The position is delta-hedged: a convertible with a 0.5 delta is hedged by shorting half the underlying shares per bond. The book then profits from rebalancing the hedge as the stock moves, from the bond's coupon, and from the short rebate, while direction is largely removed.
Residual exposures remain to credit spreads, interest rates, the borrow on the stock, and to implied volatility itself. Issuers have historically priced convertibles at a discount to theoretical value to attract these buyers, which is the structural source of the edge.
The strategy's failure mode is funding. In 2008 the trade lost heavily as credit spreads widened, short selling was restricted in some markets, and prime broker financing was withdrawn, forcing simultaneous unwinds. See convertible-bond and factor-crowding.
Related: convertible-bond, delta, volatility-arbitrage, short-rebate, prime-broker, factor-crowding