% delta vs $ delta

[wrongsaidfred]

Hi David,

In Puting VaR to Work, there is a somewhat strange convention of calling the % delta "delta" and the delta that I am more familiar with "dollar delta" or "$ delta". Is this convention used on the test? It is pretty obvious that if they tell me the delta is 5.5 that they are referring to percentage delta (a 1% drop in the price of the stock = 5.5% drop in the price of a call) but this is the first I have ever heard of actually using % delta.

Any clarification you could provide would be greatly appreciated.

Thanks,
Mike

 

[David Harper CFA FRM]

Hi Mike,

Astute, as usual! GARP hasn't yet recognized exam-wise, to my knowledge, the distinction between percentage Greek (e.g., percentage delta) and position Greek. I don't mean to flatter you, but again you've isolated on a terminology issue that I have already submitted to GARP! (I don't think i've ever had a customer identify so many of the internal coherence issues with the FRM. If you don't pass, I will fall off my chair)

The term "percentage delta" is fine but Linda Allen's usage is incorrect. You (following her, I think?) are using option delta like you would use bond duration. Understandable, but option delta is a pure 1st derivative while bond duration is the first derivative multiplied by 1/P. In the way, an option (percentage) delta of 0.5 does NOT mean: a 1% increase in stock price corresponds to a 0.5*1% increase in option price. Incorrect!

Rather, it means that a $1 increase in stock price is linearly approximated by a 0.5*$1 increase in the option value
(to remember this, just visualize the option plot: y-axis is $ option value, x-axis is $ stock price. Delta is 1st derivative, which is SLOPE, which is rise/run, which is $/$)

To show the fallacy, let S$ = stock price, c$ = call price, d = percentage delta, dS$ = change in stock price in dollar terms, dc$ = change in call price in dollar terms, dS% = % change in Stock price, dc% = % change in call price. Here is why the confusion:

percentage delta means the following:
dS$*d ~= dc$, e.g., $1.0 increase in stock where delta = 0.6 implies ~ $0.60 increase in option, which is equal to:
(S$*dS%)*d ~= (c$*dc%), and we can see that the % change in call price is given by:
dc% ~= dS%*d * (S$/c$) ... it is not given by dS%*d
... instructive? if only to illustrate the difference between option delta as 1st derivative and duration as (1st derivative/P) such that duration can be used in % terms!

In summary, if delta = d, it is not the case that dS% increase in stock price --> dS%*d increase in call price, but rather an increase of dS%*d*(S$/c$)

For this reason, we asked GARP to consider removing the L. Allen chapter on "Putting VaR to work" (on the grounds it is incorrect)

In any case, i think only two deltas matter:

• percentage delta = first derivative (yes, it's true that "percentage" can be quibbled with precisely because it may lead to the confusion, but semantically it has support; e.g., Carol Alexander)
• position delta (aka, value delta or dollar delta) = quantity of options * percentage delta

Thanks, David

 

[wrongsaidfred]

My attention to detail is both a curse and a blessing. This is a lot to take in at once.

From Allen:

"...and if the underlying fell by 1% to $99, the option value would decline to $2.58, a decline of 11 percent, hence a delta of 11."

That is what I was referring to in my last email as the terminology I have never heard of before. So is this terminology just completely wrong?

What you are refering to as percentage delta I think I am just referring to "regular delta" which is just the change in the value of 1 option with respect to the underlying. A delta of .42 means that if the underlying increases by $1 that the call would increase by $0.42. Is this what you are referring to as "percentage delta" at the bottom of your email?

Thanks again,
Mike

 

[David Harper CFA FRM]

Hi Mike,

In my opinion, Allen is completely wrong. The terminology of "percentage delta" is good and fine ( = regular delta) but her usage, the misinterpretation of "percentage" which employs it like duration, is incorrect.

Yes, your regular delta is the same as Hull's "delta" (Hull does not distinguish between percentage and position delta, but this gives many trouble because you need to hedge with position delta).

So: delta = your regular delta (exactly as you illustrated) = percentage delta = 1st derivative (change in option price w.r.t. change in stock price). It is the only delta we care about
(.... except when hedging, of course, you really have to use quantity * percentage Greek = position Greek, because neutralizing is a matter of getting the position dollars back to zero)

Thanks, David

P.S. blessing/curse, i think i know exactly what you mean, for me it's about 30% blessing/70% curse, can't speak for you

 

[wrongsaidfred]

Close to the same. Seems like I get more headaches from noticing something that looks wrong and convincing myself that I just probably dont understand what is going on. Then I find out I was right the whole time and just get frustrated. Oh well.

