# 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? – Canardini Jan 4 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? – LocalMartingale Jan 4 at 17:39
• I read your answer , exactly you got it right. The smile is a model dependent. – Canardini Jan 4 at 21:54

## 1 Answer

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.