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I'm new to local volatility model.

From Dupire's paper and most of the textbooks, they derived the local volatility $\sigma(K, T)$ in the $(K, T)$ (i.e., strike and maturity) space, from call prices or the implied volatility surface.

However, by definition, local volatility is a function in terms of $(S_t, t)$, i.e., instantaneous underlying price and time.

How to relate these two?

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    $\begingroup$ The strike prices are values that the stock price may reach. Both strike and stock prices are in the same unit of measure, but of course are different variables/parameters. I find it coherent. $\endgroup$ – Arrigo Jan 26 '15 at 11:57
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This is merely a question of notation, you should simply read $$ \sigma(K,T) = \sigma(S_t=K, t=T) $$

For an easy to follow derivation see this excellent note from Fabrice Rouah

Some intuition behind the developments:

  • The price of a European option, for instance a call, can be written in integral form: $$ C(t, S_t, K, T) = e^{-r(T-t)} \int_0^\infty (S_T-K)^+ \phi(S_T,T; S_t, t) dS_T \tag{1} $$ where $\phi(S_T=S,T;S_t,t) := f(S,T)$ figures the pdf of moving from the known current state $(S_t,t)$ to some future state $(S_T=S,T)$. This is a model free result.
  • Now, consider a local volatility dynamics $$\frac{dS_t}{S_t} = \mu(S_t,t) dt + \sigma(S_t,t) dW_t$$ it is well-known that the conditional pdf $\phi(S_T=S,T;S_t,t) = f(S,T)$ in that case solves the following equation (Kolmogorov forward or Fokker-Planck equation): $$ \frac{\partial}{\partial T}f(S,T) = -\frac{\partial}{\partial S} \left[ \mu(S,T) f(S,T) \right] + \frac{1}{2} \frac{\partial^2}{\partial S^2} \left[ \sigma^2(S,T) f(T,S) \right] \tag{2} $$ with initial condition $$f(S,t) = \phi(S_T=S,T=t; S_t, t) = \delta(S-S_t)$$ where we used the notation $$ \sigma^2(S,T) = \sigma^2(S_t=S,t=T) $$

  • Writing the time derivative of $(1)$ with respect to $T$, one can make $\frac{\partial}{\partial T}f(S,T)$ appear. This is useful, since we can now replace its expression given by $(2)$, meaning we have related some (time) derivative of the call price to our local volatility function $\sigma(.,.)$

  • The expression we get can be further simplified by identifying the spatial derivatives of the call price given by $(1)$. This involves computing space integrals, where the local volatility function ends up being evaluated at $S=K$, and finally yields the famous Dupire stripping formula $$ \sigma^2(K,T) = ... $$

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The local volatility is just a $\mathbb{R}_+\times[0,T]\mapsto \mathbb{R}_+$ function where $T$ is some time horizon. It is the solution of a simple equation so it expression is written as $\sigma(K,t)$ but here $K$ is essentially a notation to denote a strike value as the Dupire equation relates the function $\sigma$ to vanilla market prices at a given strike. Once computed (calibrated) the local volatility function is used during the diffusion of the underlying asset so we evaluate it at pair $(S_t,t)$

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