A Monte Carlo stock simulation answers a question a single forecast never can: not where the price will be, but the whole shape of where it might be. Run the same starting price, return and volatility through a few hundred random paths and instead of one number you get a distribution, a mean, a median, the percentiles that mark the optimistic and pessimistic tails, and the plain percentage chance the share finishes higher than you bought it. The Monte Carlo stock simulator does exactly that: enter a current price, an annual return, an annual volatility, a time horizon and a number of paths, and it reports the average price, the median, the 5 to 95 percent percentiles, the probability of profit and a histogram of the outcomes.
What the simulation does: one stock, five hundred possible futures
Each path is one possible life of the share. The tool draws a random shock from a normal distribution, scales it by the volatility and the length of the horizon, and compounds it onto the starting price, then repeats the whole thing five hundred times by default. No two paths are identical, and none of them is the truth, but together they trace the range of outcomes the assumptions allow. A plain growth forecast gives you only one point in that cloud, and it is usually the optimistic one, because a plain forecast assumes the return lands exactly on target every single year. The simulation shows you what happens when the return, and the volatility around it, do not.
That is why the output is a shape rather than a number. The five hundred terminal prices, sorted from worst to best, are the distribution the market assumptions imply, and every headline figure the tool reports, the mean, the median, the percentiles and the probability of profit, is just a measurement taken off that one sorted list. Change the inputs and the shape moves, but the logic of reading it stays the same no matter which stock you point it at.
The engine: geometric Brownian motion and the lognormal price
The math under the hood is geometric Brownian motion, the same continuous-time process that the Black-Scholes options model uses, and the Black-Scholes calculator is the other instrument built on exactly this engine. The terminal price is the starting price times an exponential: the drift, which is the annual return minus half the variance, times the fraction of a year, plus the volatility times the square root of that fraction times a standard normal random draw. The exponential is what keeps every path positive, a share cannot trade at a negative price, and it is what makes the distribution of the terminal price lognormal, skewed to the right with a long tail of rare big wins.
Two consequences of the lognormal shape matter for reading the result. First, the average of the paths, the mean, sits above the middle of the paths, the median, because the right tail pulls the mean upward. Second, volatility enters twice, once in the drift as the minus-half-variance drag and once in the random shock, which is why a high-volatility stock with the same expected return as a low-volatility one ends up with a fatter spread of outcomes, more of both the big losses and the big gains. The same exponential written out step by step for the option case is in the article on calculating a Black-Scholes option price, and the two tools share the drift-plus-shock line almost word for word.
The canonical case: a 150 dollar share at 10 percent return and 20 percent volatility
Run the case the tool suggests. A share at 150 dollars, an annual return of 10 percent, an annual volatility of 20 percent, a horizon of 252 trading days, which is one year, and five hundred paths. The drift for the year is 10 percent minus half of 4 percent, the square of 20, which is 8 percent, and the random term is 20 percent times one standard normal draw. Five hundred paths land with a mean of 165.78 dollars, a 10.52 percent gain over the start, and a median of 162.49 dollars, an 8.33 percent gain. The gap between the two, about 3 dollars, is the lognormal skew in dollars: the mean runs higher because a handful of paths explode upward.
The percentiles map the tails. The 5th percentile, the price that 95 percent of paths beat, is 116.94 dollars, a 22.04 percent loss. The 25th percentile is 141.99 dollars, a 5.34 percent loss, so a quarter of the futures are still underwater. The 50th percentile is the median at 162.49 dollars, up 8.33 percent. The 75th percentile is 185.96 dollars, a 23.97 percent gain, and the 95th percentile is 225.79 dollars, a 50.53 percent gain. So across the middle half of the distribution, between the 25th and 75th percentile, the share ends anywhere from a 5.34 percent loss to a 23.97 percent gain, and the probability of finishing above the 150 dollar start is 65.54 percent.
Reading the distribution: mean versus median, and the histogram
The histogram is the distribution drawn as ten bars from the worst path to the best, each bar a price range and its share of the paths. In the canonical case the bars cluster around the 160 to 175 dollar region, the middle of the distribution, and thin out toward both ends, with the left tail reaching into the 110s and the right tail past 225. The tallest bar is the most likely price region, not the mean and not the median, but the mode, and for a right-skewed lognormal the three line up in the order mode, median, mean from left to right.
Where the three split tells you the skew. If the mean were equal to the median the distribution would be symmetric, but here the mean at 165.78 runs above the median at 162.49, the right tail is doing the pulling. The probability of profit, 65.54 percent, is above 50 even though the story is a modest 10 percent expected return, because the 150 dollar start sits left of the median, so more than half the paths clear it. Read the median as what a typical single path does, the mean as what the average of many repeated years would do, and the percentiles as the honest bounds on how much better or worse the year could turn out.
Stress-testing the assumptions: longer horizons, fatter tails, and where the model breaks
Change one input at a time and the distribution moves in predictable ways. Stretch the horizon to five years at the same 10 percent return and 20 percent volatility and the median climbs to 223.77 dollars, the mean to 247.31, and the probability of profit rises to 81.45 percent, because the drift compounds over five years while the random term only grows with the square root of time. The spread also widens dramatically: the 5th percentile falls to 107.24 dollars and the 95th climbs to 466.96, so five years of 20 percent volatility is a much wider cloud than one year, and that width is the real cost of the uncertainty you are holding.
The limits of the model are as important as the output. First, it assumes the return and the volatility you type in stay constant, but real returns and volatilities are themselves random, so the true distribution is fatter than any single parameter set implies. Second, geometric Brownian motion has no jumps, no crashes, no gaps, so it under-states the left tail for a share that can gap down overnight. Third, the percentiles come from five hundred paths, which is enough to read the shape but not to quote a 1st percentile to three decimals, and the tool caps the path count at five hundred for the browser. The deterministic baseline this simulation is a fuzzing of is plain compound growth, covered in the compound interest calculator and the compound interest article: at 10 percent a year the 150 dollar share would land at exactly 165 dollars with zero spread, and the simulation mean of 165.78 sits within a dollar of that, the small gap being the continuous versus discrete compounding convention, while the entire point is the spread around it, from 116.94 to 225.79, which a single deterministic number cannot show.
Using the simulator and turning a probability into a position
The workflow is to set the inputs you actually believe, not the placeholders: a current price from the live quote, a return you are willing to defend, and a volatility that matches the recent history of the share, then read the three numbers that matter, the probability of profit, the median as the typical outcome, and the 5th percentile as the worst case in five out of a hundred runs. Once the year is over and you have a realized ending price, the CAGR calculator turns that price back into the annual rate the path actually delivered, and for how far the worst path could fall from its peak before it recovers, the maximum drawdown article is the matching risk lens on the same simulated path.
If the probability of profit and the tail risk together clear your line, the last step is size, and that is where the simulation hands off to the Kelly criterion calculator. The 65.54 percent chance of a gain is the input to a bet-sizing formula, and the shape of the distribution tells you how much you are willing to be wrong by. The simulation does not tell you the return is certain; it tells you the odds, the tails, and the size of the typical win and loss, and turning those into a stake you can survive is the final, and the only, step that depends on you.