13 research outputs found
Fractional Black-Scholes equations and their robust numerical simulations
No full textPhilosophiae Doctor - PhDConventional partial differential equations under the classical Black-Scholes approach
have been extensively explored over the past few decades in solving option
pricing problems. However, the underlying Efficient Market Hypothesis (EMH) of
classical economic theory neglects the effects of memory in asset return series, though
memory has long been observed in a number financial data. With advancements in
computational methodologies, it has now become possible to model different real life
physical phenomenons using complex approaches such as, fractional differential equations
(FDEs). Fractional models are generalised models which based on literature have
been found appropriate for explaining memory effects observed in a number of financial
markets including the stock market. The use of fractional model has thus recently
taken over the context of academic literatures and debates on financial modelling. Fractional
models are usually of a non-linear and complex nature, which pose a considerable
amount of computational and theoretical difficulties in deriving their analytical solutions.
To the best of our knowledge, currently, there exist no tractable exact/analytical
solution methods for solving fractional Black-Scholes equations, and as such, numerical
solution methods become of a vital importance in understanding nature of solutions
to such models. This thesis therefore, serves to derive some Generalised (fractional)
Black-Scholes Partial Differential Equations (fBS-PDEs), as well as, propose their
respective tractable, efficient and robust numerical simulation methods
Optimal provisioning for deposit withdrawals and loan losses in the banking industry
No full textThesis (Ph.D. (Computer, Statistical and Mathematical Sciences))--North-West University, Potchefstroom Campus, 2008.With the acceptance of the new Basel II banking regulation (implemented in South Africa in January 2008) the search for improved ways of modeling the most important banking activities has become very topical. Since the notion of Levy-process was introduced, it has emerged as an important tool for modeling economic variables in a Basel II framework. In this study, we investigate the stochastic dynamics of banking items that are driven by such processes. In particular, we discuss bank provisioning for loan losses and deposit withdrawals.
The first type of provisioning is related to the earnings that the bank sets aside in order to cover loan defaults. In this case, we apply principles from robustness to a situation where the decision maker is a bank owner and the decision rule determines the optimal provisioning strategy for loan losses. In this regard, we formulate a dynamic banking loan loss model involving a provisioning portfolio consisting of provisions for expected losses and loan loss reserves for unexpected losses. Here, unexpected loan losses and provisioning for expected losses are modeled via a compound Poisson process and an exponential Levy process, respectively. We use historical evidence from OECD (Organization for Economic Corporation and Development) countries to support the fact that the provisions for loan losses-to-total assets ratio is negatively correlated with aggregate asset prices and the private credit-to-GDP ratio.
Secondly, we construct models for provisioning for deposit withdrawals. In particular, we build stochastic dynamic models which enable us to analyze the interplay between deposit withdrawals and the provisioning for these withdrawals via Treasuries and reserves. Further insight is gained by considering a numerical problem and a simulation of the trajectory of the stochastic dynamics of the sum of the Treasuries and reserves. Since managing the risk that depositors will exercise their withdrawal option is an important aspect of this thesis, we consider the idea of a hedging provisioning strategy for deposit withdrawals in an incomplete market setting. In this spirit, we discuss an optimal risk management problem for a commercial bank whose main activity is to obtain funds through deposits from the public and use the Treasuries and reserves to cater for the resulting withdrawals. Finally, we provide a brief analysis of some of the issues arising from the dynamic models of the banking items derived.Doctora
A Classification of Fuzzy Subgroups of Finite Abelian Groups
No full textThe knowledge of fuzzy sets and systems has become a considerable aspect to apply in various mathematical systems. In this paper, we apply a knowledge of fuzzy sets to group structures. We consider a fuzzy subgroups of finite abelian groups, denoted by G = Zpn +Zqm , where Z is an integer, p and q are distinct primes and m;n are natural numbers. The fuzzy subgroups are classified using the notion of equivalence classes. In essence the equivalence relations of fuzzy subsets X is extended to equivalence relations of fuzzy subgroups of a group G. We then use the notion of flags and keychains as tools to enumerate fuzzy subgroups of G. In this way, we characterized the properties of the fuzzy subgroups of G. Finally, we use maximal chains to construct a fuzzy subgroups-lattice diagram for these groups of G
A Classification of Fuzzy Subgroups of Finite Abelian Groups
No full textThe knowledge of fuzzy sets and systems has become a considerable aspect to apply in various mathematical systems. In this paper, we apply a knowledge of fuzzy sets to group structures. We consider a fuzzy subgroups of finite abelian groups, denoted by G = Zpn +Zqm , where Z is an integer, p and q are distinct primes and m;n are natural numbers. The fuzzy subgroups are classified using the notion of equivalence classes. In essence the equivalence relations of fuzzy subsets X is extended to equivalence relations of fuzzy subgroups of a group G. We then use the notion of flags and keychains as tools to enumerate fuzzy subgroups of G. In this way, we characterized the properties of the fuzzy subgroups of G. Finally, we use maximal chains to construct a fuzzy subgroups-lattice diagram for these groups of G
