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It might help to think of the two as special cases of $$S_{i+1}-S_i = \sigma (c+S_i)^\beta \epsilon$$ which looks like a Constant Elasticity of Variance extension. Taking squares of both sides and then logs will (nearly) linearise it, allowing you to carry some basic estimation using OLS. The parameter $c$ will control the lower bound and can impose some ...


As a short summary and adaption of the question: You better redefine $\hat{r}_i= \frac{S_{i-1}}{S_1}-1$ and $\hat{S}_i = (1+\hat{r}_i)S_0$. The above definition of $\hat{S}_i$ yields a sample of potential values for $S$ for the future day. This approach is usually applied in historical simulation. The aim here is to use information of the past about the ...

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