Replicating a payoff of the larger of 3S and 15

Question

A stock is at $7$ , a bond paying $1$ at $T$ costs $0.9$ , and European calls expiring at $T$ are priced: strike $5$ at $4.2$ , strike $10$ at $1.4$ , strike $15$ at $0.7$ , strike $20$ at $0.1$ . What is the fair value today of a claim paying $\max(3S_T,15)$ at $T$ ?

Short answer

$26.1$ . The payoff equals $15+3(S_T-5)^+$ , i.e. $15$ bonds plus three $5$ -strike calls, costing $15(0.9)+3(4.2)=26.1$ .

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

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Lessons behind this question

Duration and Convexity: Measuring and Managing Interest Rate Sensitivity
Option Mechanics: Calls, Puts, Moneyness, Intrinsic Value, and Exchange vs. OTC Markets
Option Trading Strategies: Spreads, Combinations, and Payoff Engineering

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