# 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

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.