Студент Василије Никачевић одбранио је мастер рад на тему Volatility Smile , пред комисијом коју су чинили ментор проф. др Бранко Урошевић, као и чланови проф. др Милан Недељковић и проф. др Владимир Васић. Никола је одбраном мастер рада завршио студије на програму MCF (Master in Computational Finance).
У уводу свог рада Василије је истакао:
The main purpose of this thesis is to use various mathematical techniques to show how we can model volatility smiles using available option and stock prices. The starting point of our modelling is to change Black-Sholes assumption of constant volatility. Namely, according to Black and Sholes Implied volatility (volatility which equals Black and Sholes model price with market price) is constant when we plot it against the option strike. However, empirically it was shown that Implied volatility forms a smile shape if we plot it against the strike.
One possible hypothesis which can explain the existence of implied volatility is the fact that there is higher demand for in the money or out of the money options compared to the options that are at the money. Another theory suggests that deep out of the money options will have more chances to end up in the money zone if and only if there is a high volatility. Hence, more complex models than Black-Sholes generally tend to overprice out of the money options. The main reason for that is the additional risk exposure associated with deep out of the money options. Volatility smile generally serves as good indicator that allows us to see which part of the market is more likely to identify lower volatility of the price.
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We have managed to find the feasible extensions of the Black Sholes model which are able to reproduce the non-constant volatility. First, we pick up the Dupire-model. In order to avoid the various types of the arbitrages we had to transform variables. However, the reconstructed Dupire model smile was not smooth due to the finite difference approximation i.e., instability of the approximated derivatives. In the Second chapter we used two models (SVI and SABR) which were able to reproduce the empirical volatility using the parabolic fit. In the last chapter we managed to explain the stochastic nature of the volatility. Moreover, we were able to reproduce the market option prices in most cases by using the characteristic function of a given model. The calibration procedure was done fast. Model based on the distribution did a decent job. Despite the reasonable assumption regarding the stock returns distribution and calibration using the maximum likelihood method, we were just able to produce the volatility skew pattern. Since risk neutral valuation assigns the zero price to every out of the money option (spot price is lower than the strike price) we used only in or at the money options.
In conclusion, it is hard in practice to observe the volatility smile shape. The main reason in case of the call options is out of the money call options. They are responsible for the right wing of the volatility smile. In the market, the prices of the deep out of the money options are hardly available. Even if they are available the traded volume is very low. Consequently, as we move away from the more recent maturities (say one day, one week) to the more distant maturities (say one month or three months) the in the money call options which are responsible for the left wing of the volatility smile take the dominant role and the volatility skew pattern is the most likely to be observed. – закључио је Василије.
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