Before revision: Rough path analysis applies successfully to multidimensional stochastic processes. Ito diffusion processes on the other hand have provided models for stock market prices since F. Black and M. Scholes, Harrison and co-authors. In the following note, we intend to bring some controlled rough path considerations to the study of multidimensional Black-Scholes models with position-dependent volatilities. The Ito formula for reduced rough paths provides the syntactical analogy to bridge classical Mathematical Financeto the (Davie's type)RDE-based one. Once such connection is established, rough integration bounds are employed to estimate the errors arising from time-discretisation of the theoretical continuous-time trading strategies. We find that the portfolio trajectory resulting from the hedging strategy, when analysed pathwise , is sensitive to Greeks beyond the leading first-order Delta, and is Gamma-sensitive in particular.
After revision: We describe the pricing and hedging practices refraining from the use of probability. We encode volatility in an enhancement of the price trajectory and we give pathwise presentations of the fundamental equations of Mathematical Finance. In particular this allows us to assess model misspecification, generalising the so-called fundamental theorem of derivative trading (see Ellersgaard et al. 2017). Our pathwise integrals and equations exhibit the role of Greeks beyond the leading first-order Delta, and is Gamma-sensitive in particular.