The large-maturity smile for the Heston model

Reformulating the results of del Baño Rollin, Ferreiro-Castilla, and Utzet [3], we give necessary and sufficient conditions for the moments of the stock price in the Heston model to exist and extend Theorem 2.1 of [5]. Forde and Jacquier [5] provide necessary conditions for the moments to exist when κ > ρσ. Although this assumption is satisfied on Equity markets (because the correlation is generally negative), it does not hold for FX-related derivatives. Furthermore we show that the application of the Gärtner-Ellis theorem attempted in [5] fails to obtain the asymptotic behavior of calls or puts with large maturity when κ > ρσ (the case investigated in their paper). Nevertheless it can be used for put options when κ 6 ρσ. To show this, we give a detailed classification of the cases when the rate function is essentially smooth under both the original and the share measures. This classification complements and corrects [6] and [5].