Singular Perturbation and the SABR Smile-Risk Density

Summary

The document asks how Hagan’s SABR analysis derives two asymptotic forms for a transition density governed by a PDE with a small parameter. A regular perturbation around the strike produces a leading Gaussian using the local value of the diffusion function, with correction terms in the scaled displacement from the strike.

The central point is that this expansion can lose accuracy when the diffusion function changes appreciably between the strike and the point being evaluated. Reorganizing corrections into the exponential is presented as a singular-perturbation style approximation that better captures this change in the density’s tail. The document gives the equations and qualitative rationale, but no derivation, worked example, or numerical comparison; it is a question seeking clarification, so the precise steps and validity conditions are not established here.

Key ideas

• A small-parameter expansion around the strike yields a Gaussian density based on local volatility scaling.
• The regular expansion includes corrections involving displacement from the strike measured in diffusion-scaled units.
• The approximation can degrade when the diffusion coefficient differs substantially from its value at the strike.
• Reorganizing corrections within the exponential is proposed to better represent density changes away from the strike.
• The document poses the derivation question but does not supply a full derivation or validation.

Tags

# Singular Perturbation in Hagan's 2002 SABR paper "Managing Smile Risk"


# Singular Perturbation in Hagan's 2002 SABR paper "Managing Smile Risk"












I'm reading Hagan's 2002 paper Managing Smile Risk originally published on the WILMOTT magazine, and got something confusing.

The set up: $P(τ,f,α,K)$ is the solution of the problem as in Equation (A.13a) $$P_τ=\frac12 ε^2 α^2 C^2 (f) P_{ff}+ε^2 ρνα^2 C(f) P_{fα}+ \frac12 ε^2 ν^2 α^2 P_{αα}$$ for $τ>0$, and boundary condition as in Equation (A.13b) $$P=α^2 δ(f-K)$$ for $τ=0$.

Here $\varepsilon \ll 1$ is the perturbation; while $\rho$, $\nu$ are constants.

Then it says:

Using a straightforward perturbation expansion would yield a Gaussian density to leading order, as in Equation (A.15a) $$P=\frac{α}{\sqrt{2πε^2 C^2 (K)τ}} \exp⁡\left[-\frac{(f-K)^2}{2ε^2 α^2 C^2 (K)τ}\right]⋅(1+⋯)$$

Since the “+⋯” involves powers of $(f-K)/εαC(K)$, this expansion would become inaccurate as soon as $(f-K)C'(K)/C(K)$ becomes a significant fraction of 1; i.e., as soon as $C(f)$ and $C(K)$ are significantly different.

Stated differently, small changes in the exponent cause much greater changes in the probability density. A better approach is to re-cast the series as (A.15b) $$P=\frac{α}{\sqrt{2πε^2 C^2 (K)τ}} \exp⁡\left[-\frac{(f-K)^2}{2ε^2 α^2 C^2 (K)τ}⋅(1+⋯)\right]$$

Now I'm confused: Not familiar with singular perturbation, I don't quite get how (A.15a) or (A.15b) are derived. Could any one give some hint pls?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.