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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?

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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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  • $\begingroup$ 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? $\endgroup$
    – Claudio Moneo
    Jul 14, 2020 at 11:35
  • $\begingroup$ And another question: Is this the general method of finding the replicating strategy in the BS-model? Setting up two SDEs and comparing coefficients? $\endgroup$
    – Claudio Moneo
    Jul 14, 2020 at 11:36
  • $\begingroup$ What is important to realize is that no choice of the weights will do the job (apart from the delta hedge). Now that you see that the portfolio may not match the option tomorrow, you need to 'rebalance' it to get it to match the option. However, since this rebalancing is self financing, it cannot change the value of your portfolio, and thus you cannot match the option price in a self financing manner. $\endgroup$
    – Arshdeep
    Jul 14, 2020 at 11:39
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    $\begingroup$ Yes, finding the replication strategy involves getting rid of the brownian motion (risk) in your portfolio, and then invoking no arbitrage by saying that the deterministic part of the portfolio must grow at the risk free rate. $\endgroup$
    – Arshdeep
    Jul 14, 2020 at 11:42
  • $\begingroup$ Thanks, I think I got it $\endgroup$
    – Claudio Moneo
    Jul 14, 2020 at 11:47

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