Central Limit Theorem for Quant Finance

*🚀 Master Quantitative Skills with Quant Guild* https://quantguild.com *📅 Meet with me 1:1* https://calendly.com/quantguild-support *📈 Interactive Brokers for Algorithmic Trading* https://www.interactivebrokers.com/mkt/?src=quantguildY&url=%2Fen%2Fwhyib%2Foverview.php *👾 Join the Quant Guild Discord server here* https://discord.com/invite/MJ4FU2c6c3 ___________________________________________ *🪐 Jupyter Notebook* https://github.com/romanmichaelpaolucci/Quant-Guild-Library/blob/main/2025%20Video%20Lectures/61.%20Central%20Limit%20Theorem%20for%20Quant%20Finance/clt.ipynb **TL;DW Executive Summary** - Random variables define a set of outcomes with accompanying probabilities or likelihoods (discrete / continuous) - Anytime we draw a random variable we are dealing with an empirical distribution that can generate statistics - If draws are from the same population distribution (i.e. it is time invariant) empirical statistics and distributions converge by the LLN, this is not the case in practice as we are battling time variance in distributions and subsequent statistics. . . - The distribution of sample means follows a normal (Gaussian) distribution regardless of the population or data generating distribution, this is quite literally our bridge between theory and practice, statistics and probability - If the population or data generating distribution is fixed the distribution of sample means will converge to a normal distribution as the sample size becomes arbitrarily large and probabilities will be precise in the frequentist sense - In reality, population or data generating distributions are **NOT** fixed and are time variant leading us to generate incorrect probabilities and draw statistically incorrect conclusions - Though distributions in reality change over time, the CLT can offer a **snapshot** which is useful locally for generating probabilities as we saw in the trading signal example where the sentiment distribution producing a trading decision is likely to be more stable in that short region of time (minutes) required to generate a decision than it is to be stable over a series of days (which doesn't matter nearly as much as we will continue to recalibrate our model to new data) I hope you enjoyed! - Roman ___________________________________________ *📖 Chapters:* 00:00 - The Bridge Between Probability and Statistics 02:57 - Random Variables and Modeling Trade Frequency 04:48 - Theoretical vs Empirical Distributions 06:24 - Data Samples as Random Variables 07:18 - Statistics as Random Variables 09:18 - Example: Drawing Sample Means 10:36 - Law of Large Numbers (LLN) 12:14 - Calibrating Models and OOS Performance 14:11 - Key Modeling Questions in Practice 18:22 - Normal (Gaussian) Distributions 19:53 - Likelihoods vs Probabilities 22:53 - Why the Normal (Gaussian) Distribution? 24:20 - Central Limit Theorem (Rough) Proof 27:41 - Example: Distribution of Sample Means 28:35 - More Samples, More Confidence 31:02 - Example: Unknown Population Distribution 33:37 - Limitations of the Central Limit Theorem 35:55 - Application to Stock Returns 39:08 - Time Variant Population Distributions 41:44 - Application to Trading Signals 46:40 - TL;DW Executive Summary ___________________________________________ *🗣️ Shout Outs* A special thank you to my members on YouTube for supporting my channel and enabling me to continue to create videos just like this one! *⭐ Quant Guild Directors* Dr. Jason Pirozzolo ___________________________________________ *▶️ Related Videos* *Statistics and Trading Profitability Over Time (Edge) 📈* Time Series Analysis for Quant Finance https://youtu.be/JwqjuUnR8OY Quant Trader on Retail vs Institutional Trading https://youtu.be/j1XAcdEHzbU Quant on Trading and Investing https://youtu.be/CKXp_sMwPuY Why Poker Pros Make the Best Traders (It's NOT Luck) https://youtu.be/wZChBKDFFeU Quant vs. Discretionary Trading https://youtu.be/3gblERSSHXI Quant Busts 3 Trading Myths with Math https://youtu.be/wJfIk3VnubE ___________________________________________ *🗂️ Resources* *📚 Quant Guild Library:* https://github.com/romanmichaelpaolucci/Quant-Guild-Library *🌎 GitHub:* https://github.com/RomanMichaelPaolucci https://github.com/Quant-Guild *📝 Medium (Blog):* https://quantguild.medium.com/ https://medium.com/quant-guild ___________________________________________ *🛠️ Projects* *The Gaussian Cookbook:* https://gaussiancookbook.com *Recipes for simulating stochastic processes:* https://papers.ssrn.com/sol3/papers.cfm?abstract_id=5332011 ___________________________________________ *💬 Socials* *TikTok:* https://www.tiktok.com/@quantguild *Instagram:* https://www.instagram.com/quantguild/ *X/Twitter:* https://x.com/quantguild/ *LinkedIn (personal):* https://www.linkedin.com/in/rmp99/ *LinkedIn (company):* https://www.linkedin.com/company/quant-guild ___________________________________________