Here, we consider three categories of influential factors. The first is the fundamentals, which include the macroeconomic situation as well as the basic operating status of listed companies. The macroeconomic situation reflects the overall operating performance of listed companies and determines the further development of listed companies. The macroeconomic situation is closely related to listed companies and their corresponding stock prices. The basic operating status of listed companies includes their financial condition, profitability, market share, and management system.

The second category comprises psychological factors, which are mainly reflected by changes in stock prices. If people feel panic, they have negative attitudes toward the stock market, and thus prices fall. However, if people find out that they have overreacted to the pandemic or recovery rates increase, then they will regain confidence about the stock market, and stock prices will rebound.

The third category is industry factors. From our perspective, industries have different influences on stock price changes. Theoretically, after the outbreak of COVID-19, stock prices in health-related industries have risen because medical supplies are urgently needed. By contrast, stock prices in industries related to entertainment fell because people went to movie theaters, clubs, and theme parks much less often.

Hence, we introduce the following equation:

$$P = f(Fund,Psy,Ind).$$

(1)

where *P* is the firm-level stock price, *f(·)* is the function whose details are still unknown, *Fund* is firm-level fundamentals, *Psy* is psychological factors, and *Ind* is industry factors. All these functions are assumed to be continuous, and twice differentiable.

Differentiating this function with respect to time yields:

$$\begin{aligned} \mathop P\limits^{ \bullet } = f^{\prime } (Fund,Psy,Ind)\mathop {Fund}\limits^{ \bullet } + f^{\prime } (Fund,Psy,Ind)\mathop {Psy}\limits^{ \bullet } + f^{\prime } (Fund,Psy,Ind)\mathop {Ind}\limits^{ \bullet } . \\ \end{aligned}$$

(2)

As our primary research interest is in the percentage change in firm-level stock prices, we divide by *P* on both sides of the equation and obtain:

$$\begin{aligned} G_{P} = \frac{{\mathop P\limits^{ \bullet } }}{P} = f^{\prime } (Fund,Psy,Ind)\frac{{\mathop {Fund}\limits^{ \bullet } }}{P} + f^{\prime } (Fund,Psy,Ind)\frac{{\mathop {Psy}\limits^{ \bullet } }}{P} + f^{\prime } (Fund,Psy,Ind)\frac{{\mathop {Ind}\limits^{ \bullet } }}{P}. \\ \end{aligned}$$

(3)

where *G*_{P} is the growth rate or the rate of change in firm-level stock prices.

The right-hand side of the equation shows that over a short period, firm-level fundamentals do not change substantially, nor are they reflected in publicly released reports on corporations. Therefore, for simplicity, we substitute \(\mathop {Fund}\limits^{ \bullet } = 0\). However, for *Psy* and *Ind*, things are much more complicated. Based on facts during the pandemic, we assume the following:

where *Epi* represents the COVID-19 pandemic. If we differentiate these two equations with respect to time, we have:

$$\mathop {Psy}\limits^{ \bullet } = Psy^{\prime } (Epi)\mathop {Epi}\limits^{ \bullet } .$$

(6)

$$\mathop {Ind}\limits^{ \bullet } = Ind^{\prime } (Epi)\mathop {Epi}\limits^{ \bullet } .$$

(7)

Substituting Eqs. (6) and (7) into Eq. (3) and rearranging the terms, we have

$$\begin{aligned} G_{P} = \frac{{\mathop P\limits^{ \bullet } }}{P} = \frac{{f^{\prime } (Fund,Psy,Ind)}}{P}(Psy^{\prime } (Epi) + Ind^{\prime } (Epi))\mathop {Epi}\limits^{ \bullet } . \\ \end{aligned}$$

(8)

Now, we take the logarithm on both sides of the equation and obtain:

$$\begin{aligned} \ln \left( {\frac{{\mathop P\limits^{ \bullet } }}{P}} \right) = \ln (f^{\prime } (Fund,Psy,Ind)) - \ln (P) + \ln (Psy^{\prime } (Epi) + Ind^{\prime } (Epi)) + \ln (\mathop {Epi}\limits^{ \bullet } ). \\ \end{aligned}$$

(9)

As shown, this equation has very close empirical implications. However, because many of the related functional forms are still unknown, we cannot directly apply them to empirical analysis.

We further define \(Y = \ln (\frac{{\mathop P\limits^{ \bullet } }}{P})\) and then differentiate *Y* with respect to \(Ind^{\prime}(Epi)\), obtaining:

$$\frac{\partial Y}{{\partial Ind^{\prime } (Epi)}} = \frac{1}{{Psy^{\prime } (Epi) + Ind^{\prime } (Epi)}}.$$

(10)

This equation is the second-order derivative of *Y* with respect to *Ind*, thus obtaining the following relationship:

$$\begin{aligned} \int {\frac{\partial Y}{{\partial Ind^{\prime } (Epi)}}} dEpi = \frac{\partial Y}{{\partial Ind(Epi)}} = Ind^{\prime } (Epi) \\ \end{aligned}$$

(11)

Now, we integrate both sides of Eq. (10) and obtain:

$$\begin{aligned} \int {\frac{\partial Y}{{\partial Ind^{\prime } (Epi)}}} dEpi & = \int {\frac{1}{{Psy^{\prime } (Epi) + Ind^{\prime } (Epi)}}} dEpi \\ & = \frac{{\ln (Psy^{\prime } (Epi) + Ind^{\prime } (Epi))}}{{Psy^{\prime \prime } (Epi) + Ind^{\prime \prime } (Epi)}} + C_{1} . \\ \end{aligned}$$

(12)

where *C*_{1} is the integration constant.

