p, d, q) model is a time series model that generalizes ARIMAX (p, d, q) by allowing non-integer values of the differentiation parameter d. It can be defined as follows:

Second, it is the heterogeneous autoregressive model (HAR) that [

The procedure has been implemented for 3 case studies. In what follows, details about case 1 related to the USA covid-19 data and their effect on the implied volatility will be detailed. Five steps have been followed including data collection, model calibration, parameters estimation and model validation. The first step consists of collecting the data from from 31 December 2019 to 05 June 2020. The data has been divided into 80% for the model development and 20% for the model validation.

In this paper, we have proposed Hammerstein model since it includes two blocks: A linear and nonlinear in order to capture the complex dynamics of the implied volatility as affected by the number of covid-19 cases.

The proposed model has the structure as in

The main contribution of the present paper is to optimally estimate the model parameters that fit as accurately as possible the estimated infections (by the model) and the real infections (recorded by the health authorities). The problem is posed as an optimization problem solved in this study using a hybrid PSO algorithm combining discrte parameters (model orders) and real parameters (model coefficients). After running the algorithm, the results of case 1 are provided in

Our data sample consists of daily series of two variables: (i) We collect the US fear index from the Chicago Board Options Exchange (Cboe) Global Markets. (ii) Covid-19 infected cases obtained from the European Union Open Data Portal (EU ODP). The study period is from 31 December 2019 to 05 June 2020. We essentially have 158 observations divided into 2 series: The first set consists of 128 observations (NT = 128) which will be used for training the model. The second subsample is used for model validation. It contains 30 observations (NV = 30). The choice of this period is mainly motivated by the fact that it witnessed the peak of the spread of the Corona epidemic, as the World Health Organization confirmed that it was an epidemic. This is considered a huge shock to most countries in the world and has forced them to take many precautionary and proactive measures to limit its spread.

Descriptive statistics about all series of Cboe’s volatility index and the Covid-19 infected cases are presented in

Cboe’s volatility index | Infected cases of Covid-19 in USA | Infected cases of Covid-19 in world | Infected cases of Covid-19 in China | |
---|---|---|---|---|

Mean | 32.20709 | 11852.28 | 41793.22 | 532.7278 |

Median | 29.415 | 2377 | 18133.5 | 44.5 |

Maximum | 82.69 | 48529 | 127796 | 15141 |

Minimum | 12.1 | 0 | 0 | 0 |

Std. Dev. | 17.2236 | 12977 | 42357.93 | 1485.305 |

Skewness | 0.765862 | 0.442805 | 0.334072 | 6.565978 |

Kurtosis | 2.907738 | 1.70669 | 1.482167 | 60.95855 |

Jarque-Bera | 15.50173 | 16.17497 | 18.10571 | 23249.98 |

Probability | 0.00043 | 0.000307 | 0.000117 | 0.0000 |

Observations | 158 | 158 | 158 | 158 |

We now analyze the empirical evidence of the application of HPSO used to determine the parameters of the Hammerstein-ARX model. Referring to

Infected cases of Covid-19 in USA | Cboe’s volatility index | 1 | 3 | 5 | 5 | ||||

Infected cases of Covid-19 in World | Cboe’s volatility index | 1 | 1 | 4 | 9 | ||||

Infected cases of Covid-19 in China | Cboe’s volatility index | 1 | 2 | 7 | 6 |

Furthermore, we observe that the Hammerstein-ARX model run using the US COVID-19 infected cases as input provides better performance than that obtained by the international and Chinese COVID -19 infected cases. This highlight the superiority of domestic COVID-19 cases in predicting the fear index of the United States of America compared to international and Chinese COVID-19 cases. In addition, we note that the Cboe volatility index responds to the COVID-19 cases in both linear and non-linear ways. In summary,

MODEL | MAPE (%) | R^{2} (%) |
RMSE |
||
---|---|---|---|---|---|

Infected cases of Covid-19 in USA | Cboe’s volatility index | Hammerstein-ARX | 2.1013 | 0.9178 | 0.6362 |

ARFIMAX | 4.9020 | 0.3330 | 1.9246 | ||

HARX | 4.9359 | 0.3095 | 1.9582 | ||

Infected cases of Covid-19 in World | Cboe’s volatility index | Hammerstein-ARX | 2.3179 | 0.8729 | 0.7909 |

ARFIMAX | 4.9086 | 0.3299 | 1.9290 | ||

HARX | 4.6247 | 0.3564 | 1.8905 | ||

Infected cases of Covid-19 in China | Cboe’s volatility index | Hammerstein-ARX | 3.0489 | 0.6447 | 1.3227 |

ARFIMAX | 4.9113 | 0.3288 | 1.9307 | ||

HARX | 4.6909 | 0.3405 | 1.9137 |

This becomes even more evident when we look at

A further aspect consists to examine the convergence of the HPSO technique that is already well served to forecast the Cboe’s volatility index by COVID-19 cases.

Since the announcement by the World Health Organization on 11 March 2020 that COVID-19 represents a global influenza pandemic, fears have spread around the world to affect all aspects. The financial markets have taken a lion’s share of these fears, with trading volumes falling sharply in most countries. On this basis, the US financial markets are at the forefront of the trading decline and prices collapse until the end of March 2020. This fear has quickly spread to investors who are concerned about the short and long term effects of the crisis. As outlined above, many authors have attempted to study the nexus between the COVID-19 pandemic and financial markets. In response to this, our paper examines the prediction of the fear index in the US financial market using the COVID-19 infection cases in the US, the world, and China. Indeed, a non-linear Hammerstein polynomial model combined with an artificial intelligence technique was used to identify and estimate the model parameters.

Our research reveals several key findings. First, the use of HPSO as a parameter identification tool for the Hammerstein-ARX model parameters shows that the fear index in US financial market responds linearly and nonlinearly to Covid-19 infected cases in the US, Covid-19 infected cases in the world and Covid-19 infected cases in China. Performance indicators such as MAPE, RMSE and R2 provide evidence that Covid-19 infected cases are particularly dominant in forecasting the US fear index compared to global and Chinese COVID-19 cases. Second, our study showed that the Hammerstein-ARX model is the best-fitting model, which underlines its superiority over classical time series models such as ARFIMAX and HARX. Third, the combination of Hammerstein-ARX with the HPSO tool produces good convergence and better performance indicators. This further confirms the fitness and the accuracy of the proposed approach.

From a financial perspective, many practical implications arise from our results. Useful information on Covid-19 infected cases can be used by investors to forecast uncertainty in the US financial market. Similarly, the predictive power of the Covid-19 infected cases in the US for the Cboe volatility index can help individual investors to better formulate their investment strategies and effective use of hedging instruments. Thus, detecting the fear index with good accuracy allows investors to protect their portfolio at a lower cost. This will enable them to know how to manipulate the realized volatility market and the systematic risk associated with the COVID-19 pandemic.

For the US financial authorities, our study provides an opportunity to learn that the COVID-19 cases contribute to increased instability and uncertainty in the US financial market in the short term. Thus, policy makers will be obliged to intervene to limit this danger by taking measures to ensure a satisfactory level of stability and confidence in the financial market. However, these policies are likely to raise doubts among investors about market conditions, and increase questions about the ability of government interventions to limit the crisis and lead to a faster restoration of market stability in the short term [

This research has been funded by Scientific Research Deanship at University of Ha’il, Saudi Arabia through project number RG-20 210.