**2. Related works**

Traditional parametric methods have limited success in estimating and forecasting volatility as they are dependent on restrictive assumptions and difficult to estimate. Several machine learning techniques have been recently used to overcome these difficulties such as artificial neural networks and evolutionary computation algorithms. In particular, genetic programming has been successfully applied to forecast financial time series [9,10].

This paper makes an initial attempt to test whether the hedger can benefit more by using generated genetic programming implied volatilities instead of Black-Scholes implied volatilities in conducting dynamic hedging strategies.

Changes in asset prices is not the only risk faced by market participants, instantaneous changes in market implied volatility can also bring a hedging portfolio significantly out of balance. Extensive research during the last two decades has demonstrated that the volatility of stocks is not constant over time [11]. The Autoregressive Conditional Heteroskedasticiy (ARCH ) and the Generalized ARCH (GARCH) models are introduced [12,13] to describe the evolution of the asset price's volatility in discrete time. Econometric tests of these models clearly reject the hypothesis of constant volatility and find evidence of volatility clustering over time. In the financial literature, stochastic volatility models have been proposed to model these effects in a continuous-time setting [14-17]. Although these models improve the benchmark Black-Scholes model, they are complex because they require strong 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

142 Genetic Programming – New Approaches and Successful Applications

[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

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

Traditional parametric methods have limited success in estimating and forecasting volatility as they are dependent on restrictive assumptions and difficult to estimate. Several machine learning techniques have been recently used to overcome these difficulties such as artificial neural networks and evolutionary computation algorithms. In particular, genetic

This paper makes an initial attempt to test whether the hedger can benefit more by using generated genetic programming implied volatilities instead of Black-Scholes implied

Changes in asset prices is not the only risk faced by market participants, instantaneous changes in market implied volatility can also bring a hedging portfolio significantly out of balance. Extensive research during the last two decades has demonstrated that the volatility of stocks is not constant over time [11]. The Autoregressive Conditional Heteroskedasticiy (ARCH ) and the Generalized ARCH (GARCH) models are introduced [12,13] to describe the evolution of the asset price's volatility in discrete time. Econometric tests of these models clearly reject the hypothesis of constant volatility and find evidence of volatility clustering over time. In the financial literature, stochastic volatility models have been proposed to model these effects in a continuous-time setting [14-17]. Although these models improve the benchmark Black-Scholes model, they are complex because they require strong

programming has been successfully applied to forecast financial time series [9,10].

volatilities in conducting dynamic hedging strategies.

accurate hedging performance compared to that of Black-Scholes model.

and finally section 5 concludes.

**2. Related works** 

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 maturity models.

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,

etc. Most of the existing literature on hedging a target contract using other exchange-traded options focuses on static strategies, motivated at least in part by the desire to avoid the high costs of frequent trading. The goal of static hedging is to construct a buy-and-hold portfolio of exchange traded claims that perfectly replicates the payoff of a given over-the-counter product [24,25]. The static hedging strategy does not require any rebalancing and therefore, it does not incur significant transaction costs. Unfortunately, the odds of coming up with a perfect static hedge for a given over-the-counter claim are small, given the limited number of exchange listed option contracts with sufficient trading volume. In other words, the static hedge can only be efficient if traded options are available with sufficiently similar maturity and moneyness as the over-the-counter product that has to be hedged. Under a stochastic volatility, a perfect hedge can in principle be constructed with a dynamically rebalanced portfolio consisting of the underlying and one additional option. In practice, the dynamic replication strategy for European options will only be perfect if all of the assumptions underlying the Black-Scholes formula hold. For general contingent claims on a stock, under market frictions, the delta might still be used as first-order approximation to set up a riskless portfolio. However, if the volatility of the underlying stock varies stochastically, then the delta hedging method might fail severely. A simple method to limit the volatility risk is to consider the volatility sensitivity vega of the contract. The portfolio will have to be rebalanced frequently to ensure delta-vega neutrality. With transaction costs, frequent rebalancing might result in considerable losses. In practice, investors can rebalance their portfolios only at discrete intervals of time to reduce transactions costs.

Dynamic Hedging Using Generated Genetic Programming Implied Volatility Models 145

programming to train on the entire data sub samples simultaneously rather than just a single subset by changing the training Sub-sample during the run process. This permits to improve the robustness of genetic programming to generate general models adaptive to all training samples. The second thrust of this paper is to study the accuracy of the generated genetic programming implied volatility models in terms of dynamic hedging. Since the true volatility is unobservable, it is impossible to assess the accuracy of any particular model; forecasts can only be related to realized volatility. In this paper, we assume that the implied volatility is a reasonable proxy for realized volatility, to generate forecasting implied volatility models using genetic programming and then to analyze the implications of this

Figure 1 illustrates the operational procedure to implement the proposed approach.

The operational procedure consists of the following steps: The first step is devoted for the data division schemes. The second step deals with the implementation of genetic programming1 (GP), the application of training subset selection methods and the selection of the best forecasting implied volatility models. The last step is dedicated to dynamic hedging

Data used in this study consist of daily prices for the European-style S&P 500 index calls and puts options traded on the Chicago Board of Options Exchange from 02 January to 29 August 2003. The data base include the time of the quote, the expiration date, the exercise price and the daily bid and ask quotes for call and put options. Similar information for the underlying S&P 500 index is also available on a daily basis. S&P500 index options are among the most actively traded financial derivatives in the world. The minimum tick for series trading below 3 is 1/16 and for all other series 1/8. Strike price intervals are 5 points, and 25 points for far months. The expiration months are three near term months followed by three additional months from the March quarterly cycle (March, June, September, and December). Following a standard practice, we used the average of an option's bid and ask price as a stand-in for the market value of the option. The risk free interest rate is approximated by using 3 month US Treasury bill rates. It is assumed that there are no transaction costs and no

1 GP system is built around the Evolving Object library, which is an ANSI-C++ evolutionary computation Framework

predictability for hedging purposes.

results.

dividend.

(EO library).

**3.1. Data division schemes** 

**Figure 1.** Description of the proposed approach's implementation

Non parametric hedging strategies as an alternative to the existing parametric model basedstrategies, have been proposed [26,27]. Those studies estimated pricing formulas by nonparametric or semi-parametric statistical methods such as neural networks and kernel regression, and they measured their performance in terms of delta-hedging. Few researches have focused on the dynamic hedging using genetic programming, however. Chen et al. [28] have applied genetic programming to price and hedge S&P500 index options. By distinguishing the case in-the-money from the case out-of-the-money, the performance of genetic programming is compared with the Black-Scholes model in terms of hedging accuracy. Based on the post-sample performance, it is found that in approximately 20% of the 97 test paths, genetic programming has lower tracking error than the Black-Scholes formula.

Based on the literature survey, one can conclude that the genetic programming could be used to efficiently forecast volatility and implement accurate dynamic hedging strategies, which opens up an alternative path besides other data-based approaches.
