By definition:

delta = change_call / change_stock.

Rewriting

change_stock = change_call / delta.

In the example we are given delta = 0.62 and found change_call = 41,667 calls.

To find change_stock why did you *multiply* instead of *divide* change_call by delta?

the long calls increase the portfolio delta from 0 to 25,833 (41,667 * 0.62 delta/option). Now the portfolio is gamma neutral but +1,240 delta ("we fixed gamma but we broke delta")

so the trade must subtract 25,833 delta, the way to do that is short 25,833 shares (delta = 1/share). ("we fix delta” and, fortunately, shares have gamma = 0)

More thread convo here. If you like, an XLS is here. This is all based on Hull’s example.

David

David, thanks for the quick reply and I have now read Hull’s example many times over.

I know that there is something awfully trival going on here but I am still confused. 

I follow the math but not the “physics” behind it.

I am getting hungup on the *definition* of delta = dC/dS.  Maybe could you start from this definition?

Mikey,

Oh, right. I think you make a great observation (isn’t the seemingly trivial often important?) Hull uses delta with two meanings. The fabulous Carol Alexander (Chair of PRMIA, author of these) gives us the following useful distinction:

percentage delta = dC/dS
position delta = dC/dS*N, where N is the number of units, or to the source of confusion i think:
position delta = percentage delta (i.e., the first derivative) * N

So, when Hull says Gamma = -3,000, it is a position Gamma = -3000 N*d^2C/dS^2

So, using Hull’s full example

Gamma trade: -3000 Nd^2C/dS^2 + X N * d^2C/dS^2 = 0
if the percentage gamma = 1.5, then X (the position gamma) = 2000

This trade creates a POSITION DELTA (N*dC/ds) = 1240 because

2000 N*0.62 dC/dS = 1240 N*dC/dS << This is your answer

Now to neutralize the position delta of 1240 (N * dC/dS) with shares that have delta of 1.0, we need:

1240 N*dC/dS + (X) N*(1.0 dC/dS) = 0

so X = (-1240 N*dC/dS)/(1.0 N dC/dS) = -1240/1.0 = short 1240

Hope that helps, thanks for surfacing this nuance.

David

Thanks again for the fast reply but I still don’t see it.  For example in “Gamma trade:” what happened to the second partial d^2C/dS^2?

Would it help to peek in Carol Alexander’s book using Amazon’s “Look Inside”?  If so, what page is it?

Whereabouts in Hull’s book does he give the two meanings?  [I have an ancient text but since everything is pretty much the same I bet that the section headings are also the same.]

Also I apologize for torturing you more but could you rewrite your equations with Hull’s “greek” symbols using a math symbol editor and pasting them perhaps into the EditGrid spreadsheet that you created?  Maybe that might clear up my d^2C/dS^2 confusion above.

I know that we are almost there!

For example in “Gamma trade:” what happened to the second partial d^2C/dS^2?
I used the word “Gamma” instead. I have just replaced Gamma with the equation above....

Page 163 of Volume III (IIII.3.4.4 position Greeks)

Hull does not acknowledge the difference, to my knowledge, was my point. Hull uses “delta” for both “% delta” or “position delta”

How do you get math symbols into EditGrid? Please do tell?

David

Hmm… now looking at McDonald’s Derivative Markets (1st edition) and on Table 13.6 (and Figure 13.4) in the section “Gamma-Neutrality” I believe that he presents the calculations that I am looking for.  [hooray!] Unless I have trouble with McDonald I think that I’m okay so you can ignore the Carol Alexander request above. 

How do you get math symbols into EditGrid? Please do tell?

Maybe I’m wrong but haven’t I seen that in your screencasts when you pasted a graphic image into the spreadsheet?

In any case with McDonald I’m okay.  Once again, thanks!

Oh right, i can do that, i just didn’t see how that would help necessarily. But glad you are good on this, I’ll check out Culp’s text on this myself. always looking for better explains....David