**5. Conclusion**

166 Genetic Programming – New Approaches and Successful Applications

classes.

options

for all MTM classes. However, the best delta-gamma hedging performance is achieved, for genetic programming model, using out-of-the-money long term put options for all MTM

**S/K Hedging strategy Model <60 60-180 >=180 <60 60-180 >=180 <0.98 Delta hedging** BS 0,007259 0,002212 0,001189 0,015453 0,013715 0,007740

 **Gamma hedging** BS 0,000107 0,000043 0,000705 0,000383 0,000253 0,013169

 **Vega hedging** BS 0,000051 0,000715 0,000612 0,000174 0,002995 0,008527

 **Gamma hedging** BS 0,003750 0,000049 0,000027 0,032725 0,000119 0,000119

 **Vega hedging** BS 0,035183 0,000052 0,000044 0,037082 0,000329 0,000043

**>=1.03 Delta hedging** BS 0,007680 0,004469 0,000555 0,037186 0,017322 0,011739

**Table 9.** Average hedge errors of dynamic hedging strategies relative to BS and GP models for put

programming model, using at-the-money long term put options for all MTM classes.

The delta-vega hedging performance improves for BS using at-the-money and out-of-themoney put options at longer maturities and in-the-money put options at shorter maturities, at daily hedge revision frequency. However, the delta-vega hedging performance improves for all moneyness classes of put options (except in-the-money put options) at longer maturities, regarding genetic programming model at daily hedge frequency. The best delta-vega hedging performance is achieved, for BS model, using out-of-the-money long term put options for all MTM classes. However, the best delta-vega hedging performance is achieved, for genetic

The percentage of cases where the hedging error of the genetic programming model is less than the BS hedging error is around 57%. In particular, the performance of genetic

 **Gamma hedging** BS 0,000262 0,000204 0,000079 0,001196 0,001319 0,000369

**Vega hedging** BS 0,000232 0,000108 0,000025 0,000488 0,000644 0,000270

**0.98-1.03 Delta hedging** BS 0,007331 0,002267 0,001196 0,170619 0,009875 0,004265

GP 0,064397 0,002270 0,001256 0,016872 0,013933 0,007815

GP 0,000177 0,000351 **0,000676** 0,000990 0,000324 **0,009201** 

GP 0,002800 **0,000345** 0,000625 0,018351 **0,000184** 0,008979

GP **0,0073 0,002219 0,001185 0,170316 0,009715 0,004260** 

GP **0,003491 0,000031 0,000024 0,029792 0,000113 0,000103** 

GP **0,004343 0,000038 0,000043 0,037045 0,000190 0,000041** 

GP **0,006641 0,004404 0,0005 0,037184 0,017076 0,011733** 

GP 0,000548 0,000287 0,000166 0,002034 0,001323 0,001059

GP 0,000312 **0,000080 0,00002** 0,001047 0,001186 **0,000244** 

 **Rebalancing Frequency 1-day 7- days**

> This paper is concerned with improving the dynamic hedging accuracy using generated genetic programming implied volatilities. Firstly, genetic programming is used to predict implied volatility from index option prices. Dynamic training-subset selection methods are applied to improve the robustness of genetic programming to generate general forecasting implied volatility models relative to static training-subset selection method. Secondly, the implied volatilities derived are used in dynamic hedging strategies and the performance of genetic programming is compared to that of Black-Scholes in terms of delta, gamma and vega hedging.

> Results show that the dynamic training of genetic programming yields better results than those obtained from static training with fixed samples, especially when applied on time series and moneyness-time to maturity samples simultaneously. Based on the MSE total as performance criterion, three generated genetic programming volatility models are selected M4S4, MCAR and MGAR. However, the MGAR seems to be more accurate in forecasting

implied volatility than MCAR and M4S4 models because it is more general and adaptive to all time series and moneyness-time to maturity classes simultaneously.

Dynamic Hedging Using Generated Genetic Programming Implied Volatility Models 169

[1] Blair B.J, Poon S, Taylor S.J (2001) Forecasting S&P100 Volatility: The Incremental Information Content of Implied Volatilities and High Frequency Index Returns. Journal

[2] Busch T, Christensen B.J, Nielsen M.Ø (2007) The Role of Implied Volatility in Forecasting Future Realized Volatility and Jumps in Foreign Exchange, Stock, and Bond Markets. CREATES Research Paper 2007-9. Aarhus School of Business, University of

[3] Koza J.R (1992) Genetic programming: on the Programming of Computers by means of

[4] Holland J.H (1975) Adaptation in Natural and Artificial Systems. Ann Arbor: University

[6] Freund Y, Schapire R (1996) Experiments with a New Boosting Algorithm. In Proceedings of the 13th International Conference on Machine Learning. Morgan

[8] Abdelmalek W, Ben Hamida S, Abid F (2009) Selecting the Best Forecasting-Implied Volatility Model using Genetic programming. Journal of Applied Mathematics and Decision Sciences (Special Issue: Intelligent Computational Methods for Financial

[9] Tsang E, Yung P, Li J (2004) EDDIE-Automation, a Decision Support Tool for Financial

[10] Kaboudan M (2005) Extended Daily Exchange Rates Forecasts using Wavelet Temporal Resolutions. New Mathematics and Natural Computing. 1: 79-107. Available:

[11] Bollerslev T, Chou R.Y, Kroner K.F (1992) ARCH Modelling in Finance: a Review of the

[12] Engle R.F (1982) Autoregressive Conditional Heteroscedasticity with Estimates of the

[13] Bollerslev T (1986) Generalized Autoregressive Conditional Heteroscedasticity. Journal

[14] Hull J, White A (1987) The Pricing of Options on Assets with Stochastic Volatilities.

[15] Scott L (1987) Option Pricing When the Variance Changes Randomly: Theory, Estimation and an Application. Journal of Financial and Quantitative Analysis. 22: 419-

438. Available: http:// www.globalriskguard.com/resources/.../der6.pdf

Natural Selection. Cambridge, Massachusetts: the MIT Press. 819 p.

[5] Breiman L (1996) Bagging Predictors. Machine Learning. 2:123-140.

[7] Breiman L (1998) Arcing Classifiers. Annals of Statistics. 26: 801-849.

Engineering). Hindawi Publishing Corporation. Available: http://

Forecasting. Decision Support Systems. 37: 559–565.Available:

http://www.mendeley.com/.../extended-daily-... - États-Unis

Theory and Empirical Evidence. Journal of Econometrics. 52: 55-59.

www.hindawi.com/journals/jamds/2009/179230.html

http://sci2s.ugr.es/keel/pdf/specific/.../ science2\_4.pdf

Variance of U.K. Inflation. Econometrica. 50: 987-1008.

of Econometrics. 31: 307-327.

Journal of Finance. 42: 218-300.

**6. References** 

of Econometrics.105: 5-26.

Copenhagen. pp.1-39.

of Michigan Press.

Kauffman Publishers. pp. 148-156.

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

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