**Acknowledgement**

138 Genetic Programming – New Approaches and Successful Applications

obtained by PSO and DE even at the center frequency.

wide bandwidth, equal to 30%).

antennas.

higher.

**Author details** 

*Cagliari, Italy* 

Giovanni Andrea Casula and Giuseppe Mazzarella

**4. Conclusion** 

Both (Baskar et al., 2005), (Goudos et al., 2010) and (Li, 2007) decide to perform the optimization only at the center frequency, and this is a simpler task and can lead to better results than an optimization over the whole antenna bandwidth, which is the choice we made in our SED design. Nonetheless, the results obtained by SED are better than the ones

In fact we are able to get a wideband antenna with a very high gain, i.e. we both maximize antenna gain and minimize SWR and antenna size within the whole bandwidth (which is a

Therefore, SED can lead to better results if compared with PSO and DE, both in terms of performances and of overall size. This is probably due to the fact that the solution space of SED is larger than the corresponding solution spaces of PSO and DE, and hence a proper choice of the fitness function can push the evolution process to more performing

In this chapter a new design technique, namely the Structure-based Evolutionary Design (SED) has been described in detail. This is a new global random search method based on the evolutionary programming concept. The proposed technique has been compared with the standard genetic algorithms (GA), a widely used design technique, showing the numerous advantages of our approach with respect to standard ones. Its main advantage is the ability to explore a far larger solution space than standard optimization algorithms. Moreover, SED assumes no "a priori" structure, but it builds up the structure of the individuals as the procedure evolves, being able to determine both the structure shape and dimensions as an outcome of the procedure. Inclusion of input matching requirements prevents from ill-posedness, a danger always present when the solution space is so large. The described procedure has been used to design wire antennas, and several examples are presented, showing very good results. The goal of the design process is to develop wire antennas fulfilling the desired requirements for both Gain and VSWR in a frequency band as wide as possible, and with the smallest size. For each set of requirements, a suitable fitness function must be derived, and some suggestions are given to choose the best fitness for the problem at hand. The results obtained with SED are finally compared with other global search algorithms showing that both the computational cost and the complexity are of the same order of magnitude, but the performances obtained by SED are significantly

*Università degli Studi di Cagliari/Dipartimento di Ing. Elettrica ed Elettronica, Piazza d'Armi,* 

Work supported by Regione Autonoma della Sardegna, under contract number CRP1\_511, with CUP F71J09000810002, titled *"Valutazione e utilizzo della Genetic Programming nel progetto di strutture a radiofrequenza e microonde"* .

### **5. References**

	- Franceschetti, G.; Mazzarella, G.; Panariello, G. (1988). Array synthesis with excitation constraints, *Proc. IEE, pt.H*, Vol.135.

**Chapter 7** 

© 2012 Abdelmalek et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Abdelmalek et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Dynamic Hedging Using Generated** 

Fathi Abid, Wafa Abdelmalek and Sana Ben Hamida

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48148

and consequently the dynamic hedging.

**1. Introduction** 

**Genetic Programming Implied Volatility Models** 

One challenge posed by financial markets is to correctly forecast the volatility of financial securities, which is a crucial variable in trading and risk management of derivative securities. Dynamic hedging is very sensitive to volatility forecast and good hedges require accurate estimate of volatility. Implied volatilities, generated from option markets, can be particularly useful in such contents as they are forward-looking measures of the market's expected volatility during the remaining life of an option [1, 2]. Since there is no explicit formula available to compute directly the implied volatility, the latter can be obtained by inverting the option pricing model. On the contrary, the genetic programming offers explicit formulas which can compute directly the implied volatility. This volatility forecasting method should be free of strong assumptions regarding underlying price dynamics and more flexible than parametric methods. This paper proposes a non parametric approach based on genetic programming to improve the accuracy of the implied volatility forecast

Genetic Programming [3] is an optimization technique which extends the basic genetic algorithms [4] to process non-linear problem structure. In genetic programming, solutions are represented as tree structures that can vary in size and shape, rather than fixed length character strings as in genetic algorithms. This means that genetic programming can be used to perform optimization at a structural level. In the standard genetic programming, the entire population of function-trees is evaluated against the entire training data set, so the number of function-tree evaluations carried out per generation is directly proportional to both the population size and the size of the training set. Genetic programming can encounter the problem of managing training sets which are too large to fit into the memory of computers, and then the realization of predictors. In machine learning, the practiced solution to learn large data set is the application of resampling techniques, such as, bagging

