Singular Perturbation Scaling in Local Volatility Analysis

Summary

The document addresses a notation issue in a singular perturbation treatment of a local volatility model, referenced in a discussion of managing smile risk under SABR. The questioner is unsure why a coefficient A(K) is declared small, then appears independently in later expansions, and why its derivatives would also need to be small for related quantities to remain finite.

The answer says the apparent inconsistency comes from imprecise notation. The rigorous setup scales the function as epsilon times A(K), then introduces a rescaled variable x equal to (f − K) divided by epsilon. With that scaling, the subsequent analysis follows the usual singular perturbation procedure. The reply is a concise clarification of the asymptotic setup, not a derivation of the expansion or a discussion of its accuracy, assumptions, or numerical implications. Readers need the referenced model analysis for those details.

Key ideas

• The question concerns how a small parameter is used in a local volatility singular perturbation expansion.
• The answer identifies the issue as sloppy notation rather than a requirement that A(K) itself be infinitesimal.
• A clearer formulation scales A(K) by epsilon before introducing a variable rescaled by epsilon.
• The reply invokes the standard singular perturbation procedure but does not derive the expansion or evaluate its accuracy.

Tags

# Motivation of the singular perturbation solution formulation for local volatility model


# Motivation of the singular perturbation solution formulation for local volatility model












I am puzzled by the motivation of the particular choice of the (singular) perturbation method used in Equivalent Black Volatilities. Equation (A.6a) sets $$\epsilon:= A(K)\ll 1.$$ What is the motivation for this setting? I find it surprising that $A(K)$ is to be infinitesimal. However, at later expansion in Equation (A.9a), $A(K)$ seems to be treated independently from $\epsilon$ which is $A(K)$ itself. Moreover, Equation (A.9b) seems to assume $A'(K)$ and $A''(K)$ to be infinitesimal as well, if $\nu_1$ and $\nu_2$ are to be finite. This setting seems to be rather contrived.

What is going on?

This local volatility analysis is referenced in the paper Managing Smile Risk on the SABR model.

## Answer by Hans (score 4, accepted)

https://quant.stackexchange.com/a/38905

In fact, this is a confusion caused by a sloppy notation. The rigorous version of the setup should be $$A(K)\rightarrow \epsilon A(K).$$ Then we let $x:=\frac{f-K}\epsilon$. The rest is the usual singular perturbation operation.

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