Applications of Stochastic Control in Energy Real Options and Market Illiquidity

We present three interesting applications of stochastic control in finance. The first is a real option model that considers the optimal entry into and subsequent operation of a biofuel production facility. We derive the associated Hamilton Jacobi Bellman (HJB) equation for the entry and operating decisions along with the econometric analysis of the stochastic price inputs. We follow with a Monte Carlo analysis of the risk profile for the facility. The second application expands on the analysis of the biofuel facility to account for the associated regulatory and taxation uncertainty experienced by players in the renewables and energy industries. A federal biofuel production subsidy per gallon has been available to producers for many years but the subsidy price level has changed repeatedly. We model this uncertain price as a jump process. We present and solve the HJB equations for the associated multidimensional jump diffusion problem which also addresses the model uncertainty pervasive in real option problems such as these. The novel real option framework we present has many applications for industry practitioners and policy makers dealing with country risk or regulatory uncertainty---which is a very real problem in our current global environment. Our final application (which, although apparently different from the first two applications, uses the same tools) addresses the problem of producing reliable bid-ask spreads for derivatives in illiquid markets. We focus on the hedging of over the counter (OTC) equity derivatives where the underlying assets realistically have transaction costs and possible illiquidity which standard finance models such as Black-Scholes neglect. We present a model for hedging under market impact (such as bid-ask spreads, order book depth, liquidity) using temporary and permanent equity price impact functions and derive the associated HJB equations for the problem. This model transitions from continuous to impulse trading (control) with the introduction of fixed trading costs. We then price and hedge via the economically sound framework of utility indifference pricing. The problem of hedging under liquidity impact is an on-going concern of market makers following the Global Financial Crisis

On sources of risk in quadratic hedging and incomplete markets

This thesis is divided into three chapters, each dealing with a different aspect of market incompleteness and its consequences on quadratic hedging strategies and hedging errors. The first chapter studies the effects of market incompleteness due to discrete time trading. We derive the asymptotics (in trading frequency) of the quadratic hedging error of a digital option and obtain a correction to the classical granularity formula, showing that for discontinuous payoffs, the second order term driven by the Cash Gamma remains highly significant. We also show that the discrete-time quadratic hedging strategy generates the same asymptotic error as a continuous-time Black-Scholes delta-hedging strategy used on a discrete set of times. The second chapter studies the effects of market incompleteness due to jumps in cases when the discretization error from Chapter 1 is predictable. We compute the hedging error under an exponential L´evy model for a general ’L´evy contract’ that encompasses log contracts, variance swaps and higher order moment swaps. We compare two utility-based pricing approaches for incomplete markets: quadratic hedging (corresponding to quadratic utility) and exponential utility. We show that for small jumps, numerically difficult exponential utility results are well-estimated via closedform quadratic hedging formulas. We use our results on hedging errors to obtain 'good-deal bounds' for variance and skewness swaps. The third chapter studies the effects of market incompleteness due to uncertainty in the exact specification of the data generating process. We conduct quadratic hedging under a regime-switching L´evy model, which switches between a finite set of distributions based on the value of a (hidden) state variable. We solve the quadratic hedging problem in two steps. First we compute a stochastic differential equation for the filtered estimate of the hidden state. We then use it to solve the quadratic hedging problem with this additional observable variable via classic techniques. We provide Fourier Transform formulas for the mean-value process and hedging strategy, and a recursive scheme for the hedging error

An ETD method for multi-asset American option pricing under jump-diffusion model

In this paper, we propose a numerical method for American multi-asset options under jump-diffusion model based on the combination of the exponential time differencing (ETD) technique for the differential operator and Gauss-Hermite quadrature for the integral term. In order to simplify the computational stencil and improve characteristics of the ETD-scheme mixed derivative eliminating transformation is applied. The results are compared with recently proposed methods.Ministerio de Ciencia, Innovación y Universidades, Grant/Award Number: MTM2017- 89664-P; Ministerio de Economía y Competitividad, Grant/Award Number: PID2019-107685RB-I0

Multi-scale Volatility in Option Pricing

This PhD thesis investigated the influence of kaolin and bentonite clays in the ore on flotation, filtration and centrifugal concentration. The results showed that the presence of particularly bentonite in the ore had a detrimental effect on flotation and filtration. The information generated from this work will advance our knowledge as well as provide important information for plant metallurgists. The project, therefore, is essential for the mineral industry that process clay-containing ores

