Numerical Analysis on Quadratic Hedging Strategies for Normal Inverse Gaussian Models

Abstract

The authors aim to develop numerical schemes of the two representative quadratic hedging strategies: locally risk minimizing and mean-variance hedging strategies, for models whose asset price process is given by the exponential of a normal inverse Gaussian process, using the results of Arai et al. [2], and Arai and Imai [1]. Here normal inverse Gaussian process is a framework of Lévy processes frequently appeared in financial literature. In addition, some numerical results are also introduced.

A Clark-Ocone type formula via Ito calculus and its application to finance

“…, where φ * (t, z) := E P * [e iz(XT −Xt) ] for z ∈ C. Note that Assumption 3.2 (1) ensures the structure condition (SC); and MMM P * exists as an equivalent probability measure to P by the above (2) For more details on this matter, see Remark 3.5 below. Note that the formulations of ϕ * and ν * are given in [2] for Merton jump diffusion processes and VG processes, and in [1] for NIG processes, respectively. Theorem 3.4.…”

“…1,2 , the Clark-Ocone theorem is not available, but Theorem 2.7 is still available as far as Assumptions 2.1 and 2.6 are satisfied. Some examples of such cases will be discussed in Section 2.4 below.Corollary 2.9.…”