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# Local Volatility in Terms of Implied Volatility

> [Local volatility surface corresponding to the implied volatility surface](https://quant.stackexchange.com/questions/16898/local-volatility-surface-corresponding-to-the-implied-volatility-surface)
>
> Loosely speaking:
>
> * Local volatility is the instantaneous volatility at time T if the spot is S at that time.
> * Implied volatility is the expected integrated volatility from today up to time T if the spot ends up at S at that time.

1. [Lecture 1: Stochastic Volatility and Local Volatility](https://docplayer.net/17523808-Lecture-1-stochastic-volatility-and-local-volatility.html)
2. [Derivation of Local Volatility](https://docs.google.com/viewer?url=https://raw.githubusercontent.com/damonYuan/gitbook/main/docs/Dupire%20Local%20Volatility.pdf)
   1. Note that there is a typo in the article [In Dupire's paper, why is (𝑆𝑡,𝑡) in the (𝐾,𝑇) space?](https://quant.stackexchange.com/questions/16343/in-dupires-paper-why-is-s-t-t-in-the-k-t-space/71782#71782)
3. [21-256: Implicit partial differentiation](https://www.math.cmu.edu/~cnewstea/teaching/old/teaching/21-256/implicit-partial-diff.pdf)