@quantfinancetogo: ➡️ Quant Finance from scratch Part 4: why many small random steps always build the same shape Two questions guide the video: why do random steps always build this same shape, and what does it let quants say about how far a stock can move? Step 1, the board. 1,000 balls fall through 12 rows of pins and bounce right or left like a coin flip. Every ball is a random walk from part 2, and the bin where it lands counts how often it went right. Step 2, why the middle. All 4,096 paths are equally likely. Only one goes right every time, while 924 go right exactly six times and end in the middle, so most balls land there: 613 of 1,000 in the middle three bins. Step 3, part 3 again. The 1,000 walks from part 3 build the same bell after 400 steps, only wider: their spread is 19.9, the square root rule's √400 = 20. Step 4, any step shape. Add up a hundred random amounts, each anywhere between -1 and +1, and the bell appears again. That is the central limit theorem. Step 5, the rule. The bell is the normal distribution: about 68 % of results land within one standard deviation, 95 % within two and 99.7 % within three. Our 100,000 sums gave 68.2, 95.5 and 99.7 %. Step 6, a stock. A year is the sum of many daily moves, so quants model returns with the bell. With 1 % a day the yearly swing is 1 % × √252 = 15.9 %: about two years in three end within 16 % up or down, nineteen in twenty within 32 %, and the bell says a move beyond 48 % comes once in about 370 years. The honest part: real returns have fat tails. In a simulated market with fat tails and the same average swing, days beyond four standard deviations came 65 times in 40 years, where the bell expects less than one. The bell is a first model, and a later part covers the tails. This video is for education only, and nothing in it is financial advice. #quantfinance #bellcurve #normaldistribution #centrallimittheorem #python

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Wednesday 30 September 2026 12:28:17 GMT