Over the past 10 years, the market for equity derivative-linked structured products in the U.K. has grown dramatically in size and sophistication. In the tranche-based market, highly structured products are becoming increasingly commonplace. A majority of products are designed to pay an above-market coupon to the investor in return for assuming certain market risks. Many products employ structural enhancements either to provide additional safety to the investor or merely to generate additional up-front premium. Individually, none of these features are a problem to price or to hedge. However, by combining, for example, a bet option, a lookback feature to the price setting, and a knock-in on the strike, the pricing and hedging challenges are significantly greater. Finally, combining these features in a transaction of significant size, such as GBP50-100 million (USD75-149 million) presents significant exposure to market liquidity. Highlighted below are a number of the enhancements utilized in the U.K. market.
EUROPEAN BET (DIGITAL) PRICING
U.K. over-the-counter equity derivative products have often been based on annual observation periods (cliquets), where a bonus is paid should the index be higher at the end of the year than at the start. This creates a hedging problem due to the discontinuity of the payoff profile of say, either 10% paid when the final level is 100.01% or zero when the index finishes at say, 99.99%. Such a large payoff differential for a small underlying difference implies significant gearing and hence a comparable short gamma position for the option seller (Figure 1a).
To mitigate this risk the digital option may be priced as an in-the-money call spread, for example, 98%100%, with a lower gearing factor (defined as payoff/spread differential) than that implied by the theoretical price. Such a pricing methodology smoothes the delta hedging, thus dampening the short gamma position (Figure 1b).
The appropriate strike spread differential is dependant on the liquidity of the underlying market. An illiquid underlying would require a wider spread as the market would move as the derivatives house executed its hedge. For example, to cover the geared delta to hedge an underlying move from 98% to 99%, a derivatives house may move the market upwards though the 100% level without being able to purchase the required extra delta for the 'gap' move from 99% through 100%.
A further factor for consideration is the underlying volatility skew. The wider the spread the higher the differential in volatility between the short 98% and long 100% call options and consequently, the higher the premium. Given that the true volatility exposure lies at the lower volatility level of the 100% strike, pricing incorporating the 98% strike volatility is more conservative for the seller (that is, an "overhedge").
BARRIER OPTIONS
A common variation of the vanilla option employs a knock-in barrier level, for example, a vanilla put that knocks in if the level of the index trades below 80% of the initial level at any time throughout its life. Such an option is known as a barrier option, and shares strong similarities with a bet. Firstly, consider a European barrier put option with a strike of 100% that only exists if the underlying market is below 80% at maturity (Figure 2a). This can be modelled as a bet option paying 20% if the underlying closes below 80%, plus a vanilla put option struck at 80%. If the index is trading near the barrier level at maturity, the gamma position is similar to that of the bet option described above. In other words, the delta position requires the derivatives house to sell the underlying market. The barrier option would be priced similarly to the bet option: a put spread with the higher strike at 80% and the lower at, say 78%, geared by the payoff divided by the strike differential.
Barrier options are typically transacted as American style (continuous) rather than as European style (discrete) (Figure 2b). The option combination to provide such a profile is therefore long a vanilla at-the-money put and a short knock-out put option. The strike of the knock-out put option would again be lowered below the 80% barrier level for the same reasons as the bet.
The knock-in feature reduces the premium received for the put option, thereby reducing the above-market coupon payable. To compensate for the reduction in coupon additional features are added.
GEARING
It is possible to increase the gearing of the underlying put options by changing the vanilla put option (strike 100%) into a put spread (strike 100%-50%) with twice the gearing (the maximum loss remains at 100% but occurs if the reference asset falls by only 50%).
LOOKBACK OPTION
This feature picks the final maturity value as the lowest level of the reference asset during a final reference period e.g. the last calendar month. The effect of such a feature is to squeeze the terminal price distribution (fatter tails) and so increase the volatility of the option and so the premium (Figure 3).
Many of these features may be found in the Individual Saving Account and Single Premium Life Bond products issued to the retail market in the U.K.
This week's Learning Curve was written by Adam Habib, v.p., and John-Paul Booth, associate, on the U.K. marketing team for structured derivatives at Credit Suisse First Boston in London.
