By Mahavir Bhattacharya
TL;DR:
This weblog introduces retrospective simulation, impressed by Taleb’s “Fooled by Randomness,” to simulate 1,000 alternate historic value paths utilizing a non-parametric Brownian bridge methodology. Utilizing SENSEX information (2000–2020) as in-sample information, the creator optimises an EMA crossover technique throughout the in-sample information first, after which applies it to the out-of-sample information utilizing the optimum parameters obtained from the in-sample backtest. Whereas the technique outperforms the buy-and-hold strategy in in-sample testing, it considerably underperforms in out-of-sample testing (2020–2025), highlighting the chance of overfitting to a single realised path. The creator then runs the backtest throughout all simulated paths to establish probably the most ceaselessly profitable SEMA-LEMA parameter combos.
The creator additionally calculates VaR and CVaR utilizing over 5 million simulated returns and compares return extremes and distributional traits, revealing heavy tails and excessive kurtosis. This framework permits extra strong technique validation by evaluating how methods may carry out throughout a number of believable market situations.
Introduction
In “Fooled by Randomness”, Taleb says at one place, “To start with, after I knew near nothing (that’s, even lower than at the moment), I puzzled whether or not the time sequence reflecting the exercise of individuals now useless or retired ought to matter for predicting the longer term.”
This obtained me pondering. We regularly run simulations for the possible paths a time sequence can take sooner or later. Nonetheless, the premise for these simulations relies on historic information. Given the stochastic nature of asset costs (learn extra), the realised value path had the selection of an infinite variety of paths it might have taken, but it surely traversed by way of solely a kind of infinite prospects. And I assumed to myself, why not simulate these alternate paths?
In widespread apply, this strategy is known as bootstrap historic simulation. I selected to check with it as retrospective simulation, as a extra intuitive counterpart to the phrases ‘look-ahead’ and ‘walk-forward’ used within the context of simulating the longer term.
Article map
Right here’s a top level view of how this text is laid out:
Knowledge Obtain
We import the mandatory libraries and obtain the day by day information of the SENSEX index, a broad market index primarily based on the Bombay Inventory Change of India.
I’ve downloaded the information from January 2000 to November 2020 because the in-sample information, and from December 2020 to April 2025 because the out-of-sample information. We might have put in a spot (an embargo) between the in-sample and out-of-sample information to minimise, if not get rid of, information leakage (learn extra). In our case, there’s no direct information leakage. Nonetheless, since inventory ranges (costs) are identified to bear autocorrelation, like we noticed above, the SENSEX index on the primary buying and selling day of December 2020 can be extremely correlated with its stage on the final buying and selling day of November 2020.
Thus, after we prepare our mannequin on information that features the final buying and selling day of November 2020, it extracts data from that day’s stage and makes use of it to get skilled. Since our testing dataset is from the primary buying and selling day of December 2020, some residual data from the coaching dataset is current within the testing dataset.
As an extension, the coaching set accommodates some data that can also be current within the testing dataset. Nonetheless, this data will diminish over time and ultimately grow to be insignificant. Having mentioned that, I didn’t keep a spot between the in-sample and out-of-sample datasets in order that we will concentrate on understanding the core idea of this text.
You should use any yfinance ticker to obtain information for an asset of your liking. You may also regulate the dates to fit your wants.
Retrospective Simulation utilizing Brownian Bridge
The following half is the principle crux of this weblog. That is the place I simulate the doable paths the asset might have taken from January 2000 to November 2020. I’ve simulated 1000 paths. You may modify it to make it 100 or 10000, as you want. The upper the worth, the better our confidence within the outcomes, however there’s a tradeoff in computational time. I’ve simulated solely the closing costs. I saved the first-day and last-day costs the identical because the realised ones, and simulated the in-between costs.
Preserving the worth mounted on the primary day is smart. However the final day? If the costs are to observe a random stroll (learn extra), the closing value ranges of most, if not all, paths must be completely different, isn’t it? However I made an assumption right here. Given the environment friendly market speculation, the index would have a good value by the top of November 2020, and after transferring on its capricious course, it will converge again to this honest value.
