The global economy has been severely impacted during the Covid-19 period. The U.S. stock market has also experienced greater volatility. Based on data from January 2020 to June 2021, this paper studies the volatility of daily returns on the stock market in the United States. The Standard and Poor's 500 (SPX) index and eight companies traded on major exchanges such as the New York Stock Exchange and the Nasdaq are used to calculate volatility. Combining the statistical analysis methods GARCH, GARCH-M, and TARCH, the time series of each security is modeled. It is demonstrated that the conditional heteroskedasticity of stock returns depends not only on the observed historical volatility (ARCH term) but also on the conditional heteroskedasticity of prior periods (GARCH term). As expected for financial markets, the COVID-19 outbreak increased the volatility of U.S. stock market returns. After the COVID-19 outbreak, the volatility of the U.S. stock market rose dramatically. It reached an extremely high level for the first quarter of 2020 and continued to move downwards in the following quarters. The significant heteroskedasticity in the return volatility indicates that external variables significantly affect the stock. Furthermore, this study combines the Capital Asset Pricing Model (CAPM) and the research of Engle et al. (1987), which provides a way to quantify the liquidity premium. However, with the results of the GARCH-M model, this study does not find a significant liquidity premium over time. Additionally, The TARCH model reveals a significant asymmetry in stock market returns during this epidemic, suggesting that negative news has a more substantial impact on U.S. financial markets. For investors and financial institutions, this research helps identify potential volatility in the face of similar risk events. It is helpful for investors to comprehensively consider various factors when investing in special periods or consider other investment portfolios to reduce investment risks in specific periods based on research results.