Fixed-strike implied vol after the spot rallies

Question

Outside Black-Scholes, suppose implied volatility obeys a sticky-strike rule. The stock is at $12$ and the $13$ -strike option has implied volatility $0.3$ . One period later the stock is at $15$ . What is now the implied volatility of the $13$ -strike option?

Short answer

$0.3$ . Under sticky strike each strike keeps its own implied volatility regardless of where spot moves; the smile need not be flat, but it is a function of strike only. The strike has not changed, so neither has its vol.

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Lessons behind this question

Reading the Implied Vol Surface — Moneyness, Term Structure, and Quoting Conventions
Option Mechanics: Calls, Puts, Moneyness, Intrinsic Value, and Exchange vs. OTC Markets
Option Trading Strategies: Spreads, Combinations, and Payoff Engineering

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