Well, why is that so?

Why is this contribution important for volatility?

Because if the behavior of the underlying is really volatile,

then the probability that you end up at a certain point that

is far from where we are right now is actually much higher.

And therefore you can see that,

although let's say you buy an out of the money option.

So you bought a call option to buy a share of stock at 70,

while the prices right now 50.

If this is a very low volatility stock, then the probability that you will

ever reach 70 is very low, and therefore the value of this option is low too.

However, if the volatility is high then the chance that at some point

in time the price will jump over 70 or even come close to 70.

That is much higher, and therefore that contributes to the option value.

By the same token if you talk about the American option that is long,

then over this period of time, the probability of some,

let's say previously unlikely outcomes goes up.

Now there's another one very important thing which is the risk free rate.

And the risk free rate is positive,

for the call option a negative for the put option in the following way.

Because a call option basically is equivalent to, so this is right,

so you wait for some time and if you do decide to exercise that

then you will pay the price for this share of stock at the time of exercising.

So basically, you have some time to wait, or you can save this

amount over a certain period of time, and earn a risk free rate on that.

So you save this amount at in the case of a call option,

in the put option it's quite the opposite, so basically that is why these signs.

Now there are some other important things here, for example,

dividends for the stock, because if you own an option,

then you are not entitled to a dividend.

So in order to be able to receive this dividend,

you first have to exercise the option, and

to become the title owner of the share of stock, that also contributes to the value.