A Jump-Diffusion Process for Asset Price with Non-Independent Jumps

2017 ◽  
Author(s):  
Yihren Wu ◽  
Majnu John
Keyword(s):  
Diffusion Process ◽  
Asset Price ◽  
Jump Diffusion ◽  
Jump Diffusion Process

ANALYTICAL PRICING OF DOUBLE-BARRIER OPTIONS UNDER A DOUBLE-EXPONENTIAL JUMP DIFFUSION PROCESS: APPLICATIONS OF LAPLACE TRANSFORM

2004 ◽  
Vol 07 (02) ◽  
pp. 151-175 ◽  
Author(s):  
ARTUR SEPP
Keyword(s):  
Laplace Transform ◽  
Diffusion Process ◽  
Asset Price ◽  
Option Price ◽  
Jump Diffusion ◽  
Time Dependent ◽  
Barrier Option ◽  
Double Exponential ◽  
Jump Diffusion Process ◽  
Risk Parameters

We derive explicit formulas for pricing double (single) barrier and touch options with time-dependent rebates assuming that the asset price follows a double-exponential jump diffusion process. We also consider incorporating time-dependent volatility. Assuming risk-neutrality, the value of a barrier option satisfies the generalized Black–Scholes equation with the appropriate boundary conditions. We take the Laplace transform of this equation in time and solve it explicitly. Option price and risk parameters are computed via the numerical inversion of the corresponding solution. Numerical examples reveal that the pricing formulas are easy to implement and they result in accurate prices and risk parameters. Proposed formulas allow fast computing of smile-consistent prices of barrier and touch options.

scholarly journals A switching self-exciting jump diffusion process for stock prices

2018 ◽  
Vol 15 (2) ◽  
pp. 267-306 ◽  
Author(s):  
Donatien Hainaut ◽  
Franck Moraux
Keyword(s):  
Diffusion Process ◽  
Stock Prices ◽  
Jump Diffusion ◽  
Jump Diffusion Process

scholarly journals A data-driven method to learn a jump diffusion process from aggregate biological gene expression data

2021 ◽  
Author(s):  
Jia-Xing Gao ◽  
Zhen-Yi Wang ◽  
Michael Q. Zhang ◽  
Min-Ping Qian ◽  
Da-Quan Jiang
Keyword(s):  
Gene Expression ◽  
Diffusion Process ◽  
Gene Expression Data ◽  
Dynamic Models ◽  
Empirical Distribution ◽  
Jump Diffusion ◽  
Parametric Models ◽  
Expression Data ◽  
Instantaneous Rate ◽  
Jump Diffusion Process

AbstractDynamic models of gene expression are urgently required. Different from trajectory inference and RNA velocity, our method reveals gene dynamics by learning a jump diffusion process for modeling the biological process directly. The algorithm needs aggregate gene expression data as input and outputs the parameters of the jump diffusion process. The learned jump diffusion process can predict population distributions of gene expression at any developmental stage, achieve long-time trajectories for individual cells, and offer a novel approach to computing RNA velocity. Moreover, it studies biological systems from a stochastic dynamics perspective. Gene expression data at a time point, which is a snapshot of a cellular process, is treated as an empirical marginal distribution of a stochastic process. The Wasserstein distance between the empirical distribution and predicted distribution by the jump diffusion process is minimized to learn the dynamics. For the learned jump diffusion equation, its trajectories correspond to the development process of cells and stochasticity determines the heterogeneity of cells. Its instantaneous rate of state change can be taken as “RNA velocity”, and the changes in scales and orientations of clusters can be noticed too. We demonstrate that our method can recover the underlying nonlinear dynamics better compared to parametric models and diffusion processes driven by Brownian motion for both synthetic and real world datasets. Our method is also robust to perturbations of data because it only involves population expectations.

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