Spectral Representation Of Additive Subordination And ASubGBM

This paper mainly introduces the construction of the subordinate diffusion process,and uses ASubGBM as an example to calculate the pricing formulas of futures and options for crude oil and petroleum products using a spectrum decomposition method combined with the cracking spread option model.Bonchner's subordination was first proposed by S.Bonchner[1,2],which corre-sponds to a stochastic time change with respect to an independent Levy subordinator.And the new process is a time-homogeneous Markov process.In the financial mod-elling,Bonchner's subordination is a powerful tool.But since the Levy process has stable increments,the new Markov process is time homogeneous,which limits its scope of application.Thus,[8]introduces a new additive subordinator that takes part in the time-changing.It is an independent additive subordinate process without stable increments.Therefore,the new Markov process is no longer time-homogeneous and can be applied to more financial models.Thus,this paper constructs additive subor-dinate diffusions.However,in order to ensure that analytical tractability is retained for new additive subordinate diffusions,this paper combines the spectral representation method of the diffusion process provided in V.Linetsky[5].This paper uses the geometric Brownian motion as an example and uses the Liouville transform to perform spectral decomposition.Finally,this paper uses a cracking spread option introduced in[8]as a model,combined with the additive subordinate geometric Brownian motion,pricing the futures/options of crude oil and its refining products.In fact,the spectral decomposition of other diffusion processes provided by V.Linetsky[5]can be applied to different models.Of course,because it's time-inhomogeneous,ad-ditive subordinate diffusion can be better in pricing futures/options,such as electricity,crude oil,soybeans,corn,coal,and other seasonally-affected futures,compared with Bonchner's subordination.