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Deriving the Cash Account in a Black–Scholes Delta Hedge

Article Quant Q&A · Author: Sebastian Strauss Hansen
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Summary

The document asks how the money-market account appears in a Black–Scholes delta-hedging portfolio and its relationship to a hedging-error expression involving gamma and a difference between assumed and realized variance. The answer identifies the cash holding as the portfolio value remaining after financing the stock position: subtract the value of the delta shares from total portfolio value, then divide that residual by the bank-account price to express it in account units.

Applying the bank account’s growth rule, in which its value earns the continuously compounded risk-free rate, gives the portfolio dynamics as the stock contribution plus interest on the residual cash balance. This matches the stated equation for the replicating portfolio. The response clarifies that the money-market account is implicit in the self-financing construction rather than omitted. It is a focused algebraic clarification, not a derivation of the full hedging-error formula or a discussion of transaction costs, discrete rebalancing, or other practical sources of hedge error.

Key ideas

  • A delta hedge holds the option delta in shares and places the remaining portfolio value in the money-market account.
  • The cash holding in account units is the residual portfolio value divided by the account price.
  • When the bank account earns the risk-free rate, the cash position contributes interest to portfolio dynamics.
  • The stated wealth equation already incorporates the money-market account through its residual cash term.

Tags

OptionsDerivatives pricingRisk management
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# Delta hedge error black-scholes by Mark Davis


# 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?

## Answer by mmencke (score 2, accepted)

https://quant.stackexchange.com/a/63189

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 $dB_{t}=rB_tdt$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.