Integration of Fractional Order Black-Scholes Merton with Neural Network

This study enhances option pricing by presenting unique pricing model fractional order Black-Scholes-Merton (FOBSM) which is based on the Black-Scholes-Merton (BSM) model. The main goal is to improve the precision and authenticity of option pricing, matching them more closely with the financial landscape. The approach integrates the strengths of both the BSM and neural network (NN) with complex diffusion dynamics. This study emphasizes the need to take fractional derivatives into account when analyzing financial market dynamics. Since FOBSM captures memory characteristics in sequential data, it is better at simulating real-world systems than integer-order models. Findings reveals that in complex diffusion dynamics, this hybridization approach in option pricing improves the accuracy of price predictions. the key contribution of this work lies in the development of a novel option pricing model (FOBSM) that leverages fractional calculus and neural networks to enhance accuracy in capturing complex diffusion dynamics and memory effects in financial data

On Poisson constrained control of linear diffusions

The classical setting in optimal stopping and optimal control theory assumes that the agent controlling the system can operate continuously in time. In optimal stopping this setting is highly stylized for many applications, for example, in mathematical finance due to illiquid markets. In optimal stochastic control this setting often leads to optimal strategies being singular with respect to the Lebesgue measure, and thus the strategies are not feasible in practice. Hence, it is of importance to study these problems from such a perspective that their solutions are practically more implementable. In this thesis we alter the classical setting by introducing an exogenous constraint, in the form of a signal process, for the control opportunities of the agent. In order to keep the problems more tractable, especially time-homogeneous and Markovian, the signal process is assumed to be a Poisson process with constant intensity. Consequently, the agent can only have influence on the system at discrete times. We call these control problems Poisson constrained control problems and study them when the dynamics are governed by linear diffusion processes. Linear diffusions are particular enough to have a rich theory but still general enough to offer a class of interesting dynamics that are applicable in various situations. A key factor is also that many control problems with diffusions will lead to closed-form solutions. This thesis investigates to which extent the classical theory of diffusion can be applied in this class of control problems to form closed-form solutions

Stochastic optimal control and regime switching : applications in economics

Economic decisions under uncertainty generally involve a change of stochastic regime. This thesis examines the formal conditions for optimizing such decisions and looks at applications to exchange rate intervention, physical investment and consumption behaviour. Many of these economic regime switchings can be mathematically formulated as stopping problems. Global optimality is achieved by applying Hamilton-Jacobi-Bellman equations in each regime, together with the joining conditions at the switching boundaries. Chapter 1 establishes the framework for optimisation and provides various boundary conditions for different switching cases. Chapter 2 applies optimal stopping techniques to derive optimal “time-consistent” exchange rate target zones in the presence of proportional/lump sum intervention costs. It further shows that such discretionary equilibria can be improved upon by a credible commitment to an exchange rate mechanism (such as ERM). Chapter 3 characterises the irreversible oil investment decision in the North Sea as an optimal regime switching problem. In the absence of Petroleum Revenue Tax (PRT), it shows how the optimal development decision will be deferred when real oil prices follow a geometric Brownian motion. In chapter 4, an intertemporal partial equilibrium model of investment is used to assess the effects of stochastic capital depreciation on optimal investment behaviour, in a context where a sales constraint effectively decomposes the problem into two distinct regimes. The presence of the uncertainty about depreciation reduces firm’s demand for investment; and increasing the variability of capital depreciation further reduces investment. The uncertainty also makes investment “smoother” than that under certainty. Finally, chapter 5 and 6 deal with optimal consumption/portfolio decisions in a two-asset model with shortselling and borrowing restrictions imposed. Chapter 5 formulates a regime switching problem due to the presence of the borrowing constraint and specifies the corresponding boundary conditions. Chapter 6 characterises optimal solutions to various combinations of parameters for constant relative and constant absolute risk aversion utility functions. In many cases, if labour income is fully diversifiable, the borrowing constraint only binds when the wealth level falls below a threshold, and risk taking behaviour at the low level of wealth is associated with a convex portion of the indirect utility function (value function). In such regime-switch cases, the introduction of the borrowing constraint makes consumption more volatile relative to income. It also generates the precautionary motive for saving

Stochastic optimal portfolios and life insurance problems in a Lévy market

This thesis solves various optimal investment, consumption and life insurance problems described by jump-diffusion processes. In the first part of the thesis, we solve an optimal investment, consumption, and life insurance problem when the investor is restricted to capital guarantee. We consider an incomplete market described by a jump-diffusion model with stochastic volatility. Using the martingale approach, we prove the existence of the optimal strategy and the optimal martingale measure and we obtain the explicit solutions for the power utility functions. Secondly, we prove the sufficient and necessary maximum principle for the similar problem proposed in the first part. Then we apply the results to solve an investment, consumption, and life insurance problem with stochastic volatility, that is, we consider a wage earner investing in one risk-free asset and one risky asset described by a jump-diffusion process and has to decide concerning consumption and life insurance purchase. We assume that the life insurance for the wage earner is bought from a market composed of M > 0 life insurance companies offering pairwise distinct life insurance contracts. The goal is to maximize the expected utilities derived from the consumption, the legacy in the case of a premature death and the investor's terminal wealth. The third part discusses an optimal investment, consumption and insurance problem of a wage earner under inflation. Assume a wage earner investing in a real money account and three asset prices, namely: a real zero coupon bond, the inflation-linked real money account and a risky share described by jump-diffusion processes. Using the theory of quadratic-exponential backward stochastic differential equation (BSDE) with jumps approach, we derive the optimal strategy for the two typical utilities (exponential and power) and the value function is characterized as a solution of BSDE with jumps. The explicit solutions for the optimal investment in both cases of exponential and power utility functions for a diffusion case are derived.Thesis (PhD)--University of Pretoria, 2018.Mathematics and Applied MathematicsPhDUnrestricte

