Black–Scholes Implied Volatility as an Interpolation Tool
Summary
The document explains why implied volatility can still be useful even though it is calculated from an option’s market price. It presents Black–Scholes as a way to transform observed prices into volatility values, which can then be used to estimate prices for unquoted strikes or dates. For example, a volatility smile from traded options may be interpolated or extrapolated to estimate a less liquid option’s value. When market conditions change, a prior volatility estimate can also be combined with updated inputs such as the underlying price, dividends, discounting, and time to maturity.
The discussion emphasizes that Black–Scholes does not describe market option prices perfectly: different strikes commonly imply different volatilities, contrary to the model’s constant-volatility assumption. It therefore frames the model as a practical representation and estimation tool, rather than a complete explanation of prices. The post offers intuition and examples, but no calibrated volatility surface, pricing calculation, or guidance on how to extrapolate safely; estimates beyond observed data remain uncertain.
Key ideas
- Implied volatility is calculated from an observed option price and can be used to represent that price in volatility space.
- Interpolating volatility across strikes may provide better estimates than interpolating option prices directly.
- A volatility smile can be extrapolated to estimate prices for strikes without current market quotes, though those estimates are uncertain.
- Black–Scholes can help update an option estimate when inputs such as the underlying price or time to maturity change.
- Different implied volatilities across strikes show that actual option markets violate the model’s constant-volatility assumption.
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# BS model without volatility # BS model without volatility Maybe it is a naive question, I simply can't understand how the industry is using the BS model to price options, as the option pricing formula requires implied volatility as an input, which itself is derived from the option market prices. I just don't get how it all ties together given that one of the inputs is dependent on the output. EDIT Example: Assume it is Sunday, and public trading will open on Monday on a brand new equity share priced at `$s` of a new company Amazing Inc. (no dividends, 0% interest rate) and I want to issue a new call option, with strike `k`on it, expiring in a year, how can I price it in dollars without having any volatility figures at hand?? ## Answer by will (score 1, accepted) https://quant.stackexchange.com/a/54189 It's all about transposing prices into some space that changes more slowly, such that data you can garner from prices provided by someone else at some other point in time can be used to estimate value at some other point in time. Its effectively an interpolation and extrapolation tool. Say you have option prices at strikes of 10, 20, 30, 40, etc. And you want the price for a 35 strike option. You could interpolate in price space, or you could transpose the option prices into vols and then interpolate in vol space. The latter works better. Even more obvious, if you need to extrapolate to other prices, then you can take the volatility "smile" from the strikes you have, and attempt to extrapolate this, it will give you a potentially better answer. Likewise, if you have the option prices from one day, and then you need to calculate them the next day, you can adjust the underlying stock price, expected dividend yield, discounting, and time to maturity while keeping the volatility the same as your previous data point, and it will give you an idea of the option price. Black Scholes is a model that everybody knows does not work. This is evident by the fact that every option strike has a different implied volatility (despite one of the underlying assumptions of the model being that volatility is constant). What it is though is a very useful function for transforming option prices into another space which allows you to estimate the value of that option given changes to other underlying properties.
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