I understand to plot correlation and volatility smiles, we have to plot the implied normal vol vs strike and observe a U-shaped relationship. How are these smiles different? Does a vol smile plotted for a correlation product (CMS spread for example) become to be called a Correlation smile?


1 Answer 1


No, they are two different but similar things. For two assets $X$ and $Y$:

  • Implied volatility is the function (and similarly for $Y$) $$\widetilde{\sigma}_X:(X_t,K,T,V_t)\rightarrow f^{-1}(X_t,K,T,V_t),$$ that is the inverse of the Black-Scholes formula $f$ (or Black for normal distributions) given a spot price $X_t$ ($Y_t$ for the other asset), a strike $K$, term $T$ and option price $V_t$.
  • Similarly, the implied correlation $$\widetilde{\rho}:(X_t,Y_t,\widetilde{\sigma}_X,\widetilde{\sigma}_Y,K,T,V_t)\rightarrow f^{-1}(X_t,Y_t,\widetilde{\sigma}_X,\widetilde{\sigma}_Y,K,T,V_t)$$ inverts a Black-Scholes or Black pricing formula for a spread product and gives you the corresponding correlation parameter. Of course, in this case the implied correlation also depends of the individual implied volatilities of $X_t$ and $Y_t$.

Whether there is a smile or not is circumstantial (I don't know whether implied correlation exhibits a smile shape, and I know that not all asset classes have a smile volatility).

You can then implement a local volatility-correlation model for the two assets $X$ and $Y$ such that (ignoring details about the drift): $$\begin{align} dX(t)&=\color{blue}{\sigma_X(t,X_t)}dW_X(t)+\mathcal{\scriptsize O}(dt) \\ dY(t)&=\color{blue}{\sigma_Y(t,Y_t)}dW_Y(t)+\mathcal{\scriptsize O}(dt) \\ d\langle X,Y\rangle(t)&=\color{blue}{\rho(t,X(t),Y(t))}dt \end{align}$$ then $\sigma_X$ and $\sigma_Y$ are the local volatilities for the assets $X$ and $Y$, whereas $\rho$ is the local correlation for the asset pair. The 3 local functions will then be calibrated to recover implied volatilities and correlations.

  • $\begingroup$ Thanks, but it does not answer the question. What is the correlation smile, how do we plot it for cms spread options, for example? $\endgroup$
    – Bravo
    Mar 9, 2021 at 15:39
  • $\begingroup$ @Bravo I added further details. $\endgroup$ Mar 9, 2021 at 15:59
  • 1
    $\begingroup$ What is the correlation smile? The correlation smile refers to the fact that the (local) correlation $\rho(t,X(t),Y(t))$ is sensitive to the price levels of the two assets X and Y. So you cannot assume a constant correlation, just like in the vol smile case you cannot assume a constant $\sigma$ as Black Scholes did. $\endgroup$
    – nbbo2
    Mar 9, 2021 at 16:18

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