how does stochastic volatility models generate smiles?

Question

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 1

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.