Abstract
We have applied the Lie-Trotter operator splitting method to model the dynamics of both the sum and difference of two correlated constant elasticity of variance (CEV) stochastic variables. Within the Lie-Trotter splitting approximation, both the sum and difference are shown to follow a shifted CEV stochastic process, and approximate probability distributions are determined in closed form. Illustrative numerical examples are presented to demonstrate the validity and accuracy of these approximate distributions. These approximate probability distributions can be used to valuate two-asset options, e.g. spread options and basket options, where the CEV variables represent the forward prices of the underlying assets. Moreover, we believe that this new approach can be extended to study the algebraic sum of N CEV variables with potential applications in pricing multi-asset options.
Highlights
Lo [1] proposed a new simple approach to tackle the long-standing problem: “Given two correlated lognormal stochastic variables, what is the stochastic dynamics of the sum or difference of the two variables?”; or equivalently, “What is the probability distribution of the sum or difference of two correlated lognormal stochastic variables?” The solution to this problem has wide applications in many fields including financial modelling, actuarial sciences, telecommunications, biosciences and physics [2-15]
By means of the Lie-Trotter operator splitting method [16], Lo showed that both the sum and difference of two correlated lognormal stochastic processes could be approximated by a shifted lognormal stochastic process, and approximate probability distributions were determined in closed form
In terms of the approximate solutions, Lo presented an analytical series expansion of the exact solutions, which can allow us to improve the approximation systematically. In this communication we extend Lo’s approach to study the dynamics of both the sum and difference of two correlated constant elasticity of variance (CEV) stochastic processes
Summary
Lo [1] proposed a new simple approach to tackle the long-standing problem: “Given two correlated lognormal stochastic variables, what is the stochastic dynamics of the sum or difference of the two variables?”; or equivalently, “What is the probability distribution of the sum or difference of two correlated lognormal stochastic variables?” The solution to this problem has wide applications in many fields including financial modelling, actuarial sciences, telecommunications, biosciences and physics [2-15]. By means of the Lie-Trotter operator splitting method [16], Lo showed that both the sum and difference of two correlated lognormal stochastic processes could be approximated by a shifted lognormal stochastic process, and approximate probability distributions were determined in closed form. By the Lie-Trotter operator splitting method, we show that both the sum and difference of two correlated CEV processes can be modelled by a shifted CEV process. Approximate probability distributions of both the sum and difference are determined in closed form, and illustrative numerical examples are presented to demonstrate the validity and accuracy of these approximate distributions
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