Singular Perturbation in SABR Implied Volatility Derivation

Summary

The document raises a mathematical question about deriving an asymptotic option-price expression used in the Hagan SABR implied volatility formula. It focuses on a partial differential equation for option value, with terms involving the forward price, volatility level, correlation, and a small perturbation parameter. The question asks how singular perturbation expansion leads from that equation to a Gaussian-shaped leading-order expression.

Its practical relevance is to quantitative option pricing and understanding the approximation behind SABR implied volatility. However, the document is a request for explanation rather than a worked derivation: it supplies no expansion steps, boundary conditions, error analysis, or references beyond naming the Hagan paper. Readers should treat it as identifying a difficult step in the derivation, not as providing a complete method or evidence for the formula’s accuracy.

Key ideas

• The question concerns a singular perturbation approach to solving the SABR pricing equation.
• The displayed leading-order expression has a Gaussian form centered on the strike.
• The equation includes volatility dynamics and correlation between the underlying and volatility.
• The document poses the problem but does not provide the derivation or assess approximation error.

Tags

# In-depth derivation of implied volatility in the SABR model


# In-depth derivation of implied volatility in the SABR model












I'm working through the derivation of Hagan's formula (Hagan et al, 2002) for the implied volatility of an option in the SABR model. I'm finding it pretty confusing. Most of my hang-ups are coming from the singular perturbation expansion. Specifically, how does one use singular perturbation expansion to find the solution of, $$ P_\tau = \frac{1}{2}\varepsilon^2\alpha^2C^2(f)P_{ff} + \varepsilon^2\rho\nu\alpha^2C(f)P_{f\alpha} + \frac{1}{2}\varepsilon^2\nu^2\alpha^2P_{\alpha\alpha} $$ to be, $$ P = \frac{\alpha}{\sqrt{2\pi\varepsilon^2C^2(K) \tau }} e^{-\frac{(f-K)^2}{2\varepsilon^2\alpha^2C^2(K)\tau}}\{1+...\}. $$ If anyone has any resources/textbooks that they think might help that would be greatly appreciated!

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