# Financial modelling with 2-EPT probability density functions

 dc.contributor.advisor Hanzon, Bernard en dc.contributor.author Sexton, Hugh Conor dc.date.accessioned 2014-03-05T16:35:57Z dc.date.available 2014-03-05T16:35:57Z dc.date.issued 2013 dc.date.submitted 2013 dc.identifier.citation Sexton, H. C. 2013. Financial modelling with 2-EPT probability density functions. PhD Thesis, University College Cork. en dc.identifier.endpage 184 dc.identifier.uri http://hdl.handle.net/10468/1430 dc.description.abstract The class of all Exponential-Polynomial-Trigonometric (EPT) functions is classical and equal to the Euler-d’Alembert class of solutions of linear differential equations with constant coefficients. The class of non-negative EPT functions defined on [0;1) was discussed in Hanzon and Holland (2010) of which EPT probability density functions are an important subclass. EPT functions can be represented as ceAxb, where A is a square matrix, b a column vector and c a row vector where the triple (A; b; c) is the minimal realization of the EPT function. The minimal triple is only unique up to a basis transformation. Here the class of 2-EPT probability density functions on R is defined and shown to be closed under a variety of operations. The class is also generalised to include mixtures with the pointmass at zero. This class coincides with the class of probability density functions with rational characteristic functions. It is illustrated that the Variance Gamma density is a 2-EPT density under a parameter restriction. A discrete 2-EPT process is a process which has stochastically independent 2-EPT random variables as increments. It is shown that the distribution of the minimum and maximum of such a process is an EPT density mixed with a pointmass at zero. The Laplace Transform of these distributions correspond to the discrete time Wiener-Hopf factors of the discrete time 2-EPT process. A distribution of daily log-returns, observed over the period 1931-2011 from a prominent US index, is approximated with a 2-EPT density function. Without the non-negativity condition, it is illustrated how this problem is transformed into a discrete time rational approximation problem. The rational approximation software RARL2 is used to carry out this approximation. The non-negativity constraint is then imposed via a convex optimisation procedure after the unconstrained approximation. Sufficient and necessary conditions are derived to characterise infinitely divisible EPT and 2-EPT functions. Infinitely divisible 2-EPT density functions generate 2-EPT Lévy processes. An assets log returns can be modelled as a 2-EPT Lévy process. Closed form pricing formulae are then derived for European Options with specific times to maturity. Formulae for discretely monitored Lookback Options and 2-Period Bermudan Options are also provided. Certain Greeks, including Delta and Gamma, of these options are also computed analytically. MATLAB scripts are provided for calculations involving 2-EPT functions. Numerical option pricing examples illustrate the effectiveness of the 2-EPT approach to financial modelling. en dc.description.sponsorship Science Foundation Ireland (07/MI/008) en dc.format.mimetype application/pdf en dc.language.iso en en dc.publisher University College Cork en dc.rights © 2013, H. Conor Sexton en dc.rights.uri http://creativecommons.org/licenses/by-nc-nd/3.0/ en dc.subject RARL2 en dc.subject Rational approximation en dc.subject Wiener-Hopf factorization en dc.subject Variance gamma en dc.subject 2-EPT probability density function en dc.subject.lcsh Options (Finance)--Prices--Mathematical models en dc.subject.lcsh Lévy processes en dc.subject.lcsh Distribution (Probability theory) en dc.subject.lcsh Probabilities en dc.title Financial modelling with 2-EPT probability density functions en dc.type Doctoral thesis en dc.type.qualificationlevel Doctoral en dc.type.qualificationname PhD (Financial Mathematics) en dc.internal.availability Full text available en dc.check.info No embargo required en dc.description.version Accepted Version dc.contributor.funder Science Foundation Ireland en dc.description.status Peer reviewed en dc.internal.school Mathematics en dc.check.type No Embargo Required dc.check.reason No embargo required en dc.check.opt-out Not applicable en dc.thesis.opt-out false * dc.check.embargoformat Not applicable en ucc.workflow.supervisor b.hanzon@ucc.ie * dc.internal.conferring Autumn Conferring 2013 en
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