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What is Delta?

You can interpret the delta number by way of the following simple example:

Set up as the default case a short maturity IBM call option which is approximately at-the-money.  For the Y-Axis, click beside Delta; and for the X-axis, click beside Asset Price.  Next, click on the button - Plot.

Now you see in the display window how the delta for this IBM call option changes as the underlying price for IBM changes.  

At the time of writing this lesson IBM was trading at 123 1/4.  The delta of the approximately at-the-money short maturity call option is 0.427.  

This means that for the case of one option on one IBM share, if IBM increased by $1 the approximate  increase in the option price is $0.427.  

In the US markets one option contract for IBM controls 100 shares of IBM. Thus, if IBM stock increased by $1, delta gives you the approximate change in the option contract's value, which is 100*0.427 = $42.70 for the current example.  It is approximate because delta is a linear approximation to the change in the option value.

The value of delta is that, at a glance, it gives a trader important hedging information.  For example, a delta neutral position is a position that has delta equal to zero.  That is, if you add up the deltas for each component of the position the sum would equal zero.  

Suppose a trader has written 25 at-the-money call options on IBM.  In the above example, this trader is now exposed to price increases in IBM.  This is because if IBM increases by $1, the option holder may exercise the contract against the option writer for $42.70*50 = $2135 more than just prior to the increase.

In other words, an option writer hopes that the stock declines in price, or at least does not increase in price by more than the premium collected from writing the option.  

In summary, the option's delta is a number that gives by how much the option price is predicted to change, given a small change in the underlying asset price.  Thus, delta is a measure of the option's exposure to the risk that the underlying asset's price changes.

The sign of delta (positive for call options and negative for put options) gives you the direction of the exposure. That is, a call option increases in value if the underlying asset price changes; a put option decreases in value if the underlying asset price changes.

Delta and Hedging 

Suppose the writer wants to hedge all risk.

How many IBM stock would the writer need to hold?

To answer this question we will construct a delta neutral position.  A delta neutral position is a position that has a delta equal to 0.

For example, suppose we have written 50 call option contracts.  Our current position delta is -50*0.427 = -2135.  

It is minus because the delta of a call option is positive, but writing a call option is equivalent to shorting the call option.  As a result, we have a negative position number multiplied by a positive delta number.

To make our position delta equal to zero we need to add 2135 to this number to make it zero.

How can we do this?

Recall that delta is a number that measures by how much the price changes, given a small change in the underlying asset price.  Thus, there is nothing special about options and the term delta.  We can talk about the delta of the underlying asset itself and we can even talk about the delta of a riskless bond.  

The riskless bond has a delta equal to zero because the riskless bond price does not change if the underlying asset price changes.  On the other hand, the delta of the underlying asset equals one because, by definition, if the underlying asset price changes by one - then the underlying asset price changes by one!

Lets now re-consider the above IBM example.  The option's delta is 0.427 and IBM's delta equals 1.  

The answer, therefore, to the question 'How many IBM stock do we need to hedge our option writing risk?' is easy:  it must equal the delta of our option position.

That is, we would hold a long position 2135 IBM stock to hedge this price risk.

This is because the delta of writing 50 call options = -2135, and the number of IBM stock is n*IBM Stock delta = 2135.  This implies that n = 2135 because the delta of the underlying itself is always equal to 1.

Delta is not Constant

One problem that immediately arises when applying delta to a hedging strategy is that delta changes with a change in the underlying.  When this happens the number of units of the underlying must also change.

That is, maintaining a delta neutral position requires re-trading.  

You can see how delta is predicted to change by clicking on the graph in the display window when you plot delta against the underlying.

For example, suppose IBM stock increases to $131.48.  Now the delta of the call option has increased 0.591.  This means that our hedger now needs to increase the number of the underlying IBM stock required to maintain a delta neutral position.

In this case it is:  0.591*100*50 = 2955 or an additional 820 shares over the original example.

Delta Interpreted as a Hedge Ratio

From the current example you can see why delta is referred to as a "hedge ratio."  It is a number that you can immediately use to determine how many of the underlying asset must be held to hedge yourself against the risk that the underlying asset price changes over the life of the option.   Of course, the underlying number must be opposite in sign to make the position delta equal to zero (i.e. be delta neutral).

Option sensitivities lets you ask and answer important "what-if" questions before changes occur.

Definition of Delta

Formally, delta (d) is the change in the option price with respect to changes in the underlying asset price.

Graphical Interpretation of Delta

In the display window now plot Option Value against the Underlying Asset Price.  Formally, delta is the slope of this graph when evaluated at a particular point.  This is why it is initially fairly flat (i.e. 0) then rises sharply until it approaches 1.  It then becomes flat again as the slope gets closer and closer to 1.  

To see how this works you should first plot Option Value against Asset Price, and then Delta against Asset Price.  You can observe how these two graphs are related.

Relating the Graphical Interpretation to the Terminal Option Value

Recall that the terminal value of a call option is Max{0, S-X} where S is the underlying asset price, X is the strike or exercise price, and Max is an operator.  That means if S-X is negative, the terminal value equals 0; otherwise it equals S-X.

From this definition of the terminal option price you should convince yourself that the delta of a call option (when defined relative to 1 unit of the underlying asset) can never exceed 1.  This would imply that the option price increases by an amount that is greater than the amount by which the underlying asset price changes.  Clearly, this is impossible because X is a constant (contractually specified).

Similarly, the change in the call option price can never be negative given a change in the underlying asset price because at the minimum it stays at zero.


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