On refined volatility smile expansion in the Heston model

Abstract

It is known that Heston's stochastic volatility model exhibits moment explosion, and that the critical moment s+ can be obtained by solving (numerically) a simple equation. This yields a leading order expansion for the implied volatility at large strikes: σBS(k, T ) 2 T ∼ Ψ(s+ − 1) × k (Roger Lee's moment formula). Motivated by recent "tail-wing" refinements of this moment formula, we first derive a novel tail expansion for the Heston density, sharpening previous work of Drȃgulescu and Yakovenko [Quant. Finance 2, 6 (2002), 443-453], and then show the validity of a refined expansion of the type σBS(k, T ) 2 T = (β1k 1/2 + β2 + . . . ) 2 , where all constants are explicitly known as functions of s+, the Heston model parameters, spot vol and maturity T . In the case of the "zero-correlation" Heston model such an expansion was derived by Gulisashvili and Stein [Appl. Math. Optim. 61, 3 (2010), 287-315]. Our methods and results may prove useful beyond the Heston model: the entire quantitative analysis is based on affine principles: at no point do we need knowledge of the (explicit, but cumbersome) closed form expression of the Fourier transform of log ST (equivalently: Mellin transform of ST ); what matters is that these transforms satisfy ordinary differential equations of Riccati type. Secondly, our analysis reveals a new parameter ("critical slope"), defined in a model free manner, which drives the second and higher order terms in tail-and implied volatility expansions.

Asymptotic Formulas with Error Estimates for Call Pricing Functions and the Implied Volatility at Extreme Strikes

“…1 Formula (82) was recently established for the stock price density in the correlated Heston model, and a better error estimate O((log x) −1/2 ) was obtained in this formula (see [16]). Moreover, using the methods developed in the present paper, formula (105) (see section 7) was extended to the implied volatility in the correlated Heston model.…”

“…Let C ∈ P F ∞ , and suppose C is a positive function satisfying the following condition: There exist ν > 0 and K 0 > 0 such that (16) log 1…”

Large-Maturity Regimes of the Heston Forward Smile

“…Roger Lee [55] was the first to study extreme strike asymptotics, and further works on this have been carried out by Benaim and Friz [6,7] and in [39,40,41,31,23,19]. Large-maturity asymptotics have only been studied in [67,27,46,45,29] using large deviations and saddlepoint methods.…”

Shapes of Implied Volatility with Positive Mass at Zero

“…Note that the logarithm of the cdf F (K x ) appears in the formula, instead of the cdf itself as in (3.4). In many stochastic volatility models, such as Heston and Stein-Stein, the cdf of the stock price satisfies [21,13],…”