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Heston and SABR: Calibration Fit and Arbitrage Constraints

Article Quant Q&A · Author: user6703592
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Summary

The note distinguishes fitting observed option prices from ensuring that a pricing model is free of arbitrage. It explains that Heston and SABR each have a limited number of parameters and restrict the volatility shapes they can produce, so neither is guaranteed to match every market quote exactly, even when fitting a small set of quotes at one maturity.

Both underlying stochastic volatility models are described as arbitrage-free, but the commonly used SABR volatility formula is an approximation and may introduce arbitrage. The note points to arbitrage-free interpolation methods, including Andreasen-Huge and Le Floc'h's C2 method. It cautions that using enough parameters, such as a spline with one degree of freedom per quote, can improve fit but does not by itself ensure arbitrage freedom. No empirical comparison or implementation details are provided.

Key ideas

  • A model's arbitrage properties and its ability to fit market quotes exactly are separate questions.
  • Heston and SABR have limited parameters and cannot guarantee an exact fit to every set of option prices.
  • The standard SABR volatility formula is approximate and may not preserve the arbitrage properties of the full model.
  • A flexible spline can fit quotes closely, but is not generally arbitrage-free.
  • The note names Andreasen-Huge and Le Floc'h C2 interpolation as arbitrage-free alternatives.

Tags

OptionsVolatilityDerivatives pricingArbitrageStatistics
Full text
# Does Heston and SABR match market vol smile perfectly (arbitrage free)?


# Does Heston and SABR match market vol smile perfectly (arbitrage free)?












What I saw in the references is

> Heston model can matches market option prices perfectly and SABR cannot.

Is it correct? But for my understanding, a model matches market option prices perfectly only when its equivalent local vol model should strictly follow Dupire equation (Dupire local vol model).

## Answer by jherek (score 2, accepted)

https://quant.stackexchange.com/a/73857

The question mixes up various concepts:

- Exact fit of model to market prices. Neither Heston nor SABR can claim to fit exactly market prices. There are two reasons: both have a limited number of parameters (5 for Heston, 4 for SABR), and their corresponding models further restrict the shapes attainable with those parameters. Even for a small number (let's say 3) quotes at a given maturity, they are not guaranteed to match those exactly.

- Arbitrages in the parameterization: both stochastic volatility models do not allow arbitrage. But, for SABR, the practice is to use an approximation formula, which is not always arbitrage-free (it is only an approximation of the actual SABR stochastic volatility model).



For a perfect fit, you typically need at least as many parameters as there are quotes (an approach sometimes called non-parametric). You could use a spline but this will not be arbitrage free in general. In terms of arbitrage-free methods, there is:

- Andreasen Huge volatility interpolation

- Le Floc'h arbitrage free interpolation of class C2

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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