One-step tree price of a call struck at 5

Question

A stock now at $10$ will, after one period, either double or halve. Interest rates are zero. Using a one-period binomial model, find today's price of a European call with strike $5$ .

Short answer

$5$ . The risk-neutral up probability solves $20q+5(1-q)=10$ , so $q=\tfrac13$ . The call pays $15$ after an up move and $0$ after a down move, worth $\tfrac13\cdot15=5$ (here the call is equivalent to the stock less a riskless $5$ ).

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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