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I am following along with the paper linked here: https://math.nyu.edu/~avellane/thesis_Zhang.pdf .

In section 4.4, equation (4.4.2) makes the claim:

$$\sigma(k) = |\beta|\sigma_s(S_0k^*)$$

where: $$S_0k^*=S_0\left ( e^{\frac{\beta^2-\beta}{2}V_t} \right )^\frac{1}{\beta}$$ and: $$V_t=\int_{0}^{t}\sigma^2ds$$

I am confused how to use this relation to go from an implied vol on the base S to a corresponding implied vol on the leveraged ETF. What is mainly confusing me is I do not see any dependence on k on the RHS of this equation. Perhaps I am looking at it all wrong?

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