 Values of delta and gamma for calls and puts

So im really confused right now
I was reading on options greeks and learnt that calls have positive delta and gamma and puts have negative delta and positive gamma

which was right up until i was browsing through an options course which showed that short puts have positive delta and negative gamma.

How does delta and gamma differ for calls and puts ? is the “delta positive for calls and negative for puts” rule dependent on whether or not youre long/short an option ? could somebody explain the rationale behind why these greek values are positive/negative so that i dont have to remember them like rules and can derive the reasoning as and when i need ?

I think there are more intuitive ways to think about this but I think of it as, if the price of the underlying goes up is my Option more valuable (Positive Delta) or worth less (Negative Delta). So if you are short a Put let’s say, you want the stock price to go up – so Positive Delta.

As for Gamma, I just think “how does the slope of my Delta change, as the price of the underlying increases?” so if it goes from flat “–“ (0) to positive sloped “/” (+1) then it is Positive Gamma. Going back to the Short Put, you go from a +1 Delta “/” to a 0 Delta “–“ so Negative Gamma.

And “\” is a -1 slope. When I say slope I am talking about the payoff. So a long call looks like _/, going from 0 to 1. A short put looks like /¯, so goes from +1 to 0.

Um . . . of course.

It’s a zero sum game. If the value of the long option increases, then the value of the short option decreases, and vice versa.

Try drawing pictures: the price of a long call, the price of a long put, the price of a short call, the price of a short put.

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I partly got that

So as my underlying increases in value, i as a short put holder benefit . Which means the value of my option increases as prices of the underlying rise. But wouldnt that mean gamma or the rate change of delta is also positive ? And since as a short put holder my max gains are capped , i could also make the case that gamma would increase slightly as i approach expiry if my underlying price increases ?

I know im making a mistake somewhere, but this is how im thinking and cant figure out where it is im wrong

Well the Gamma would be Negative actually. Let’s say you are ATM so your Delta is 0.5 then as the underlying increases in value the Delta gets closer to 0 (remember Delta goes from +1 to 0 “/¯” for a Short Put), so now the 0.5 Delta might be something like 0.4.

i guess i cant quite grasp why the delta goes from +1 to 0.

The only things i can understand is short put, so i will benefit if underlying prices rise = options value increases = delta increases. why does the delta go from +1 to 0 ? cannot intuitively grasp that

No.

When the (short) put is far out of the money, its delta is roughly +1; when it’s far in the money, its delta is roughly zero. If it goes from +1 to zero, it’s decreasing, not increasing.

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I guess a couple things that might help. The absolute Delta gives you the approximate probability the Option will finish in the money. So think 0.50 is ATM, 0 is OTM, 1 is ITM (ignore the signs +/- for now)

Going back to the Short Put, if the underlying share price is lower than the exercise price, it is a Put so it is ITM. Then the question is, what is the sign. If you want the share price to increase, the Delta is +, if you want the underlying share price to decrease then Delta is -

When you think about Gamma, think Left to Right. In the case of a Short Put, all the way to the left (when underlying share price is low) option is ITM, so +1, then go all the way to the right, the option is OTM so 0. You are moving from +1 to 0 to Gamma is Negative.

Long Call: You want upward movement in the underlying: Delta is +
When share price is low you are OTM, when high you are ITM
so it goes from 0 to +1

Long Put: You want downward price movement so Delta is -
When the share price is low you are ITM, when high you are OTM
So left to right, -1 to 0, Gamma is positive

Short Call: You want underlying share price to go down, so Delta is -
When share price is low, OTM when share price is high, ITM
So from left to right, 0 to -1, so Gamma is Negative

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Okay, got a couple things, but ITM and OTM is determined by whether or not im the buyer or seller right ?

so for a buyer of a call, id want price to go up, so high price is ITM and low is OTM

but then similarly, if im a seller of a put, short put, id want the price to go up as well so shouldnt high price be ITM and low OTM as well ? so im moving from 0 to +1 again ?

I think you have it backwards

Here is a description from Fidelity re: short puts

Put options change in price based on their “delta,” and long put options have negative deltas. Short put option positions, therefore, have positive deltas. At-the-money short puts typically have deltas of approximately +50%, so a \$1 rise or fall in stock price causes an at-the-money short put to make or lose approximately 50 cents. In-the-money short puts tend to have deltas between +50% and +100%. Out-of-the-money puts tend to have deltas between zero and +50%.

im still not clear on this.

okay ill ask a very simple question.

Take a call option at 99. underlying is trading at 100. that makes the option ITM. If im a buyer of said call option, anything below 100 is ITM to me, and anything above is OTM. Now i want the price of the underlying to increase right , then im profitable. So wouldnt that mean i am moving from an ITM region (less than 100) to an OTM region (more than 100) ? wouldnt that mean going from a +1 region to 0 ?

This is where im getting confused

I think you might have it backwards
ITM = option will be exercised
OTM = option will not be exercised
Call ITM is when underlying price is higher than strike price
Put ITM when underlying price is below strike price
Definition of ITM or OTM doesn’t change whether long or short

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i see. so call ITM means underlying > strike

so, when you say option being exercised, you mean the buyer of the option exercising it ?

then , in a long call, since im the holder of the call, ill exercise it only when underlying price increases , im moving towards ITM. if price reduces , im OTM. id want to get to ITM from OTM, so 0 to +1

conversely, in a short call, the buyer of my call option wont exercise it if price goes down, so my option price increases as underlying price goes down , so as a short put holder id want movement from ITM to OTM. Since as a short put holder i want price to go down, delta is negative.

i hope ive gotten it right this far.

the final question i have is, as a short call holder id want price to reduce right ? so shouldnt i be moving from -1(ITM) to 0(OTM) ?

Think left to right though
Underlying share price low (left of the strike) the OTM because it’s a call
Then moved to the right to ITM but negative Delta
So 0 to -1

is it a convention to always think from left to right ?

i was thinking we should think in terms of what direction is favorable to the option holder so by that logic, id ( as a call seller) want to go from high price to low price hence form iTM to OTM so -1 to 0

I was trying to explain how to think of Gamma
At first I think your questions were around positive and negative Gamma

it is it is

im just trying o understand why it is we think of it from left to right. i mean why not right to left or directional , as i said above ?

You can think of it either way, but it’s easier to think about it when the underlying value is increasing.

By definition,

∆_{option} = \frac{∂P_{option}}{∂P_{underlying}} ≈ \frac{∆P_{option}}{∆P_{underlying}}

If you move from left to right, so that that change in the underlying’s price is positive, then the sign on delta is the same as the sign on the change in the option’s price. If you move from right to left, those two signs are opposite each other.

Similarly for gamma.

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Alright i think i have somewhat of an idea, conceptually, now

Thanks a ton, magician and GAT. I know it isnt easy clearing strangers’ doubts online , so i really appreciate it.