Why Option Values Are Convex Before Expiration

# Why would options have gamma?


# Why would options have gamma?












I don't understand on an intuitive level why options have convexity. The payoff of ITM options is completely linear which is why I can get my head around the fact that there is convexity in options.

Let's say the following example. I have a call on $C$ with underlying stock $S$. $C$ has gamma: $\Gamma_1$ and delta $\Delta_1$. Now if the stock moves up by $1bp$ we have:

$C = C + \Delta_1 \cdot 0.001 + \frac{1}{2} \cdot \Gamma_1 \cdot 0.001^2$

Now the options has a new $\Delta_2 = \Delta_1 + 0.001 \cdot \Gamma_1$ and a new $\Gamma_2$, now if the stock moves down by $1$bp we get that the price of the option is:

$C = C + \Delta_1 \cdot 0.001 + \frac{1}{2} \cdot \Gamma_1 \cdot 0.001^2 - \Delta_2 \cdot 0.001 + \frac{1}{2}\Gamma_2 \cdot 0.001^2 = C - \frac{1}{2} \Gamma_1 \cdot 0.001^2 + \frac{1}{2} \Gamma_2 \cdot 0.001^2$

After the stock moves up by $1$bp and then down by $1$bp we should expect the price of the call remain the same right?

So basically we should have: $\Gamma_2 = \Gamma_1$? which is obviously false. If the option is OTM then $\Gamma_2 > \Gamma_1$ and so we are sure to make money.

Which means that for an OTM call option if the stock goes an infinite amount of times up and down by one bp then the value for the call option is infinite even if the call is OTM. And if the call is ITM then the value of the call goes to 0.It doesn't make any sense.

So I guess w should loose somewhere else. Is it because $\Gamma$ is somehow related to $\theta$ and thus $\Gamma$ reflects the fact that if the stock goes up by $1$ and then goes down by $1$ bp then during than up and down move time passed (for example $1s$) which is bad for OTM option so you loose what you gain because of theta, and for ITM option you gain money because if the stock doesn't move it is good for you so what you loose on Gamma you regain it on theta?

But then if I make these up and down move by the same amount with a time interval for these moves equal to $\delta \to 0$, then I make infinite money on OTM options?

## Answer by D Stanley (score 4)

https://quant.stackexchange.com/a/80311

I see two incorrect assumptions that may answer your questions:

> The payoff of ITM options is completely linear which is why I can get my head around the fact that there is convexity in options.

The payoff at expiry is piecewise linear (linear if the option is ITM, constant 0 otherwise). But prior to expiry the value is non-linear with convexity. An option that is currently OTM has some probability of expiring ITM, so it has value and does not have the same discontinuity of the slope at the strike that the ultimate payoff graph does.

You can search for option value images and see that it is indeed a curve.

> After the stock moves up by 1bp and then down by 1bp we should expect the price of the call remain the same right?

Yes, but the Taylor expansion with Delta and Gamma do not give the exact new option price after the up and down moves, only an approximation, because Gamma is not constant either. There would be third, fourth, etc. order factors that would account for the difference between the up and down changes.

I'm sure there are more sophisticated mathematical explanations, but the bottom line is that your "equality" does not necessarily hold because of the Taylor approximation.

## Answer by JoshK (score 2)

https://quant.stackexchange.com/a/80335

Think about it this way. Take a \$50 call on a \$1 stock. There's little delta to this call, of course. Now, imagine that spot moves from \$1 to \$2. The stock doubled and yet the delta is still basically 0 for the call.

Now, imagine the stock rallies to \$40. All of a suddent spot moves will have some meaningful increase in delta. That's the gamma. The increase in delta from \$1 to \$2 is much less than from \$40 to \$41 (even though as a % of spot it's not as much as from \$1 to \$2).

## Answer by Hasselhoff (score 1)

https://quant.stackexchange.com/a/80334

I don't know a ton about options, but if the spot price went up and then down, that means time passes, and options are sensitive to time. So the option price will not be the same holding all greeks constant as time passes, even if the spot price doesn't move.