One of the most common ways to estimate risk is the use of a Monte Carlo simulation (MCS). For example, to calculate the value at risk (VaR) of a portfolio, we can run a Monte Carlo simulation that attempts to predict the worst likely loss for a portfolio given a confidence interval over a specified time horizon - we always need to specify two conditions for VaR: confidence and horizon. (For related reading, see The Uses And Limits Of Volatility and Introduction To Value At Risk (VAR) - Part 1 and Part 2.)

In this article, we will review a basic MCS applied to a stock price. We need a model to specify the behavior of the stock price, and we'll use one of the most common models in finance: geometric Brownian motion (GBM). Therefore, while Monte Carlo simulation can refer to a universe of different approaches to simulation, we will start here with the most basic.

Where to Start
A Monte Carlo simulation is an attempt to predict the future many times over. At the end of the simulation, thousands or millions of "random trials" produce a distribution of outcomes that can be analyzed. The basics steps are:

1. Specify a model (e.g. geometric Brownian motion)
2. Generate random trials
3. Process the output

1. Specify a Model (e.g. GBM)
In this article, we will use the geometric Brownian motion (GBM), which is technically a Markov process. This means that the stock price follows a random walk and is consistent with (at the very least) the weak form of the efficient market hypothesis (EMH): past price information is already incorporated and the next price movement is "conditionally independent" of past price movements. (For more on EMH, read Working Through The Efficient Market Hypothesis and What Is Market Efficiency?)

The formula for GBM is found below, where "S" is the stock price, "m" (the Greek mu) is the expected return, "s" (Greek sigma) is the standard deviation of returns, "t" is time, and "e" (Greek epsilon) is the random variable:


If we rearrange the formula to solve just for the change in stock price, we see that GMB says the change in stock price is the stock price "S" multiplied by the two terms found inside the parenthesis below:


The first term is a "drift" and the second term is a "shock". For each time period, our model assumes the price will "drift" up by the expected return. But the drift will be shocked (added or subtracted) by a random shock. The random shock will be the standard deviation "s" multiplied by a random number "e". This is simply a way of scaling the standard deviation.

That is the essence of GBM, as illustrated in Figure 1. The stock price follows a series of steps, where each step is a drift plus/minus a random shock (itself a function of the stock's standard deviation):

Figure 1

2. Generate Random Trials
Armed with a model specification, we then proceed to run random trials. To illustrate, we've used Microsoft Excel to run 40 trials. Keep in mind that this is an unrealistically small sample; most simulations or "sims" run at least several thousand trials.

In this case, let's assume that the stock begins on day zero with a price of $10. Here is a chart of the outcome where each time step (or interval) is one day and the series runs for ten days (in summary: forty trials with daily steps over ten days):

CT-MonteCarlo4ra1.gif CT-MonteCarlo45ra2.gif
Figure 2: Geometric Brownian Motion

The result is forty simulated stock prices at the end of 10 days. None has happened to fall below $9, and one is above $11.

3. Process the Output
The simulation produced a distribution of hypothetical future outcomes. We could do several things with the output. If, for example, we want to estimate VaR with 95% confidence, then we only need to locate the thirty-eighth-ranked outcome (the third-worst outcome). That's because 2/40 equals 5%, so the two worst outcomes are in the lowest 5%.

If we stack the illustrated outcomes into bins (each bin is one-third of $1, so three bins covers the interval from $9 to $10), we'll get the following histogram:

Figure 3

Remember that our GBM model assumes normality: price returns are normally distributed with expected return (mean) "m" and standard deviation "s". Interestingly, our histogram isn't looking normal. In fact, with more trials, it will not tend toward normality. Instead, it will tend toward a lognormal distribution: a sharp drop off to the left of mean and a highly skewed "long tail" to the right of the mean. This often leads to a potentially confusing dynamic for first-time students:

  • Price returns are normally distributed.
  • Price levels are log-normally distributed.