Why solely November 2020?
Was the extent of the index at its fairest value at the moment? No approach of realizing. Nonetheless, one date is nearly as good as some other, and we have to work with a particular date, so I selected this one.
One other consideration right here is on what foundation we enable the simulated paths to meander. Ought to or not it’s parametric, the place we assume the time sequence to observe a particular distribution, or non-parametric, the place we don’t make any such assumption? I selected the latter. The monetary literature discusses costs (and their returns) as belonging roughly to sure underlying distributions. Nonetheless, with regards to outlier occasions, reminiscent of extremely risky value jumps, these assumptions start to interrupt down, and it’s these occasions {that a} quant (dealer, portfolio supervisor, investor, analyst, or researcher) must be ready for.
For the non-parametric strategy, I’ve modified the Brownian bridge strategy. In a pure Brownian bridge strategy, the returns are assumed to observe a Gaussian distribution, which once more turns into considerably parametric (learn extra). Nonetheless, in our strategy, we calculate the realized returns from the in-sample closing costs and use these returns as a pattern for the simulation generator to select from. We’re utilizing bootstrapping with substitute (learn extra), which signifies that the realized returns aren’t simply being shuffled; some values could also be repeated, whereas some might not be used in any respect. If the values are merely shuffled, all simulated paths would land on the final closing value of the in-sample information. How will we be sure that the simulated costs converge to the ultimate shut value of the in-sample information? We’ll use geometric smoothing for that.
One other consideration: since we use the realized returns, we’re priming the simulated paths to resemble the realized path, appropriate? Kind of, but when we have been to generate pseudo-random numbers for these returns, we must make some assumption about their distribution, making the simulation a parametric course of.
Right here’s the code for the simulations:
Word that I didn’t use a random seed when producing the simulated paths. I’ll point out the rationale at a later stage.
Let’s plot the simulated paths:
The above graph reveals that the beginning and ending costs are the identical for all 1,000 simulated paths. We should always observe one factor right here. Since we’re working with information from a broad market index, whose ranges rely upon many interlinked macroeconomic variables and components, it is extremely unlikely that the index would have traversed many of the paths simulated above, given the identical macroeconomic occasions that occurred through the simulation interval. We’re making an implicit assumption right here that the required macroeconomic variables and components differ in every of the simulated paths, and the interactions between these variables and components outcome within the simulated ranges that we generate. This holds for some other asset class or asset you resolve to switch the SENSEX index with, for retrospective simulation functions.
Exponential Shifting Common Crossover Technique Growth and Backtesting on In-Pattern Knowledge, and Parameter Optimisation
Subsequent, we develop a easy buying and selling technique and conduct a backtest utilizing the in-sample information. The technique is a straightforward exponential transferring common crossover technique, the place we go lengthy when the short-period exponential transferring common (SEMA) of the shut value goes above the long-period exponential transferring common (LEMA), and we go quick when the SEMA crosses the LEMA from above (learn extra).
Via optimisation, we’ll try to seek out the very best SEMA and LEMA mixture that yields the utmost returns. For the SEMA, I exploit lookback durations of 5, 10, 15, 20, … as much as 100, and for the LEMA, 20, 30, 40, 50, … as much as 300.
The situation is that for any given SEMA and LEMA mixture, the LEMA lookback interval must be better than the corresponding SEMA lookback interval. We might carry out backtests on all completely different combos of those SEMA and LEMA values and select the one which yields the very best efficiency.
We’ll plot:
the fairness curve of the technique with the best-performing SEMA and LEMA lookback values, plotted in opposition to the buy-and-hold fairness,the purchase and promote indicators plotted together with the shut costs of the in-sample information and the SEMA and LEMA strains,the underwater plot of the technique, and,a heatmap of the returns for various LEMA and SEMA calculations.
We’ll calculate:
the SEMA and LEMA lookback values for the best-performing mixture,the whole returns of the technique,the utmost drawdown of the technique, and,the Sharpe ratio of the technique.
We may even assessment the highest 10 SEMA and LEMA combos and their respective performances.
