WEBVTT 00:00:00.000 --> 00:00:02.140 Welcome back to the Deep Dive, the show dedicated 00:00:02.140 --> 00:00:04.759 to turning dense financial and scientific source 00:00:04.759 --> 00:00:08.019 material into thoroughly digestible yet sophisticated 00:00:08.019 --> 00:00:12.939 knowledge. Today, we are diving deep into what 00:00:12.939 --> 00:00:15.339 is probably the single most important and perhaps 00:00:15.339 --> 00:00:19.760 most feared metric in all of finance, volatility. 00:00:20.059 --> 00:00:22.460 It really is. It's the engine of market movement, 00:00:22.620 --> 00:00:24.699 you know, the source of both panic and opportunity. 00:00:25.100 --> 00:00:28.079 But so many people still use the term really 00:00:28.079 --> 00:00:31.250 loosely. equating volatility simply with risk. 00:00:31.429 --> 00:00:33.869 Right. Just a general sense of danger. And that's 00:00:33.869 --> 00:00:35.969 I mean, that's a very reductive view. Volatility 00:00:35.969 --> 00:00:38.329 is far more precise than that. It is fundamentally 00:00:38.329 --> 00:00:41.109 the degree of statistical variation in trading 00:00:41.109 --> 00:00:44.240 prices over a specified period of time. And if 00:00:44.240 --> 00:00:45.799 you've ever been an investor, whether you're 00:00:45.799 --> 00:00:48.140 managing a small personal portfolio or running 00:00:48.140 --> 00:00:50.640 a massive hedge fund, you have absolutely felt 00:00:50.640 --> 00:00:53.579 volatility. That surge of anxiety when you check 00:00:53.579 --> 00:00:55.460 your account right before bed or the euphoria 00:00:55.460 --> 00:00:58.259 when a stock unexpectedly pops 10%. That's it 00:00:58.259 --> 00:01:01.039 in action. That feeling is the emotional response 00:01:01.039 --> 00:01:04.439 to volatility. This one concept governs pretty 00:01:04.439 --> 00:01:06.920 much everything, from how financial institutions 00:01:06.920 --> 00:01:09.799 price incredibly complex derivatives like options 00:01:09.799 --> 00:01:14.239 to the emotional discipline required just to 00:01:14.239 --> 00:01:16.519 stay invested for the long term. So our mission 00:01:16.519 --> 00:01:18.920 for this deep dive is to give you, the listener, 00:01:19.099 --> 00:01:22.459 a really... comprehensive and actionable understanding 00:01:22.459 --> 00:01:25.439 of this concept. Exactly. We're going to go well 00:01:25.439 --> 00:01:27.579 beyond the textbook definition, you know, the 00:01:27.579 --> 00:01:30.000 standard deviation of returns, to explore the 00:01:30.000 --> 00:01:32.140 precise financial language that practitioners 00:01:32.140 --> 00:01:34.659 actually use. We'll analyze the different types 00:01:34.659 --> 00:01:37.560 of volatility. We'll look at the surprising mathematical 00:01:37.560 --> 00:01:40.239 rules that dictate how price dispersion expands 00:01:40.239 --> 00:01:43.219 over time. And finally, we will tackle the profound 00:01:43.219 --> 00:01:45.739 philosophical critiques raised by industry legends 00:01:45.739 --> 00:01:49.000 about why even the most complex forecasting models 00:01:49.870 --> 00:01:52.810 well, why they often fall short. OK, so let's 00:01:52.810 --> 00:01:55.129 nail down the big picture definition first. What 00:01:55.129 --> 00:01:58.109 exactly are we measuring? We are measuring dispersion. 00:01:58.109 --> 00:02:02.269 That's the keyword. In finance, volatility, which 00:02:02.269 --> 00:02:04.489 is nearly always denoted by the Greek letter 00:02:04.489 --> 00:02:07.689 sigma, is the degree of variation of a trading 00:02:07.689 --> 00:02:11.050 price series over time. And crucially, the standard 00:02:11.050 --> 00:02:13.650 industry measure is the standard deviation of 00:02:13.650 --> 00:02:16.250 the asset's logarithmic return. OK, let's pause 00:02:16.250 --> 00:02:19.060 there. Why? logarithmic returns specifically, 00:02:19.360 --> 00:02:22.360 not just standard percentage returns? It's a 00:02:22.360 --> 00:02:24.479 great question because it gets right to the heart 00:02:24.479 --> 00:02:27.639 of financial math. Finance deals with compounding. 00:02:27.659 --> 00:02:30.240 When you calculate returns with standard arithmetic, 00:02:30.580 --> 00:02:34.819 a move from 100 to 110 is 10%. But a move from 00:02:34.819 --> 00:02:38.699 110 to 120 is only 9 .09%. Even though they're 00:02:38.699 --> 00:02:41.199 both $10 increases. Exactly. Logarithmic return 00:02:41.199 --> 00:02:43.539 or log returns, they normalize these movements. 00:02:43.780 --> 00:02:46.900 It allows you to treat returns as additive over 00:02:46.900 --> 00:02:49.939 time, which makes the statistical analysis, particularly 00:02:49.939 --> 00:02:52.080 when you're trying to annualize things, much, 00:02:52.099 --> 00:02:53.979 much simpler. It's just a more accurate picture 00:02:53.979 --> 00:02:57.219 of continuous compounding. And for a real world 00:02:57.219 --> 00:02:58.960 anchor, a concept the listener has certainly 00:02:58.960 --> 00:03:02.530 heard of, we have the VIX. The CBOE Volatility 00:03:02.530 --> 00:03:06.009 Index or VIX is probably the most famous real 00:03:06.009 --> 00:03:08.650 time representation of this concept, but it's 00:03:08.650 --> 00:03:11.069 not calculated from historical prices. Instead, 00:03:11.250 --> 00:03:13.490 it measures the market's expectation of volatility 00:03:13.490 --> 00:03:16.530 over the next. 30 calendar days. And gets that 00:03:16.530 --> 00:03:18.770 from option prices? It's derived from the prices 00:03:18.770 --> 00:03:21.990 of S &P 500 index options. Yes. It's often called 00:03:21.990 --> 00:03:24.289 the fear gauge because it acts as a real time 00:03:24.289 --> 00:03:26.969 index of implied volatility. When that number 00:03:26.969 --> 00:03:29.689 spikes, it signals a market consensus of extreme 00:03:29.689 --> 00:03:33.270 uncertainty and high expected variation. It's 00:03:33.270 --> 00:03:37.169 fear, math and opportunity all wrapped up in 00:03:37.169 --> 00:03:39.979 a single famous ticker. Okay, let's unpack the 00:03:39.979 --> 00:03:41.860 terminology because this is where many people 00:03:41.860 --> 00:03:44.960 get lost. Volatility is often used as this kind 00:03:44.960 --> 00:03:47.180 of catch -all term, but our sources show it's 00:03:47.180 --> 00:03:49.039 rigidly categorized. It splits into two main 00:03:49.039 --> 00:03:51.400 branches, actual and implied. And entertaining 00:03:51.400 --> 00:03:54.400 the split is, well, it's foundational to any 00:03:54.400 --> 00:03:56.840 kind of advanced trading. It really is. And the 00:03:56.840 --> 00:03:58.699 key differentiating factor is just the direction 00:03:58.699 --> 00:04:00.979 of observation in time. Are you looking backward 00:04:00.979 --> 00:04:02.719 or are you looking forward? Okay, let's start 00:04:02.719 --> 00:04:04.759 with the rearview mirror. Actual volatility, 00:04:06.060 --> 00:04:08.039 which is often just called historical volatility, 00:04:08.300 --> 00:04:10.360 is purely backward looking. It's a measure of 00:04:10.360 --> 00:04:14.000 fact. It's based on a time series of past market 00:04:14.000 --> 00:04:16.500 prices. So if you are calculating the standard 00:04:16.500 --> 00:04:18.779 deviation of Apple's daily returns over the last 00:04:18.779 --> 00:04:21.639 year, you are calculating its actual volatility. 00:04:21.800 --> 00:04:23.439 It's something that has already happened. A known 00:04:23.439 --> 00:04:25.839 quantity. Conversely, we have the predictive 00:04:25.839 --> 00:04:28.519 element. And that's implied volatility. It looks 00:04:28.519 --> 00:04:30.930 forward in time. It is not calculated from the 00:04:30.930 --> 00:04:33.569 asset's past price movements at all. Instead, 00:04:33.829 --> 00:04:36.990 it is derived or implied from the current market 00:04:36.990 --> 00:04:39.230 price of a specific derivative, specifically 00:04:39.230 --> 00:04:41.730 a traded option. How does that work? How is it 00:04:41.730 --> 00:04:44.160 implied? Well, the price of an option inherently 00:04:44.160 --> 00:04:46.860 contains the collective market's consensus expectation 00:04:46.860 --> 00:04:49.699 of how volatile the underlying asset will be 00:04:49.699 --> 00:04:51.720 between now and the option's expiration date. 00:04:51.920 --> 00:04:54.680 So what we do is we take the famous black skulls 00:04:54.680 --> 00:04:56.639 formula, we plug in the current option price, 00:04:56.879 --> 00:05:00.360 and we solve it backwards to find the one unknown 00:05:00.360 --> 00:05:03.480 variable, the implied volatility that justifies 00:05:03.480 --> 00:05:07.500 that price. So, actual volatility is a calculated 00:05:07.500 --> 00:05:11.500 certainty, a known historical fact. whereas implied 00:05:11.500 --> 00:05:14.300 volatility is the market's current best guess. 