# 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.

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,
}

• I am wondering about the bias and the sensitivity to small changes of other parameters. I'll see the article and follow up. – alexbougias Sep 27 '19 at 5:33