Peculiarities Of Volatility Ernest Chan Quantopian

TITLE: “The Peculiarities of Volatility” by Dr Ernest Chan CHANNEL: Quantopian DATE: 2017-03-06 ---TRANSCRIPT--- My name is Dan Dunn. I am the VP of Product and Marketing at Quantopian. It is my pleasure to introduce Ernie Chan. Ernie Chan, for this crowd, is of course somebody who doesn’t need an introduction. He spoke at QuantCon last year and he was very well received, and we’re very excited to have him back. He’s— I’m happy— when I first started at Quantopian 4 years ago, one of the very first things that I did is I did a phone interview with Dr. Chan, which you can find on blog.quantopian.com. and you can see how little I knew about what I was talking about when I started asking him questions. I read his book, as I’m sure we all have, Quantitative Trading: How to Build Your Own Algorithmic Trading Business, and its follow-on book. And so without any further, I’m very pleased to have Ernie Chan here with us today. Thank you very much for your kind introduction, Dan, and thanks for inviting me again back. So, yeah, today I’m going to talk about volatilities, which is always a hot topic in finance. Oh, let me actually— I pressed the wrong button. Let me make sure that the— Press the same button again. I’m sorry? I pressed the arrow instead of actually this arrow. Press the same one again. Oh, the same one, okay. Ah, okay, very good. All right. Yeah, so, For those who don’t know me, I actually started off as a physicist and, as many physicists do, got into finance later on. But right now I’m managing a small hedge fund as well as a number of managed accounts. But let’s go back to volatility. There are 3 main points I want to make about volatility. The first thing is that predicting volatility, that’s easy. And this might come a little bit surprise to many of us, but that’s actually not the main problem, especially if you are trying to predict realized volatility. And I will talk a little bit about why it’s actually so easy to predict volatility. The second point is that, unfortunately, though you may be quite successful in predicting realized volatility, it doesn’t help you very much in predicting implied volatility. And unfortunately, you can only trade implied volatility. You can’t really trade realized volatility. So then the third point is that fortunately, you can arbitrage between a predicted realized volatility and the current implied volatility. So not all is lost. There’s some use for predicting realized volatility. Even though it’s not as good, as useful as you might think it is. So these I will elaborate later in this talk. Let me quote. There are many thousands of papers on forecasting volatility using a host of increasingly sophisticated, even Nobel Prize-winning statistical techniques. And that’s a quote from Ahmed and Paul Wilmott in 2005. Note the year. It has been 11 years since they made that quote in the paper, and no doubt there are many more thousands of very sophisticated papers on volatility prediction being published since. So probably now it’s 2,000 or 3,000 of such papers. And in fact, about a month ago, I went to listen to a seminar by Professor Gatharel. He is a professor of finance at CUNY, though he was a head quant at Merrill Lynch before he joined academia. In this talk, he presented an incredibly complicated and but brilliant paper on volatility prediction based on the notion of fractional Brownian motion. I must admit that I had my mouth open throughout the talk. I could not close it in awe, and I can probably only understand 10% of it. But thank goodness we are practical traders. We are not in academia to prove theories. So we actually can just stick to the simplest volatility prediction model for hopefully it will work for us. And that would be the tried and true GARCH model. So the GARCH model is exceedingly simple, actually. The principle is a linear model. It’s just saying that the— expected volatility, the expected realized volatility, is just a sum of past prediction. Oh, I see that my slide had— it used to be very nicely formatted Greek and mathematical formula, but once it’s transformed into this format, it doesn’t make any sense anymore. But anyway, forget about the actual formulas. I’m just talking through it. It’s just a sum of past predicted conditional volatilities, as well as a sum of past realized volatility. So it’s purely a— you can think of it as linear regression model of past realized volatility and past predicted volatilities. And it tells you what’s the expected next period conditional variance, it is, for your price movement. So, OK, that’s all there is. But there are, as every linear regression model, there are many regression coefficients you have to fit and so forth, and also the number of lookbacks you need for the past conditional variance and the number of lookbacks you need for