Title: Functional Expansions, Signature and Claim Decomposition
Abstract: Functionals are omnipresent in finance, whether it be the payoff of a claim, a hedging strategy, or path-dependent volatility. However, problems involving functionals are often infinite-dimensional and thus challenging from a conceptual and computational perspective. In the first part of the talk, we present a simulation algorithm based on the Karhunen-Loève expansion to efficiently price exotic options. In the second, we compare static expansions (Volterra/Wiener series as well as the novel intrinsic value expansion) and promote the functional Taylor expansion (FTE). The latter combines the Functional Itô Calculus with the signature to quantify the effect in a functional when a “shock" path is concatenated with the source path. The notions of analytic functionals and radius of convergence in the path space are then defined. We finally apply the FTE to decompose exotic claims. This is joint work with Bruno Dupire (Bloomberg LP).