**4.1. Selection of the best genetic programming-implied volatility forecasting models**

Selection of the best generated genetic programming volatility model, relative to each training set, for TS, MTM, and both TS and MTM classifications, is made according to the training and test MSE. For static training-subset selection method, nine generated genetic programming volatility models are selected for TS (M1S1…M9S9) and similarly nine generated genetic programming volatility models are selected for MTM classification (M1C1…M9C9). The performance of these models is compared according to the MSE Total, computed using the same formula as the basic MSE for the enlarged data sample.

Table 4 reports the MSE total and the standard deviation (in parentheses) of the generated genetic programming volatility models, using static training-subset selection method, relative to the TS samples and the MTM classes.


**Table 4.** Performance of the generated genetic programming volatility models using static trainingsubset selection method, according to MSE total for the TS samples and the MTM classes

Table 4 shows that, the generated genetic programming volatility models M4S4, M4C4 and M6C6 present the smallest MSE on the enlarged sample for TS and MTM samples respectively. Comparison between these models reveals that the TS model M4S4 seems to be more performing than MTM models M4C4 and M6C6 for the enlarged sample. Furthermore, results show that the performance of TS models is more uniform than that of MTM models. MTM models are not able to fit appropriately the entire data sample as well as the TS models as they have large Total MSE. Indeed, the MSE total exceed 1 with some MTM classes, however it does not reach 0.006 for all TS samples. Figure 12 describes the evolution's pattern of the squared errors given by TS models and MTM models for all observations in the enlarged data sample. Some extreme MSE values for MTM data are not shown in this figure.

It appears throughout Figure 12 that, the TS models are adaptive not only to training samples, but also to the enlarged sample. In contrast, the MTM models such as M1C1 are adaptive to training classes, but not all to the enlarged sample. A first plausible explanation of these unsatisfied results is an insufficient search intensity inducing difficulty to obtain general model suitable for the entire benchmark input data. To enhance exploration intensity during learning and thus improve the genetic programming performance, we introduced to the evolution procedure the dynamic subset selection, which aims to obtain a general model that can be adaptive to both TS and MTM classes simultaneously.

160 Genetic Programming – New Approaches and Successful Applications

maturity class and *<sup>i</sup>*

initial option price *V* 0 .

*P*

**models** 

 

**4. Result analysis and empirical findings** 

relative to the TS samples and the MTM classes.

*rT i*

Where, n is the number of options corresponding to a particular moneyness-time to

over the observation path N (as a function of rebalancing frequency), divided by the

*n*

 

1

*i*

 

*e*

**4.1. Selection of the best genetic programming-implied volatility forecasting** 

computed using the same formula as the basic MSE for the enlarged data sample.

**TS Models MSE Total MTM Models MSE Total**  M1S1 0,002723 (0,004278) M1C1 2,566 (20,606) M2S2 0,005068 (0,006213) M2C2 0,006921 (0,032209) M3S3 0,003382 (0,004993) M3C3 0,030349 (0,076196) **M4S4 0,001444 (0,002727) M4C4 0,001710 (0,004624)**  M5S5 0,002012 (0,003502) M5C5 1,427142 (33,365115) M6S6 0,001996 (0,003443) **M6C6 0,002357 (0,004096)**  M7S7 0,001901 (0,003317) M7C7 0,261867 (0,303256) M8S8 0,002454 (0,004005) M8C8 0,004318 (0,008479) M9S9 0,002419 (0,004095) M9C9 0,002940 (0,010490)

**Table 4.** Performance of the generated genetic programming volatility models using static training-

subset selection method, according to MSE total for the TS samples and the MTM classes

Selection of the best generated genetic programming volatility model, relative to each training set, for TS, MTM, and both TS and MTM classifications, is made according to the training and test MSE. For static training-subset selection method, nine generated genetic programming volatility models are selected for TS (M1S1…M9S9) and similarly nine generated genetic programming volatility models are selected for MTM classification (M1C1…M9C9). The performance of these models is compared according to the MSE Total,

Table 4 reports the MSE total and the standard deviation (in parentheses) of the generated genetic programming volatility models, using static training-subset selection method,

*n i i M*

 

*P*

*N V*

0

is the present value of the absolute hedge error of the portfolio

(16)

(a) MSE pattern for TS samples (b) MSE pattern for MTM classes

**Figure 12.** Evolution of the squared errors for total sample of the best generated GP volatility models, using static training-subset selection method, relative to TS samples(a) and MTM classes (b).

