I was reading something when the author said that "x converges to a dirac delta function". Was wondering if someone could explain what it means. I work in IB and am unsure what this means.
The exact comment was "The longer the maturity, the more and more gaussian the gamma of your option is and therefore the jump costs you less. Whereas for short term maturities, the gamma converges to a dirac delta function. This implies that if there were a jump, the slippage is much higher for shorter to maturity functions."
Link to the explanation: https://quant.stackexchange.com/a/5976/46192
The Dirac delta function is not really a function. Roughly speaking, it is a very tall, very narrow function centered on $0$ with area $1$ beneath it. One can often think of it as the "limit" of a series of functions that get taller and taller and narrower and narrower. They could be Gaussians, they could be step functions
$$\delta_n(x)=\begin {cases} n&-\frac 1{2n} \lt x \lt \frac 1{2n}\\ 0&\text{otherwise} \end {cases}$$
so $\delta (x) =0$ whenever $x \neq 0$ but if you integrate from any negative number to any positive number the area is $1$. The usual use is to pick out the value of some other function $f(x)$ at $0$, so $\int_{-a}^a f(x)\delta(x) dx=f(0)$, which you can prove by integrating by parts. I don't know what gamma and slippage are in your use.