transcripts/glWFuobqIqQ.json

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  "video_id": "glWFuobqIqQ",
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  "segments": [
    {
      "start": 0.05,
      "end": 5.77,
      "text": "300 years of calculus, and nobody saw this. Why is that? Well, my guess is"
    },
    {
      "start": 5.77,
      "end": 11.05,
      "text": "nobody ever thought about using the hockey sticks, right? Who would use that?"
    },
    {
      "start": 11.05,
      "end": 20.85,
      "text": "Option traders would use that. Option quants would use that. Hi, everybody. I'm"
    },
    {
      "start": 20.85,
      "end": 25.97,
      "text": "Doug Costa. I'm a quant here at Susquehanna. My background is in mathematics. I was"
    },
    {
      "start": 25.97,
      "end": 31.77,
      "text": "an undergraduate math major. I went on to get a PhD in mathematics specializing in"
    },
    {
      "start": 31.77,
      "end": 38.33,
      "text": "abstract algebra. When I finished my PhD in 1974, I was lucky enough to get a"
    },
    {
      "start": 38.33,
      "end": 42.33,
      "text": "position as a professor in the Department of Mathematics at the University of"
    },
    {
      "start": 42.33,
      "end": 50.05,
      "text": "Virginia, where I taught and did research for a number of years in 1993 or so"
    },
    {
      "start": 50.05,
      "end": 57.09,
      "text": "through a former grad student. I became a consultant for Susquehanna, helping"
    },
    {
      "start": 57.09,
      "end": 62.97,
      "text": "with mathematical questions, and that led to a very good relationship."
    },
    {
      "start": 62.97,
      "end": 69.57,
      "text": "In 1997, I joined the firm full-time as the head of the Quantitative Research"
    },
    {
      "start": 69.57,
      "end": 77.41,
      "text": "Department. I held that position until mid-2015 when I transitioned into a"
    },
    {
      "start": 77.41,
      "end": 82.65,
      "text": "role in our education department. We have a very highly developed education"
    },
    {
      "start": 82.65,
      "end": 90.21,
      "text": "department, and I now do a lot of teaching in that group. So today, I'm going"
    },
    {
      "start": 90.21,
      "end": 98.61,
      "text": "to show you some of what we talk about here concerning the VIX. This is the"
    },
    {
      "start": 98.61,
      "end": 107.01,
      "text": "SIBO, the Chicago Board Options Exchange's S&P 500 Volatility Index. So it's"
    },
    {
      "start": 107.01,
      "end": 113.63,
      "text": "an index that was designed to try to measure volatility, which means standard"
    },
    {
      "start": 113.63,
      "end": 123.51,
      "text": "deviation of returns of the S&P 500 index. Now, the exciting thing about this"
    },
    {
      "start": 123.51,
      "end": 130.07,
      "text": "is that the financial engineering technology that was used to create the"
    },
    {
      "start": 130.07,
      "end": 134.07,
      "text": "new VIX, there's an old VIX and a new VIX, we'll talk about that. This"
    },
    {
      "start": 134.07,
      "end": 138.43,
      "text": "financial engineering technology that was invented in the 1990s to model"
    },
    {
      "start": 138.43,
      "end": 143.71,
      "text": "variance swaps as an effective way to model variance swaps, and it really is"
    },
    {
      "start": 143.71,
      "end": 148.95,
      "text": "quite quite pretty mathematically speaking. So that's what we'll be talking about"
    },
    {
      "start": 148.95,
      "end": 156.71,
      "text": "today. Okay, so the story begins with the old VIX. So this is pre-2002."
    },
    {
      "start": 156.71,
      "end": 161.19,
      "text": "Sometime in the 1980s, I'm not quite sure when. The SIBO decided that they"
    },
    {
      "start": 161.19,
      "end": 165.87,
      "text": "would like to publish a volatility index, and it was supposed to represent"
    },
    {
      "start": 165.87,
      "end": 171.75,
      "text": "nominally the volatility over the next 30 days, so kind of predicting future"
    },
    {
      "start": 171.75,
      "end": 178.95,
      "text": "volatility. And the tools they had available then were not as well developed as"
    },
    {
      "start": 178.95,
      "end": 187.59,
      "text": "tools we have now in financial engineering. So here's what they did. Since the S&P 500"
    },
    {
      "start": 187.59,
      "end": 195.23,
      "text": "index futures trade on the SIBO, that this was their product, those, the"
    },
    {
      "start": 195.23,
      "end": 200.39,
      "text": "options on those futures are European style. So an important thing about being"
    },
    {
      "start": 200.39,
      "end": 205.71,
      "text": "European style is that they can be priced using the Black Sholes formula. Now,"
    },
    {
      "start": 205.71,
      "end": 211.55,
      "text": "doing so means that you're going to use the Black Sholes formula. So for example,"
    },
    {
      "start": 211.55,
      "end": 216.35,
      "text": "for a call option with strike K and maturity T, which I have written here, the"
    },
    {
      "start": 216.35,
      "end": 221.15,
      "text": "inputs to the Black Sholes call formula are the current price of the index,"
    },
    {
      "start": 221.15,
      "end": 225.59,
      "text": "which I've written S not, and typically when we use Black Sholes, we use S here"
    },
    {
      "start": 225.59,
      "end": 229.75,
      "text": "because it's generically for a stock price. But in this case, it's the index"
    },
    {
      "start": 229.75,
      "end": 237.63,
      "text": "value, the S&P 500, the strike price K or exercise price, the maturity T, the"
    },
    {
      "start": 237.63,
      "end": 244.67,
      "text": "risk-free rate R, this Q, we use as the dividend yield if there is a dividend"
    },
    {
      "start": 244.67,
      "end": 250.19,
      "text": "yield, and sigma is the volatility of the index. And in Black Sholes, the"
    },
    {
      "start": 250.19,
      "end": 254.83,
      "text": "assumption is that the volatility is some constant. The Black Sholes formula is"
    },
    {
      "start": 254.83,
      "end": 261.59,
      "text": "this formula here. It's S not e to the minus Qt, big n of d1, big n is the standard"
    },
    {
      "start": 261.59,
      "end": 267.15,
      "text": "cumulative normal distribution function, minus the strike K e to the minus RT,"
    },
    {
      "start": 267.15,
      "end": 273.03,
      "text": "big n of d2, and d1 and d2 can be computed this way. I'm using the notation that"
    },
    {
      "start": 273.03,
      "end": 278.31,
      "text": "you will find in John Hull's wonderful textbook options, futures, and other"
    },
    {
      "start": 278.31,
      "end": 284.47,
      "text": "derivative securities, and that's a standard book that we refer to. Okay, so the"
    },
    {
      "start": 284.47,
      "end": 293.79,
      "text": "idea of Black Sholes implied volatility is that you have this formula, and it"
    },
    {
      "start": 293.79,
      "end": 298.03,
      "text": "postulates that if you know the volatility, then that should give you the price."
    },
    {
      "start": 298.03,
      "end": 303.15,
      "text": "Well, the idea of implied volatility is to say, hmm, I definitively know the"
    },
    {
      "start": 303.15,
      "end": 307.19,
      "text": "price of the index, I definitively know the strike of my option, the maturity of my"
    },
    {
      "start": 307.19,
      "end": 311.55,
      "text": "option, the risk-free rate, and the dividend yield. I don't necessarily know the"
    },
    {
      "start": 311.55,
      "end": 315.99,
      "text": "volatility, but I can look at the market price, where is the option trading,"
    },
    {
      "start": 315.99,
      "end": 320.83,
      "text": "where's the call option trading, and then I can try to do a search, you can do a"
    },
    {
      "start": 320.83,
      "end": 326.95,
      "text": "simple binary search, to find the sigma, the unique sigma, that will produce"
    },
    {
      "start": 326.95,
      "end": 332.11,
      "text": "that market price, and then that is called the Black Sholes implied volatility"
    },
    {
      "start": 332.11,
      "end": 336.67,
      "text": "for this particular option with this particular strike and this particular"
    },
    {
      "start": 336.67,
      "end": 344.19,
      "text": "maturity. So the people at the SIBO, I guess, thought, well, yeah, we can use"
    },
    {
      "start": 344.19,
      "end": 348.91,
      "text": "Black Sholes implied volatilities as our index. Now, this raises the question, what"
    },
    {
      "start": 348.91,
      "end": 352.59,
      "text": "strike and what maturity? Well, they wanted the maturity to be"
    },
    {
      "start": 352.59,
      "end": 362.07,
      "text": "nominally 30 days. So there's also the Black Sholes formula for put options, so we"
    },
    {
      "start": 362.07,
      "end": 365.71,
      "text": "have call options and put options, so they wanted to incorporate both of those."
