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By Lawrence G. McMillan

This article was originally published in The Option Strategist Newsletter Volume 12, No. 8 on April 24, 2003. 

The concept of “delta neutral” is an intriguing one – especially to traders who have had a hard time predicting the market or to those who don’t believe the market can be predicted (random walkers). The concept is even sometimes “sold” to novice investors as a sort of “can’t-lose” trading method, even though that isn’t true at all. While the idea of having a position that can make money without predicting the direction of the underlying stock seems attractive, in practice the strategy is difficult, if not impossible, to apply – at least in terms of keeping a position delta neutral.


The delta of an option describes how responsive it is to the movement of the underlying instrument. It is a number that ranges between 0 and 1 for calls and between –1 and 0 for puts. “Delta Neutral” positions can be constructed with any combination of options and/or the underlying instrument. One merely calculates the market exposure (“position delta”) of one leg of the position and offsets it with other, opposite, legs. For example, suppose that one is considering establishing a simple call ratio spread in a stock, using two options. The delta neutral ratio – how many to buy and how many to sell – is simply determined by dividing the deltas of the two options in question.

Example: XYZ: 48
  July 50 call delta: 0.50
  July 60 call delta: 0.25
Delta neutral position:
  Buy 1 July 50 call & Sell 2 July 60 calls

Since the ratio of the deltas is 2.0-to-1 (50 divided by 25), then the indicated spread is delta neutral.

A delta neutral position may be constructed out of any number of options and the underlying – from as few as 2 options as in this example to very complex positions, such as market makers might hold. No matter how many options are in one’s position, the position delta can be computed by multiplying the quantity of each option held by the delta of that option and summing over all the options held. That sum will generally not be zero, but it is a simple matter to neutralize the position by taking an offsetting position in the underlying stock. Here is more a complicated example, showing this concept:

Example: Suppose the following prices exist:
    XYZ: 48
    July 60 call delta: 0.25
    Oct 55 call delta: 0.40
    July 50 call delta: 0.50
And further suppose that this is a position that you hold:
    Long 20 July 60 calls
    Long 30 Oct 55 calls
    Short 10 July 50 calls
    The position delta of your holding can be
    computed by multiplying the quantity of each
    option in your position by its delta (and then
    multiplying by 100, assuming the option is for
    100 shares of stock). This computation is also
    sometimes called the “equivalent stock position
    (ESP)” because it shows how many shares of
    stock are equivalent to the given option position:

 Position   Qty   Delta   Position Delta (ESP)
  Jul 60    +20   0.25          + 500
  Oct 55    +30   0.40          +1200
  July 50   –10   0.50           –500
    Total                       +1200

    Hence the total “exposure” of all these options
    combined is equivalent to being long 1200 shares
    of XYZ. To neutralize this position – i.e., to
    make it “delta neutral,” one could simply short
    1200 shares of XYZ while still keeping the option
    holdings the same.

When a delta neutral position is established, the position has no delta – that is, it does not profit (on the very short term) if the underlying moves up or down. That’s all “delta neutral” really means. It does not mean, for example, that the position will profit no matter which way the stock moves. This simple statement may come as a big surprise to more novice traders who thought that delta neutral was more of a “lock” in terms of profits.

This misconception usually arises from a concept introduced in the first papers written about the Black- Scholes model by its authors, Fisher Black and Myron Scholes. They concluded that, if one bought or sold “mispriced” options and hedged them in a delta neutral manner, arbitrage-like profits would be made because of the mispricing. This concept is true, of course, but only in a theoretical sense. It is impossible to keep a position delta neutral at all times, especially when one considers the commission costs and bid-asked spread costs involved with “constantly” adjusting a position.

So, why bother with delta neutral at all, in the real world? It has some merit because it allows one to establish a position in which the trader truly doesn’t care which way the stock moves, but then – as the trade develops and the stock begins to move – the trader must take actions based on the price of the underlying stock in order to make money.

The purchase of a straddle is a perfect example. Most straddle buyers will attempt to find cheap options to buy and then set up the position in a delta neutral manner. That is, they don’t care which way the stock moves – as long as it moves somewhere. In reality, there are two ways to profit from a long straddle that has been established by buying “cheap” options: 1) the stock moves farther than the straddle price, or 2) the options become more expensive. The latter way is what Black and Scholes were talking about. The first way, however, is much more likely to be the deciding factor and so the trader must decide how he’s going to handle stock price movements.

Is he going to let the stock move and then follow the break-out by “going with the flow” – holding the long options as the stock breaks out? Or is he going to neutralize the position every time the stock moves towards its breakeven points? There is merit in both approaches, and we normally use a combination of the two approaches in our straddle positions. If the stock moves we often take a partial profit on the winning side (this is a means of “reneutralizing” if you want to think of it that way), but we also sell out the losing side and hold the remainder of the winning side, using a trailing stop (this is “going with the flow”).

Just how neutral is “Delta Neutral?”

The concept of delta neutral is a short-term one. As stated above, it really only means that the position will neither profit nor lose from stock price movements – but only over the very short term and for very small stock price movements. As soon as time passes or as soon as the stock moves far enough or if volatility changes – any or all of these things – the deltas will change, and the position will no longer be delta neutral.

In fact, what may be delta neutral might not even correspond with one’s own longer-term concepts of what delta neutral means. Let’s use a straddle purchase to demonstrate what I’m talking about here.

If we buy a delta neutral straddle, we can compute the upside and downside breakeven points. Since the straddle is neutral to begin with, one would think that the probabilities of reaching the breakeven points would be equal. In other words, what’s the point of being delta neutral if the position has a longer-term bias – if there is a greater chance of reaching one breakeven point than the other?

Example: Suppose the following data exist:
XYZ: 50 Volatility = 70%

Option        Price   Delta
Oct 50 call   9.80    0.60
Oct 50 put    9.30    0.40
(The above figures are theoretical price and delta
as determined by the Black-Scholes model)

A “delta neutral” straddle would thus require one
to buy 3 puts and 2 calls (the ratio of the deltas is
60/40, or 3/2). Buying 2 calls and 3 puts would
require a debit of 47.50 points for the entire
position. Thus, the breakeven points would be:

Upside breakeven: 73.75
Downside breakeven: 34.17

Probability of closing above upside: 24%
Probability of closing below downside: 26%

Probability of ever hitting upside: 46%
Probability of ever hitting downside: 48%

Standard deviations to reach the upside: 0.67
Standard deviations to reach the downside: –0.66

So these probabilities are fairly even – meaning that in this case, delta neutral corresponds with one’s longer-term concepts of what neutrality should be – an unbiased position, with about equal chances of hitting the upside or the downside breakeven points.

However, this example is dependent on the volatility of the underlying (and it’s also somewhat dependent on interest rates, as well). Without going into as much detail, if interest rates are higher and volatility is lower, there is a bias that creeps into “delta neutral,” so that the probability of hitting the downside breakeven is greater than that of hitting the upside.


The point is, once again, that delta neutrality is really only descriptive of short-term movements by the underlying. To attempt to apply it to longer-term projections or to expect a delta neutral position to “magically” be capable of producing profits would be wrong. What it is good for is allowing one to set up a position that has approximately equal chances of profit on the upside or the downside (in the case of a straddle buy, for example). This is important for many traders who are looking to buy cheap options without necessarily having to factor in an outlook for the underlying stock, index, or futures contract.


This article was originally published in The Option Strategist Newsletter Volume 12, No. 8 on April 24, 2003.  

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