Fast binomial procedures for pricing Parisian/ParAsian options

The discrete procedures for pricing Parisian/ParAsian options depend, in general, by three dimensions: time, space, time spent over the barrier. Here we present some combinatorial and lattice procedures which reduce the computational complexity to second order. In the European case the reduction was already given by Lyuu-Wu \cite{WU} and Li-Zhao \cite{LZ}, in this paper we present a more efficient procedure in the Parisian case and a different approach (again of order 2) in the ParAsian case. In the American case we present new procedures which decrease the complexity of the pricing problem for the Parisian/ParAsian knock-in options. The reduction of complexity for Parisian/ParAsian knock-out options is still an open problem.Les méthodes à temps discret pour le pricing des options parisienne et parAsian dépendent généralement de trois paramètres : Le temps, l'espace et le temps écoulé proche de la barrière. Dans ce travail, nous présentons des procédures combinatoires et de treillis qui permettent de réduire d'ordre 2 la complexité du calcul. Ces simplifications ont déjà été utilisées par Lyuu-Wu et Li-Zhao dans le cas des options européennes. Dans cet article, une technique plus efficace est employée pour les options parisienne et parAsian. Nous introduisons aussi de nouvelles méthodes rapides pour les options américaines applicables aux parisiennes et parAsians knock-in. La généralisation de ce type de procédures aux options parisienne/parAsian knock-out reste un problème ouvert

Applications of optimization to sovereign debt issuance

This thesis was submitted for the degree of Doctor of Philosophy and was awarded by Brunel UniversityThis thesis investigates different issues related to the issuance of debt by sovereign bodies such as governments, under uncertainty about the future interest rates. Several dynamic models of interest rates are presented, along with extensive numerical experiments for calibration of models and comparison of performance on real financial market data. The main contribution of the thesis is the construction and demonstration of a stochastic optimisation model for debt issuance under interest rate uncertainty. When the uncertainty is modelled using a model from a certain class of single factor interest rate models, one can construct a scenario tree such that the number of scenarios grows linearly with time steps. An optimization model is constructed using such a one factor scenario tree. For a real government debt issuance remit, a multi-stage stochastic optimization is performed to choose the type and the amount of debt to be issued and the results are compared with the real issuance. The currently used simulation models by the government, which are in public domain, are also reviewed. Apparently, using an optimization model, such as the one proposed in this work, can lead to substantial savings in the servicing costs of the issued deb

A study in the financial valuation of a topping oil refinery

Oil refineries underpin modern day economics, finance and engineering – without their refined products the world would stand still, as vehicles would not have petrol, planes grounded without kerosene and homes not heated, without heating oil. In this thesis I study the refinery as a financial asset; it is not too dissimilar to a chemical plant, in this respect. There are a number of reasons for this research; over recent years there have been legal disputes based on a refiner's value, investors and entrepreneurs are interested in purchasing refineries, and finally the research in this arena is sparse. In this thesis I utilise knowledge and techniques within finance, optimisation, stochastic mathematics and commodities to build programs that obtain a financial value for an oil refinery. In chapter one I introduce the background of crude oil and the significance of the refinery in the oil value chain. In chapter two I construct a traditional discounted cash flow valuation often applied within practical finance. In chapter three I program an extensive piecewise non linear optimisation solution on the entire state space, leveraging off a simulation of the refined products using a set of single factor Schwartz (1997) stochastic equations often applied to commodities. In chapter four I program an optimisation using an approximation on crack spread option data with the aim of lowering the duration of solution found in chapter three; this is achieved by utilising a two-factor Hull & White sub-trinomial tree based numerical scheme; see Hull & White (1994) articles I & II for a thorough description. I obtain realistic and accurate numbers for a topping oil refinery using financial market contracts and other real data for the Vadinar refinery based in Gujurat India

Three Essays on Stochastic Optimization Applied in Financial Engineering and Inventory Management

Stochastic optimization methods are now being widely used in a multitude of applications. This dissertation includes three essays on applying stochastic optimization methods to solve problems in inventory management and financial engineering. Essay one addresses the problem of simultaneous price determination and inventory management. Demand depends explicitly on the product price p, and the inventory control system operates under a periodic review (s, S) ordering policy. To minimize the long-run average loss, we derive sample path derivatives that can be used in a gradient-based algorithm for determining the optimal values of the three parameters (s, S, p) in a simulation-based optimization procedure. Numerical results for several optimization examples via different stochastic algorithms are presented, and consistency proofs for the estimators are provided. Essay two considers the application of stochastic optimization methods to American-style option pricing. We apply a randomized optimization algorithm called Model Reference Adaptive Search (MRAS) to pricing American-style options through parameterizing the early exercise boundary. Numerical results are provided for pricing American-style call and put options written on underlying assets following geometric Brownian motion and Merton jump-diffusion processes. We also price American-style Asian options written on underlying assets following geometric Brownian motion. The results from the MRAS algorithm are compared with the cross-entropy (CE) method, and MRAS is found to be an efficient method. Essay three addresses the problem of finding the optimal importance sampling measure when simulating portfolios of credit risky assets. We apply a gradient-based stochastic approximation method to find the parameters in the minimum variance problem when importance sampling is used. The gradient estimator is obtained under the original measure. We also employ the CE method to solve the same variance minimization problem. Numerical results illustrating the variance reduction are presented for the estimation of the portfolios' expected loss, unexpected loss and quantiles

