When you use a convertible hedge, you buy a convertible bond, which you can exchange under certain circumstances for shares of the company's common stock. At the same time, you sell short the common stock of the same company.
As in any hedge, your goal is to make more money on one of the transactions than you lose on the other. For example, if the price of the stock falls, you're in a position to make money on the short sale while at the same time knowing that the convertible bond will continue to be at least as valuable as other bonds the company has issued.
On the other hand, if the stock gains value, you hope to be able to realize more profit from either selling the convertible or exchanging it for shares you can sell than it costs you to have borrowed and repaid the shares you sold short.
There are no guarantees this strategy or any other hedging strategy will work, especially for an individual investor who faces the challenge of identifying an appropriate security to hedge and the appropriate time to act.