In 2011, Forde et al. proposed a second order small time expansion around the money, which I found to work well for calibration. More recently, Lorig et al. proposed different expansions up to order-3 around the money. I already looked at the later in my previous post, applying the idea to Schobel-Zhu.
I noticed, however, that on some surfaces, the Lorig expansion was quickly very inaccurate (LPP1 for order-1, LPP2 for order-2, LPP3 for order-3). Those surfaces seem to be the ones were the Feller condition is largely violated. In practice, in my set of volatility surfaces for 10 different equities/indices, the best fit is always produced by Heston parameters where the Feller condition is violated.
|T=0.5, Feller condition largely violated|
|T=0.5, Feller condition slightly violated|
Out of curiosity, I calibrated my surfaces feeding the order-1 approximation to the differential evolution, in order to find my initial guess, and it worked for all surfaces.
The order-3 formula, even though it is more precise at the money, was actually more problematic for calibration: it failed to find a good enough initial guess in some cases, maybe because the reference data should be truncated, to possibly keep the few shortest expiries, and close to ATM strikes.