00:05:14.759 --> 00:05:17.360 about future price dispersion. Exactly. A perfect 00:05:17.360 --> 00:05:19.519 summary. And even within that backward looking 00:05:19.519 --> 00:05:22.959 data, the term actual volatility has three specific 00:05:22.959 --> 00:05:24.959 dimensions, depending on the timeframe we're 00:05:24.959 --> 00:05:27.639 looking at. Practitioners need this level of 00:05:27.639 --> 00:05:29.699 precision. Right. So let's detail the first of 00:05:29.699 --> 00:05:32.620 those three actual current volatility. This refers 00:05:32.620 --> 00:05:34.920 to volatility that's calculated based on historical 00:05:34.920 --> 00:05:38.279 prices over a very specified recent period, say 00:05:38.279 --> 00:05:42.240 the last 30 or 90 days. The crucial element here 00:05:42.240 --> 00:05:44.319 is that the last observation in the data set 00:05:44.319 --> 00:05:46.319 is the most recent price, like today's close. 00:05:46.660 --> 00:05:49.480 You're trying to gauge the immediate past market 00:05:49.480 --> 00:05:51.439 environment leading right up to this moment. 00:05:51.620 --> 00:05:53.600 Like a moving window that ends right now. Exactly. 00:05:53.879 --> 00:05:56.240 Now, the second one is actual historical volatility, 00:05:56.540 --> 00:05:58.779 which you'll also hear called realized volatility. 00:05:59.279 --> 00:06:00.740 Okay. How is that different from what you just 00:06:00.740 --> 00:06:03.600 said? It's conceptually identical, but the specified 00:06:03.600 --> 00:06:06.560 period ends on a date in the past. So maybe you 00:06:06.560 --> 00:06:08.379 want to analyze market behavior specifically 00:06:08.379 --> 00:06:12.319 during the 2020... crash for the period ending 00:06:12.319 --> 00:06:15.899 on, say, April 1st, 2020. That's historical volatility. 00:06:16.279 --> 00:06:19.319 The term realized volatility is used a lot, especially 00:06:19.319 --> 00:06:21.600 in academic papers, and it's technically defined 00:06:21.600 --> 00:06:23.800 as the square root of the realized variance. 00:06:24.079 --> 00:06:26.699 It's just the measure of how much price variation 00:06:26.699 --> 00:06:30.000 actually occurred over some specific past interval. 00:06:30.259 --> 00:06:33.839 Got it. And the third, which must always be stated 00:06:33.839 --> 00:06:35.759 with the caveat that we can't actually know it 00:06:35.759 --> 00:06:38.839 yet, is actual future volatility. This is the 00:06:38.839 --> 00:06:40.860 unknown target. This is what everyone's trying 00:06:40.860 --> 00:06:43.120 to predict. It's the measure of the volatility 00:06:43.120 --> 00:06:45.660 of a financial instrument over a specified period 00:06:45.660 --> 00:06:47.779 starting now and ending at a future date, say, 00:06:47.879 --> 00:06:50.259 the next option expiration cycle three months 00:06:50.259 --> 00:06:52.319 from now. We can only know this number when that 00:06:52.319 --> 00:06:54.220 future date finally arrives, which is why we 00:06:54.220 --> 00:06:56.060 spend so much time and energy trying to predict 00:06:56.060 --> 00:06:58.459 it using all the other types of volatility. Right. 00:06:58.540 --> 00:07:01.879 Now we flip to the forward -looking side. Implied 00:07:01.879 --> 00:07:05.209 volatility. So if actual volatility looks at 00:07:05.209 --> 00:07:07.949 past, present, and future periods using historical 00:07:07.949 --> 00:07:11.189 prices, then implied volatility looks at those 00:07:11.189 --> 00:07:13.930 same three time points, but through the lens 00:07:13.930 --> 00:07:17.189 of options prices. Precisely. Implied volatility 00:07:17.189 --> 00:07:20.250 is observed from the options market, and we categorize 00:07:20.250 --> 00:07:22.269 it simply by the point in time we observe the 00:07:22.269 --> 00:07:24.910 options prices themselves. So the first is historical 00:07:24.910 --> 00:07:27.829 implied volatility. This is the implied volatility 00:07:27.829 --> 00:07:29.629 you would get while looking at options prices 00:07:29.629 --> 00:07:32.430 that existed on some historical date. It doesn't 00:07:32.430 --> 00:07:34.069 tell you what the stock actually did. It tells 00:07:34.069 --> 00:07:35.769 you what the market expected the stock to do 00:07:35.769 --> 00:07:37.810 back then. For example, if you wanted to see 00:07:37.810 --> 00:07:40.089 the level of fear right before the 2008 election, 00:07:40.389 --> 00:07:42.589 you look at the historical implied volatility 00:07:42.589 --> 00:07:45.209 of options that were trading in, say, September 00:07:45.209 --> 00:07:47.490 of 2008. That's fascinating. And the second and 00:07:47.490 --> 00:07:49.410 probably the most operational term for traders, 00:07:49.629 --> 00:07:52.189 current implied volatility. This is what most 00:07:52.189 --> 00:07:54.529 people mean when they just say implied vol -ish. 00:07:54.750 --> 00:07:57.149 It's the implied volatility derived from the 00:07:57.149 --> 00:07:59.189 options prices that are trading right now, at 00:07:59.189 --> 00:08:02.370 this very moment. This is the VIX number, or 00:08:02.370 --> 00:08:04.670 the standard figure you'd see in a trader's model. 00:08:04.910 --> 00:08:07.790 It represents the market's real -time consensus 00:08:07.790 --> 00:08:10.850 forecast for future price dispersion over the 00:08:10.850 --> 00:08:13.189 life of the option. Okay, and finally, the most 00:08:13.189 --> 00:08:17.019 abstract one, future implied volatility. This 00:08:17.019 --> 00:08:19.540 refers to the implied volatility observed from 00:08:19.540 --> 00:08:22.060 options prices that won't even exist until some 00:08:22.060 --> 00:08:24.339 point in the future. We often observe this indirectly. 00:08:24.720 --> 00:08:27.939 If you buy a complex derivative known as a volatility 00:08:27.939 --> 00:08:30.759 swap or a swaption, you are essentially making 00:08:30.759 --> 00:08:33.460 a trade today on the implied volatility that 00:08:33.460 --> 00:08:35.879 the market expects to see at some future date. 00:08:36.179 --> 00:08:38.639 So it's the market's expected expectation of 00:08:38.639 --> 00:08:40.720 future fear. A great way to put it. It's a second 00:08:40.720 --> 00:08:43.320 order derivative. That is a fascinating taxonomy. 00:08:43.480 --> 00:08:45.879 But why is it so crucial for our listener to... 00:08:46.120 --> 00:08:48.539 Hold these six terms straight. What is the practical 00:08:48.539 --> 00:08:51.659 application? The application is the entire prediction 00:08:51.659 --> 00:08:54.399 game and the search for value for mispricing. 00:08:55.210 --> 00:08:57.669 You see, investors often make the basic mistake 00:08:57.669 --> 00:09:00.649 of using historical actual volatility as their 00:09:00.649 --> 00:09:03.429 only input to predict actual future volatility. 00:09:03.750 --> 00:09:06.210 They just assume inertia. They think, well, the 00:09:06.210 --> 00:09:08.830 past 90 days were stable, so the next 90 days 00:09:08.830 --> 00:09:10.750 will probably be stable, too. Which is often 00:09:10.750 --> 00:09:13.090 a terrible assumption right before a major event 00:09:13.090 --> 00:09:15.570 like an election or an earnings call. Exactly. 00:09:15.789 --> 00:09:18.230 But if you look at the current implied volatility 00:09:18.230 --> 00:09:20.990 and you see that it's significantly higher than 00:09:20.990 --> 00:09:23.830 the actual current volatility, that divergence 00:09:23.830 --> 00:09:27.559 is a massive flashing warning sign. The market 00:09:27.559 --> 00:09:29.899 consensus priced into the options is telling 00:09:29.899 --> 00:09:33.279 you we expect things to get much, much wilder 00:09:33.279 --> 00:09:35.700 than they have been recently. That signal gives 00:09:35.700 --> 00:09:38.700 you a crucial risk management shortcut and it 00:09:38.700 --> 00:09:41.039 helps you identify potential market mispricing 00:09:41.039 --> 00:09:43.600 or an unappreciated upcoming event. You're essentially 00:09:43.600 --> 00:09:47.179 comparing objective realized data against the 00:09:47.179 --> 00:09:48.980 collective forward looking market sentiment. 00:09:49.340 --> 00:09:51.179 Precisely. And the difference between the two 00:09:51.179 --> 00:09:53.570 is where traders make their money. Now we move 00:09:53.570 --> 00:09:56.350 into the cold hard math. This section is where 00:09:56.350 --> 00:09:58.570 we strip away the psychology and establish the 00:09:58.570 --> 00:10:01.389 fundamental rules governing price movement. At 00:10:01.389 --> 00:10:04.909 its core, we said volatility is the statistical 00:10:04.909 --> 00:10:07.110 standard deviation of the logarithmic returns, 00:10:07.350 --> 00:10:09.870 but we have to standardize this measure to make 00:10:09.870 --> 00:10:11.909 it comparable across different assets and time 00:10:11.909 --> 00:10:14.519 frames. And the standard comparable measure that 00:10:14.519 --> 00:10:17.399 we use is annualized volatility. That's the key 00:10:17.399 --> 00:10:19.519 term. You'll see it written as sigma annually. 00:10:19.840 --> 00:10:22.480 This is defined as the standard deviation of 00:10:22.480 --> 00:10:25.179 an instrument's expected yearly logarithmic returns. 00:10:25.639 --> 00:10:28.519 This annualization process is absolutely essential 00:10:28.519 --> 00:10:31.080 because, you know, daily volatility looks tiny, 00:10:31.259 --> 00:10:33.860 monthly volatility is small, but the yearly measure 00:10:33.860 --> 00:10:36.480 provides a consistent benchmark for risk management. 