In the present work, a novel second-order approximation for ATM option prices is derived for a large class of exponential L\'evy models with or without Brownian component. The results hereafter shed new light on the connection between both the volatility of the continuous component and the jump parameters and the behavior of ATM option prices near expiration. In the presence of a Brownian component, the second-order term, in time-t, is of the form d_{2}  t^{(3-Y)/2}, with the coefficient d_{2 on Y, the degree of jump activity, on \sigma, the volatility of the continuous component, and on an additional parameter controlling the intensity of the "small" jumps (regardless of their signs). /2\pi. } In contrast, under a pure-jump model, the dependence on Y and on the separate intensities of negative and positive small jumps are already reflected in the leading term, which is of the form d_{1}t^{1/Y}. The second-order term is shown to be of the form d _{2} t and, therefore, its order of decay turns out to be independent of Y. The asymptotic behavior of the corresponding Black-Scholes implied volatilities is also addressed. Our method of proof is based on an integral representation of the option price involving the tail probability of the log-return process under the share measure and a suitable change of probability measure under which the pure-jump component of the log-return process becomes a Y-stable process. Our approach is sufficiently general to cover a wide class of L\'evy processes which satisfy the latter property and whose L\'evy densitiy can be closely approximated by a stable density near the origin. Our numerical results show that  first-order term typically exhibits rather poor performance and that the second-order term significantly improves the approximation's accuracy  .
In the present work, a novel second-order approximation for ATM option prices is derived for a large class of exponential L\'{e models with or without Brownian component. The results hereafter shed new light on the connection between both the volatility of the continuous component and the jump parameters and the behavior of ATM option prices near expiration. In the presence of a Brownian component, the second-order term, in time-t, is of the form d_{2} \, t^{(3-Y)/2}, with d_{2 on Y, the degree of jump activity, on \sigma, the volatility of the continuous component, and on an additional parameter controlling the intensity of the "small" jumps (regardless of their signs). This extends the well known result that the leading first-order term is \sigma t^{1/2/2\pi. } In contrast, under a pure-jump model, the dependence on Y and on the separate intensities of negative and positive small jumps are already reflected in the leading term, which is of the form d_{1}t^{1/Y}. The second-order term is shown to be of the form _{2} t and, therefore, its order of decay turns out to be independent of Y. The asymptotic behavior of the corresponding Black-Scholes implied volatilities is also addressed. Our approach is sufficiently general to cover a wide class of L\'{e processes which satisfy the latter property and whose L\'{e densitiy can be closely approximated by a stable density near the origin. Our numerical results show that the first-order term typically exhibits rather poor performance and that the second-order term can significantly improve the approximation's accuracy , particularly in the absence of a Brownian component .
R
L\'evy
L\'{e
A
\,
R
the coefficient d_{2
d_{2
A
This extends the well known result that the leading first-order term is \sigma t^{1/2
D
d
D
method of proof is based on an integral representation of the option price involving the tail probability of the log-return process under the share measure and a suitable change of probability measure under which the pure-jump component of the log-return process becomes a Y-stable process. Our
R
L\'evy
L\'{e
R
L\'evy
L\'{e
A
the
R
significantly improves
can significantly improve
A
, particularly in the absence of a Brownian component