Pricing European Calls with Multi-Step Binomial Trees

Summary

The document extends binomial-tree option pricing from a small tree to a finite N-step model. It explains backward propagation from known terminal payoffs and presents risk-neutral valuation as an alternative: calculate the probabilities of ending at each terminal stock price, multiply by the corresponding option payoff, and take the expected value. In its worked example, the stock starts at 100, moves in increments of 2.5 over four steps, and has a strike of 100 with a zero interest rate, producing equal up and down probabilities.

For the general case, terminal prices depend on the count of up moves, while binomial coefficients account for the number of paths reaching each node. The resulting derivative value is a probability-weighted sum of terminal payoffs. The model assumes a finite tree with exactly two children per node and uses simplified fixed moves and zero rates in the example; it does not address calibrating the tree to market dynamics or early exercise.

Key ideas

• A multi-step binomial tree can be valued by propagating option values backward from terminal payoffs.
• Under risk-neutral pricing, the derivative value is the expected terminal payoff using risk-neutral probabilities.
• Binomial coefficients count the distinct paths that reach a terminal node with a given number of up moves.
• The worked example assumes zero interest rates, making the up and down probabilities equal.
• The model described uses a finite tree with two possible successor states at each node.

Tags

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.