**Author details**

Fathi Abid and Wafa Abdelmalek *Research Unit MODESFI, Faculty of Economics and Business, Sfax, Tunisia* 

Sana Ben Hamida

*Research Laboratory SOIE (ISG Tunis), Paris West University, Nanterre, France* 

#### **6. References**

168 Genetic Programming – New Approaches and Successful Applications

all time series and moneyness-time to maturity classes simultaneously.

Black-Scholes delta-gamma hedging error is 100% for at-the-money put options.

interest for application and will be object of our future works.

*Research Unit MODESFI, Faculty of Economics and Business, Sfax, Tunisia* 

*Research Laboratory SOIE (ISG Tunis), Paris West University, Nanterre, France* 

**Author details** 

Sana Ben Hamida

Fathi Abid and Wafa Abdelmalek

Finally, improving the accuracy of implied volatility forecasting using genetic programming can lead to well hedged options portfolios relative to the conventional parametric models.

Our results suggest some interesting issues for further investigation. First, the genetic programming can be used to hedge options contracts using implied volatility of other models than Black-Scholes model, notably stochastic volatility models and models with jump, as a proxy for genetic programming volatility forecasting. Further, the hedge factors can be computed numerically not analytically. Second, this work can be reexamined using data from individual stock options, American style index options, options on futures, currency and commodity options. Third, as the genetic programming can incorporate known analytical approximations in the solution method, parametric models such as GARCH models can be used as a parameter in the genetic programming to build the forecasting volatility model and the hedging strategies. Finally, the genetic programming can be extended to allow for dynamic parameter choices including the form and the rates of genetic operators, the form and pressure of selection mechanism, the form of replacement strategy and the size of population. This dynamic genetic programming method can improve the performance without extra calculation costs. We believe these extensions are of

implied volatility than MCAR and M4S4 models because it is more general and adaptive to

The main conclusion concerns the importance of implied volatility forecasting in conducting hedging strategies. Genetic programming forecasting volatility makes hedge performances higher than those obtained in the Black-Scholes world. The best genetic programming hedging performance is achieved for in-the-money call options and at-the-money put options in all hedging strategies. The percentage of cases where the hedging error of the genetic programming model is less than the Black-Scholes hedging error is around 59% for calls and 57% for puts. The performance of genetic programming is pronounced essentially in terms of delta hedging for call and put options. The percentage of cases where the delta hedging error of the genetic programming model is less than the Black-Scholes delta hedging error is 100% for out-of-the money and in-the-money call options as well as for at-the-money and out-of-the-money put options. The percentage of cases where the delta-vega hedging error of the genetic programming model is less than the Black-Scholes delta-vega hedging error is 100% for in-the-money call options as well as for at-the-money put options. The percentage of cases where the delta-gamma hedging error of the genetic programming model is less than the

	- [16] Wiggins J (1987) Option Values under Stochastic Volatility: Theory and Empirical Evidence. Journal of Financial Economics. 19: 351-372.

Dynamic Hedging Using Generated Genetic Programming Implied Volatility Models 171

[31] Merton R.C (1973) Theory of Rational Option Pricing. Bell Journal of Economics and

[32] Cai W, Pacheco-Vega A, Sen M, Yang K.T (2006) Heat Transfer Correlations by Symbolic Regression. International Journal of Heat and Mass Transfer. 49: 4352-

[33] Gustafson S, Burke E.K, Krasnogor N (2005) On Improving Genetic programming for Symbolic Regression. In Proceedings of the IEEE Congress on Evolutionary

[34] Keijzer M (2004) Scaled Symbolic Regression. Genetic programming and Evolvable

[35] Lew T.L, Spencer A.B, Scarpa F, Worden K (2006) Identification of Response Surface Models Using Genetic programming. Mechanical Systems and Signal Processing. 20:

[36] Black F, Scholes M. (1973) The Pricing of Options and Corporate Liabilities. Journal of

[37] Kraft D. H, Petry F. E, Buckles W. P, Sadasivan T (1994) The Use of Genetic Programming to Build Queries for Information Retrieval. In Proceedings of the 1994

[39] McKay B, Willis M.J, Barton G.W (1995) Using a Tree Structural Genetic Algorithm to Perform Symbolic Regression. In First International Conference on Genetic Algorithms in Engineering Systems: Innovations and Applications (GALESIA). 414:

[40] Schwefel H.P (1995) Numerical Optimization of Computer Models. John Wiley & Sons,

[41] Cavaretta M.J, Chellapilla K. (1999) Data Mining Using Genetic Programming: The Implications of Parsimony on Generalization Error. In Proceedings of the 1999 Congress

[42] Gathercole C, Ross P (1994) Dynamic Training Subset Selection for Supervised Learning in Genetic Programming. Parallel Problem Solving from Nature III. 866 of LNCS: 312-

[43] Leland H.E. (1985) Option Pricing and Replication with Transaction Costs. Journal of

[44] Kabanov Y.M, Safarian M.M (1997) On Leland Strategy of Option Pricing with

[45] Ahn H, Dalay M, Grannan E, Swindle G (1998) Option Replication with Transactions

IEEE World Congress on Computational Intelligence. IEEE Press. pp. 468–473. [38] Angeline P. J (1996) An Investigation into the Sensitivity of Genetic Programming to the Frequency of Leaf Selection during Subtree Crossover. In: Koza J. R et al., editors. Genetic Programming 1996: Proceedings of the First Annual Conference. MIT Press.

pp. 21–29. Available: www.natural-selection.com/Library/1996/gp96.zip.

on Evolutionary Computation (CEC' 99). IEEE Press. pp. 1330-1337.

