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I am reading Derman's article/notes regarding local volatilty: http://www.emanuelderman.com/writing/entry/the-local-volatility-surface. I am examining the graph on page 13.

The Implied volatility (IV or $\hat{\sigma}$) is a function of strike and maturity and the Local volatility (LV or $\sigma$) is a function of spot and time to maturity (or simply just time). His rule of thumb on page 13 states: Local volatility varies with market level about twice as rapidly as implied volatility varies with strike.

Assuming linear skew (as the graph on the article), todays spot given as $S_0$ and ATM LV = ATM IV; I can construct the equations for IV and LV as $$IV(K)=\hat{\sigma}(K)=skew*K+(\hat{\sigma}_{ATM}-skew*S_0)$$ $$LV(spot)=\sigma(spot)=2*skew*spot+(\hat{\sigma}_{ATM}-2*skew*S_0)$$ Is this correct interpreted?

But how should LV be understood? For instance let's assume tomorrow ($t=1$) the price rises to $S_1$. How does my LV graph look tomorrow and at time $t=1$ what is my Local Vol? I have included a graph in which my confusion is demonstrated. From that graph: how do I determine the $LV(t=1,spot=S_1)$

enter image description here

And since LV doesn't vary with strike: does that mean that using a LV model the volatility is constant through different strikes?

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