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...AMS subject classifications. 60F10, 91B70 DOI. 10.1137/090745465 1. Introduction. Large deviations theory provides a natural framework for approximating the exponentially small probabilities associated with the behavior of a diffusion process over a small time interval. In the context of financial mathematics, large deviations theory arises in the computation of small-maturity, out-of-the-money (OTM) call or put option prices or the probability of reaching a default level in a small time period. The theory of large deviations has recently been applied to local and stochastic volatility models [3, 4, 5, 6, 14, 23] and has given very interesting results on the behavior of implied volatilities near maturity. (An implied volatility is the volatility parameter needed in the Black–Scholes formula in order to match a call option price. It is common practice to quote prices in volatility through this transformation.) In the context of stochastic volatility models, the rate function involved in the large deviation estimates is given in terms of a distance function, which in general cannot be calculated in closed form. For particular models, such as the SABR model [13, 15], approximations obtained by expansion ...

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...ity as harmonic mean local volatility In certain regimes, the representation of implied volatility as an average expected volatility can be made precise. Specifically, Berestycki, Busca, and Florent (=-=[7]-=-; BBF henceforth) show that in the short-maturity limit, implied volatility is the harmonic mean of local volatility. The PDE that relates implied volatility I(x, T ) to local volatility σ(x, T ) is, ...

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...ery particular Lévy processes. Literature. Due to their importance for model calibration and testing, small-time asymptotics of option prices have received considerable attention in recent years; see =-=[2, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 22, 24]-=-. A survey of recent literature is given in the introduction of Forde et al. [16]. We shall review only the works most closely related to our study; in particular, we focus on the at-the-money case. H...

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...nificantly with strong emphasis to consider either a purely-continuous model or a purely Lévy model. In case of stochastic volatility models and local volatility models, we can mention, among others, =-=[7]-=-, [8], [12], [17], [18], [20], [23], [33]. For Lévy processes, [36] and [39] show that OTM option prices are generally1 asymptotically equivalent to the time-to-maturity τ as τ → 0. In turn, such a be...

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...ot volatility value is recovered from the implied volatility surface. This has been known for some time in the financial literature, see for example, [21], [6], and [3]. Rigorous proofs were given in =-=[1]-=- and [2] in a different context. Theorem 1. Assume Assumptions 1–5 hold true. The behavior of the implied volatility at expiry is given as follows. For every x ∈ (−x̄, x̄), ∂θXt(0, x) = |σt − xηt(0, x...

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... LIPTON while RRs and BFs we have the following expressions: (8.15) RR (τ) ∼ − √ pi 8 ξ0ξ1τ 1/2, (8.16) BF (τ) ∼ pi 32 ξ0 ( −1 2 ξ0ξ1 + 4 3 ξ0ξ2 + 1 3 ξ21 ) τ . Proof. Formula (8.13) is given in [5], =-=[19]-=-, [20], [40]. Rewriting it in the form (8.17) k σimp (k) = ∫ k 0 dξ σloc (ξ) . and differentiating (8.17) with respect to k three times, we obtain (8.14). Substituting (8.14) in (8.8), (8.9), we obtai...

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...ating function computation in the particular case of the Heston model. 1. Introduction. On one hand, the theory of large deviations has been recently applied to local and stochastic volatility models =-=[1, 2, 4, 5, 20]-=- and has given very interesting results on the behavior of implied volatilities near maturity. (An implied volatility is the volatility parameter needed in the Black–Scholes formula in order to match ...