How far must spot move for a straddle to pay off

Question

The underlying trades at $25, and you buy a straddle (one call and one put) both struck at $25 for a total of $5. How large a move in the underlying do you need by expiry to break even?

Short answer

At least $5 in either direction, i.e. a $20\%$ move. At expiry the straddle pays $|S_T - 25|$ , so it recovers the $5 premium only if $S_T \le 20$ or $S_T \ge 30$ (ignoring financing of the premium).

Adapted from Crack, Heard on the Street, Q2.24, p.28; 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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