Calibrate a model parameter with an error function

Question

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 1

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,
}