    },
    {
      "start": 365.71,
      "end": 371.31,
      "text": "So what they decided to do was find strikes nearest the forward for a given"
    },
    {
      "start": 371.31,
      "end": 377.75,
      "text": "maturity, and then let sigma k be the, at the forward implied volatility. Now"
    },
    {
      "start": 377.75,
      "end": 382.47,
      "text": "they're going to find that by calculating the call and the put volatility and"
    },
    {
      "start": 382.47,
      "end": 388.31,
      "text": "dividing by two, so that'll be the average of those two will be the implied"
    },
    {
      "start": 388.31,
      "end": 392.95,
      "text": "volatility they use for that particular maturity. The problem with the 30-day"
    },
    {
      "start": 392.95,
      "end": 399.15,
      "text": "nominal maturity is that on any given day, there may or may not be an"
    },
    {
      "start": 399.15,
      "end": 403.99,
      "text": "expiration exactly falling 30 days in the future, so typically they're not, so"
    },
    {
      "start": 403.99,
      "end": 409.03,
      "text": "what they decided to do is find the two expirations, capital T1 and capital T2,"
    },
    {
      "start": 409.03,
      "end": 416.15,
      "text": "surrounding the 30-day point in time, and then calculate an implied volatility"
    },
    {
      "start": 416.15,
      "end": 421.47,
      "text": "for T1 and implied volatility for T2, and now you have the problem of how do"
    },
    {
      "start": 421.47,
      "end": 428.95,
      "text": "you mush those together to get a 30-day implied volatility. So here's what"
    },
    {
      "start": 428.95,
      "end": 434.35,
      "text": "they did. They said, okay, these volatilities in black sholes, let's go back"
    },
    {
      "start": 434.35,
      "end": 438.19,
      "text": "to that for a second, the black sholes formula, these volatilities that we get"
    },
    {
      "start": 438.19,
      "end": 442.59,
      "text": "are annualized volatilities. That's the input to the black sholes formula, and"
    },
    {
      "start": 442.59,
      "end": 447.39,
      "text": "that means that volatility corresponds to the volatility you would see over the"
    },
    {
      "start": 447.39,
      "end": 457.47,
      "text": "course of one year, or for annualized annual returns. So in the time period T1, the"
    },
    {
      "start": 457.47,
      "end": 464.19,
      "text": "actual variance realized is sigma T1 squared times T1. Sigma T1 squared would be"
    },
    {
      "start": 464.19,
      "end": 470.99,
      "text": "the variance for a full year, T1 is part of a year, or some fraction of a year,"
    },
    {
      "start": 470.99,
      "end": 477.67,
      "text": "so this product is going to be the total variance indicated by that option price,"
    },
    {
      "start": 477.67,
      "end": 484.67,
      "text": "or the put and call price averaged. Similarly, calculate sigma 2 squared T2, that"
    },
    {
      "start": 484.67,
      "end": 490.75,
      "text": "should be a 2 there. Sorry about that. And then use time linear interpolation of"
    },
    {
      "start": 490.75,
      "end": 495.59,
      "text": "variance. So what's the idea behind time linear interpolation of variance? Well,"
    },
    {
      "start": 495.59,
      "end": 503.35,
      "text": "the idea behind that is a belief, basically. If you believe that disjoint"
    },
    {
      "start": 503.35,
      "end": 510.79,
      "text": "intervals of time produce independent changes in the index, then the"
    },
    {
      "start": 510.79,
      "end": 516.59,
      "text": "variances should add, which means that over time, if you chop a time interval into,"
    },
    {
      "start": 516.59,
      "end": 523.39,
      "text": "say, n equal size intervals that are disjoint, then in each interval you have"
    },
    {
      "start": 523.39,
      "end": 528.49,
      "text": "independent changes in the index, and those variables give you independent"
    },
    {
      "start": 528.49,
      "end": 532.95,
      "text": "returns. The variance of the sum of those returns is the sum of the"
    },
    {
      "start": 532.95,
      "end": 537.55,
      "text": "variances because of the independence, and that means that the total variance"
    },
    {
      "start": 537.55,
      "end": 541.75,
      "text": "should be proportional to time, and that would justify time linear interpolation of"
    },
    {
      "start": 541.75,
      "end": 549.35,
      "text": "variance. Well, you could reasonably disagree with that, and think, well, maybe"
    },
    {
      "start": 549.35,
      "end": 554.63,
      "text": "those movements in the index are not independent from one time interval to another."
    },
    {
      "start": 554.63,
      "end": 560.03,
      "text": "Maybe this is an auto correlation. Well, yes, we can, we can worry about that. However,"
    },
    {
      "start": 560.03,
      "end": 566.15,
      "text": "this assumption gives us a nice way to get to a good formula. So for the"
    },
    {
      "start": 566.15,
      "end": 572.39,
      "text": "30 day, if t, if my t now, I set at 30 days, then I can do this time linear"
    },
    {
      "start": 572.39,
      "end": 575.55,
      "text": "interpolation. Here you have standard linear interpolation. We're"
    },
    {
      "start": 575.55,
      "end": 580.23,
      "text": "interpolating the variances. So we put the variances there, and then the"
    },
    {
      "start": 580.23,
      "end": 585.75,
      "text": "interpolation coefficients are t2 minus t over t2 minus t1, and t minus t1"
    },
    {
      "start": 585.75,
      "end": 592.59,
      "text": "over t2 minus t1, and you can verify that that works to give you the first one,"
    },
    {
      "start": 592.59,
      "end": 599.91,
      "text": "if the 30 day mark falls at t1, and the second one, if the 30 day mark is t2, and"
    },
    {
      "start": 599.91,
      "end": 604.95,
      "text": "this is obviously linear and t, so that does the job. And then, once you've done"
    },
    {
      "start": 604.95,
      "end": 609.83,
      "text": "this, just solve this equation for sigma t. So you have to divide this by t, then"
    },
    {
      "start": 609.83,
      "end": 615.71,
      "text": "take the square root, and then the old vix is 100 times sigma t. So it's"
    },
    {
      "start": 615.71,
      "end": 619.91,
      "text": "expressed in volatility points, what traders think of as volatility points. The"
    },
    {
      "start": 619.91,
      "end": 625.19,
      "text": "sigma is typically something number like 0.35, and multiplying by 100 makes it"
    },
    {
      "start": 625.19,
      "end": 632.27,
      "text": "35, and people talk about 35 val for volatility. Okay, so that's the the old"
    },
    {
      "start": 632.27,
      "end": 644.69,
      "text": "vix. Now, here's an issue with the old vix. I believe that the CBO was hoping"
    },
    {
      "start": 644.69,
      "end": 649.57,
      "text": "that there would be so much interest in the in the vix as a measure of future"
    },
    {
      "start": 649.57,
      "end": 657.81,
      "text": "volatility that there would be a demand for futures on the vix and options on the"
    },
    {
      "start": 657.81,
      "end": 665.57,
      "text": "futures to be traded. But that demand never materialized, because if we go back"
    },
    {
      "start": 665.57,
      "end": 672.57,
      "text": "to this picture here, the old vix is an at the forward implied volatility"
    },
    {
      "start": 672.57,
      "end": 676.77,
      "text": "essentially. And one thing we know about blacksholes implied"
    },
    {
      "start": 676.77,
      "end": 682.21,
      "text": "volatilities is the blacksholes model is really not correct, and so there's"
    },
    {
      "start": 682.21,
      "end": 687.13,
      "text": "what's called a volatility smile. So if you look at a given maturity, strike by"
    },
    {
      "start": 687.13,
      "end": 692.65,
      "text": "strike, and find the blacksholes implied volatilities, they're all different. And so"
    },
    {
      "start": 692.65,
      "end": 699.05,
      "text": "what's special about the at the forward volatility? Why is that one supposedly"
    },
    {
      "start": 699.05,
      "end": 706.17,
      "text": "predictive of the future? So it was never clear what the value of that"
    },
    {
      "start": 706.17,
      "end": 713.17,
      "text": "particular number was in terms of indicating something about markets. Although"
    },
    {
      "start": 713.17,
      "end": 719.49,
      "text": "there's this phenomenon that when markets go down, people are panic, volatility"
    },
    {
      "start": 719.49,
      "end": 724.89,
      "text": "goes up, and when markets go up, volatility goes down. So it became useful as a"
    },
    {
      "start": 724.89,
      "end": 729.89,
      "text": "talking point, a lot of financial advisors and Wall Street commentators used to"
    },
    {
      "start": 729.89,
      "end": 734.41,
      "text": "call it the fear index, they probably still call it that, meaning when the vix is"
    },
    {
      "start": 734.41,
      "end": 741.73,
      "text": "high, people are kind of panicked, and they would give advice such as, well,"
    },
    {
      "start": 741.73,
      "end": 746.09,
      "text": "they're overreacting. It's a good time to buy because the market is"
    },
    {
      "start": 746.09,
      "end": 751.97,
      "text": "irrationally low, and when the vix is low, then people are complacent. So, you"
    },
    {
      "start": 751.97,
      "end": 755.73,
      "text": "know, maybe don't do anything, but now in the 1990s, as I said, this"
    },
    {
      "start": 755.73,
      "end": 760.81,
      "text": "financial engineering technology was developed, which is really quite"
    },
    {
      "start": 760.81,
      "end": 767.85,
      "text": "beautiful and sophisticated, as a way to model variance swaps. But this turned"
    },
    {
      "start": 767.85,
      "end": 772.93,
      "text": "out to be something that the SIBO could use to create a new vix, and they did"
