Worst of Up and Out Put Option – Vega Risk and the Correlation Skew
June 11, 2009
In the previous article on Worst of Up and Out Put option we discussed the need for such a structure and how to analyse the risk. In this article, we continue with that discussion and talk about how to hedge the Vega risks and the correlation skew risk of such a structure.
Figure 1. Vega of asset A & asset B v/s spot of asset A for different levels of correlation. Volatility of asset A & asset B at 40%, Spot of asset B at 100%.
For similar volatility levels of asset A & asset B (Both at 40%):
At 0% correlation: (Refer figure 1) the relative position of asset B with respect to that of asset A has a major effect on the vega of asset B. Vega on asset B dies down quickly with spot of asset A (spot of asset B fixed at 100%). In case the asset A moves up rapidly a trader is left with a shorter vega position on asset B which he has to buyback. The vega hedge rebalancing P&L will depend on asset B’s volatility movement. At +99% correlation, the vega is not that significant.
At -99% correlation: (Refer figure 1), considering a case where asset A is in near range of asset B (which is at 100%), the probability of structure knocking out is very low with almost equal probability of asset A and asset B being the worst performer. Hence, asset A and B have the same vega for spot levels of asset A which are in the near range of asset B.
But at higher spot levels of asset A, the probability of asset A landing below the barrier is low, consecutively the probability of structure knocking out increases as spot of asset A increases. Hence at high spot levels of asset A, vega of asset A shoots up and since asset B becomes the worst performer, vega on asset B decreases. The constant vega on asset A and B can be hedged by variance swaps, but incase asset A moves up drastically, the effective maturity of the structure decreases, hence long term vega has to be unwounded and short term vanillas would have to be traded on asset A and asset B to replicate the vega profile for higher spot price levels.
Figure 2. Vega of asset A and B v/s spot of asset A at different correlation levels. Spot of asset B at 110%, Volatility of asset A & asset B at 40%.
During the life of the trade incase asset B moves up massively the vega profiles of asset A and B are reversed. (Refer figure 2). Again trading vanillas can be used to replicate the vega profile at highly negative correlation levels.
The vega hedge in general will consist of a portfolio of vanillas of different maturities. We call those maturities as “Vega Buckets”. Since for every time instant in future there is a definite probability of the structure knocking out, the overall vega for an asset can therefore be seen as being distributed among time buckets. It’s intuitive that as spots and volatilities in general rise, the probability of structure knocking out in the near term increases leading to a significant increase in magnitude of the short term vega along with a decrease in the magnitude of long term vega. At the inception of the structure, each underlying will have a vega in each bucket. Moreover, it can be seen that if an asset outperforms all others, two effects will occur: it’s own vega will considerably decrease in magnitude in all buckets because it is not the worst performer, the vega of other assets will increase in magnitude since the outperforming asset is already near the barrier and increasing their volatility would cause the knock out probability to increase considerably. The vega hedge must be dynamically adjusted after considerable market movements.
The rebalancing interval is a trading judgment which will depend on the trade notional and current spot levels. A market move of say 3% might cause the vega to reverse and trigger a re-hedge. The vega hedge will approximately offset the gamma of the position. In practice the option is likely to be hedged as part of a book rather than being hedged individually causing a mismatch between the maturities chosen for vega hedge and actual time buckets for the option.
Figure 3. . Price v/s spot of asset A. Spot of asset B at 100% and its volatility at 20%. Spot of asset A varies from 90% to 110%. The variation in price depends strongly on the ratio between volatilities. The effect is displayed for different correlation levels: (a) 0 correlation ;(b) +0.75 correlation ;(c) -0.75 correlation
Figure 4. . . Dividend exposure of asset A v/s spot A for different correlation levels. Spot of asset B at 100%, Volatility of asset A &B at 40%. Dividend exposure is the change in option price with 1% change in dividend yield.
Overall the structure is long dividends. Significantly high values of drho can be seen at high negative correlation levels (Refer Figure 4). Dividends can be hedged using dividend swaps, jelly rolls depending on the expected maturity of the structure.
Correlation skew Risk
Figure 5. Correga (Change in price / 5% change in correlation) v/s spot of asset A for different levels of (Vol A, Vol B, Correlation). Spot of asset B at 100%.
Correlation skew resembles volatility skew since correlation in general is high when volatility is high and vice-versa. The structure is short correlation-skew i.e. When markets fall, correga decreases and a trader needs to buy correlation and vice-versa when the market rises (Refer Figure 5). In case the pricing model doesn’t take correlation skew into account; correga rebalancing cost can be calculated by having an estimate of the changes in correlation levels with spot movement and should be directly added to the flat correlation price of the structure.
It’s a toxic structure to trade, especially at highly negative correlation levels. Because of the complexities involved (path dependency and multiple assets), this structure is difficult (computationally expensive) to price. Moreover, as we have seen, the risks can change markedly in volatile market conditions as the trade progresses (vega, cross-gamma reversals) and hence incurring a significant cost during the hedge rebalancing. Also there is need to consider the high magnitude of dividend and correlation exposures which could be hard to hedge due to illiquidity in dividend and correlation trading markets.
*Nikit Kothari is an alumnus of Indian Institute of Technology, 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 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.
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