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140 Genetic Programming – New Approaches and Successful Applications

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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 and consequently the dynamic hedging.

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

© 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.

[5], boosting [6] and arcing [7]. However, these techniques require that the entire data sets be stored in the main memory. When applied to large data sets, this approach could be impractical. In this paper, we proposed to split data into smaller subsets. First, the genetic programming is run separately on all training sub-samples. Such approach is called static training-subset selection method [8]; it might provide local solutions not adaptive to the entire enlarged data set. Alternatively, a dynamic training approach is developed. It allows genetic programming to learn simultaneously on all training sub-samples and it implies a new parameter added to the basic genetic programming algorithm which is the number of generations to change sample. This approach lightens the training task for the genetic programming and favors the discovery of solutions that are more robust across different learning data samples and seem to have better generalization ability. Comparison between generated models using static and dynamic selection methods reveals that, the dynamic approach improves the forecasting performance of the generated models using genetic programming. The best forecasting implied volatility models are selected according to total MSE criterion. They are used to compute hedge factors and implement dynamic hedging strategies. According to the average hedging errors, the genetic programming presented accurate hedging performance compared to that of Black-Scholes model.

Dynamic Hedging Using Generated Genetic Programming Implied Volatility Models 143

assumptions and computational effort to estimate parameters and stochastic process. As mentioned in [18], traditional financial engineering methods based on parametric models such as the GARCH model family, seem to have difficulty to improve the accuracy in volatility forecasting due to their rigid as well as linear structure. Using its basic and flexible tree-structured representation, genetic programming is capable of solving non-linear problems. In the context of forecasting volatility, most of research papers have focused on forecasting historical volatility based on past returns in different markets. Using historical returns of Nikkei 225 and S&P500 indices, Chen and Yeh [19] have applied a recursive genetic programming approach to estimate volatility by simultaneously detecting and adapting to structural changes. Results have shown that the recursive genetic programming is a promising tool for the study of structural changes. Using high frequency foreign exchange USD-CHF and USD-JPY time series, Zumbach et al. [20] have compared the genetic programming forecasting accuracy to that of historical volatilities, the GARCH (1,1), FIGARCH and HARCH models. According to the root-mean squared errors, the generated genetic programming volatility models did consistently outperform the benchmarks. Similarly, Neely and Weller [21] have tested the forecasting performance of genetic programming for USD-DEM and USD-YEN daily exchange rates against that of GARCH (1,1) model and a related RiskMetrics volatility forecast over different time horizons, using various accuracy criteria. While the genetic programming rules did not usually match the GARCH (1,1) or RiskMetrics models' MSE or <sup>2</sup> *R* , its performance on those measures was generally close. But, the genetic programming did consistently outperform the GARCH model on mean absolute error (MAE) and model error bias at all horizons. Overall, on some dimensions the genetic programming has produced significantly superior results. Applying a combination of theory and techniques such as wavelet transform, time series data mining, Markov chain based discrete stochastic optimization, and evolutionary algorithms genetic algorithms and genetic programming, Ma et al. [22,23] have proposed a systematic approach to address specifically non linearity problems in the forecast of financial indices using intraday data of S&P100 and S&P500 indices. As a result, accuracy of forecasting has reached an average of over 75% surpassing other publicly available results on the forecast of any financial index. Abdelmalek et al. [8] have extended the studies mentioned earlier by forecasting the implied volatility of Black-Scholes from the S&P500 index call options instead of historical volatility using a static training of genetic programming. The performance of generated genetic programming volatility forecasting models is compared between time series samples and moneyness-time to maturity classes. Using Total and outof-sample mean squared errors (MSE) as forecasting performance measures, the time series model seems to be more accurate in forecasting implied volatility than moneyness-time to

Option contracts prices are affected by new information and changes in expectations as much as they are by changes in the value of the underlying index. If traders have perfect foresight on forward volatility, then dynamic hedging would be essentially riskless. In practice, continuous hedging is impossible, but the convexity of option contract allows for adjustments in the exposure to higher-order sensitivities of the model, such as gamma, vega,

maturity models.

The rest of the paper is organized as follows: section 2 provides background information regarding related works in forecasting volatility and dynamic hedging, section 3 describes research design and methodology used in this paper, section 4 reports experimental results and finally section 5 concludes.