Linear state models for volatility estimation and prediction

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.This thesis concerns the calibration and estimation of linear state models for forecasting stock return volatility. In the first two chapters I present aspects of financial modelling theory and practice that are of particular relevance to the theme of this present work. In addition to this I review the literature concerning these aspects with a particular emphasis on the area of dynamic volatility models. These chapters set the scene and lay the foundations for subsequent empirical work and are a contribution in themselves. The structure of the models employed in the application chapters 4,5 and 6 is the state-space structure, or alternatively the models are known as unobserved components models. In the literature these models have been applied in the estimation of volatility, both for high frequency and low frequency data. As opposed to what has been carried out in the literature I propose the use of these models with Gaussian components. I suggest the implementation of these for high frequency data for short and medium term forecasting. I then demonstrate the calibration of these models and compare medium term forecasting performance for different forecasting methods and model variations as well as that of GARCH and constant volatility models. I then introduce implied volatility measurements leading to two-state models and verify whether this derivative-based information improves forecasting performance. In chapter 6I compare different unobserved components models' specification and forecasting performance. The appendices contain the extensive workings of the parameter estimates' standard error calculations

Mathematical control theory and Finance

Control theory provides a large set of theoretical and computational tools with applications in a wide range of fields, running from ”pure” branches of mathematics, like geometry, to more applied areas where the objective is to find solutions to ”real life” problems, as is the case in robotics, control of industrial processes or finance. The ”high tech” character of modern business has increased the need for advanced methods. These rely heavily on mathematical techniques and seem indispensable for competitiveness of modern enterprises. It became essential for the financial analyst to possess a high level of mathematical skills. Conversely, the complex challenges posed by the problems and models relevant to finance have, for a long time, been an important source of new research topics for mathematicians. The use of techniques from stochastic optimal control constitutes a well established and important branch of mathematical finance. Up to now, other branches of control theory have found comparatively less application in financial problems. To some extent, deterministic and stochastic control theories developed as different branches of mathematics. However, there are many points of contact between them and in recent years the exchange of ideas between these fields has intensified. Some concepts from stochastic calculus (e.g., rough paths) have drawn the attention of the deterministic control theory community. Also, some ideas and tools usual in deterministic control (e.g., geometric, algebraic or functional-analytic methods) can be successfully applied to stochastic control. We strongly believe in the possibility of a fruitful collaboration between specialists of deterministic and stochastic control theory and specialists in finance, both from academic and business backgrounds. It is this kind of collaboration that the organizers of the Workshop on Mathematical Control Theory and Finance wished to foster. This volume collects a set of original papers based on plenary lectures and selected contributed talks presented at the Workshop. They cover a wide range of current research topics on the mathematics of control systems and applications to finance. They should appeal to all those who are interested in research at the junction of these three important fields as well as those who seek special topics within this scope.info:eu-repo/semantics/publishedVersio

Optimal portfolio allocation of commodity related assets using a controlled forward-backward algorithm

In the first part of this thesis we develop an investment consumption model with convex transaction costs and optional stochastic returns for a finite time horizon. The model is a simplified approach for the investment in a portfolio of commodity related assets like real options or production facilities. In contrast to common models like [Awerbuch, Burger 2003] our model is a multi time step approach that optimizes the investment strategy rather then calculating a static imaginary optimal portfolio. On one hand, our numerical results are consistent with the well-known investment-consumption theory in the literature. On the other hand, this is the first in-depth numerical study of a case with convex transaction costs and optional returns. Our focus in the analyses is the form of the investment strategy and its behavior with respect to model parameters. In the second part, an algorithm for solving continuous-time stochastic optimalcontrol problems is presented. The numerical scheme is based on the Stochastic Maximum Principle (SMP) as an alternative to the widely studied dynamic programming principle (DPP). By using the SMP, [Peng 1990] obtained a system of coupled forward-backward stochastic differential equations (FBSDE) with an external optimality condition. We extend the numerical scheme of [Delarue, Menozzi 2005] by a Newton-Raphson method to solve the FBSDE system and the optimality condition simultaneously. This is the first fully implemented algorithm for the solution of stochastic optimal control problems through the solution of the corresponding extended FBSDE system. We show that the key to its success and numerical advantage is the fact that it tracks the gradient of the value function and an adjusted Hessian backwards in time. The additional information is then exploited for the optimization