In the present work, a novel second-order approximation for ATM option prices under an exponential tempered stable model, a rich class of L\'evy processes with desirable features for financial modeling, is derived and, then, extended to a model with an additional independent Brownian component . 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   process becomes stable . Our approach is sufficiently general to cover a wide class of L\'evy processes which satisfy the latter property and whose L\'evy densities can be closely approximated by a stable density near the origin. 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 an additional Brownian component, the second-order term, in time-t, is of the form d_{2}t^{(3-Y)/2}, with the coefficient d_{2} depending only on the overall jump intensity of the process and the tail-heaviness parameter Y. This extends the  known result that the leading term is \sigma t^{1/2}/2\pi, where \sigma is the volatility of the continuous component. In contrast, under  a pure-jump tempered stable model, the dependence on the overall jump intensity and Y is already reflected in the leading term, which is of the form d_{1} t^{1/Y}. The information on the relative frequency of negative and positive jumps appears only in the second-order term, which is shown to be of the form d_{2} t and whose order of decay turns out to be independent of Y.  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\'evy models . 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 densities can be closely approximated by a stable density near the origin. 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 an additional Brownian component, the second-order term, in time-t, is of the form d_{2}t^{(3-Y)/2}, with the coefficient d_{2} depending only on the overall jump intensity of the process and the tail-heaviness parameter Y. This extends the already known result that the leading term is \sigma t^{1/2}/2\pi, where \sigma is the volatility of the continuous component. In contrast, under such a pure-jump model, the dependence on the overall jump intensity and Y is already reflected in the leading term, which is of the form d_{1} t^{1/Y}. The information on the relative frequency of negative and positive jumps appears only in the second-order term, which is shown to be of the form d_{2} t and whose order of decay turns out to be independent of Y. The asymptotic behavior of the corresponding Black-Scholes implied volatilities is also addressed. 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.
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under an exponential tempered stable model, a rich class of
is derived for a large class of exponential
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processes with desirable features for financial modeling, is derived and, then, extended to a model with an additional independent Brownian component
models
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pure-jump component of the log-return
A
process becomes a Y-stable
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becomes stable
A
already
A
such
D
tempered stable
A
The asymptotic behavior of the corresponding Black-Scholes implied volatilities is also addressed.
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rather
rather