the past realized volatility that you need. All these Greek, all these coefficients can be estimated quite readily with simple maximum likelihood estimation with some penalty for model complexity. And you don’t have to worry about any of these because there are packages, toolboxes galore for doing that. You merely need to find the right MATLAB, R, and Python packages and feed it all in, and they will compute these coefficients for you. You don’t have to worry about a thing. Let’s try this on some of our most familiar ETFs. SPY, the stock index ETF, USO, the oil futures ETF, GLD, gold spot price ETF, and AAPL, Apple stock, and finally the EUR/USD currency pair and see how the SCART model works. Okay, so obviously we need to estimate the coefficients using the in-sample period, which is about the same size as this out-of-sample period where we test how well the prediction works. The out-of-sample period is from 2010 to 2016. And our objective, our measurement here is actually measuring how accurate it can predict the sign of a risk-reward change. This is, to be very specific, we are only interested to see whether volatility is going to increase or decrease, because most often that’s the most important trading signal we are interested in. Applying that to SPY, we find that 66% of the times that this prediction was made on SPY volatility, it works. It correctly predicts 66% of the times that which direction volatility moves. For USO, it’s 67%. For GLD, it’s 59%. For Apple, it’s 60%. For Euro, it’s 62%. All very, very statistically significant numbers, perhaps to our surprise. Now, if we have this kind of accuracy in predicting the sign of one-day price change, not volatility change, but price change, we would all be rich. We do not need to sit here for the rest of the day. We should all go home and start coding. But unfortunately, it is much harder to predict when the price change, even the sign of it, as opposed to volatility change. And actually, I have a student who, a brilliant student who asked me this philosophical question. This student already had a PhD in mathematics, but he’s now trying to learn something about finance. And he asked me, Why is it that it is so hard to predict the sign of price change as opposed to the sign of volatility change? What’s the fundamental reason? What’s the philosophical reason behind it? And the answer has to do with arbitrage. Most things, most questions in finance can be answered by an arbitrage argument. So here it is. If you can actually predict very easily the sign of price change, you would be raising a hedge fund of about $1 trillion and bet on it. And in no time, this predictive accuracy would disappear after this $1 trillion has been put onto this trade. And that’s applied to any trade that you can find that has very high accuracy in prediction, in predicting price change. On the other hand, if you can accurately predict the sign of volatility change, it is not that easy to create a trade that can benefit from it, perhaps a little bit counterintuitively. And that is actually the main point I want to address in the rest of this talk, is that despite the fact that you can accurately predict the sign of realized volatility change, there is no ready-made trading strategy that can allow you to bet on it. And therefore, the predictive accuracy remains. There is no arbitrage that you can employ to exploit predicted-realized volatility change, at least accurately. OK, so, but you might— this statement might seem a bit shocking because, after all, why can’t we trade— a whole bunch of volatility ETFs. There are a whole alphabet soup of volatility ETFs out there. There’s VXX, VXV, VXZ, XIV, XVZ, and of course the VIX future, and so on and so forth. All of them, by the way, this involves V for obvious reasons. But note my omission of the fixed index, and that’s important. You can’t trade a fixed index, but you can trade all these volatility ETFs that track, supposedly track, the fixed index. And so you might ask, why can’t we just deploy this volatility GARCH prediction on trading these ETFs? For example, if GARCH predicts an increase in realized volatility, it would seem sensible to buy VXX and vice versa. That should work, right? If volatility increases, why not buy volatility ETF? So what’s the result of this very simple intuitive strategy? It’s a relentless loser. My goodness. The only time that it ever made any money is during August of 2011, around that time. And if you remember, August of 2011, that was when the US Treasury debt was downgraded by the S&P. That was a historically unique event, and it resulted in one week of market chaos. I couldn’t understand why. Who cares about S&P anyway? But a lot of people seemed to care, and it resulted in this chaos. And that’s the only time that this strategy worked. And we’ll get back to that actually later on