Short put and negative gamma

Since delta of short puts is positive (short put, stock ↑, value of option ↑), when stock price rises, delta of short puts will decrease and moves from +1 to 0.

Since it is moving from +1 to 0, why does a short put have negative gamma?

Gamma is negative because . . . delta is decreasing.

Okay for short put, when stock price ↓, value of option ↓, delta of short puts will still decrease but now from 0 to - 1. Is that correct? (Hence why gamma for shorts is negative) And similarly

For long call when stock price ↓, value of option ↓, delta long calls will increase from -1 to 0. Is that correct? (Hence why gamma for longs is positive).

Long options ALWAYS = long gamma.

Short options ALWAYS = short gamma.

–*Also this may be outside the scope of curriculum but it’s worth noting: - being long gamma forces rebalancing by selling stock high and buying stock low so delta hedging long gamma works best in oscilating, flat markets.

-being short gamma forces you to buy more shares higher or sell more shares lower and works best in a trending market (you will keep buying more stocks all the way up)

^— that is just for delta hedging (rebalancing). In general being long gamma is rewarded by large movements and being short gamma you are hurt by large movements in the underlying.

Let’s look at all option positions and see what happens when the price of the underlying increases:

• Long call: delta increases from 0 to +1; gamma is positive.
• Long put: delta increases from −1 to 0; gamma is positive.
• Short call: delta decreases from 0 to −1; gamma is negative.
• Short put: delta decreases from +1 to 0: gamma is negative.

Thanks s2000magician. Can we please confirm what happens when price of the underlying decreases. Is the below correct?

· Long call: delta increases from -1 to 0; gamma is positive.

· Long put: delta increases from 0 to +1; gamma is positive.

· Short call: delta decreases from -1 to 0; gamma is negative.

· Short put: delta decreases from 0 to -1: gamma is negative.

Nope.

When the price of the underlying decreases:

• Long call: delta decreases from +1 to 0; gamma is positive.
• Long put: delta decreases from 0 to −1; gamma is positive.
• Short call: delta increases from −1 to 0; gamma is negative.
• Short put: delta increases from 0 to +1: gamma is negative.

So if we stick to the rule: - All short strategies have negative gamma. All long strategies have positive gamma. When underlying is decreasing, how does a long call have positive gamma, when delta is decreasing?

A better way to think of it is in absolute |delta| terms.

So an ATM call has a |50d| and so does an ATM put.

Say you are short a 50d put. The stock falls 10%, what is the new put delta? well it’s got to be bigger because puts are worth more the lower the stock. Say the delta goes up to 65d.

If you sold 1 put contract you’d have to sell 50 x 100 = 5000 shares to be hedged. Now the delta is 65 so you need to sell more stock 65 x100 = 6500 (less 5000 already sold) = 1500 more shares.

*if you are short puts you are “long the market” so you need to sell shares to be delta (directional) neutral

___

now for your question about long call and +gamma.

Now you own a 1 contract of a 50d call. You need to sell 50 x 100 = 5000 shares to be delta neutral. Now the stock drops 10%. The option of the delta will decrease because calls are worth less when the underlying goes down. Say the delta falls to 40d. To be delta neutral you would need be short 40 x 100 = 4000 shares. You are already short 5000 shares so you need to buy 1000 shares back.

*if you are long calls you are “long the market” so you must sell shares to hedge in order to be delta neutral.

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Gamma = change in delta / change in underlying price

If

change in delta < 0

and

change in underlying price < 0

then

Gamma = negative number / negative number = positive number

Hi Magician…
When we say delta is between 0 and 1, its for a long call option, right?
A short call option will have delta between 0 and -1?

Yup.

thank u…

You’re welcome.

Why are they saying that - A collar position is, economically, intermediate between pure equity and fixed-income exposure.

Below is the snapshot from the curriculum, page 265, R15.
As the chosen put and call exercise prices move successively farther in opposite directions from the current price, the combined collar position begins to replicate the underlying gain/loss pattern of a long position in the underlying security. Conversely, as the chosen strike prices approach and meet each other, the expected returns and volatility become less and less equity-like and eventually converge on those of a risk-free, fixed income return to the time horizon. Thus, a collar position is, economically, intermediate between pure equity and fixed-income exposure.