Calibration of options on a reduced basis

∂tC + 1 2 σS∂SSC + rS∂SC − rC = 0, C(S, T ) = (S −K) (1) where r is the interest rate and σ the volatility. While in the original model σ is constant, it is common practice to calibrate (adjust) σ to allow (??) to reproduce market observations Ck at S0, t0, namely known calls with maturities Tk and strikes Kk, k=1..K. The calibration problem (also called the “smile”) is quite stiff. Avellaneda et al [?] proposed a way around by minimizing an entropy function instead of using the Black-Scholes model; other authors have used optimization and control theory (see Achdou et al[?] and the references therein) at the cost of large computing time somehow unfit to trading.