also. So the lesson in this, horrible equity curve is that just being able to predict realized volatility does not imply that you can predict implied volatility. In fact, if you study the correlation between the two kinds of volatility, they only move in the same direction 51% of days. Effectively, statistically speaking, the correlation is exactly zero. There is no relationship whatsoever between the movement, the direction of change of realized volatility and implied volatility. That might come also as a surprise to some of us. No correlation whatsoever. If you, in fact, if you focus your attention on only those days where the market move up, where the market has a positive one-day return and did the same correlation test, you will find that they move in opposite direction. When market goes up, we probably understand, many of us understand that the implied vol will go down because the option trader perceived the risk is decreased and there’s less demand perhaps to buy portfolio insurance. So the implied vol goes down when the market went up. But that doesn’t mean that realized volatility goes down as well. In fact, S&P, let’s say, go up 2% a day, that’s a big move. And that shows up as an increase in realized volatility while the implied vol goes down. So in those days where the market goes up, you actually have an anti-correlation between realized and implied vol. The upshot of all this is that being able to predict realized volatility is no help whatsoever in predicting the change in implied volatility. But that doesn’t mean that predicting realized vol is useless. That’s not the conclusion we’re trying to make. You can use the predicted realized vol from GARCH model, for example. Let’s call it RV. That’s the realized volatility at the next time period, which we can predict. We use our GARCH model to predict that. And we take the difference of that with the current implied vol, which we can use the fixed as a proxy. I mean, this formula can be applied to any instrument. You can apply it to ETFs. You can apply it to stocks. You can apply it to currency. But for now, let’s focus on just the S&P 500 index, so that the implied vol is readily given by looking up the fixed index on Yahoo Finance, for example. So, if you take that as a trading signal, and you say that, okay, whenever I predict that the realized vol is going to be higher than the current implied vol, I buy VXX, I buy implied vol, and vice versa. How does that work? Now, before I tell you how that works, let us contemplate for a little bit the implications of this trading model. The implication of this trading model is to say that the GARCH-predicted and realized volatility, the GARCH prediction is better in predicting future implied vol than the current implied vol. That’s the implication. And that, again, is counterintuitive to a lot of traders. A lot of traders think that, wow, who cares about this? Time series models and whatnot. I have my implied vol, and that is the consensus for all these very smart options traders who typically look down on stock traders. And the options traders rule, right? I mean, they know what they’re doing. So if they say that implied vol is high, no doubt that they must be right on average. Well, if this trading model works, that’s telling you that that is not correct. Actually in pi4 is not very good in predicting future in pi4. In fact, the time series models like GARCH is much better in predicting future realized vol than the current in pi4. And so, okay, so let’s see how this model works. But again, before I show you the result, let’s take a look at the trading signal. This is the trading signal. This trading signal is the GARCH predicted in pi4 on the next time period minus the current in pi4. And you can see that sometimes positive, sometimes it’s negative. And particularly, the interesting period, again, is in 2011 August, where you had this US Treasury debt downgrade. That’s the time where you had a huge difference. The GARCH-predicted volatility is much lower than what the current implied vol is. What that means is that, What this model would imply is that options traders are vastly overvaluing options in that time. The time series model is predicting much lower volatility than what the options traders are willing to pay for. They are overvaluing options. Similarly, there’s another period where that happened. And that happened quite recently in September of 2015, where this difference is again very pronounced. Remember, September 2015 is when we have the oil crash, we have the Chinese stock market crash, we have the yuan crash, we have the emerging market panic. Everything seems to be coming apart at that time. Wow, that seems like ages ago, but yeah, that was only in September 2015. And that’s again when you had this options trader vastly overpaying for protection. So that’s the trading signal