00:10:36.759 --> 00:10:39.440 So you can compare the risk of a stock to a bond 00:10:39.440 --> 00:10:41.820 to a currency all on the same scale. Exactly. 00:10:41.899 --> 00:10:44.519 regardless of how frequently it trades or the 00:10:44.519 --> 00:10:46.539 length of the holding period you're analyzing. 00:10:46.799 --> 00:10:49.720 And this standardization process brings us directly 00:10:49.720 --> 00:10:52.480 to a principle that often surprises people who 00:10:52.480 --> 00:10:55.120 are new to finance, the square root of time rule. 00:10:56.039 --> 00:10:58.639 Our sources confirm that if an asset's returns 00:10:58.639 --> 00:11:01.799 follow a Gaussian random walk, which is the big 00:11:01.799 --> 00:11:04.340 simplifying assumption in the classic black schools 00:11:04.340 --> 00:11:07.919 model known as a wiener, process volatility does 00:11:07.919 --> 00:11:10.399 not increase linearly with time. This is one 00:11:10.399 --> 00:11:14.179 of the most powerful and counterintuitive ideas 00:11:14.179 --> 00:11:17.220 in quantitative finance. While the price dispersion 00:11:17.220 --> 00:11:19.259 does increase over time, making it more likely 00:11:19.259 --> 00:11:21.120 for the price to stray farther from its starting 00:11:21.120 --> 00:11:24.039 point, that dispersion increases only with the 00:11:24.039 --> 00:11:26.159 square root of time. Okay, we need to spend some 00:11:26.159 --> 00:11:28.879 time on the intuition behind this. Why the square 00:11:28.879 --> 00:11:31.299 root? Why isn't 10 days of volatility simply 00:11:31.299 --> 00:11:34.000 10 times the daily volatility? Think about the 00:11:34.000 --> 00:11:36.639 nature of randomness. Price movements, especially 00:11:36.639 --> 00:11:39.179 in very liquid markets, are somewhat independent 00:11:39.179 --> 00:11:42.059 from day to day. If the price goes up 1 % today, 00:11:42.279 --> 00:11:44.679 it has a roughly equal chance of going down 1 00:11:44.679 --> 00:11:47.679 % tomorrow. So if you aggregate 100 days of movement, 00:11:47.899 --> 00:11:50.399 you shouldn't assume that... all 100 days moved 00:11:50.399 --> 00:11:52.460 in the same direction. Right. There's a high 00:11:52.460 --> 00:11:54.679 probability that some of those fluctuations will 00:11:54.679 --> 00:11:57.779 offset or cancel each other out over longer periods. 00:11:57.840 --> 00:12:00.620 It's the expected chance of reversal that dampens 00:12:00.620 --> 00:12:02.620 the aggregate movement. It's like flipping a 00:12:02.620 --> 00:12:05.700 coin repeatedly. Well, if you flip a coin four 00:12:05.700 --> 00:12:08.519 times, the expected range of outcomes for heads 00:12:08.519 --> 00:12:11.000 minus tails is much, much less than four times 00:12:11.000 --> 00:12:13.960 the range of flipping it just once. Why? Because 00:12:13.960 --> 00:12:15.940 the results are likely to partially negate each 00:12:15.940 --> 00:12:18.710 other. The net effect is that the most likely 00:12:18.710 --> 00:12:21.629 deviation after twice the time will be only the 00:12:21.629 --> 00:12:23.970 square root of two times the original deviation, 00:12:24.289 --> 00:12:27.570 not twice the deviation. That makes sense. And 00:12:27.570 --> 00:12:29.549 the generalized formula reflects this elegant 00:12:29.549 --> 00:12:33.190 nonlinear scaling. The volatility over a time 00:12:33.190 --> 00:12:36.169 period t is the annualized volatility multiplied 00:12:36.169 --> 00:12:38.669 by the square root of time t, where t is measured 00:12:38.669 --> 00:12:40.870 in years. This is the essential bridge between 00:12:40.870 --> 00:12:43.720 short -term noise and long -term risk. And this 00:12:43.720 --> 00:12:46.220 principle is applied operationally every single 00:12:46.220 --> 00:12:49.200 day when we perform the necessary step of annualizing 00:12:49.200 --> 00:12:51.440 daily returns. So we take our daily volatility, 00:12:51.779 --> 00:12:54.019 let's call it sigma daily, and we convert it 00:12:54.019 --> 00:12:56.379 into the standard yearly number, sigma annually. 00:12:56.639 --> 00:12:59.580 The formula is sigma annually sigma daily squirt 00:12:59.580 --> 00:13:02.460 PSP, where P is the number of trading days in 00:13:02.460 --> 00:13:04.889 a year. And the source highlights the common 00:13:04.889 --> 00:13:07.230 assumption for P. In North American and many 00:13:07.230 --> 00:13:09.289 global markets, the standard assumption for P 00:13:09.289 --> 00:13:13.110 is 252 trading days per year. This accounts for 00:13:13.110 --> 00:13:15.590 52 weekends and the major national holidays. 00:13:15.929 --> 00:13:19.529 You'll sometimes see 250 or 260 U's, but 252 00:13:19.529 --> 00:13:21.990 is the most robust industry convention. Let's 00:13:21.990 --> 00:13:23.909 run the concrete example provided in our source 00:13:23.909 --> 00:13:25.789 material, because this really drives home the 00:13:25.789 --> 00:13:28.169 power of the square root rule. Say we observe 00:13:28.169 --> 00:13:30.889 an average daily volatility, a standard deviation 00:13:30.889 --> 00:13:35.350 of daily log return. of 0 .01 or 1%, what does 00:13:35.350 --> 00:13:37.529 that look like when it's annualized? So using 00:13:37.529 --> 00:13:40.490 our formula, sigma annually equals 0 .01 times 00:13:40.490 --> 00:13:43.529 the square root of 252. The result is approximately 00:13:43.529 --> 00:13:48.389 0 .1587. So that seemingly tiny 1 % daily fluctuation 00:13:48.389 --> 00:13:50.909 translates into an annualized volatility of 15 00:13:50.909 --> 00:13:54.909 .87%. That is a staggering amplification of risk 00:13:54.909 --> 00:13:56.870 perception based purely on the time horizon. 00:13:57.289 --> 00:14:00.809 If someone just naively multiplied 1 % by 252 00:14:00.809 --> 00:14:03.169 days, they would erroneously conclude the annual 00:14:03.169 --> 00:14:06.809 volatility was 252%. Which is absurd, right? 00:14:06.909 --> 00:14:09.610 That linear calculation vastly overestimates 00:14:09.610 --> 00:14:12.009 the expected dispersion, and it really underscores 00:14:12.009 --> 00:14:14.509 why the square root rule is absolutely mandatory. 00:14:14.970 --> 00:14:17.350 It's the difference between assuming that every 00:14:17.350 --> 00:14:19.750 single day will compound perfectly in one direction 00:14:19.750 --> 00:14:22.929 without any reversals versus the statistical 00:14:22.929 --> 00:14:25.970 reality of a random walk. Now, for the real world, 00:14:26.029 --> 00:14:27.659 where we often... and need quick estimates without 00:14:27.659 --> 00:14:30.120 a calculator, we have the famous shortcut known 00:14:30.120 --> 00:14:32.879 as the rule of 16. The rule of 16 is fantastic 00:14:32.879 --> 00:14:35.259 for back of the envelope calculations, but it's 00:14:35.259 --> 00:14:37.580 critical to understand its limitations. The rationale 00:14:37.580 --> 00:14:40.440 is purely convenience. 16 is the square root 00:14:40.440 --> 00:14:43.639 of 256, which is an extremely close round number 00:14:43.639 --> 00:14:46.700 approximation of 252 trading days. So if you 00:14:46.700 --> 00:14:49.100 read a news story that an asset index moves about 00:14:49.100 --> 00:14:51.700 100 points on average each day and the index 00:14:51.700 --> 00:14:55.289 is near 10 ,000, which is a 1 % daily move. You 00:14:55.289 --> 00:14:58.970 multiply 1 % by 16 and quickly estimate 16 % 00:14:58.970 --> 00:15:03.450 annual volatility. It's a fast, simple, and satisfyingly 00:15:03.450 --> 00:15:07.350 close to the mathematically precise 15 .87%. 00:15:07.350 --> 00:15:09.570 It's great for that. But the source material 00:15:09.570 --> 00:15:13.049 provides a crucial warning. This is a crude estimation. 00:15:13.169 --> 00:15:16.009 It is not a statistical calculation. Why is it 00:15:16.009 --> 00:15:18.570 crude specifically? What does it miss? Well, 00:15:18.590 --> 00:15:20.970 a few things. First, it often relies on the average 00:15:20.970 --> 00:15:23.850 magnitude of observations. essentially the average 00:15:23.850 --> 00:15:25.950 daily price range, rather than the statistically 00:15:25.950 --> 00:15:28.649 derived standard deviation of returns. And the 00:15:28.649 --> 00:15:31.629 standard deviation, by its very definition, captures 00:15:31.629 --> 00:15:34.169 the influence of those extreme, infrequent daily 00:15:34.169 --> 00:15:36.529 movements, the outliers, because it squares the 00:15:36.529 --> 00:15:39.090 differences from the mean. The rule of 16 often 00:15:39.090 --> 00:15:41.750 fails to properly account for those massive single 00:15:41.750 --> 00:15:44.350 -day moves. So the shortcut is structurally biased 00:15:44.350 --> 00:15:46.830 against capturing tail risk. It misses the big 00:15:46.830 --> 00:15:49.570 crashes and the big rallies. Exactly. And furthermore, 00:15:49.629 --> 00:15:51.950 just from a pure math perspective, assuming a 00:15:51.950 --> 00:15:54.250 normal distribution with a mean of zero, the 00:15:54.250 --> 00:15:56.370 expected value of the magnitude of observations 00:15:56.370 --> 00:15:58.690 is actually slightly less than the true standard 00:15:58.690 --> 00:16:02.049 deviation. The combined result of these approximations 00:16:02.049 --> 00:16:04.929 is that the crude rule