Mastering Monte Carlo’s

Years ago, I had made it to the final round in an interview for a Senior Delta One/Quantitative Futures position at a HF firm (unnamed for privacy). Things were going well, I had answered two out of three of those ridiculous questions that are only applicable in Subsaharan Africa or Finance interviews (Like how to get 5 gallons from a 6 and 4 gallon jug); I was feeling good. They asked me about my optimization process — a layup compared to most — and I went through my process and ended with Monte Carlo Simulation, where their Head of Quant asked me How I run Monte Carlo Simulations, and what parameters I use.

The easy answer is “I run it in Multicharts”, I click Monte Carlo — but I decided to try to explain my Python code. I got so wrapped up in it, by the end of it I had lost my place and forgotten what Monte Carlo is really doing at it’s core. What should have been a home run became a sloppy drawn out mess of an answer that missed the key points. I essentially explained the worlds most confusing backtest/parameter optimization, and blanked on what was unique by the time I got there. I want to make a point now to emphasize that there’s alot more to Monte Carlo’s than colorful line plots.

Luckily I did later realize what I was grasping for — and I used my favorite analogy; If a backtest is a Ladder, Monte Carlo is randomly rearranging the rungs on that ladder, and determining the likelihood of possible outcomes. THAT is an answer — if only it were my first answer. Needless to say I wasn’t offered this job, but it taught me an important lesson — knowing what your models and code are doing is just as important as being able to write them.

After reading this article, I will ensure you don’t fall victim to coding yourself into a corner with a model like Monte Carlo. First of all, let me clarify that there are a few different types of Monte Carlo optimizations — they are not all created equally. First of all, there is entirely random MC’s, Random within a Normal Distribution MC’s, and simple Random Trade Order. Random can be further sectioned into with or without replacement, but I will leave it at these three types — which should make more sense to you as we continue. I will primarily focus on entirely (pseudo) Random MC’s, as I find them to be the most useful / least prone to error (for more info on limitations of Normal Distributions, I encourage you to read the Incerto series by Nassim Nicholas Taleb).

So what is a Monte Carlo simulation? Many of you have either heard of or extensively used Monte Carlo methods of optimization or simulation — it can be an invaluable tool in measuring the unpredictable. They are not only useful in optimization problems, but great for forecasting things like Max DD, or complex scenarios like the probability of your savings being sufficient for retirement expenses. I primarily use them for two key parts of development; Portfolio Selection/ Optimization, and System/Portfolio Stress Testing. Before we dig into applications, let me explain what Monte Carlo simulations are.

Monte Carlo (MC) simulations are models used to model the probability of complex events by compiling thousands - millions of various outcomes with a pre-determined ‘random’ (changing) variable. Essentially you run 10k iterations with random values for a specific variable, in hopes of finding an optimum value, or determining a range of possible outcomes — i.e. using randomness to solve a complex problem. A simple example is modeling the Maximum Sharpe Ratio of a Portfolio, based on ‘random’ security weights — so you have a Portfolio comprised of AAPL, AMZN, AMD, & ADBE and you want to determine the ideal weighting of these securities to maximize Sharpe.

The other more common scenario is using MC to determine the probability of outcomes — for example % Risk of Ruin with a portfolio, given its return characteristics (Mean, Std), and initial balance. This is where MC sims have applications in virtually every field from Finance, and Engineering to Logistics or Social Sciences. Many common metrics such as VaR and CVaR (Conditional Value at Risk) are derived at their core from MC sims, and have proven to be a valuable tool in a Quant’s toolkit.

The most important thing to take away from this is that Monte Carlo Sims are endlessly flexible — if there’s ever a problem that you need to solve that you cannot figure out, chances are MC can be used to get you pretty close to correct. Recently I was trying to find a way to optimize a complex strategy that utilizes spreads of spreads in Futures markets, and this optimization algorithm was killing me. I decided that I could set the spread ratio as a random variable, and run it as a MC sim and at least get in the right direction — in 5 minutes & 100k iterations I had a simple 15 line solution to a problem that had taken me maybe 350 lines of Python when I initially tried to use a minimization function. This is the adjustable wrench of your toolbox.

Let’s dive in, and I’m going to over comment this code so it could not be clearer what’s doing what. First off, we need your security returns. This can be pulled from system performance (Like my example) or pulled with quantrautil using simple price data. The DF should be Securities Tickers as columns, with date rows and daily return values. Technically, it should be log returns so I’ve included that calculation.

Begin by initializing the arrays for saving the performance values from the MC runs, and set up the MC loop, defining the # of runs as high as your PC will allow (start with 1k or so, scale up). The important thing to remember is the weights, this is the MC magic, as this will select a new value each time and provide the power to the optimization, making each run unique. Also notice how each value your saving (ret, vol, Sharpe) is really an array being saved with indexing per each run — so run 0 begins, randomizes weights, and calculates the return, saving it to index 0 in ret arr, and then vol saves in 0 index, and finally SR. Once you’ve looped through, you run an argmax() function in the SR arr (or whatever value your maximizing), which will give you a weight, 477 in my example. This means run 477 gave you the highest SR value, and that is your ideal weight! You can find the optimal values in allweights, indexing with your max run number (I also included a helper function to save the weights and tickers into a DF, and Pkl it for reference).

If you’re a visual person like myself, you can plot it with a quick pyplot, using the retarr and volarr max values, and can select the max Sharpe to be highlighted as shown. That wasn’t so bad !

Remember, you can make these return and volatility arrays maximize anything you’d like — Correlation, Beta, anything. You can also randomize anything you’d like to optimize for within reason — you just need to ensure the logic works, and that it’s incorporated properly (Hint: I’ve taken entire strategies, and simply put a MC loop at the VERY end when calculating return, and added weights or thresholds in to be randomized and multiplied by the returns to optimize for them — it could even theoretically be randomized before the entries, just put a loop in there and randomize your entry characteristics.). My hope is to open you up a world of possibilities for MC sims to solve equations you never thought possible.

My next example is a more common MC method, using Portfolio characteristics to predict expected returns, variance and worst case scenarios. I’ll use the same data in my example, and plot them out for visualization. Don’t worry, this one is MUCH simpler.

This example simply requires you to pull the mean daily (log) return, and the daily standard deviation of your system/portfolio. Once you plug those values in, you have everything you need, just plug in a number of iterations in the range(), and ensure your plot is INSIDE the loop, with the .show() outside.

This chart, while somewhat pretty, is not very useful; my preferred method of utilizing this is to plot it as a distribution, and take various metrics of the universe of portfolio runs. Keep in mind, this is simply using your daily mean, and standard deviation to run 1000’s of 1 yr (T value) performance trajectories. I used normal distribution for randomness factor here to make the histogram cleaner, but you can use any random distribution or entirely random value/sample you would like here! Play around with various models, and see how they vary.

I like to calculate various percentiles, and track minimums along with various common metrics. The histogram provides a much clearer picture — in absolute terms, it can just seem large so I like to divide it by initial account value to make it percent values instead. This should be straightforward, aside from maybe the list comp, which simply is takings each runs result as a percentage of initial account value.

So there we have it — Monte Carlo Simulations are one of the most flexible models we have at our disposal, becoming comfortable with the inner workings of these models can make all the difference in optimizing complex problems. I hope you’ve also learned not to answer a MC interview question with a complex answer that misses the point, dig in to the basic moving parts as that’s where the magic really is in these models. Mastering Monte Carlo’s will provide you with the tools to solve otherwise insurmountable equations and untenable problems — or of course make really colorful line plots.

Happy Trading!

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-ZO

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