In , the authors state that "Although some studies apply the curve-fitting method directly to option prices, the severely nonlinear relationship between option price and strike price often leads to numerical difficulties. [...], we apply the curve fitting method to implied volatilities instead of option prices".
I've seen this kind of statement made in many other places before, but I haven't yet seen an explanation of why this is the case. Precisely what kind of numerical difficulties can one encounter when fitting a curve to option prices? Isn't the relationship between implied volatilities and strike prices also severely nonlinear?
 Jiang, E J. and Tian, G S., The Model-Free Implied Volatility and Its Information Content ( 2005). The Review of Financial Studies, Vol. 18, Issue 4, pp. 1305-1342, 2005. www.u.arizona.edu/~gjiang/JiangTian-RFS05.pdf