# 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?

## 1 Answer

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 $$dB_{t}=rB_tdt$$