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Black-Scholes volatility is average of local volatilities.

It is from: https://bookdown.org/maxime_debellefroid/MyBook/all-about-volatility.html

First what's the meaning of the average of all the paths between spot and the maturity and strike of the option? Here does path mean the underlying path satisfying $$S_0=\text{Spot}\rightarrow S_T=\text{Strike}?$$ and does average means for each time $T',$ the BS implied vol $$\sigma_{BS}(T',K) = average\Big[\sigma_{local}(S_i(T'))|S_i(0)=\text{Spot},S_i(T)=K\Big]?$$ here $i$ is the index of i-th path. And I saw the similar result that local vol is the average of forward instantaneous vol: $$\sigma^2(K,T)=E[\sigma^2_T|S_T=K],$$ do they say the same thing?

Also how to understand ATM volatility is the same for both Black-Scholes and Local Volatility?

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user6703592
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  • What is a path? The picture is helpful, it shown three different paths, in red. Of course there is an infinite number of paths between two endpoints, not just three. – nbbo2 Nov 20 '22 at 17:15
  • @nbbo2 what's the value of x-axis? time to maturity? I confused with right-down side labels (high local volatility). Seems it is the path of underlying price what I said above – user6703592 Nov 20 '22 at 17:19
  • @nbbo2 could I understand that average means for each time $T',$ the BS implied vol $\sigma_{BS}(T',K) = average\Big[\sigma_{local}(S_i(T'))|S_i(0)=Spot,S_i(T)=K\Big]?$ here $i$ is the index of i-th path. – user6703592 Nov 20 '22 at 17:26
  • The x axis shows time (t) from now until expiration. The y axis show price S. Local volatility is a function of Price and (optionally) Time $\sigma_t=\sigma(t,S)$. Usually this function assigns low volatility to high prices and vice versa, which explains the labels low volatility and high volatility on the left of the figure. – nbbo2 Nov 21 '22 at 06:52
  • @nbbo2 I updated understanding, am I correct? – user6703592 Nov 21 '22 at 09:46

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