    },
    {
      "start": 772.93,
      "end": 781.17,
      "text": "that and started publishing it in 2002. The new vix depends on two financial"
    },
    {
      "start": 781.17,
      "end": 789.29,
      "text": "discoveries, or formulas. The first is Lewis Scott's log contract paper, which"
    },
    {
      "start": 789.29,
      "end": 796.37,
      "text": "is, I think, about 1987, and a new calculus formula. This is kind of amazing and"
    },
    {
      "start": 796.37,
      "end": 801.73,
      "text": "wonderful that there's a new formula in the 300-year-old history of the"
    },
    {
      "start": 801.73,
      "end": 808.05,
      "text": "mathematics of calculus. Something new pops up in the 1990s. I call it the Car"
    },
    {
      "start": 808.05,
      "end": 813.33,
      "text": "Medan formula, and this is my personal name for it. This name comes from the"
    },
    {
      "start": 813.33,
      "end": 817.41,
      "text": "fact that the first place I read about it was in a little article in a"
    },
    {
      "start": 817.41,
      "end": 823.97,
      "text": "quant journal written by Peter Carr and Philip Medan. But I asked Peter about it"
    },
    {
      "start": 823.97,
      "end": 828.33,
      "text": "one time, and he said, oh, everybody knew about that in the 90s. It was just in the"
    },
    {
      "start": 828.33,
      "end": 837.01,
      "text": "air. So, lots of people apparently realized that this formula existed, and had"
    },
    {
      "start": 837.01,
      "end": 840.81,
      "text": "discovered it simultaneously. So, we're going to talk about these two"
    },
    {
      "start": 840.81,
      "end": 845.73,
      "text": "components, and then put them together to see how to find the new vix, how to"
    },
    {
      "start": 845.73,
      "end": 849.69,
      "text": "calculate the new vix. So, the first thing is the log contract. So, what was"
    },
    {
      "start": 849.69,
      "end": 854.45,
      "text": "Lewis Scott doing? He was saying, he was just posing the hypothetical question, what"
    },
    {
      "start": 854.45,
      "end": 863.17,
      "text": "is the value of a European-style derivative security, which pays as its payoff to"
    },
    {
      "start": 863.17,
      "end": 868.77,
      "text": "the holder of the contract, the log of the stock price at maturity. So, this"
    },
    {
      "start": 868.77,
      "end": 873.45,
      "text": "contract, if you own this contract, it does nothing until maturity, and then it"
    },
    {
      "start": 873.45,
      "end": 877.45,
      "text": "pays you the log of the stock price. It's a little weird. I've never seen"
    },
    {
      "start": 877.45,
      "end": 880.77,
      "text": "everybody trading such things, but he was asking this question more or less"
    },
    {
      "start": 880.77,
      "end": 892.05,
      "text": "abstractly. Now, in no arbitrage pricing, we know that finding the no arbitrage"
    },
    {
      "start": 892.05,
      "end": 896.13,
      "text": "price, which is the correct price for a derivative security, is the same as"
    },
    {
      "start": 896.13,
      "end": 902.93,
      "text": "finding the discounted expected payoff under the risk-neutral probability"
    },
    {
      "start": 902.93,
      "end": 908.45,
      "text": "distribution. So, this little hat over the E is my reminder to myself and to"
    },
    {
      "start": 908.45,
      "end": 913.85,
      "text": "you always, that this is not the expectation using any kind of real"
    },
    {
      "start": 913.85,
      "end": 921.05,
      "text": "probability for the future stock price. It's the expectation using the so-called"
    },
    {
      "start": 921.05,
      "end": 925.97,
      "text": "risk-neutral probability density function for this future stock price, or in this"
    },
    {
      "start": 925.97,
      "end": 932.21,
      "text": "case, the index price ST. And that risk-neutral distribution is, there's a lot"
    },
    {
      "start": 932.21,
      "end": 935.69,
      "text": "of math behind that, but it's the distribution that's equivalent to finding the"
    },
    {
      "start": 935.69,
      "end": 942.45,
      "text": "no arbitrage price. So, this should be what we're after, if we want a price, the"
    },
    {
      "start": 942.45,
      "end": 947.85,
      "text": "Lewis Scott log contract. So, the key question is, how do we find E hat of log"
    },
    {
      "start": 947.85,
      "end": 954.93,
      "text": "of ST? Okay, so, in order to do that, we have to make some assumptions about the"
    },
    {
      "start": 954.93,
      "end": 961.33,
      "text": "form of that risk-neutral distribution. Or in other words, where does the risk-neutral"
    },
    {
      "start": 961.33,
      "end": 967.13,
      "text": "distribution come from? It comes from the stochastic process for the stock. So, Lewis Scott"
    },
    {
      "start": 967.13,
      "end": 975.77,
      "text": "is going to make assumptions about the underlying process for the index or the stock, in this"
    },
    {
      "start": 975.77,
      "end": 981.61,
      "text": "case, the S&P 500 index. He's going to represent the stochastic process, stochastic, by the way,"
    },
    {
      "start": 981.61,
      "end": 987.17,
      "text": "just means random. That's all it is, random process. He's going to represent it by what's"
    },
    {
      "start": 987.17,
      "end": 993.09,
      "text": "called a stochastic differential equation, or SDE. So, that's why I wrote SDE here. And"
    },
    {
      "start": 993.09,
      "end": 1000.89,
      "text": "this is the differential. So, the DST is the differential of the index price over a little"
    },
    {
      "start": 1000.89,
      "end": 1006.69,
      "text": "infinitesimal amount of time, DT. So, literally, it's just, well, when you get to time T, you"
    },
    {
      "start": 1006.69,
      "end": 1014.61,
      "text": "know the index price is ST. We don't know what it's going to be, DT time later, infinitesimally,"
    },
    {
      "start": 1014.61,
      "end": 1021.61,
      "text": "infinitesimal increment later. But we'll just call that difference, DST. So, this is a random"
    },
    {
      "start": 1021.61,
      "end": 1029.81,
      "text": "variable. And the model for this is going to be, well, DST is going to look like R minus Q. So,"
    },
    {
      "start": 1029.81,
      "end": 1034.13,
      "text": "this is what makes it risk-neutral. That's why R minus Q is here. Ordinarily, there would be"
    },
    {
      "start": 1034.13,
      "end": 1041.33,
      "text": "some unknown drift that includes some risk premium. But here, we're doing risk-free, risk-neutral"
    },
    {
      "start": 1041.33,
      "end": 1048.29,
      "text": "pricing times STDT. And if you look at this and forget about the random shock over here, this"
    },
    {
      "start": 1048.29,
      "end": 1054.81,
      "text": "says DST over ST is just R minus QDT. And that's the differential, the ordinary differential"
    },
    {
      "start": 1054.81,
      "end": 1061.09,
      "text": "equation for exponential growth. But then we add randomness to it. So, the DWT is a standard"
    },
    {
      "start": 1061.09,
      "end": 1069.57,
      "text": "brownie in motion, standard brownie in motion increment. And you make this proportional to the"
    },
    {
      "start": 1069.57,
      "end": 1075.89,
      "text": "volatility that you want and the stock price so that it's still modeling the return DST over"
    },
    {
      "start": 1075.89,
      "end": 1084.37,
      "text": "ST. Now, the question is, what do you assume about SIGMA? Well, Lewis Scott wanted this"
    },
    {
      "start": 1084.37,
      "end": 1089.57,
      "text": "to be very, very general. So, he said, well, let's assume a stochastic volatility model, in"
    },
    {
      "start": 1089.57,
      "end": 1097.77,
      "text": "which case, you need a stochastic process for the volatility SIGMA. So, you have to have"
    },
    {
      "start": 1097.77,
      "end": 1103.29,
      "text": "another stochastic differential equation for SIGMA. So, this is going to be D SIGMA T. So, SIGMA T"
    },
    {
      "start": 1103.29,
      "end": 1108.49,
      "text": "itself is random, which gives you a more realistic representation of the way things work in the world."
    },
    {
      "start": 1109.77,
      "end": 1118.33,
      "text": "Now, there are infinitely many choices possible for how you construct the SDE or the process for SIGMA T."
    },
    {
      "start": 1119.29,
      "end": 1123.45,
      "text": "Anything can be the beauty of this analysis is that anything works here,"
    },
    {
      "start": 1124.17,
      "end": 1130.01,
      "text": "provided that the SIGMA T sample paths are square integral. That's really all you need."
    },
    {
      "start": 1130.01,
      "end": 1135.93,
      "text": "That's the only restriction you need. So, it doesn't matter what form this takes. That's a wonderful"
    },
    {
      "start": 1135.93,
      "end": 1143.53,
      "text": "feature here. Okay. So, in other words, we require that these integrals exist for each path"
    },
    {
      "start": 1143.53,
      "end": 1149.05,
      "text": "followed by SIGMA T. And those paths will be continuous because brownie in motion is a continuous"
    },
    {
      "start": 1149.05,
      "end": 1157.21,
      "text": "process. Okay. So, given this stochastic process for ST and SIGMA T, then the question became,"
    },
    {
      "start": 1157.21,
      "end": 1162.89,
      "text": "well, what's the process for log ST? Because if we can find the random process for log ST,"
    },
    {
      "start": 1162.89,
      "end": 1167.85,
      "text": "then we can find the probability distribution for log ST and find its expectation."
    },
    {
      "start": 1169.13,
      "end": 1173.45,
      "text": "And that's all going to be risk neutral because we put the R minus Q in there as the drift."
    },
    {
      "start": 1173.61,
      "end": 1183.05,
      "text": "Okay. So, now I'm going to just use a Taylor expansion for the log function around the fixed point ST."
