2012/08/27 by José E. Figueroa‐López, José E. Figueroa-López, Ruoting Gong +4
Economics, Econometrics and Finance · Mathematics · #60F99 #60G51 #91G20 #91G60 #Complex Systems and Time Series Analysis #FOS: Economics and business #FOS: Mathematics #Financial Risk and Volatility Modeling #Pricing of Securities (q-fin.PR) #Probability (math.PR) #Stochastic processes and financial applications #math.PR #msc:60F99 #msc:60G51 #msc:91G20 #msc:91G60 #q-fin.PR
paper · pdf · doi:10.48550/arxiv.1208.5520
35 pages, 8 figures. This is an extension of our earlier submission arXiv:1112.3111. To appear in Mathematical Finance
openalex publication_date 2012/08/27 · arxiv created 2014/04/05 · arxiv updated 2014/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the present work, a novel second-order approximation for 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 d2 t(3-Y)/2, with d2 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 σt1/2/√(2π). 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 d1t1/Y. The second-order term is shown to be of the form d2 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 Lévy processes which satisfy the latter property and whose Lévy 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.