One of the difficulties in managing equity and foreign exchange barrier option books is handling potentially unstable gamma positions when the spot is near the barrier and/or the option near expiry. The figures below illustrate such an example using a down-and-out put. The option matures in one month, is struck at-the-money, and has a barrier at 80% of the strike. The top figure plots the option price versus spot and the large curvature near the barrier indicates a large gamma. This is clearly seen in the lower figure, which illustrates the corresponding gamma. The gamma profile near the barrier is quite unstable and the delta hedge will be inefficient and often costly. This is one of the main barrier risks. One technique to manage such risks is to use exponential soft barriers.
The Exponential Soft Barrier Approach
The exponential soft barrier approach serves to "soften" barrier risks by splitting the hard contractual barrier into soft barriers that decay exponentially. One can also apply exponential decay boundary conditions near the barrier within the partial differential equation solver to achieve a similar effect. The barrier option can then be booked as a weighted average of soft barrier options and managed and hedged using the exponential soft barriers. This allows for more conservative booking. Thus, the soft barrier approach includes some reserves for the barrier risks. Let us examine two barrier option examples. Both have a maturity of one month, and at-the-money strikes. A representative set of FTSE-100 market data including a skew implied volatility surface is used in the examples.
Down-and-Out Put
Suppose we short a down-and-out put with a hard contractual barrier at 80%. We can split the hard barrier trade into several exponential soft barriers. The first soft barrier is at 80%, and the others range from 80% to 70%. The weighted average of the soft barriers forms the soft barrier booking. With soft barriers below 80%, the option is less likely to be knocked out than with the hard barrier at 80%. Hence the soft barrier price is more expensive. As we are short the position, the soft barrier booking is more conservative. The figure below illustrates this fact using a smile/skew partial differential equation barrier option model for the down-and-out put. Comparing the overall price profiles, we see that the soft barrier profiles are broadened. If the soft barrier profiles are used to hedge, once
the hard barrier is touched, the real barrier position jumps down from the soft barrier curve to the hard barrier curve. This jump theoretically represents a hedging windfall, which is the implicit reserve in the soft barrier booking.
The figure below shows the comparison of gamma profiles of the hard and soft barrier for the down-and-out put. When spot is near the barrier, the hard barrier gamma shows an unstable nature characterised by a large jump, while the soft barrier gamma is more stable and manageable.
Down-and-In Put
Suppose we buy a down-and-in put with a hard contractual barrier at 80%. We could choose one soft barrier at 80% and decay the rest exponentially to 70%. The soft barrier booking is the weighted average of these soft barrier options. With the soft barriers below 80% the down-and-in put is less likely to be knocked in than with the hard barrier at 80%. Given that we are long the option, the soft barrier booking is more conservative. The figure below shows the down-and-in put price versus spot using a smile/skew PDE barrier option model for both hard and soft barriers. Comparing the overall price profiles, the soft barrier profiles are smoothed around the real barrier (at 80%) region. This represents a less sudden price
change, and hence a smoother delta, more stable gamma and reduced barrier risks. If traders follow the soft barrier profile to hedge, once the real barrier (at 80%) is hit, their position jumps up from the soft barrier curve to the real barrier curve. This jump theoretically represents a hedging windfall, which is the implicit reserve in the soft barrier booking.
The figure below compares the gamma profiles of the hard and soft barrier for the down-and-in put. As can be seen in the figure, when the spot is near the barrier, the hard barrier gamma profile has a large jump and is unstable, while the soft barrier gamma profile is much more stable and manageable.
Conclusion
The exponential soft barrier methodology is an effective way to manage equity and foreign exchange barrier option books. The methodology can be refined to ensure a very smooth gamma profile. There are other soft barrier approaches--one could, for example, use different barrier-splitting schemes or indeed alter the barrier duration along the time axis. All these approaches serve the same purpose of softening and reducing the barrier risks. The extra premium required for adopting the soft barrier approaches at the outset is not too much. When the extra premium is included in the initial price, it effectively represents the reserved hedging cost.
It is important to remember that soft barrier methodology serves only to soften the impact of the barrier risks. They do not eliminate barrier risks altogether. In practice, the size of the trade, the volatility of the underlying and liquid of the hedging instruments all contribute toward how soft barriers are determined and how barrier risks are managed.
This week's Learning Curve was written by Dong Qu, head of equity derivatives quantitative research and applications at Abbey National Treasury Servicesin London.
