# 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?

To see that, try to write $$dV(t,S(t))$$ from Ito's lemma and from your equation, see if they match.
• I used Ito and now I understand. Also, my choice of $\phi$ doesn´t work because we have $\phi(t)^{B} B_{t} + \phi(t)^{S} S_{t} = \phi(t)^{B} B_{t} + V(t,S_{t})$ which is only equal to $V(t,S_{t})$ if $\phi(t)^{B}=0$ - contradicting the property of being self-financing... is that correct? Jul 14, 2020 at 11:35