    },
    {
      "start": 1183.05,
      "end": 1190.65,
      "text": "Remember, once you're at time T, ST is known. ST plus DT is the random piece, and so these"
    },
    {
      "start": 1190.65,
      "end": 1198.33,
      "text": "differentials are random, but ST is known. Now, remember from calculus that the derivative of"
    },
    {
      "start": 1198.33,
      "end": 1203.37,
      "text": "log is 1 over x. The second derivative is therefore the derivative of 1 over x, which is x to the"
    },
    {
      "start": 1203.37,
      "end": 1208.81,
      "text": "minus first power, so it's minus x to the minus 2 or minus 1 over x squared, and the derivatives keep"
    },
    {
      "start": 1208.81,
      "end": 1216.01,
      "text": "going. The Taylor expansion is the first derivative times the change, this difference, plus half the"
    },
    {
      "start": 1216.01,
      "end": 1221.37,
      "text": "second derivative times the square of the change, plus 1 over three factorial third derivative times"
    },
    {
      "start": 1221.45,
      "end": 1228.01,
      "text": "the cube of the change, change, and so on, add and finite them. And if I plug these two into the"
    },
    {
      "start": 1228.01,
      "end": 1235.69,
      "text": "first two terms, we see that we get 1 over ST, that's 1 over x, DST, minus 1 half, that's the half"
    },
    {
      "start": 1235.69,
      "end": 1243.13,
      "text": "there, and times the 1 over x squared is 1 over s squared, DST squared, and so on. And now we're"
    },
    {
      "start": 1243.13,
      "end": 1250.41,
      "text": "going to ask ourselves, what do these terms look like? And a key point here is how many of them are"
    },
    {
      "start": 1250.41,
      "end": 1259.45,
      "text": "truly large enough to be considered. So we need to focus on a rule in stochastic calculus, so this is"
    },
    {
      "start": 1259.45,
      "end": 1268.97,
      "text": "kind of a computational rule in stochastic calculus. Since DT is infinitesimal, DT to a power that's"
    },
    {
      "start": 1268.97,
      "end": 1276.17,
      "text": "bigger than 1 is 0, we wipe it out. The basic intuitive understanding of this is if I were calculating"
    },
    {
      "start": 1276.17,
      "end": 1285.85,
      "text": "derivatives, so I'm taking DS over DT of this random quantity, and I'm calculating the derivative"
    },
    {
      "start": 1285.85,
      "end": 1293.05,
      "text": "of some function of S. Well, when I calculate the numerator, the DS, the differential of S, or"
    },
    {
      "start": 1293.05,
      "end": 1298.73,
      "text": "differential of the function of S, if there are terms in it with DT to a higher power than 1,"
    },
    {
      "start": 1298.73,
      "end": 1305.93,
      "text": "DT to the epsilon, then when I find the derivative by dividing by DT, that divided by DT is going to"
    },
    {
      "start": 1305.93,
      "end": 1311.85,
      "text": "be DT to the epsilon minus 1, which is still a positive power, and in the limit, because these are"
    },
    {
      "start": 1311.85,
      "end": 1318.49,
      "text": "infinitesimal, they're really zeros, the DTs, that goes to zero. So those terms disappear, they don't count."
    },
    {
      "start": 1319.61,
      "end": 1326.33,
      "text": "So this is our rule of thumb. Now we have to ask, well, what's the order of magnitude in terms of DT"
    },
    {
      "start": 1326.33,
      "end": 1333.21,
      "text": "of these terms we're looking at? What's the size of DST? What's the size of DWT? So remember, the"
    },
    {
      "start": 1333.21,
      "end": 1341.45,
      "text": "increment of Brownian motion at time T is the future WT plus DT minus WT, and the reason we call it"
    },
    {
      "start": 1341.45,
      "end": 1350.57,
      "text": "standard Brownian motion is that it's normal with mean zero, standardized to mean zero, and standard"
    },
    {
      "start": 1350.57,
      "end": 1354.81,
      "text": "deviation, I'm going to put the standard deviation as the second parameter of the normal distribution"
    },
    {
      "start": 1354.81,
      "end": 1362.97,
      "text": "here, square root of DT. WT itself is normal with mean zero, standard deviation squared"
    },
    {
      "start": 1362.97,
      "end": 1370.09,
      "text": "of T. So the variance is T, notice that W1 for one unit of time, W1 is standard normal and of"
    },
    {
      "start": 1370.09,
      "end": 1377.29,
      "text": "zero one, that's why we call it standard Brownian motion. Okay, so knowing that, we can ask about"
    },
    {
      "start": 1377.29,
      "end": 1385.21,
      "text": "sizes. So what is the size of a random variable like DWT? Well, a random variable you can think of"
    },
    {
      "start": 1385.21,
      "end": 1391.77,
      "text": "as something in function space or just a vector, and how do you measure the size? Well, the L2"
    },
    {
      "start": 1391.77,
      "end": 1398.09,
      "text": "norm is a standard measure of size. And what is the L2 norm of a random variable? Well, it's"
    },
    {
      "start": 1398.81,
      "end": 1404.25,
      "text": "the square root of the mean of the squares, the expected value of the squares, that's the standard"
    },
    {
      "start": 1404.25,
      "end": 1410.17,
      "text": "deviation. So the standard deviation is the measure of size, it's the L2 norm of DWT, and so"
    },
    {
      "start": 1410.17,
      "end": 1417.21,
      "text": "that's the square root of DT or DT to the one half. So that in stochastic calculus, DWT stays in"
    },
    {
      "start": 1417.21,
      "end": 1427.93,
      "text": "all the equations. Okay, but now we can start looking at things like DST squared. So notice in our"
    },
    {
      "start": 1427.93,
      "end": 1435.93,
      "text": "stochastic process, our random process for the stock price, DST is R minus QST times DT to the"
    },
    {
      "start": 1435.93,
      "end": 1443.21,
      "text": "first power, that's not a prime, that's the first power, plus sigma TST, DWT, and this is"
    },
    {
      "start": 1443.21,
      "end": 1450.01,
      "text": "a border DT to the one half power. So the overall size of DST is, or the order of it, is DT to the"
    },
    {
      "start": 1450.01,
      "end": 1457.05,
      "text": "one half power. Okay, so what about DST squared? Well, if you square this, notice the first term"
    },
    {
      "start": 1457.05,
      "end": 1463.13,
      "text": "squared is DT squared, which goes to zero, that disappears. The cross terms, when you square it,"
    },
    {
      "start": 1463.13,
      "end": 1468.49,
      "text": "are DT to the three halves, bigger than power, bigger than one, that goes to zero, so those are gone."
    },
    {
      "start": 1469.37,
      "end": 1475.21,
      "text": "So DST squared actually reduces to just the square of the last term, which is going to be"
    },
    {
      "start": 1475.21,
      "end": 1483.21,
      "text": "of order DT and should survive. Okay, what about the cubes? Well, luckily for us, and wonderfully,"
    },
    {
      "start": 1483.69,
      "end": 1489.37,
      "text": "all the higher powers go to zero, so we don't need all those extra terms in the Taylor series expansion."
    },
    {
      "start": 1490.41,
      "end": 1494.57,
      "text": "They just disappear. We only need those first two terms, which is why I only wrote those two"
    },
    {
      "start": 1496.25,
      "end": 1503.05,
      "text": "down here. These are the only two terms we get in DLog ST. So now we're going to compute those."
    },
    {
      "start": 1505.94,
      "end": 1511.54,
      "text": "So, DWT is normal, with mean zero, standard deviation squared of T."
    },
    {
      "start": 1512.98,
      "end": 1517.94,
      "text": "If I take a normal random variable, minus its mean, which in this case is zero, and divide by the"
    },
    {
      "start": 1517.94,
      "end": 1523.38,
      "text": "standard deviation, that standardizes it, so it becomes a standard normal, which we can call Z,"
    },
    {
      "start": 1523.38,
      "end": 1528.5,
      "text": "because everybody thinks of Z as the name of a standard normal. Well, the Z squared is always"
    },
    {
      "start": 1528.58,
      "end": 1533.14,
      "text": "chi squared with one degree of freedom. And we know about the chi squared distribution."
    },
    {
      "start": 1534.34,
      "end": 1540.66,
      "text": "Its mean is one, its variance is two. So now we know the mean and variance of the square of"
    },
    {
      "start": 1540.66,
      "end": 1550.74,
      "text": "DWT over the square root of DT. The mean of DWT over square root of DT squared is one, and if you do"
    },
    {
      "start": 1550.74,
      "end": 1557.38,
      "text": "the algebra here, you get that the mean, the expectation of DWT quantity squared is DT."
    },
    {
      "start": 1557.7,
      "end": 1564.42,
      "text": "When you square this, the denominator is just DT. Likewise here, you have DWT quantity squared"
    },
    {
      "start": 1564.42,
      "end": 1570.1,
      "text": "divided by DT. Now, with the variance, when you factor out a constant, it's squared, because the"
    },
    {
      "start": 1570.1,
      "end": 1578.02,
      "text": "variance is the expectation of the square of something. That means that the variance of DWT"
    },
    {
      "start": 1578.02,
      "end": 1585.38,
      "text": "squared is going to be two DT squared, but this is stochastic calculus, DT squared is zero."
    },
    {
      "start": 1586.1,
      "end": 1591.94,
      "text": "This thing has variance zero, variance of DWT squared is zero. That means it's a constant."
    },
    {
      "start": 1593.38,
      "end": 1598.66,
      "text": "And if it's a constant, then the mean is that constant. So we now know DWT squared,"
    },
    {
      "start": 1599.38,
      "end": 1605.7,
      "text": "speaking in stochastic calculus calculations, is DT. So we can now substitute this back into our"
    },
    {
      "start": 1605.7,
      "end": 1615.06,
      "text": "formulas, because DST squared is now sigma squared S squared DT. So when these two"
    },
    {
      "start": 1615.06,
      "end": 1620.66,
      "text": "terms, the only surviving terms, we have one over ST times the first term DST. That just"
    },
    {
      "start": 1621.3,
      "end": 1628.02,
      "text": "cancels out the S's from the stochastic differential equation for S, leaving R minus QDT plus"
    },
    {
      "start": 1628.02,
      "end": 1636.5,
      "text": "sigma TDWT. And then here, we're dividing by ST squared. We have the minus 1 half, and that gets"
    },
    {
      "start": 1636.5,
      "end": 1644.58,
      "text": "rid of the S's, the ST's in the DST squared. Those are squares. And that's just sigma T squared DT."
    },
    {
      "start": 1644.58,
      "end": 1652.9,
      "text": "So we get minus 1 half sigma T squared DT. So this is the SDE for log ST."
    },
    {
      "start": 1654.34,
      "end": 1658.9,
      "text": "Now there are two random pieces here. This part is random, and that part is random."