Think about it this way: A stock can return up or down 5% or 10%, but after a certain period of time, the stock price cannot be negative. Further, price increases on the upside have a compounding effect, while price decreases on the downside reduce the base: lose 10% and you are left with less to lose the next time. Here is a chart of the lognormal distribution superimposed on our illustrated assumptions (e.g. starting price of $10):

Figure 4

A Monte Carlo simulation applies a selected model (a model that specifies the behavior of an instrument) to a large set of random trials in an attempt to produce a plausible set of possible future outcomes. In regard to simulating stock prices, the most common model is geometric Brownian motion (GBM). GBM assumes that a constant drift is accompanied by random shocks. While the period returns under GBM are normally distributed, the consequent multi-period (for example, ten days) price levels are lognormally distributed.

Check out David Harper's movie tutorial, Monte Carlo Simulation with Geometric Brownian Motion, to learn more on this topic.

Related Articles
  1. Mutual Funds & ETFs

    ETF Analysis: PowerShares S&P 500 Low Volatility

    Find out about the PowerShares S&P 500 Low Volatility ETF, and learn detailed information about this fund that provides exposure to low-volatility stocks.
  2. Mutual Funds & ETFs

    ETF Analysis: SPDR Barclays Short Term Corp Bd

    Learn about the SPDR Barclays Short-Term Corporate Bond ETF, and explore detailed analysis of the exchange-traded fund tracking U.S. short-term corporate bonds.
  3. Mutual Funds & ETFs

    ETF Analysis: Vanguard Intermediate-Term Bond

    Find out about the Vanguard Intermediate-Term Bond ETF, and delve into detailed analysis of this fund that invests in investment-grade intermediate-term bonds.
  4. Investing Basics

    How AQR Places Bets Against Beta

    Learn how the bet against beta strategy is used by a large hedge fund to profit from a pricing anomaly in the stock market caused by high stock prices.
  5. Fundamental Analysis

    Is India the Next Emerging Markets Superstar?

    With a shift towards manufacturing and services, India could be the next emerging market superstar. Here, we provide a detailed breakdown of its GDP.
  6. Term

    What are Metrics?

    Metrics are tools that measure a company’s performance.
  7. Term

    Estimating with Subjective Probability

    Subjective probability is someone’s estimation that an event will occur.
  8. Investing Basics

    Understanding the Modigliani-Miller Theorem

    The Modigliani-Miller (M&M) theorem is used in financial and economic studies to analyze the value of a firm, such as a business or a corporation.
  9. Economics

    Explaining Kurtosis

    Kurtosis describes the distribution of data around an average.
  10. Fundamental Analysis

    Calculating Free-Float Methodology

    Free-float methodology is used to calculate the total market capitalization of an index’s underlying companies.
  1. Discount Bond

    A bond that is issued for less than its par (or face) value, ...
  2. Compound Annual Growth Rate - CAGR

    The Compound Annual Growth Rate (CAGR) is the mean annual growth ...
  3. Internal Rate Of Return - IRR

    A metric used in capital budgeting measuring the profitability ...
  4. Financial Singularity

    A financial singularity is the point at which investment decisions ...
  5. Revenue-based Financing

    Revenue-based financing, also known as royalty based financing, ...
  6. Precedent Transaction Analysis

    A valuation method in which the prices paid for similar companies ...
  1. What assumptions are made when conducting a t-test?

    The common assumptions made when doing a t-test include those regarding the scale of measurement, random sampling, normality ... Read Full Answer >>
  2. What is the utility function and how is it calculated?

    In economics, utility function is an important concept that measures preferences over a set of goods and services. Utility ... Read Full Answer >>
  3. What are some of the more common types of regressions investors can use?

    The most common types of regression an investor can use are linear regressions and multiple linear regressions. Regressions ... Read Full Answer >>
  4. What types of assets lower portfolio variance?

    Assets that have a negative correlation with each other reduce portfolio variance. Variance is one measure of the volatility ... Read Full Answer >>
  5. When is it better to use systematic over simple random sampling?

    Under simple random sampling, a sample of items is chosen randomly from a population, and each item has an equal probability ... Read Full Answer >>
  6. What are some common financial sampling methods?

    There are two areas in finance where sampling is very important: hypothesis testing and auditing. The type of sampling methods ... Read Full Answer >>

You May Also Like

Trading Center

You are using adblocking software

Want access to all of Investopedia? Add us to your “whitelist”
so you'll never miss a feature!