By combining Eqs. (11) and (12), we obtain:

$$\frac{{\ln (Psy^{\prime } (Epi) + Ind^{\prime } (Epi))}}{{Psy^{\prime\prime}(Epi) + Ind^{\prime\prime}(Epi)}} + C_{1} = Ind^{\prime } (Epi).$$

(13)

After rearranging terms, we obtain:

$$\begin{aligned} \ln (Psy^{\prime } (Epi) + Ind^{\prime } (Epi)) = (Ind^{\prime } (Epi) - C_{1} )(Psy^{\prime\prime}(Epi) + Ind^{\prime\prime}(Epi)). \\ \end{aligned}$$

(14)

Now, we get to the “tricky” part. If we take the exponential form on both sides of Eq. (14), we obtain

$$\begin{aligned} Psy^{\prime } (Epi) + Ind^{\prime } (Epi) = \exp ((Ind^{\prime } (Epi) - C_{1} )(Psy^{\prime\prime}(Epi) + Ind^{\prime\prime}(Epi))). \\ \end{aligned}$$

(15)

Thus, we obtain:

$$\begin{aligned} Psy^{\prime } (Epi) = \exp ((Ind^{\prime } (Epi) - C_{1} )(Psy^{\prime\prime}(Epi) + Ind^{\prime\prime}(Epi))) - Ind^{\prime } (Epi). \\ \end{aligned}$$

(16)

To determine the key function of *Psy(Epi)*, we need to integrate Eq. (16) on both sides as follows:

$$\begin{aligned} \int {Psy^{\prime } (Epi)} dEpi & = \int {(\exp ((Ind^{\prime } (Epi) - C_{1} )(Psy^{\prime\prime}(Epi)} + Ind^{\prime\prime}(Epi))) - Ind^{\prime } (Epi))dEpi \\ & = \int {\exp ((Ind^{\prime } (Epi) - C_{1} )(Psy^{\prime\prime}(Epi)} + Ind^{\prime\prime}(Epi)))dEpi - \int {Ind^{\prime } (Epi)} dEpi \\ & = \int {\exp ((Ind^{\prime } (Epi) - C_{1} )(Psy^{\prime\prime}(Epi)} + Ind^{\prime\prime}(Epi)))dEpi - (Ind(Epi) + C_{2} ). \\ \end{aligned}$$

(17)

Focusing on the term \(\int {\exp ((Ind^{\prime}(Epi) - C_{1} )(Psy^{\prime\prime}(Epi) + Ind^{\prime\prime}(Epi))} )dEpi\) and considering all the second-order terms constants for simplicity, we have

$$\begin{aligned} & \int {\exp ((Ind^{\prime}(Epi) - C_{1} )(Psy^{\prime\prime}(Epi) + Ind^{\prime\prime}(Epi)))} dEpi \\ & \quad = \frac{{\exp ((Ind^{\prime}(Epi) - C_{1} )(Psy^{\prime\prime}(Epi) + Ind^{\prime\prime}(Epi)))}}{{(Psy^{\prime\prime}(Epi) + Ind^{\prime\prime}(Epi))Ind^{\prime\prime}(Epi)}} + C_{3} . \\ \end{aligned}$$

(18)

where *C*_{2} and *C*_{3} are the integration constants in these steps. Substituting Eq. (18) into Eq. (17), we obtain:

$$\begin{aligned} & Psy(Epi) = \frac{{\exp ((Ind^{\prime}(Epi) - C_{1} )(Psy^{\prime\prime}(Epi) + Ind^{\prime\prime}(Epi)))}}{{(Psy^{\prime\prime}(Epi) + Ind^{\prime\prime}(Epi))Ind^{\prime\prime}(Epi)}} - Ind(Epi) + C_{4} . \\ \end{aligned}$$

(19)

where \(C_{4} = C_{3} - C_{2}\).

After some complex derivations (see the mathematical appendix for details), we obtain:

$$\begin{aligned} Psy(Epi) + Ind(Epi) &= \exp (((Psy^{\prime\prime}(Epi) + Ind^{\prime\prime}(Epi))Ind^{\prime\prime}(Epi))Epi) \\&= (\exp (Epi))^{{((Psy^{\prime\prime}(Epi) + Ind^{\prime\prime}(Epi))Ind^{\prime\prime}(Epi))}} . \\ \end{aligned}$$

(20)

Although we have made a relatively strong assumption that all the second-order terms are constants, Eq. (20) offers a novel perspective for depicting the quantitative relationship between stock prices and the psychological response to a pandemic. If the specific functional forms of *Psy* and *Ind* are in quadratic form, then the assumption of constant second-order terms is reasonable, because the first-order condition of a typical quadratic function is linear.

