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How Bartlett’s Delta Changes SABR Delta Interpretation

Article Quant Q&A · Author: Dejean
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Summary

The document clarifies how beta, the backbone, and volatility skew relate in the SABR model. It explains that beta is selected to represent prior beliefs about the underlying’s distribution and determines how at-the-money volatility changes as the underlying level changes. The volatility skew is mainly shaped by the model’s vol-of-vol and correlation parameters, rather than beta alone.

Bartlett’s delta accounts for correlation between the forward and the SABR volatility parameter alpha. Because alpha depends on the forward, the relevant sensitivity includes an adjustment to the partial derivative of volatility with respect to the forward. This adjustment can reduce the effect of beta choice on delta and other Greeks, especially near at-the-money strikes. The explanation is conceptual and gives no calibration example or empirical comparison; it does not imply that different beta assumptions produce identical skew or identical risk away from the stated setting.

Key ideas

  • SABR beta represents a prior about the underlying distribution and shapes the at-the-money volatility backbone.
  • The vol-of-vol and correlation parameters primarily determine the volatility skew.
  • Bartlett’s delta incorporates the dependence of alpha on the forward through their correlation.
  • The adjustment can attenuate beta’s influence on Greeks, particularly near at-the-money strikes.

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Derivatives pricingVolatilityOptions
Full text
# How Does Bartlett’s Delta Affect the Interpretation of Beta in the SABR Model?


# How Does Bartlett’s Delta Affect the Interpretation of Beta in the SABR Model?












In the SABR model, my understanding is that beta essentially determines the backbone of ATM volatility and is usually pre-specified to reflect prior beliefs about the ATM vol skew.

However, after studying Bartlett’s delta, I’m confused about whether my understanding of beta is correct. Bartlett’s delta essentially adds one adjustment so that delta is less affected by the chosen beta. But isn’t the influence of different betas on delta exactly what we want?

For instance, if I specify beta ≈ 0, I assume Black volatility to rise as rates drop, whereas with beta ≈ 1, the behavior is different. How can Bartlett’s delta yield similar deltas when the skew assumptions are fundamentally different?

## Answer by user35980 (score 2)

https://quant.stackexchange.com/a/81875

In SABR, $\beta$ is chosen and fixed to reflect prior beliefs about the distribution of the underlying (e.g. the forward rate $F$ in the rates world), not about the vol skew. The backbone traces out the ATM vols for different levels of the underlying - again has nothing to do with the vol skew, which is mainly determined by the $\nu,\rho$ parameters.

Barlett's delta simply captures forward/alpha correlation inherent in SABR i.e. since the vol $\sigma$ is a function of $\alpha$, and $\alpha$ is correlated with the forward $F$, the derivative $\frac{d\sigma}{dF}$ will be a partial derivative. Barlett provides an exact expression for this impact in terms of the other SABR parameters, namely: $$ \frac{d\sigma}{dF} \to \frac{d\sigma}{dF}+\frac{\rho\nu}{F^\beta}\frac{d\sigma}{d\alpha}.$$ A consequence of this adjustment is that the effect of the choice of $\beta$ on delta (and other greeks) is attenuated, particularly near the ATM strikes.

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.