# What is the difference between a volatility smile and a correlation smile?

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?

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 $$Y_t$$.
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.
• 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. Mar 9, 2021 at 16:18