Lee’s Moment Formula in the Heston Model

Lee’s moment formula fixes the slopes of the implied total variance at extreme strikes through the largest finite moments of the price. In the Heston model these follow from the moment-explosion time, a closed form with three branches that we write in one convention, so the wing slopes $\beta_R$ and $\beta_L$ are explicit at every maturity. As the maturity falls the critical moments grow like $1/T$ and the ratio of the two slopes tends to $\arccos\rho/(\pi – \arccos\rho)$; as it grows they tend to the roots of a quadratic at the rate $T^{-2}$, and the slopes to closed forms in the ratio $(1-\rho)/(1+\rho)$. For an equity-like set ($v_0 = \theta = 0.04$, $\kappa = 1.5$, $\xi = 0.5$, $\rho = -0.7$, $T = 1$), $\beta_R = 0.024222$ and $\beta_L = 0.085741$. Damped Fourier prices down to $4.9 \times 10^{-70}$, agreeing with 34-digit arithmetic to $8.5 \times 10^{-14}$, show how slowly the smile gets there: at a strike $2981$ times the forward the call wing is still $9.16$ per cent above its slope, and at $1/2981$ of the forward the put wing is $16.2$ per cent above. Lee’s slope bounds the wings; it describes no quoted strike.