15,779 research outputs found

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

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    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

    Pricing American Options by Exercise Rate Optimization

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    We present a novel method for the numerical pricing of American options based on Monte Carlo simulation and the optimization of exercise strategies. Previous solutions to this problem either explicitly or implicitly determine so-called optimal exercise regions, which consist of points in time and space at which a given option is exercised. In contrast, our method determines the exercise rates of randomized exercise strategies. We show that the supremum of the corresponding stochastic optimization problem provides the correct option price. By integrating analytically over the random exercise decision, we obtain an objective function that is differentiable with respect to perturbations of the exercise rate even for finitely many sample paths. The global optimum of this function can be approached gradually when starting from a constant exercise rate. Numerical experiments on vanilla put options in the multivariate Black-Scholes model and a preliminary theoretical analysis underline the efficiency of our method, both with respect to the number of time-discretization steps and the required number of degrees of freedom in the parametrization of the exercise rates. Finally, we demonstrate the flexibility of our method through numerical experiments on max call options in the classical Black-Scholes model, and vanilla put options in both the Heston model and the non-Markovian rough Bergomi model

    Real Option Valuation of a Portfolio of Oil Projects

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    Various methodologies exist for valuing companies and their projects. We address the problem of valuing a portfolio of projects within companies that have infrequent, large and volatile cash flows. Examples of this type of company exist in oil exploration and development and we will use this example to illustrate our analysis throughout the thesis. The theoretical interest in this problem lies in modeling the sources of risk in the projects and their different interactions within each project. Initially we look at the advantages of real options analysis and compare this approach with more traditional valuation methods, highlighting strengths and weaknesses ofeach approach in the light ofthe thesis problem. We give the background to the stages in an oil exploration and development project and identify the main common sources of risk, for example commodity prices. We discuss the appropriate representation for oil prices; in short, do oil prices behave more like equities or more like interest rates? The appropriate representation is used to model oil price as a source ofrisk. A real option valuation model based on market uncertainty (in the form of oil price risk) and geological uncertainty (reserve volume uncertainty) is presented and tested for two different oil projects. Finally, a methodology to measure the inter-relationship between oil price and other sources of risk such as interest rates is proposed using copula methods.Imperial Users onl

    Efficient Numerical Methods for Pricing American Options under LĂ©vy Models

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    Two new numerical methods for the valuation of American and Bermudan options are proposed, which admit a large class of asset price models for the underlying. In particular, the methods can be applied with LĂ©vy models that admit jumps in the asset price. These models provide a more realistic description of market prices and lead to better calibration results than the well-known Black-Scholes model. The proposed methods are not based on the indirect approach via partial differential equations, but directly compute option prices as risk-neutral expectation values. The expectation values are approximated by numerical quadrature methods. While this approach is initially limited to European options, the proposed combination with interpolation methods also allows for pricing of Bermudan and American options. Two different interpolation methods are used. These are cubic splines on the one hand and a mesh-free interpolation by radial basis functions on the other hand. The resulting valuation methods allow for an adaptive space discretization and error control. Their numerical properties are analyzed and, finally, the methods are validated and tested against various single-asset and multi-asset options under different market models

    Evaluating Callable and Putable Bonds: An Eigenfunction Expansion Approach

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    We propose an efficient method to evaluate callable and putable bonds under a wide class of interest rate models, including the popular short rate diffusion models, as well as their time changed versions with jumps. The method is based on the eigenfunction expansion of the pricing operator. Given the set of call and put dates, the callable and putable bond pricing function is the value function of a stochastic game with stopping times. Under some technical conditions, it is shown to have an eigenfunction expansion in eigenfunctions of the pricing operator with the expansion coefficients determined through a backward recursion. For popular short rate diffusion models, such as CIR, Vasicek, 3/2, the method is orders of magnitude faster than the alternative approaches in the literature. In contrast to the alternative approaches in the literature that have so far been limited to diffusions, the method is equally applicable to short rate jump-diffusion and pure jump models constructed from diffusion models by Bochner's subordination with a L\'{e}vy subordinator

