Reading 15 - Exhibit 8

CFA Level III Derivatives
#1

I’m looking over exhibit 8 on page 237. I understand that gamma is largest ATM. Exhibit 8 (book 3), the largest gamma value is at share price 15 which is not ATM. ATM is 16. Is this a mistake?

Thanks anybody in advance.

#2

It looks like a mistake to me.

Is it listed in the errata?

If not, e-mail CFA Institute (info@cfainstitute.org) and mention it to them.

#3

Thanks S2000.

#4

Can you give us a snippet of the actual exhibit and or problem? There could be a lot of missing context here.

#5

Gamma is always highest at St=X. Period. But obvious at St=15, it will still have very high gamma but not the peak one. No need for further explanation

#6

Interestingly, I was just creating a graph of gamma from the BSM model and it appears that this isn’t true, period or not.

I have a call option with a strike price of USD 25 and one year to expiry; gamma is highest at USD 21.40, not USD 25.

Note that this is consistent with the graph in the exhibit: the maximum gamma occurs when the option is somewhat out of the money.

I’m e-mailing CFA Institute about this.

#7

So the graph is correct but the written logic in the text is incorrect?

#8

According to what I worked out in Excel, it appears so.

We’ll see what the authors of that reading have to say.

#9

Did CFAI reply to the gamma at St=X vs St<X?

#10

I believe that the explanation “Gamma is largest at ATM” is just an approximation. In general, the maximum gamma depends on several factors, such as volatility, time to expiration, rate of interest, etc. To get an accurate value for the maximum gamma, you would need to caculate the partial derivative of gamma with respect to S and find S such that the partial derivative of gamma with respect to S (i.e. speed) is zero, and that the derivative of speed is negative. This seems difficult to do in an exam setting, and arguably unfair to candidates if Gamma turns out to be largest when it’s slightly OTM.

I tried taking the derivative of gamma, and after simplifying, I found that, for speed to be zero, and assuming I didn’t make a mistake,
S = K*exp[-(r - q + 1.5*\sigma^2)*T )]
where q is the continuously compounded dividend yield
so gamma is not highest ATM in general.

edit: i can only imagine how much fun it must be to try and find S such that the gamma of a gap call option is largest…

edit: fixed

when T = 1 year, then above eq suggests that
S = K*exp[-(r - q + 1.5*\sigma^2)]
maximizes european call gamma

as an example, suppose T = 1 year, K = 25, r = 10% (continusouly compounded), q = 0, and annualized \sigma = 30%. Then
S = 25*exp[-(.10 + 1.5 *.30^2)] = 19.76427124 = about 19.76
so in this case, S = 19.76 maximizes gamma.

#11

Save all the spadework. Gamma is highest at St=x AND at T-t~0. Period. It’s mathematics.Just remember both wil have to hold. Volatility per se has no EXTRA effect on Gamma. Volatility is anyway highest at St=X…

#12

can you prove mathematically that’s the case? because both my work and @S2000magician’s seem to contradict this.

edit: (yes, I’m sort of answering my own question)

from my previous equation,

when T = 0, this simplifies to S = K.
so only in the special case where T = 0 will S = K hold.
when T is very close to 0, S will be approximately equal to K as well.

edit: my textbook says that short-term options peak near the strike price, while longer-lived options peak further to the left and less steeply. my work here seems to confirm this. S = K only when the option expires immediately, and the longer the time to expiration, the gamma peaks further to the left.

edit: saying that Gamma is highest ATM when T = 0 seem as useful as saying that the price of a call option (well, any option, really) exactly equal its payoff when it expires immediately. interesting, but not very useful in practice.

#13

This makes no sense, of course.

Volatility doesn’t know your strike price.

#14

Do revisit. You n me are not quite apart .BSM is an award winning model and I have tested it. Nothing to doubt really.

#15

You are right. I stand corrected. Volatility is that of underlying and any given time there are n no. of strike price existing.

#16

Thanks, HD!

It doesn’t happen often, but sometimes I’m right.

Stay safe and healthy!

#17

if you were referring to my last comment, it doesn’t contradict BSM.
Essentially, in my closing comments, I was saying that an option that gives you d dollars immediately with no uncertainty should cost you d dollars (so that you neither gain nor lose from this transaction). Otherwise, there would be an arbitrage opportunity.

#18

I am a big fan of the Magician as thousands others. Equally, there’s one more guy who I hold in very high regard. David Harper of Bionic Turtle. I sometimes feel guilty that I used his free resources available on YouTube that I banked upon to pass my FRM and subsequently acquire my Charter.

However the need for prologue is to to urge and perhaps suggest to see his videos on Option pricing models . I believe they are still freely available on the YouTube. You may thank me later.

#19

Thanks for the suggestion, although I feel comfortable enough with the basics of Option Pricing, after working through 3 different books on option pricing (well, more like breeze reading two and working actively on one). I still have two other books, but these turned out to be a little too advanced for me (Stochastic Calculus by Shreve). Shreve’s books are far more rigorous than any other book I’ve seen. I believe that working through challenging problems is the way to really learn the material, although tutorial videos certainly help. I tried looking David Harper on Youtube, and the platform thinks he’s a medical doctor and suggests videos completely unrelated to finance.

I get the impression though, that I might have made a mistake on a previous post, judging from your post. If you, or anyone else, can be precise on where I made a mistake, I would be thankful.

#20

Try searching Bionic Turtle. I would not know any Dr. David Harper