scholarly journals Including Jumps in the Stochastic Valuation of Freight Derivatives

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 154
Author(s):  
Lourdes Gómez-Valle ◽  
Julia Martínez-Rodríguez
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Diffusion Process ◽  
Option Price ◽  
Jump Diffusion ◽  
Freight Rate ◽  
Parametric Models ◽  
Geometric Process ◽  
Rate Processes ◽  
Jump Diffusion Model

The spot freight rate processes considered in the literature for pricing forward freight agreements (FFA) and freight options usually have a particular dynamics in order to obtain the prices. In those cases, the FFA prices are explicitly obtained. However, for jump-diffusion models, an exact solution is not known for the freight options (Asian-type), in part due to the absence of a suitable valuation framework. In this paper, we consider a general jump-diffusion process to describe the spot freight dynamics and we obtain exact solutions of FFA prices for two parametric models. Moreover, we develop a partial integro-differential equation (PIDE), for pricing freight options for a general unifactorial jump-diffusion model. When we consider that the spot freight follows a geometric process with jumps, we obtain a solution of the freight option price in a part of its domain. Finally, we show the effect of the jumps in the FFA prices by means of numerical simulations.

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.

scholarly journals The ruin problem for a Wiener process with state-dependent jumps

2020 ◽  
Vol 16 (1) ◽  
pp. 13-23
Author(s):  
M. Lefebvre
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Diffusion Process ◽  
Wiener Process ◽  
Jump Diffusion ◽  
Moment Generating Function ◽  
Continuous Part ◽  
State Dependent ◽  
The Moment ◽  
First Time

AbstractLet X(t) be a jump-diffusion process whose continuous part is a Wiener process, and let T (x) be the first time it leaves the interval (0,b), where x = X(0). The jumps are negative and their sizes depend on the value of X(t). Moreover there can be a jump from X(t) to 0. We transform the integro-differential equation satisfied by the probability p(x) := P[X(T (x)) = 0] into an ordinary differential equation and we solve this equation explicitly in particular cases. We are also interested in the moment-generating function of T (x).

scholarly journals Optimal Strategies for an Ambiguity-Averse Insurer under a Jump-Diffusion Model and Defaultable Risk

2020 ◽  
Vol 2020 ◽  
pp. 1-26 ◽  
Author(s):  
Man Li ◽  
Yingchun Deng ◽  
Ya Huang ◽  
Hui Ou
Keyword(s):  
Diffusion Process ◽  
Default Risk ◽  
Optimal Strategies ◽  
Jump Diffusion ◽  
Corporate Bond ◽  
Value Functions ◽  
Case Scenario ◽  
Worst Case ◽  
Special Cases ◽  
Jump Diffusion Model

In this paper, we consider a robust optimal investment-reinsurance problem with a default risk. The ambiguity-averse insurer (AAI) may carry out transactions on a risk-free asset, a stock, and a defaultable corporate bond. The stock’s price is described by a jump-diffusion process, and both the jump intensity and the distribution of jump amplitude are uncertain, i.e., the jump is ambiguous. The AAI’s surplus process is assumed to follow an approximate diffusion process. In particular, the reinsurance premium is calculated according to the generalized mean-variance premium principle, and the reinsurance type has to follow a self-reinsurance function. In performing dynamic programming, both the predefault case and the postdefault case are analyzed, and the optimal strategies and the corresponding value functions are derived under the worst-case scenario. Moreover, we give a detailed proof of the verification theorem and give some special cases and numerical examples to illustrate our theoretical results.

scholarly journals Joint distribution of a Lévy process and its running supremum

2018 ◽  
Vol 55 (2) ◽  
pp. 488-512 ◽  
Author(s):  
Laure Coutin ◽  
Monique Pontier ◽  
Waly Ngom
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Weak Solution ◽  
Diffusion Process ◽  
Lebesgue Measure ◽  
Joint Distribution ◽  
Random Variable ◽  
Jump Diffusion ◽  
Marginal Density ◽  
The Law

