In this module, we're going to spend a little bit of time discussing how to

risk-manage option portfolios. We're going to briefly discuss two

methods. First method is based on the Greeks.

We're going to use a Delta-Gamma-Vega approximation to hedge against relatively

small changes in the underlying security price and the volatility parameter

signal. However that method will not work well

when these changes in the underlying security price and the volatility

parameter are substantial. In that situation, we would use scenario

analysis instead and we, we will spend a little bit of time discussing scenario

analysis as well. Once again we have here the Black-Scholes

formula. I just want to emphasize that the

Black-Scholes formula gives us a closed form or analytic expression.

For the price of European call and put options in the Black-Scholes framework.

That is where it is assumed the stock price follows a geometric Brownian motion

with these dynamics where we can trade continuously in time with no transactions

costs, and where short selling of the stock is allowed.

So Black-Scholes using a replicating argument that we also used in the case of

the binomial model showed how to compute the prices of call input options in this

framework. And indeed they came up with this, the

Black-Scholes formula. In early modules, we saw how we can

compute the delta. We know that delta is equal to delta c,

delta s. We also saw how to compute gamma.

Which is equal to delta 2c, delta s squared.

We also saw how to compute vega. So vega was equal to delta c, delta

sigma. So these are just partial derivatives of

the option price, with respect to the parameters s, and sigma.

We also saw indeed how to compute theta which is equal to the negative of the

partial derivative of the option price with respect to time to maturity.

So it's very straightforward to compute these quantities just by taking

derivatives appropriately inside here. And indeed, in the case of a put option,

we can also compute these expressions very easily sometimes simply using put

call parity. In fact, by put call parity, it is easy

to see that the gamma and vega of call and put options are identical.

So at this point, we have the Black-Scholes formula and I'm going to

assume that we know how to calculate these quantities as well.

These are easy to calculate programatically one can do them in Excel,

or indeed in Or, Python in any programming language that you like.

Let's consider some approximations. We're going to view the option price as a

function of s and sigma only. Then the simple application of Taylor's

Theorem. Now Taylor's Theorem is a theorem I hope

you saw in your undergraduate mathematics class.

If you haven't don't worry about it. What it does is the following.

It enables us to see what happens to the call auction price.

For small changes in s and small changes in Sigma.

In particular, suppose we let s go to s plus Delta s, so Delta s represents the

change in the underlying stock price. And sigma goes to sigma plus delta sigma.

So delta sigma represents the change In sigma, the volatility parameter.

Well then Taylor's Theorem allows us to say that this option price at the new

parameters, s plus delta s and sigma plus delta sigma is approximately equal to the

option price at the original parameters s and sigma plus delta s times delta c

delta s. Plus a half delta s squared plus delta 2c

delta s squared plus delta sigma times delta c delta sigma.

So we recognize that delta c delta s is equal delta.

Delta 2c delta s squared is equal to our gamma term and delta c delta sigma is

equal to our vega term. So we therefore get that the P&L,

remember the P&L would therefore be this term minus c, s sigma so I will bring

this over to this side and I get the P&L on the left hand side, P&L standing for

Profit and Loss. So P&L would be with the delta times the

change in the stock price, plus gamma over 2 times the change in the stock

price squared plus vega times delta sigma.

And so what we have here, is that the profit and loss on the option price when

the stock changes by an amount of delta s and the volatility changes by an amount

delta sigma. When that profit and loss is equal to

the, a delta component, which is this, a gamma component, which is this, and a

vega component which is this. Now I should mention as well, if I wanted

to I could also include time to maturities, another parameter.

So I could have t and t plus delta t in here.

And then I would also have a theta component, so that's perfectly fine as

well, and indeed people do this. But to keep things simple, I just want to

stick with delta, gamma and vega here. And if fact sometimes people just work

with delta and gamma. So if I assume delta sigma equals 0, I

will obtain a delta-gamma approximation. So, the p now, in this case, will just be

due to delta and gamma. And, this is often used, for example, in

historical Value-at-Risk calculations. Now, we go, won't go, anymore, into, into

Value-at-Risk, for options portfolios here, but I know you've seen

Value-at-Risk elsewhere in the course. Now something else I can do is I can

actually go back to this expression here. And just do some simple algebraic

manipulations to get the following. I can also say that the P&L is equal to

delta s times delta s over s plus gamma s squared over 2 times delta s over s all

to be squared, plus vega times delta sigma.

Now I can write this. So this is my return, delta s over s, is

the return on the stock price. I'm going to call delta, this is, this

delta here is delta c delta s. So, delta times s, is often called the

esp, standing for equivalent stock position or the dollar delta.

Delta s over s all to be squared well this is my return squared and so this

quantity here gamma s squared over 2 is sometimes called dollar gamma.

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.

So this is an alternative approach to risk management.

It gives you a more global approach than the approach given to us by the Greeks

that we saw in the previous slide Where we just use delta, vega, and gamma, and

theta and so on, to analyze the risk of small changes in the underlying

parameters. Here we're looking at much larger changes

in the underlying parameters, either the volatility, or the underlying security.

And then we figure out what the P&L is in these scenarios.

It is important to choose the risk factors and stress levels carefully.

It's pretty straightforward to do this with a vanilla options portfolio.

By vanilla I mean where the options are pretty standard or straightforward like

European call options. But if you're trying to do scenario

analysis with very complex portfolios, portfolios containing complex derivative

securities, understanding what these risk factors are can be a challenge in and of

itself. Moreover, figuring out what the

appropriate stress levels are, and by stress level, I mean a 2% or 10% or 5

volatility points, they're examples of stress levels.

So, with very complex derivatives portfolios, figuring out what appropriate

stress levels are can also be very challenging.

And if you don't believe me, you can just think of what went down during the

financial crisis when people were working with CDOs and Asset Backed Securities and

ABS-CDOs and so on. In these situations the portfolios were

very complex. With many many risk factors and

understanding how to do scenario analysis with these portfolios was more or less

impressive. And this is one of the many reasons that

explain what went wrong during the financial crisis and the difficulties

with these exotic structure products during the financial crisis.

Risk-Management of Derivatives Portfolios

金融工程与风险管理，第 2 部分

与讲师见面