Smile Asymptotics II: Models with Known Moment Generating Functions

Abstract

In a recent article the authors obtained a formula which relates explicitly the tail of risk neutral returns with the wing behavior of the Black Scholes implied volatility smile. In situations where precise tail asymptotics are unknown but a moment generating function is available we first establish, under easy-to-check conditions, tail asymptoics on logarithmic scale as soft applications of standard Tauberian theorems. Such asymptotics are enough to make the tail-wing formula work and we so obtain a version of Lee's moment formula with the novel guarantee that there is indeed a limiting slope when plotting implied variance against log-strike. We apply these results to time-changed Lévy models and the Heston model. In particular, the term-structure of the wings can be analytically understood.

Asymptotics of Forward Implied Volatility

“…PDE methods for continuous-time diffusions [9,32,51], large deviations [15,18], saddlepoint methods [20], Malliavin calculus [7,41] and differential geometry [26,33] are among the main methods used to tackle the small-maturity case. Extreme strike asymptotics arose with the seminal paper by Roger Lee [43] and have been further extended by Benaim and Friz [4,5] and in [30,31,23,15]. Comparatively, large-maturity asymptotics have only been studied in [55,19,36,35,21] using large deviations and saddlepoint methods.…”

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

“…The theory of regularly varying functions has interesting applications in financial mathematics. Besides [2,3,4], such functions appear in the study of Pareto-type tails of distributions of stock returns (see, e.g, [10] and the references therein). We refer the reader to [1] for more information on Pareto-type distributions and their applications.…”

“…Here r ≥ 0 denotes the interest rate. It follows from the martingale condition that (3) x 0 = e −rt E * [X t ] for all t ≥ 0.…”

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.…”