I wouldn't characterize the person asking this question as an idiot. It's actually a thoughtful question that demonstrates some understanding of options pricing, even if there's a misunderstanding about the relationship between implied volatility (IV) and option strikes.
Let's break down the concepts and address the misconception:
1. Volatility smile: This is a real phenomenon in options markets where out-of-the-money (OTM) options often have higher implied volatility than at-the-money (ATM) options.
2. Linearity of price: While the underlying asset's price is indeed linear across strikes, option prices are not. Options have non-linear payoffs, which is reflected in their pricing.
3. Implied volatility: IV is not a direct input to option pricing but rather a derived value that makes the theoretical price match the market price. It's not necessarily constant across strikes.
4. Probability of expiring OTM: While OTM options do have a higher probability of expiring worthless, this is already factored into their lower absolute prices.
The misconception here is the assumption that IV should be constant across strikes. In reality, the volatility smile exists for several reasons:
1. Market demand: OTM options are often used for hedging tail risks, which can increase their demand and price.
2. Black-Scholes model limitations: The model assumes constant volatility and log-normal distribution of returns, which doesn't always hold in real markets.
3. Crash fears: After major market crashes, investors often price in higher probabilities of extreme events, especially on the downside.
4. Skewed return distributions: Asset returns often show skewness and kurtosis, which aren't accounted for in basic option pricing models.
The higher IV for OTM options doesn't necessarily mean you're "paying relatively more" for them. The absolute price of OTM options is still lower than ATM or in-the-money (ITM) options. The higher IV reflects the market's assessment of the probability of large moves in the underlying asset.
In conclusion, this question shows curiosity about a complex topic in finance. It's a good starting point for learning more about advanced option pricing concepts and market behavior.