we are trying to arbitrage on, see how well it does. Well, if you use that trading signal, you’ll find a cumulative average growth rate, compounded return of 26% annualized. Pretty decent. So I must admit that the Sharpe ratio is a bit low, it’s 0.7, so it doesn’t have a lot of statistical significance, and that is again on performance on the out-of-sample period. So it does put some justification to say that the GARCH model is actually a better model than predicting future implied vol than current implied vol. But that’s not the only model. That’s not the only way you can use that signal. The way that I use the signal is the simplest, perhaps most intuitive one. But referring to that paper by Ahmed and Wilmot in 2005, they actually suggested using the same signal to apply not necessarily to buy straddles or strangles, but just to buy any option, put or call, and delta hedge that option until it’s expired. So it’s, again, a very simple delta hedging strategy. If you predict the realized vol is higher than the current implied vol of the option, buy the option and delta hedge it. That’s it. And they have proved that under certain assumptions, amongst them, that Black-Scholes model work, that the volatility remains constant throughout the rest of the life of the option, and so forth. A whole bunch of assumptions. But under those assumptions, you can actually theoretically prove that this strategy is profitable. And not only is it profitable, in some circumstance, you can exactly derive what the profit is going to be. There’s no uncertainty about the profit under these assumptions and for this particular delta hedging strategy. So it’s quite interesting. And as I said, it doesn’t only work on the SPX option. It works on stock options, currency options, whatever option that you can apply Black-Scholes model on. So let’s switch our attention to consider whether we can— we have been talking about trading VXX, and we want to turn our attention to to the fact of what exactly we are doing by trading VXX. When we are trading VXX, we are definitely not trading the fixed index. That’s one of the key points in the last part of this talk I want to make. That is quite different from trading the SPY. So, for example, if you want to benefit from the return of the SPX index, the S&P 500 index, Everybody knows that the easiest way to do that is to buy the SPY. That’s a no-brainer. And that you actually get a pretty good tracking of the SPX index if you just buy and hold SPY. So, no problem. And the reason for that is that the SPX index is pretty stable. The portfolio of stocks in the SPX index is fairly constant. It might change a few times a month, 2 times, 3 times a month, when you have addition of stock or subtraction of stock, but even with that, it doesn’t affect the portfolio of stocks in the SPX very much. It’s very stable, and therefore, by spying the SPY, you are effectively, you know, re-tracking the SPX return. The situation is completely different if you are trying to track the VIX index. It is completely different. There’s no analogy whatsoever from that. When you’re trading the VXX, it is completely different from— from trying to track the return of VIX. And why is that? Well, to understand a bit better, you have to understand how the VIX index is constructed. The VIX index, as defined by the CBOE, is a portfolio of options, okay? Just like SPX is a portfolio of stocks, VIX is also a portfolio of real options, not extrapolated, not mathematical. It’s really a portfolio of options. But the definition of this portfolio is such that they have to be OTM options. They have to have a tenor, the time to maturity, between 23 and 37 days. And there are a whole bunch of criteria for inclusion in this portfolio. And also, the weightings are quite peculiar. You have to read their formula, which is available on this white paper, to exactly understand how they define the weightings of each option. All these definitions mean, as they admit, as the CBOE admit, that it could result in a portfolio that differs from minute to minute. Now, that’s very different from the SPX Index, right? SPX Index doesn’t change for at least a few weeks, and even if it changes, it changes very little. But the VIX Index changes every minute. You cannot possibly hold a portfolio and replicate the, uh, the return of the VIX index, because it is not the same from minute to minute. Uh, for— to just give a quick example, let’s say the VIX index is $20 at one moment, at t, and the VIX index at t 1 is $21. Let’s say increase the VIX index. Does that mean that the portfolio we held replicating the VIX portfolio of options at time t, does that mean that portfolio has appreciated by $1? Not so, because the VIX at t 1 is completely different. It could be a completely different portfolio from the VIX at t. So, because you’re holding the portfolio