Right here’s the code for all the above:
And listed here are the outputs of the above code:
Finest SEMA: 5, Finest LEMA: 40
Complete Return: 873.43%
Most Drawdown: -41.28 %
Sharpe Ratio: 0.59
Prime 10 Parameter Combos:
SEMA LEMA Return
2 5 40 8.734340
3 5 50 7.301270
62 15 60 6.021219
89 20 50 5.998316
116 25 40 5.665505
31 10 40 5.183363
92 20 80 5.071913
32 10 50 5.022373
58 15 20 4.959147
27 5 290 4.794400
The heatmap reveals a gradual change in shade from one adjoining cell to the following. This means that slight modifications to the EMA values don’t result in drastic modifications within the technique’s efficiency. After all, it will be extra gradual if we have been to scale back the spacing between the SEMA values from 5 to, say, 2, and between the LEMA values from 10 to, say, 3.
The technique outperforms the buy-and-hold technique, as proven within the fairness plot. Excellent news, proper? Word right here that this was in-sample backtesting. We ran the optimisation on a given dataset, took some data from it, and utilized it to the identical dataset. It’s like utilizing the costs for the following yr (that are unknown to us now, besides for those who’re time-travelling!) to foretell the costs over the following yr. Nonetheless, we will utilise the knowledge gathered from this dataset to use it to a different dataset. That’s the place we use the out-of-sample information.
Backtesting on Out-of-Pattern Knowledge
Let’s run the backtest on the out-of-sample dataset:
Earlier than we see the outputs of the above codes, let’s record what we’re doing right here.
We’re plotting:
The fairness curve of the technique plotted alongside that of the buy-and-hold, and,The underwater plot of the technique.
We’re calculating:
Technique returns,Purchase-and-hold returns,Technique most drawdown,Technique Sharpe ratio,Purchase-and-hold Sharpe ratio, and,Technique hit ratio.
For the Sharpe ratio calculations, we assume a risk-free fee of return of 0. Listed below are the outputs:
Out-of-Pattern Technique Complete Return: 15.46%
Out-of-Pattern Purchase-and-Maintain Complete Return: 79.41%
Out-of-Pattern Technique Most Drawdown: -15.77 %
Out-of-Pattern Technique Sharpe Ratio: 0.30
Out-of-Pattern Purchase-and-Maintain Sharpe Ratio: 0.56
Out-of-Pattern Hit Ratio: 53.70%
The technique underperforms the underlying by a big margin. However that’s not what we’re primarily excited by, so far as this weblog is worried. We have to think about that we ran an optimisation on solely one of many many paths that the costs might have taken through the in-sample interval, after which extrapolated that to the out-of-sample backtest. That is the place we use the simulation we carried out originally. Let’s run the backtest on the completely different simulated paths and test the outcomes.
Backtesting on Simulated Paths and Optimising to Extract the Finest Parameters
This is able to preserve printing the corresponding SEMA and LEMA values for the very best technique efficiency, and the efficiency itself for the simulated paths:
Accomplished optimization for column 0: SEMA=65, LEMA=230, Return=1.8905
Accomplished optimization for column 1: SEMA=45, LEMA=140, Return=4.4721
……………………………………………………………
Accomplished optimization for column 998: SEMA=10, LEMA=20, Return=3.6721
Accomplished optimization for column 999: SEMA=15, LEMA=20, Return=9.8472
Right here’s a snap of the output of this code:
Now, we’ll kind the above desk in order that the SEMA and LEMA mixture with the very best returns for probably the most paths is on the prime, adopted by the second-best mixture, and so forth.
Let’s test how the desk would look:
Right here’s a snapshot of the output:
Of the 1000 paths, 47 confirmed the very best returns with a mixture of SEMA 5 and LEMA 20. Since I didn’t use a random seed whereas producing the simulated paths, you’ll be able to run the code a number of instances and acquire completely different outputs or outcomes. You’ll see that the very best SEMA and LEMA mixture within the above desk would almost definitely be 5 and 20. The frequencies can change, although.
How do I do know?
