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The process {Z(t), t ≥ 0}, called in [14] the Normal Compound Poisson (NCP) process, is therefore a process with independent stationary increments, whose distribution (the NCP distribution) over unit time interval, is given by: X|V ∼ N (μ, θ + σ 2 V ), where V ∼ Poisson(λ). ) of X are given by ∞ F (x) = λn e−λ Φ n! n=0 x−μ θ + σ2 n , 2 φX (u) = exp iμu − u2 θ/2 + λ(e−u (5) σ2 /2 − 1) . f. of a standard normal distribution. The distribution of X is thus a normal with mixing on the variance, is symmetric about μ, and has the same form irrespective of the size of time increment t.

Variance-Gamma and Monte Carlo Michael C. Fu Robert H. edu Summary. The Variance-Gamma (VG) process was introduced by Dilip B. Madan and Eugene Seneta as a model for asset returns in a paper that appeared in 1990, and subsequently used for option pricing in a 1991 paper by Dilip and Frank Milne. This paper serves as a tutorial overview of VG and Monte Carlo, including three methods for sequential simulation of the process, two bridge sampling methods, variance reduction via importance sampling, and estimation of the Greeks.

N i=1 Δti = T . Initialization: Set X0 = 0; σε2 = +ε −ε x2 k(x)dx, λ+ ε = ∞ +ε −ε k(x)dx, λ− ε = −∞ k(x)dx, − − kε+ (x) = k(x)1{x≥ε} /λ+ ε , kε (x) = k(x)1{x≤−ε} /λε . Loop from i = 1 to N : 1. , Ni− , Xi,j Zi ∼ N (0, 1). √ 2. Return Xti = Xti−1 + Zi σε Δti + Ni+ j=1 + Xi,j + Ni− j=1 − Xi,j . Fig. 1. Algorithms for sequentially simulating VG process on [0, T ]. Variance-Gamma and Monte Carlo 27 Instead of sequential sampling, which progresses chronologically forward in time, an alternative method for simulating asset price paths is to use bridge sampling, which samples the end of the path first, and then “fills in” the rest of the path as needed.