By differentiating both sides of Eq. (20), we obtain

$$\begin{aligned} Psy^{\prime}(Epi) + Ind^{\prime}(Epi)& = ((Psy^{\prime\prime}(Epi) + Ind^{\prime\prime}(Epi))Ind^{\prime\prime}(Epi))\exp (((Psy^{\prime\prime}(Epi) \\ & \quad + Ind^{\prime\prime}(Epi))Ind^{\prime\prime}(Epi))Epi) = ((Psy^{\prime\prime}(Epi) \\ & \quad + Ind^{\prime\prime}(Epi))Ind^{\prime\prime}(Epi))(\exp (Epi))^{{((Psy^{\prime\prime}(Epi) + Ind^{\prime\prime}(Epi))Ind^{\prime\prime}(Epi))}} . \\ \end{aligned}$$

(21)

Then, we take the logarithm on both sides of Eq. (21), and we obtain

$$\begin{aligned} \ln (Psy^{\prime}(Epi) + Ind^{\prime}(Epi)) &= \ln (((Psy^{\prime\prime}(Epi) + Ind^{\prime\prime}(Epi))Ind^{\prime\prime}(Epi)) \\ & \quad + ((Psy^{\prime\prime}(Epi) + Ind^{\prime\prime}(Epi))Ind^{\prime\prime}(Epi))Epi. \\ \end{aligned}$$

(22)

Now, if we substitute Eq. (22) into Eq. (9), we obtain:

$$\begin{aligned} \ln \left( {\frac{{\mathop P\limits^{ \bullet } }}{P}} \right) & = \ln (f^{\prime}(Fund,Psy,Ind)) - \ln (P) + \ln (((Psy^{\prime\prime}(Epi) + Ind^{\prime\prime}(Epi))Ind^{\prime\prime}(Epi)) \\ & \quad + ((Psy^{\prime\prime}(Epi) + Ind^{\prime\prime}(Epi))Ind^{\prime\prime}(Epi))Epi + \ln (\mathop {Epi}\limits^{ \bullet } ). \\ \end{aligned}$$

(23)

Finally, by rearranging the terms, we obtain:

$$\begin{aligned} \ln \left( {\frac{{\mathop P\limits^{ \bullet } }}{P}} \right) & = \ln (f^{\prime}(Fund,Psy,Ind)((Psy^{\prime\prime}(Epi) + Ind^{\prime\prime}(Epi))Ind^{\prime\prime}(Epi)) \\ & \quad - \ln (P) + ((Psy^{\prime\prime}(Epi) + Ind^{\prime\prime}(Epi))Ind^{\prime\prime}(Epi))Epi + \ln (\mathop {Epi}\limits^{ \bullet } ). \\ \end{aligned}$$

(24)

The constant term in the regression is:

\(\ln (f^{\prime}(Fund,Psy,Ind)((Psy^{\prime\prime}(Epi) + Ind^{\prime\prime}(Epi))Ind^{\prime\prime}(Epi))\), and the estimation coefficient for *Epi* is \(((Psy^{\prime\prime}(Epi) + Ind^{\prime\prime}(Epi))Ind^{\prime\prime}(Epi))\), which is easy to understand and has a meaningful economic interpretation.

Moreover, to use the logarithm of real-world data in which some of the numerical values for the rate of change are negative, we multiply both sides of Eq. (24) by 2 and obtain:

$$\begin{aligned} 2\ln \left( {\frac{{\mathop P\limits^{ \bullet } }}{P}} \right) & = 2\ln (f^{\prime}(Fund,Psy,Ind)((Psy^{\prime\prime}(Epi)+ Ind^{\prime\prime}(Epi))Ind^{\prime\prime}(Epi)) \\ & \quad - 2\ln (P) + 2((Psy^{\prime\prime}(Epi) + Ind^{\prime\prime}(Epi))Ind^{\prime\prime}(Epi))Epi + 2\ln (\mathop {Epi}\limits^{ \bullet } ). \\ \end{aligned}$$

(25)

which is,

$$\begin{aligned} \ln \left( {\left( {\frac{{\mathop P\limits^{ \bullet } }}{P}} \right)^{2} } \right) & = 2\ln (f^{\prime}(Fund,Psy,Ind)((Psy^{\prime\prime}(Epi) + Ind^{\prime\prime}(Epi))Ind^{\prime\prime}(Epi)) \\ & \quad - \ln (P^{2} ) + 2((Psy^{\prime\prime}(Epi) + Ind^{\prime\prime}(Epi))Ind^{\prime\prime}(Epi))Epi + \ln ((\mathop {Epi}\limits^{ \bullet } )^{2} ). \\ \end{aligned}$$

(26)

Equation (26) suggests the combined use of the quadratic form of certain variables with the log form. This is the final expression that we obtain theoretically, which has meaningful practical empirical implications.