of 16 generally underestimates 00:16:04.929 --> 00:16:08.450 the true volatility by about 20%. It's a great 00:16:08.450 --> 00:16:10.450 starting point for intuition, but you should 00:16:10.450 --> 00:16:14.590 never use it for, say, pricing an option. And 00:16:14.590 --> 00:16:16.809 this brings us to the most significant flaw in 00:16:16.809 --> 00:16:18.850 traditional financial modeling, which is the 00:16:18.850 --> 00:16:21.070 assumption we've been relying on this whole time. 00:16:21.250 --> 00:16:23.850 The Gaussian model. Yeah. We have to go beyond 00:16:23.850 --> 00:16:25.429 the Gaussian model because the real world does 00:16:25.429 --> 00:16:27.710 not follow a perfect bell curve. This is the 00:16:27.710 --> 00:16:31.850 source of endless frustration and risk for quants 00:16:31.850 --> 00:16:34.529 and investors alike. All the standard formulas, 00:16:34.769 --> 00:16:36.570 the square root rule, the calculation of the 00:16:36.570 --> 00:16:39.610 95 % return range, they all rely on the assumption 00:16:39.610 --> 00:16:41.649 that price changes follow a Gaussian or normal 00:16:41.649 --> 00:16:44.169 distribution. But empirical evidence confirms 00:16:44.169 --> 00:16:46.570 over and over again that observed price changes 00:16:46.570 --> 00:16:49.549 are often leptokurtotic. Let's dedicate an exchange 00:16:49.549 --> 00:16:51.629 to defining that term clearly for the listener. 00:16:51.809 --> 00:16:54.629 What does leptokurtotic mean? It's a fancy term 00:16:54.629 --> 00:16:57.009 for having fat tails. Think of a standard bell 00:16:57.009 --> 00:16:59.700 curve. The tails on the far left and far right 00:16:59.700 --> 00:17:02.700 represent the extremely rare events, the two, 00:17:02.759 --> 00:17:06.299 three, or five standard deviation moves. A normal 00:17:06.299 --> 00:17:08.460 distribution predicts that a five sigma event, 00:17:08.700 --> 00:17:11.599 a massive catastrophic move, should happen something 00:17:11.599 --> 00:17:14.299 like once every several million days, maybe once 00:17:14.299 --> 00:17:16.559 every few thousand years. And what do the fat 00:17:16.559 --> 00:17:18.940 tails of real -world financial data tell us? 00:17:19.230 --> 00:17:22.150 They tell us that those extreme, infrequent movements 00:17:22.150 --> 00:17:24.650 happen much, much more often than the normal 00:17:24.650 --> 00:17:27.890 distribution predicts. Crashes, bubbles, massive 00:17:27.890 --> 00:17:30.490 single -day jumps or drops. These events are 00:17:30.490 --> 00:17:32.769 fundamentally more probable in financial markets 00:17:32.769 --> 00:17:35.490 than the clean Gaussian math allows for. So if 00:17:35.490 --> 00:17:37.529 the model says a 10 % drop should happen once 00:17:37.529 --> 00:17:40.210 every century, the fat tails are telling us financially 00:17:40.210 --> 00:17:42.529 that it might happen once every 10 or 20 years. 00:17:42.750 --> 00:17:45.549 This mismatch between theory and reality is why 00:17:45.549 --> 00:17:48.609 WISC management models often underestimate catastrophes. 00:17:48.519 --> 00:17:51.160 It's a crucial insight. So how do sophisticated 00:17:51.160 --> 00:17:53.779 models try to capture this reality? Well, they 00:17:53.779 --> 00:17:56.240 have to turn to alternative statistical frameworks. 00:17:56.519 --> 00:17:59.440 For instance, the Levy distribution, specifically 00:17:59.440 --> 00:18:02.000 the Levy alpha stable distribution, is often 00:18:02.000 --> 00:18:05.000 used. These models are built to accommodate processes 00:18:05.000 --> 00:18:08.000 where large jumps are more frequent and the standard 00:18:08.000 --> 00:18:10.579 deviation might even be technically infinite. 00:18:11.609 --> 00:18:14.029 The famous mathematician Benoit Mendelbrot found 00:18:14.029 --> 00:18:16.329 that when he was analyzing cotton prices, the 00:18:16.329 --> 00:18:19.230 data perfectly fit a Levy alpha -stable distribution 00:18:19.230 --> 00:18:22.769 with an alpha value of 1 .7. And you need an 00:18:22.769 --> 00:18:25.680 alpha value of 2. for the standard predictable 00:18:25.680 --> 00:18:28.960 Wiener process, the random walk. Correct. The 00:18:28.960 --> 00:18:30.980 fact that financial markets often exhibit an 00:18:30.980 --> 00:18:33.940 alpha of less than two suggests that market movement 00:18:33.940 --> 00:18:36.400 is significantly less predictable, potentially 00:18:36.400 --> 00:18:39.279 infinitely more volatile, and certainly much, 00:18:39.420 --> 00:18:41.740 much messier than the clean classical financial 00:18:41.740 --> 00:18:44.980 formulas suggest. This realization is what drives 00:18:44.980 --> 00:18:46.980 the constant search for better modeling techniques. 00:18:47.440 --> 00:18:49.579 We've established the math, but the sources we 00:18:49.579 --> 00:18:52.279 reviewed noted a critical point. Despite decades 00:18:52.279 --> 00:18:54.690 of research and complex models, modeling, very 00:18:54.690 --> 00:18:57.490 few theoretical models truly explain how volatility 00:18:57.490 --> 00:18:59.750 originates in the first place. What actually 00:18:59.750 --> 00:19:02.710 causes price variation day after day. Yeah, the 00:19:02.710 --> 00:19:05.470 focus has shifted over time from macroeconomics 00:19:05.470 --> 00:19:07.529 to what's called market microstructure, which 00:19:07.529 --> 00:19:09.450 is the study of the mechanics of trading itself. 00:19:10.259 --> 00:19:13.579 Roll, back in 1984, provided some of the early 00:19:13.579 --> 00:19:15.859 evidence that volatility isn't just an inherent 00:19:15.859 --> 00:19:18.619 property of an asset, but it's actively affected 00:19:18.619 --> 00:19:20.920 by the specific rules and mechanics of how the 00:19:20.920 --> 00:19:23.799 market operates, how orders are placed, how prices 00:19:23.799 --> 00:19:27.160 are quoted, the timing of information flow. And 00:19:27.160 --> 00:19:30.180 then Glossin and Milgram in 1985 gave us a very 00:19:30.180 --> 00:19:32.700 granular explanation for one specific source, 00:19:32.920 --> 00:19:35.849 the liquidity provision process. This is where 00:19:35.849 --> 00:19:38.650 we see the fear element quantified. Liquidity 00:19:38.650 --> 00:19:41.109 is provided by professional market makers who 00:19:41.109 --> 00:19:43.490 have to constantly stand ready to buy at their 00:19:43.490 --> 00:19:46.029 bid price or sell at their ask price. They are 00:19:46.029 --> 00:19:48.690 the essential lubricant of the market, but they 00:19:48.690 --> 00:19:50.549 operate under the constant threat of adverse 00:19:50.549 --> 00:19:52.769 selection. And that's the fear of being taken 00:19:52.769 --> 00:19:54.849 advantage of by someone with better information, 00:19:55.150 --> 00:19:57.789 someone who knows for sure the stock is truly 00:19:57.789 --> 00:20:01.519 undervalued or overvalued. Precisely. If a market 00:20:01.519 --> 00:20:03.579 maker suspects there's a high probability they 00:20:03.579 --> 00:20:06.039 are trading with a better informed party, which 00:20:06.039 --> 00:20:07.960 is common during periods of political turmoil, 00:20:08.160 --> 00:20:10.240 regulatory uncertainty, or right before an earnings 00:20:10.240 --> 00:20:13.180 report, they have to protect themselves. They 00:20:13.180 --> 00:20:15.299 do this by widening the difference between their 00:20:15.299 --> 00:20:18.200 bid and ask price. They increase the bid ask 00:20:18.200 --> 00:20:20.900 spread. And that widening of the bid ask spread 00:20:20.900 --> 00:20:23.319 directly translates to an increase in volatility. 00:20:23.640 --> 00:20:26.099 Absolutely. The wider the spread, the larger 00:20:26.099 --> 00:20:28.180 the band of price oscillation is for the next 00:20:28.180 --> 00:20:31.500 trade to occur within. Volatility in this context 00:20:31.500 --> 00:20:34.359 becomes less an abstract mathematical measure 00:20:34.359 --> 00:20:37.180 and more a quantifiable symptom of information 00:20:37.180 --> 00:20:39.579 asymmetry and collective nervousness among the 00:20:39.579 --> 00:20:42.000 professionals who facilitate trading. It's their 00:20:42.000 --> 00:20:44.299 insurance premium against being blindsided. And 00:20:44.299 --> 00:20:46.779 sometimes that information asymmetry or just 00:20:46.779 --> 00:20:49.339 general uncertainty comes from truly external 00:20:49.339 --> 00:20:52.099 sources. We have a fantastic modern anecdote 00:20:52.099 --> 00:20:54.960 that captures this exogenous nature of some volatility. 00:20:55.259 --> 00:20:59.079 The Volvo Index. Yeah. Oh, this story is fantastic 00:20:59.079 --> 00:21:01.880 because it bridges political science and quantitative 00:21:01.880 --> 00:21:06.339 finance. In September of 2019, analysts at JPMorgan 00:21:06.339 --> 00:21:09.519 Chase developed the Vol -Fet Index, a play on 00:21:09.519 --> 00:21:12.680 volatility and the Ka -Fet typo meme, specifically 00:21:12.680 --> 00:21:15.019 to track the correlation and effect of U .S. 00:21:15.019 --> 00:21:17.539 President Donald Trump's tweets on market volatility. 00:21:17.920 --> 00:21:19.619 What were they actually quantifying? What did 00:21:19.619 --> 00:21:21.640 the index measure? They quantified two things. 