at t, it doesn’t mean that it appreciates by $1. In fact, it usually doesn’t. In fact, it might go down a lot by t 1, despite the fact that the VIX index increased by $1. So, to drive home the point, even if the implied vol remains constant in time, so let’s say the VIX is constant, no change, the market value of the VIX portfolio will go down for sure, guaranteed, from t to t+1. Why? It’s obvious why, because you know that options has a time decay of premium, and it has negative theta. So if you’re holding a bunch of options, it’s going to decay in value, no matter if all remain constant, everything remain constant in the market. So negative theta, or the time decay of options value, lead to, if you are trying to trade the VIX index by trading the fixed futures, which VXX is a vehicle of. VXX contains nothing but the VXX future, so we might as well focus our attention to the VXX future. The VXX future has a negative roll return. Roll return is basically the difference between the spot price and the future price divided by the time to maturity. I will discuss that in a little bit. The reason why the VIX future has a negative roll return most of the time, even when the pi_4 is constant, even when the fixed index is constant. And that’s because, ultimately, because of the time decay of the option premium. However, that’s actually interesting, because we can use the roll return of the VIX future as a trading signal as well. As I said, the roll return is proportional to the difference between the fixed index and the future. That’s the rule. Actually, I have a— I’m sorry, I have a minus sign error in this slide, I just realized. So the roll return is actually the index value minus the future value, not the other way around. So if you can use that as the trading signal, buy when the roll return is positive and sell when the roll return is negative. That is to say, if you expect that the volatility implied vol is constant, but therefore you only want to benefit from the decay of premium, just short the VX. And vice versa. If you think that there are possibilities that actually the volatility is going to increase more than the decay by the VX future, how does that work? That’s another intuitive strategy. That strategy returns 60%. And analyzed, and has a Sharpe ratio of 1 for the period of 2004 and 2015. So it’s statistically significant, and it’s a very attractive return. But— there’s always a but— if you look at the equity curve, indeed, this strategy performs superbly up to 2013, or thereabouts. This equity curve is like an exponential increase in wealth. It’s quite shocking. After that, it— doesn’t work anymore. So you might ask, what was going on with 2013 or 2012? What happened? I, you know, we can’t remember any big, big market dislocation at that time. What’s the possible trigger for this, you know, falling apart of this strategy? Well, in 2012, Professor Simon and Carpesano published a paper that described this strategy that’s become very popular, and I wrote about it in my blog to refer to this paper. I wrote about in my in my book, and many, many people start to trade this strategy in 2013. And perhaps, now I’m not saying this is proof, but perhaps it’s no coincidence that after it’s widely publicized, this strategy has completely gone away. Indeed, the arbitrageur has removed this opportunity for everybody. But again, that’s an interesting implication, is that This trading of the VIX future is like the tail wagging the dog, because the trading of this VIX future has actually implied changes in the option prices themselves, because otherwise it couldn’t have possibly destroyed the strategy. So by trading the VIX future, you have changed, actually, the prices of the underlying SPX options prices. So that brings an interesting question for us to contemplate. If that’s the case, is the VIX index still a good predictor of future realized volatility, given that it seems that it has been arbitraged away by trading this VIX future? And if it is not a good predictor of future realized volatility, what’s the use of the VIX index? Thank you very much, and please visit my website, my blog, for, you know, to have a further discussion. Thank you. We’re going to take a few questions. Who’s got a question about implied volatility or realized? On one of your slides you said— oh, we’re good. On one of your slides you showed that the movement of the VIX and volatility only aligns like 51% of the time, I believe. So basically there’s no correlation at all. Have you or has anybody explored maybe there being a time lag there so that the VIX maybe is predicting volatility for the market a day or two out, and maybe the correlation would be stronger there. Uh, that— I don’t know, I haven’t explored other time frames, but yes, certainly possible that shorter time frames might be better predictive. You mentioned that