As a result of I’ve finished so, and have gotten the mixture of 5 and 20 within the first place each time (adopted by 100 and 300 within the second place). After all, it’s not that there’s a zero likelihood of getting another mixture within the prime row.
Out-of-Pattern Backtesting utilizing Optimised Parameters primarily based on Simulated Knowledge Backtesting
We’ll extract the SEMA and LEMA look-back mixture from the earlier step that yields the very best returns for many of the simulated paths. We’ll use a dynamic strategy to automate this choice. Thus, if as an alternative of 5 and 20, we have been to acquire, say, 90 and 250 because the optimum mixture, the identical can be chosen, and the backtest can be carried out utilizing that.
Let’s use this mix to run an out-of-sample backtest:
Listed below are the outputs:
Out-of-Pattern Technique Complete Return: -7.73%
Out-of-Pattern Purchase-and-Maintain Complete Return: 79.41%
Out-of-Pattern Technique Most Drawdown: -23.70 %
Out-of-Pattern Technique Sharpe Ratio: -0.05
Out-of-Pattern Purchase-and-Maintain Sharpe Ratio: 0.56
Out-of-Pattern Hit Ratio: 52.50%
Dialogue on the Outcomes and the Strategy
Right here, the technique not solely underperforms the underlying but in addition generates damaging returns. So what’s the purpose of all this effort that we put in? Let’s observe that I employed the transferring common crossover technique to illustrate the applying of retrospective simulation utilizing a modified Brownian bridge. This strategy is extra appropriate for testing complicated methods with a number of circumstances, and machine studying (ML)-based and deep studying (DL)-based methods.
We’ve approaches reminiscent of walk-forward optimisation and cross-validation to beat the issue of optimising or fine-tuning a technique or mannequin on solely one of many many doable traversable paths.
Nonetheless, this strategy of retrospective simulation ensures that you just don’t should depend on just one path however can make use of a number of retrospective paths. Nonetheless, since working an ML-based technique on these simulated paths can be too computationally intensive for many of our readers who don’t have entry to GPUs or TPUs, I selected to work with a easy technique.
Moreover, for those who want to modify the strategy, I’ve included some strategies on the finish.
Analysis of VaR and C-VaR
Let’s transfer on to the following half. We’ll utilise the retrospective simulation to calculate the worth in danger and the conditional worth in danger (learn extra: 1, 2, 3).
Output:
Worth at Threat – 90%: -0.014976172535594811
Worth at Threat – 95%: -0.022113806787530325
Worth at Threat – 99% -0.04247765359038646
Anticipated Shortfall – 90%: -0.026779592114352924
Anticipated Shortfall – 95%: -0.035320511964199504
Anticipated Shortfall – 99% -0.058565593363193474
Let’s decipher the above output. We first calculated the day by day p.c returns of all 1000 simulated paths. Each path has 5,155 days of knowledge, which yielded 5,154 returns per path. When multiplied by 1,000 paths, this resulted in 5,154,000 values of day by day returns. We used all these values and located the bottom ninetieth, ninety fifth, and 99th percentile values, respectively.
From the above output, for instance, we will say with 95% certainty that if the longer term costs observe paths much like these simulated paths, the utmost drawdown that we will face on any given day can be 2.21%. The anticipated drawdown can be 3.53% if that stage will get breached.
Let’s discuss concerning the extremes now. Let’s evaluate the utmost and minimal day by day returns of the simulated paths and the realised in-sample path.
Realized Lowest Each day Return: -0.1315258002691394
Realized Highest Each day Return: 0.17339334818061447
The utmost values from each approaches are shut, at round 17.4%. Similar for the minimal values, at round -13.2%. This makes a case for utilizing this strategy in monetary modelling.
Distribution of Simulated Knowledge
Let’s see how the simulated returns are distributed and evaluate them visually to a traditional distribution. We’ll additionally calculate the skewness and the kurtosis.
Skewness: -0.11595652411010503
Kurtosis: 9.597364213156881
The argument ‘kde’, when set to ‘True’, smooths the histogram curve, as proven within the above plot. Additionally, if you’d like a extra granular (coarse) visible of the distribution, you’ll be able to improve (cut back) the worth within the ‘bins’ argument.