Abstract Let X be a jump-diffusion process and X* its running supremum. In this paper we first show that for any t > 0, the law of the pair (X*t, Xt) has a density with respect to the Lebesgue measure. This allows us to show that for any t > 0, the law of the pair formed by the random variable Xt and the running supremum X*t of X at time t can be characterized as a weak solution of a partial differential equation concerning the distribution of the pair (X*t, Xt). Then we obtain an expression of the marginal density of X*t for all t > 0.

scholarly journals Adaptive nonparametric drift estimation of an integrated jump diffusion process

2018 ◽  
Vol 22 ◽  
pp. 236-260 ◽  
Author(s):  
Benedikt Funke ◽  
Émeline Schmisser
Keyword(s):  
Diffusion Process ◽  
High Frequency ◽  
Jump Diffusion ◽  
High Frequency Data ◽  
Compact Set ◽  
Penalized Least Squares ◽  
Adaptive Estimator ◽  
Drift Estimation ◽  
Jump Diffusion Process ◽  
Jump Diffusion Model

In the present article, we investigate nonparametric estimation of the unknown drift function b in an integrated Lévy driven jump diffusion model. Our aim will be to estimate the drift on a compact set based on a high-frequency data sample. Instead of observing the jump diffusion process V itself, we observe a discrete and high-frequent sample of the integrated process Xt := ∫0t Vsds Based on the available observations of Xt, we will construct an adaptive penalized least-squares estimate in order to compute an adaptive estimator of the corresponding drift function b. Under appropriate assumptions, we will bound the L2-risk of our proposed estimator. Moreover, we study the behavior of the proposed estimator in various Monte Carlo simulation setups.

scholarly journals Valuation of Game Options in Jump-Diffusion Model and with Applications to Convertible Bonds

2009 ◽  
Vol 2009 ◽  
pp. 1-17 ◽  
Author(s):  
Lei Wang ◽  
Zhiming Jin
Keyword(s):  
Free Boundary ◽  
Diffusion Process ◽  
Stock Price ◽  
Free Boundary Problems ◽  
Jump Diffusion ◽  
Convertible Bonds ◽  
Simple Application ◽  
Boundary Problems ◽  
Game Options ◽  
Jump Diffusion Model

Game option is an American-type option with added feature that the writer can exercise the option at any time before maturity. In this paper, we consider some type of game options and obtain explicit expressions through solving Stefan(free boundary) problems under condition that the stock price is driven by some jump-diffusion process. Finally, we give a simple application about convertible bonds.

scholarly journals Pricing of American Put Option under a Jump Diffusion Process with Stochastic Volatility in an Incomplete Market

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Shuang Li ◽  
Yanli Zhou ◽  
Xinfeng Ruan ◽  
B. Wiwatanapataphee
Keyword(s):  
Diffusion Process ◽  
Stochastic Volatility ◽  
American Options ◽  
Option Price ◽  
Jump Diffusion ◽  
Incomplete Market ◽  
American Put Option ◽  
Put Option ◽  
Jump Diffusion Process ◽  
American Put

We study the pricing of American options in an incomplete market in which the dynamics of the underlying risky asset is driven by a jump diffusion process with stochastic volatility. By employing a risk-minimization criterion, we obtain the Radon-Nikodym derivative for the minimal martingale measure and consequently a linear complementarity problem (LCP) for American option price. An iterative method is then established to solve the LCP problem for American put option price. Our numerical results show that the model and numerical scheme are robust in capturing the feature of incomplete finance market, particularly the influence of market volatility on the price of American options.

