Bank of America Stock Price Research

The stock price is always an interesting topic. On the one hand, from intuition, the stock price of bank is relatively stable since bank usually does not take risk behavior. From 2007 to 2014, the price of stock of Bank of American is relatively stable by checking the RRV (relative realized volatility). There is only one day has relatively high RRVs during 2007 to 2014. The date May 6 th 2010, which called crash day, is a special day that needs to analyze separately. On the other hand, to find the pattern of frequency of trading, there are different sample sizes tested and compared with Poisson distribution. The result is that we can use Poisson distribution to predict probability of no arrival trade when second gap is relatively small. In addition, when plotting the daily 100 seconds accumulated RV (realized volatility) and daily average RV, there was found strong linear relationship between these two variables. In the end, using the Heston model to verify if there exist linear relationship between daily average and mean reversion rate. Then comparing trend of alpha with weekly VIX from Yahoo Finance. When using 5 days as a period to calculate the daily average RV and mean reversion rate, the significance of linear relationship is stronger. It proved the statistic intuition that larger sample size tends to decrease the volatility. The overall trend of VIX from Yahoo Finance is similar to the shape of five-day period alpha.

the first clash from September 1929 to June 1932 caused SP500 loss 86.1 percent.
Then there were 10 other historical crashes before 2020. Excluding the year 2020, the most recent crash was from October 2007 to March 2009 which caused SP500 loss 56.4 percent (Hristova, n.d.). Therefore, analyzing the volatility and price index trend is one of the critical parts of predicting stock price. People can learn from the past and try to forecast the future to avoid the loss from crash.
Bank of America, which found in 1998, is one of the greatest banks in the United States. Researching on stock price of Bank of America is intuitive because bank stock price is considered as the most stable stock and Bank of America is an icon of conservatism (Johnston, 2000). While some people do prefer risk behavior in order to gain greater return, there are some people who like to stay safe and have stable return. Researching on stocks which have relative low volatility is significant and fitful for people who want to play safe.

Stability of Stock Price
To get RRV (Relative Realized Volatility), the first step to calculate log return and then calculate realized volatility by using the following formulas: In this experiment, the 100 second period realized volatility is used to calculate RRV. By using the equation below, it is straight forward to get RRV:

RV RRV
Daily Median RV = From Table 1 and Table 2, there is only 1 day that has RRV greater than 1000.
The day is May 6 th , 2010. From the output we can see that the interval 0 to 2.5 has 95.1% days and it means that most of days do not have large realized volatility. The interval 0 to 2.5 and 2.5 to 5 contains more than 99% of days. It indicates the stock prices of Bank of American are relatively stable and will not change too much in a day.  Table 3) indicates that if choose small seconds to count gaps, the result will be much closer to the Poisson distribution probability. However, when choose larger seconds to count gaps, the difference between empirical probability and Poisson model probability will be relatively large. This trend can be clearly seen in Figure 1 in every year.

Linear Relationship between Daily Accumulated RV and Daily Average RV
Thirdly, the daily accumulated RV and daily average RV are really close to each other and they have approximately linear relationship. Except just a few outliers, many points are approximately in linear relationship (as shown in Figures 1-7). The followings are the graphs for each year.