00:21:21.859 --> 00:21:24.440 The frequency of the tweets and also the subject 00:21:24.440 --> 00:21:26.579 matter. For example, if it was about trade, the 00:21:26.579 --> 00:21:29.180 Fed or China. And they found a statistically 00:21:29.180 --> 00:21:31.799 significant link between the frequency and content 00:21:31.799 --> 00:21:34.420 of the president's social media activity and 00:21:34.420 --> 00:21:37.859 immediate short -term market variation. A sudden, 00:21:37.980 --> 00:21:40.619 unexpected tweet on trade policy caused market 00:21:40.619 --> 00:21:43.079 makers to instantly widen their spreads due to 00:21:43.079 --> 00:21:45.799 this new, unexpected information. And that translated 00:21:45.799 --> 00:21:48.599 directly into a measurable spike in both realized 00:21:48.599 --> 00:21:50.880 and implied volatility. It showed that a single 00:21:50.880 --> 00:21:53.140 political figure's social media feed could become 00:21:53.140 --> 00:21:55.299 a primary source of microstructure volatility. 00:21:56.140 --> 00:21:58.759 forcing quantitative models to incorporate real 00:21:58.759 --> 00:22:00.980 -time linguistic analysis into their forecasts. 00:22:01.380 --> 00:22:03.680 That's incredible. Okay, so that's the genesis 00:22:03.680 --> 00:22:06.400 of volatility. Let's pivot and look at the consequence 00:22:06.400 --> 00:22:09.500 for the listener. Why does this deep dive matter 00:22:09.500 --> 00:22:12.180 for the individual investor? Our source lists 00:22:12.180 --> 00:22:15.420 seven distinct, profound reasons why volatility 00:22:15.420 --> 00:22:18.220 is central to investor decisions and portfolio 00:22:18.220 --> 00:22:21.069 construction. These points move the discussion 00:22:21.069 --> 00:22:23.329 from the theoretical realm of the quant desk 00:22:23.329 --> 00:22:26.089 right into the practical implications for every 00:22:26.089 --> 00:22:28.049 single person managing capital. Okay, number 00:22:28.049 --> 00:22:31.410 one, the emotional impact. This is pure behavioral 00:22:31.410 --> 00:22:34.829 finance. An asset with high volatility tests 00:22:34.829 --> 00:22:37.029 the discipline of the investor. It's that simple. 00:22:37.329 --> 00:22:39.990 Wider, faster price swings make it profoundly 00:22:39.990 --> 00:22:42.670 harder emotionally to stick to a long term plan. 00:22:42.970 --> 00:22:45.390 People are naturally loss averse and volatility 00:22:45.390 --> 00:22:48.069 exploits that aversion, often leading to panic 00:22:48.069 --> 00:22:50.369 selling at the absolute bottom or chasing gains 00:22:50.369 --> 00:22:52.710 at the top. Number two is about position sizing. 00:22:53.049 --> 00:22:55.630 Volatility is the primary determinant of risk. 00:22:56.200 --> 00:22:58.400 Therefore, it dictates how much capital should 00:22:58.400 --> 00:23:01.259 be allocated to any given security within a diversified 00:23:01.259 --> 00:23:05.380 portfolio. The principle is really clear. If 00:23:05.380 --> 00:23:08.000 an asset has high volatility, you must hold a 00:23:08.000 --> 00:23:10.680 smaller position in it to maintain the same overall 00:23:10.680 --> 00:23:13.579 risk contribution to your total portfolio as 00:23:13.579 --> 00:23:16.000 a more stable asset would have. This is risk 00:23:16.000 --> 00:23:18.819 budgeting in action. Number three, shortfall 00:23:18.819 --> 00:23:21.490 risk. This is critical for anyone with a known 00:23:21.490 --> 00:23:23.869 fixed liability coming up in the future. If you 00:23:23.869 --> 00:23:26.309 need cash flow at a specific future date to pay 00:23:26.309 --> 00:23:29.490 for, say, a child's college tuition, higher volatility 00:23:29.490 --> 00:23:31.369 means there is a much greater chance that the 00:23:31.369 --> 00:23:33.769 portfolio's value will fall short of that required 00:23:33.769 --> 00:23:36.470 amount when the deadline arrives. Volatility 00:23:36.470 --> 00:23:39.089 introduces uncertainty into meeting fixed obligations. 00:23:39.609 --> 00:23:42.250 Number four, retirement savings during the accumulation 00:23:42.250 --> 00:23:44.839 phase. While you're saving and contributing to 00:23:44.839 --> 00:23:47.319 your retirement accounts, higher volatility means 00:23:47.319 --> 00:23:50.119 a wider distribution of possible final portfolio 00:23:50.119 --> 00:23:53.480 values. Now, while you do have a higher chance 00:23:53.480 --> 00:23:56.339 of ending up significantly wealthy due to large 00:23:56.339 --> 00:23:59.279 positive swings, you also face a higher probability 00:23:59.279 --> 00:24:03.380 of ending up significantly poor. Volatility increases 00:24:03.380 --> 00:24:05.279 the planning uncertainty for your retirement 00:24:05.279 --> 00:24:07.380 goals. And that leads right into number five. 00:24:08.059 --> 00:24:10.299 Post -retirement impact, which is sometimes called 00:24:10.299 --> 00:24:12.980 sequence of return risk. This is perhaps the 00:24:12.980 --> 00:24:15.460 most devastating practical impact. Once you are 00:24:15.460 --> 00:24:17.660 retired and taking regular withdrawals, high 00:24:17.660 --> 00:24:20.480 volatility, particularly early in your retirement, 00:24:20.680 --> 00:24:23.220 can permanently damage your portfolio. If the 00:24:23.220 --> 00:24:25.799 market suffers a sharp drop high volatility and 00:24:25.799 --> 00:24:27.839 you are forced to withdraw funds at that low 00:24:27.839 --> 00:24:29.900 point, you compound the loss and significantly 00:24:29.900 --> 00:24:32.660 reduce the capital base needed for future recovery. 00:24:33.289 --> 00:24:35.269 This sequence of return risk is primarily driven 00:24:35.269 --> 00:24:37.690 by volatility. Number six is the other side of 00:24:37.690 --> 00:24:40.109 the coin, the concept of informed opportunity. 00:24:40.269 --> 00:24:42.890 Right. For professionals and those with superior 00:24:42.890 --> 00:24:46.000 information, volatility is a feature. not a bug. 00:24:46.220 --> 00:24:49.299 It presents opportunities to acquire assets cheaply 00:24:49.299 --> 00:24:51.200 when the price is depressed due to temporary 00:24:51.200 --> 00:24:54.200 noise and fear, or to sell them when short -term 00:24:54.200 --> 00:24:56.839 enthusiasm pushes them way above their intrinsic 00:24:56.839 --> 00:25:00.079 value. Volatility creates the pricing inefficiencies 00:25:00.079 --> 00:25:02.460 that active traders are always looking to exploit. 00:25:02.700 --> 00:25:05.420 And finally, number seven, we come back to options 00:25:05.420 --> 00:25:08.819 pricing. This is the hard mathematical cornerstone. 00:25:09.240 --> 00:25:12.140 Volatility is the single most critical unobserved 00:25:12.140 --> 00:25:15.039 parameter in option pricing models, most famously 00:25:15.039 --> 00:25:18.220 the black skulls model. A higher expected volatility 00:25:18.220 --> 00:25:21.079 directly means a higher option premium because 00:25:21.079 --> 00:25:23.220 the increased price dispersion increases the 00:25:23.220 --> 00:25:25.180 statistical probability that the option will 00:25:25.180 --> 00:25:27.480 finish in the money and be valuable at expiration. 00:25:27.819 --> 00:25:30.059 To really cement the separation of this concept 00:25:30.059 --> 00:25:32.500 from the simple idea of price movement, let's 00:25:32.500 --> 00:25:34.480 focus on the crucial distinction, volatility 00:25:34.480 --> 00:25:37.859 versus direction. Volatility measures dispersion, 00:25:37.859 --> 00:25:40.480 not the direction of the price change. This requires 00:25:40.480 --> 00:25:42.640 a clear understanding of the standard deviation 00:25:42.640 --> 00:25:45.400 calculation itself. Because the formula involves 00:25:45.400 --> 00:25:48.160 squaring all the differences from the mean, all 00:25:48.160 --> 00:25:50.779 movements, negative and positive, are converted 00:25:50.779 --> 00:25:53.460 into a single positive non -directional quantity. 00:25:54.299 --> 00:25:57.779 An asset that rockets up 10 % one day and then 00:25:57.779 --> 00:26:00.619 plunges 10 % the next with a net return of near 00:26:00.619 --> 00:26:04.319 zero is highly volatile. The illustrative example 00:26:04.319 --> 00:26:06.240 provided in the source makes this distinction 00:26:06.240 --> 00:26:09.420 incredibly clear. Let's compare two hypothetical 00:26:09.420 --> 00:26:12.640 stocks, stock A and stock B. Both have the exact 00:26:12.640 --> 00:26:15.660 same expected average return, let's say 7 % annually. 00:26:15.900 --> 00:26:18.279 But their risk profiles are wildly different. 00:26:18.339 --> 00:26:22.059 Stock A has a low annual volatility of 5%, while 00:26:22.059 --> 00:26:25.599 stock B is highly volatile with 20 % annual volatility. 00:26:25.920 --> 00:26:27.859 Now, we use the normal distribution assumption 00:26:27.859 --> 00:26:31.240 to calculate the 95 % return range, which means 00:26:31.240 --> 00:26:33.539 that the expected outcome falls within two standard 00:26:33.539 --> 00:26:35.519 deviations of the mean, and the difference is 00:26:35.519 --> 00:26:39.079 staggering. For stock A, the 95 % range runs 00:26:39.079 --> 00:26:42.420 from approximately 7 % minus 2 standard deviations, 00:26:42.480 --> 00:26:46.339 so 10%, to 7 % plus 2 standard deviations. That's 00:26:46.339 --> 00:26:49.359 a very contained range, from negative 3 % to 00:26:49.359 --> 00:26:52.660 positive 17%. The returns are relatively predictable. 