# Implied volatility functions on Finnish warrant market

##### Ahoranta, Veli-Matti (2008)

**Kokoteksti luettavissa vain Tritonian asiakaskoneilla.**

Ahoranta, Veli-Matti

2008

#### Kuvaus

Opinnäytetyö kokotekstinä PDF-muodossa.

##### Tiivistelmä

Evidences that there are volatility smiles and smirks in various financial markets sug-gest that Black and Scholes (1973) valuation formula is not completely valid. This thesis investigates implied volatility patterns and –functions on Finnish warrant market. The intention of the thesis is to find answers to the three following questions: what is the form of the volatility structure in Finnish warrant markets? Does there exist a better method to estimate volatilities than basic Black-Scholes constant volatility model? In case that there exist a superior method to estimate volatilities, is the method constantly best with every level of moneyness and time to expiration?

This study focuses on Finnish warrant markets which has not been studied earlier from the point of view of volatility functions. The study uses the most recent data and the sample period includes over 7000 observations from May 2006 to December 2006. The sample period has been divided into six months estimation period and one month evaluation period. On the estimation period the implied volatilities are first estimated by using Newton-Raphson method and then the means for different subclasses are calcu-lated in order to investigate volatility patterns. Moreover, the parameter estimates and explanatory powers of volatility functions are calculated based on the observations of the estimation period by using OLS-regression. Finally, the performances of volatility functions are evaluated on evaluation period with five measurements.

The empirical results indicate that the volatility patterns for both call and put warrants are U-shaped in most of the cases. However, for longer term put warrants the volatility pattern is a frown. The result that the volatilities are not constant across moneyness and time to expiration suggest that it might be possible to achieve more accurate volatility estimates by using alternative volatility functions. The results indicate, however, that in most of the cases alternative volatility models perform relatively poorly compared to the Black-Scholes constant volatility model. Nevertheless, it appears that a specification which include both linear and quadratic variables of standardized moneyness perform especially well for short term warrants which are not at-the-money.

This study focuses on Finnish warrant markets which has not been studied earlier from the point of view of volatility functions. The study uses the most recent data and the sample period includes over 7000 observations from May 2006 to December 2006. The sample period has been divided into six months estimation period and one month evaluation period. On the estimation period the implied volatilities are first estimated by using Newton-Raphson method and then the means for different subclasses are calcu-lated in order to investigate volatility patterns. Moreover, the parameter estimates and explanatory powers of volatility functions are calculated based on the observations of the estimation period by using OLS-regression. Finally, the performances of volatility functions are evaluated on evaluation period with five measurements.

The empirical results indicate that the volatility patterns for both call and put warrants are U-shaped in most of the cases. However, for longer term put warrants the volatility pattern is a frown. The result that the volatilities are not constant across moneyness and time to expiration suggest that it might be possible to achieve more accurate volatility estimates by using alternative volatility functions. The results indicate, however, that in most of the cases alternative volatility models perform relatively poorly compared to the Black-Scholes constant volatility model. Nevertheless, it appears that a specification which include both linear and quadratic variables of standardized moneyness perform especially well for short term warrants which are not at-the-money.