cross gamma term

[shanlane]

Hello,

You said that the cross gamma term for the other counterparty represented wrong way risk. What does your own cross gamma term represent? Since the counterparty's cross gamma term is -1 and our own term is +1 does this somehow mean that it represents right way risk? Its a bit far fetched, but its all I can come up with.

Thanks!

Shannon

 

[David Harper CFA FRM]

Hi Shannon (I assume "you" is me?)

That is interesting. To re-cap Canabarro's example ("cross gamma" isn't my term, it's Canabarro's, I'm not yet comfortable with it):

From A's perspective (Canabarro p 120):

• Mid-market value (from A's perspective) = -$50 million
• E(B)*s(B) = $100 * 5% = $5 million; i.e., exposure faced by A with respect to B
• E(A)*s(A) = $200 * 2% = $4 million; i.e., exposure faced by B with respect to A
• CVA (from A's perspective) = E(A)*s(A) - E(B)*s(B) = $200 * 2% - $100 * 5% = $4 million - $5 million = -$1 million
• mid-market value, net of CVA adjustment = -$50 - $1 = -$51 million

As E(B)*s(B) = E(B)*PD(B)*LGD(B), this implicitly assumes no correlation between exposure and PD or EL.

In the video, it occurred to me that, from A's perspective, wrong-way risk would be unfavorable correlation between E(B) and s(B).
In the case above, A's wrong-way risk manifests as positive correlation between E(B) and s(B), which implies:
CVA (from A's perspective) = E(A)*s(A) - E(B)*s(B) = $4 million - [greater than $5 million] = [less than - $1 million]

It seems to me, from A's perspective, right-way risk would be favorable correlation[E(B), s(B)] such that in this case:
CVA (from A's perspective) = E(A)*s(A) - E(B)*s(B) = $4 million - [<$5 million] = [greater than $1 million]

From B's perspective, we can consider similarly the right-way/wrong-way risk as it impacts E(B)*s(B).

(the "cross-gamma" phrase is a new term to me/FRM. I haven't read all of Canabarro's book. I am not entirely comfortable with the brief partial derivative he shows, I don't see an example of his 2nd derivative. Suffice to say, he clearly means "cross-gamma" term to capture the right/wrong way risk)

To your point, from A's perspective, it is interesting that:
An increase in B's wrong-way risk (an increase in the correlation between E(A) and s(A)) has the same directional impact as an increase in A's right-way risk; both increase the CVA adjustment (from A's perspective) and consequently increase the CVA-adjusted mid-market value of the position. But, I would not semantically refer to B's wrong-way risk as the same thing as A's right-way risk. (It would be akin to me writing you a put on my stock and referring to your wrong-way risk and my own right-way risk ... mathematically true but semantically doesn't work for me). Thanks!

 

[shanlane]

Sorry, I just watched the video before I wrote this so by "you" I meant I heard your voice saying it, even though it is from Canabarro.

So A's right way risk would end up lowering E(a)s(a), because this would mean a lower loss rate as exposure rises (or less exposure as loss rate increases). This is making me a little dizzy but I think it makes sense.

Thanks!

Shannon