American Call Options Under Jump-Diffusion Processes - A Fourier Transform Approach

We consider the American option pricing problem in the case where the underlying asset follows a jump-diffusion process. We apply the method of Jamshidian to transform the problem of solving a homogeneous integro-partial differential equation (IPDE) on a region restricted by the early exercise (free) boundary to that of solving an inhomogeneous IPDE on an unrestricted region. We apply the Fourier transform technique to this inhomogeneous IPDE in the case of a call option on a dividend paying underlying to obtain the solution in the form of a pair of linked integral equations for the free boundary and the option price. We also derive new results concerning the limit for the free boundary at expiry. Finally, we present a numerical algorithm for the solution of the linked integral equation system for the American call price, its delta and the early exercise boundary. We use the numerical results to quantify the impact of jumps on American call prices and the early exercise boundary

Coagulation disorders in coronavirus infected patients: COVID-19, SARS-CoV-1, MERS-CoV and lessons from the past

© 2020 Elsevier B.V. Coronavirus disease 2019 (COVID-19) or severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), a novel coronavirus strain disease, has recently emerged in China and rapidly spread worldwide. This novel strain is highly transmittable and severe disease has been reported in up to 16% of hospitalized cases. More than 600,000 cases have been confirmed and the number of deaths is constantly increasing. COVID-19 hospitalized patients, especially those suffering from severe respiratory or systemic manifestations, fall under the spectrum of the acutely ill medical population, which is at increased venous thromboembolism risk. Thrombotic complications seem to emerge as an important issue in patients infected with COVID-19. Preliminary reports on COVID-19 patients’ clinical and laboratory findings include thrombocytopenia, elevated D-dimer, prolonged prothrombin time, and disseminated intravascular coagulation. As the pandemic is spreading and the whole picture is yet unknown, we highlight the importance of coagulation disorders in COVID-19 infected patients and review relevant data of previous coronavirus epidemics caused by the severe acute respiratory syndrome coronavirus 1 (SARS-CoV-1) and the Middle East Respiratory Syndrome coronavirus (MERS-CoV)

Pricing American options using Fourier analysis

University of Technology, Sydney. Faculty of Business.The analytic expression for an American option price under the Black-Scholes model requires the early exercise boundary as one of its inputs, and this is not known a priori. An implicit integral equation can be found for this free boundary, but it has no known closed-form solution, and its numerical solution is highly non-trivial. This has given rise to a number of analytical solution methods and numerical techniques designed to handle the early exercise feature. The aim of this thesis is to explore Fourier-type solution methods for pricing American options. The price is defined as a free boundary value problem, whose solution satisfies the Black-Scholes PDE with certain final and boundary conditions. This problem is solved using the incomplete Fourier transform method of McKean (1965). The method is generalised to American options with monotonic and convex payoffs in a systematic way, and is further extended by applying it to solve the PIDE for the American call option under Merton’s (1976) jump-diffusion model. In this case numerical integration solutions require an intense level of computation. The thesis considers the Fourier-Hermite series expansion method as an alternative approach. This is extended to allow for jump-diffusion with log-normally distributed jump sizes. The main contributions of the thesis are: • Evaluation of American Options under Geometric Brownian Motion - Chapters 2 and 3. The details of McKean’s (1965) incomplete Fourier transform are provided for a monotonic payoff function, and several forms for the price and free boundary are reproduced in the case of an American call. A numerical scheme for implementing the equations is given, along with a comparison of several existing numerical solution methods. The applicability of the transform technique to more general payoff types is demonstrated using an American strangle position with interdependent component options. A coupled integral equation system for the two free boundaries is found and solved using numerical integration. The resulting free boundaries are consistently deeper in-the-money than those for the corresponding independent American call and put. • Pricing American Options under Jump-Diffusion - Chapter 4. The incomplete Fourier transform method is applied to the jump-diffusion model of Merton (1976). The PIDE for an American call is solved, and the results are simplified to replicate the integral equations of Gukhal (2001) for the price and free boundary. An implicit expression for the limit of the free boundary at expiry is derived, and an iterative algorithm is presented for solving the integral expressions numerically. The results are demonstrated to be consistent with existing knowledge of American options under jump-diffusion, and display behaviour that is consist with market-observed volatility smiles. • Fourier-Hermite Series Expansions for Options under Jump-Diffusion - Chapter 5. The Fourier-Hermite series expansion method is extended to the jump- diffusion model of Merton (1976) in the case where the jump sizes are lognormally distributed. With the aid of a suitably calibrated scaling parameter, the method is used to evaluate American call options. The pricing accuracy of this approach is shown to be comparable to both the iterative numerical integration method, and the method of lines technique by Meyer (1998). The series expansion method displays a high computation speed in exchange for some loss of accuracy in the free boundary approximation

The evaluation of american option prices under stochastic volatility and jump-diffusion dynamics using the method of lines

