Financial modelling with 2EPT probability density functions 
Sexton, Hugh Conor



The class of all ExponentialPolynomialTrigonometric (EPT) functions is classical and equal to the Eulerd’Alembert class of solutions of linear differential equations with constant coefficients. The class of nonnegative 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 2EPT 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 2EPT density under a parameter restriction. A discrete 2EPT process is a process which has stochastically independent 2EPT 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 WienerHopf factors of the discrete time 2EPT process. A distribution of daily logreturns, observed over the period 19312011 from a prominent US index, is approximated with a 2EPT density function. Without the nonnegativity 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 nonnegativity 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 2EPT functions. Infinitely divisible 2EPT density functions generate 2EPT Lévy processes. An assets log returns can be modelled as a 2EPT Lévy process. Closed form pricing formulae are then derived for European Options with specific times to maturity. Formulae for discretely monitored Lookback Options and 2Period 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 2EPT functions. Numerical option pricing examples illustrate the effectiveness of the 2EPT approach to financial modelling.

Keyword(s):

RARL2; Rational approximation; WienerHopf factorization; Variance gamma; 2EPT probability density function; Options (Finance)PricesMathematical models; Lévy processes; Distribution (Probability theory); Probabilities 
Publication Date:

2013 
Type:

Doctoral thesis 
PeerReviewed:

Yes 
Language(s):

English 
Institution:

University College Cork 
Funder(s):

Science Foundation Ireland 
Citation(s):

Sexton, H. C. 2013. Financial modelling with 2EPT probability density functions. PhD Thesis, University College Cork. 
Publisher(s):

University College Cork 
File Format(s):

application/pdf 
Supervisor(s):

Hanzon, Bernard 
First Indexed:
20140308 05:30:55 Last Updated:
20161014 06:13:59 