Monte Carlo analysis is a statistical method that involves repeatedly running simulations of a model or system to understand the possible outcomes and probabilities of different events occurring. It is often used in financial modeling and risk analysis, as well as in other fields where probabilistic analysis is needed.

By analyzing the results of these simulations, the trader can gain insights into the potential risks and returns of the strategy or portfolio under different trading environments.

**Monte Carlo analysis** is a statistical method named after the city of Monte Carlo in Monaco, which is known for its casinos and gambling.

**Stanislaw Ulam** was a Polish mathematician who played a key role in the development of Monte Carlo analysis.

Ulam’s work on Monte Carlo analysis involved developing algorithms for generating random numbers, which are essential for running simulations. He realized that by generating random numbers and using them as input for mathematical models, he could simulate the behavior of complex systems and understand the probabilities of different outcomes occurring.

Ulam’s work on Monte Carlo analysis laid the foundation for the development of a wide range of probabilistic modeling techniques that are used in many fields today, including finance, engineering, and physics.

Monte Carlo simulation involves using randomized simulated trade sequences to evaluate the performance of a trading system.

Monte Carlo simulation can be used to evaluate a wide range of characteristics of a trading system, including its expected return, risk, and volatility. It can also be used to understand the distribution of possible outcomes and the probability of different events occurring, such as the likelihood of achieving a certain level of return or incurring a loss.

One of the key benefits of Monte Carlo simulation is that it allows traders to analyze a trading system under a wide range of conditions, rather than relying on historical data or assumptions about future events. This can provide a more realistic and comprehensive view of the potential risks and returns of the system.

**The Bootstrap test**

The bootstrap test is a statistical method that can be used in trading to assess the reliability of statistical estimates and to test hypotheses about the performance of a trading system or strategy.

For example, a trader might use the bootstrap test to evaluate the reliability of a trading system’s historical returns and to test hypotheses about the system’s expected performance in the future. This could be done by sampling with replacement from the system’s historical trades to create a number of new simulated trade sequences, and then calculating the returns for each of the simulated sequences. The resulting distribution of returns can then be used to estimate the sampling distribution of the system’s returns and to test hypotheses about the system’s expected performance.

The bootstrap test can be particularly useful for traders who are trying to evaluate the performance of a trading system or strategy that is based on limited data or that has not been tested over a long period of time. By using the bootstrap test to estimate the sampling distribution of the system’s returns, traders can gain a better understanding of the system’s potential risks and returns and can make more informed decisions about whether to invest in the system.

**Monte Carlo Simulation in Amibroker**

AmiBroker is a technical analysis software that is used by traders and investors to analyze financial markets and to design and test trading strategies. Monte Carlo analysis is a statistical method that can be used in conjunction with AmiBroker to evaluate the performance and risk profile of a trading system or strategy.

To use Monte Carlo analysis in AmiBroker, a trader would first define the parameters of the analysis, including the inputs (such as market data and assumptions about future events) and the outputs (such as the expected return on a portfolio). The trader would then run the simulation multiple times, each time generating a different set of possible outcomes based on the input variables.

AmiBroker includes a built-in Monte Carlo simulation tool that can be used to run simulations and analyze the results. The tool allows traders to specify the parameters of the simulation, including the number of simulations to run and the duration of each simulation. It also provides a range of graphical and statistical tools for analyzing the results of the simulations, such as graphs and tables that show the distribution of possible outcomes and statistical measures such as the mean and standard deviation of the returns.

**How to Perform Monte Carlo Analysis in Amibroker**

In Amibroker Monte Carlo Analysis is readily available in the backtest report once the backtest is done. The output statistics are a set of CDF Charts (Cumulative Distribution Function Charts). **MC Min/Max Equity, MC Final Equity, MC CAR, MC Max Drawdown $ and %, Lowest Equity** are the output statistics from Monte Carlo Analysis. And one can access these reports from the Backtesting report from **Amibroker v5.96 beta onwards.**

**Steps to Perform Monte Carlo Simulation**

1)Goto File -> New -> New Analysis

2)Select the Trading System to perform Monte-Carlo Simulation and ensure that the fixed position size is defined properly in the trading system.

3)Now open backtester settings and set the backtesting parameters accordingly.

4)Goto the Monte Carlo tabs and Ensure Monte Carlo Simulation is enabled and set the broom chart plots to 100.

5)Now press backtest to perform the backtest + Monte Carlo simulation.

**How to Interpret Monte Carlo Analysis**

The results of the Monte Carlo simulation are displayed on the “Monte Carlo” page of Backtest report. To interpret the results of a Monte Carlo analysis, it is important to consider the specific goals and objectives of the analysis. Some common ways to interpret the results of a Monte Carlo analysis include:

**Expected value:**The expected value is the average outcome of the simulations, and it represents the most likely outcome of the system or strategy under the assumptions used in the analysis.**Probability of achieving a certain level of return:**By evaluating the distribution of returns generated by the simulations, it is possible to understand the probability of achieving a certain level of return or exceeding a certain threshold.**Risk measures:**Common risk measures such as the standard deviation or variance of the returns can be used to understand the potential volatility of the system or strategy.**Probability of incurring a loss:**By evaluating the distribution of returns, it is possible to understand the probability of incurring a loss and the potential magnitude of such a loss.**Sensitivity analysis:**By changing the input assumptions used in the analysis, it is possible to understand how the results of the simulations are sensitive to different factors, such as changes in market conditions or the performance of individual securities.

Here are sample monte Carlo results (highlights are added manually for the purpose of illustration). Starting equity was 30000 in this example. The test was done over 2 years (intraday 1min data).

To Understand the Key statistics from Monte Carlo Simulated results one must understand cumulative distribution charts (CDFs)

A cumulative distribution function is a statistical tool that is used to understand the probability distribution of risk/return of a trading system and to understand the probability of different outcomes occurring. It is a valuable tool for analyzing data and making informed decisions based on probabilistic analysis.

**MC Min/Max Equity**

Note that green and red lines (min/max equity) are not really single “best” and “worst” equities. They are bar-by-bar highest (max) and lowest( min) points of ALL equities generated during Monte carlo.So they are actually best points from all equities and worst points from all equities. And blue line (avg) is the average from all equity lines (all runs).

**MC Final Equity**

Above MC Final Equity is the cumulative result of 1000 simulated runs with a sample strategy. From the CDF charts, 30% of the trial runs generated equity returns between 0 and 3,00,000 returns. And rest of the 70% of the simulated runs are able to generate more than 3,00,000 profits with an Initial Capital of 3,00,000. Every time you do backtest you will get different output statistics due to the random generation of the Equity Curve.

**MC Drawdown %**

below MC drawdown% is the cumulative result of 1000 simulated runs with a sample strategy. From the CDF charts, 70% of the trial runs generated drawdown more than 30%. And rest of the 30% of the simulated runs are able to generate lesser than 30% drawdown with an Initial Capital of 3,00,000. which outlines that probabilistic risk of running the trading system is very high.

Overall Monte Carlo will help you in understand the robustness of the trading system and identify the problem with the trading strategy but Monte Carlo analysis will not solve the problem in your trading system. To put it in simple words it is just a validation process. To create a better trading system the trader should adapt better trading design principles to improve his/her trading system performance.