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This summary is machine-generated.

We analyze the behavior of implied volatility

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Area of Science:

  • Quantitative Finance
  • Stochastic Calculus
  • Financial Derivatives

Background:

  • Implied volatility's behavior near expiration is crucial for option pricing.
  • Understanding the at-the-money (ATM) strike derivative provides insights into market expectations.
  • Lévy models offer a flexible framework for capturing asset price dynamics.

Purpose of the Study:

  • To quantify the ATM strike derivative of implied volatility as maturity approaches zero.
  • To analyze the slope of this derivative for specific Lévy models.
  • To connect the ATM slope behavior to the steepness of option price "smiles".

Main Methods:

  • Asymptotic analysis of option pricing formulas.
  • Mellin transform techniques for deriving short-maturity expansions.
  • Consideration of infinite activity exponential Lévy models with a Brownian component.

Main Results:

  • The study quantifies the behavior of the ATM implied volatility slope for a class of Lévy models.
  • Asymptotic expansions for short-maturity ATM digital call options are derived.
  • Conditions are discussed for the ATM slope's consistency with smile steepness.

Conclusions:

  • The ATM implied volatility slope exhibits specific behaviors in Lévy models as maturity vanishes.
  • Mellin transform asymptotics provide a powerful tool for analyzing short-dated derivatives.
  • The relationship between the ATM slope and smile wings offers a deeper understanding of volatility surfaces.