    },
    {
      "start": 1661.61,
      "end": 1665.05,
      "text": "Now we're going to integrate both sides of that equation. So if you look at this equation,"
    },
    {
      "start": 1665.05,
      "end": 1670.41,
      "text": "we're going to integrate this, and we're going to integrate that, and we're going to integrate"
    },
    {
      "start": 1670.41,
      "end": 1678.57,
      "text": "them as time goes from 0 to to big T. Well, what's the integral from 0 to big T of D log ST? Well,"
    },
    {
      "start": 1678.57,
      "end": 1687.61,
      "text": "now we're just adding up differentials. And the differentials, each one is like log ST plus"
    },
    {
      "start": 1687.61,
      "end": 1698.65,
      "text": "DT minus the previous ST log ST. That's a collapsing sum. And so it's just the last term minus the"
    },
    {
      "start": 1698.65,
      "end": 1705.37,
      "text": "first term. Even though this is a random quantity in here, we can still add these up because the"
    },
    {
      "start": 1706.33,
      "end": 1712.97,
      "text": "the differentials form this collapsing sum, and we just get this. It's still random because S"
    },
    {
      "start": 1712.97,
      "end": 1719.69,
      "text": "in the future ST is random. Okay, then the other three on the other side, we have the three terms"
    },
    {
      "start": 1719.69,
      "end": 1725.53,
      "text": "to integrate. This is deterministic. So the integral of this is just R minus QT. That's just"
    },
    {
      "start": 1725.53,
      "end": 1733.13,
      "text": "simple calculus. This is an integral of a random path, and this is an integral of a random path."
    },
    {
      "start": 1733.13,
      "end": 1739.21,
      "text": "So those are just random. So we have the random log ST. We could write as the log of S naught plus"
    },
    {
      "start": 1739.21,
      "end": 1746.49,
      "text": "this, and then plus these two stochastic terms, random terms. So the next step in the technology"
    },
    {
      "start": 1746.49,
      "end": 1751.05,
      "text": "development here is we're going to take risk neutral expectations of all these things."
    },
    {
      "start": 1753.91,
      "end": 1761.51,
      "text": "Okay, well, what we have so far rewriting this is the log of ST is the log of S0 plus R minus QT"
    },
    {
      "start": 1761.51,
      "end": 1767.83,
      "text": "plus these two integrals. One thing I want to note right off the bat here is that if you look at"
    },
    {
      "start": 1767.83,
      "end": 1775.83,
      "text": "log of S0 plus R minus QT, that is the log of the forward price. So we can just summarize that as"
    },
    {
      "start": 1775.83,
      "end": 1783.27,
      "text": "log of F. And then we have these other two terms. Now we're going to take the risk neutral"
    },
    {
      "start": 1783.27,
      "end": 1789.59,
      "text": "expectation of this. This is a constant. So it's just that constant, and then we have the risk"
    },
    {
      "start": 1789.59,
      "end": 1795.03,
      "text": "neutral expectation of this integral, because expectation is an integral, and we have Fubini's"
    },
    {
      "start": 1795.03,
      "end": 1798.95,
      "text": "theorem that says we can switch the order of integration or in calculus, you know, that with"
    },
    {
      "start": 1798.95,
      "end": 1803.99,
      "text": "double integrals, you can switch the order of integration. We can move the expectation inside"
    },
    {
      "start": 1803.99,
      "end": 1810.31,
      "text": "the integral, and here I'm going to leave the expectation outside for a certain reason."
    },
    {
      "start": 1811.03,
      "end": 1817.19,
      "text": "And now Lewis Scott is kind of stuck. He says, okay, I've got this integral here of this"
    },
    {
      "start": 1817.19,
      "end": 1822.63,
      "text": "expectation of this product. I don't know what to do, and you can see what's going to happen here."
    },
    {
      "start": 1822.63,
      "end": 1826.55,
      "text": "He's going to make another assumption. In addition to stochastic volatility,"
    },
    {
      "start": 1827.51,
      "end": 1833.67,
      "text": "he's going to assume that sigma t, the sigma t process is non-anticipating with respect to"
    },
    {
      "start": 1834.63,
      "end": 1838.95,
      "text": "dw t. And essentially what that means, we just need to"
    },
    {
      "start": 1840.15,
      "end": 1848.15,
      "text": "know that it means that sigma t is independent of dw t. The thinking being sigma t is determined"
    },
    {
      "start": 1848.15,
      "end": 1855.11,
      "text": "at time t. It becomes a constant. And so this, which is entirely in the future from time t,"
    },
    {
      "start": 1855.11,
      "end": 1859.91,
      "text": "it comes after t shouldn't really depend on this. They should be independent."
    },
    {
      "start": 1860.87,
      "end": 1867.91,
      "text": "Now, again, this is something you could argue with. But it's not crazy to assume this. And since"
    },
    {
      "start": 1867.91,
      "end": 1875.11,
      "text": "it's not crazy, it gives us an avenue for simplifying the result here. And you can see what happens."
    },
    {
      "start": 1875.11,
      "end": 1880.07,
      "text": "Once these are independent, the expectation of the product is the product of the expectations"
    },
    {
      "start": 1880.07,
      "end": 1886.23,
      "text": "from basic probability. And the expectation of Brownian motion is zero. So this whole integral"
    },
    {
      "start": 1886.23,
      "end": 1895.67,
      "text": "disappears under this assumption. And now we have this beautiful formula. E hat log st is the"
    },
    {
      "start": 1895.67,
      "end": 1904.58,
      "text": "log of the forward minus one half the expectation of this integral. And now suddenly, okay, so this is"
    },
    {
      "start": 1904.58,
      "end": 1913.06,
      "text": "kind of the end of the discussion of the log contract for Lewis Scott, I guess. But it's"
    },
    {
      "start": 1913.06,
      "end": 1920.1,
      "text": "an incredible discovery because of this term. If we rewrite this to express that term in terms"
    },
    {
      "start": 1920.1,
      "end": 1927.62,
      "text": "of the other two, what do we have? We have an expectation of future realized total variance."
    },
    {
      "start": 1927.62,
      "end": 1936.98,
      "text": "Notice sigma t squared dt. That's since volatility is stochastic. That's the infinitesimal volatility"
    },
    {
      "start": 1936.98,
      "end": 1943.3,
      "text": "or infinitesimal variance that's going to occur in the time interval interval from t to t plus"
    },
    {
      "start": 1943.3,
      "end": 1950.34,
      "text": "dt. And if we add up all those infinitesimal variances, we get the total variance that it is"
    },
    {
      "start": 1950.34,
      "end": 1956.18,
      "text": "going to be realized from zero to t. It's random. And here, we're calculating an actual mathematical"
    },
    {
      "start": 1956.18,
      "end": 1963.54,
      "text": "expectation of future realized variance. It's the risk neutral expected future realized variance."
    },
    {
      "start": 1963.54,
      "end": 1970.66,
      "text": "This is amazing. This is amazing. And we can express it as two times log of the forward minus"
    },
    {
      "start": 1970.66,
      "end": 1976.9,
      "text": "the risk neutral expectation of the log of st. So this is really tantalizing because it says we"
    },
    {
      "start": 1976.9,
      "end": 1983.38,
      "text": "could get a really mathematically sound estimate of future realized volatility during a fixed time"
    },
    {
      "start": 1983.38,
      "end": 1994.1,
      "text": "period. If only we could calculate the log contract future value e hat log of st. So that's brilliant."
    },
    {
      "start": 1995.14,
      "end": 2000.5,
      "text": "Now, once we are able to do that, which we're able to do in a few minutes,"
    },
    {
      "start": 2001.54,
      "end": 2009.86,
      "text": "I want to think of this risk neutral expected future realized variance as the annualized the"
    },
    {
      "start": 2009.86,
      "end": 2022.1,
      "text": "average annualized variance sigma t squared times t. If I had an annualized variance for that time"
    },
    {
      "start": 2022.1,
      "end": 2028.66,
      "text": "period, this would be the total variance in that time period. So sigma t bar would be the average"
    },
    {
      "start": 2028.66,
      "end": 2033.7,
      "text": "annualized volatility that's going to be realized in the future time period from zero to t."
    },
    {
      "start": 2034.42,
      "end": 2042.82,
      "text": "So substituting that in here, we have this equation and solving for sigma t bar squared, we get this"
    },
    {
      "start": 2043.86,
      "end": 2049.94,
      "text": "and this is really the key to variance swaps. But we need to be able to fill this in."
    },
    {
      "start": 2052.39,
      "end": 2059.35,
      "text": "Okay, so the next part is the karma Dan formula because that's the key to filling in the expectation"
    },
    {
      "start": 2059.59,
      "end": 2067.91,
      "text": "of the log of st. Okay, so this, as I said, I don't know exactly the origin of this. It seems to"
    },
    {
      "start": 2067.91,
      "end": 2077.43,
      "text": "have arisen naturally in the 1990s. But it's basically a theorem in calculus. And it really"
    },
    {
      "start": 2077.43,
      "end": 2084.15,
      "text": "belongs in two textbooks as an exercise in this section on integration by parts because that's"
    },
    {
      "start": 2084.15,
      "end": 2090.95,
      "text": "how we're going to prove it. So here's the theorem. The theorem says let f of x be a twice"
    },
    {
      "start": 2090.95,
      "end": 2096.55,
      "text": "piecewise continuously differentiable function on an interval a b. It can be any interval."