Although the histogram resembles a bell curve, it’s removed from a traditional distribution. It reveals heavy kurtosis, which means there are important possibilities of discovering returns which might be many normal deviations away from the imply. And this isn’t any shock, since that’s how fairness and equity-index returns are inherently.
The place This Strategy Can Be Most Helpful
Whereas the technique I used right here is straightforward and illustrative, this retrospective simulation framework comes into its personal when utilized to extra complicated or nuanced methods. It’s useful in circumstances the place:
You are testing multi-condition or ML-based fashions that may overfit on a single realized path.You need to stress take a look at a technique throughout alternate historic realities—ones that didn’t occur, however very effectively might have.Conventional walk-forward or cross-validation methods don’t appear to be sufficient, and also you need an added lens to judge generalisability.You are exploring how a technique may behave (or might need behaved had the worth taken on any alternate value path) below excessive market strikes that aren’t current within the precise historic path.
In essence, this methodology lets you transition from “what occurred” to “what might have occurred,” a delicate but highly effective shift in perspective.
Instructed Subsequent Steps
If you happen to discovered this strategy fascinating, listed here are a number of methods you’ll be able to prolong it:
Attempt extra refined methods: Apply this retrospective simulation to mean-reversion, volatility breakout, or reinforcement learning-based methods.Introduce macro constraints: Anchor the simulations round identified macroeconomic markers or regime modifications to check how methods behave in such environments.Use intermediate anchor factors: As a substitute of simply fixing the beginning and finish costs, strive anchoring the simulation at quarterly or annual ranges to raised management drift and convergence.Practice ML fashions on simulated paths: If you happen to’re working with supervised studying or deep studying fashions, prepare them on a number of simulated realities as an alternative of 1.Portfolio-level testing: Use this framework to judge VaR, CVaR, or stress-test a whole portfolio, not only a single technique.
That is just the start—the way you construct on it depends upon your curiosity, computing assets, and the questions you are attempting to reply.
In Abstract
The weblog launched a retrospective simulation framework utilizing a non-parametric Brownian bridge strategy to simulate alternate historic value paths.We employed a easy EMA crossover technique to illustrate how this simulation might be built-in into a standard backtesting loop.We extracted the very best SEMA and LEMA combos after working backtests on the simulated in-sample paths, after which used these for backtesting on the out-of-sample information.This simulation methodology permits us to check how methods would behave not solely in response to what occurred, but in addition in response to what might have occurred, serving to us keep away from overfitting and uncover strong indicators.The identical simulated paths can be utilized to derive distributional insights, reminiscent of tail threat (VaR, CVaR) or return extremes, providing a deeper understanding of the technique’s threat profile.
Regularly Requested Questions
1. Curious why we simulate value paths in any respect?Actual market information reveals just one path the market took, amongst many doable paths. However what if we need to perceive how our technique would behave throughout many believable realities sooner or later, or would have behaved throughout such realities previously? That’s why we use simulations.
2. What precisely is a Brownian bridge, and why was it used?A Brownian bridge simulates value actions that begin and finish at particular values, like actual historic costs. This helps guarantee simulated paths are anchored in actuality whereas nonetheless permitting randomness in between. The primary query we ask right here is “What else might have occurred previously?”.
3. What number of simulated paths ought to I generate to make this evaluation significant?We used 1000 paths. As talked about within the weblog, when the variety of simulated paths will increase, computation time will increase, however our confidence within the outcomes grows too.
4. Is that this solely for easy methods like transferring averages?In no way. We used the transferring common crossover simply for example. This framework might be (and must be) used whenever you’re testing complicated, ML-based, or multi-condition methods that will overfit to historic information.
5. How do I discover the very best parameter settings (like SEMA/LEMA)?For every simulated path, we backtested completely different parameter combos and recorded the one which gave the very best return. By counting which combos carried out finest throughout most simulations, we recognized the mixture that’s almost definitely to carry out effectively. The concept is to not depend on the mixture that works on only one path.
