Modeling Correlated Measurement Error in Option Surfaces

# Covariance structure of call option surface


# Covariance structure of call option surface












Assume the observed call option prices $C(K_i,T_i)$ for $i = 1,\dots,N$ are disturbed by some unknown measurement noise $\epsilon$. What would an appropriate covariance structure be for $\epsilon$?

In literature I often see authors making the simplified assumption that $\epsilon_i$ are independent and identically distributed Gaussian with some scaling variance that could depend on for example the bid-ask spread. This seems very unreasonable since if one considers two options $C(K,T)$ and $C(K + \delta,T)$ then in the limit $\delta \to 0$ they should be perfectly correlated.

Has anyone read any literature that discusses these types of modelling choices in more detail?

## Answer by Yulia V (score 2)

https://quant.stackexchange.com/a/17241

You need to see the deals on these options and/or have deep knowledge of how these prices are marked to be able to have a better model.

First thing first, I believe that the prices that you see are usually either "marked" (set) by one or several treaders, or they are the prices on last transaction before the close of the market/first transaction of the day etc. (and if there was no transaction during the day, either some interpolation or the price observed the day/week before etc.) In the former case, the person(s) who mark the prices may introduce subjective bias. In the latter case, if the trades did not happen at the same moment, you will have asynchronisity bias. There is also some error introduced by rounding of the prices. All these errors ideally should be modeled in a different way, but I guess it is pretty hard to get data/information to do so, thus independent Gaussian variables.

Some strikes are more liquid than the others, and the error on those should be smaller. I guess this one is easier to model.

## Answer by AFK (score 2)

https://quant.stackexchange.com/a/21004

Most practitioners think of option prices in terms of implied volatility. It is easier to interpret and to model. One can consider the implied volatility surface as a random field : $\Sigma : \Omega \times \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$ and apply PCA. The first 3 eigenmodes correspond to absolute level (ATM vol), slope in the strike direction (skew) and curvature in strike direction (smile). See Dynamic of the implied volatility surface by Cont & de Fonseca. Considering pertubation of the coefficients of the modes by a Gaussian noise and then applying the BS formula probably gives you a more realistic and insightful pertubation of the option prices than trying to model directly the movements of the prices.