So dollar gamma is this expression here. and what I should emphasize, is that in

practice, market participants, option traders, or just investors who happen to

invest in options, as well as underlying securities, stocks, and futures and so

on. What they will do is they will often know

what the ESP is of their option. So they will often know, so they will

typically know the ESP , the equivalent stock position, they will know their

dollar gamma and they will know their vega.

And knowing these quantities will help them understand how their portfolio

behaves as the underlying stock moves and as the volatility parameter's sigma

changes. So, for example, let's consider the

following situation. Suppose the ESP, the equivalent stock

position is equal to 1 million dollars, now this might come about because maybe S

is $100, delta's a half so a half times 100 is 50.

But maybe I've got thousands of these options, and altogether they combine to

give you the ESP of $1,000,000. Maybe my dollar gamma is equal to let's

say 500k, $500,000. And supposed my vega is equal to $100,000

for 1% change in sigma. Now suppose delta s over s is equal to

10%. So, suppose the underlying stock has

increased by 10%, may be this is over the next day and suppose sigma goes to

whatever the previous value was plus 2 percentage points.

Well, then I can use this expression and these quantities here to approximate my

P&L. So, in this case, my P&L, my Profit and

Loss on my option position will be approximately equal to my ESP which is 1

million dollars times 10%. So it's going to be 1 million times 10%.

Plus, my dollar gamma, which is 500k times the return squared.

My return is 10%, so 10% squared is 1%, plus my vega which is 100k, per 1%.

So it's plus 100k and volatility might doubt, the sigma has changed by 2% so

that's times 0.02 and this is equal to the, let's see, it's a 100k plus 5k plus

2k. So that's equal to 107k.

And, of course, I should mention that one can get very different options, so one

can get options with very different ESP's gamma's, and vega's.

What also interesting is that people have used this not just for a single option,

but for an entire portfolio of options. What can have an entire portfolio, it can

compute the ESP for the entire portfolio, and the dollar gamma for the entire

portfolio and indeed the vega for the entire portfolio.

And so one will then understand there maybe represent the exposure of the

portfolio in terms of the ESP, the dollar gamma and the vega.

Typically of course they will also know the data for the portfolio and maybe some

of the other Greeks as well, that we have the time to go into.

So, the Greeks are very important. People understand their sensitivities,

their risk-sensitivities, in terms of these Greeks.

Now, as I've shown you here, understanding your Greeks and typically

writing them in terms of quantities like ESP Equivalent Stock Position or dollar

gamma and so on. Here's a very good idea and market

participants do use these approximations all the time, but it is also worth

pointing out that for very large moves in s or sigma can, can break down, it will

no longer work. And the reason that it will no longer

work It's because Taylor's Theorem isn't valid for very large moves in s or sigma.

Now I won't go into any further details on this but you have to understand that

Taylor's Theorem it gives you a good approximation for relatively small

changes in s and relatively small changes in sigma.

If you get very large moves or extreme moves, then these approximations break

down, and they aren't very accurate. In that case, what people often use is

scenario analysis. So, here's one slide giving you an ideal

of what is going on with scenario analysis.

So what I've shown here, is the following, it's an example of a pivot

table, that I've constructed in Excel. If you don't know what a Pivot table is

when then you can use the help facilities in Excel to figure out what they are and

how to use them, they can be very useful in manage situations.

we're not going to say anything more about them here.

what we've assumed here is that we've got an options portfolio, the options

portfolio is written on the s and p 500s, we've got lots of options and maybe we've

also got futures in our portfolio. And what we done is we've considered two

stresses. We've stressed the underlying security

price. In this case, the underlying security

price, as I said, is the s and p 500. And down on this axis we're considering

stresses where the s and p 500 falls by 1% or it falls by 2% or 5% up as much as

20%. We're also considering situations where

the s and p 500 increase by 1%, 2%, 5%, 10%, and 20%.

Across the x axis up here, we're considering stresses and volatility.

So this vol here refers to the sigma parameter we've been discussing.

So this is the sigma that enters into the black shoals formula.

What we're doing is we're considering sigma going to sigma plus 1 percentage

point, sigma plus 2 percentage points, up to sigma plus 10 percentage points.

And down to minus 10 percentage points as well and then in anyone one cell we can

see what the profit or loss is on the portfolio at that particular scenario.

So for example down here, this corresponds to the s and p increasing 5%,

and implied volatility's increasing 5 percentage points.

Well in that case I'm going to see a loss of 4, 3, 2, 2.

Now, if this is in units of dollars, then it represents $4,322.

But maybe it's in units of 1 thousand, in which case it represents a loss of 4.3

million dollars. So, this is an example of a scenario

analysis. People do this all the time in finance

with derivatives portfolios. To stress their risk factors, in this

case the risk factors are the prices of the underlying security and the

volatility of the options. And then they reevaluate their portfolios

in these new scenarios and compute the profit or loss in these scenarios.