# Time-shifted power law in path dependent volatility

I can't understand a function which is part of a volatility model.

This is all explained in an open access paper titled "Volatility is (mostly) path-dependent" by Guyon and Lekeufack. My understanding after reading the paper is that we can model volatility as a simple regression of

$$\sigma_t = \beta_0 + \beta_1 R_{1,t} + \beta_2 \sqrt{R_{2,t}}$$

The paper fits this equation on realized volatility (RV) and implied volatility (IV). This means that $$\sigma_t$$ is RV or IV and

$$r_{t_i} = \frac{S_{t_i} - S_{t_{i-1}}}{S_{t_{i-1}}}$$

$$R_{1,t} = \sum_{t_i \le t} K_1(t-t_i)r_{t_i}$$

and

$$R_{2,t} = \sum_{t_i \le t} K_2(t-t_i)r_{t_i}^2$$

Now, $$K_1(t)$$ and $$K_2(t)$$ can be one of a number of decay function but the one the model uses is a time-shifted power law (see p. 8 of paper):

$$K(\tau) = K_{\alpha, \delta}(\tau) = Z^{-1}_{\alpha, \delta}(\tau + \delta)^{-\alpha}, \quad \tau \ge 0, \quad \alpha > 1, \delta > 0$$

where in the continuous time limit

$$Z_{\alpha, \delta} = \int_0^\infty (\tau + \delta)^{-\alpha} d \tau = \frac{\delta^{1-\alpha}}{\alpha -1}$$

Now, I can't understand how to implement the $$K(\tau)$$ functions. I just don't understand what $$Z^{-1}_{\alpha, \delta}(\tau)$$ is and how to implement it numerically. Is $$Z(t)$$ connected to the normal distribution pdf?

The time-shifted power-law kernels $$K_1 (t)$$ and $$K_2 (t)$$ assign a weight to past returns and squared returns, respectively. Each kernel is a function of the following parameters: lag parameter $$\tau>0$$, time-shift $$\delta>0$$ that ensures that the kernel does not blow up when $$\tau$$ becomes very small, and $$\alpha>1$$ is the scaling exponent of the power-law distribution. The term $$Z_{\alpha,\delta}$$ is a component of this kernel, which integrates the time-shifted power-law function over the time domain. As the author uses a continuous limit, there is no need for numerical integration. To implement the kernel functions in practice, one needs to calibrate the parameters $$\alpha$$ and $$\delta$$. I will treat them as known, since the paper describes how these parameters can be calibrated from market data.
To calculate the $$K(\tau)$$ function in practice, I created a toy example in Excel, considering a random path for the stock price $$S_t$$. As we can see, the kernel function assigns higher weights to the most recent returns, and vanishes over time. 