the, uh, VIX trading strategy crashed in 2013 after the paper was published in 2012. Your argument was that this is probably arbitrage to me. Could you also comment that was the paper just overfit? That there was no effect, it was just looking at data. That is a bit not as likely because the— The paper has very few parameters. So, and in fact, in the version that I described, it’s actually simpler than what the paper described. I’m only using the sign of the roll return. There’s no threshold, there’s no fancy lookback or holding period. So, they’re practically a parameterless trading strategy. So, I think it’s quite unlikely that it’s a result of overfitting. Hi, Ernie. Hi. The returns you described on some of these strategies were quite impressive. Could you tell us a little bit about the effects of trading friction, spreads? Well, these strategies are not really high frequency. They are only end of day. Trading strategy. And obviously for the VXX, you can use market on close order, so you would pretty much get exactly the price, so you’re not paying the bid-ask spread at the close. But there might be a bit of a lag in the time you get a signal and the time you can submit your market on close order. But generally speaking, it is an extremely liquid instrument. VXX is, you know, favorite trading vehicle for many people. So there’s hardly any trading cost involved. And the VIX future, there is a bit of a bid-ask because their bid-ask in terms of basis points is a bit significant because their minimum increment is a bit significant. But still, you know, being a daily trading strategy, that does not have a big impact. Impact on the return. The biggest impact of the return is that it collapsed completely. It’s not a consequence of trading costs. Yeah. Once these strategies get arbitraged away, presumably people stop trading them, so does the relationship reappear? That might well be the case, yes. I have seen strategies that have been dead for a few years and suddenly come back to life. So that is something that we should always monitor from time to time. That’s a very good question. Hi, Andy. Have you tested this strategy for similar— for other mixed products, for example, the Brazil or the Apple, where there is a mixed equivalent in X? Yes, I try to— Not the last strategy to talk about, but I tried to test the delta hedge strategy that I mentioned at one point, following the recommendation of Ahmed and Wilmot. It didn’t work out for me, but I didn’t do a lot of work, but I just tested it on Apple. I did it once, and it wasn’t profitable. And that’s probably because it violated some of the assumptions of that paper, so I didn’t get the guaranteed profit that they promised. Yes. So you actually show two strategies here. First strategy is RVt+1 minus IVt, which is something like a variance risk premium in a sense. Yes. And that one you didn’t show that it doesn’t average out the way, or not average out the way. Right. Is that strategy related to a risk premium, or do you think that’s a mispricing mispricing? That’s my first question. Second question, the second one, it looks like the real yield versus some kind of real yield that’s implied by the contract itself. So I think that is more like a mispricing. So when arbitrageurs get onto that, is this arbitrageable? Well, to answer your second question first, I don’t think that the second strategy is really a matter of mispricing. The second strategy is you can think of it as a factor model. The return you get is a factor return, not so much an arbitrage return, not an alpha. Because when you are shorting the VIX future because it has a negative roll return, you are shorting volatility. And you know what happened to people who short volatility. They don’t have a very long, happy life. And so you are taking factor risk in that case. So yeah. So it’s definitely not arbitrage in the second case. In the first case, where you are— yeah, you can look at it as an arbitrage between realized and implied, or you can look at it as shorting the variance risk premium. I’m not convinced that that is a factor return. I think it has more of an arbitrage return component to it, because there’s no obvious reason why one or the other has a more— Follow up on the first. Have you looked at the cross-sectional? Does that work only for, let’s say, index, or it works for individual stocks? Because we know variance premium exists somewhere, but there’s not strong evidence. Well, yes. The only study I’ve done here is for the index. So it’s a, you might call it, time series. Variance risk premium study. So it’s, you know, looking at only one instrument in time. What you mentioned, the cross-sectional study, there are many papers on that, on studying, you know, like ranking stocks in terms of their variance risk premium and using that as a cross-sectional factor to determine