I am trying to implement a simple delta-hedging strategy. The idea is that I want to verify that the covered position "1 option long + delta stocks short" is actually evolving as $e^{rt}$, the bond, as I see from the Black-Scholes model.

I am currently considering a stock starting from $S_0=100$, with $\mu=0.1$ and $\sigma=0.2$, and a European call with $K=100$ and $r=0.05$.

I actually see something much more random, even if the portfolio evolves in a large number of time steps, say $10^5$, nowhere close to an exponential, and that suddenly reaches 0 at the option expiration. Is there anything wrong?


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