Estimating Option Price Changes from Implied Volatility Changes

Summary

The document frames a problem in option valuation: given an implied volatility surface indexed by moneyness and maturity, the observer has changes in implied volatility over time but not the volatility levels. The goal is to estimate the corresponding change in an option’s Black–Scholes price. It notes that vega multiplied by the volatility change is a natural first-order approximation, while also identifying the obstacle: evaluating vega requires a volatility level.

The text is only a question and gives no proposed estimator, derivation, example, or empirical evidence. In particular, it does not establish whether the volatility level can be reconstructed from the observed changes, or what assumptions or initial data would be needed. It also leaves unstated how underlying price, rates, dividends, and other inputs evolve between periods. The topic is useful as a prompt about the distinction between price sensitivity and price change, but the document alone does not supply a complete pricing method.

Key ideas

• A volatility surface can be indexed by option moneyness and maturity.
• Vega times the implied volatility change gives a first-order price change approximation.
• Calculating vega requires an implied volatility level, which the questioner does not observe.
• The question provides no method or evidence for recovering missing volatility levels.
• Changes in other Black–Scholes inputs are not addressed.

Tags

# Change in Option Price given Change in Implied Volatliity, Moneyness, and Maturity


# Change in Option Price given Change in Implied Volatliity, Moneyness, and Maturity












I have an implied volatility surface parametrized into moneyless-maturity coordinates. At each period of time, I only have access to an option's moneyness (K/S), maturity, and change in implied volatility (defined as $\Delta I_t(\xi,\tau)=I_{t+1}(\xi,\tau)-I_t(\xi,\tau)$), where $\xi$ is the moneyness, $\tau$ is maturity. I want to find the change in an option's price (from period t going to t+1) via the Black-scholes formula. I know that the Vega of the formula times the change in implied vol gives the change in price, but to calculate Vega you need the actual volatility, in this setting we only have access to the changes in implied volatility. How can I find the change in price under this setting?

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