    },
    {
      "start": 2097.51,
      "end": 2106.47,
      "text": "Pick any fixed point x naught in a b. Then for any x in a b, we can represent the value of the"
    },
    {
      "start": 2106.47,
      "end": 2112.15,
      "text": "function f of x as f of x naught plus f prime x naught times x minus x naught. So this is like"
    },
    {
      "start": 2112.15,
      "end": 2117.19,
      "text": "the Taylor expansion, the beginning of it, but we cut it off there and replace everything else"
    },
    {
      "start": 2117.19,
      "end": 2125.19,
      "text": "by these two integrals. The integral from a to our fixed cutoff point times y minus x plus y minus"
    },
    {
      "start": 2125.19,
      "end": 2133.35,
      "text": "x plus is the positive part. That's the maximum of that number and zero. So it's y minus x if y"
    },
    {
      "start": 2133.35,
      "end": 2140.47,
      "text": "minus x is positive and it's zero if y minus x is negative. So it's the integral of y minus x plus"
    },
    {
      "start": 2140.47,
      "end": 2147.19,
      "text": "times f double prime of y dy plus the integral from x naught to b of x minus y plus the positive"
    },
    {
      "start": 2147.19,
      "end": 2153.19,
      "text": "part of that times f double prime of y dy. So notice we're using the first, the function,"
    },
    {
      "start": 2153.19,
      "end": 2158.47,
      "text": "the first derivative and the second derivatives. And you can see why we need, we basically need f double"
    },
    {
      "start": 2158.47,
      "end": 2166.12,
      "text": "prime to be integral for this to work. Okay, that's the theorem. We're going to give a quick"
    },
    {
      "start": 2166.2,
      "end": 2172.2,
      "text": "proof of it in a second. But first, let's take a look at what it says that's really incredible."
    },
    {
      "start": 2173.96,
      "end": 2180.76,
      "text": "So what is x minus y positive part as a function of x? Because if you go back to this, you can think"
    },
    {
      "start": 2180.76,
      "end": 2186.76,
      "text": "of this as saying, okay, I've got this function of f of x. I can express it as a constant plus a"
    },
    {
      "start": 2186.76,
      "end": 2194.04,
      "text": "linear function of x. And then I can think of the y minus x plus as a function of x for each y,"
    },
    {
      "start": 2194.04,
      "end": 2198.92,
      "text": "I have a different one and I'm adding up this intervals like taking a big linear combination of"
    },
    {
      "start": 2198.92,
      "end": 2204.36,
      "text": "these functions with these coefficients. And over here, I've got a big linear combination of the"
    },
    {
      "start": 2204.36,
      "end": 2211.0,
      "text": "x minus y plus functions of x with these coefficients. So this says f of x can be represented as a"
    },
    {
      "start": 2211.0,
      "end": 2218.76,
      "text": "constant plus a linear function plus linear combinations of those functions, those y minus x and x"
    },
    {
      "start": 2218.76,
      "end": 2226.04,
      "text": "minus y plus functions. Well, as a function of x, y is fixed, and this is what that function looks"
    },
    {
      "start": 2226.04,
      "end": 2233.64,
      "text": "like as a function of x, y minus x plus is this one. And immediately, you see that this looks like"
    },
    {
      "start": 2233.64,
      "end": 2239.24,
      "text": "the call payoff function for the payoff of a call at maturity, and this is the put payoff function."
    },
    {
      "start": 2240.84,
      "end": 2247.4,
      "text": "Peter Carr, by the way, who was Canadian, was a huge hockey fan, and he used to always refer to"
    },
    {
      "start": 2247.4,
      "end": 2253.4,
      "text": "these two pictures as the hockey sticks. So the call and put payoff functions are the hockey sticks."
    },
    {
      "start": 2255.16,
      "end": 2261.96,
      "text": "So we've learned that every twice piecewise continuously differentiable function can be represented"
    },
    {
      "start": 2261.96,
      "end": 2267.48,
      "text": "as a big linear combination of hockey sticks and a constant. And a linear function can always be"
    },
    {
      "start": 2267.48,
      "end": 2274.52,
      "text": "written as two hockey sticks, call minus a put with the same strike. Okay, so here's the proof."
    },
    {
      "start": 2275.48,
      "end": 2282.6,
      "text": "We're going to divide this into two cases. Notice in the theorem, we only have to verify this"
    },
    {
      "start": 2282.6,
      "end": 2288.52,
      "text": "as an identity for a fixed x for each x separately. So we're going to do this one x at a time."
    },
    {
      "start": 2289.8,
      "end": 2295.0,
      "text": "Case one is when x is less than or equal to x0, case two is x bigger than x0."
    },
    {
      "start": 2296.2,
      "end": 2302.84,
      "text": "So in case one, let's notice that the second integral goes from x0 to b. Well,"
    },
    {
      "start": 2302.92,
      "end": 2309.8,
      "text": "x is less than or equal to x0. So y is the variable of integration. It goes from x0 to b. If x is"
    },
    {
      "start": 2309.8,
      "end": 2316.6,
      "text": "less than or equal to x0, y is always bigger than x, because x is below x0, for this fixed x we're"
    },
    {
      "start": 2316.6,
      "end": 2321.88,
      "text": "talking about. And therefore, this is always zero. So the second integral disappears. It's gone."
    },
    {
      "start": 2323.0,
      "end": 2329.32,
      "text": "What about the first integral? Well, the first integral goes from a to x0, and it's y minus x plus"
    },
    {
      "start": 2329.4,
      "end": 2335.88,
      "text": "f double prime y dy. Now x is less than or equal to x0. It's bigger than a, because it's in the"
    },
    {
      "start": 2335.88,
      "end": 2343.48,
      "text": "interval. But notice, for the y is below x, this is zero. So we only have to integrate from x to"
    },
    {
      "start": 2343.48,
      "end": 2350.84,
      "text": "x0. And between x and x0, y minus x plus is just y minus x. And now this integral,"
    },
    {
      "start": 2350.84,
      "end": 2356.36,
      "text": "integral of y minus x f double prime of y dy is a perfect candidate for integration by parts."
    },
    {
      "start": 2356.36,
      "end": 2361.4,
      "text": "What is integration by parts? It's the reverse of the product rule for derivatives. Here's the"
    },
    {
      "start": 2361.4,
      "end": 2369.72,
      "text": "product rule for derivatives written in differential form. If we rewrite that as udv is duv minus vdu,"
    },
    {
      "start": 2369.72,
      "end": 2375.24,
      "text": "and integrate both sides, we get integration by parts. Integral of udv is uv minus vdu. Well,"
    },
    {
      "start": 2375.88,
      "end": 2384.12,
      "text": "in this integral, we can choose u to be y minus x as a function of y, because we're doing an"
    },
    {
      "start": 2384.12,
      "end": 2391.24,
      "text": "integral of a function of y. The x is fixed. dv will take to be this. Well, for v, we need an"
    },
    {
      "start": 2391.24,
      "end": 2397.24,
      "text": "anti-derivative of that. Obviously an anti-derivative of f double prime is f prime. The second derivative"
    },
    {
      "start": 2397.24,
      "end": 2404.68,
      "text": "anti-derivative is the first derivative. And then the derivative of y minus x with respect to y is"
    },
    {
      "start": 2404.68,
      "end": 2410.2,
      "text": "du is just dy, because that's linear. So now applying integration by parts,"
    },
    {
      "start": 2410.76,
      "end": 2417.0,
      "text": "that's going to be equal to y minus x f prime of y evaluated from x to x naught minus the"
    },
    {
      "start": 2417.0,
      "end": 2424.28,
      "text": "integral from x to x naught of vdu, which is f prime y dy. Well, this integral is just f of y"
    },
    {
      "start": 2424.28,
      "end": 2431.0,
      "text": "evaluated from x to x naught. So we get this. Here, if I substitute in x naught for y,"
    },
    {
      "start": 2431.0,
      "end": 2435.48,
      "text": "I get this. When I substitute x in, it's zero, because this is y minus x."
    },
    {
      "start": 2435.48,
      "end": 2441.96,
      "text": "So here are the terms. Well, notice this is just f of x minus the first two terms"
    },
    {
      "start": 2443.32,
      "end": 2448.04,
      "text": "in the carbon-dan formula. So they cancel out and the identity holds. We just get f of x on"
    },
    {
      "start": 2448.04,
      "end": 2457.32,
      "text": "the right-hand side. Done. This is so cool. This is amazing. Case two is similar. In that case,"
    },
    {
      "start": 2457.32,
      "end": 2462.28,
      "text": "the first integral becomes zero. The second integral gives you this identity."
    },
    {
      "start": 2462.28,
      "end": 2469.16,
      "text": "I'm going to let you figure that out for yourself if it isn't already clear to you, but you can work"
    },
    {
      "start": 2469.16,
      "end": 2477.24,
      "text": "this out. It's a good exercise. But so cool. Really so cool. 300 years of calculus and nobody"
    },
    {
      "start": 2477.24,
      "end": 2483.48,
      "text": "saw this. Why is that? Well, my guess is nobody ever thought about using the hockey sticks."
    },
    {
      "start": 2484.68,
      "end": 2489.32,
      "text": "Right? Who would use that? Option traders would use that. Option quants would use that."
    },
    {
      "start": 2489.32,
      "end": 2495.24,
      "text": "Okay. So now we're going to apply the carbon-dan formula to derivative securities."
    },
    {
      "start": 2496.6,
      "end": 2505.32,
      "text": "So in order to make it mentally more absorbable to facilitate grasping this,"
    },
    {
      "start": 2506.2,
      "end": 2511.96,
      "text": "I'm going to change the notation in the formula. I'm going to let x be the future index or stock price."
    },
    {
      "start": 2512.68,
      "end": 2521.24,
      "text": "I'm going to let y be a generic strike, strike price, k, and here's a little mistake."
    },
    {
      "start": 2522.04,
      "end": 2526.92,
      "text": "But there it is. X naught is going to be a fixed strike, which we're free to choose."
