How is delta defined as a unit?

This is going to be a embarrassingly basic question. But the answer seems to be hard to find.

What does, say, selling, $$d$$ delta of calls mean? How is the "delta" defined? I am not asking about Greek as in $$\frac{\partial V}{\partial S}$$ where $$V$$ is the price of the option and $$S$$ that of the underlying stock.

You'll find the same issue with all of the greeks.

I would say that the standard rule for delta is the following:

If you're talking about a single option/strategy, then people will talk about delta percentage*.

If you're talking about a single option/strategy, and you don't plan to hedge the delta when it tades (i.e. you want to trade it live), then you'll talk about the delta in an absolute term, or some other term that you're familiar with - i.e. \$35m delta, or 500 lots).

If you're talking about delta of something else - say a portfolio, where the total notional of positions is not obvious/known well by everyone you're talking to, then i would say most people again talk in absolute terms as above.

Some people also prefer to talk about delta in terms of spot, and others forward delta. This is again going to be desk specific.

The delta is normally spoken about scaled to be for a 100% move in the underlying**. Where you have non linear risk, this will be the instantaneous greek scaled to be as if it's the TV change for a 100% move (i.e. $$\frac{TV(S+\mathrm{d}S) - TV(S)}{\mathrm{d}S}$$)

*There are some cases where the delta will be used to describe the strikes - i.e. a 25d Risk Reversal is a trade where you buy(sell) a put with 25d, and sell(buy) a call with 25d also.

**unless you're talking about dv01, in which case it's for a 1bp move in rates, and is your rates delta.

In answer to Hans' comment -

You can just use finite difference to calculate the difference in the value. Again there are conventions though - do you care about the change in volatility caused by the change in the spot price? Are you valuing your derivatives in a mean reverting model such that moving the spot price moves the forward in a non linear way? Do you have non linear dividends? Are there other implications in your model resulting from changing the spoot price? It is not a clear cut answer, and conventions need to be decided on. For me, there are three kinds of delta i care about:

1. Partial delta - i.e. $$\frac{\partial \mathrm{TV}}{\partial S}$$, that is, the change in price from moving only the spot price, and nothing else.
2. Simple Delta Partial delta + vol delta: $$\frac{\partial \mathrm{TV}}{\partial S} + \frac{\partial \sigma}{\partial S}\cdot\frac{\partial \mathrm{TV}}{\partial \sigma}$$, i.e. the partial delta + the impact on value from the move in vol expected to result from a move in spot price.
3. Full Delta: $$\frac{\mathrm{d}\mathrm{TV}}{\mathrm{d}S}$$ - i.e. the full delta after accounting for all changes in the model as a result of moving the spot price.
• Thank you, Will. I understand how the Greek is defined. I am asking about it being used as a unit of the option as is mentioned in your 4'th paragraph. Would you please give a mathematical definition of that?
– Hans
Aug 30, 2021 at 20:46
– will
Aug 30, 2021 at 21:03
• Will, like I said before, I am familiar with the definition of the Greek delta. So that is not what I am after. What you have defined is precisely the Greek delta as opposed to the definition of delta as a unit of option. So you do not seem to answer my question. But of course, maybe my question is not the right one because this is what confuses me.
– Hans
Aug 31, 2021 at 5:44
• @hans Can you give an example with actual numbers (and preferably units too)?
– will
Aug 31, 2021 at 5:47
• @noob2's answer seems to me to pertains more to what I am asking. Do you agree? I am just asking, as I am not completely sure.
– Hans
Aug 31, 2021 at 13:31

If you want to split hairs, as I like to do, there are 2 ways to express Delta.

"Pure Delta" is a fraction, i.e. a number between 0 and 1 (for a Call). In terms of units, it is a pure number.

"Position Delta" is equal to Delta times the size of your Call position. If you have Calls on 100 shares and Delta is 0.5 then Position Delta is 100 x 0.5 = 50 shares. As you can see the units for Position Delta is "shares". This is what you actually buy or sell when you hedge or replicate the option by trading in the underlying. (In practice the size of the Call position is often expressed in terms of Contracts, where 1 Contract is equal to 100 shares. So you have to convert from contracts to shares before multiplying by pure delta).

When a textbook says "buy Delta shares" they are assuming you have calls on 1 share, which is fine as an example, but lacks generality and applicability to the real world.

HTH

• I sometimes hear traders say an option priced at 5 delta. Which definition are they using?
– Hans
Sep 2, 2021 at 13:56
• They leave out the decimal point but they are talking about a pure delta of 0.5 Sep 2, 2021 at 16:56
• So that is equivalent to saying the option is priced where its delta is equal to the given number, right?
– Hans
Sep 3, 2021 at 0:18
• I disagree on that, 5 delta is 0.05 or 5%. The only one that follows slightly ambiguous rules is "delta 1" and "1 delta" - in this case, the former would mean 100% while the later would mean 1%. The sign is also ignored and assumed to be known by all.
– will
Sep 3, 2021 at 6:46
• That's right, 0.5 would be "fifty delta". Sorry for mistake. Sep 3, 2021 at 8:36