Citation Context

...spect to k exists for k 6= k(τ), and for all k ≤ 0 (14) D−V (k, τ) > −4; (iii) Whenever both one-sided derivatives exist, D−V (k, τ) ≤ D+V (k, τ). (iv) Provided (15) St → 0 in distribution as t ↑ ∞ 8 then for all M > 0 the following inequalities hold: (16) lim sup τ↑∞ sup k∈[−M,M ] max{|D−V (k, τ)|, |D+V (k, τ)|} ≤ 4. (v) The bound (16) is sharp in the sense that there exists a martingale (St)t≥0 such that for all M > 0, DV (k, τ) → −4 as τ ↑ ∞ uniformly for k ∈ [−M, M ]. Remark 5. The flattening of implied volatility as τ ↑ ∞ has been noticed before in the context of specific models, and Lee [10] has shown that the gradient of the implied volatility smile tends to zero pointwise as τ ↑ ∞ under some additional smoothness and finiteness assumptions. The uniform bounds and sharp constant contained in Theorem 5.1 appear to be new. The flattening phenomenon has been incorrectly attributed to the central limit theorem; for instance, see Section 7.3 of Rebonato’s book [12]. We would like to emphasize that the flattening of the implied volatility smile is a universal property of all martingale models. Indeed, suppose that the log stock price is a spectrally negative α-stable Levy process with...

Citation Context

...f implicit volatility, the heuristics is the following: the larger the implied volatility of OTM options, the larger the right tail of ST . This is similar for small strikes K, using Put options. Lee =-=[10]-=- has been the first one to quantify these features relating the behavior of implied volatility to the tails of ST , with an encoding of the tails through the existence of positive/negative moments. Th...

Citation Context

... C(S,K,T). Implied volatility allows to compare option prices for different strikes, maturities, underlyings, and valuation times. For entry points to the literature on implied volatility, see, e.g., =-=[9, 10, 11, 16, 21]-=-. The compute the implied volatility from a call price, equation (2) has to be solved numerically; many authors have added that the inversion cannot be done in closed form. The latter statement has be...

Citation Context

...ally, we discuss the implied volatility skew. Suppose that the underlying S generates the call price surface C(K, T ): C(K, T ) = e−rT E[(ST −K)+], K > 0, T > 0. Then the implied volatility (see e.g. =-=[25]-=-) for strike K and maturity T is the volatility σimp(K, T ) that makes the Black-Scholes call price equal to C(K, T ): CBS(K, σimp, T ) = C(K, T ). The map K → σimp(K, T ) is called the volatility smi...

Citation Context

...easing and convex function of strike to exclude static arbitrage opportunities. Other work has been done on the necessary conditions on implied volatility for the absence of arbitrage, as surveyed in =-=[33]-=-. The main results include the moment formula for implied volatility at extreme strikes by Lee [32], and the implied volatility expansion under fast mean reversion by Fouque, Papanicolaou and Sircar [...