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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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The local volatility graph tomorrow doesn't change, unless the implied volatility surface tomorrow is not the same as today. LV takes the implied vol surf today as input, and outputs a instantaneous volatility function of spot and time, which can price vanilla options today exactly the same as the market prices.

In local volatility world, you assume the implied vol surf does not change. If there is a change in the implied vol surface tomorrow, you have to remark the implied volatilities, and create a new local vol graph, which is alike to remarking implied vol in Black Scholes. If you do not remark, you will have un-explained P&L, because the old local vol function can no longer perfectly calibrate to your vanillas options anymore, which gives you wrong vanilla prices and un-explained P&L.

In your picture, you only have one red line for implied volatility, which i assume your implied vol surface at $t=1$ is same as $t=0$. Also assuming $dt$ is small, $$\sigma_{LV}(t=1,spot=S_1)=Lower\ one\ of\ "LV(t=1)?"$$

Loosely speaking, implied variance is the average integrated local variance from today up to time T weighted by dollar gamma. For more, you should get familiar with skew stickiness ratio (ratio of ATMF instantaneous vol's slope over log spot over ATMF implied vol's slope over log strike) by Bergomi, which tells you that $SSR_{LV}=2$

"And since LV doesn't vary with strike"

  • It's because LV describes the instantaneous vol of the spot, not the implied vol of any option.

"LV model the volatility is constant through different strikes"

  • Not sure if i understood correctly. What volatility are you referring to?

  • If you are saying the instantaneous vol of the spot, the function $\sigma_{LV}(S,t)$ does not depend on strike, since it describe the spot price's instantaneous volatility not an option's implied vol

  • If you are saying the evolution of implied vol of an option ($K, T$) under LV dynamics, it is a function of spot price and time $t$

$$\hat\sigma_{K,T}(S,t)=C_{BS}^{(-1)}(S,K,T-t)=f(S,t|\Sigma)$$, where spot dynamics is

$$dS=(r-q)Sdt+\sigma(S,t)SdW$$

  • By Ito's lemma, $$d\hat\sigma_{K,T}=\frac{\partial f}{\partial S}dS+\frac{\partial f}{\partial t}dt+\frac{1}{2}\frac{\partial^2 f}{\partial S^2}dS^2$$

$$d\hat\sigma_{K,T}=\frac{\partial f}{\partial S}[(r-q)Sdt+\sigma(S,t)SdW]+\frac{\partial f}{\partial t}dt+\frac{1}{2}\frac{\partial^2 f}{\partial S^2}\sigma(S,t)^2S^2dt$$

$$d\hat\sigma_{K,T}=[\frac{\partial f}{\partial S}(r-q)S+\frac{\partial f}{\partial t}+\frac{1}{2}\frac{\partial^2 f}{\partial S^2}\sigma(S,t)^2S^2]dt + \frac{\partial f}{\partial S}\sigma(S,t)SdW$$

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