Heston Model
In the end, the final goal is to measure the alpha (the daily average realized volatility for 100 seconds) and Beta (the mean-reversion rate) by linear regression.
To derive the linear format, the following equation is the initial equation for Heston model:     After switching terms and some calculations: here the formula looks more like a simple linear regression. After calculating one-day period (as shown in Figures 8-11) and five-day period (as shown in Figures 12-15) RV t and RV t-1 , then using the lm function in R studio to generate alphas and betas. It was found that the five-day period obviously has smaller variance. It makes sense because larger sample size tends to make variance smaller. The most obvious is the outlier point date May 6 th , 2010 which is the Crash Day. On that day, the alpha of that day is more than 4 but in five-day period, the alpha is less than 4. After computing the alpha, we will use 5-day period to compare with the VIX from yahoo finance from 2007 to 2014. In addition, since the alpha for the crash day is not significant anyway, therefore it was excluded from the alpha plots.
The following is one day period alphas and betas plots.
The following is five-day period. After calculating the alphas and betas from one-day period and five-day period, in order to compare with the movement of the market volatility, the resource that this paper chooses is VIX from yahoo finance. By plotting VIX with time from 2007 to 2014 (as shown in Figure 16), then the goal is compared VIX distribution and distribution of significant alphas. The following are the plot about VIX and plot about significant alpha.
The VIX values are calculated from the implied volatility of stock options. However, our RVs are from actual stock prices. Comparing the four plots from VIX, their shapes are approximately the same, but surprisingly, our significant alphas with five-day period have close shape to the plots from VIX by ignoring the missing values in year 2013.

Conclusion
Through the research, each question in the beginning of the paper is answered.         Secondly, if defining the gap as small number of seconds, for example, 5 seconds or 10 seconds as a gap, then Poisson distribution is a good prediction about probability of no trade arrival. However, when unit of gaps increases, Poisson distribution may not be a good choice.
Thirdly, the linear relationship between the Daily Accumulated Realized Volatility and Daily Average Realized Volatility is strong. Through all the plots from 2007 to 2014, there is obvious linear relationship in plots.
In the end, by transformation of Heston model and linear regression function of R studio, the daily average realized volatility and mean reversion rate were calculated. Then selecting the significant daily average realized volatility and mean reversion rate and plot them versus time. The result is that their distribution is closed to the distribution of VIX from Yahoo Finance. The Heston model can be used for stock price of Bank of America. Journal of Financial Risk Management There are definitely some limitations of the method. First of all, although the RRV is relatively stable since there is only one RRV which is greater than 1000. However, there are still some RRVs which are approximately 100. It means that there is no guaranteed winning in stock exchange. Secondly, the stock price is dependent on many different variables and circumstances. Sometimes a breaking news can dramatically change the stock price and no one can always predict the future.

Discussion
From the previous conclusion, there are some points which can be easily summarized. First of all, the Bank of America stock price is relative stable during long period. This is reasonable since it is a huge international bank and has thousands of clients. People cannot live without bank and bank is a significant pivot point of transactions.
Secondly, when we use small number of seconds, the Poisson distribution may be a good choice to predict no trade arrival. It is reasonable that the stock of Bank of America has lots of trading volume every day. Therefore, it is preferred to use small number of seconds to determine the number of gaps between two trades. Thirdly, Heston model is one of significant model to predict stock price. Comparing to Black Scholes Model, it does not have constant volatility. Therefore, the outcomes of Heston mode are better performed. Overall, the stock price of Bank of America is reliable and worth to invest.

Theory
Log return: the formula of log return is following: Realized Volatility (Shreve, 2010): the realized volatility is calculated by taking the sum over the past squared return Journal of Financial Risk Management    The definition and use of RRV: in order to observe pronounced volatility jumps in high-frequency data instead of in frequent data. To better analyze the jump effects in RV100s, it is helpful to calculate the relative realized volatility to normalize RV100s. RRV has the formula as follows:

RV RRV
Daily Median RV = From the mean-reversion feature of Heston model, we have the following: We have the alpha which is daily average and beta as mean-reversion rate.
Since the equation is similar to linear equation formula, we decide to test if there is strong linear relationship between mean reversion rate and daily average.

Additional Section
From the analysis of data of stock price, the most interesting values are RV in 2010/05/06. There was a significant flash crash happened on that day. It started at 14:32 pm and lasted about 36 minutes. Some research indicated that the most obvious reason is the debt crisis from Greece. The equity market began to fall rapidly and followed one 300-point drop and two 600-point drop (Treanor, 2015). It was the most significant crash on year 2010. The following are the RV versus time, log return versus time and price versus time (Treanor, 2015) (as shown in Figures 17-19).

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.