Delta of an underlying is the rate of change of an underlying with respect to itself

Last week, we discussed about delta as a measure of probability of an option ending in the money. This week, we discuss how individuals can use delta to understand options as equivalent underlying positions.

Underlying equivalent

Delta is the change in the option price for a one-point change in the underlying. So, if a call option has a delta of 0.50, the option price will approximately change by 0.50 for a one-point change in the underlying. Note that delta is only an approximation because the delta also changes as the underlying changes. This change is captured by the option gamma.

If delta is the rate of change of option with respect to the underlying, does the underlying have a delta? The answer is yes! Delta of an underlying is the rate of change of an underlying with respect to itself. So, the delta of an underlying is always one. For the sake of convenience, we will convert delta into percentage format. So, a call delta of 0.50 will be 50 (0.50 multiplied by 100) and the underlying’s delta of one will be 100. It is now easy to understand options in terms of equivalent underlying. For example, having two shares is equivalent to 200 delta, given that each share has a delta of 100. That would be approximately the same as having four call options of 50 delta each.

You must multiply the delta of the option that you are buying with the appropriate permitted lot size to arrive at the equivalent underlying position. Suppose you buy call option with a delta of 50 and the permitted lot size of 500, you are long on 25,000 delta, which is equivalent to 250 shares of the underlying (25,000 divided by 100 delta for each underlying).

Does this argument work for long puts? Yes, because put delta is interpreted the same way as call delta. The only difference is that put delta has a negative sign to indicate that puts move in the opposite direction to the underlying. Note that the above explanation for calls and puts simply offers a perspective (not a trading strategy) on your option position as equivalent shares. An option is not only sensitive to the movement in the underlying, but also to factors such as interest rate and time to maturity.

Take note

Delta of an underlying is always one

Long call positions have positive gamma

Short positions have negative gamma

Optional reading

Suppose you short a call with a delta of 50. If the permitted lot size is 100, you have minus 5,000 delta for each contract- equivalent of short position in 50 shares. You must be, however, cautious about using delta to arrive at equivalent underlying when you are setting up short option positions. Why?

Long call positions have positive gamma, whereas short positions have negative gamma. Therefore, the delta of at-the-money and out-of-the money calls typically increases at an increasing rate when the underlying moves up and decreases at a decreasing rate when the underlying moves down. This could be harmful for short call positions if you do not acknowledge the associated risk.

Suppose a call delta is 52 and its gamma is 1 (in percentage format). If the underlying moves by 10 points, the new delta is 62 (you must add gamma of 1 time 10 to the delta of 52). If you short one option contract and the permitted lot size is 250, your position has minus 15,500 delta (62 times 250) as opposed to minus 13,000 delta (52 times 250) when you set up the position. The adverse movement in the underlying by 10 points increased your risk by 2,500 delta or 25 shares!

What does this mean? If you trade stocks and are a recent entrant to the options market, you could use delta to get a perspective on the equivalent long position in an underlying. Because of gamma risk, this argument may not work well for short option positions.

(The author offers training programmes for individuals to manage their personal investments)

Published on May 07, 2022