We need mathematical models to capture both the future evolution of the underlying and the probabilistic nature of the contingent cash flows we encounter in financial derivatives.
Regarding the contingent cash flows, these can be represented in terms of the payoff function H(S(T)) for the specific derivative we are considering. Because S(T) is a stochastic variable, the value of H(S(T)) ought to be computed as an expectation E[H(S(T))]. And in order to compute this expectation, we need techniques that allow us to predict or simulate the behavior of the underlying S(T) into the future, so as to be able to compute the value of ST and finally be able to compute the mean value of the payoff E[H(S(T))].
Regarding the behavior of the underlying, typically, this is formalized using Stochastic Differential Equations (SDEs), such as Geometric Brownian Motion (GBM), as follows:
Equation 2
The previous equation fundamentally says that the change in a stock price (dS), can be understood as the sum of two effects—a deterministic effect (first term on the right-hand side) and a stochastic term (second term on the right-hand side). The parameter 
is called the drift,
and the parameter 
is called the volatility.
S is the stock price, dt is a small time interval, and dW is an increment in the Wiener process.
This model is the most common model to describe the behavior of stocks, commodities, and foreign exchange. Other models exist, such as jump, local volatility, and stochastic volatility models that enhance the description of the dynamics of the underlying.
Regarding the numerical methods, these correspond to ways in which the formal expression described in the mathematical model (usually in continuous time) is transformed into an approximate representation that can be used for calculation (usually in discrete time). This means that the SDE that describes the evolution of the price of some stock index into the future, such as the FTSE 100, is changed to describe the evolution at discrete intervals. An approximate representation of an SDE can be calculated using the Euler approximation as follows:
Equation 3
The preceding equation needs to be solved in an iterative way for each time interval between now and the maturity of the contract. If these time intervals are days and the contract has a maturity of 30 days from now, then we compute tomorrow's price in terms of todays. Then we compute the day after tomorrow as a function of tomorrow's price and so on. In order to price the derivative, we require to compute the expected payoff E[H(ST)] at maturity and then discount it to the present. In this way, we would be able to compute what should be the fair premium 
associated with a European option contract with the help of the following equation:
Equation 4