Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

A survey of electricity spot and futures price models for risk management applications

The occurrence of prolonged dry spells and the shortage of precipitation are two different hazardous factors affecting rainfed agriculture. This study investigates a multi-state Markov chain model with the states of dry spell length coupled with a probability distribution of positive rainfall depths. The Nineveh Plains of Northern Iraq is chosen as the study site, where the rainfed farmers are inevitably exposed to drought risks, for demonstration of applicability to real-time drought risk assessment. The model is operated on historical data of daily rainfall depths observed at the city Mosul bordering the Nineveh Plains during the period 1975–2018. The methodology is developed in the context of contemporary probability theory. Firstly, the Kolmogorov–Smirnov tests are applied to extracting two sub-periods where the positive rainfall depths obey to respective distinct gamma distributions. Then, empirical estimation of transition probabilities determining a multi-state Markov chain results in spurious oscillations, which are regularized in the minimizing total variation flow solving a singular diffusion equation with a degenerating coefficient that controls extreme values of 0 and 1. Finally, the model yields the statistical moments of the dry spell length in the future and the total rainfall depth until a specified terminal day. Those statistical moments, termed hazard futures, can quantify drought risks based on the information of the dry spell length up to the current day. The newly defined hazard futures are utilized to explore measures to avert drought risks intensifying these decades, aiming to establish sustainable rainfed agriculture in the Nineveh Plains.

A multi-state Markov chain model to assess drought risks in rainfed agriculture: a case study in the Nineveh Plains of Northern Iraq

This paper develops a new class of models for pricing dual-expiry options that are characterized by two expiry dates. The underlying asset price is modeled by a time changed exponential Ornstein Uhlenbeck (OU) process, where the time change process is a Levy subordinator. The new models can capture both mean reversion and jumps often observed in various types of underlying assets of exotics. The pricing method exploits the observation that dual expiry options have payoffs that can be perfectly replicated by a particular set of first and second order binary options. The novelty of the paper is that we are able to derive the analytical solutions to the prices of these binaries through eigenfunction expansion method. Based on that, we can obtain the formulas for dual-expiry exotics through static replication. We also numerically investigate the sensitivities of prices of chooser, compound and extendable options with respect to the parameters of the models.

/pdf/the-pricing-of-dual-expiry-exotics-with-mean-reversion-and-2gtznsduto.pdf

The Pricing of Dual-Expiry Exotics with Mean Reversion and Jumps

In this paper we develop a new class of models for pricing autocallables based on multivariate subordinate Orstein Uhlenbeck (OU) processes. Starting from d independent OU processes and an independent d-dimensional Levy subordinator, we construct a new process by time changing each of the OU processes with a coordinate of the Levy subordinator. The prices of underlying assets are then modeled as an exponential function of the subordinate processes. The new models introduce state-dependent jumps in the asset prices and the dependence among jumps is governed by the Levy measure of the d-dimensional subordinator. By employing the eigenfunction expansion technique, we are able to derive the analytical formulas for the worst-of autocallable prices. We also numerically implement a specific model and test its sensitivity to some of the key parameters of the model.

A recursive pricing method for autocallables under multivariate subordination

Starting from CIR process, we build a new model for pricing discrete arithmetic Asian options with nonlinear transformation and stochastic time change. The new model introduces the nonlinearity in ...

Analytical pricing of discrete arithmetic Asian options under generalized CIR process with time change

We propose a new class of models for pricing generalized variance swaps. We assume that, in the most general form, the process for the asset price is a function of a general time-homogeneous diffusion process belonging to a symmetric pricing semigroup, time changed by a composition of a Levy subordinator and an absolutely continuous process. We derive the analytical pricing formulas for various types of generalized variance swaps based on eigenfunction expansion method. We also numerically implement the model and test its sensitivity to some of the key parameters of the model.

Analytical pricing formulas for discretely sampled generalized variance swaps under stochastic time change

In this paper, we first develop a one-factor diffusion model for electricity prices, which is based on a power transformation of CIR process. We show that the new model is tractable and we are able to derive the analytical solutions for future and future option prices. To enhance the model's ability to capture the prices spikes, we extend it to a time-changed model where the price is modelled by a nonlinear CIR process time changed by Levy subordinators. We employ the eigenfunction expansion methods to obtain the closed-form solutions for the derivatives. Our empirical study indicates the new models have the potential to capture the main features of electricity data better than the competing models.

A nonlinear diffusion model for electricity prices and derivatives

In this paper, we propose a new class of term structure of interest rate models which is built on the subdiffusion processes. We assume that the spot rate is a function of a time changed diffusion process belonging to a symmetric pricing semigroup for which its spectral representation is known. The time change process is taken to be an inverse Levy subordinator in order to capture the stickiness feature observed in the short-term interest rates. We derive the analytical formulas for both bond and bond option prices based on eigenfunction expansion method. We also numerically implement a specific subdiffusive model by testing the sensitivities of bond and bond option prices with respect to the parameters of time change process.

Term structure of interest rates with stickiness: a subdiffusion approach

In this paper, we propose to model rainfall based on a continuous time latent process. In this model, both parts of rainfall process — occurrence and intensity, are determined by a censored power-transformed Ornstein–Uhlenbeck (OU) process. When the latent variable takes negative values, the rainfall is censored and takes the value of zero. The new model is tractable and we are able to derive the analytical formulas for rainfall future and future option prices by employing the eigenfunction expansion method. We also carry out an empirical study where the parameters of the model are estimated using the maximum likelihood. The estimation results demonstrate the superior goodness-of-fit of the proposed model. To further enhance the model’s ability to capture the extreme rainfalls, we extend the censored OU model to a censored subordinate OU model where the latent process is modeled by the OU process time changed by Levy subordinators.

Zhigang Tong

Papers

Analytical pricing of discrete arithmetic Asian options under generalized CIR process with time change

Analytical pricing formulas for discretely sampled generalized variance swaps under stochastic time change

A nonlinear diffusion model for electricity prices and derivatives

Term structure of interest rates with stickiness: a subdiffusion approach

A censored Ornstein–Uhlenbeck process for rainfall modeling and derivatives pricing