Now that I have your ear can I ask you something else about this reading?

The section on "Delta normal" vs full revaluation seems really bad.

First they say full revaluation involves assuming a one-for-one or one-for-delta move with the underlying factor. Am I reading something incorrectly or isnt this actually the delta normal approach?

And then there is this, referring to full revaluation:

"The VaR of the option is just the value of the option evaluated at the value of the index after reducing it by the appropriate percentage decline that was calculated as the 1 percent VaR of the index itself."

What in the world is this actiually trying to say?

Thanks again,
Mike

 

[David Harper CFA FRM]

yea, it's a mess, agreed it reads too much like delta-normal. I think she just means:

if c(I) = value of option portfolio based on underlying index (I),
to re-value 9x% VaR of c(I), reduce the index to (I-) and re-price ("the value of the option evaluated at the value of the index") the option portfolio:
c(I-) = reduced value of option portfolio based on loss of underlying; the difference being that we fully recompute the derivative based on the different value of the underlying
And that is full revaluation; i.e., VaR = c(I) - c(I-)

David

 

[wrongsaidfred]

Perfect.

Thanks,
Mike

 

[David Harper CFA FRM]

Answer to the today's "pop quiz" newsletter question:

Question: A European call option with a delta of 0.60 has a price of $3.60 when the underlying stock is $20.00.If the stock price increases by 5%, by what percentage (%) does the option price increase?

Answer: ~16.7%
(if you answered 3%, then I caught you! I hope you don't mind ...)


+5% means the stock increases by $1.00 to $21.00 (i.e., +1/20 = +5%)
Option delta of 0.60 means that, as a linear (imprecise) approximation, a $1.00 increase in the underlying stock price is associated with a $0.60 increase in the call price.
So the option price increases ~+$0.60 to $4.20.
Percentage increase in option price ~= +$0.60/$3.60 = 16.67%

Proposed moral of the story:
Option delta is not exactly analogous to bond modified duration. Option delta is a pure first derivative (delta = dc/dS), a pure slope; but bond modified duration is a function of the first derivative (modified duration = dP/dy* -1/P), so it is "infected by price". Related, the units the not the same either: option delta is unitless (dy/dx is $/$) but bond duration (Mac and Mod) are both denoted in years.

(but of course, it is critical to understand what they have in common: both are linear approximation and therefore inaccurate. We are using the slope of a line when the underlying relationship in both cases in non-linear)

Add-on question: what do we call the pure first derivative (dP/dy) in the bond price-yield relationship--i.e., the slope of the tangent line--and why is it important?

 

[wrongsaidfred]

Nice!

This may be a really dumb question, but if the stock went up by, say 1% to 20.20 we would just say that the option went up by 0.2*.6=.12 or 12 cents and then we would say the percent increase was .12/3.6=3.33%. Correct?

Thanks again for all of your help.

Mike

 

[David Harper CFA FRM]

Mike,

Exactly, I agree!

If you don't mind, I thought it would be fun to pop-quiz the above question on our newsletter (later today) because I think most of us would get it wrong ... not that the following next step is useful, but as your example and mine both illustrate, the %-based analog to duration is, under these assumptions, appears to be 3.33 (your +1% S --> +3.33% c, just as my +5% corresponds to 3.33*5% = +16.67%). This is shown by dc% ~= dS%*d * (S$/c$), where here, dc% ~= dS%*0.60* ($20.00/$3.60) = dS%*0.60* ($20.00/$3.60) ---> dc%/dS% = 3.33. This 3.33 is the "analog" to duration, meaning it is "infected" by the price. Not useful except to highlight the difference. Thank you for exploring this with me! David

 

[wrongsaidfred]

That is a good one. Go for it. It will be interesting to see how people react to it.

Take care,
Mike

 

[David Harper CFA FRM]

Mike, thanks ... i asked you this earlier but can i collect some of your best queries under a "Mike's" sub-heading in the newsletter, just to share (i wanted to share the most helpful threads from the last two weeks). It's okay if you'd prefer not to be cited by name, in which case i will just mix them in ... i don't usually do it that way, but you've had several that cut right to key points and it just feels more transparent to credit you explicitly. No big deal either way. Thanks, David

 

[wrongsaidfred]

Absolutely. You can post anything of mine that you would like.

