Barrier options are the most actively traded options in the OTC derivatives market, often structured in accordance with the investor’s risk appetite. There are various structures that can be used by bullish investors to hedge their portfolio ranging from simple vanilla options like plain puts to barrier options like down and in puts. Incorporating barriers in a vanilla put provides flexibility to the client in terms of the different barrier levels he can choose and makes it cheaper as well.

An investor is forced to look beyond vanilla put as a hedging alternative when the market volatility is very high – a scenario in which buying *vega* is very expensive. The situation demands for a structure with low *vega* and this is where modifying vanilla options by introducing barriers can prove to be useful. Barriers can be of two types: *In barrier* – when stock price hits a particular barrier, the option is activated (for example a down and in put option), *Out Barrier* – when stock price hits the barrier the option is knocked out or the terms and conditions of the structure no longer hold true (for example an up and out put). Clearly an option (having potential upside) with a knock out barrier will have a low *vega risk* as compared to that on a vanilla put or a down and in put option. Hence, an up and out put option fits the hedging requirements of a bullish investor in a high volatility environment. Moreover, in a scenario where resistance-support reversal is expected (as shown in figure 1) for stock price levels, worst of up and out put option proves to be a good hedge for the portfolio.

**Figure 1**. *Resistance becomes support*

To further elaborate on this, consider a case where an investor has a long position on STOXX50E with a view that if the index level crosses a certain resistance level, the initial resistance level will become the new support level. In simple words the investor expects that after crossing the initial resistance level, STOXX50E has very low chances of falling below the initial resistance level. In that case the investor doesn’t care much about the put option (which is a hedge for the bullish portfolio) knocking out since index crossing the resistance level is a sign of market recovery. Clearly the barrier should be chosen around the initial resistance level. Appropriate baskets for WOUP (worst of up and out put) options can be designed in accordance to the composition of the portfolio that a fund manager needs to hedge. Such multi asset WOUP options are attractive to fund managers in tumultuous market conditions because:

- High volatility prevents investors from buying worst of put and worst of down and in put options;

- The high correlation scenario makes WOUP options cheaper

**Pricing of the Structure**

The presence of barrier indicates that the structure will have an asymmetric *vega* profile with respect to spot, hence giving rise to the *vega skew* exposure. The strong path dependency (because of continuous barrier) and significant *skew exposure* would require this structure to be priced in a local volatility model framework using Monte Carlo simulations.

What about the barrier shift?

There’s no barrier shift required for a single asset up and out put option because long an up and out put option always means a short delta position. In case the barrier is hit and stock price jumps aberrantly beyond the barrier, trader sells the delta at a higher price, which he had been holding to hedge a long up and out put position. Clearly there is no risk of losing money - contrary to a situation that a trader faces in case of down and in put options.

Now, let’s consider the case of a two asset (asset A and asset B) worst of up and out put option with a 105% barrier on the worst of. Observe the delta profile of asset A with respect to its spot in a high negative correlation scenario and asset B at its initial spot level (Refer Figure 2).

**Figure 2.** *Delta of asset A v/s spot of asset A for different correlation levels. Asset B spot at 100%, Volatility of asset A and B at 40%, Maturity: 1 yr *

The delta profile of asset A at higher spots (110% performance level) is similar to that of a short position in a digital option. The delta falls and rises up steeply increasing the risk of losing money on gamma in the delta hedging process. To account for that hedging cost a barrier shift is required; for the same reason a barrier shift is incorporated when pricing digitals.

Spot Exposures: Understanding delta and higher order risks

**At 99% Correlation **- The option is just like a single asset up & out put as both the assets (having similar volatilities) move together. In such a case the steepness of the delta profile can be replicated by trading short term maturity put options.

*At -99% (minus 99%) Correlation:* Consider a case where spot of asset A is less than spot of asset B and asset B is at its initial level as shown in the figure below.

During simulations there are two possible cases -

**Case I**: Both move away from each other (i.e. spot of asset A decreases and spot of asset B increases)

Delta on asset B is close to zero and delta on asset A is close to minus one since asset A becomes the worst performer and moves in the money.

Case II: Both move towards each other.

