When Linear Hedges Work and Where They Can Fail
Summary
The document considers whether linear hedges, such as delta hedges, can meet most clients' needs without other derivatives. It notes that Black–Scholes theory can use a rebalanced linear hedge to replicate a nonlinear option payoff. For linear exposures, or when a hedge can be adjusted frequently enough, a linear hedge may also be a reasonable first approximation amid broader business and economic uncertainty.
The answers identify situations where this approach can fall short: sudden events may prevent timely rebalancing, currency movements can affect a multinational's sales as well as its exchange-rate exposure, and the hedge instrument may be correlated with another important variable. A local delta hedge can also become inadequate after a large price gap because option sensitivities change as the underlying moves. These are conceptual examples, not a comparative study; hedge sufficiency depends on exposure, market access, and how often positions can be adjusted.
Key ideas
- A rebalanced linear hedge can replicate a nonlinear option payoff in Black–Scholes theory.
- Linear hedges may suit linear exposures or risks that can be hedged with frequent adjustments.
- Gaps and unavailable liquidity can prevent timely rebalancing and make a local hedge ineffective.
- Business effects and correlations with other material variables can create risks a simple linear hedge misses.
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Full text
# Is a linear hedge sufficient for most purposes? # Is a linear hedge sufficient for most purposes? I was in a lecture of Bruno Dupire's when he said something along the lines of a linear hedge being sufficient for most purposes. He gave a counter example as well: a corporation producing something and multiple currencies, but it went over my head at the time. (I wish I had taken notes) So my question is: is it true that a linear hedge is sufficient for most clients, ie. there is no inherent need for derivatives (except futures/swaps)? If so, where can I read more about this and see a possible counter example? ## Answer by GodLovesATrier (score 2) https://quant.stackexchange.com/a/29510 There are many times when a linear hedge is sufficient, and indeed the Black/Scholes option pricing theory shows that a linear hedge can even hedge a highly non-linear payoff. However it's not hard to think of many situations where a linear hedge would not be sufficient eg: - company has high sensitivity to one-off event, market might not be open or liquid for hedge rebalancing - multinational company attempts to hedge currency positions but as rates of exchange change so do sales (2nd order effects) - any correlation between the price of the hedging instrument and another material variable If it's a linear risk or if the hedge can be rebalanced at any interval then probably linear hedges are sufficient, and to first approximation given the many uncertainties in the economy and business this might well be a good assumption. ## Answer by wchyk-cyw (score 1) https://quant.stackexchange.com/a/29627 When you talking about linear hedge you are talking about flattening the Delta on the Option. However, all greeks are local, as the underlying travels a long way away, your greeks will change too. Therefore, in a scenario where the underlying suddenly gaps up or down by a long way, your local linear hedge will have too little or too much effect. All optionalities of an Option are local phenomenons. A real world example would the EURCHF event in early 2015.
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