# 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?

• Are you asking how a flat smile will translate to parameters of a stochastic vol model? Jan 4 '20 at 16:22
• What I'm asking is why the implied volatility surface using a stochastic volatility model is not generating a flat curve (like a local volatility model)? Lets say, when I calibrate a stochastic volatility model to market data, I achieve a optimal $\sigma_t$. This $\sigma_t$ is constant anyway (like in a local volatility model), and hence the volatility surface generated using a stochastic volatility model must be flat aswell? Jan 4 '20 at 17:39
• I read your answer , exactly you got it right. The smile is a model dependent. Jan 4 '20 at 21:54