cross gamma term

Discussion in 'P2.T6. Credit Risk (25%)' started by shanlane, Apr 10, 2012.

  1. shanlane

    shanlane Active Member

    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
  2. 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!
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  3. shanlane

    shanlane Active Member

    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
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