# A Look at Small Time Expansions for Heston

Feb 12, 2014 · 2 minute read · CommentsSmall time expansions for Heston can be useful during the calibration of the implied volatility surface, in order to find an initial guess for a local minimizer (for example, Levenberg-Marquardt). Even if they are not so accurate, they capture the dynamic of the model parameters, and that is often enough.

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.