    },
    {
      "start": 2526.92,
      "end": 2533.32,
      "text": "Notice in carbon-dan, you can choose X naught to be any point. Now, what am I going to choose for"
    },
    {
      "start": 2533.32,
      "end": 2539.56,
      "text": "A and B? Well, I'm working with a variable that represents a stock price or an index price. Those"
    },
    {
      "start": 2539.56,
      "end": 2546.12,
      "text": "prices just go from zero to infinity. So I'm going to choose A to be zero and B to be infinity."
    },
    {
      "start": 2547.24,
      "end": 2554.92,
      "text": "So having done that, the carbon-dan formula is rewritten as F of ST is F of k naught our fixed strike"
    },
    {
      "start": 2556.04,
      "end": 2562.6,
      "text": "plus F prime at k naught times ST minus k naught. That's our x minus y. I'm sorry, x minus x naught"
    },
    {
      "start": 2563.48,
      "end": 2570.04,
      "text": "plus the integral from zero to k naught of k minus ST plus. That's the put payoff function,"
    },
    {
      "start": 2570.92,
      "end": 2576.2,
      "text": "F double prime of k dk plus the integral from k naught to infinity of the call payoff function,"
    },
    {
      "start": 2576.84,
      "end": 2585.4,
      "text": "F double prime of k dk. So that is amazing. It says, if you have a European style derivative"
    },
    {
      "start": 2585.4,
      "end": 2591.08,
      "text": "security, which is entirely defined by what it's going to pay you at maturity only,"
    },
    {
      "start": 2591.88,
      "end": 2599.08,
      "text": "so that's F of ST, it's of that form, then you can express that payoff function as a combination"
    },
    {
      "start": 2599.08,
      "end": 2605.56,
      "text": "of a constant linear payoff. This is like a forward contract payoff and put payoffs and call payoffs."
    },
    {
      "start": 2606.84,
      "end": 2614.04,
      "text": "So puts and calls are fundamental. So now we want to take the risk neutral expectation of both sides,"
    },
    {
      "start": 2614.04,
      "end": 2619.24,
      "text": "and this is where we're going to do pricing of things, option pricing or, you know,"
    },
    {
      "start": 2619.24,
      "end": 2625.88,
      "text": "derivative security pricing of general derivative securities. So e hat the risk neutral expectation"
    },
    {
      "start": 2625.88,
      "end": 2632.52,
      "text": "of F of ST is then the expectations of these. Well, this is a constant, so that's its expectation."
    },
    {
      "start": 2633.24,
      "end": 2640.12,
      "text": "ST in the risk neutral world has expectation the forward price. So this is F prime of k"
    },
    {
      "start": 2640.92,
      "end": 2649.32,
      "text": "k naught, F minus k naught. Now, as before, using the ability to switch the order of integration,"
    },
    {
      "start": 2649.32,
      "end": 2654.84,
      "text": "we can push the expectation inside the other two intervals. If we integrate the put payoff function,"
    },
    {
      "start": 2654.84,
      "end": 2662.04,
      "text": "what we're going to get is the expected payoff of the put at maturity. That's the forward value"
    },
    {
      "start": 2662.04,
      "end": 2668.04,
      "text": "of the put. So I can express that as e to the RT, the forward value of the current put price."
    },
    {
      "start": 2668.44,
      "end": 2676.36,
      "text": "So that expectation becomes that. And similarly, in the second integral, it's the call payoff function."
    },
    {
      "start": 2677.32,
      "end": 2685.0,
      "text": "And so when I take the expectation of that, I get the value of the call at maturity. That's the"
    },
    {
      "start": 2685.0,
      "end": 2690.6,
      "text": "forward value of the call price today. So I can express this in terms of put and call prices."
    },
    {
      "start": 2690.6,
      "end": 2700.84,
      "text": "So just looking at this line here, this is just stunning. It says, look, if you want a price,"
    },
    {
      "start": 2700.84,
      "end": 2706.92,
      "text": "some weird derivative security, the weird payoff function that's a European style security,"
    },
    {
      "start": 2706.92,
      "end": 2712.28,
      "text": "all you need to know is the market prices of puts and calls, because you can just add those up,"
    },
    {
      "start": 2712.28,
      "end": 2717.64,
      "text": "weighted by the second derivative of that function, and add these two terms, you need to know"
    },
    {
      "start": 2717.72,
      "end": 2723.24,
      "text": "the forward price. And boom, you're done. You don't need to go through any big contortions."
    },
    {
      "start": 2725.0,
      "end": 2732.04,
      "text": "So now finally, we apply that to the log of ST. And we can achieve the Lewis Scott vision."
    },
    {
      "start": 2732.84,
      "end": 2739.24,
      "text": "e had a log of ST. Well, we have to put in the derivatives of the log function. Well,"
    },
    {
      "start": 2739.24,
      "end": 2750.44,
      "text": "the derivative of the log function is 1 over. So we're going to get 1 over K0. So the first"
    },
    {
      "start": 2750.44,
      "end": 2756.68,
      "text": "term is just the log of K0, because the function is log. This is 1 over K0. I'm going to write that"
    },
    {
      "start": 2756.68,
      "end": 2762.44,
      "text": "whole thing as f over K0 minus 1, for a particular reason. That'll be good to think of it this way."
    },
    {
      "start": 2762.44,
      "end": 2770.2,
      "text": "And then the derivatives here are minus 1 over K squared. So in these integrals, we get minus"
    },
    {
      "start": 2770.2,
      "end": 2778.12,
      "text": "integral from 0 to K0 of e to the RT times the put divided by K squared dK and similarly for"
    },
    {
      "start": 2778.12,
      "end": 2785.88,
      "text": "the calls. This is the famous 1 over K squared weighted portfolio of puts and calls that's involved"
    },
    {
      "start": 2785.88,
      "end": 2795.32,
      "text": "in variance swaps and in the new VIX. And now we're ready to put this into our formula for"
    },
    {
      "start": 2795.32,
      "end": 2803.24,
      "text": "expected future realized volatility or variance. So here's our expected future realized variance."
    },
    {
      "start": 2803.8,
      "end": 2809.56,
      "text": "I left the bar off that. There should be a bar on there for averaged annualized volatility. But"
    },
    {
      "start": 2810.52,
      "end": 2818.52,
      "text": "it's 2 over t log f minus this expectation. So substituting in our newly found value for this via"
    },
    {
      "start": 2818.52,
      "end": 2826.36,
      "text": "Carmedan, it becomes 2 over t times the log of f minus the first term, which is log of K0. I've"
    },
    {
      "start": 2826.36,
      "end": 2832.28,
      "text": "summarized that here as log of f over K0, because the log of a quotient is the difference of the"
    },
    {
      "start": 2832.28,
      "end": 2838.52,
      "text": "logs. And then I have to change the sign of the other term. So I'm subtracting this instead of"
    },
    {
      "start": 2838.52,
      "end": 2845.16,
      "text": "adding it and adding these instead of subtracting them. So here's our nice formula. And this is what"
    },
    {
      "start": 2845.16,
      "end": 2854.2,
      "text": "you can use to calculate sigma t bar squared ti ti for any time. And formulating the new VIX,"
    },
    {
      "start": 2854.2,
      "end": 2861.64,
      "text": "we're going to choose two explorations, two S&P 500 option explorations, surrounding the 30-day"
    },
    {
      "start": 2861.64,
      "end": 2867.16,
      "text": "point. And this 30-day point is going to be our t that we're after. But we have to separately,"
    },
    {
      "start": 2867.16,
      "end": 2872.68,
      "text": "because we have, we only have option prices for fixed maturities, we have to separately calculate"
    },
    {
      "start": 2873.4,
      "end": 2881.8,
      "text": "the future expected future realized variances for t1 and t2 and then interpolate time linearly"
    },
    {
      "start": 2881.8,
      "end": 2892.12,
      "text": "to get the one in the middle. That's how you do the new VIX. So the new VIX is then 100 times"
    },
    {
      "start": 2892.12,
      "end": 2900.36,
      "text": "sigma t. You just do the time linear interpolation just as we did in the old VIX. And then you solve"
    },
    {
      "start": 2900.36,
      "end": 2909.64,
      "text": "that equation that gives you sigma t squared t. You solve it for sigma t. Notice it involves 100"
    },
    {
      "start": 2909.64,
      "end": 2916.52,
      "text": "times the square root. So this is the new VIX. That's what they publish. Now I want you to just take"
    },
    {
      "start": 2916.52,
      "end": 2924.04,
      "text": "a minute to notice something. You could also have a bone to pick with this, which is perfectly"
    },
    {
      "start": 2924.04,
      "end": 2932.04,
      "text": "reasonable. We've gone through all this and realized that this is the actual expected risk-neutral"
    },
    {
      "start": 2932.04,
      "end": 2941.72,
      "text": "expected future realized variance. So sigma t squared is the actual expected future realized"
    },
    {
      "start": 2941.72,
      "end": 2950.02,
      "text": "and annualized variance, the annualized value of that expected variance. So that's sigma t squared"
    },
    {
      "start": 2950.02,
      "end": 2956.98,
      "text": "as an expectation. And now we're expressing the VIX as 100 times the square root of an expectation."
    },
    {
      "start": 2956.98,
      "end": 2961.7,
      "text": "Well, you could worry about the square root of an expectation, not being the expectation of the"
    },
    {
      "start": 2961.7,
      "end": 2966.9,
      "text": "square root. And that's perfectly reasonable. But this is the choice the SIBO made in how to"
    },
    {
      "start": 2966.9,
      "end": 2972.26,
      "text": "calculate this and publish it. So it is this 100 times the square root of the expectation."
    },
    {
      "start": 2972.9,
      "end": 2978.9,
      "text": "Now, if we're going to talk about the new VIX, there are various computational details."
