Signature and the Functional Taylor Expansion

This chapter links the path signature to the functional Itô calculus (Dupire, Functional Itô calculus. SSRN (2009). Republished in Quantitative Finance 19(5), 721–729, 2019). The junction is the functional Taylor expansion (FTE), discussed in Sect. 5, which is a powerful tool to approximate functionals after an observed path. In particular, the FTE decomposes the functional from future scenarios. In risk analysis, the FTE leads to new Greeks, one of particular importance being the Lie bracket between the space and time functional derivatives, which we call Libra. Once paired with the Lévy area, the Libra can be used to speed up the computation of risk measures such as Value at Risk and Expected Shortfall as explained in Sect. 6.1. In Sect. 6.3, we explain how the FTE can quantify the hedging error of replicating portfolios in a robust manner. In the context of financial derivatives, we demonstrate in Sect. 6.4 how the FTE generates approximations of exotic payoffs. The relevance of the FTE for cubature methods is finally outlined in Sect. 6.5.