A mean-variance Portfolio Optimizing Trading Algorithm using regime-switching Economic Parameters

In this master's thesis a model of algorithmic trading is constructed. The model aims to create an optimal investment portfolio consisting of a risk-free asset and a risky asset. The risky asset is in the form of a stock generated using regime-switching parameters with a Markov chain explaining the state of the economy. The optimization of the portfolio is carried out under certain assumptions and reasonable constraints on risk, transaction costs and amount traded. The constraint on nancial risk is implemented through the recognized mean-variance criterion, balancing the expected value of the portfolio against the variance of the portfolio after every time period. The algorithm is implemented using quadratic programming techniques in Matlab. By varying parameters of the model a sensitivity analysis is performed. Simulated scenarios and the behaviour of the algorithm is presented in graphs. The algorithm is found to be rational and outperforms a static portfolio in every scenario

Risk-Based Indifference Pricing in Jump Diffusion Markets with Regime-Switching

This paper is concerned with risk indifference pricing of a European type contingent claim in an incomplete market, where the evolution of the price of the underlying stock is modeled by a regime-switching jump diffusion. The rationale of using such a model is that it can naturally capture the inherent randomness of a prototypical stock market by incorporating both small and big jumps of the prices as well as the qualitative changes of the market. While the model provides a realistic description of the real market, it does introduces substantial difficulty in the analysis. In particular, in contrast with the classical Black-Scholes model, there are infinitely many equivalent martingale measures and hence the price is not unique in our incomplete market. In particular, there exists a big gap between the commonly used sub- and super-hedging prices.\\ We approach this problem using the framework of risk-indifference pricing. By transforming the pricing problem to an equivalent stochastic game problem, we solve this problem via the associated Hamilton-Jacobi-Bellman-Issac equations. Consequently we obtain a new interval which is smaller than the interval from super- and sub-hedging

Further applications of higher-order Markov chains and developments in regime-switching models

We consider a higher-order hidden Markov models (HMM), also called weak HMM (WHMM), to capture the regime-switching and memory properties of financial time series. A technique of transforming a WHMM into a regular HMM is employed, which in turn enables the development of recursive filters. With the use of the change of reference probability measure methodology and EM algorithm, a dynamic estimation of model parameters is obtained. Several applications and extensions were investigated. WHMM is adopted in describing the evolution of asset prices and its performance is examined through a forecasting analysis. This is extended to the case when the drift and volatility components of the logreturns are modulated by two independent WHMMs that are not necessarily having the same number of states. Numerical experiment is conducted based on simulated data to demonstrate the ability of our estimation approach in recovering the “true” model parameters. The analogue of recursive filtering and parameter estimation to handle multivariate data is also established. Some aspects of statistical inference arising from model implementation such as the assessment of model adequacy and goodness of fit are examined and addressed. The usefulness of the WHMM framework is tested on an asset allocation problem whereby investors determine the optimal investment strategy for the next time step through the results of the algorithm procedure. As an application in the modelling of yield curves, it is shown that the WHMM, with its memory-capturing mechanism, outperforms the usual HMM. A mean-reverting interest rate model is further developed whereby its parameters are modulated by a WHMM along with the formulation of a self-tuning parameter estimation. Finally, we propose an inverse Stieltjes moment approach to solve the inverse problem of calibration inherent in an HMM-based regime-switching set-up

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