Consider a portfolio with exactly one share of a stock, value S. How do you protect it from small moves in S? That is, how do you keep the value of your portfolio stable when S changes by a small amount? The answer: You buy insurance. Translated: You take a position (usually, in an option) which moves in the opposite direction of S. Then when S moves one way, the option will offset it with a move in the other way. With the right number of units of the option, your portfolio will now be stable. But what is the right number of units of the option? There's where delta hedging comes in.

In an earlier post, we saw that delta is defined by:

$$\Delta =\frac{\partial f\left(S,\mathrm{...}\right)}{\partial S}$$

Writing this in the form of infinitesimals, we get:

$$\Delta =\frac{\delta f}{\delta S}$$

Transposing the infinitesimal of S with delta:

$$\delta S=\frac{1}{\Delta}\delta f$$

Or, on moving the right-hand-side term to the left:

$$\delta S-\frac{1}{\Delta}\delta f=0$$

In other words, when S changes by a small amount, a number of units of [-1 divided by delta] of f will ensure that your portfolio does not change in value. It is protected. (That's what the 0 means. The change in portfolio value is zero.)

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As an example, suppose you have one share of IBM and you want to delta-hedge it. You are thinking about hedging it with IBM call options each with a delta of 0.5. How many call options do you need? Answer: You need (-1/0.5) or -2. Thus, you need to short 2 call options. Easy, right?

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To summarize: To delta-hedge each share of stock that you hold, you need a position of [-1 divided by delta] units of the option. If the minus sign throws you off, think of the number of units as [1 divided by delta] and set the sign mentally to make the net change equal to 0. Thus, in the IBM example, you need to short the option because you need a negative sign to offset the positive position in the stock.

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