Transaction Costs. Finance Stochastic. 1: 239-250.

Costs: General Diffusion Limits. Ann. Appl. Prob. 8: 676-707.

Management Science. 4: 141-183.

Computation. 1: 912-919.

Political Economy. 81: 637-659.

Machines. 5: 259-269.

1819-1831.

487-492.

New York.

321.

Finance. 40: 1283-1301.

4359.


[31] Merton R.C (1973) Theory of Rational Option Pricing. Bell Journal of Economics and Management Science. 4: 141-183.

170 Genetic Programming – New Approaches and Successful Applications

Review of Financial Studies. 6: 327-344.

Morgan Kaufmann Publishers. pp. 58-63.

Finance. Kluwer Academic Publishers. pp. 557-581.

567-572.

Press.

1165-1190.

851-889.

1551-1561.

Derivatives. 2: 78-95.

Management. 4: 237-251.

Evidence. Journal of Financial Economics. 19: 351-372.

[16] Wiggins J (1987) Option Values under Stochastic Volatility: Theory and Empirical

[17] Heston S.L (1993) A Closed-Form Solution for Options with Stochastic Volatility.

[18] Ma I, Wong T, Sankar T, Siu R (2004) Volatility Forecasts of the S&P100 by Evolutionary Programming in a Modified Time Series Data Mining Framework. In: Jamshidi M, editor. Proceedings of the World Automation Congress (WAC2004). 17:

[19] Chen S.H, Yeh C.H (1997) Using Genetic programming to Model Volatility in Financial Time Series. In: Koza J.R, Deb K, Dorigo M, Fogel D.B, Garzon M, Iba H, Riolo R.L, editors. Genetic programming 1997, Proceedings of the Second Annual Conference.

[20] Zumbach G, Pictet O.V, Masutti O (2002) Genetic programming with Syntactic Restrictions Applied to Financial Volatility Forecasting. In: Kontoghioghes E.J, Rustem B, Siokos S, editors. Computational Methods in Decision-Making, Economics and

[21] Neely C.J, Weller P.A (2002) Using a Genetic Program to Predict Exchange Rate Volatility. In: Chen S.H, editor. Genetic Algorithms and Genetic programming in

[24] Derman E, Ergener D, Kani I (1995) Static Options Replication. The Journal of

[25] Carr P, Ellis K, Gupta V (1998) Static Hedging of Exotic Options. Journal of Finance. 53:

[26] Hutchinson J.M, Lo A.W, Poggio T (1994) A NonParametric Approach to Pricing and Hedging Derivative Securities via Learning Network. Journal of Finance. 49:

[27] Aït-Sahalia Y, Lo A (1998) Nonparametric Estimation for State-Price Densities Implicit

[28] Chen S.H, Lee W.C, Yeh C.H (1999) Hedging Derivative Securities with Genetic Programming. International Journal of Intelligent Systems in Accounting, Finance and

[29] Harvey C.R, Whaley R.E (1991) S&P 100 Index Option Volatility. Journal of Finance. 46:

[30] Harvey C.R, Whaley R.E (1992) Market Volatility Prediction and the Efficiency of the

S&P100 Index Option Market. Journal of Financial Economics. 31: 43-73.

in Financial Asset Prices. The Journal of Finance. 53: 499-547.

Computational Finance, Chapter 13. Kluwer Academic Publishers. pp. 263-279. [22] Ma I, Wong T, Sanker T (2006) An Engineering Approach to Forecast Volatility of Financial Indices. International Journal of Computational Intelligence. 3: 23-35. [23] Ma I, Wong T, Sanker T (2007) Volatility Forecasting using Time Series Data Mining and Evolutionary Computation Techniques. In Proceedings of the 9th Annual Conference on Genetic and Evolutionary Computation (GECCO 07). ACM New York

	- [46] Grandits P, Schachinger W (2001) Leland's Approach to Option Pricing: The Evolution of Discontinuity. Math Finance. 11: 347-355.

**Chapter 8** 

© 2012 Dubrovski and Brezočnik, 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

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Dubrovski and Brezočnik, 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,

properly cited.

**The Usage of Genetic Methods** 

**for Prediction of Fabric Porosity** 

Advanced fabric production demands developing strategies with regard to new fabric constructions in which sample-production is reduced to a minimum. It is clear that a new fabric construction should have the desired end-usage properties pre-specified as project demands. Achieving such a demand is a complex task based on our knowledge of the relations between the fabric constructional parameters and the predetermined fabric endusage properties that fit the desired quality. Individual fabric properties are difficult to predict when confronting the various construction parameters, which can be separated into the following categories: raw materials, fabric structure, design, and manufacturing

Many attempts have been made to develop predictive models for fabric properties with different modelling tools. There are essentially two types of modelling tools: deterministic (mathematical models, empirical models, computer simulation models) and nondeterministic (models based on genetic methods, neural network models, models based on chaos theory and theory of soft logic), and each of them has its advantages and

Deterministic modelling tools present the heart of conventional science and have their basis in first principles, statistical techniques or computer simulations. Mathematical models offer a deep understanding of relations between constructional parameters and predetermined fabric property, but due some simplifying assumptions large prediction errors occur. Empirical models based on statistical techniques show a much better agreement with the real values but the problems with samples preparing, process repeatability, measurements errors and extrapolation occur. They usually refer to the one type of testing method of particular fabric property. The advantage of computer simulation models is their ability to capture the randomness inherent in fabric structure so the predicted values are very near the

Polona Dobnik Dubrovski and Miran Brezočnik

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48188

**1. Introduction** 

parameters.

disadvantages [1].