Delta on asset A is close to zero and delta on asset B is close to minus one since asset B moves in the money and becomes the worst performer and asset A moves out of the money (assuming sufficiently high volatility levels). Hence net expected delta on asset A is close to minus half. (Refer Figure 2).

For higher spot levels (above the barrier) of asset A, the probability of asset A landing below the barrier decreases as spot of asset A increases. To get a clear picture of this, consider a case where spot of asset A is above the barrier level of 105% and asset B is at its initial spot level. Now there are two possible cases during the pricing simulations:

Case I: Both assets move away from each other - delta on asset A is close to zero and delta on asset B is close to minus one and further there is no possibility of structure knocking out.

**Case II**: Both assets move towards each other. Now during pricing simulations, assuming asset B rises by x%, then asset A must also fall by x% (since volatility of asset A equals the volatility of asset B). For the structure to knock out, both asset A and asset B should be greater than the barrier level, so asset B must move up by at least 5% from its initial spot level of 100% and asset A should correspondingly fall by 5%. This explains the negative peak of delta of asset A at 110% spot level (Refer Figure 2).

In case, spot of asset B moves to 101%, the negative peak of delta on asset A shifts leftwards to the 109% mark from the previous 110% mark (when spot of asset B was at 100%) (Refer figure 3). This symmetry is also consistent with the explanation given above. Also the magnitude of delta rises since probability of knock out is higher (as spot of asset B is nearer to the barrier).

At high negative correlation levels deltas change by a factor of nearly four or above depending on the volatility and spot levels. This can be a drastic change in risk in case of high notional trades. A practical hedge in this instance which greatly mitigates the problem is trading short term call spreads in the region of peak of delta.

**Figure 3**. *Delta of asset A v/s spot A for different spot levels of asset B, Volatility of asset A & asset B at 40%, Correlation at -99%. Observe the movement in peak of delta of asset A as spot of asset B changes.*

**Cross-Gamma**

Consider a case where both the assets are highly negatively correlated and asset B is at its initial spot level. To hedge the delta risk, a trader buys short term call spreads on asset A with strike levels near the peak of delta of asset A (110% level). Now if the spot of asset B moves to 101%, the peak of delta of asset A moves towards the barrier (to 109% as discussed in the previous section). The magnitude of peak of delta of asset A also increases, hence more call spreads around the 109% strike level need to be bought as compared to the number of call spreads that have to be unwound at the 110% strike level. Money is lost on rebalancing this hedge of call spreads since a trader has to buy 109% strike call spreads (in greater quantity) & sell 110% strike ones. The cost of rebalancing this hedge of call spreads will also depend on the spot level of asset A as elucidated below:
Case I: Spot of asset A is less than 110% (peak of delta point, Refer figure 3 & 4) and spot of asset B is at 100%. Now if spot of asset B moves by 1% to 101%, more money is lost on rebalancing the hedge of call spreads if the realized correlation is 100% as opposed to when its -100%. Also the magnitude of this loss is more when asset A is nearer to the 110% mark.

Case II: Spot of asset A is more than 110% (peak of delta point, Refer figure 3 & 4) and spot of asset B is at 100%. Now if spot of asset B moves by 1% to 101%, more money is lost on rebalancing the hedge of call spreads if the realized correlation is -100% as opposed to when its 100%. Also the magnitude of this loss is more when asset A is nearer to the 110% mark.

Similarly when spot of asset B moves down by 1% to 99%, peak of delta of asset A moves away from the barrier to 111%. In this case the trader makes money on rebalancing the hedge as he sells the 110% strike call spreads and buys the 111% strike ones. The difference in money made follows the same dynamics as described above for the up move of asset B. Further this also explains the cross gamma reversal around the 110% spot level of asset A (Refer figure 4). This reversal in cross gamma can be hedged by trading outperformance put spreads near the cross gamma reversal zone as the hedge replicates the cross gamma profile.

**Figure 4**. *Gamma & Cross-Gamma of Asset A v/s spot of asset A for different levels of correlation: spot of asset B at 100%, Volatility of asset A & asset B at 40%.*

In the next article we will try to intuitively understand the nature of other risks such as *vega, correlation skew* etc. and discuss relevant hedging strategies for the same.

**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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