# Black-Scholes-Merton and alternatives as interpolation tools

This is a not very quantitative question, but is nevertheless related to quantitative methods in Finance.

I was reading the following paragraph from Hull's Options, Futures, and other Derivatives:

It can be argued that the Black–Scholes–Merton model is no more than a sophisticated interpolation tool used by traders for ensuring that an option is priced consistently with the market prices of other actively traded options. If traders stopped using Black–Scholes–Merton and switched to another plausible model, then the volatility surface and the shape of the smile would change, but arguably the dollar prices quoted in the market would not change appreciably. Greek letters and therefore hedging strategies do depend on the model used. An unrealistic model is liable to lead to poor hedging. Models have most effect on the pricing of derivatives when similar derivatives do not trade actively in the market.

If I understand correctly what Hull means, the point of his sentence is:

• we have a model for pricing derivatives which is based on the no-arbitrage principle and on the assumption that the underlying asset follows a geometric Brownian motion with constant drift and volatility.
• If we try to apply the model, we discover that we lack one crucial parameter, which we infer from actual market prices (the implied volatility).
• The volatility smile tells us that the constant volatility assumption is not completely justified, but nevertheless we can still use the model and the market data to price new options.

But then, why is Hull writing that if we used another plausible model, the volatility surface would change? If the prices of the options do not change, when we use this data to compute implied volatilities (using the Black-Scholes-Merton model), the volatilities would still be the same, right?

• I think he meant that the volatility surface that uses the other model's pricing to back out an implied volatility could have a different shape than the Black Scholes volatility surface. For example, if we had a 'dumb' new model that always output a price equal to the input volatility, we can make a volatility surface using market prices and that shape would look different than the black scholes volatility surface. – Slade Nov 10 '19 at 0:46
• @Slade Huh, so Hull is probably assuming that the alternative model to Black-Scholes-Merton would still involve an underlying asset with a constant volatility. Thanks. – J. D. Nov 10 '19 at 9:31