**Barrier shift** is mainly used as a pricing mechanism for down-in-puts and other barrier structures to go conservative on the price of the structure over its model price. It is necessary in the pricing mechanism to account for the real world difficulty in executing large deltas at barrier that the model does not consider.

To elucidate, while pricing a DIP the model assumes that the entire delta can be executed at the barrier price B when the barrier is hit. But in the markets it’s seldom possible to execute the whole delta (i.e. sell shares in this case when trader is long a DIP) at a particular price & hence if the trader ends up executing the delta below the barrier price he/she makes a loss. To compensate for this loss a trader gives a barrier shift. The barrier shift is also the parameter in down an in barrier options that can be played around with to go conservative or aggressive on the price, with all other parameters being directly implied by the market.

Here, it is proposed to study the behavior of the option around the barrier, aiming to track the various factors that affect the barrier shift and also the effect of each on the price of option. Finally the intent is to get to a systematic way and create a model to assess our barrier risk and also incorporate this barrier risk into the option price.

To understand the need for this shift, let us first look at the behavior of a dip (or wodip) around the barrier.**I. Profile: To start with, the let us look at the profile of a DIP structure, or simply the premium of the structure at different spot levels.
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The above graph shows the profile of a dip, this is taken at low time to maturity and at low volatility levels. The higher the vol and time to maturity, the lower the effect seen of the American barrier.

II. **Delta Profile:** The delta profile of the DIP, before the barrier is hit can be divided into three major regions of delta values with respect to spot levels: Until the barrier point of 70%, from a value of 0% the value of the delta is ~ -1. The delta sharply dips to a peak delta value. And then starts increasing slowly towards a value of 0. The region where the delta dips towards the peak delta would be a negative gamma region. At higher values of the spot, the delta moves towards 0, and is constant there. After the barrier is hit, as expected the delta resembles the same profile as that of a put

The above graph generated from 50 closely taken values illustrates the point where at spots greater than 75% the delta is 0. As for the premium profile itself, the vol levels taken have been low so too the time to maturity.

Based on the above preliminary analysis & common sense, the barrier shift of an option should depend on the following factors

1. Trade Notional and Daily volume of shares traded

2. Distance between Strike & Barrier

3. Implied Volatility at the barrier (not ATM volatility)

4. Time to maturity

To take into account the above four factors into workable parameters for the barrier shift, two independent variables have been devised that sum up to give the net shift measure:**1. Liquidity and peak delta based barrier shift2. Volatility based shift**

The calculation of each of the above two is discussed as below.**A) Estimating the Barrier shift based on liquidity factor:**

The methodology for finding the liquidity factor would be based on two major factors, that is, the volume on market of the underlying’s and peak delta based on the product itself. The liquidity factor which is a measure of how easy it is to execute the delta is simply the ratio of the two. Let us look at each of the factors for this in further detail.**Volume estimation:** Incase the stock is illiquid throughout the year, there is a huge liquidity risk & trader might not be able to execute the orders at the barrier price, hence volume is an important factor that needs to be estimated.**Critical Volume = W1×(Average of weekly minimum for last 252 trading days) + W2×(X% percentile value: below which a certain percent of observations fall)**W1, W2 & X% are adjusted by the trader. W1 & W2 have been set to 50% giving equal weight-age to both the factors). X has been set to 10% which gives a number below which 10% of the 1 year (252 trading days) observations fall. The adjustments are based on how conservative or aggressive the trader wants to be, to account for the inability to execute deltas. Since it takes weekly minimum volumes the critical volume factor adequately captures the tough periods in which it might be difficult for a trader to execute big delta orders. In case of harsh conditions (with low volumes, like currently in the market), the trader would adjust X to a lower value, say 5%.

**2. Delta Estimation:**indicator of expected delta to be executed on hitting the barrier.

Given that the barrier can hit anytime during the maturity of the DIP, the total maturity has been divided into certain number of periods & deltas are calculated near the barrier for those maturity periods. Expected delta can be calculated as the average of the deltas to be executed for different maturity periods.

The problem with taking an overall average of the deltas is that it gives a very conservative value of the expected delta because deltas take very high values incase barrier is hit near the maturity of the option. To get around this problem an average of the last N period deltas should be considered. This N can be decided by the trader. Preferably it should be greater than 8 so as to cover the deltas of the last two months since last two months show markedly higher delta as compared to previous months & for the same reason the 2 month period has been divided into 8 maturity periods as shown in the below figure.

The above figure shows the $ deltas to be executed in case the barrier is hit in the corresponding maturity period. (Trade Notional is 10 m EUR, Maturity 2 years). The third row shows the average delta for the last N observations which differs markedly from the overall average. N is 8 in this case.

**3. Liquidity factor**: Liquidity factor has been defined as = **Estimated delta / Critical Volume**. This factor can be used as a benchmark for the amount of shift that should be given to the barrier. It’s intuitive that the shift should be an increasing function of liquidity factor. The function can be estimated by back calculating the barrier shift for different Down & in puts cleared in market trading. Such function would give an idea of how market is pricing the barrier risk. This implied function can be modified in case we believe that market has mispriced the barrier risk & then could be contributed with suitable bumps.

**B) Barrier Shift based on implied volatility at barrier: **To account for this particular factor, we need to take the sigma at the barrier, which will be implied from the market.

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**Conclusion: **as discussed above we have successfully accounted for the factors affecting the barrier shift. The net barrier shift comprises of two parts; one part accounts for Liquidity & Expected delta to execute on hitting the barrier, the other part is volatility based shift to account for the one day movement around the barrier & the distance between peak delta point and the barrier level

gr8 work .. keep it up

Comment by saurabh agarwal — October 14, 2009 @ 12:22 pm |