# Delta hedge error black-scholes by Mark Davis

I'm currently reading a paper by Mark Davis in which he talks about a delta hedging error in the Black-Scholes formula. The delta hedging error is given expressed as $$Z_t$$ with the formula: $$Z_t = \int_{0}^{T} e^{r(T-s)} \frac{1}{2} S_t^2 \Gamma_t(\hat{\sigma}- \beta_t^2)dt$$ Where $$\beta$$ is the realized volatility. My question is not wether is is true, as I understand the hedging error quite well, especially after reading Interpertation of delta hedge error in Black Scholes. However, in the linked article the answer express a replicated portfolio given by: $$\Pi_t = -V_t + \Delta_tS_t + \frac {(V_t - \Delta_t)}{B_t}B_t$$ Where the latter is the residual cash position / money market account. However, I can't seem to derive the money market account from Davis portfolio construction given by: $$dX_t = \frac{\partial C}{\partial s}dS_t + (X_t- \frac{\partial C}{\partial s} S_t) r dt$$ Where $$X_0=C(0,S_0)$$. Can anyone explain if Davis just ignore the money market account or is it an implicit derivation of X, which reaveals this?

On page 119 in Björk (3rd edition) we have the replicating portfolio (equations 8.20 and 8.21): Hold $$\frac{\partial C}{\partial s}$$ of the stock and $$\frac{X_{t}-S_{t}\frac{\partial C}{\partial s}}{B_{t}}$$ in the bank-account. The dynamics of this portfolio is given by $$dX_{t}=\frac{\partial C}{\partial s}dS_{t}+\frac{X_{t}-S_{t}\frac{\partial C}{\partial s}}{B_{t}}dB_t=\frac{\partial C}{\partial s}dS_{t}+(X_{t}-S_{t}\frac{\partial C}{\partial s})rdt$$ as $$dB_{t}=rB_tdt$$