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

Question

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?

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.