OPTIMAL PORTFOLIO AND CONSUMPTION FOR A MARKOVIAN REGIME-SWITCHING JUMP-DIFFUSION PROCESS

2021 ◽  
pp. 1-25
Author(s):  
CAIBIN ZHANG ◽  
ZHIBIN LIANG ◽  
KAM CHUEN YUEN
Keyword(s):  
Diffusion Process ◽  
Diffusion Model ◽  
Regime Switching ◽  
Jump Diffusion ◽  
Optimal Portfolio ◽  
Dynamic Programming Principle ◽  
Pure Diffusion ◽  
Special Cases ◽  
Jump Diffusion Process ◽  
Jump Diffusion Model

Abstract We consider the optimal portfolio and consumption problem for a jump-diffusion process with regime switching. Under the criterion of maximizing the expected discounted total utility of consumption, two methods, namely, the dynamic programming principle and the stochastic maximum principle, are used to obtain the optimal result for the general objective function, which is the solution to a system of partial differential equations. Furthermore, we investigate the power utility as a specific example and analyse the existence and uniqueness of the optimal solution. Under the constraints of no-short-selling and nonnegative consumption, closed-form expressions for the optimal strategy and the value function are derived. Besides, some comparisons between the optimal results for the jump-diffusion model and the pure diffusion model are carried out. Finally, we discuss our optimal results in some special cases.

Optimal portfolio and consumption for a Markovian regime-switching jump-diffusion process

2021 ◽  
Vol 63 ◽  
pp. 308-332
Author(s):  
Caibin Zhang ◽  
Zhibin Liang ◽  
Kam Chuen Yuen
Keyword(s):  
Diffusion Process ◽  
Diffusion Model ◽  
Regime Switching ◽  
Jump Diffusion ◽  
Optimal Portfolio ◽  
Dynamic Programming Principle ◽  
Pure Diffusion ◽  
Special Cases ◽  
Jump Diffusion Process ◽  
Jump Diffusion Model

We consider the optimal portfolio and consumption problem for a jump-diffusion process with regime switching. Under the criterion of maximizing the expected discounted total utility of consumption, two methods, namely, the dynamic programming principle and the stochastic maximum principle, are used to obtain the optimal result for the general objective function, which is the solution to a system of partial differential equations. Furthermore, we investigate the power utility as a specific example and analyse the existence and uniqueness of the optimal solution. Under the constraints of no-short-selling and nonnegative consumption, closed-form expressions for the optimal strategy and the value function are derived. Besides, some comparisons between the optimal results for the jump-diffusion model and the pure diffusion model are carried out. Finally, we discuss our optimal results in some special cases.   doi:10.1017/S1446181121000122

Productivity Assessment Based on Jump Diffusion Model Considering the Effort Management for OSS Project

2019 ◽  
Vol 26 (05) ◽  
pp. 1950022 ◽  
Author(s):  
Yoshinobu Tamura ◽  
Hironobu Sone ◽  
Shigeru Yamada
Keyword(s):  
Project Management ◽  
Diffusion Process ◽  
Process Model ◽  
Jump Diffusion ◽  
Equation Model ◽  
Operation Performance ◽  
Jump Diffusion Process ◽  
Jump Diffusion Model ◽  
Productivity Assessment

Various open source software (OSS) projects are in action around the world. Many OSS are developed and maintained under these OSS projects. Considering the characteristics of OSS, the operation performance of OSS development will take an irregular fluctuation in the long term of operation, because several developers and many users are closely related to the maintenance of OSS. This paper focuses on the irregular fluctuation of the operation performance of OSS. We apply the jump diffusion process model to the noisy cases in the operation of OSS. In particular, the maintenance effort is estimated by the stochastic differential equation model in terms of OSS project management. Moreover, we discuss the method of maintenance effort management based on jump diffusion process model considering the irregular fluctuation of performance for OSS projects. In particular, we propose the method of productivity assessment based on the proposed jump diffusion models. Thereby, it is helpful for the OSS development managers to understand the effort status of OSS from the standpoint of OSS project management. Also, we analyze actual data to show numerical examples of the proposed method considering the characteristics of OSS projects.

Sign in / Sign up

Export Citation Format

Share Document