Dupire Local Volatility and the Need for an Arbitrage-Free Surface

Summary

The note distinguishes the Dupire method from the broader concept of local volatility. One answer describes Dupire as a method for deriving a local volatility surface from an implied volatility surface; another says the famous Dupire model is commonly used as another name for the local volatility model. The distinction is that local volatility is a modeling concept, while Dupire’s equation is one approach to constructing its surface from market option prices.

A key implementation caveat is that the input implied volatility surface should be arbitrage-free. Otherwise, the derived instantaneous variance can be negative, making the resulting local volatility invalid. The note recommends smoothing the implied volatility surface before applying the method. It mentions cubic splines as a smoothing choice that does not guarantee arbitrage freedom and contrasts this with a method that does. It gives conceptual guidance, but no derivation, calibration example, or comparison of model performance.

Key ideas

• Dupire’s equation can derive a local volatility surface from an implied volatility surface.
• Local volatility is a broader concept than any one construction method.
• The implied volatility input should be arbitrage-free to avoid invalid negative instantaneous variance.
• Smoothing before deriving local volatility matters, and not every smoothing method preserves arbitrage freedom.

Tags

# Dupire model and Local Volatility model


# Dupire model and Local Volatility model












In the context of Option pricing model.

Is there a difference between the Dupire Model and the Local volatility model ?

Thanks Achal

## Answer by Karthik Balasubramaniam (score 2, accepted)

https://quant.stackexchange.com/a/15630

Dupire model is just one way of generating a local volatility surface from an implied volatility surface. There are many other ways to generate a local volatility surface. One critical aspect of Dupire model is that the input implied volatility (IV) surface should be arbitrage free. If not, you will negative instantaneous variance when generating the local volatility surface. Consequently your local volatility (square root of variance) would be invalid.

There are many ways of smoothing the implied volatility surface. Cubic spline is a good, but does not guarantee an aribtrage free IV surface. Fengler model does. http://www.econbiz.de/archiv1/2008/58072_arbitrage-free_smoothing.pdf Also remember to smooth the IV surface before generating a local volatility surface.

## Answer by Drew (score 1)

https://quant.stackexchange.com/a/15622

No, if you are referring to the famous Dupire Model (there are others), then they are the same. It is usually referred to as the Local Volatility Model and the Dupire Equation. I would disentagle those with the concept of Local Volatility, which is model independent and a fairly deep result.

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