Why Stochastic Volatility Produces Nonflat Black–Scholes Implied Volatility

Summary

The note explains why option prices generated by a stochastic volatility model can imply a volatility smile when those prices are converted using the Black–Scholes formula. Under Black–Scholes pricing with one fixed volatility, the prices of calls across strikes and maturities return that same volatility when inverted, yielding a flat implied volatility surface. The key distinction is that implied volatility is defined relative to a pricing model: prices from a different model generally do not invert to one constant Black–Scholes volatility.

The discussion is a conceptual clarification rather than a derivation or empirical study. It does not specify a stochastic volatility model, show how its parameters are calibrated, or provide numerical evidence. It also does not establish that every stochastic volatility model produces a particular smile shape; the result depends on the model and option characteristics. The closing point is that inverting prices with the generating model itself would recover its own parameters, which is different from reporting Black–Scholes implied volatility.

Key ideas

• Black–Scholes prices generated with a single fixed volatility imply a flat surface when inverted in the same model.
• Implied volatility depends on the pricing model used for the inversion.
• Stochastic volatility model prices can map to varying Black–Scholes implied volatilities across options.
• The note gives a conceptual explanation but no model-specific smile shape or empirical evidence.

Tags

# how does stochastic volatility models generate smiles?


# how does stochastic volatility models generate smiles?












When calibrating call price with the BS-model, we achieve some parameters and especielly we achieve $\sigma^*$. Now, lets say I will price call options using these parameters. Then we achieve, lets say $C_1^{BS},...,C_n^{BS}$. Now, the claim is that if I want to calculate the implied volatility surface of $C_1^{BS},...,C_n^{BS}$, then I get a flat surface since $\sigma_{IV}(C_i^{BS};...) = \sigma^*$ for all $i$.

But when coming to stochastic volatility models, even though the volatility is now stochastic, when we calibrate parameters, the calibration is just an deterministic optimization problem and $\sigma_t$ is still constant? How to capture smiles then?

## Answer by LocalMartingale (score 3)

https://quant.stackexchange.com/a/50545

Nevermind, i'm just confusing myself. Now I understand what I misunderstood. The implied volatility surface of a prices of calls generated by a stochastic volatility model will not be constant since the implied volatility is found using the Black-Scholes model. The Black-Scholes model and a stochastic volatility model of course disagree on prices, and hence fixing prices generated by a stochastic volatility model into the Black-Scholes formula to find implied volatilities gives different volatilities for each call.

The implied volatility surface of a prices of calls generated by a stochastic volatility model will only be constant if I use the model itself to find the implied volatilities, obviously.

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