Use of the Monte Carlo Simulation in Valuation of European and American Call Options

This thesis examines the valuation methods used for pricing European and American call options. Options are financial instruments that play an important role in the financial industry and are used in hedging, speculating and arbitraging. Because options are widely used in investing, there is a need for valuation methods that are as precise as possible. Options have been perceived as obscure financial instruments due to the lack of valuation techniques in the past. However, with the discovery of Black-Scholes Model in 1973, the first option valuation method, option trading escalated. In this thesis, the fair market value of S&P 500 index with European exercise style, The Google Option Contract and Apple Option Contract will be obtained by using the Black-Scholes Model, the General Monte Carlo Simulation, The Combined Method and the Least-Squares Monte Carlo. The results from three models will be compared and contrasted in order to determine the best valuation method

Default Risk, Bankruptcy Procedures and the Market Value of Life Insurance Liabilities

The topic of insolvency risk in connection with life insurance companies has recently attracted a great deal of attention. In this paper, the question is investigated of how the value of the equity and of the liability of a life insurance company are affected by the default risk and the choice of the relevant bankruptcy procedure. As an example, the U.S. Bankruptcy Code with Chapter 7 and Chapter 11 bankruptcy procedures is used. Grosen and Jørgensen's (2002) contingent claim model, implying only a Chapter 7 bankruptcy procedure, is extended to allow for more general bankruptcy procedures such as Chapter 11. Thus, more realistically, default and liquidation are modelled as distinguishable events. This is realized by using so-called standard and cumulative Parisian barrier option frameworks. It is shown that these options have appealing interpretations in terms of the bankruptcy mechanism. Furthermore, a number of representative numerical analyses and comparative statics are performed in order to investigate the effects of different parameter changes on the values of the insurance company's equity and liability, and hence on the value of the life insurance contract. To complete the analysis, the shortfall probabilities of the insurance company implied by the proposed models are computed and compared.Equity--Linked Life Insurance, Default Risk, Liquidation Risk, Contingent Claims Pricing, Parisian Options, Bankruptcy Procedures

A study in the financial valuation of a topping oil refinery

Oil refineries underpin modern day economics, finance and engineering – without their refined products the world would stand still, as vehicles would not have petrol, planes grounded without kerosene and homes not heated, without heating oil. In this thesis I study the refinery as a financial asset; it is not too dissimilar to a chemical plant, in this respect. There are a number of reasons for this research; over recent years there have been legal disputes based on a refiner's value, investors and entrepreneurs are interested in purchasing refineries, and finally the research in this arena is sparse. In this thesis I utilise knowledge and techniques within finance, optimisation, stochastic mathematics and commodities to build programs that obtain a financial value for an oil refinery. In chapter one I introduce the background of crude oil and the significance of the refinery in the oil value chain. In chapter two I construct a traditional discounted cash flow valuation often applied within practical finance. In chapter three I program an extensive piecewise non linear optimisation solution on the entire state space, leveraging off a simulation of the refined products using a set of single factor Schwartz (1997) stochastic equations often applied to commodities. In chapter four I program an optimisation using an approximation on crack spread option data with the aim of lowering the duration of solution found in chapter three; this is achieved by utilising a two-factor Hull & White sub-trinomial tree based numerical scheme; see Hull & White (1994) articles I & II for a thorough description. I obtain realistic and accurate numbers for a topping oil refinery using financial market contracts and other real data for the Vadinar refinery based in Gujurat India

A Brief Review on Mathematical Tools Applicable to Quantum Computing for Modelling and Optimization Problems in Engineering

Since its emergence, quantum computing has enabled a wide spectrum of new possibilities and advantages, including its efficiency in accelerating computational processes exponentially. This has directed much research towards completely novel ways of solving a wide variety of engineering problems, especially through describing quantum versions of many mathematical tools such as Fourier and Laplace transforms, differential equations, systems of linear equations, and optimization techniques, among others. Exploration and development in this direction will revolutionize the world of engineering. In this manuscript, we review the state of the art of these emerging techniques from the perspective of quantum computer development and performance optimization, with a focus on the most common mathematical tools that support engineering applications. This review focuses on the application of these mathematical tools to quantum computer development and performance improvement/optimization. It also identifies the challenges and limitations related to the exploitation of quantum computing and outlines the main opportunities for future contributions. This review aims at offering a valuable reference for researchers in fields of engineering that are likely to turn to quantum computing for solutions. Doi: 10.28991/ESJ-2023-07-01-020 Full Text: PD