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dc.contributor.advisor Mhlanga, F. J. Nkosi, Siboniso Confrence 2021-07-26T08:54:10Z 2021-07-26T08:54:10Z 2019
dc.description Thesis (M.Sc. (Applied Mathematics)) -- University of Limpopo, 2019 en_US
dc.description.abstract Functional Itˆo calculus is based on an extension of the classical Itˆo calculus to functionals depending on the entire past evolution of the underlying paths and not only on its current value. The calculus builds on F¨ollmer’s deterministic proof of the Itˆo formula Föllmer (1981) and a notion of pathwise functional derivative recently proposed by Dupire (2019). There are no smoothness assumptions required on the functionals, however, they are required to possess certain directional derivatives which may be computed pathwise, see Cont and Fournié (2013); Schied and Voloshchenko (2016a); Cont (2012). In this project we revise the functional Itô calculus together with the notion of quadratic variation. We compute the pathwise change of variable formula utilizing the functional Itô calculus and the quadratic variation notion. We study the martingale representation for the case of weak derivatives, we allow the vertical operator, rX, to operate on continuous functionals on the space of square-integrable Ft-martingales with zero initial value. We approximate the hedging strategy, H, for the case of path-dependent functionals, with Lipschitz continuous coefficients. We study some hedging strategies on the class of discounted market models satisfying the quadratic variation and the non-degeneracy properties. In the classical case of the Black-Scholes, Greeks are an important part of risk-management so we compute Greeks of the price given by path-dependent functionals. Lastly we show that they relate to the classical case in the form of examples. en_US
dc.description.sponsorship NRF and AIMS-SA en_US
dc.format.extent v, 59 leaves en_US
dc.language.iso en en_US
dc.relation.requires PDF en_US
dc.subject Stochastic calculus en_US
dc.subject Functional calculus en_US
dc.subject Horizontal derivative en_US
dc.subject Vertical derivative en_US
dc.subject Martingale representation en_US
dc.subject Euler approximation en_US
dc.subject Path-dependence en_US
dc.subject Greeks en_US
dc.subject.lcsh Functional analysis en_US
dc.subject.lcsh Malliavin calculus en_US
dc.subject.lcsh Finance -- Mathematical models en_US
dc.title Pathwise functional lto calculus and its applications to the mathematical finance en_US
dc.type Thesis en_US

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