Let positions size for the instrument be
$$K_t \frac{V_t \Sigma^{target}}{\tilde{N_t}} \frac{1}{\sigma_t P_t}$$
where $$K_t = \bar{K}_{t-1} \frac{\Sigma^{target}}{\Sigma^{realized}_{t-1}}$$
where
$V_t$ is portfolio capital
$\Sigma^{target}$ is target annualized volatility
$\tilde{N}_t$ is the number of positions to be in
$\sigma_t$ is annualized volatility of instrument
$P_t$ is price of instrument
$\bar{K}_{t-1}$ is the average of the previous $K_t$'s for the past M periods and
$\Sigma^{realized}_{t-1}$ is the realized annualized volatility of the portfolio.
I found this implementation in a blog (https://hangukquant.substack.com/p/volatility-targeting-the-strategy and https://hangukquant.substack.com/p/volatility-targeting-the-asset-level). I have changed the notation a bit for simplicity.
The math adds up for me but I'm unsure about using a moving average of previous values to calculate $K_t$. He seems to reason that it's an empirical approach that bypasses the need for calculating the covariance matrix, but I would think that simply setting $$K_t = \frac{\Sigma^{target}}{\Sigma^{realized}_{t-1}}$$ would take care of that; while multiplying by $\bar{K}_{t-1}$ would simply stablize/smooth out the values of $K_t$ through time.
I ran some experiments and it appears that not multiplying by the $\bar{K}_{t-1}$ term tends to undershoot the vol target by about 40%, while multiplying by the $\bar{K}_{t-1}$ term actually gets you pretty close to the vol target.
Is this a valid implementation? Is there a better implementation I should look at? Did this guy invent the use of the moving average for $K_t$ in this context or did he take it from somewhere else? The papers he's cited don't use this implementation and I haven't seen it anywhere else either.
