Why Delta Hedging Produces a Replicating Option Portfolio

Summary

The document asks why a Black–Scholes option replication uses delta hedging rather than simply holding enough risky asset to match the option’s current value, with the bond position chosen to make the portfolio self-financing. The key distinction is between matching value at one instant and matching the option’s changing value as the underlying asset moves. A portfolio that equals the option value now need not continue to do so at a later time.

The accepted response suggests comparing the portfolio’s change in value with the option’s change derived using Itô’s lemma. In the Black–Scholes framework, delta specifies the risky-asset holding needed to match the option’s local sensitivity, while the bond position accounts for the remaining value. The excerpt gives no full derivation or numerical example, so it offers a diagnostic rather than a complete proof. The replication claim also relies on the idealized assumptions of the stated model and continuous rebalancing.

Key ideas

• Matching a portfolio’s value to an option at one moment does not guarantee future replication.
• The proposed value-over-price asset holding does not ensure that portfolio changes match option changes.
• Itô’s lemma provides a way to compare the option’s value dynamics with those of a candidate portfolio.
• Delta determines the risky-asset holding used to match the option’s local sensitivity in the Black–Scholes model.
• The excerpt gives a conceptual test but not a full derivation.

Tags

# Why do replicating strategies delta hedge?


# Why do replicating strategies delta hedge?












We have a simple BS-market of one risky asset $S_{t}$, a bond $B_{t}$ and a digital option $X$ on the risky asset with value process $V(t,S_{t})$. I was able to derive $V(t,S_{t})$ using risk-neutral valuation. Now, I am supposed to set up a replicating strategy for this option, i.e. find a self-financing trading strategy $\phi(t)=(\phi(t)^{B},\phi(t)^{S})$ such that for all $t$: $$\phi(t)^{B} B_{t} + \phi(t)^{S} S_{t} = V(t,S_{t})$$

The book says that I should delta hedge the option. Now I am wondering why I couldn´t just construct the replicating strategy as $\phi(t)^{S}=\frac{V(t,S_{t})}{S_{t}}$ and choose $\phi(t)^{B}$ such that the strategy is self-financing? Why is delta-hedging necessary?

## Answer by Arshdeep (score 1, accepted)

https://quant.stackexchange.com/a/55655

You have not achieved replication here. The idea is that, tomorrow, I must end up with portfolio value equal to the value of the option. That is not guaranteed with this setup.

To see that, try to write $dV(t,S(t))$ from Ito's lemma and from your equation, see if they match.

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