Why Option Models Must Be Calibrated to Market Prices

# Why do mathematical model prices still differ from actual market option prices


# Why do mathematical model prices still differ from actual market option prices












As a student of quantitative finance, I’ve been exploring the concept of implied volatility and came across a fundamental question regarding the persistent difference between actual market option prices and the prices predicted by mathematical models.

I’m aware that the classical Black-Scholes model is built on several simplifying assumptions, such as constant volatility, no jumps in asset prices, frictionless markets, and continuous trading, which understandably contribute to discrepancies between model and market prices.

However, over the years, many advanced extensions of the Black-Scholes framework have been developed to address these limitations, including:

- Generalized Black-Scholes Equations,

- Stochastic volatility models (e.g., Heston, SABR),

- Jump-diffusion models (e.g., Merton, Kou),

- Stochastic interest rate models,

- Fractional Models,

- Models incorporating transaction costs, liquidity effects, and other real-world frictions.

Considering all these advancements, my question is:

Why do we still observe a significant difference between model-based option prices and actual traded prices, even when these more sophisticated models are used?

Moreover, I understand that the market-observed option prices and Greeks (such as delta, gamma, vega, theta, etc.) as the data generated by exchanges, are typically derived using mathematical models (often based on Black-Scholes or its variants).

So I’m also curious:

Is the discrepancy solely due to restrictions on disclosing advanced mathematical models used in generating this data (by the authority), or are there other contributing factors?

Any insights on how practitioners reconcile model outputs with observed prices, or whether this discrepancy is simply accepted as part of model risk, would be greatly appreciated.

## Answer by Quantuple (score 6, accepted)

https://quant.stackexchange.com/a/83827

There are fundamental misconceptions in your question, in my opinion. Market prices are determined by supply and demand, a process known as price discovery, not by a model. That being said:

- Marking the model to the market: If the model you use for P&L reporting purposes does not align with observed market prices, you are simply not compliant from an accounting perspective (e.g. fair value of trading books under IFRS9) with important consequences for your activity. For vanilla options, the Black-Scholes model is more than sufficient to achieve your goal. For exotic options (assuming there is no market for those), if your model does not recover observed prices, it will a fortiori misprice the hedge instruments involved in your dynamic replication strategy. In that case, you will then misprice the exotics (i.e. either be too aggressive and loading your books with significant risk, or too conservative hence noncompetitive).

- Using a model to trade/hedge: There is no such thing as an actual 'traded' price. There are effective bid and offered prices for a given timestamp... and a whole order book behind. There cannot be a single model reflecting a universal truth as market participants have different views and also face different financial realities (e.g. funding costs, fiscal/tax laws etc.). This variability ultimately translates to supply and demand mechanisms. While it is true that any model could be made more elaborate to account for, say, discrete re-hedging, trading costs and market microstructure (liquidity, market impact, and the order book), it does not alter that fundamental truth. To conclude, let's have a thought experiment. Assume you managed to identify the 'best' model there is. You use this model to pin down the 'true' price you should bid for a specific instrument. When you go to the market, you find that there is no market participant willing to sell at that price: was your model useful? Rebonato has an excellent paper on this topic is freely available here.

So to answer your question: no it's not at all a question of identifying more sophisticated models. Sophisticated models are mostly useful to model quantities we cannot directly imply from the market (e.g. risk-neutral pricing of hedgeable risks vs. absolute pricing of non-observable risks). If the goal is to fit exchange-listed instruments' prices, a simple model à la Black-Scholes whose parameters are implied from the market is sufficient, by design.