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.