00:26:52.920 --> 00:26:55.700 But for stock B, the high volatility asset, the 00:26:55.700 --> 00:27:00.160 95 % range runs from 7 % minus 40 % to 7 % plus 00:27:00.160 --> 00:27:02.940 40%. Which gives you a 95 % return range spanning 00:27:02.940 --> 00:27:07.059 from negative 33 % to positive 40%. 47%. A range 00:27:07.059 --> 00:27:10.059 that large means a high volatility asset is essentially 00:27:10.059 --> 00:27:13.019 a coin flip between euphoria and absolute despair. 00:27:13.680 --> 00:27:16.099 Volatility is the measure of that range of possible 00:27:16.099 --> 00:27:18.140 outcomes, independent of whether the expected 00:27:18.140 --> 00:27:20.759 average outcome is positive or negative. And 00:27:20.759 --> 00:27:22.480 we must immediately remind the listener of the 00:27:22.480 --> 00:27:25.059 caveat we covered earlier. This calculation assumes 00:27:25.059 --> 00:27:28.180 a clean, normal distribution. In financial reality, 00:27:28.500 --> 00:27:32.099 those returns are leptoketotic fat -tailed. Therefore, 00:27:32.180 --> 00:27:34.900 the probability of those extreme downside events, 00:27:35.140 --> 00:27:39.059 like a 33 % loss or even worse, is actually more 00:27:39.059 --> 00:27:41.059 likely than this simple calculation suggests. 00:27:41.759 --> 00:27:44.180 Volatility, as measured by standard deviation, 00:27:44.579 --> 00:27:47.279 is thus often an optimistic estimate of the true 00:27:47.279 --> 00:27:50.440 tail risk. We now enter the territory of modern 00:27:50.440 --> 00:27:53.079 quantitative finance, which was built entirely 00:27:53.079 --> 00:27:56.240 on the failure of a fundamental assumption. The 00:27:56.240 --> 00:27:58.720 classic black skulls equation, the cornerstone 00:27:58.720 --> 00:28:01.940 of options pricing, assumes predictable, constant 00:28:01.940 --> 00:28:04.500 volatility. Empirically, we know this is completely 00:28:04.500 --> 00:28:07.380 false. Prices do not move with a consistent sigma. 00:28:07.579 --> 00:28:10.420 That failure necessitated a revolution in modeling. 00:28:10.859 --> 00:28:13.099 Volatility changes, and it changes dynamically, 00:28:13.400 --> 00:28:15.839 meaning the risk profile of an asset is not fixed. 00:28:16.079 --> 00:28:18.480 This led to all sorts of models designed to address 00:28:18.480 --> 00:28:21.039 that reality. Local volatility models developed 00:28:21.039 --> 00:28:23.960 by Derman, Connie, and Dupier, models incorporating 00:28:23.960 --> 00:28:25.859 what are called Poisson processes to account 00:28:25.859 --> 00:28:28.559 for sudden, unexpected volatility jumps. And 00:28:28.559 --> 00:28:30.740 the widely used stochastic volatility models, 00:28:30.920 --> 00:28:33.000 like the Heston model, where volatility itself 00:28:33.000 --> 00:28:35.480 is treated as a random variable, exhibiting its 00:28:35.480 --> 00:28:37.880 own unpredictable movements. And the most commonly 00:28:37.880 --> 00:28:40.079 observed empirical pattern in market dynamics. 00:28:40.200 --> 00:28:43.440 is the concept of volatility clustering. This 00:28:43.440 --> 00:28:45.859 is a fundamental law of finance discovered by 00:28:45.859 --> 00:28:48.980 Robert Engel, who won a Nobel for it. Assets 00:28:48.980 --> 00:28:51.420 experience distinct periods of high volatility 00:28:51.420 --> 00:28:54.720 where prices whipsaw quickly up and down, followed 00:28:54.720 --> 00:28:57.180 by distinct periods of low volatility where prices 00:28:57.180 --> 00:28:59.940 barely move at all. And crucially, these periods 00:28:59.940 --> 00:29:03.019 are sticky. High volatility tends to follow high 00:29:03.019 --> 00:29:05.480 volatility and low volatility tends to follow 00:29:05.480 --> 00:29:08.200 low volatility. It implies that the magnitude 00:29:08.200 --> 00:29:10.720 of recent price movement contains information 00:29:10.720 --> 00:29:13.980 about the magnitude of near future price movement. 00:29:14.180 --> 00:29:15.960 So the market gets into a rhythm, whether that 00:29:15.960 --> 00:29:19.339 rhythm is chaos or stability. Yes, exactly. And 00:29:19.339 --> 00:29:21.099 our sources also note that in certain markets, 00:29:21.279 --> 00:29:23.819 these dynamic changes exhibit seasonality and 00:29:23.819 --> 00:29:26.279 presaging movements. Where is seasonality most 00:29:26.279 --> 00:29:29.230 evident? In the foreign exchange or forex market, 00:29:29.450 --> 00:29:31.910 price changes are seasonally heteroscedastic, 00:29:32.150 --> 00:29:34.049 which is a long way of saying the volatility 00:29:34.049 --> 00:29:36.349 patterns follow predictable recurring riddlings 00:29:36.349 --> 00:29:39.369 over one day in one week. For instance, volatility 00:29:39.369 --> 00:29:41.589 often spikes right after the opening of the London 00:29:41.589 --> 00:29:44.130 or New York trading session when new information 00:29:44.130 --> 00:29:47.130 hits the market. And this seasonality feeds into 00:29:47.130 --> 00:29:50.690 the theoretical concept of autoregressive conditional 00:29:50.690 --> 00:29:55.410 heteroscedasticity or ARCH. ARCH is the formal 00:29:55.410 --> 00:29:58.210 recognition that extreme market movements, a 00:29:58.210 --> 00:30:01.230 crash or a bubble, don't just appear spontaneously 00:30:01.230 --> 00:30:04.089 out of nowhere. They are often presaged by a 00:30:04.089 --> 00:30:06.170 period of larger than usual preceding movements 00:30:06.170 --> 00:30:09.210 or by a buildup of known uncertainty regarding 00:30:09.210 --> 00:30:11.869 specific future events. Think of the week leading 00:30:11.869 --> 00:30:14.190 up to a major geopolitical vote or a Federal 00:30:14.190 --> 00:30:16.230 Reserve interest rate announcement. Volatility 00:30:16.230 --> 00:30:18.650 usually creeps up beforehand as the market prices 00:30:18.650 --> 00:30:21.529 in that uncertainty. So ARCH models can tell 00:30:21.529 --> 00:30:23.450 you when the market is about to get significant. 00:30:23.559 --> 00:30:26.000 more active or volatile. They can tell you when 00:30:26.000 --> 00:30:27.839 the distribution of outcomes is about to get 00:30:27.839 --> 00:30:31.980 wider. But this is the critical limitation. ARCH 00:30:31.980 --> 00:30:35.640 and its more advanced cousin, GRCH, are fundamentally 00:30:35.640 --> 00:30:38.940 models of dispersion. They tell you very little 00:30:38.940 --> 00:30:41.119 about whether the subsequent large movement will 00:30:41.119 --> 00:30:44.460 be a sharp move up or a sharp move down. Prediction 00:30:44.460 --> 00:30:46.400 of volatility is far easier than the prediction 00:30:46.400 --> 00:30:48.940 of direction. We also have the subtle but important 00:30:48.940 --> 00:30:52.009 phenomenon known as the resolution problem. This 00:30:52.009 --> 00:30:54.430 reminds us that how we measure volatility is 00:30:54.430 --> 00:30:57.190 highly dependent on our instrumentation. A volatility 00:30:57.190 --> 00:30:59.609 measure depends not only on the total period 00:30:59.609 --> 00:31:02.589 measured, say, the last 30 days, but profoundly 00:31:02.589 --> 00:31:05.150 on the time resolution you use. Are you calculating 00:31:05.150 --> 00:31:08.230 returns from minute by minute ticks, hourly intervals, 00:31:08.410 --> 00:31:11.089 or just daily closing prices? And why does that 00:31:11.089 --> 00:31:13.609 choice of interval, the tick size, create such 00:31:13.609 --> 00:31:15.670 a measurable difference in the final volatility 00:31:15.670 --> 00:31:18.650 number? Because the flow of information is asymmetric 00:31:18.650 --> 00:31:21.369 across different timescales, short -term high 00:31:21.369 --> 00:31:24.029 -frequency traders focus intensely on minute 00:31:24.029 --> 00:31:26.109 -by -minute noise and order book fluctuations, 00:31:26.450 --> 00:31:28.869 and that generates high -resolution volatility. 00:31:29.529 --> 00:31:32.009 Long term institutions might only care about 00:31:32.009 --> 00:31:35.029 daily or weekly movements. High resolution volatility 00:31:35.029 --> 00:31:38.329 contains information, the noise of intraday trading, 00:31:38.450 --> 00:31:41.950 that low resolution daily volatility misses entirely. 00:31:42.369 --> 00:31:45.150 And conversely, low resolution volatility might 00:31:45.150 --> 00:31:47.869 smooth out intraday noise, revealing longer term 00:31:47.869 --> 00:31:50.089 structural trends that the minute by minute data 00:31:50.089 --> 00:31:52.910 just obscures. So volatility is fundamentally 00:31:52.910 --> 00:31:55.859 a scale dependent measure. There's no one true 00:31:55.859 --> 00:31:58.700 number. Exactly. This inherent complexity forced 00:31:58.700 --> 00:32:01.460 options traders to develop tools that move beyond 00:32:01.460 --> 00:32:04.299 simple standard deviation for forecasting. They 00:32:04.299 --> 00:32:07.000 introduced a crucial distinction, clean vol versus 00:32:07.000 --> 00:32:09.400 dirty vol. OK, this sounds like a practitioner's 00:32:09.400 --> 00:32:11.680 tool. It is. It's a sophisticated segmentation 00:32:11.680 --> 00:32:14.000 used for accurate option pricing, which divides 00:32:14.000 --> 00:32:16.420 the forecast of future volatility into two specific 00:32:16.420 --> 00:32:18.559 components based on the cause of the dispersion. 