Is Delta-Gamma valuation a Local-valuation method?

[Liming]

Dear David,

Is Delta-Gamma valuation a Local-valuation method? I think it is definitely not a full valuation as it doesn't involve full repricing of portfolio in question. So to me, it seems to be a local valuation, just like delta-normal, with the exception of the added non-linearity. Thanks!

Cheers!
Liming
30/09/2009

 

[David Harper CFA FRM]

Hi Liming,

Yes, I absolutely agree that delta-gamma is local valuation.
And, as Jorion shows, I think it's helpful to generalize to the Taylor series approximation such that delta-gamma is merely "truncated" Taylor that stops at the first two terms of the infinite Taylor:

Taylor: f(x) ~ f(a) + f'(a)(x-a) + f''(a)/2!*(x-a)^2 + f'''(a)/3!*(x-a)^3 + ...

so both the delta-gamma (e.g., options) are using the first two terms:

..for an option:
f(x) ~ f(a) + f[delta is 1st derivative] + f[gamma is 2nd derivative] + ignore the rest b/c it is too small to matter?

...and for a bond, the bond duration-convexity is really the same thing!
f(x) ~ f(a) + f[dollar duration is 1st derivative] + f[dollar convexity is 2nd derivative] + ignore the rest ..

it's not full revaluation because we are not re-pricing the instrument/portfolio, rather we are using the derivatives to estimate (approximate) the change
links here from ajsa's related query: http://forum.bionicturtle.com/viewthread/1670/
here somebody submitted a delta-gamma query example: http://forum.bionicturtle.com/viewthread/1840/
(or this is a delta-gamma-vega as two first derivatives and one 2nd derivative are employed to estimate change. But still a "local" as we did not re-price the option )

...and, just as you say, the gamma (or convexity) correction (or adjustment) will introduce non-linearity into the approximation. This is a good point b/c some mistakenly assume local = linear, but Jorion (p. 360 handbook) appropriately divides local valuation into two parts:
Local > Linear, and
Local > Non-linear (i.e., adds gamma and convexity)


i.e., delta by itself implies linearity, but local does not, just as you suggest!
Why local? because if we think about delta-gamma visually, the accuracy is better if our our "shock" is small

(finally, it's more than you asked but be careful about confusing delta-gamma with delta-normal; the "normal" in delta-normal refers to the assumption that the underlying risk factor is normally distributed, which is convenient but not required).

Thanks, David

 

[Ryan S]

from above, these links appear to no longer work. Can these be udpated?

inks here from ajsa's related query: http://forum.bionicturtle.com/viewthread/1670/
here somebody submitted a delta-gamma query example: http://forum.bionicturtle.com/viewthread/1840/

Thanks,
Ryan

 

[David Harper CFA FRM]

Hi Ryan, sorry I can't currently find them (they appear to be over four years old, well before our forum was upgraded to Xenforo such that I can't use the numbers). Thanks,