Note 04: My understanding to Volatility Smile and it's Models

It is well known for almost everyone in finance that volatility surface is not flat.
In my understanding, volatility smile is a result of market multi-interaction: Hedging(fear of risk) , Stochastic Volatility/Jump Diffusion , Different opinions on future trend and so on. But the main reason is that the emergence of the new jobs:Quant. The reason is simple. Before 1987, the surface is flat. After 1987, with the increase of the usage of complex and sophisticated financial tools, the surface become curved. In this sense, both Stochastic Volatility/Jump Diffusion and Different opinions on future trend are not the critical idea for this effect since even before 1987, jump occurs frequently and volatility is still not a constant (eg, volatility cluster).

In this sense, the evolution of volatility surface from flat plane to curved one mainly comes from the hedging side boosted by quants. If someone(some companies) began to use the sophisticated model (eg, Garch model,local volatility models, SV/JD models) instead of BS model to model the options, the volatility surface appear naturally. And then more and more people noticed this effect and then they began to use new tools and then .... It is a feedback loop like electronic amplification circuit.

Within this philosophy, how can we model the volatility surface? The answer is that we need adaptive model can survive as the volatility surface itself is emergent effect due to market feedback of human knowledge. But that can not solve anything. Our model does not have intelligence, they can not evolve dynamically ...

So here, let me just analyze those basic models one by one , and thinking is still going on...

1) local volatility model
The most disadvantage of this model is that the asymptotic surface when maturity time period is large is flat. The reason is simple, when the time period is large, one can go almost all kind of path of different price of stocks, then there is no advantage of usage of local volatility. According to ito33.com, we can not local "everything", interest rate, volatility, correlation can not be parameterized at all, they are dynamically interacted and changed with the evolution of markets.

2) Stochastic volatility model
At first time, it is promising as it is adaptive in some degree. But first, maybe the fatal point, is that option is just the reflection of the human prediction for the market in the future. It is not true that everyone use the SV model or use simply vanilla option to calibration. In this sense, people's prediction can not agree each other toward a option. Secondly, jump happens. One can not mimic large jump with the usage of brownian motion. In mathematic language is that one can compensate the small jump by correcting the model but can not compensate the large jump (That's why we need compensated poisson process to mimic a levy process). It is true that Stochastic volatility can correctly reflect volatility cluster in different economical period but the jump nature determined SV model is not a perfect one . Third, it is not easy to hedge out the risk of uncertainty of volatility.

3) Jump diffusion model
Despite that jump model cannot predict any real jump like the impact of 911, the jump model statistically model the jump in a reasonable sense. And they can correctly mimic the fat tail effect and also fit the distribution well like other sophisticated models.

(to be continued...)