I understand that the delta of long puts is negative (if you are long a put, if stock ↑, value of option ↓). But why does this mean positive gamma?

As the price of the underlying *increases*, the delta ** increases** (from −1 to 0), so gamma is positive.

So would it be correct to say that the only way to have a negative (“short”?) gamma position is to short options? (assuming my interpretation of the above explanation is that a positive relationship between px of underlying and delta implies positive gamma and a negative relationship implies short gamma)

Long Put - (laid out above)

Long Call - price of underlying increases, delta increases (in the range of 0 to 1)

Short Put - price of underlying increases, delta decreaases (in the range of 1 to 0)

Short Call - price of undrlying increases, delta decreases (in the range of 0 to -1)

Thanks vm!

Correct.

Yes. Gamma is negative for both: Short Calls and Short Puts

With short puts now: I understand that delta of short puts is positive (short put, stock ↑, value of option ↑). But when stock price rises (which is benefit to a short put), why does the delta of a short put decrease in the range of 1 to 0? I thought since it is a benefit, delta increases?

Just a practical way of approaching this (regardless of whether its a call or put):

**BUYING an option = Long Gamma and Long Vol** (you want to see volatility and fluctuations in the underlying increase)

**SELLING an option = Short Gamma and Short Vol** (if you sold an option, the last thing you want is volatlity or fluctuations in the underlying - the more it moves, the higher the chances that it might be called/put. You want a more stable underlying)

Draw a picture of the payoff: /¯.

The slope goes from +1 to 0.

Okay i understand the diagram and slope.

But conceptually, when stock price rises (which is a benefit to a short put), why does the delta decrease? I thought that since it is a benefit, the delta would increase, so the short put option is more valuable given the stock price rise?

The value of a short put is negative.

As the stock price rises, it becomes less negative (and delta decreases); less negative is more valuable.

So is delta the change in the value of the option or the hange in the number from -1/1 to 0? Because with long puts, if stock ↑, value of option ↓. At the same time when when stock price ↑, the delta number increase, i.e move from -1 to 0.

It’s both.

Change in option price ≈ option delta × change in underlying price

s2000 thank you for your persistence on this, but i am still confused with the short put.

I get that delta of short puts is positive (stock ↑, value of option ↑). When stock price rises, the delta number of short puts will become less negative, but how is a move from +1 to 0 less negative?

confused

Long gamma = Long Vol

No. Gamma and Vol are different measures of risk. Although directions look same in long/short positions.

puts move from -1 to 0. So less negative. Also, when stock goes up, value of puts go down…meaning out of money, meaning towards zero, meaning less negative, meaning good for short position - the guy who sells puts.

Yes i think long volatility = long vega?

I believe there is no causal relationship between gamma and vol measures, the input to the absolute gamma is the underlying px relative to strike (implies the senstivity of the option’s delta to change in underlying px) while the input to vega is the vol of the underlying, however it is correct to say that if you are short an option you are short both vega and gamma (and vice versa).