In this brief module, we're going to show you some videos of how the volatility surface moves. We will look at examples during the financial crisis and we will see how the volatility surface spiked up during that period. We will also see that the volatility surface is not flat. It has a certain shape, i.e. the skew is very noticeable. We will also see the inverted term structure of the volatility surface. Particularly during times of market stress and market panic, for example, during the financial crisis. So hopefully this will emphasize to you the importance of the volatility surface and how stochastic it is, how far it is from being constant and in particular how that means the Black Scholes model is far from being a good approximation. What we have here is a three dimensional figure. It is a plot of the implied volatility surface of the S&P 500 on August 1st, 2007. Now what we're going to do is actually play a video showing how this implied volatility surface varies through time, right up until January 2009. So before we play let's just make a few observations first of all. Along this axis here we have what is called the moneyness. Now the moneyness is nothing more than the strike divided by the current security price. In this case, the security is the S&P 500 Index. Over on this axis, we have time to maturity measured in days, so in fact you've got times to maturity ranging from just a few days depending on the next option maturity at a given time out until 800 days. Now, on the z axis, the vertical axis here, we are plotting the implied volatility, so we've plotted it from 10% up as far as 70%. Now, one thing to keep in mind is that on this scale, from 10% to 70%, the implied volatility surface on August 1, 2007, looks very flat. It looks like that there's no implied volatility skew here. Well that is not true. That only appears to be the case because we are plotting it against a wide range, 0.1 up to 0.7. If we were to plot this surface in the range 0.1 to 0.3, we actually would see the implied skew. We would see implied volatilities corresponding to these strikes here, 1.1, 1.15, and so on. Being maybe 4 or 5 volatility points below, the implied volatilities for strikes corresponding to .85, and .9 and so on. So you can take my word for it, that there is a skew here, it's just hard to see because we're plotting it against a range of 0.1 up to 0.7. Another thing to keep in mind here is that we have the date up here. We have August 1, 2007. So this is the start date for the video. We're going to run this up until January 2009. Now the volatility data we're using is coming from the option metrics applied volatility database. This database has all the implied volatilities on US options, index options, ETFs, and so on. It gives you the implied volatilities at the close of the day, and so that's another important point. This video is just showing the implied volatility surface at the close of each day. Within each day the implied volatility surface will also be moving around, but we don't have that data. So we just going to plot the close of day implied volatility surface. So the first thing we'll do is we'll just play the video from the start to the end and then we'll go back and look at a few periods in time. So here's the video, you can see it moving, you can see sometimes the implied volatility surface becomes inverted, sometimes it's not inverted. OK, so you saw during the middle there that the implied volatility surface got very, very high indeed. In fact it actually rose well above 0.7 or 70%. let's go back a few moments, and see what was going on here. So one moment to look at is back around March, 2008. so if we, we can play it one day at a time here. So you can see that the volatility surface is around, hanging around 20%. Obviously, as I said, there is a skew there, so it's not flat. But, it's around 20% at this point, at this point in time. If we move on we see it starting to rise some more, now if we go on to March 13th, March 14th we see a big jump up. Actually what happened there was that Bear Sterns stock started to plummet. there were worries that it wasn't viable, that it was going to go bankrupt. Then in fact that's what was going on around here. round this time as well is acquired for $2 a share by J P Morgan. and then the volatility surface came back down. So now let's roll it forward until September 2008 which is where the financial crisis really took off. Lehman Brothers went bankrupt, Merrill Lynch was bought by Bank of America and in response to all of this the United States Congress attempted to pass some emergency legislation to effectively bail out the financial system. This was rejected initially by Congress. Let's see what happened that day, let's play. OK, so now we are into September 8th. Again the volatility surface isn't too high. It's around 20% ish, it looks at that level maybe a little bit higher. It's hard to see the exact levels in this three-dimensional surface. so let's see what happens. September 15th was the date when Lehman filed for bankruptcy. So, this is the 11th, so this is the 12th, and I think this was a Friday. So, the next date we're going to see is September 15th, which was a Monday. That was the day that Lehman Brothers filed for bankruptcy, and we see that the volatility surface moves up entirely. But also we see the short term vols really spike up. So, these are climbing up to levels maybe around 40 or 40% plus, again it's hard to see exactly what levels these are on this surface, but we can certainly see what is happening. let's keep going on. So, it calms down a little bit, it's climbing, so this is sort of, getting into the middle of the financial crisis. 25th, 26th, 29th, so what happened here was that the Emergency Economic Stabilization Act was defeated in the United States House of Representatives, so this was the day when Congress defeated a bill that was proposed. The idea was that this bill would help bail out the financial system. Congress defeated it, and so initially there was a huge panic in the market on this date. The following date we see it falls back down again. Note, however, that these are still pretty high levels of volatility, up to 40% and, and beyond. On some of the days, we are seeing the highest volatility levels that had ever been seen. So now, we're in October 14th, October 15th, October 16th, and so on. It's worth mentioning that on October 6th to the 10th, that was the worst week for the stock market in 75 years. The Dow Jones lost 22.1%. the S&P 500 lost 18.2% that week. So in fact we can go back and take a look at that week. So it started on the 6th. So this was the implied volatility surface on the 6th of October. This was the week, as I said, where the stock market had its worst returns in over 75 years. So all in all a pretty, intense, period of time in the financial markets. letting the video run we see things calming down somewhat, by the end of January 2009, but still these volatility levels are much higher then they were before the financial crisis took hold. it's worth also mentioning the Vix index, the Vix index is sometimes called the Fear index. It's a widely used market indicator that's constructed actually using the options for the S&P 500. So the Vix index is actually constructed using very near terms option. So typically one month after two months. It's also worth pointing out that these, this is what happened to the implied volatility surface of the S&P 500, this is the most important equity index in the world. If we were to plot the implied volatility surface for individual stocks, individual banking or financial stocks, or even industry indices, we would see far more extreme moves in the implied volatility surface. Anyway, so this is just a video showing you what happened to the volatility surface, the implied volatility surface of the S&P 500 during the financial crisis. One of the points to take home is that it is not a constant, as it would be implied by geometric Brownian motion model of Black and Scholes. Moreover, it is very stochastic, it moves about an awful lot. In periods of high market stress, obviously it increases. With the short end or, if you like, the short time to maturity, implied volatility is rising, much more than the implied volatilities of longer time to maturity options.