We see that for the option that is out of the money, it is 10% out of the money, so

0.9k, the current stock price is less than K.

So if the option were to expire today, you'd get nothing.

And therefore, what we see is that the delta decreases, and it decreases toward

0 as the time to maturity decreases toward 0.

Similarly, the in the money option, where S0 equals 1.1k.

So therefore, remember the payoff of the option is equal to the maximum of ST

minus K and 0. So if S, T, is equal to 1.1 k, well this

would be equal to 1 K, so it's in the money.

And what we see here is, that, as the time to majority goes towards 0, the

delta of this in the money option, goes towards 1.

And that be, and that is because as the time to maturity goes towards 0 would

become more and more likely to exercise the option.

And so the option behaves more and more like ST minus K, because this maximum is

going to be equal to ST minus K as the time to maturity goes to 0.

And of course, the partial derivative of this expression here is equal to 1 and

that's why the delta goes to 1. On the other hand, down here in the 10%

out of the money case well then this is going to behave like 0.

If we're out of the money, it's going to behave like 0.

As the time to maturity goes to 0 and therefore, the partial derivative of this

will be equal to 0 and that's what we're seeing here.

Perhaps the more interesting case is when the option is at the money and K equals

S0. Well, then in that case, and I'm talking

approximately here, the chances of exercising approach 50%.

So, basically, there's a 50% chance of exercising and 50% chance of not

exercising. And it turns out and it can be confirmed

by differentiating the Black-Scholes formula, or calculate the expression we

saw on the earlier slide that the delta actually approaches 0.5.

The gamma of an option is the partial derivative of the options delta with

respect to the price of the underlying security.

So, the gamma measures the sensitivity of the option delta to the price of the

underlying security. The gamma of a call option is therefore

given to us by delta 2C delta S squared, and again it's somewhat tedious but it

can easily be calculated using basic calculus.

We can take the partial derivatives of the BlackâScholes formula to calculate

the gamma. If we do that we will find a sequel to

this expression here. E to the minus C time T, N of d1 divided

sigma S square root T. How about the gamma of the European put

option? Well that's easily calculated from

put-call parity. So put-call parity is given to us here.

So therefore, we can actually say that P, is equal to C, plus e to the minus rt

times K, minus e to the minus cT times S. So we can therefore, see the delta 2p,

delta s squared is equal to delta 2c delta s squared.

Well, plus 0 minus 0 because the second partial derivative of this is equal to 0.

And the second partial derivative of this with respect to S is equal to 0.

So therefore, we see that, see that the gamma for put option is equal to the

gamma for a call option. So, once we know the gamma for European

call option we therefore we have the gamma for European put option.

And in fact, you can see that this expression is always greater than are

equal to 0. So the gamma for European options is

always positive. this is due to what's called option

convexity. Here is the plot of the gamma for

European options is, as time to maturity varies.

So, the gamma here is a function of the stock price, and we've got three

different times to maturity. 0.05 years, 0.25 years and 0.5 years as

we saw before. Notice that the gamma is steepest for the

shortest maturity. So, in this case, for T equals 0.05 years

i.e approximately two and half weeks to maturity, we see that the gammas are very

steep around the strike of K equals 100. But we also see that it falls away to 0

much faster than the option when T equals 0.25 or the option when T equals 0.5

years. The reason is as follows, the delta of

the European option when T equals 0.05 years, well, it's going to be a half, or

approximately a half when the stock prices at the strike K.

But as the stock price goes up, the delta's going to move towards 1 and it's

going to move towards 1 much faster than the options with the higher time to

maturity. Similarly, as the stock price falls below

the strike of 100, the delta of the call option is going to move towards 0.

And it's going to move towards 0 much faster than the delta of the options with

times to maturity of 0.25 and 0.5 years. So this actually is, another, this plot

here is just another way of looking at this plot.