An Independent Delta

Posted January 14th, 2011 by Josip Causic in Trading & Analysis

An Independent Delta

In this article, I will discuss the concept that Delta is the same regardless of which side of the trade a trader is on.

In one my classes, as I was covering the concept of Delta value, some of the students really got confused. As I worked on explaining the concept, I created the following table - see Figure 1 below.

However, prior to diving into the explanation of Figure 1, I will briefly define Delta, for it has multiple levels of meaning. One of the most commonly quoted definitions is that Delta is a measure of the change in an option's value with respect to a change in the price of the underlying; the percentage of any stock price change which is directly related to the option price. The second: As a Hedge Ratio, Delta tells how many underlying shares of the options are required for a neutral hedge. The third and last: As an Approximate Measure of Probability, how much of a chance does the option have of expiring ITM (in-the-money)? The second meaning of Delta is not a part of the discussion of this article, and it can be learned in-depth by clicking here: Delta as a Hedge.

Now, let us turn our attention to the figure below. I have selected a call with a Delta of 0.86 to explain how Delta works. If a buyer purchases a particular call with a Delta of 0.86 for a premium of $4.25, then the trader's account will be debited because the trader has paid money out of his account for this buying transaction. In Figure 1 below, this is visually displayed in the middle column. The buyer has BTO (bought to open) an ITM (in-the-money) call.

In the right column, I have the very same call with the very same Delta and the premium value. A quick disclaimer: We are assuming the premium of $4.25 is the mid-price between the Bid and Ask, and to keep things as simple as possible, we are keeping it fixed. The seller's action involved STO (selling to open) the very same ITM (in-the-money) call.

So far, this was clear to all the students in the class. The following rows, Increase of Delta as well as the New Premium, were still somewhat clear. When the underlying asset goes up one whole point, our call is gaining 0.86 cents in value on top of the existing $4.25.

Figure 5: Increase in the premium of the 55 call
without the change in implied volatility

The new premium value of 4.25 + 0.86 becomes $5.11.

Yet, what confused the students is not the math behind this, but what the outcome was to each side. To the buyer, the underlying going up in value was a good thing since they purchased a deep (in-the-money) call because of a bullish forecast. As the product increased in value by one whole point, the buyer gained not a 100% of the move, which would have been penny for penny, but only about 86% of the same one dollar move.

At the same time, the seller of that very same deep (in-the-money) call has received the opposite effect. The seller sold with the anticipation of seeing the sold call expiring OTM (out-of-the-money). This result could not take place due to the fact that the instrument has increased in value by a whole point; therefore, the seller needs to buy back the obligation for more money than what they had sold it for. Figure 1 shows that the call was sold for $4.25 and the amount was colored in green for it was a credit. The closing action involved BTC (buying to close) at the higher premium price from what had been sold, namely at $5.11 colored in red. The difference of the selling price and the purchasing price to close is exactly 0.86 cents, which is the Delta value.

Having seen both sides of the trade, the buyer and the seller, the students were able to grasp the main difference between the two. To make the things even simpler, at the bottom of the table, I have included a row that specifies what the trader's goal was. The buyer wanted to see the option increasing in value, and that is exactly what has taken place. Whereas the seller did not want to see an increase but a decrease in the premium and obviously, that did not happen. In terms of probability, this outcome might have been obvious from the beginning had the trader really understood that Delta is also an approximate Measure of Probability; how much of a chance does the option have of expiring ITM (in-the-money)?

The last row of Figure 1 shows that the buyer has an approximate probability of seeing his option expiring ITM by 86%, due to the Delta being at 0.86, while the seller had the flip side of this, which means that the seller did not want to see the premium increase but decrease. To better understand this concept, which is a different topic, I suggest the reader click here: Delta Part 2: Delta as an Approximate Measure of Probability.

In conclusion, in this article, I have discussed the concept of Delta being the same regardless of the trader's action. Whether the trader is a buyer or a seller has absolutely no impact on the Delta's value since it remains the same. One can be on either side of the trade while the Delta is the Delta no matter what. The Delta is truly independent of a trader's action of buying or selling. Know your Greeks well and as we say at Online Trading Academy, Plan the Trade and Trade the Plan. Have green trading.

- Josip Causic

"