This paper considers the problem of numerically evaluating American option prices when the dynamics of the underlying are driven by both stochastic volatility following the square root process of Heston [18], and by a Poisson jump process of the type originally introduced by Merton [25]. We develop a method of lines algorithm to evaluate the price as well as the delta and gamma of the option, thereby extending the method developed by Meyer [26] for the case of jump-diffusion dynamics. The accuracy of the method is tested against two numerical methods that directly solve the integro-partial differential pricing equation. The first is an extension to the jump-diffusion situation of the componentwise splitting method of Ikonen and Toivanen [21]. The second method is a Crank-Nicolson scheme that is solved using projected successive over relaxation and which is taken as the benchmark for the price. The relative efficiency of these methods for computing the American call option price, delta, gamma and free boundary is analysed. If one seeks an algorithm that gives not only the price but also the delta and gamma to the same level of accuracy for a given computational effort then the method of lines seems to perform best amongst the methods considered. © 2009 World Scientific Publishing Company

Nonparametric Adjustment for Measurement Error in Time to Event Data

Measurement error in time to event data used as a predictor will lead to inaccurate predictions. This arises in the context of self-reported family history, a time to event predictor often measured with error, used in Mendelian risk prediction models. Using a validation data set, we propose a method to adjust for this type of measurement error. We estimate the measurement error process using a nonparametric smoothed Kaplan-Meier estimator, and use Monte Carlo integration to implement the adjustment. We apply our method to simulated data in the context of both Mendelian risk prediction models and multivariate survival prediction models, as well as illustrate our method using a data application for Mendelian risk prediction models. Results from simulations are evaluated using measures of mean squared error of prediction (MSEP), area under the response operating characteristics curve (ROC-AUC), and the ratio of observed to expected number of events. These results show that our adjusted method mitigates the effects of measurement error mainly by improving calibration and by improving total accuracy. In some scenarios discrimination is also improved

THE EVOLUTION OF CRITERIA FOR LIVER TRANSPLANTATION FOR HEPATOCELLULAR CARCINOMA: FROM MILAN TO SAN FRANCISCO AND ALL AROUND THE WORLD!

Introduction: Hepatocellular carcinoma (HCC) is the fifth most common malignancy and the third most common cancerrelated cause of death in the world. According to the stage of the disease, each patient is allocated to a different treatment option. Liver transplantation, along with surgical resection, is the only totally therapeutic option and is primarily indicated in HCC patients with underlying cirrhosis. However, the restricted number of liver grafts imposes difficulties in selecting the most suitable patients to receive those limited grafts and therefore certain criteria have been proposed. The Milan criteria are currently the most widely accepted and utilized criteria around the world, despite their restrictiveness. In an attempt to assist HCC patients exceeding them, but with a potential to display acceptable survival outcomes, undergo liver transplantation, research teams worldwide suggest expanded criteria based on their findings. Some of the most broadly known are the University of California, San Francisco (UCSF), Kyoto, Tokyo, Hangzhou and up-to-7 criteria. On the other hand, in order to expand the liver donor pool, grafts may be accepted from living, non-heart beating, elderly, steatotic, or even HCV-infected donors, in addition to the use of split livers with both advantages and disadvantages. The aim of this review is to thoroughly present the current situation of liver transplantation for HCC patients, with a focus on the criteria used and emerging challenges presented. Core tip: Hepatocellular carcinoma (HCC) is the third most common malignancy worldwide and liver transplantation represents the treatment of choice, particularly in the setting of cirrhosis. Lack of grafts led to the utilization of certain criteria in order to determine the eligibility of an HCC patient to access the waiting list. The most widely accepted are the Milan criteria, even though they are thought off as too restrictive. Consequently, transplant research groups all over the world published their own criteria, which showed acceptable outcomes. Living donor liver transplantation and other extended-criteria grafts have been proposed as an alternative to reduced donations. Ziogas IA, Tsoulfas G. The evolution of criteria for liver transplantation for hepatocellular carcinoma: from Milan to San Francisco and all around the world! DOI: https://doi.org/10.25176/RFMH.v17.n3.119

Extending Mendelian Risk Prediction Models to Handle Misreported Family History

Mendelian risk prediction models calculate the probability of a proband being a mutation carrier based on family history and known mutation prevalence and penetrance. Family history in this setting, is self-reported and is often reported with error. Various studies in the literature have evaluated misreporting of family history. Using a validation data set which includes both error-prone self-reported family history and error-free validated family history, we propose a method to adjust for misreporting of family history. We estimate the measurement error process in a validation data set (from University of California at Irvine (UCI)) using nonparametric smoothed Kaplan-Meier estimators, and use Monte Carlo integration to implement the adjustment. In this paper, we extend BRCAPRO, a Mendelian risk prediction model for breast and ovarian cancers, to adjust for misreporting in family history. We apply the extended model to data from the Cancer Genetics Network (CGN)