An elementary approach to delta and gamma hedging, and the Black--Scholes PDE

Abstract

We show how the robustness of gamma hedging can be understood just using the concepts of $p^{th}$ variation along a partition sequence and Taylor's theorem. There is no need to define an integral or proving a version of Itô's lemma. The same approach allows classical results on delta-hedging to be proved without defining an integral and without the need to define the concept of self-financing in continuous time. We show that the approach can also be applied to barrier options and Asian options. Finally, we extend the framework to path-dependent derivatives. We introduce a Gubinelli-type control along a sequence of partitions, and use it to define a pathwise notion of arbitrage. Finally, we show that in any arbitrage-free pricing model the greeks must satisfy a linear relation. If one additionally assumes time and scaling symmetries, this yields the Black--Scholes PDE.