Distinguishing Implied Volatility Smiles from Correlation Smiles
Summary
The document explains that a volatility smile and a correlation smile are related concepts but measure different implied quantities. An implied volatility is recovered by inverting an option-pricing formula using an asset’s price, strike, maturity, and option value. For a spread or other product tied to two assets, implied correlation is instead inferred by inverting a pricing formula that also depends on both assets’ implied volatilities, as well as strike, maturity, and product value.
A plotted pattern across strikes may resemble a smile in either quantity, but the shape is market-dependent; the response does not claim that implied correlation always has a smile. It also distinguishes implied quantities from local model inputs. In a local volatility-correlation model, each asset has its own local volatility function, while the asset pair has a local correlation function; these functions can be calibrated to recover observed implied volatilities and correlations. No empirical example or calibration procedure is supplied.
Key ideas
- Implied volatility is inferred from a single-asset option price using an option-pricing model.
- Implied correlation is inferred from a two-asset product price and depends on each asset’s implied volatility.
- A smile describes how an implied quantity varies with strike, but neither quantity must exhibit one.
- Local volatility functions describe individual assets, while local correlation describes the relationship between an asset pair.
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Full text
# What is the difference between a volatility smile and a correlation smile?
# 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?
## Answer by Daneel Olivaw (score 3)
https://quant.stackexchange.com/a/61601
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.