High-order compact finite difference schemes for option pricing in stochastic volatility jump-diffusion models

This thesis focusses on the derivation and implementation of high-order compact finite difference schemes to price a variety of options under various stochastic volatility and jump models, with the inclusion of further studies relating to the derivatives of options and the practice of hedging. First, we derive a implicit-explicit high-order compact finite difference scheme for pricing European options under the Bates model. The resulting scheme is fourth order accurate in space and second order accurate in time. In the numerical study this scheme is compared to both a second order finite difference scheme and high-order finite element methods, where it outperforms both in terms of convergence, computational speed and required memory allocation. A numerical stability study is conducted which indicates unconditional stability of the scheme. Second, we introduce the practice of hedging and give examples of hedging strategies created from a combination of option payoffs, we show the important role the derivatives of the option price play in forming profitable strategies. We go on to complete a study of the convergence of derivatives of the option price, the so-called Greeks. We conduct studies into Delta, vega and gamma hedging, where the derived high-order compact scheme outperforms a second-order finite difference method. Examples are provided to display how this increase in computational efficiency may assist financial practictoners. Third, we extend the high-order compact scheme to price European options under the stochastic volatility with comtemporaneous jumps model. The derived scheme is fourth order accurate in space and second order accurate in time. We conduct numerical studies to test the new high-order compact schemes convergence, computational speed and required memory allocation against a second-order finite difference scheme, where the results show improvements in convergence at the expense of computational time. Further studies of numerical stability indicate unconditional stability of the high-order compact scheme

Essays in Risk Management and Asset Pricing with High Frequency Option Panels

The thesis investigates the information gains from high frequency equity option data with applications in risk management and empirical asset pricing. Chapter 1 provides the background and motivation of the thesis and outlines the key contributions. Chapter 2 describes the high frequency equity option data in detail. Chapter 3 reviews the theoretical treatments for Recovery Theorem. I derive the formulas for extracting risk neutral central moments from option prices in Chapter 4. In Chapter 5, I specify a perturbation theory on the recovered discount factor, pricing kernel, and the physical probability density. In Chapter 6, a fast and fully-identified sequential programming algorithm is built to apply the Recovery Theorem in practice with noisy market data. I document new empirical evidence on the recovered physical probability distributions and empirical pricing kernels extracted from both index and single-name equity options. Finally, I build a left tail index from the recovered physical probability densities for the S&P 500 index options and show that the left tail index can be used as an indicator of market downside risk. In Chapter 7, I uniquely introduce the higher dimensional option-implied average correlations and provide the procedures for estimating the higher dimensional option-implied average correlations from high frequency option data. In Chapter 8, I construct a market average correlation factor by sorting stocks according to their risk exposures to the option-implied average correlations. I find that (a) the market average correlation factor largely enhances the model-fitting of existing risk-adjusted asset pricing models. (b) the market average correlation factor yields persistent positive risk premiums in cross-sectional stock returns that cannot be explained by other existing risk factors and firm characteristic variables. Chapter 9 concludes the thesis

Robust Spectral Methods for Solving Option Pricing Problems

Doctor Scientiae - DScRobust Spectral Methods for Solving Option Pricing Problems by Edson Pindza PhD thesis, Department of Mathematics and Applied Mathematics, Faculty of Natural Sciences, University of the Western Cape Ever since the invention of the classical Black-Scholes formula to price the financial derivatives, a number of mathematical models have been proposed by numerous researchers in this direction. Many of these models are in general very complex, thus closed form analytical solutions are rarely obtainable. In view of this, we present a class of efficient spectral methods to numerically solve several mathematical models of pricing options. We begin with solving European options. Then we move to solve their American counterparts which involve a free boundary and therefore normally difficult to price by other conventional numerical methods. We obtain very promising results for the above two types of options and therefore we extend this approach to solve some more difficult problems for pricing options, viz., jump-diffusion models and local volatility models. The numerical methods involve solving partial differential equations, partial integro-differential equations and associated complementary problems which are used to model the financial derivatives. In order to retain their exponential accuracy, we discuss the necessary modification of the spectral methods. Finally, we present several comparative numerical results showing the superiority of our spectral methods

Stochastic calculus and derivatives pricing in the Nigerian stock market

Led by the Central Bank of Nigeria (CBN) and the Nigerian Stock Exchange (NSE), policy makers, investors and other stakeholders in the Nigerian Stock Market consider the introduction of derivative products in Nigerian capital markets essential for their investment and risk management needs. This research foregrounds these interests through detailed theoretical and empirical review of derivative pricing models. The specific objectives of the research include: 1) To explore the key stochastic calculus models used in pricing and trading financial derivatives (e.g. the Black-Scholes model and its extensions); 2) To examine the investment objectives fulfilled by derivatives; 3) To investigate the links between the stylized facts in the Nigerian Stock Market (NSM), the risk management techniques to be adopted, and the workings of the pricing models; and 4) To apply the research results to the NSM, by comparing the investment performance of selected derivative pricing models under different market scenarios, represented by the stylized facts of the underlying assets and market characteristics of the NSM. The foundational concepts that underpin the research include: stochastic calculus models of derivative pricing, especially the Black-Scholes (1973) model; its extensions; the practitioners’ Ad-Hoc Black Scholes models, which directly support proposed derivative products in the NSM; and Random Matric Theory (RMT). RMT correlates market data from the NSM and Johannesburg Stock Exchange (JSE) and facilitates possible simulation of non-existing derivative prices in the NSM, from those in the JSE. Furthermore, the research explores in detail the workings of different derivative pricing models, for example various structures for the Ad-Hoc Black Scholes models, using selected underlying asset prices, to determine the applicability of the models in the NSM. The key research findings include: 1) ways to estimate the parameters of the stochastic calculus models; 2) exploring the benefits of introducing pioneer derivative products in the NSM, including risk hedging, arbitrage, and price speculation; 3) using NSM stylized facts to calibrate selected derivative pricing models; and 4) explaining how the results could be used in future experimental modelling to compare the investment performance of selected models. By way of contributions to knowledge, this is the first study known to the researcher that provides in-depth review of the theoretical and empirical underpinnings of derivative pricing possible in the NSM. This forms the basis for the Black Scholes approach to asset pricing of European option contract, which is the kind of call/put option contract that is being adopted in the NSM. The research provides the initial foundations for effective derivatives trading in the NSM. By explaining the heuristics for developing derivative products in the NSM from JSE information, the research will support future work in this important area of study

Three Risky Decades: A Time for Econophysics?

Our Special Issue we publish at a turning point, which we have not dealt with since World War II. The interconnected long-term global shocks such as the coronavirus pandemic, the war in Ukraine, and catastrophic climate change have imposed significant humanitary, socio-economic, political, and environmental restrictions on the globalization process and all aspects of economic and social life including the existence of individual people. The planet is trapped—the current situation seems to be the prelude to an apocalypse whose long-term effects we will have for decades. Therefore, it urgently requires a concept of the planet's survival to be built—only on this basis can the conditions for its development be created. The Special Issue gives evidence of the state of econophysics before the current situation. Therefore, it can provide excellent econophysics or an inter-and cross-disciplinary starting point of a rational approach to a new era

Energy: A continuing bibliography with indexes, issue 32

This bibliography lists 1316 reports, articles, and other documents introduced into the NASA scientific and technical information system from October 1, 1981 through December 31, 1981

Aeronautical Engineering: A special bibliography with indexes, supplement 73, August 1976

This bibliography lists 206 reports, articles, and other documents introduced into the NASA scientific and technical information system in July 1976