Risk Analysis of Multi-asset options
Traders keep using the terms cross-gamma and correlations when dealing with multi-asset options. It is hard to bring out the intuition for such derivatives without a solid quantitative approach. The analysis below brings out explicitly the dependence of the Profit and Loss (P&L) on the realized and implied co-variances (and hence correlations).
Let X and Y be two random variables representing the prices of stock X and stock Y.
We have to model the returns on X and Y as Brownian motions:
Construct a portfolio "" that is long the multi-asset option and short delta units of X and delta units of Y.
Applying Ito's Lemma in two dimensions (V is value of the multi-asset option) ,
In order to remove the randomness (i.e. to hedge), the terms containing dX and dY need to be eliminated from the above expression. This can be done by choosing
No Arbitrage (when interest rates are zero) requires that , i.e. we cannot make profits for free.
This is the differential equation governing the price process of any two asset derivative. The above analysis outlines the recipe for hedging any multi-asset derivative: short delta units of stock X and delta units of stock Y and dynamically re-balance the portfolio till maturity.
But delta hedging only removes the first order risk. After hedging the deltas, the higher order risks still remain and are of concern from a risk management perspective.
From Taylor series expansion in two variables, the actual pnl realized by the portfolio is
But the price process differential equation prescribes
The last line is the P&L expression which shows that the risks in the delta-hedged portfolio are the second order risks driven by the gamma and cross-gamma terms. The analysis also brings out the dependence of the P&L on the realized and implied covariances (and hence correlations). If the cross-gamma term is positive and realized covariance is less than the implied covariance a large negative P&L will be realized from the cross gamma term.
Therefore, to breakeven,
To highlight the correlation intuition, assume realized volatility is equal to the implied volatility. The P&L expression then simplifies to
To breakeven the realized correlation between the stocks should be equal to the implied correlation.
*About the Author: Kamesh Chilukuri is an alumnus of Indian Institute of Technology Madras, India and Massachusetts Institute of Technology, Cambridge MA, USA. He is currently an Associate Director, UBS Investment Bank in Stamford, CT. The ideas expressed in this article are his own.
Disclaimer:The ideas and opinions expressed in this article are solely those of the authorís 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 nor will it 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. The author of this article has confirmed to Risk Latte Company that he holds the intellectual property rights to the content outlined here.
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