# 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?

• Why does your argument about limit contradict independent and identically distributed gaussians. Also remember for models one usual looks for simplicity mathematically if the simplicity is justified or the alteration is not a sensitive aspect of the reality so I think you want to ask if this assumption, though most likely isn't true, affects the outcome drastically or not – Kamster Apr 5 '15 at 11:10
• Well there should not be any arbitrage, so if $\delta$ is arbitrarily small $C(K,T)$ and $C(K+\delta,T)$ should not differ too much. – Lotus3000 Apr 5 '15 at 13:32
• I don't think your comment about the limit makes any sense. In an idealized world yes, the two options would converge because of no arbitrage. But if the world was ideal then you wouldn't need to model any error at all because everyone would be trading on Black Scholes prices. – analystic Apr 6 '15 at 8:40

## 2 Answers

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.

• Thank you for the answer. I'm looking to model the measurement noise in some sort of intuitive way that does not require much delving into market microstructure effects. – Lotus3000 Apr 4 '15 at 22:35
• Lotus300: in general, when you want to model anything extra, you need either some extra data (as mentioned above) and/or some sort of criterion (like some smoothness, smallest distance to some target curve/surface etc.; I do not know what it would be in this case). If you gave neither additional information nor criterion, independent Gaussians appear to be the best default choice. – Yulia V Apr 10 '15 at 16:58

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.