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...orter. For instance, in the FX option markets, which are among the most liquid derivatives markets in the world, options with fixed values of the Black-Scholes delta are quoted for each maturity (see =-=[2]-=- for the details on the conventions in FX option markets and a natural parameterisation of the smile using the Black-Scholes delta). The market data in Figure 1 therefore suggests that, in order to un...

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...ey options converges to the volatility of the diffusion component as the maturity tends to zero. Small-maturity asymptotics for models with jumps (including Lévy processes) have been investigated in =-=[2, 3, 48, 45, 44, 16]-=-. However, these asymptotics do not provide any information on the forward smile or forward-start-based payoffs, such as cliquets and forward-start options and the literature on this topic is sparse. ...

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...epness of the smile wings, as given by Lee’s moment formula. 1. Introduction Recent years have seen an explosion of the literature on asymptotics of option prices and implied volatilities (see, e.g., =-=[3, 17]-=- for many references). Such results are of practical relevance for fast model calibration, qualitative model assessment, and parametrization design. The small-time behavior of the level of implied vol...

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...[10]) is constructed. 2. A duality formula for Black–Scholes call prices The main result of this section is the following duality formula: Lemma 2.1. For all k ∈ R and v ≥ 0 we have CBS(k, v) = sup u∈=-=[0,1]-=- [ Φ ( Φ−1(u) + √ v )− eku] . Proof. Fix k ∈ R and v ≥ 0 and let g(u) = Φ ( Φ−1(u) + √ v )− eku. If v = 0 then g(u) = u(1 − ek) ≤ (1 − ek)+ with equality if u = 1{k<0}. So suppose v > 0. Note that the...

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...volatility. 1 Introduction Stemming in part from its importance for model calibration and testing, small-time asymptotics of option prices have received a lot of attention in recent years (see, e.g., =-=[3]-=-, [5], [6], [10], [11], [12], [13], [15], and references therein). In the present paper, we study the small-time behavior for at-the-money (ATM) call (or equivalently, put) option prices [ Π(t) := E (...

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...volatility. 1 Introduction Stemming in part from its importance for model testing and calibration, small-time asymptotics of option prices have received a lot of attention in recent years (see, e.g., =-=[3]-=-, [5], [10], [11], [18], [19], [20], [22], [25], and references therein). The fact that option prices and implied volatilities exhibit sharply different behaviors under different model assumptions pro...

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...thout jumps in [15] and for the Kou model in [29] and [16]. These formulas can be extended to include five terms and an error estimate. We would also like to bring the reader’s attention to the paper =-=[3]-=- concerning the asymptotic behavior of the implied volatility in exponential Lévy models. We will next briefly overview the contents of the present paper. In Subsection 2.1, we define the Mellin conv...

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...2 ANTOINE JACQUIER AND PATRICK ROOME (including Lévy processes), studied in the above references for large maturities and extreme strikes, ‘explode’ in small time, in a precise sense investigated in =-=[1, 2, 61, 55, 54, 20]-=-. On the other hand the literature on asymptotics of forward-start options and the forward smile is sparse. Glasserman and Wu [33] use different notions of forward volatilities to assess their predict...