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I'm trying to replicate, with a binomial model, the price of an option obtained with a PDE. It doesn't really work, so I was wondering, if there are some caveats when doing that.

The PDE model use a continuous interest rate of r and the dynamic is $$ dX_t = (r-c)dX_t + \sigma X_t dW_t, $$ where $c$ is a continuously deducted fee, $\sigma$ the volatility and $W_t$ a standard Brownian motion. The goal is to find $c^*$ for which $$ E\left(e^{-10r}X_{10} - c \int_0^{10}e^{-r s} X_s ds\right)=0. $$

For the binomial model, I use 10 periods with \begin{align} u &=e^{\sigma/\sqrt{N}}=1/d \\ \pi &= \frac{e^{r/N}-d}{u-d} \end{align} and, for an up move, I calculate \begin{align} X_{t+1} &=X_t u e^{-c}, \\ c \int_0^{t}e^{-r s} X_s ds &\approx c \sum_{s=1}^t e^{-rs}X_s \tag{1} \end{align} with $X_0=1$.

With the PDE, $c^*$ is in basis point, while in the binomial model it is in percent. Even for large $N$.
I suspect (1) is not a good approximation of the integral.
What am I missing?

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