Local Volatility Parameterizations and the Dupire Surface

Summary

The document asks how local volatility, defined as a function of spot and time, can appear in Dupire’s equation as a function of strike and maturity. It also asks what functional forms are used to represent local volatility in practice and whether a quadratic or cubic form is adequate. The reply lists several proposed parameterizations from the literature, including cubic polynomials, piecewise quadratics, cubic splines, hyperbolic trigonometric functions, and Hermite polynomials.

The response says a quadratic or cubic specification may be sufficient as a starting point, while emphasizing that the parameterization should preserve the static smile property. It does not derive the connection between spot-time local volatility and strike-maturity quantities in the Dupire equation, nor does it compare the listed forms empirically. The guidance is therefore a brief starting point, not a calibration recipe or evidence that one functional form is generally best.

Key ideas

• The question contrasts spot-time local volatility with strike-maturity inputs in Dupire’s equation.
• Proposed forms include polynomials, splines, hyperbolic trigonometric functions, and Hermite polynomials.
• Quadratic and cubic forms are suggested as possible starting choices.
• A chosen parameterization should preserve the static smile property.
• The response offers no empirical comparison or detailed calibration procedure.

Tags

# Question on Local Volatility


# Question on Local Volatility












In Gatheral's book The Volatility Surface (Wiley, 2006), a local volatility model is defined as... $$ dS_t =S_t \mu_tdt + S_t \sigma(S_t, t)dZ $$ The famous Dupire Equation is given by... $$ \sigma^2(K, T, S) = \frac{\partial C/\partial T}{\frac{1}{2} K^2\partial^2C/\partial K^2} $$

I have two questions...

(a) according to the definition of local vol, $\sigma$ is a function of t and S. How come it also depends on K in the second equation?

(b) In practice, what kind of functions do people use for $\sigma(t, S_t)$? Would a quadratic function or cubic function suffice?

## Answer by Karl L (score 4)

https://quant.stackexchange.com/a/45505

To answer your second question, per "Pricing, Hedging and Trading Financial Instruments" by Carol Alexander, the following approaches have been proposed in literature:

- cubic polynomials (Dumas et al., 1998)

- piecewise quadratic functions (Beaglehole and Chebanier, 2002)

- cubic splines (Coleman et al., 1999)

- hyperbolic trigonometric functions (Brown and Randall, 1999)

- Hermite Polynomials (McIntyre, 2001)

Hopefully this list can give you a place to start. A quadratic or cubic function would likely suffice however be sure that the parameterization you choose preserves the static smile property of the local volatility(i.e. the underlying price of the security should not change the volatility at any point on the surface)

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