Choosing Error Functions for Option Model Calibration
Summary
The document asks how to estimate one implied volatility from market prices across several call option strikes at a fixed maturity. It frames calibration as minimizing aggregate pricing errors between theoretical and observed prices, using squared errors as one possible objective and mean absolute deviation as another. The central lesson is that the chosen distance function defines what the calibration prioritizes, so it can affect the fitted parameter when one volatility cannot match every strike exactly.
The answer points to research on calibration risk for exotic options and on heuristic methods for calibrating option pricing models. These references suggest two related lines of inquiry: how the objective function influences calibrated results, and how to solve the resulting optimization problem. The document does not compare loss functions empirically, prescribe a preferred method, or discuss weighting prices by liquidity, scale, or uncertainty. Its example is explicitly hypothetical and concerns a single parameter, so the discussion does not establish guidance for more complex models or datasets.
Key ideas
- A single volatility fitted across strikes can be estimated by minimizing aggregate pricing errors.
- Squared error and mean absolute deviation are alternative calibration objectives.
- The objective function may influence the fitted parameter when market prices cannot all be matched.
- The cited literature concerns calibration risk and heuristic optimization for option pricing models.
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Full text
# Calibrate a model parameter with an error function
# Calibrate a model parameter with an error function
Suppose I want to find the implied volatility using an option model from market prices. Surely I can find the implied volatility for each strike price ($k$ different strike prices) for a given maturity, but this will give me $k$ different implied volatilites. I want using the market values of (e.g calls) to get a single implied volatility. Let me state the problem more formally.
Suppose I want to find a parameter $\sigma_{IV}$. Consider the time to maturity as given. For time to maturity $T$, we have $k$ call option prices $c(K_i), i \in \{1,2,...k\}$. One can find $\sigma_{IV,i} ,i \in \{1,2,...k\}$ but I am not concerned with this. I want to find $\sigma_{IV}$ given these constraints
$C_{Theoretical} (K_i)=C_{Market} (K_i), i \in \{1,2,...k\}$. I am thinking of minimizing an error function, such as the sum of squares of the errors
$$\min_{\sigma_{IV}} \sum_i^k \bigg(C_{Theoretical} (K_i)-C_{Market} (K_i)\bigg)^2$$
Someone however can find a different distance function (e.g Mean Absolute Deviation). Any related literature treating this type of problem?
Note: The example is a hypothetical case for the shake of argument.
## Answer by Enrico Schumann (score 2, accepted)
https://quant.stackexchange.com/a/48936
You mean what influence has the objective function on the results of the calibration? Perhaps look at this paper (there are free versions on the web).
```
@ARTICLE{Detlefsen2007,
author = {Kai Detlefsen and Wolfgang K. H{\"a}rdle},
title = {Calibration Risk for Exotic Options},
journal = {Journal of Derivatives},
year = 2007,
volume = 14,
pages = {47--63},
number = 4,
}
```
If this is more about optimization, perhaps this is useful, too. (I am a co-author; there are free versions on the web as well.)
```
@INCOLLECTION{Gilli2011,
author = {Manfred Gilli and Enrico Schumann},
title = {Calibrating Option Pricing Models with Heuristics},
booktitle = {Natural Computing in Computational Finance},
publisher = {Springer},
year = 2011,
editor = {Brabazon, Anthony and O'Neill, Michael and Maringer,Dietmar},
volume = 4,
}
```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.