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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.

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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.