Two-step tree value of a squared-stock payoff

Question

A stock starts at $5$ and in each of two periods either doubles or halves, with zero interest. A contract pays $S_2^2$ at the end of the second period. Using risk-neutral pricing on the binomial tree, what is it worth today?

Short answer

$56.25$ . The risk-neutral up probability solves $2q+\tfrac12(1-q)=1$ , giving $q=\tfrac13$ . Terminal prices $20,5,1.25$ occur with probabilities $\tfrac19,\tfrac49,\tfrac49$ , so the value is $(400+4\cdot25+4\cdot1.5625)/9=56.25$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Market Model and Risk-Neutral Valuation
Option Mechanics: Calls, Puts, Moneyness, Intrinsic Value, and Exchange vs. OTC Markets
Option Trading Strategies: Spreads, Combinations, and Payoff Engineering

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