A limit theorem for generalized tempered stable processes and their quadratic variations with stable index tending to two

Abstract: We study the limit of the joint distribution of a multidimensional Generalized Tempered Stable (GTS) process and its quadratic covariation process when the stable index tends to two. Under a proper scaling, the GTS processes converges to a Brownian motion that is a stable process with stable index two. We renormalize their quadratic covariation processes so that they have a nondegenerate limit distribution. We show that the limit is a stable process with stable index one and is independent of the limit Brownian motion of the GTS processes. In addition, we apply this convergence result to finance. By using the scaled GTS process defined above, we construct a pure jump asset price model approaching to the Black-Scholes model. To evaluate how $\alpha$-stable jumps affect the implied volatility, we obtain the asymptotic expansion of the at-the-money implied volatility skew when the model approaches to the Black-Scholes model.