Put butterfly price from call quotes

Question

A stock is at $15$ . European calls with a common expiry are quoted at: strike $10$ for $8.1$ , strike $15$ for $4.2$ , strike $20$ for $1.4$ . What is the time- $0$ value of a $10/15/20$ butterfly built with puts?

Short answer

$1.1$ . A put butterfly and a call butterfly on the same strikes have identical payoffs (by put–call parity the stock and bond pieces cancel), so price it with calls: buy both wings and sell two bodies for $1.4+8.1-2\times4.2=1.1$ .

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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