High-order short-time expansions for ATM option prices under the CGMY model

The short-time asymptotic behavior of option prices for a variety of models with jumps has received much attention in recent years. In the present work, a novel second-order approximation for ATM option prices under the CGMY Lévy model 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. This approach is sufficiently efficient to produce the third-order asymptotic behavior of the option prices and, moreover, is expected to apply to many other popular classes of Lévy processes which satisfy the fundamental property of being stable under a suitable change of probability measure. Our results 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 case of an additional Brownian component, the second-order term, in time-t, is of the form d2 t (3−Y , with the coefficient d2 depending only on the overall jump intensity parameter C and the tail-heaviness parameter Y . This extends the known result that the leading term is (σ/ √ 2π)t, where σ is the volatility of the continuous component. In contrast, under a pure-jump CGMY model, the dependence on the two parameters C and Y is already reflected in the leading term, which is of the form d1t 1/Y . 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 d2t and whose order of decay turns out to be independent of Y . The third-order asymptotic behavior of the option prices as well as the asymptotic behavior of the corresponding Black-Scholes implied volatilities are also addressed. Our numerical results show that first-order term typically exhibits extremely poor performance and that the second-order term significantly improves the approximation’s accuracy. AMS 2000 subject classifications: 60G51, 60F99, 91G20, 91G60.