# Intuition behind local volatility curve shapes in interest rate environments

I have some questions regarding the intuition behind shapes for the local volatility (LV) curve as seen in quite popular models. Let's say we have the following generalized stochastic-local volatility (SLV) model for modeling forward rates $$dF(t)=\alpha(t)\sigma(F(t),t))dW_1(t)$$ $$d\alpha(t)=\nu\alpha(t)dW_2(t)$$ where $$dW_1(t)dW_2(t)=\rho dt$$ for some $$\rho\in[-1,1]$$. Then we could recover the standard SABR model by setting $$\sigma(F(t),t)=F^\beta(t)$$ for some $$\beta\in[0,1]$$. However, what is the intuition behind the shape of the curve admitted by this LV specification for, e.g. $$\beta=0.5$$? This would yield an LV curve that has a square root function type shape, but this feels counter-intuitive as an LV curve shape. I can generally understand the shapes of LV curves that follow from application of Dupire's formula, as they make sense intuitively, but I cannot say the same about this function.

Another example would be LV "curve" you get if you would set $$\beta=1$$. This isn't even a curve but just a straight monotonically increasing line. What is the sense behind selecting this as your LV function?

I understand that for the SABR example the selection for $$\beta$$ is in direct relation to the backbone of the distribution of $$F(t)$$ and hence influences the (log)normality of this distribution, but what can be said about the shapes of the selected LV curve?

• Note that for a stochastic-local volatility (SLV) model that is perfectly calibrated to the vanilla prices, you must apply Dupire's formula to get the LV component, which is a result due to Gyongy's lemma. For $\beta=0.5$, you assume vol increases in square root of $F$, while for $\beta=1$, you assume vol increases in linearly with $F$. The LV curve (surface) is basically a function of volatility against spot (and time).
– ryc
Jul 18 '20 at 14:39