Mike

 

[David Harper CFA FRM]

Great, thanks, we are sending it tonight, David

 

[[email protected]]

I think it is the Modified duration. The change in bond prices, if expressed in terms of Taylor series expansion, then it also contains a 2nd degree derivative term. Though for small changes in interest rates, it is not significant, it becomes significant for large changes due to convexity effect.
Thats why when we use the formula -dp/P = Md * dy, we say that it only works for small values of changes in rates. For large changes, the equation has to consider the convexity term as well.

Answer to the today's "pop quiz" newsletter question:

Question: A European call option with a delta of 0.60 has a price of $3.60 when the underlying stock is $20.00.If the stock price increases by 5%, by what percentage (%) does the option price increase?

Answer: ~16.7%
(if you answered 3%, then I caught you! I hope you don't mind ...)

+5% means the stock increases by $1.00 to $21.00 (i.e., +1/20 = +5%)
Option delta of 0.60 means that, as a linear (imprecise) approximation, a $1.00 increase in the underlying stock price is associated with a $0.60 increase in the call price.
So the option price increases ~+$0.60 to $4.20.
Percentage increase in option price ~= +$0.60/$3.60 = 16.67%

Proposed moral of the story:
Option delta is not exactly analogous to bond modified duration. Option delta is a pure first derivative (delta = dc/dS), a pure slope; but bond modified duration is a function of the first derivative (modified duration = dP/dy* -1/P), so it is "infected by price". Related, the units the not the same either: option delta is unitless (dy/dx is $/$) but bond duration (Mac and Mod) are both denoted in years.

(but of course, it is critical to understand what they have in common: both are linear approximation and therefore inaccurate. We are using the slope of a line when the underlying relationship in both cases in non-linear)

Add-on question: what do we call the pure first derivative (dP/dy) in the bond price-yield relationship--i.e., the slope of the tangent line--and why is it important?

ink

 

[David Harper CFA FRM]

Hi ink,

It is tempting to call the first derivative modified duration, and many texts do. But dP/dy is actually the dollar duration.
As you have: -dp/P = Md * dy, dp/dY = -Md*P. So, the slope of the tangent line (i.e., dP/dy) is modified duration * Price = dollar duration.
For a $100 bond with modified duration of, say, 6.0, we might expect a slope (dP/dy) of -600.
(seems like a startling large number?! but it is the price change for a ONE unit change in the x-axis, and one unit here happens to be 1.0 = 100% = 10,000 basis points!)

Why is it important? Because we would hedge not with modified duration itself, but the dollar duration. If we want to "neutralize" the duration, we would look to sum the dollar durations to zero.

Thanks! David

 

[[email protected]]

hi David

i have been following this thread with keen interest. it was an issue i was deliberating on before Mike posted the question. this thread has addressed my problem and i thank Mike for that.

to Mike, you are a star. i appreciate you.

thanks

 

[David Harper CFA FRM]

Hi baffour, I agree, he is a star, he obviously has some preexisting aptitude but his due diligence with regard to the concepts is really sharp. But his approach contributes to our challenge to GARP to improve the rigor and consistency in the exam: just on Mike's feedback alone, I would estimate we've identified over a dozen definitional issues (inconsistencies), many repeats from last year and beyond. As I have argued to GARP, many of these will never be noticed by the "average" candidate, such that inconsistencies at the margin are, ironically and unfairly, a penalty to more careful candidates.

For example, if your only goal is to pass the exam, you are currently better off (I'd argue) with a superficial understanding of "alpha" than a detailed view b/c the detailed view may cause you to parse the question too finely. Another (eg): maybe you are better not to know relative vs absolute VaR since they have not used those terms in the exam (?!).

I personally like to learn, so deeper is better, but the exam methodology should not incur marginal cost (w.r.t. passing) as knowledge increases... it's not quite right that the concern with somebody like Mike is "they get to know too much" (e.g., that really doesn't happen with the CFA: the rigor ensures it. For CFA, knowing more is pretty much exam-wise always better). Thanks David

 

[[email protected]]

sure David, GARP need someone like Mike in order to unveil some of the loopholes in the AIMS. this is an indeniable fact.
as some one who really want to enter into risk management and investment banking i definitely do not learn just to pass the papers but understanding is the most important thing for me.

i really will want to know Mike and you very well, even after this course. for me i am just an "apprentice" who is hungry to learn more from you. and i appreciate you too much.

thank you David. May God keep you for me.