Why Delta Hedging Requires Self-Financing Trades
Summary
The document addresses an apparent inconsistency in Black–Scholes delta hedging: the hedge ratio is chosen by differentiating portfolio value with respect to the stock price, yet that ratio itself changes with the stock price. The resolution is that the hedge position is selected using information available before the next price movement, so it is held fixed during that instantaneous change. A trader cannot adjust the position in response to a movement that has not yet occurred.
The answer frames replication as a self-financing strategy: changes in portfolio value come from gains or losses on the stock and riskless asset, without outside cash flows. Applying this condition and matching the portfolio’s change to the option’s change leads to the pricing PDE. This is a conceptual explanation rather than a numerical demonstration, and it assumes the Black–Scholes setting and continuous rebalancing framework.
Key ideas
- A hedge ratio is chosen using information available at the time of the trade.
- The hedge position is held fixed while calculating the effect of an instantaneous stock price move.
- A self-financing portfolio changes value through its asset holdings rather than external cash injections or withdrawals.
- Matching the self-financing portfolio dynamics to the option dynamics produces the pricing PDE.
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# A simple question on Delta hedging
# A simple question on Delta hedging
In the Black and Scholes model, when it is needed to immunize the portfolio from variations in the stock the argument given is the following. If $\alpha_t$ is the amount of invested in the stock, $\beta_t$ the amount in the bond, we construct a portfolio whose value is
$$ V_t = -C_t+\alpha_t\,S_t+\beta_t\,B_t, $$
where $C_t=f\left(t,S_t\right)$ is the price of a Call option and where $S_t$ and $B_t$ are, respectively, the price of the stock and the price of the bond at time $t$. So we are short selling one unit of the call and buying the portfolio whose $\alpha_t\,S_t+\beta_t\,B_t$. Now one chooses $x$ such that
$$ \frac{\partial V_t}{\partial S_t} = 0\Leftrightarrow -\frac{\partial C_t}{\partial S_t}+\alpha_t=0\Leftrightarrow \alpha_t = \frac{\partial C_t}{\partial S_t}\equiv \Delta_t. $$
What puzzles me is the fact that when deriving the value of the portfolio we assume that $\alpha_t$ does not depend from $S_t$ whereas the final solution does depend. So in principle one should do this computation (assuming that $\beta_t$ does not depend on $S_t$)
$$ \frac{\partial V_t}{\partial S_t} = -\frac{\partial C_t}{\partial S_t}+\alpha_t+\frac{\partial \alpha_t}{\partial S_t}\,S_t = 0 $$
whose solution is of course different from $\alpha_t=\Delta_t$. Where am I wrong?
## Answer by user9403 (score 2, accepted)
https://quant.stackexchange.com/a/24616
$\alpha_t$ must be chosen prior to stock price movements so the expression $S_t d\alpha $ does not make sense: we can't take a position in a stock based off information that we don't know yet.
The missing step is that the replicating portfolio is required to be self financing: that is, for all $t$ the following equations hold: $$X_t=\Delta S_t+\Gamma M_t$$ $$dX=\Delta dS+\Gamma dM$$ Where $X$ is the portfolio value and $S$ and $M$ are the stock and riskless asset. The first equation states that no external asset is injected or removed at any time. The second states that we cannot take a position in an asset based off information that is not available at time $t$ (since naively applying Ito's lemma to $X_t$ would yield a $d\Delta$ and $d\Gamma$ term).
Combining the two equations yields $$dX=\Delta dS +r(X_t-\Delta S_t)dt$$
Matching this equation with $df(S, t)$ yields the correct PDE.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.