In Dupire's paper, why is $(S_t, t)$ in the $(K, T)$ space?

Question

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?

Answer 1

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 $$ \sigma^2(K,T) = ... $$

Answer 2

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 $(S_t,t)$

Answer 3

It is probably worth pointing out that the cited article by Rouah has a serious typo/error. Namely, the quoted result in equation 3 is wrong: \begin{equation*} v_L=\frac{\frac{\partial w}{\partial T}}{\left[1-\frac{y}{w}\frac{\partial w}{\partial y}+\frac{1}{2}\frac{\partial^2 w}{\partial y^2}+\frac{1}{4}\left(-\frac{1}{4}-\frac{1}{w}+\frac{y^2}{w}\right)\left(\frac{\partial w}{\partial y}\right)^2\right]} \end{equation*} instead of \begin{equation*} v_L=\frac{\frac{\partial w}{\partial T}}{\left[1-\frac{y}{w}\frac{\partial w}{\partial y}+\frac{1}{2}\frac{\partial^2 w}{\partial y^2}+\frac{1}{4}\left(-\frac{1}{4}-\frac{1}{w}+\frac{y^2}{w^2}\right)\left(\frac{\partial w}{\partial y}\right)^2\right]}. \end{equation*} Unfortunately the wrong equation is also the equation which is "derived" in the body of the article.