The number of jumps is represented by a Poisson process, and the size of each jump by a random variable. The distribution chosen for the jump size gives the model its name. Merton's model gives the ratio of the price just after a jump to the price just before it a lognormal distribution1.

If the number and size distribution of the jumps are fixed, some models let the price of a European option be computed as a weighted sum of prices, one for each number of jumps. The formula and a numerical example are on the related article page.

Notes

  1. 久田祥史 (Yoshifumi Hisata), "ジャンプ拡散過程を用いたオプション価格付けモデルについて" (On option pricing models using jump-diffusion processes), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (June 2003), section 3 (pages 55–57) (in Japanese). A process made of a diffusion term and a jump term, and the lognormal distribution for the jump ratio in Merton's model. ↩