Pricing a plus-or-minus-one digital payoff

Question

A stock trades at $S_0=25$ . A contract pays $+1$ at expiry if $S_T\ge 24$ and $-1$ if $S_T<24$ . Digital calls paying $1$ when $S_T\ge K$ are available at these prices: strike $25$ at $0.6$ , strike $24$ at $0.73$ and strike $23$ at $0.88$ . A bond paying $1$ at expiry costs $0.9$ . What is the contract worth today?

Short answer

$0.56$ . The payoff equals two digital calls struck at $24$ minus one bond: it is $2-1=1$ above the strike and $0-1=-1$ below. Its price is $2(0.73)-0.9=0.56$ .

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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