    },
    {
      "start": 2978.9,
      "end": 2987.62,
      "text": "How exactly do you do this? Because after all, these integrals assume"
    },
    {
      "start": 2989.3,
      "end": 2998.98,
      "text": "a continuum of put and call prices from the market for all strikes K over all real numbers K."
    },
    {
      "start": 2998.98,
      "end": 3004.82,
      "text": "Well, with those that don't exist, puts and calls only exist at discrete well-defined strikes."
    },
    {
      "start": 3004.82,
      "end": 3010.02,
      "text": "They're only a finite number of them. You can't literally do these integrals, so you have to do"
    },
    {
      "start": 3010.02,
      "end": 3016.58,
      "text": "numerical approximations to the integrals. And there are other numerical choices to be made here."
    },
    {
      "start": 3016.58,
      "end": 3022.42,
      "text": "So I'm just going to go through some of the SIBO details. One of the interesting first details is"
    },
    {
      "start": 3023.22,
      "end": 3028.34,
      "text": "they decided for some reason to replace these two terms by a single term."
    },
    {
      "start": 3029.3,
      "end": 3036.34,
      "text": "And what they're doing is using the approximation of log of 1 plus x, that function by x minus x"
    },
    {
      "start": 3036.34,
      "end": 3041.3,
      "text": "squared over 2. These are the first two terms of the Taylor expansion of log of 1 plus x."
    },
    {
      "start": 3043.06,
      "end": 3047.06,
      "text": "They cut it off. The next one would be plus x cubed over 3. It's alternating."
    },
    {
      "start": 3047.06,
      "end": 3053.7,
      "text": "And then let x equal f over k naught minus 1. That's why I wrote it that way before. Because if"
    },
    {
      "start": 3053.7,
      "end": 3059.78,
      "text": "you do that, the log of 1 plus x is the log of f over k naught. And then this expansion says,"
    },
    {
      "start": 3059.78,
      "end": 3065.86,
      "text": "well, that is approximately f over k naught minus 1 minus 1 half f over k naught minus 1 squared."
    },
    {
      "start": 3066.58,
      "end": 3072.5,
      "text": "So the log of f over k naught minus this term that we saw in that formula is really just this term"
    },
    {
      "start": 3073.3,
      "end": 3077.94,
      "text": "if you assume this approximation. And so the SIBO rewrites it as"
    },
    {
      "start": 3079.86,
      "end": 3088.34,
      "text": "expected future realized variance in the time period 0 to t is 2 over t times this term. So it's one"
    },
    {
      "start": 3088.34,
      "end": 3095.46,
      "text": "term instead of 2. And then they write the integral as the integral from 0 to infinity of e to the"
    },
    {
      "start": 3095.46,
      "end": 3102.42,
      "text": "rt q of k over k squared, where q of k is the midquote. So I have to choose what is the price?"
    },
    {
      "start": 3102.42,
      "end": 3107.46,
      "text": "What is the market price of the option? Well, they're saying, I don't know, let's make it the midquote."
    },
    {
      "start": 3107.46,
      "end": 3113.62,
      "text": "After all, in the market, you don't get a single price. You get a bid in an offer. And there's"
    },
    {
      "start": 3113.62,
      "end": 3120.82,
      "text": "always a gap there. The market maker's spread. So they say, let's use midquotes. And then the"
    },
    {
      "start": 3120.82,
      "end": 3128.18,
      "text": "question is, how do you numerically do this integral? And by the way, at the forward for the"
    },
    {
      "start": 3128.18,
      "end": 3136.98,
      "text": "at the forward strike, they replace. So so below the forward, these are midquotes for the puts."
    },
    {
      "start": 3136.98,
      "end": 3141.22,
      "text": "And those are the sort of the out of the money puts. Above the forward, it's the midquote for the"
    },
    {
      "start": 3141.22,
      "end": 3146.58,
      "text": "calls, the out of the money calls. At the forward, they actually take both the call and the put price"
    },
    {
      "start": 3146.58,
      "end": 3152.42,
      "text": "and average them. So that's what q of k stands for. But how are you going to do the numerical"
    },
    {
      "start": 3152.42,
      "end": 3159.14,
      "text": "integral? Well, they're going to do a numerical integral over all the existing strikes, except for"
    },
    {
      "start": 3159.14,
      "end": 3164.98,
      "text": "some will exclude in a second. So it's going to look like the value at that strike times a delta"
    },
    {
      "start": 3164.98,
      "end": 3171.14,
      "text": "k i. Now here's their choice for delta k i. And their white papers don't explain this choice."
    },
    {
      "start": 3171.78,
      "end": 3176.34,
      "text": "But if you understand the trapezoidal rule, it'll jump out at you immediately that"
    },
    {
      "start": 3176.34,
      "end": 3180.5,
      "text": "that implies what they're using. That's that that's what they're using. So they're using the trapezoidal"
    },
    {
      "start": 3180.5,
      "end": 3186.74,
      "text": "rule. That means their delta k i should be this quantity k i plus one minus k i minus one over two."
    },
    {
      "start": 3188.74,
      "end": 3194.9,
      "text": "Okay, so they've chosen the trapezoidal rule. Another choice they make is to choose k naught as"
    },
    {
      "start": 3194.9,
      "end": 3200.9,
      "text": "basically the first strike below the forward. And I put a star here, they use a very clever way"
    },
    {
      "start": 3200.9,
      "end": 3207.38,
      "text": "of calculating the forward price. With the S&P 500 index, the forward price can be kind of tricky"
    },
    {
      "start": 3207.38,
      "end": 3212.34,
      "text": "to calculate because there's this stream of dividends that comes with it and you'd have to"
    },
    {
      "start": 3212.34,
      "end": 3222.58,
      "text": "kind of statistically estimate what that's going to be. Instead, the SIBO basically leverages"
    },
    {
      "start": 3222.58,
      "end": 3228.02,
      "text": "the market and says, well, we know these are European options. So put call parity applies."
    },
    {
      "start": 3228.66,
      "end": 3235.46,
      "text": "And so what we'll do is just say the forward is the strike plus e to the RT times the call minus"
    },
    {
      "start": 3235.46,
      "end": 3241.54,
      "text": "put. This is put call parity. And we'll choose the k close to the basically close to the forward"
    },
    {
      "start": 3241.54,
      "end": 3248.5,
      "text": "price itself. In other words, they're just assuming that the market makers have got the call"
    },
    {
      "start": 3248.5,
      "end": 3253.7,
      "text": "and a put price is right. In other words, they've done all the work to find out what the dividends"
    },
    {
      "start": 3253.7,
      "end": 3259.22,
      "text": "are and so on. And they'll just piggyback off that and say, okay, the market must have the forward"
    },
    {
      "start": 3259.22,
      "end": 3265.06,
      "text": "right. So here it is. That's a very clever approach. They also have rules that say, well,"
    },
    {
      "start": 3266.1,
      "end": 3272.82,
      "text": "if a call or put option has a zero bid, that means the bid price is zero, that really means we"
    },
    {
      "start": 3272.82,
      "end": 3279.06,
      "text": "don't know what the price is. So don't use those. Throw those out. And if you go up the marching up"
    },
    {
      "start": 3279.06,
      "end": 3284.26,
      "text": "the call strikes, if you hit two zero bids in a row, stop. Don't do any. Don't include any"
    },
    {
      "start": 3284.26,
      "end": 3289.06,
      "text": "strikes above that and the same thing going down with the puts. Once you hit two zero bid puts in a"
    },
    {
      "start": 3289.06,
      "end": 3295.86,
      "text": "row, stop. Don't include any further strikes. Anyway, at this point, you have all the information"
    },
    {
      "start": 3295.86,
      "end": 3301.3,
      "text": "you need to go to the SIBO website, read their white papers on the VIX. And by the way, they have"
    },
    {
      "start": 3301.3,
      "end": 3307.22,
      "text": "now proliferated VIX. They're VIX on many, many different indices, many products."
    },
    {
      "start": 3308.66,
      "end": 3314.34,
      "text": "And I wanted to mention that after they started publishing this in 2002, there was demand for"
    },
    {
      "start": 3314.34,
      "end": 3321.78,
      "text": "trading derivatives based on this. So in 2004, the SIBO launched the VIX futures with Susquehanna"
    },
    {
      "start": 3321.78,
      "end": 3329.46,
      "text": "as the specialist, the primary market maker. And we had to get all of this stuff encoded in a"
    },
    {
      "start": 3330.26,
      "end": 3338.58,
      "text": "model and a trading application. And then later in, I think around 2006, they also introduced"
    },
    {
      "start": 3338.58,
      "end": 3343.46,
      "text": "options on the VIX futures. So we now have VIX options, VIX futures. And by the way, that's"
    },
    {
      "start": 3343.46,
      "end": 3348.34,
      "text": "really interesting because once you have options on something, you can do this whole calculation to"
    },
    {
      "start": 3348.34,
      "end": 3354.58,
      "text": "calculate the VIX of that thing. So there can be a VIX of the VIX and there is. Anyway,"
    },
    {
      "start": 3355.54,
      "end": 3360.42,
      "text": "now you should go out, read the VIX white papers. And you can also read about variant swaps."
    },
    {
      "start": 3360.42,
      "end": 3365.06,
      "text": "There are plenty of papers and white papers about those. So that's all for today. Thanks for"
    },
    {
      "start": 3365.06,
      "end": 3370.1,
      "text": "watching. And if you'd like to see some of my other lectures, they will be posted on Susquehanna's"
    },
    {
      "start": 3370.1,
      "end": 3371.86,
      "text": "YouTube channel. Thank you."
    }
  ]
}