[论文解读] At-the-money short-time call-price asymptotics for new classes of exponential Lévy models

Allen Kyle Hoffmeyer, Christian Houdré|arXiv (Cornell University)|Mar 16, 2026

Stochastic processes and financial applications被引用 0

一句话总结

本文推导出对数收益在 alpha-stable 分布领域的第一阶 ATM 期权看涨价与隐含波动率在短时间极限定理下的渐近表达，模型为指数 Lévy 过程；Lévy 流量的正则变异性控制结果，并引入超越经典 t^{1/alpha} 标度的新收敛速度。

ABSTRACT

We develop at-the-money call-price and implied volatility asymptotic expansions in time to maturity for a class of asset-price models whose log returns follow a Lévy process. Under mild assumptions placing the driving Lévy process in the small-time domain of attraction of an $α$-stable law with $α\in (1,2)$, we give first-order at-the-money call-price and implied volatility asymptotics. A key observation is that both the stable domain of attraction and the finiteness of the centering constant $\barμ$ are preserved under the share measure transformation, so that all of the distributional input needed for the call-price expansion can be read off from the regular variation of the Lévy measure near the origin. When the Lévy process has no Brownian component, new rates of convergence of the form $t^{1/α} \ell(t)$ where $\ell$ is a slowly varying function are obtained. We provide an example of an exponential Lévy model exhibiting this behavior, with $\ell$ not asymptotically constant, yielding a convergence rate of $(t / \log(1/t))^{1/α}$. In the case of a Lévyprocess with Brownian component, we show that the jump contribution is always lower order, so that the leading $\sqrt{t}$ behavior of the at-the-money call price is universal and driven entirely by the Gaussian part of the characteristic triplet.

研究动机与目标

• Motivate and develop ATM short-time expansions for exponential Lévy models with log-prices following Lévy processes.
• Characterize how small-time regular variation of the Lévy measure determines stable-domain attraction and scaling.
• Derive first-order ATM call-price and implied volatility asymptotics under mild assumptions, including Brownian and pure-jump components.
• Show invariance of stable domain and centering constants under share measure transformation used for pricing.
• Provide examples illustrating new convergence rates beyond power-law behavior.

提出的方法

• Use share measure transformation to express ATM call prices via X_t and an independent exponential variable (Carr–Madan representation).
• Characterize DOA to alpha-stable laws via regular variation of the Lévy measure near zero (gamma and V functions).
• Prove that stable domain and centering constant finite when transformed to the share measure (nu^* = e^{x} nu).
• Derive first-order ATM price c(t,0) = (S0 E^*[Z_+]) B_t + o(B_t) with B_t tied to beta_t via t ~ beta_t^{-eta} and slowly varying functions (Theorem 3.2).
• Obtain universal sqrt(t) leading behavior when a Brownian component is present (Theorems 3.4–3.5).
• Provide an explicit example where the convergence rate involves a logarithmic correction (Section 4).

实验结果

研究问题

• RQ1 What are the first-order ATM call-price and implied-volatility asymptotics for exponential Lévy models when X_t is in the small-time domain of attraction of an alpha-stable law with alpha in (1,2)?
• RQ2 How does the regular variation of the Lévy measure near the origin determine the scaling B_t and the centering constant, and is this structure preserved under share measure transformation?
• RQ3 What is the impact of a Brownian component on the leading ATM behavior of call prices and implied vol?
• RQ4 Can new convergence rates beyond t^{1/alpha} be generated via slowly varying modifiers in the Lévy measure near zero?
• RQ5 Under what conditions can we explicitly characterize the rate of convergence for ATM prices and implied vol, including logarithmic corrections?

主要发现

• ATM call prices admit first-order expansions c(t,0) = (S0 E^*[Z_+]) B_t + o(B_t) where Z is alpha-stable and B_t is linked to the regular variation of the Lévy measure near zero.
• Implied volatility at-the-money satisfies hat{sigma}(t) ~ sqrt(2π) B_t / sqrt(t) E^*[Z_+] as t → 0.
• If a Brownian component is present, the jump contribution is o(sqrt(t)) and the leading ATM behavior is governed by the Gaussian part (leading to universal sqrt(t) scaling).
• Share measure transformation preserves both the stable domain of attraction and the finiteness of the centering constant, so the limiting Z and inputs come from the original Lévy measure near zero.
• The paper provides an explicit model showing a convergence rate of (t / log(1/t))^{1/alpha} due to a logarithmic slowly varying term in the Lévy density.
• Section 4 demonstrates how different slowly varying modifiers yield new first-order rates beyond the classical t^{1/alpha} scaling.