Exotic Derivatives Trading

February 19, 2009

Gamma/Cross-Gamma Pnl – “Rebalancing deltas”

Filed under: Risk Management of Exotic Derivatives — Exotics Trader @ 4:52 pm

How do we make money on Gamma?
Considering you are long an Option, you make money when realized volatility is greater than the implied volatility & vice versa when you are short an Option.

Is it always true?
1.Answer is No, let’s consider a 1 yr Long call, If realized volatility at the end of the 1 yr is greater than the implied volatility, It’s not necessary that you will make money. Let’s see how.
Daily Pnl for a delta hedged Long call position incase implied vols dont move is : 0.5×Gamma×Spot2 {Realized Variance – Implied Variance}

With this expression, we can clearly see that the final P&L is the sum of the daily Variance Spread weighted by the Dollar Gamma. Thus, considering those days when Dollar Gamma is high will tend to dominate the final P&L & Hence it’s the variance spread {Realized Variance – Implied Variance} in those high dollar gamma days that will determine the Pnl, Not the variance spread at the end of 1 yr. For the final P&L to be path-independent, the Dollar Gamma must be constant.

Intuitive explanation of cross-gamma Pnl: Assuming we are long an ATM 2-asset worst of put which is near expiry.

Delta of asset 1 @ 99% implied correl: 0.25
Case 1: They both move OTM, In that case delta on both is 0
Case 2: They both move ITM, In that case delta on both is 0.5
So expected delta is 0.5×0 + 0.5×0.5 ~ 0.25
Delta of asset 2 @ 99% implied correl: 0.25
Assuming the volatilities of both the assets are similar in magnitude, Now in case the realized correlation is -1, i.e. asset A moves OTM & asset B becomes the WOP, delta on both assets change & to re-balance the delta, we buy 0.75 of B at a lower price since it has gone down & sell 0.25 of asset A (to flatten the delta on A) at a higher price. Hence making money…
Conclusion: (Short cross-gamma ~ Short correlation ~ Short covariance) & Cross gamma Pnl would depend on : Cross-Gamma *(Realized covariance – Implied Covariance ), A similar equation with that of Gamma Pnl.
In the above case we are short cross-gamma, i.e. negative cross-gamma & realized covariance(negative) is less than the implied covariance(positive) & hence we make money…




  1. I would expect the crossgamma way the way correl sensi is.

    Visually, it is quite intuitive that holding a worst-of, you want asset to decorrelate as much as possible, so that at least one is in in the money.

    Comment by nicolas — December 29, 2009 @ 9:58 am | Reply

  2. ” Case 1: They both move OTM, In that case delta on both is 0
    Case 2: They both move ITM, In that case delta on both is 0.5
    So expected delta is 0.5×0 + 0.5×0.5 ~ 0.25″

    If the option is near expiry and it’s in-the-money….exactly why would its delta be 0.5?

    Comment by Aquarius — January 7, 2010 @ 10:02 am | Reply

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