Python-Monte-Carlo-Simulation

Monte Carlo simulations are a powerful computational technique used to estimate and analyze complex systems or processes that involve random variables. It relies on repeated random sampling to obtain numerical results, making it suitable for problems that are difficult or impossible to solve analytically. Monte Carlo simulation is particularly useful in scenarios where there are multiple uncertain variables interacting with each other, leading to complex and unpredictable outcomes. By repeating the simulation with different random inputs, it provides a distribution of possible results, offering a comprehensive understanding of the system's behavior and aiding in making better-informed decisions. One specific scenario is finance. Financial markets are inherently uncertain, and numerous factors influence their behavior. Monte Carlo simulation allows one to incorporate this uncertainty by using random sampling of input variables, such as interest rates, asset returns, and economic indicators, to model a wide range of possible scenarios.

The Process

In this case, the simulation will be applied to the S&P 500 to generate one year (252 trading days) of returns. The reason for this is that the S&P represents the broader market and is less subject to volatility in any specific equity which can help improve the accuracy of our model considering the predictions are drawn from a normal distribution (for now).

Historical Price of the S&P 500 Index from 2013 to 2023

Generating a Monte Carlo Simulation in Python can be done in only a few lines of code. First acquire the financial data you want to simulate (SPY in our case):

import yfinance
start = datetime.datetime(2013,6,1)
end = datetime.datetime(2023,6,1)
data = yfinance.download('SPY', start, end)

Next, calculate the past volatility using Numpy's standard deviation function. This standard deviation will be used in our distribution.

returns = data['Adj Close'].pct_change()
vol = returns.std()

Next intialize the first generated value:

lastprice = data['Adj Close'][-1]
prices = []
price = lastprice * (1 + np.random.normal(0, vol))
prices.append(price)

Finally, generate the next years worth of daily close prices:

for x in range(251):
  price = prices[x] * (1 + np.random.normal(0, vol))
  prices.append(price)

Here is an example of a single year of generated prices:

However, one simulation is not very useful. Instead, we can generate many different years of simulated prices to view a wide variety of results:

10 different years of simulated prices of the S&P 500

100 different years of simulated prices of the S&P 500

1000 different years of simulated prices of the S&P 500

Analysis

After simulating these prices, we can see how Monte Carlo can provide a wide range of possible scenarios. However, one vital assumption we are making is that the market returns are normally distributed. With extreme events like financial crashes and recessions happening somewhat often, a more fat-tailed distribution may be more suitable for simulating market returns. Using other distributions readily available in Numpy, we can analyze and compare how they can make a difference in the accuracy of our simulations. By plotting the prices after one year, we can see how the returns across the different simulations are distributed:

Here, we can see how the returns are mostly normally distributed with a mean of $416.45 (denoted by the dashed red line), very near to the actual closing price of $421.82 (01/06/23). Additionally, our simulation shows a single standard deviation change of about 18%. Meaning that our simulation shows that 67% of the time SPY will increase or decrease by 18% in one year (denoted by the dashed black lines).
Using different distributions, we can try to better represent the actual distribution of the returns of the market:

10000 simulated years of SP500 returns drawing randomly from the Laplace distribution

Here we can see how the fatter tailed Laplace distribution affects the returns. Here the mean final price is $417.23 and shows a single standard deviation change of about 25%.

10000 simulated years of SP500 returns drawing randomly from the Logistic distribution

Here, we can see how the fatter tailed Laplace distribution affects the returns. In this case, the mean final price is $415.93 and shows a single standard deviation change of about 32%.

Conclusion

Ultimately, Monte Carlo Simulation is a potent analytical tool with diverse applications. Its capabilities extend to estimating and forecasting future volatility, offering valuable insights in intricate and uncertain domains, notably financial markets. These simulations play a significant role in multiple fields, including risk management, portfolio optimization, derivative pricing, stress testing, and various others.