the future return of the stock itself or the future change in the implied volatility. There are many studies of that also, but I might have written it somewhere in my new book, but I can’t remember the results. Kind of following up on that, so, do these strategies only make money when you’re shortfall, or do they also make money when you’re long call as well? Yeah, it’s supposed to make money both ways. But this last strategy, however, because the VIX future is most of the time, maybe 90% of the time, has been in contango. So that means that most of the time you’re shorting volatility. So if you analyze whether it’s deriving the profit from long or short, well, yeah, 90% of the time it’s because you’re shorting. Well. Have you tested this strategy with some other time series models, especially asymmetrical GARCH? No. See, the thing is that I know that there are about 2,000 to 3,000 volatility prediction models, and I have no intention of testing all of them. So I started with the simplest, and already it gives me 67% accuracy. I’m not I’m not so keen to add another 3% because I don’t believe that it will do anything to the actual trading. See, I think the big problem with this kind of model is not how well you predict realized volatility, but how do you deal with the difference of behavior between implied vol and realized vol. Yeah? I have a question about the implications of the last point you made. So I know traditionally spot VIX is used usually is comprised of puts. Portfolio managers traditionally just use puts to hedge their positions. How are the puts, or how are the options that go into spot VIX being affected by trading in VIX futures? Well, let me think about this. There are people, again, arbitraging the price of SPX options and the price of VIX futures. So if the VIX futures price is grossly distorted from the current price of the fixed option portfolio, that will present an attractive arbitrage opportunity as well. So the options cannot have an implied vol that is so different from the VIX future price, or what’s implied by the VIX future price. So there’s arbitragers doing that for us as well. So that’s my thinking, yeah. We’ve got time for 2 or 3 more. Ernie, on the roll return strategy, what if you sliced out the long and the short side into 2 different strategies? Because when people panic, the VIX is gonna spike still. Maybe not as high as it used to, but I think on that roll return strategy on the long side of buying volatility, even if it’s arbitrage away, when people panic, the VIX is still gonna spike. Mm-hmm. So how would that be arbitrage away on the long side? I can see how it can be arbitrage because it won’t go as high because the arbitrage acts as a reality check and doesn’t let the VIX get into skyrocket territory. It puts a lid on it, so it’s taking away some of that profit because it’s not going so high and it’s not coming back down on the short side. Well, it’s not a question of whether people panic and whether the VIX will go up. Yes, like you said, for sure VIX is going to skyrocket when there’s panic. The question is, whether you can base your prediction on the current value of the fix to predict whether it’s going up further the next day. So maybe they overreact. So a lot of times, as I’ve shown in an earlier signal plot, option traders have overreacted by pushing the fix too high. And so you are better off actually shorting a fix, shorting in private at that time, rather than following the the model and buying it because it’s in backwardation. For the first strategy you mentioned, where you buy the VIX on the signal when the predicted is higher than the implied, were you saying, was that hypothetical return based on imagining that you were tracking the VIX index over that period, or was it buying the VIX futures? Or the actual ETF? Let me think. Yes. The realized forward prediction is made by GARCH, just one-day prediction. And the implied forward is actually the fixed index, not the VXX ETF. But the instrument that you actually trade is the VXX ETF, of course. Yes, so the signal is generated by the index, but the actual instrument you trade is the ETF. And that actually had a positive return even though you were talking about how the actual ETF— I mean, you have transaction costs, and it can’t actually match its composition exactly. Right. Right, because the reason it has a positive return is that the realized volatility prediction is actually better than what implied vol can tell you. So this is not dependent on the fact that we are not using the implied vol by itself as a signal. So we don’t care that we cannot track the implied vol. We are saying that the realized vol prediction can better predict the future in vivo than the current in vivo. Yeah. Yeah. Okay. I think that’s it. We’re about out of time. Thank you, Dr. Chan, very much. Thank you very much.