There are quite a few problems with delta hedging, on both the practical and the theoretical side. In practice, hedging must be done at discrete times and is costly. Sometimes one has to buy or sell a prohibitively large number of the underlying in order to follow the theory. This is a problem with barrier options and options with discontinuous payoff. On the theoretical side, the model for the underlying is not perfect, at the very least we do not know parameter values accurately. Delta hedging alone leaves us very exposed to the model, this is model risk. Many of these problems can be reduced or eliminated if we follow a strategy of static hedging as well as delta hedging; buy or sell more liquid traded contracts to reduce the cashflows in the original contract. The static hedge is put into place now, and left until expiry. In the extreme case where an exotic contract has all of its cashflows matched by cashflows from traded options then its value is given by the cost of setting up the static hedge; a model is not needed. (But then the option wasn't exotic in the first place.)
In incomplete markets you cannot eliminate all risk by classical dynamic delta hedging. But sometimes you can superhedge meaning that you construct a portfolio that has a positive payoff whatever happens to the market. A simple example of this would be to superhedge a short call position by buying one of the stock, and never rebalancing. Unfortunately, as you can probably imagine, and certainly as in this example, superhedging might give you prices that differ vastly from the market.
Often what causes banks, and other institutions, to suffer during volatile markets is not the change in the paper value of their assets but the requirement to suddenly come up with a large amount of cash to cover an unexpected margin call. Examples where margin has caused significant damage are Metallgesellschaft and Long Term Capital Management. Writing options is very risky. The downside of buying an option is just the initial premium, the upside may be unlimited. The upside of writing an option is limited, but the downside could be huge. For this reason, to cover the risk of default in the event of an unfavourable outcome, the clearing houses that register and settle options insist on the deposit of a margin by the writers of options. Margin comes in two forms: the initial margin and the maintenance margin. The initial margin is the amount deposited at the initiation of the contract. The total amount held as margin must stay above a prescribed maintenance margin. If it ever falls below this level then more money (or equivalent in bonds, stocks, etc.) must be deposited. The amount of margin that must be deposited depends on the particular contract. A dramatic market move could result in a sudden large margin call that may be difficult to meet. To prevent this situation it is possible to margin hedge. That is, set up a portfolio such that a margin calls on one part of the portfolio are balanced by refunds from other parts. Usually over-the-counter contracts have no associated margin requirements and so won't appear in the calculation.
Crash (Platinum) hedging
The final variety of hedging is specific to extreme markets. Market crashes have at least two obvious effects on our hedging. First of all, the moves are so large and rapid that they cannot be traditionally delta hedged. The convexity effect is not small. Second, normal market correlations become meaningless. Typically all correlations become one (or minus one). Crash or Platinum hedging exploits the latter effect in such a way as to minimize the worst possible outcome for the portfolio. The method, called CrashMetrics, does not rely on parameters such as volatilities and so is a very robust hedge. Platinum hedging comes in two types: hedging the paper value of the portfolio and hedging the margin calls.
References and Further Reading
Taleb, NN 1997 Dynamic Hedging. John Wiley & Sons Ltd
Wilmott, P 2006 Paul Wilmott on Quantitative Finance, second edition. John Wiley & Sons Ltd