# Binomial model and delta hedging

I've got a question about theory which is probably a one line answer. I use to understand it but I'm stuck right now.

In the Binomial model, we define the progression of the price as:

$$S_k = S_{k-1} e^{\alpha X_k}$$

where $$P(X_k = 1) = p$$ and $$P(X_k = -1) = 1-p = q$$

Now, to determine $$\alpha$$ and $$p$$ using the delta hedging argument (rather than risk-neutrality) we define a small time increment $$h$$ and at time $$t_{n-1} = (n-1)h$$ a portfolio $$\Pi$$

$$\Pi_{n-1} = f(S_{n-1}, t_{n-1}) - \Delta_{n-1} S_{n-1}$$

where $$S_t$$ is the asset price and $$f(S_t, t)$$ is the payout. To determine $$\alpha$$ and $$p$$ we choose $$\Delta_{n-1}$$ so that the evolution of $$\Pi_{n-1}$$ is deterministic. My lecture notes define $$\Pi_n$$ at $$t=nh$$ like so

$$\Pi_n = f(S_{n}, t_{n}) - \Delta_{n-1} S_{n}$$

Substitute $$S_k = S_{k-1} e^{\alpha X_k}$$

$$\Pi_n = f(S_{n-1} e^{\alpha X_n}, t_{n}) - \Delta_{n-1} S_{n-1} e^{\alpha X_n}$$

Then we calculate $$\Delta_{n-1}$$ so that $$\Pi_n$$ is deterministic etc.

What I do not understand is why do we have $$\Delta_{n-1}$$ in the formula for $$\Pi_n$$? Shouldn't the formula for $$\Pi_n$$ be:

$$\Pi_n = f(S_{n}, t_{n}) - \Delta_{n} S_{n}$$

Say there are just two periods: Payoff at n, and premium/price at $$n-2$$.
We know the current stock price, say $$S_{n-2}$$, and we know in the next epoch, it will either be: $$S_{n-1}^u=uS_{n-2}$$ (up state), or $$S_{n-1}^d=dS_{n-2}$$ (down state). We need to make the decision at epoch $$n-2$$ as to how many units of the stock to buy or sell to hedge the option, and assume we decided to buy $$\Delta_{n-2}$$ units of stock.
We then wait a bit to find out the true state of nature. Price has gone up: Our stock is worth $$\Delta_{n-2} S_{n-1}^u$$. Price has gone down: Our stock is worth $$\Delta_{n-2} S_{n-1}^d$$. So hoping the hedging has worked, we need to get ready for the next move, so we rebalance the portfolio, which means decide at $$n-1$$ how many units of the stock to hold, $$\Delta_{n-1}$$, to hedge the option position against the next move.
So the one liner could be: We decide at $$n-1$$ how many units of the stock to buy/sell to hedge against the next up/down move of the stock price.
Unlike $$S$$ and $$f$$ which are driven by the market that are out of your control, $$\Delta$$ is the amount of stocks $$S$$ that you have decided to short in the previous time step for this portfolio $$\Pi$$. It is not an intrinsic time dependent quantity. So of course it is staying fixed until you decide to change or not to change it in the next time step.