When terms are combined we find that those involving dz cancel out. This is the Black-Scholes differential equation for call option value. Had we considered the put value P instead of the call value we would have come up with the same equation. The price of the stock one period ahead has a log-normal distribution with mean and volatility q, which are both constant over the life of the option. There exists a risk-free interest rate which is constant over the life of the option.
Individuals can borrow as well as lend at the risk-free interest rate.