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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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... 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 behavior impl...

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...the previous paragraph, a problem that has received a great deal of attention in the last few years is the study of small-time asymptotics for option prices and implied volatilities (see, e.g., [17], =-=[10]-=-, [15], [6], [11], [32], [35], [16], [25], [13]). As a byproduct of the asymptotics for the tail distributions (1.4), we derive here the leading term of the small-time expansion for the arbitrage-free...

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... D we check that I2(x;x0,t) = t¯ ( x0−x L0 t endre transform of ¯ |x0−x| 2 H0 defined in (3.15); and I4 = 2¯σ 2 . t 6.2. Option prices. Proof of Corollary 2.1. We follow the proof of Corollary 1.3 in =-=[11]-=- and show that lim ǫ→0 +ǫlogE[ (Sǫ,t −K) +] is bounded above and below by −Ir(logK;x0,t). Recall that we are considering out-of-the-money call options and hence x0 < logK (see (2.4)). Since our rate f...

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...in (13). Fortunately, this hurdle can be overcome by the asymptotic approximation of the joint default probability (see Equation (9)) because its leading-order approximation term has a constant correlation ρ . As a result, our methodology to estimate the two-name joint default probability with stochastic correlation is simply to apply the efficient importance sampling scheme associated with the effective correlation, derived from the singular perturbation analysis. Detailed variance analysis for this methodology is left as a future work. A recent large deviation theory derived in Feng et al. [8] can be a 8 Chuan-Hsiang Han valuable source to provide a guideline for solving this theoretical problem. Table 1 Two-name joint default probability estimations under a stochastic correlation model are calculated by the basic Monte Carlo (BMC) and the homogenized importance sampling (HIS), respectively. Several time scales ε are given to compare the effect of stochastic correlation. The total number of simulations is 104 and an Euler discretization scheme is used by taking time step size T/400, where T is one year. Other parameters include S10 = S20 = 100,σ1 = 0.4,σ2 = 0.4,B1 = 50,B2 = 40,Y0 =...

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...inance, where the increasingly high frequency of trades in financial markets requires pricing models behaving reasonably both on very short and on long time horizons. In the works [2], [3], [4], [6], =-=[13]-=-, [14], [21] and the references therein the authors study the behavior of the random variables E[f(Xt0+δ)|FXt0 ] for small values of δ > 0, where X is a finite-dimensional (jump-)diffusion process, a ...

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...modeling (a): rare 1 transition events between equilibrium states of a rough energy landscape [8, 15, 26], or (b): short time maturity asymptotics for fast mean reverting stochastic volatility models =-=[10, 11]-=-. See also [13, 18] for a thorough discussion on different mathematical and statistical modeling aspects of perturbations of dynamical systems by small noise. Parameter estimation in multiscale models...