Questions: Implied Volatility Extraction and Interpretation

Question 1

An option trader observes that a deep out-of-the-money put on a stock index has higher implied volatility than an at-the-money put on the same index with the same expiration. What does this 'volatility skew' most directly reveal?

• A. The deep OTM put is mispriced and represents an arbitrage opportunity
• B. The market assigns higher probability to large downward moves than Black-Scholes assumes, reflecting crash insurance demand
• C. Historical volatility has been higher when the market is down, proving past asymmetry
• D. The deep OTM put has more time value, which always converts to higher implied vol

Answer: (B)

The volatility skew reflects market beliefs about the tails of the return distribution — specifically, that large drops are more likely or more feared than a normal distribution would imply. Investors pay a premium for OTM puts as crash insurance, driving up their prices and thus their implied volatilities. The skew is not a pricing error but a rational premium for downside risk. It is forward-looking, not simply a reflection of past returns. Time value and implied vol are related but distinct concepts.

Question 2

A quant says 'the stock has realized 20% volatility over the past year, so its options are overpriced at 25% implied vol.' What is the conceptual flaw in this reasoning?

• A. Historical volatility should always be annualized before comparing to implied volatility
• B. Implied volatility is forward-looking and includes a risk premium for uncertainty, so it rationally exceeds historical volatility
• C. Options are always fairly priced in efficient markets, so 25% implied vol must be correct
• D. Realized volatility is calculated incorrectly and should use log returns

Answer: (B)

Historical and implied volatility measure fundamentally different things. Historical volatility looks backward at actual price movements. Implied volatility looks forward — it reflects market participants' expectations about future uncertainty and a risk premium they demand for bearing that uncertainty. In calm markets, options often trade at implied vols above recent realized vol precisely because uncertainty about the future exceeds the past's tidiness. Treating them as interchangeable is the core misconception.

Question 3

Implied volatility is extracted by solving for the volatility input that makes a pricing model's theoretical price match the observed market price.

• T. True
• F. False

Answer: True

This is the defining procedure. All other Black-Scholes inputs (stock price, strike, time to expiration, risk-free rate) are directly observable. Volatility is not, so implied volatility inverts the relationship: given the market price, what sigma makes the model price match? Because Black-Scholes has no closed-form inverse for sigma, this requires numerical methods (typically Newton-Raphson iteration using vega as the derivative). The result is a market-consensus forward-looking volatility estimate.

Question 4

According to Black-Scholes theory, most options on the same underlying asset with the same expiration date should have the same implied volatility.

• T. True
• F. False

Answer: False

Black-Scholes assumes constant volatility across all strikes, so in theory all options on the same underlying at the same expiration should have the same implied vol. But in practice they do not — the volatility smile and skew are empirically pervasive, especially in equity markets where lower strikes carry higher implied vols. This reveals that Black-Scholes' constant-volatility assumption is wrong: real markets price in heavier tails and asymmetric crash risk that a single sigma cannot capture.

Question 5

Why is implied volatility more useful than historical volatility for an options trader pricing a new contract, and what information does the volatility surface convey that a single number cannot?

Model answer: Historical volatility is backward-looking — it summarizes past price movements. An options trader needs a forward estimate of future uncertainty over the option's life. Implied volatility extracts the market's current collective forecast of future uncertainty, incorporating current information and risk preferences. The volatility surface (implied vol across all strikes and maturities) conveys the full shape of market beliefs: the skew reveals how much crash protection costs relative to upside calls, and the term structure shows whether near-term or long-term uncertainty is greater. A single historical vol number cannot capture any of these dimensions.

The VIX is itself an implied volatility measure aggregated across S&P 500 option strikes — it is 'implied' not 'historical' precisely because forward-looking estimates are more useful for pricing and hedging. The surface is the rich multi-dimensional version of this single-number summary.