Risk-Neutral Approach

The risk-neutral approach assumes a risk-neutral probability, often denoted as p, to value options. Under this approach, the expected return on the underlying asset is equal to the risk-free rate, making the discounted expected payoffs of the option equivalent to its present value.

European Call and Put Options: In the binomial model, the risk-neutral probabilities are used to calculate the expected payoffs of the option at each node of the option price tree. These expected payoffs are then discounted to the present value using the risk-free rate. By working backward through the tree, the option’s value can be determined.
American Call and Put Options: The risk-neutral approach is also applied to American options within the binomial model. At each node, the option’s value is compared with the expected payoff from early exercise. If the early exercise payoff is higher, it may be optimal to exercise the option at that node. The option’s value is then determined by taking the maximum of the expected payoff and the early exercise value.