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The Lazy Man's Guide to Delta-Hedging

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Something every options trader needs to consider.

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In my last post on this topic, Why Delta Hedging Matters, I argued that an essential aspect of options trading is hedging away unwanted risks. For most traders, the unwanted risk is usually to directional price movement, or delta risk. This issue is important whether you're short one call contract or managing a book of hundreds of positions. Again, the motivation for hedging away some or all of your delta exposure is if the purpose of your trade(s) is to gain exposure to changes in volatility, rather than price.

Delta hedging is the practice of buying or selling options and/or the underlying asset in order to reduce the net deltas in an open position. For example, let's say I thought implied volatility in August XLE options was overpriced and that I therefore wanted to be short that volatility. (I don't have a view on XLE either way at the moment; this is just an example.)

The August XLE 45 call is at the money and has a delta of about 50; if I were short 2 of those calls, my net delta would be about -100. In order to delta-hedge the position, I'd buy 100 shares of XLE at the current price. Let's say I did all of those things at once -- bought the shares and sold the calls all at the same time, giving me a nearly delta-neutral opening position.

Because all options have some gamma -- because the delta of an option changes in response to changes in the price of the underlying -- that initial hedge is only the first step. As the underlying moves around over time, the net delta of the position will change as well.

The obvious response to any changes would be to adjust the hedge continuously, buying and selling the underlying asset tick-by-tick in order to stay completely neutral at all times. But that's not possible for several reasons: Transaction costs would eat up any profits, contract sizes might prevent sufficiently granular hedges, and the discrete nature of market prices means a continuous hedge is as impossible as counting all the real numbers using only the natural ones. Hedging continuously represents one extreme approach; the other extreme would be to never hedge, or to do so infrequently.

So the key to dynamic hedging is to navigate the 2 extremes, avoiding undue delta exposure on the one hand while keeping transaction costs as low as possible on the other.

Here are some methods of delta-hedging to consider. I'll refer to a "book," which should be understood as the set of all option contracts and positions that a trader wishes to hedge.

1. Time-Based

Review the net delta exposure of the book at every time (t) and re-balance the hedge. The length of t will vary with the time to expiration of the options being hedged: A short out-of-the-money contract expiring 2 years from now may require very little attention at first -- only a weekly examination and monthly re-balancing, perhaps -- but a weekly review may be inappropriate when that same contract has only a month until expiration.

2. Price-Based

Since it's the changes in the prices of underlying assets that create the need for delta-hedging in the first place, there's something intuitive about flattening out the deltas of your book in response to price movement of a certain amount.

So the method here is to re-balance the hedge whenever the underlying price moves by a certain amount. How to determine that amount is, of course, the whole issue: Some traders rely on intuition and/or subjective technical analysis to determine appropriate price levels, but those approaches raise more problems than they solve. A better approach would be to define a price range in terms of recent historical or implied volatility.

3. Fixed Delta Bands

When the delta exposure of the book exceeds a given level, rebalance the hedge. The level of the bands -- the amount of deltas above and below zero that you're willing to tolerate -- will depend in part on the nature of your book and your risk tolerance.

A general rule of thumb here would be to determine a rough dollar amount that you're comfortable losing due to price movement. Then, determine the number of points the underlying is likely to move in a given day (assuming you're willing to adjust your hedge on a daily basis if needed), and divide your tolerable loss by that likely range to get your delta bands. In the example above, the expected daily range for XLE is about 1.10 points (about 2% at current prices).

So if your tolerable daily loss was $300, you'd want to keep your deltas for that trade roughly between -270 and 270.

4. Variable Delta Bands

The fixed-band approach isn't sensitive to changes in gamma or implied volatility. That's important because, all else being equal, a book with substantial short gamma has more risk: As the underlying moves further against you, your deltas will increase adversely as well.

By contrast, a book with substantial long gammas can afford to let the underlying run since it will become more neutral during an adverse move. The point here isn't that traders should be short or long gamma; it's that the hedging method employed should account for these risks wherever possible.

The chart below, from the Zakamouline article linked below, plots a delta band for a short butterfly spread with strikes at 80, 100, and 120. As you can see, the hedging bandwidth moves in line with the absolute value of the portfolio gamma, and reaches a fixed level as gamma approaches zero.



5. Academic Models

There's a substantial academic literature on optimal techniques for delta hedging, and some of it even deigns to consider whether the techniques under discussion could ever be deployed in the real world. (I'm calling this the lazy guide to delta hedging precisely because there are no formulas involved. Unless you're managing a very large book and/or institutional money, you needn't necessarily bother with the ivory tower approaches. If you do want to geek out, the bibliographies in the links above offer ample opportunity.) I found Zakamouline 2006 helpful, and chapter 4 of Volatility Trading by Euan Sinclair offers a thorough and accessible survey of this topic.

I've intentionally avoided endorsing one method over another, because, as is so often the case in options trading, the appropriate method for an individual will depend on her book, her experience, her risk tolerance, the nature of her strategy, etc.

The smart way to approach the topic of delta hedging isn't as a search for some one, right answer, but rather for the best fit given your situation. One issue I haven't discussed yet is the matter of books with multiple underlying assets: A trader holding options on Google (GOOG) and US Steel (X) shouldn't treat their deltas equally.

It's not a problem to hedge at the level of individual products, i.e. buying/selling the relevant shares of GOOG and X as needed. But for larger books, this quickly becomes less desirable. One solution to this problem is to beta-weight individual assets to some smaller set, i.e. weighting Apple (AAPL), Google, and Research in Motion (RIMM) options to the NASDAQ 100 and hedging with index products.

Inexperienced traders sometimes tend to pay too much attention to up-front transaction costs and minimize the costs of carrying unhedged risk. Whatever method you use, the most important thing is that you use one. Delta hedging is a fairly advanced topic, but it's something that every options trader needs to consider.
No positions in stocks mentioned.

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