Regular Variation and Smile Asymptotics

Abstract

We consider risk-neutral returns and show how their tail asymptotics translate directly to asymptotics of the implied volatility smile, thereby sharpening Roger Lee's celebrated moment formula. The theory of regular variation provides the ideal mathematical framework to formulate and prove such results. The practical value of our formulae comes from the vast literature on tail asymptotics and our conditions are often seen to be true by simple inspection of known results.

Options with Extreme Strikes

“…Note that (2.6) is implied by Theorem 1 and 2 in Benim and Friz [24]. Formula (2.6) will be used repeatedly in the proofs later in the paper.…”

“…In Benaim and Friz [24], they studied the price of the vanilla European options, the implied volatility at extreme strikes for different underlying stock price processes and sharpened Lee's moment formulas [23]. There is a one-to-one correspondence between the tail probabilities of P(S T ≥ K) for K → ∞ and the asymptotic tails of the option price C(K) = e −rT E[(S T − K) + ] and the implied volatility.…”

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

“…as K → 0, where K → P (K) is the put pricing function corresponding to C. Formulas (1) and (2) are valid under very mild restrictions on call and put pricing function. For special stochastic volatility models, sharp asymptotic formulas for the implied volatility were established in the papers of E. M. Stein and the author (see [20,22,19,21]).…”

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.…”