Limits of Predicting Delta-Gamma Approximation Error

Summary

The document raises the question of whether the sign of the error in a delta-gamma estimate of a warrant's price change can be predicted. The estimate combines the current price with delta times the underlying move and a gamma-based quadratic adjustment. It is presented as a local approximation that assumes a small underlying price change while other inputs, such as volatility and interest rates, remain fixed.

The author observes that larger moves can increase the absolute approximation error and asks whether the estimate will systematically overstate or understate the realized price change. No answer or supporting analysis is included, so the document does not establish a rule for predicting the error's direction. It also notes a practical data limitation: the warrant source does not report gamma, requiring it to be inferred from recent price changes. That estimate may be noisy, and changes in other pricing inputs further limit conclusions from the approximation.

Key ideas

• The delta-gamma approximation estimates price changes with first- and second-order sensitivity to the underlying.
• Its stated setup assumes a small underlying move and unchanged volatility, rates, and other inputs.
• The document asks whether approximation error has a predictable sign but supplies no resolution.
• Gamma may need to be estimated from observed warrant price changes when it is not published.
• Estimation noise and changing market inputs can limit the reliability of the approximation.

Tags

# Is the sign of the delta-gamma approximation error predictable?


# Is the sign of the delta-gamma approximation error predictable?












I self-study quantitative finance, but I have a hard time connecting the textbook formula with the market reality and available data.

I use delta-gamma approximation to estimate the price change of warrants on the French market after a small change $dS$ of the underlying asset:

$P_{est} \approx P_0 + \Delta * dS + \frac12 * \Gamma * (dS)^2 $

This is an assumption based on the fact that $dS$ is small all other parameters (volatility, interest rate, ...) will remain constant. I understand that when $|dS|$ grows, the error ${err} = P_{real} - P_{est}$ grows in absolute value. But can we know for sure the sign of the error? In other words, is the estimate always wrong by excess (resp. default), or is this something we can't know?

FWIW, below is an example of the typical data available for a Warrant on the emitter website. Notice the gamma isn't specified, and I have to estimate it from recent price changes (which is sub-optimal, I suppose).

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.