Interpreting SABR Local Volatility Shapes Through the Distribution Backbone

Summary

The document asks how to interpret the local volatility shapes implied by a stochastic-local volatility specification related to SABR. In this setup, the forward rate is scaled by a power of its level, with the exponent controlling the backbone; the author focuses on the square-root shape associated with an exponent of one half and the linearly increasing shape associated with an exponent of one. The central question is why these functions are sensible choices for local volatility.

The author connects the exponent to the distribution's normal or lognormal character, but does not supply an answer or empirical evidence explaining the shapes. The material is therefore a conceptual question rather than a resolved modeling explanation. It identifies a distinction between a parametric volatility backbone and the strike-dependent local volatility surface recovered through Dupire's formula, while leaving calibration implications and practical model choice open.

Key ideas

• In the described SABR setup, the exponent determines how volatility scales with the forward rate.
• An exponent of one half produces a square-root dependence, while an exponent of one gives a linearly increasing dependence.
• The author relates the exponent to the distribution backbone and its normal or lognormal character.
• The document raises, but does not resolve, the intuition behind choosing these local volatility shapes.

Tags

# Intuition behind local volatility curve shapes in interest rate environments


# 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?

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.