High-order short-time expansions for ATM option prices of exponential Lévy models

The short-time asymptotic behavior of option prices for a variety of models with jumps has received much attention in recent years. In this work, a novel second-order approximation for at-the-money (ATM) option prices is derived for a large class of exponential Lévy 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 , with d 2 only depending on Y, the degree of jump activity, on σ, 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 . 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. The second-order term is shown to be of the form 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 Lévy processes, which satisfy the latter property and whose Lévy density 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\u27s accuracy, particularly in the absence of a Brownian component