Where's my delta?

15 December 2011

By Jor Molchan, director, Sapient Global Markets,
Fabrice Douglas Rouah, director, Sapient Global Markets

The increased use of over-the-counter (OTC) derivatives, such as barrier options, is presenting new challenges for writers of these options who wish to hedge their positions.

Hedging involves eliminating risk in one investment by taking an offsetting position in another. For example, shorting index futures contracts with the equivalent, notional principal to an underlying index investment or buying put options with a strike close to the current market price in a quantity matching the portfolio will each hedge against a decline. In both of these cases, matching the hedge size to the portfolio size minimises the need for adjustments. Dynamic hedging, by comparison, refers to a hedging strategy that requires periodic adjustment. The benefit of this additional effort is that the hedge can be established and maintained with a lower cost, less upside, compromised or both.

Several strategies can mitigate the risk the seller of an options contract faces of a loss from the exercise of the option. The first is to buy exactly the number of underlying shares or what is known as the covered strategy. For example, if one vanilla call is sold, the writer immediately buys 100 shares. Then if the call expires in the money (ie, when the underlying instrument price surpasses the strike price), the writer sells the shares at a profit, but hands over the profit to the buyer. This strategy is not optimal for many reasons. First, there is the opportunity cost of tying up cash to buy 100 shares. Second, if the call expires out of the money (ie, when the strike price is higher than the price of the underlying asset), the writer will be freed of his obligation on the call but will hold shares that have depreciated in value. Finally, if the call expires in the money, a better strategy for the writer would have been to not write the call but to buy the shares as an investment. The second strategy is to buy 100 shares as soon as the call goes in the money, then sell the shares immediately when the option goes out of the money, and do this continually over the life of the call. However, this stop-loss strategy could add substantial churn and generate small but regular losses. The preferable strategy, called delta hedging, is to hold only the optimal number of shares at every instance of the call option’s life.

Delta represents the amount of the underlying security to be held during the call’s life in order for the writer’s portfolio to be insensitive to changes in the price of the underlying. Delta is not fixed but changes in response to changes in the underlying price. This change is measured by gamma. When delta changes rapidly, gamma is high and the portfolio must be rebalanced more often, which can lead to high transaction costs. The typical remedy is to gamma hedge the portfolio so that it is insensitive to changes in delta. Gamma hedging eliminates the constant adjustments required by pure delta hedging. Proper risk management of options therefore depends on how well delta, gamma, and other option price sensitivities are calculated. For vanilla options, delta and gamma are usually straightforward to calculate and well behaved.

However, barrier options present an entirely different challenge. Barrier options are similar to vanilla options, except that a trigger is specified so that the option can either cease to exist (a knock-out barrier), or can become alive (a knock-in barrier) when the stock price crosses the barrier. A barrier option that is knocked out becomes void. Thus, a trader writing barrier call options can potentially increase volume by writing contracts more frequently.

One appeal of barrier options is their ability to provide a more finely matched and targeted strategy. Vanilla options are useful when investors anticipate either upward or downward future movements in the stock price, but barrier options are preferable when these anticipations are more detailed. An investor who is bullish overall can buy a vanilla call, but one who is bullish and expects a floor on prices can buy a down-and-out call. Barrier options do not force investors to pay for price scenarios that they believe to be improbable, and barrier options are usually cheaper than their vanilla counterparts. Thus, barrier options are popular because they can better match investor beliefs about future price movements than vanilla options.

There is a cost beyond the lesser premium obtained from selling knock-out calls as compared to standard calls, however, as delta and gamma can behave poorly for barrier options, which makes dynamic hedging of these options both impractical and costly. In particular, the behaviour of delta and gamma becomes increasingly erratic as the stock price approaches the barrier. This is especially true for reverse barrier options, which are barrier options for which the barrier is located in the region of the stock price when h the option is in-the-money region. The erratic behaviour makes the risk management of barrier options much more difficult than their vanilla counterparts. While market makers have welcomed the growth of OTC derivatives markets and barrier options, they in particular are being faced with ineffective delta and gamma hedges in these markets.

The difficulty of hedging barrier options has prompted researchers to design static and semi-static hedging strategies specifically for barrier options. These strategies require little or no rebalancing, so liquidity and transaction costs associated with dynamic hedging are much less of a consideration. Delta hedging is appropriate for hedging vanilla options, but in its basic form it is usually ineffective for hedging barrier options. Hence, while the price and market opportunity for barrier options is attractive, writers of these options must exercise care because the hedging strategies they require are completely different.

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