00:32:18.579 --> 00:32:21.180 So clean volatility is the baseline noise. Yes, 00:32:21.240 --> 00:32:24.180 it represents the routine expected noise. It's 00:32:24.180 --> 00:32:26.920 the volatility caused by standard daily transactions, 00:32:27.319 --> 00:32:30.460 general market flow, regulatory noise, and the 00:32:30.460 --> 00:32:33.279 constant predictable stream of small, non -event 00:32:33.279 --> 00:32:36.200 driven information. It is the persistent underlying 00:32:36.200 --> 00:32:39.039 level of dispersion that would exist even without 00:32:39.039 --> 00:32:41.539 a major announcement looming. And dirty volatility, 00:32:41.819 --> 00:32:45.619 or event vol, is the explicit risk factor for 00:32:45.619 --> 00:32:48.539 a specific event. This is the volatility caused 00:32:48.539 --> 00:32:50.680 by specific scheduled or sometimes unexpected 00:32:50.680 --> 00:32:53.519 events. For instance, consider a major pharmaceutical 00:32:53.519 --> 00:32:56.190 company. Their clean vol is caused by routine 00:32:56.190 --> 00:32:58.750 trading, but their dirty vol is caused by the 00:32:58.750 --> 00:33:01.430 date of a major FDA drug trial result announcement, 00:33:01.730 --> 00:33:04.269 a sudden patent ruling, or a quarterly earnings 00:33:04.269 --> 00:33:06.990 call. These events are known to cause massive 00:33:06.990 --> 00:33:09.190 price jumps or drops. So let's walk through a 00:33:09.190 --> 00:33:11.910 scenario. If I am pricing an option that expires 00:33:11.910 --> 00:33:14.029 the day after a pharmaceutical company's massive 00:33:14.029 --> 00:33:16.269 drug trial results are announced, the option 00:33:16.269 --> 00:33:18.849 price must factor in a disproportionately large 00:33:18.849 --> 00:33:22.119 amount of dirty vol. correct absolutely the option 00:33:22.119 --> 00:33:24.640 pricing model has to isolate the clean bowl for 00:33:24.640 --> 00:33:26.740 the days leading up to the announcement and then 00:33:26.740 --> 00:33:29.440 spike the volatility input dramatically for the 00:33:29.440 --> 00:33:32.220 period covering the event itself it's essentially 00:33:32.220 --> 00:33:34.980 isolating the jump risk into the dirty vole component 00:33:34.980 --> 00:33:38.019 the core job of fundamental analysts at option 00:33:38.019 --> 00:33:40.660 trading firms is to translate qualitative news 00:33:40.660 --> 00:33:43.119 like the expected success or failure of a drug 00:33:43.119 --> 00:33:46.099 trial into a precise numeric value for the dirty 00:33:46.099 --> 00:33:48.519 vole component so they can accurately price options 00:33:48.519 --> 00:33:51.869 that straddle that That shows how volatility 00:33:51.869 --> 00:33:54.390 modeling moves from pure statistics into the 00:33:54.390 --> 00:33:57.240 analysis of fundamental company -specific or 00:33:57.240 --> 00:34:00.160 even geopolitical events. We now transition to 00:34:00.160 --> 00:34:02.099 perhaps the most humbling and expensive concept 00:34:02.099 --> 00:34:04.279 for the individual investor. The mathematical 00:34:04.279 --> 00:34:06.220 reality of the volatility doesn't just mean bigger 00:34:06.220 --> 00:34:08.880 swings. It acts as a constant measurable drag 00:34:08.880 --> 00:34:11.420 on long -term portfolio performance. This is 00:34:11.420 --> 00:34:13.619 the volatility tax. This is often referred to 00:34:13.619 --> 00:34:15.900 as a drag on the compound annual growth rate, 00:34:15.980 --> 00:34:20.219 or CAGR. It's a mathematical consequence of compounding 00:34:20.219 --> 00:34:23.139 returns. If you achieve an average arithmetic 00:34:23.139 --> 00:34:26.440 return of, say, 10 % per year, year, but that 00:34:26.440 --> 00:34:28.980 return is achieved with high volatility, your 00:34:28.980 --> 00:34:31.820 actual compounded return over time will always 00:34:31.820 --> 00:34:35.460 be lower than 10%. Volatility imposes a financial 00:34:35.460 --> 00:34:38.119 cost. Let's explain the mechanism intuitively. 00:34:38.440 --> 00:34:41.599 Why is the impact of a loss greater than the 00:34:41.599 --> 00:34:43.980 impact of an equal gain when you're compounding? 00:34:44.019 --> 00:34:46.989 It's asymmetry. If you start with $100 and you 00:34:46.989 --> 00:34:49.809 lose 50%, you're left with $50. To get back to 00:34:49.809 --> 00:34:52.050 your original $100, you don't need a 50 % gain. 00:34:52.170 --> 00:34:54.829 You need a 100 % gain on your remaining $50. 00:34:55.250 --> 00:34:57.750 The mathematical impact of a negative percentage 00:34:57.750 --> 00:35:00.190 change is always greater than the impact of an 00:35:00.190 --> 00:35:02.610 equal positive percentage change. The higher 00:35:02.610 --> 00:35:04.650 the volatility, the more frequently you experience 00:35:04.650 --> 00:35:07.130 these large swings, and the more often you are 00:35:07.130 --> 00:35:09.150 mathematically forced to earn a larger return 00:35:09.150 --> 00:35:12.050 just to get back to even. That asymmetry is the 00:35:12.050 --> 00:35:16.340 volatility tax in action. for quantifying this 00:35:16.340 --> 00:35:19.050 drag uses the Taylor series expansion. Right. 00:35:19.130 --> 00:35:21.909 The standard estimation is that the CAGR is approximately 00:35:21.909 --> 00:35:25.550 equal to the average return, or AR, minus the 00:35:25.550 --> 00:35:28.230 volatility tax term, which is 1 half sigma squared. 00:35:28.510 --> 00:35:32.510 That term, TFRAC 1 -2 -SIGRA, is the direct quantification 00:35:32.510 --> 00:35:35.969 of the tax. So if an asset averages a 15 % return 00:35:35.969 --> 00:35:39.869 with 20 % volatility, the tax is 0 .5 times 0 00:35:39.869 --> 00:35:44.289 .2 squared, or 0 .5 times 0 .04, which is 2%. 00:35:44.289 --> 00:35:48.030 So the effect of CGR drops from 15 % to 13%. 00:35:48.250 --> 00:35:49.849 And two percentage points a year doesn't sound 00:35:49.849 --> 00:35:52.710 like much, but over a 10 or 20 year investment 00:35:52.710 --> 00:35:56.449 horizon, that 2 % annual drag compounds significantly, 00:35:56.789 --> 00:35:59.210 eating into the total final portfolio value. 00:35:59.530 --> 00:36:02.309 It's devastating over the long term. And unfortunately, 00:36:02.429 --> 00:36:05.250 that simple Taylor series approximation is, once 00:36:05.250 --> 00:36:08.389 again, often overly optimistic. It is. The formula 00:36:08.389 --> 00:36:10.889 is too optimistic in the real world. And that's 00:36:10.889 --> 00:36:13.030 because of those fat tails and negative skewness 00:36:13.030 --> 00:36:15.920 we discussed. Exactly. Most financial assets 00:36:15.920 --> 00:36:18.519 tend to have more large negative movements than 00:36:18.519 --> 00:36:20.400 large positive ones. That's negative skewness. 00:36:20.559 --> 00:36:22.920 And the tails are thicker than normal, which 00:36:22.920 --> 00:36:26.599 is leptokurtosis. Therefore, the actual drag 00:36:26.599 --> 00:36:29.219 imposed by real -world volatility and tail risks 00:36:29.219 --> 00:36:31.820 is usually greater than the simple half sigma 00:36:31.820 --> 00:36:34.579 squared calculation suggests. So practitioners 00:36:34.579 --> 00:36:37.260 need to adjust this formula for reality. Yes. 00:36:37.559 --> 00:36:39.840 Industry practitioners often use an empirical 00:36:39.840 --> 00:36:42.860 factor, let's call it K, in the formula to adjust 00:36:42.860 --> 00:36:45.440 the sigma squared term. And given the observed 00:36:45.440 --> 00:36:48.260 tendency of real assets to be riskier than predicted, 00:36:48.619 --> 00:36:51.659 K values typically range from 5 to 10, which 00:36:51.659 --> 00:36:54.000 vastly increases the calculated volatility tax 00:36:54.000 --> 00:36:56.780 to account for the true cost of those large frequent 00:36:56.780 --> 00:36:59.800 downside shocks. Moving on, we tackle a highly 00:36:59.800 --> 00:37:02.340 contentious area, critiques of forecasting models. 00:37:02.800 --> 00:37:04.820 Despite the massive intellectual investment in 00:37:04.820 --> 00:37:07.940 developing models like JRCH, RHE, and stochastic 00:37:07.940 --> 00:37:10.159 volatility frameworks, there is an ongoing and 00:37:10.159 --> 00:37:12.420 profound debate about their true predictive power. 00:37:12.780 --> 00:37:15.679 The skepticism comes from a very simple yet powerful 00:37:15.679 --> 00:37:19.460 claim that these incredibly complex, highly parameterized 00:37:19.460 --> 00:37:22.300 models often have predictive power that's similar 00:37:22.300 --> 00:37:25.360 to just plain vanilla measures like simply using 00:37:25.360 --> 00:37:29.039 the past or actual volatility we calculated back 00:37:29.039 --> 00:37:32.219 in Section 1. So all that complexity yields no 00:37:32.219 --> 00:37:34.829 measurable edge. That is the skeptics argument, 00:37:35.010 --> 00:37:36.670 especially when they're tested out of sample. 