Pricing an european option using a $n$-step binomial model is done through the following formula:
$$\pi_i (S_i) = \frac{1}{e^{r \Delta t}} \left( q \pi_{i+1} (S_i \cdot u) + (1 - q) \pi_{i+1}(S_i \cdot d) \right) \quad \text{for i = 0, 1, ..., n - 1}$$
where $\pi_i$ is the price of option and $S_i$ is the price of underlying asset at time step $i$. $u$ is the factor which price rises and $d$ the factor which the price falls. $r$ is the risk-free rate and $\Delta t$ the time (in years) per time step. Lastly, $q$ is the martingale measure.
For last time step (step $n$ in time $T$), the option can be exercised and the value of the option is therefore:
$$\pi_n (S_T) = \phi(S_T)$$ where $\phi$ is the payoff function, positive if the option is in-the-money and 0 otherwise. In a binomial model, the price in next time step can either go up (with a factor $u$) or down (with a factor $d$).
Black-Scholes is a well-known formula for pricing of options, and descibes the relation between the input parameters (current stock price, strike price, volatiility, risk-free rate and time to expiration). The plot to the left visualises the relation between underlying stock price, time to expiration and the price of an European call option.
European options are options that gives the buyer right, but not the obligation to buy (call option) or sell (put option) an underlying asset at a specific price (strike price) on a specific date (expiration date).
European options are options that gives the buyer right, but not the obligation to buy (call option) or sell (put option) an underlying asset at a specific price (strike price) at any point in time before a specific date (expiration date).