Macroeconomic Volatility and Sovereign Asset-Liability Management

For most developing countries, the predominant source of sovereign wealth is commodity related export income. However, over-reliance on commodity related income exposes countries to significant terms of trade shocks due to excessive price volatility. The spillovers are pro-cyclical fiscal policies and macroeconomic volatility problems that if not adequately managed, could have catastrophic economic consequences including sovereign bankruptcy. The aim of this study is to explore new ways of solving the problem in an asset-liability management framework for an exporting country like Ghana. Firstly, I develop an unconditional commodity investment strategy in the tactical mean-variance setting for deterministic returns. Secondly, in continuous time, shocks to return moments induce additional hedging demands warranting an extension of the analysis to a dynamic stochastic setting whereby, the optimal commodity investment and fiscal consumption policies are conditioned on the stochastic realisations of commodity prices. Thirdly, I incorporate jumps and stochastic volatility in an incomplete market extension of the conditional model. Finally, I account for partial autocorrelation, significant heteroskedastic disturbances, cointegration and non-linear dependence in the sample data by adopting GARCH-Error Correction and dynamic Copula-GARCH models to enhance the forecasting accuracy of the optimal hedge ratios used for the state-contingent dynamic overlay hedging strategies that guarantee Pareto efficient allocation. The unconditional model increases the Sharpe ratio by a significant margin and noticeably improves the portfolio value-at-risk and maximum drawdown. Meanwhile, the optimal commodities investment decisions are superior in in-sample performance and robust to extreme interest rate changes by up to 10 times the current rate. In the dynamic setting, I show that momentum strategies are outperformed by contrarian policies, fiscal consumption must account for less than 40% of sovereign wealth, while risky investments must not exceed 50% of the residual wealth. Moreover, hedging costs are reduced by as much as 55% while numerically generating state-dependent dynamic futures hedging policies that reveal a predominant portfolio strategy analogous to the unconditional model. The results suggest buying commodity futures contracts when the country’s current exposure in a particular asset is less than the model implied optimal quantity and selling futures contracts when the actual quantity exported exceeds the benchmark.Open Acces

Strategically-Timed Actions in Stochastic Differential Games

Financial systems are rich in interactions amenable to description by stochastic control theory. Optimal stochastic control theory is an elegant mathematical framework in which a controller, profitably alters the dynamics of a stochastic system by exercising costly control inputs. If the system includes more than one agent, the appropriate modelling framework is stochastic differential game theory — a multiplayer generalisation of stochastic control theory. There are numerous environments in which financial agents incur fixed minimal costs when adjusting their investment positions; trading environments with transaction costs and real options pricing are important examples. The presence of fixed minimal adjustment costs produces adjustment stickiness as agents now enact their investment adjustments over a sequence of discrete points. Despite the fundamental relevance of adjustment stickiness within economic theory, in stochastic differential game theory, the set of players’ modifications to the system dynamics is mainly restricted to a continuous class of controls. Under this assumption, players modify their positions through infinitesimally fine adjustments over the problem horizon. This renders such models unsuitable for modelling systems with fixed minimal adjustment costs. To this end, we present a detailed study of strategic interactions with fixed minimal adjustment costs. We perform a comprehensive study of a new stochastic differential game of impulse control and stopping on a jump-diffusion process and, conduct a detailed investigation of two-player impulse control stochastic differential games. We establish the existence of a value of the games and show that the value is a unique (viscosity) solution to a double obstacle problem which is characterised in terms of a solution to a non-linear partial differential equation (PDE). The study is contextualised within two new models of investment that tackle a dynamic duopoly investment problem and an optimal liquidity control and lifetime ruin problem. It is then shown that each optimal investment strategy can be recovered from the equilibrium strategies of the corresponding stochastic differential game. Lastly, we introduce a dynamic principal-agent model with a self-interested agent that faces minimally bounded adjustment costs. For this setting, we show for the first time that the principal can sufficiently distort that agent’s preferences so that the agent finds it optimal to execute policies that maximise the principal’s payoff in the presence of fixed minimal costs

Portfolio Choice with Stochastic Investment Opportunities: a User's Guide

This survey reviews portfolio choice in settings where investment opportunities are stochastic due to, e.g., stochastic volatility or return predictability. It is explained how to heuristically compute candidate optimal portfolios using tools from stochastic control, and how to rigorously verify their optimality by means of convex duality. Special emphasis is placed on long-horizon asymptotics, that lead to particularly tractable results.Comment: 31 pages, 4 figure