    Pricing American options by exercise rate optimization

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    We present a novel method for the numerical pricing of American options based on Monte Carlo simulation and the optimization of exercise strategies. Previous solutions to this problem either explicitly or implicitly determine so-called optimal exercise regions, which consist of points in time and space at which a given option is exercised. In contrast, our method determines the exercise rates of randomized exercise strategies. We show that the supremum of the corresponding stochastic optimization problem provides the correct option price. By integrating analytically over the random exercise decision, we obtain an objective function that is differentiable with respect to perturbations of the exercise rate even for finitely many sample paths. The global optimum of this function can be approached gradually when starting from a constant exercise rate. Numerical experiments on vanilla put options in the multivariate Black-Scholes model and a preliminary theoretical analysis underline the efficiency of our method, both with respect to the number of time-discretization steps and the required number of degrees of freedom in the parametrization of the exercise rates. Finally, we demonstrate the flexibility of our method through numerical experiments on max call options in the classical Black-Scholes model, and vanilla put options in both the Heston model and the non-Markovian rough Bergomi model

    A local radial basis function method for high-dimensional American option pricing problems

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    In this work, we apply the local Wendland radial basis function (RBF) for solving the time-dependent multi dimensional option pricing nonlinear PDEs. Firstly, cross derivative terms of the PDE are removed with a change of spatial variables based in LDLT factorization of the diffusion matrix. Then, it is discussed that the valuation of a multi-asset option up to 4D can be computed using a modified shape parameter algorithm. In fact, several experiments containing of three and four assets are worked out showing that the results of the presented method are in good agreement with the literature and could be much more accurate once the shape parameter is chosen carefully.This work has been partially supported by the European Union in the FP7-PEOPLE-2012-ITN program under Grant Agreement Number 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance) and the Ministerio de EconomĂ­a y Competitividad Spanish grant MTM2013-41765-P

    Implications and Ramifications of a Sample-Size Approach to Intuition

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    [...from the chapter] In the present article, we delineate a different approach, which is by no means inconsistent, but largely overlaps with the aforementioned definitions. However, our approach is simpler and refrains from a number of rather strong assumptions to which other conceptions subscribe. Using a simple and straightforward criterion, we define intuition in terms of the size of the sample used in reaching a decision: Judgments and decisions are intuitive to the extent that they rest on small samples.

    Valuation of American Options and Employee Stock Options

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    Options play an important role in the financial world and are actively traded with huge trading volume. Most of the options traded on exchanges are American options. Spanning over a few decades, the American option pricing problem continues to intrigue scholars and practitioners in finance. The employee stock options (ESOs), a variant of American options, has been increasingly popular for firms to compensate, motivate and retain employees. ESOs importantly do not trade in markets nevertheless fair value must be determined – often by accountants. Unique features of ESOs however complicate the valuation. Our research, consisting of three papers, focuses on the improved lattice techniques for valuing American options and ESOs. Research paper 1 (Chapter 2) introduces an intelligent lattice search algorithm to efficiently locate the optimal exercise boundary for American options. The computational runtime can be reduced from over 18 minutes down to less than 3 seconds to estimate a 15,000-step CRR binomial tree. Research paper 2 (Chapter 3) introduces a set of lattice techniques to the Leisen-Reimer and Tian binomial models for American options pricing. A level of accuracy and efficiency combined can be achieve that surpass analytical solution models prominent in the literature. Moreover, lattices importantly afford an explicit trade-off locus between accuracy and speed that can be navigated according to predetermined precision tolerance levels and option types. These should have practical relevance to trading platforms that require real-time estimates of implied volatility. Research paper 3 (Chapter 4) proposes adjustments to the Hull-White ESO pricing model, based on insights developed by Boyle-Lau and Tian specifications. The proposed Hull-White-Boyle-Lau and Hull White-Tian revamps expand the practicable menu choice available to stakeholders tasked with the valuation of these ESOs. Accountants, across many jurisdictions, are subjected to higher demands for disclosure and fair valuation. The streamlined valuation approaches developed here may prove ii useful in expanding the tool kit of practicable/workable models. This improved efficiency can be harnessed even at the level of a basic spreadsheet and this this should assist in testing, validating and benchmarking valuation in lattices and in evaluating the newer generation of closed-form solutions emerging in the literature
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