Despite the fact that the Black-Scholes model was built on a platform that one is able to continuously manage a delta-neutral portfolio, there are practical shortcomings associated with this strategy. Examples of these are discussed in the following subsections.

Buy High and Sell Low

In the examples discussed earlier, the hedger shorted an option and then went on to delta-hedge the risk by buying (selling) shares when the share price went up (down). This is also referred to as the buy-high-and-sell- low strategy. The consequence of this is that the risk-manager would be exposed to a lower hedge P&L if the stock price path turns out to be more volatile than what was anticipated when selling the option.^{[1]} To reduce this, one in practice would need to create a strategy that would be robust to changes in delta as the underlying stock price changes. More precisely, by immunizing against the changes in sensitivity of option delta to the underlying stock price movements, one can avoid the need to constantly delta hedge. Called gamma hedging,^{[2]} practitioners use this approach to reduce the frequency of trading associated with rebalancing of the deltas. Denotingfor a vanilla European-style call and

for a vanilla European-style put, it can be easily shown that for dividend paying stocks,

(7.2a)

(7.2b)

[1] The intuition underlying this stems from the fact that the higher the volatility, the greater the premium. Hence, by selling an option for an implied volatility that turns out to be lower than that realized during the life of the option, the hedger incurs a lower hedge P&L. The converse of this happens when the hedger goes long an option. More precisely, the hedger would want the realized volatility to be higher than the implied volatility paid. In this instance, ignoring transaction costs, to generate as much profit as possible, the hedger would buy-low-and-sell-high on every trade in the hope of generating as much revenue as possible during the life of the option.

[2] While delta hedging involves the use of stocks, gamma hedging involves the use of stock options (simply because stocks only have delta and stock options have both delta and gamma). Having said that, it is imperative for the reader to note that while it is theoretically possible to neutralize deltas using stock options as opposed to using stocks, this is not done in practice simply due to the transaction costs and the fact that the consequence of such a hedge might increase the total gamma/vega/rho of the portfolio.

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