6. How do I do know which parameter combo to make use of within the markets?The concept is to choose the combo that the majority ceaselessly yielded the very best outcomes throughout many simulated realities. This helps keep away from overfitting to the one historic path and as an alternative focuses on broader adaptability. The precept right here is to not let our evaluation and backtesting be topic to likelihood or randomness, however quite to have some statistical significance.
7. What occurs after I discover that “finest” parameter mixture?We run an out-of-sample backtest utilizing that mixture on information the mannequin hasn’t seen. This assessments whether or not the technique works outdoors of the information on which the mannequin is skilled.
8. What if the technique fails within the out-of-sample take a look at?That’s okay, and on this instance, it did! The purpose is to not “win” with a fundamental technique, however to point out how simulation and strong testing reveal weaknesses earlier than actual cash is concerned. After all, whenever you backtest an precise alpha-generating technique utilizing this strategy and nonetheless get underperformance within the out-of-sample, it possible signifies that the technique isn’t strong, and also you’ll must make modifications to the technique.
9. How can I exploit these simulations to grasp potential losses?We adopted the strategy of flattening the returns from all simulated paths into one huge distribution and calculating threat metrics like Worth at Threat (VaR) and Conditional VaR (CVaR). These present how dangerous issues can get, and the way usually.
10. What’s the distinction between VaR and CVaR?
VaR tells us the worst anticipated loss at a given confidence stage (e.g., “you’ll lose not more than 2.2% on 95% of days”).CVaR goes a step additional and says, “If you happen to lose greater than that, right here’s the typical of these worst days.”.
11. What did we study from the VaR/CVaR outcomes on this instance?We noticed that 99% of days resulted in losses no worse than ~4.25%. However when losses exceeded that threshold, they averaged ~5.86%. That’s a helpful perception into tail threat. These are the uncommon however extreme occasions that may extremely have an effect on our buying and selling accounts if not accounted for.
12. Are the simulated return extremes life like in comparison with actual markets?Sure, they matched very intently with the utmost and minimal day by day returns from the true in-sample information. This validates that our simulation isn’t simply random however is grounded in actuality.
13. Do the simulated returns observe a traditional distribution?Not fairly. The returns confirmed excessive kurtosis (fats tails) and slight damaging skewness, which means excessive strikes (each up and down) are extra widespread than a traditional distribution would have. This mirrors actual market behaviour.
14. Why does this matter for threat administration?If our technique assumes regular returns, we’re closely underestimating the chance of serious losses. Simulated returns reveal the true nature of market threat, serving to us put together for the sudden.
15. Is that this simply an educational train, or can I apply this virtually? This strategy is extremely helpful in apply, particularly whenever you’re working with:
Machine studying fashions which might be susceptible to overfittingStrategies designed for high-risk environmentsPortfolios the place stress testing and tail threat are crucialRegime-switching or macro-anchored fashions
It helps shift our mindset from “What labored earlier than?” to “What would have labored throughout many alternate market situations?”, and that may be one latent supply of alpha.
Conclusion
Hope you discovered at the very least one new factor from this weblog. In that case, do share what it’s within the feedback part under and tell us for those who’d wish to learn or study extra about it. The important thing takeaway from the above dialogue is the significance of performing simulations retrospectively and making use of them to monetary modelling. Apply this strategy to extra complicated methods and share your experiences and findings within the feedback part. Comfortable studying, blissful buying and selling 🙂
Credit
José Carlos Gonzáles Tanaka and Vivek Krishnamoorthy, thanks to your meticulous suggestions; it helped form this text!
Chainika Thakar, thanks for rendering and publishing this, and making it obtainable to the world, that too in your birthday!
Disclaimer: All investments and buying and selling within the inventory market contain threat. Any determination to position trades within the monetary markets, together with buying and selling in inventory or choices or different monetary devices is a private determination that ought to solely be made after thorough analysis, together with a private threat and monetary evaluation and the engagement {of professional} help to the extent you consider needed. The buying and selling methods or associated data talked about on this article is for informational functions solely.