A Classification of Fuzzy Subgroups of Finite Abelian Groups
No full textThe knowledge of fuzzy sets and systems has become a considerable aspect to apply in various mathematical systems. In this paper, we apply a knowledge of fuzzy sets to group structures. We consider a fuzzy subgroups of finite abelian groups, denoted by G = Zpn +Zqm , where Z is an integer, p and q are distinct primes and m;n are natural numbers. The fuzzy subgroups are classified using the notion of equivalence classes. In essence the equivalence relations of fuzzy subsets X is extended to equivalence relations of fuzzy subgroups of a group G. We then use the notion of flags and keychains as tools to enumerate fuzzy subgroups of G. In this way, we characterized the properties of the fuzzy subgroups of G. Finally, we use maximal chains to construct a fuzzy subgroups-lattice diagram for these groups of G
A study of fuzzy sets and systems with applications to group theory and decision making
No full textIn this study we apply the knowledge of fuzzy sets to group structures and also to decision-making implications. We study fuzzy subgroups of finite abelian groups. We set G = Z[subscript p[superscript n]] + Z[subscript q[superscript m]]. The classification of fuzzy subgroups of G using equivalence classes is introduced. First, we present equivalence relations on fuzzy subsets of X, and then extend it to the study of equivalence relations of fuzzy subgroups of a group G. This is then followed by the notion of flags and keychains projected as tools for enumerating fuzzy subgroups of G. In addition to this, we use linear ordering of the lattice of subgroups to characterize the maximal chains of G. Then we narrow the gap between group theory and decision-making using relations. Finally, a theory of the decision-making process in a fuzzy environment leads to a fuzzy version of capital budgeting. We define the goal, constraints and decision and show how they conflict with each other using membership function implications. We establish sets of intervals for projecting decision boundaries in general. We use the knowledge of triangular fuzzy numbers which are restricted field of fuzzy logic to evaluate investment projections
An Efficient Numerical Scheme for a Time-Fractional Black–Scholes Partial Differential Equation Derived from the Fractal Market Hypothesis
No full textSince the early 1970s, the study of Black–Scholes (BS) partial differential equations (PDEs) under the Efficient Market Hypothesis (EMH) has been a subject of active research in financial engineering. It has now become obvious, even to casual observers, that the classical BS models derived under the EMH framework fail to account for a number of realistic price evolutions in real-time market data. An alternative approach to the EMH framework is the Fractal Market Hypothesis (FMH), which proposes better and clearer explanations of market behaviours during unfavourable market conditions. The FMH involves non-local derivatives and integral operators, as well as fractional stochastic processes, which provide better tools for explaining the dynamics of evolving market anomalies, something that classical BS models may fail to explain. In this work, using the FMH, we derive a time-fractional Black–Scholes partial differential equation (tfBS-PDE) and then transform it into a heat equation, which allows for ease of implementing a high-order numerical scheme for solving it. Furthermore, the stability and convergence properties of the numerical scheme are discussed, and overall techniques are applied to pricing European put option problems
An efficient numerical method for pricing double-barrieroptions on an underlying stock governed by a fractal stochastic process
No full textAfter the discovery of the fractal structures of financial markets, enormous effort has been dedicated to finding accurate and stable numerical schemes to solve fractional Black-Scholes partial differential equations. This work, therefore, proposes a numerical scheme for pricing double-barrier options, written on an underlying stock whose dynamics are governed by a non-standard fractal stochastic process. The resultant model is time-fractional and is herein referred to as a time-fractional Black-Scholes model. The presence of the time-fractional derivative helps to capture the time-decaying effects of the underlying stock while capturing the globalized change in underlying prices and barriers. In this paper, we present the construction of the proposed scheme, analyse it in terms of its stability and convergence, and present two numerical examples of pricing double knock-in barrier-option problems. The results suggest that the proposed scheme is unconditionally stable and convergent with order O(h2 + k2)
Identifying Factors for Integrating Math and Music Education at Primary Schools in Namibia
No full textAn Efficient Numerical Method for Pricing Double-Barrier Options on an Underlying Stock Governed by a Fractal Stochastic Process
No full textAfter the discovery of the fractal structures of financial markets, enormous effort has been dedicated to finding accurate and stable numerical schemes to solve fractional Black-Scholes partial differential equations. This work, therefore, proposes a numerical scheme for pricing double-barrier options, written on an underlying stock whose dynamics are governed by a non-standard fractal stochastic process. The resultant model is time-fractional and is herein referred to as a time-fractional Black-Scholes model. The presence of the time-fractional derivative helps to capture the time-decaying effects of the underlying stock while capturing the globalized change in underlying prices and barriers. In this paper, we present the construction of the proposed scheme, analyse it in terms of its stability and convergence, and present two numerical examples of pricing double knock-in barrier-option problems. The results suggest that the proposed scheme is unconditionally stable and convergent with order O(h2+k2)
