Monte Carlo Simulation of Stock Price

Monte Carlo methods are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other mathematical methods. In this research we will apply it for somilating stock price over time with given volatility.

On high level, many simulations performed in order to see what would be most likely the worst case scenario and the best case scenario.

• Starting price - dollar amount of the stock price. By default $200.00 is used (which is about the price of S&P 500 in the beggining of 2015)
• Drift - normal growth rate. For example, use 0.03 if the growth rate is expected to be close to 3% average annual inflation rate (in the United States).
• Volatility - defines how volatile and how random this stock prices change over time. For example, the value 0.12 would stand for 12% which is close to S&P 500 volatility. Actually, this is standard deviation of return rates from the recent periods.

The following formula defines general idea: \[ dS_{t} = \mu S_{t} d_{t} + \sigma S_{t} dW_{t} \]

This is the actual formula used in the code (the formula is simplified in sake of example): \[ Price_{t} = Price_{t-1}(1 + \frac{drift}{N} + \frac{(Random-\frac{1}{2}) * Volatility}{\sqrt{N}}) \]

where N is the total number of days in the given period. In the application I use N=255 since there are 255 working days in a year.

By using this formula we indeed have chaotic enough movements of stock price around the median values determined by constant steady growth defined by drift rate.

The application is published on R-Studio Shiny server: https://stas.shinyapps.io/MonteCarloSimulation/ The full code is not enclosed here, but you can see extremely simplified example of chaotic movements of a price over time below:

prices <- rep(200,255)
for (i in 2:255){prices[i] <- prices[i-1]+(runif(1)-0.5)*5}
plot(prices, type="l")