# Options Trading IV, Velocity of the Delta

*This publication is part of a series of posts on the Greeks. These posts are a short summary of a part of the content of my book How to Calculate Options Prices and Their Greeks. *

In the previous two articles, we’ve discussed how the Delta of an option changes in relation to Volatility and Time to Maturity. As a conclusion there were two rules of thumb (for around the at the money options):

The lower the Volatility the faster the delta changes:

The shorter the Time to Maturity the faster the Delta changes:

So how does this knowledge contribute to trading options?

Let’s talk briefly about Delta neutrality:

A private investor being long a call option is not bothered by the Delta of his position, most probably he is just aiming for an increase of the market and to earn a profit on the back of higher levels in the underlying. He will not hedge his Delta long position (by selling stocks or Futures), because it will only result in a lower profit (opportunity loss) when the market moves in his anticipated direction.

A market maker on the other hand will hedge his Delta to create a Delta neutral portfolio. He’s earning a profit on the back of the bid- ask spread he is applying in his markets, his target is not being impacted by the market direction.

Some traders don’t use options for a directional play in the underlying (a Delta play), but for generating a profit on the back of other Greeks like Gamma, Vega and Rho. These traders can trade their strategy having a Delta neutral position.

It is this third group who is the most interested in the (velocity of the) Delta changes. Let’s assume a trader who is long 100 Call options (100 underlying units per option), 50 Strike, Underlying at 50, Volatility at 10% and Maturity 1 year. We will monitor his hedges being done in four very volatile days.

On day one the market will move from 50 to 54.

On day two, the market will retrace to 50

On day three, the market will move from 50 to 48

On day four the market will retrace again to 50

The distribution of the Delta (at the specific parameters as mentioned above) of the 50 call will look as follows:

At 50 in the Underlying, the Delta of the 50 call will be 52%

At 54 in the Underlying, the Delta of the 50 call will be 79%

At 48 in the Underlying, the Delta of the 50 call will be 36%

The trader, maintaining a Delta neutral portfolio will perform the following trades: when buying the call option, he will need to sell 5,200 Deltas/ stocks/ underlying units (52% x 10,000) at $50 (Please note: one Stock, Future, Underlying unit represents 1 Delta).

When the market moves to $54, he is long 7,900 Deltas on the back of his options, however he only sold 5,200 underlying units at inception of the trade. Thus he will need to sell another 2,700 Deltas at 54 in order to neutralize the Delta position of his portfolio. When on day two the market retraces to $50, he is long 5,200 Deltas on the back of his options, but he is short 7,900 underlying units, so he is net short 2,700 Deltas which he will have to buy back (at $50) to become Delta neutral again.

On day three his options have an aggregate Delta of 3,600 and his short underlying position is 5,200 Deltas, hence he will need to buy 1,600 Deltas/ underlying units. When the market retraces to 50 on day four, he will sell these 1,600 Deltas again and will have a neutral Delta position again.

In a table, his hedges will look as follows:

The trader made a profit of $14,000 in just a few days trading/ scalping on the back of the change of his Delta position. Performing these kind trades is called Gamma trading!

(the changes in the underlying are extremes, usually, as you may have understood from previous articles I’ve written, these extremes don’t occur in a market where Volatility is at 10%)

The Delta each time, with a different level in the Underlying, changes. The change of the Delta of an option compared to a one dollar change in the Underlying is called the Gamma.

So when we now start philosophising about Gamma trading, we know that when the Delta distribution is much steeper one can trade more Deltas in and out. Looking at the first chart of the article again one can clearly see this feature:

At 10% Volatility, the 50 Call, when trading at $54 in the Underlying has a Delta of 79%, while at 20% Volatility it has a Delta of 69% (versus 54% for the 50 call at $50 in the Underlying). This means that at 20% Volatility one only had to sell 1,500 Deltas at $54 instead of 2,700 and at $48 one only had to buy 800 Deltas instead of 1,600 at 10% Volatility. Therefore, the P&L will be much lower at a higher Volatility, as shown below.

The same will apply for the Time to maturity. The longer the Time to Maturity the more flattish the Delta distribution will be, the shorter the Time to Maturity the steeper the Delta distribution will be (see the second chart of the article).

So the steeper the Delta distribution, the more Gamma the position has. In other words, low Volatility and short Time to Maturity options (when being around the at the money) generate the most Gamma (and thus the most optionality when running a gamma book) and the more Gamma one has the more P&L one can generate when the market starts moving.

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