Pricing a Call Option with a Two-Step Binomial Tree

Summary

The document extends one-step binomial option pricing to a two-step stock tree, where the initial price of 100 can move through intermediate values of 105 or 95 and finish at 110, 100, or 90. It works through a call with strike 100, determining values at intermediate nodes by replicating the option payoff with a stock position and applying the no-arbitrage principle. Backward induction gives a value of 5 at the up node and zero at the down node, then a current option price of 2.5.

The article reaches the same result through risk-neutral valuation. It chooses probabilities that make expected stock values match the node prices, then discounts expected option payoffs across the tree. The example illustrates agreement between replication and risk-neutral pricing in this setting. It is a deliberately small, frictionless model: the excerpt does not discuss interest rates, transaction costs, early exercise, or calibration to market data, and its specific probabilities and price apply only to the stated tree.

Key ideas

• A multi-step binomial tree can be priced by working backward from terminal option payoffs.
• At each node, a stock position can replicate the option payoff across the next possible states.
• No-arbitrage requires equivalent hedged portfolios to have equal value at a node.
• Risk-neutral probabilities provide an alternative calculation that agrees with the hedging method in the example.
• The resulting option price depends on the tree’s assumed stock moves and payoffs.

Tags

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