00:37:36.829 --> 00:37:40.050 And that testing method is crucial. You train 00:37:40.050 --> 00:37:42.409 the model on one period of data, and then you 00:37:42.409 --> 00:37:44.929 test its forecast accuracy on a completely separate 00:37:44.929 --> 00:37:47.389 later period of data that has never seen before. 00:37:47.750 --> 00:37:50.409 When subjected to this rigorous testing, critics 00:37:50.409 --> 00:37:53.230 argue that the complex GRCH models often fail 00:37:53.230 --> 00:37:56.110 to provide a statistically significant out -of 00:37:56.110 --> 00:37:58.550 -sample forecast improvement over simply using, 00:37:58.650 --> 00:38:00.849 say, the last 30 days of realized volatility 00:38:00.849 --> 00:38:02.630 as your prediction. But I understand this debate 00:38:02.630 --> 00:38:06.219 is far - Oh, it's not. The counterargument from 00:38:06.219 --> 00:38:08.320 the model proponents is that the studies claiming 00:38:08.320 --> 00:38:11.309 failure are flawed. They suggest that the critics 00:38:11.309 --> 00:38:13.730 failed to correctly implement the more complicated 00:38:13.730 --> 00:38:16.730 models in their testing. The implication is that 00:38:16.730 --> 00:38:19.670 GRCH models do offer superior predictive capability, 00:38:19.889 --> 00:38:21.989 but only when they're correctly specified and 00:38:21.989 --> 00:38:24.889 used by experienced professionals. The debate 00:38:24.889 --> 00:38:27.250 really boils down to whether the complexity is 00:38:27.250 --> 00:38:30.369 inherently flawed or whether it's just too difficult 00:38:30.369 --> 00:38:32.730 to implement correctly and consistently across 00:38:32.730 --> 00:38:35.469 all market conditions. Regardless of who is right 00:38:35.469 --> 00:38:38.730 in the GRCH debate, the sources reveal a deep 00:38:39.099 --> 00:38:41.800 philosophical disillusionment with volatility 00:38:41.800 --> 00:38:44.119 modeling among some of the sharpest minds in 00:38:44.119 --> 00:38:46.610 the field. This brings us back to the human element 00:38:46.610 --> 00:38:48.949 and the limits of our knowledge. This skepticism 00:38:48.949 --> 00:38:51.269 is perhaps the most humbling takeaway for anyone 00:38:51.269 --> 00:38:54.050 entering quantitative finance. Nassim Taleb, 00:38:54.289 --> 00:38:56.269 who dedicated his career to understanding high 00:38:56.269 --> 00:38:59.130 impact, low probability events, famously titled 00:38:59.130 --> 00:39:01.050 one of his foundational papers with the statement, 00:39:01.250 --> 00:39:03.309 we don't quite know what we are talking about 00:39:03.309 --> 00:39:06.309 when we talk about volatility. That title alone 00:39:06.309 --> 00:39:08.809 suggests that even the most rigorous mathematical 00:39:08.809 --> 00:39:12.130 definition of volatility only captures a fraction 00:39:12.130 --> 00:39:14.969 of the underlying market reality. What was his 00:39:14.969 --> 00:39:18.269 central critique? Taleb argues that traditional 00:39:18.269 --> 00:39:20.869 measures of volatility are too reliant on the 00:39:20.869 --> 00:39:23.409 Gaussian framework, precisely because they fail 00:39:23.409 --> 00:39:26.309 to correctly account for those fat tails. We 00:39:26.309 --> 00:39:28.469 use a metric that assumes predictability and 00:39:28.469 --> 00:39:31.030 bounded risk. But the financial world, which 00:39:31.030 --> 00:39:33.429 is driven by human behavior, is fundamentally 00:39:33.429 --> 00:39:36.389 subject to sudden, extreme, nonlinear shocks. 00:39:37.070 --> 00:39:39.250 He contends that our mathematical definitions 00:39:39.250 --> 00:39:41.670 give us a false sense of security and control. 00:39:41.929 --> 00:39:44.409 And Emanuel Derman, another figure central to 00:39:44.409 --> 00:39:46.949 the creation of modern Wall Street models, shared 00:39:46.949 --> 00:39:49.309 a similar disillusionment, focusing on the distinction 00:39:49.309 --> 00:39:51.789 between theory and metaphor. Derman expressed 00:39:51.789 --> 00:39:54.329 fatigue with the enormous supply of empirical 00:39:54.329 --> 00:39:57.010 models that lack robust underlying theoretical 00:39:57.010 --> 00:39:59.809 support. He argues that we must remember the 00:39:59.809 --> 00:40:01.610 fundamental difference between the two concepts. 00:40:02.210 --> 00:40:04.989 Theories, like the laws of physics, seek to uncover 00:40:04.989 --> 00:40:07.110 hidden, immutable principles of the universe. 00:40:07.329 --> 00:40:10.599 They aim for universal truth. Models, Derman 00:40:10.599 --> 00:40:13.239 states, are merely metaphors. They're analogies 00:40:13.239 --> 00:40:15.039 that describe one thing relative to another. 00:40:15.400 --> 00:40:17.840 The black schools model is a wonderful metaphor 00:40:17.840 --> 00:40:20.599 for option pricing, but it should never be mistaken 00:40:20.599 --> 00:40:22.760 for a fundamental truth about future price movements. 00:40:23.260 --> 00:40:26.659 It's a tool for estimation, not a perfect predictor. 00:40:26.780 --> 00:40:29.739 The danger lies in confusing the illusion, the 00:40:29.739 --> 00:40:32.300 perfect fit of a model to pass data with the 00:40:32.300 --> 00:40:34.880 actual unpredictable reality of future price 00:40:34.880 --> 00:40:38.159 movement. That intellectual humility is the ultimate 00:40:38.159 --> 00:40:41.190 insight. We have completed a comprehensive deep 00:40:41.190 --> 00:40:43.789 dive into volatility, moving from simple definition 00:40:43.789 --> 00:40:46.809 to complex mathematical critique. Let's briefly 00:40:46.809 --> 00:40:48.710 recap the essential takeaways for the learner. 00:40:48.929 --> 00:40:51.530 We established the necessary taxonomy, the six 00:40:51.530 --> 00:40:53.610 categories that split volatility into backward 00:40:53.610 --> 00:40:55.769 -looking actual volatility, current, historical, 00:40:55.789 --> 00:40:57.929 realized, and future, and forward -looking consensus 00:40:57.929 --> 00:41:01.130 -based implied volatility, historical, current, 00:41:01.309 --> 00:41:03.989 and future. Understanding the divergence between 00:41:03.989 --> 00:41:07.010 actual and implied is a key to identifying market 00:41:07.010 --> 00:41:10.739 signals. the math, showing the price dispersion 00:41:10.739 --> 00:41:13.360 grows not linearly, but by the square root of 00:41:13.360 --> 00:41:16.500 time, based on the Wiener process. This translates 00:41:16.500 --> 00:41:20.340 a 1 % daily move into nearly 16 % annualized 00:41:20.340 --> 00:41:23.460 volatility. We also identified the critical problem 00:41:23.460 --> 00:41:26.400 of fat tails, that real -world extreme market 00:41:26.400 --> 00:41:29.360 events are far more probable than our clean statistical 00:41:29.360 --> 00:41:32.460 models allow. We looked at the origin, recognizing 00:41:32.460 --> 00:41:35.199 that volatility is fundamentally driven by market 00:41:35.199 --> 00:41:37.500 microstructure and the market maker's collective 00:41:37.500 --> 00:41:40.619 fear of adverse selection, which causes them 00:41:40.619 --> 00:41:43.719 to widen their trading spreads. We also understood 00:41:43.719 --> 00:41:46.119 the practical consequences, from the devastating 00:41:46.119 --> 00:41:48.519 impact of sequence of return risk in retirement 00:41:48.519 --> 00:41:50.940 to its mathematical role in the black school's 00:41:50.940 --> 00:41:53.280 options pricing model. Crucially, we separated 00:41:53.280 --> 00:41:56.400 volatility from direction, recognizing that dispersion 00:41:56.400 --> 00:41:58.800 imposes a constant, often underestimated financial 00:41:58.800 --> 00:42:01.800 cost. The volatility tax on compound returns 00:42:01.800 --> 00:42:04.300 due to the mathematical asymmetry of losses and 00:42:04.300 --> 00:42:06.739 gains. And finally, we ended with the intellectual 00:42:06.739 --> 00:42:08.980 humility required in this field, reminded by 00:42:08.980 --> 00:42:11.519 quant masters that even the most complex forecasting 00:42:11.519 --> 00:42:14.480 models are often only useful metaphors, not fundamental 00:42:14.480 --> 00:42:17.280 laws. We spent this deep dive establishing that 00:42:17.280 --> 00:42:20.059 volatility is a mathematical measure of dispersion, 00:42:20.059 --> 00:42:22.250 independent of whether the prices move up or 00:42:22.250 --> 00:42:25.269 down. Now consider this as you go forward and 00:42:25.269 --> 00:42:28.630 watch the markets move. If volatility is just 00:42:28.630 --> 00:42:30.650 a measure of how spread out the potential returns 00:42:30.650 --> 00:42:33.289 are, how much of market behavior, the frantic 00:42:33.289 --> 00:42:35.610 buying, the sudden selling, the panic, the greed, 00:42:35.769 --> 00:42:38.190 is actually driven by the emotional response 00:42:38.190 --> 00:42:40.989 to volatility itself, rather than the fundamental 00:42:40.989 --> 00:42:43.050 information that caused the dispersion in the 00:42:43.050 --> 00:42:45.449 first place. It makes you question if volatility 00:42:45.449 --> 00:42:47.969 is ultimately less a measure of objective risk 00:42:47.969 --> 00:42:50.730 and more a measure of raw human reaction to uncertainty. 00:42:51.280 --> 00:42:53.420 Something profound to mull over until our next 00:42:53.420 --> 00:42:54.880 deep dive. Thank you for joining us.