More on Risk Analysis of Multi-Asset Options
January 2, 2010
Further, elaborating on the article on Risk Analysis of Multi-asset Options.The last line shows that the P&L of the risks in the delta-hedged portfolio are the second order risks driven by the gamma and cross-gamma terms.
The final equation gives two very important pointers on how to hedge the correlation risk and covariance risk. It has to be noted that the two risks are different but related.
- If we make portfolio gamma neutral, the first two terms vanishes and we get a portfolio having a pure covariance exposure.
- If we make the portfolio vega neutral - which means no sensitivity to implied volatility or rather realized volatility, we get a portfolio which has pure correlation exposure
In practice, multi asset exotic books have covariance exposure which is generally hedged by trading basket of calls v/s call on a basket. The problem with such an approximate hedge is that the covariance exposure is significant near ATM only.
Intuitive explanation of cross-gamma Pnl for a Worst of Put Option:
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 1 moves OTM & asset 2 becomes the worst performer, delta on both assets change & to re-balance delta, we buy 0.75 of 2 at a lower price since it has gone down & sell 0.25 of asset 1 (to flatten the delta on A) at a higher price. Hence making money.
Therefore, we conclude that Short cross-gamma implied Short correlation which in turn implies Short covariance; and the Cross gamma P&L would depend on the expression:
Cross-Gamma *(Realized covariance – Implied Covariance)
The above equation is a similar equation with that of Gamma P&L.
In the above case we are short cross-gamma, i.e. negative cross-gamma & realized covariance (negative) is less than the implied covariance (positive) and hence we make money.
*Nikit Kothari is an alumnus of Indian Institute of Technology (IIT), Kharagpur, India and has two years of experience in trading exotic derivatives. Most recently, he was with Lehman Brothers.
Disclaimer:The ideas and opinions expressed in this article are solely those of the author’s and does not reflect the ideas and opinions of Risk Latte Company or any of its member and/or employees. Risk Latte Company and/or any of its employees or members will not be responsible for any inaccuracies, errors or omissions and/or any factual inconsistencies in the article and will NOT be liable for any losses, direct or indirect, arising out of the usage and applications of any of these ideas and opinions by anyone. This article appears solely for educational purposes only
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