For dynamic training-subset selection methods (RSS, SSS, ASSS and ARSS), four generated genetic programming volatility models are selected for TS classification (MSR, MSS, MSAS and MSAR). Similarly, four generated genetic programming volatility models are selected for MTM classification (MCR, MCS, MCAS and MCAR) and four generated genetic programming volatility models are selected for global classification, both TS and MTM classes (MGR, MGS, MGAS and MGAR). Table 5 reports the best generated genetic programming volatility models, using dynamic training-subset selection, relative to TS samples, MTM classes and both TS and MTM data.

162 Genetic Programming – New Approaches and Successful Applications


Dynamic Hedging Using Generated Genetic Programming Implied Volatility Models 163

The best generated genetic programming volatility models selected, relative to dynamic training-subset selection method, are compared to the best generated genetic programming volatility model, relative to static training-subset selection method. Results are reported in

> M4S4 0,001444 (0,002727) MCAR 0.001424 (0.003527) **MGAR 0.001599 (0.003590)**

**Table 6.** Comparison between best models generated by static and dynamic selection methods for call

Comparison between models reveals that the best models generated respectively by static (M4S4) and dynamic selection methods (MCAR and MGAR) present total MSE small and very close. While the generated genetic programming volatility models M4S4 and MCAR have total MSE smaller than the MGAR model, the latest seems to be more accurate in forecasting implied volatility than the other models. This can be explained by the fact that, on one hand, the difference between forecasting errors is small, and on the other hand, the MGAR model is more general than MCAR and M4S4 models because it is adaptive to all TS and MTM classes simultaneously. In fact, the MGAR model, generated using ARSS method, is trained on all TS and MTM classes simultaneously. Whereas, the MCAR model, generated using ARSS method, is trained only on MTM classes simultaneously; and the M4S4 model, generated using

As the adaptive-random training subset selection method is considered the best one to generate implied volatility model for call options, it is applied to put options. The decoding of volatility forecasting formulas generated for call and put options as well as their

A detailed examination of the formulas in Table 7 shows that the implied volatilities generated by genetic programming are function of all the inputs used, namely the option

be negative since they are computed using the square root and the normal cumulative distribution functions as the root nodes. Furthermore, the performance of models is uniform

The performance of the best genetic programming forecasting models is compared to the Black-Scholes model in delta, gamma and vega hedging strategies. Table 8 reports the

*<sup>K</sup>* for puts), the index price divided by strike

. The implied volatilities generated for calls and puts cannot

**Models MSE total**

static training-subset selection method, is trained separately on each subset of TS.

for calls and *<sup>P</sup>*

forecasting errors are reported in Table 7.

*K*

as they present near MSE on the enlarged sample.

price divided by strike price ( *<sup>C</sup>*

*K* and time to maturity

**4.2. Dynamic hedging results:** 

price *<sup>S</sup>*

Table 6.

options

**Table 5.** Performance of the generated genetic programming volatility models, using dynamic trainingsubset selection method, according to MSE total for the TS samples, the MTM classes and both TS and MTM samples

Based on the MSE total as performance criterion, the generated genetic programming volatility models MSS, MCAR and MGAR are selected. They seem to be more accurate in forecasting implied volatility than the other models because they have the smallest MSE in enlarged sample. However, the MTM model MCAR and the global model MGAR outperform the TS model MSS. Figure 13 describes the evolution's pattern of the squared errors for these generated volatility models.

Figure 13 shows that almost all models relative to each data's group are performing on the enlarged sample and present forecasting errors which are small and very closed. Forecasting errors are higher for the MTM classes than for the TS samples and both TS and MTM samples. Comparison between models generated using static training-subset selection method (Figure 12) and dynamic training-subset selection methods (Figure 13) respectively, reveals that the amplitude of forecasting errors relative to TS and MTM classes respectively is lower for the models generated using dynamic training-subset selection methods than for the models generated using static training-subset selection method. Actually, the quality of the generated genetic programming forecasting models has been improved with the dynamic training, in particular for MTM classes.

**Figure 13.** Evolution of the squared errors for total sample of the best generated GP volatility models, using dynamic training-subset selection methods, relative to TS samples (a), MTM classes (b) and both TS and MTM samples (c).

The best generated genetic programming volatility models selected, relative to dynamic training-subset selection method, are compared to the best generated genetic programming volatility model, relative to static training-subset selection method. Results are reported in Table 6.


**Table 6.** Comparison between best models generated by static and dynamic selection methods for call options

Comparison between models reveals that the best models generated respectively by static (M4S4) and dynamic selection methods (MCAR and MGAR) present total MSE small and very close. While the generated genetic programming volatility models M4S4 and MCAR have total MSE smaller than the MGAR model, the latest seems to be more accurate in forecasting implied volatility than the other models. This can be explained by the fact that, on one hand, the difference between forecasting errors is small, and on the other hand, the MGAR model is more general than MCAR and M4S4 models because it is adaptive to all TS and MTM classes simultaneously. In fact, the MGAR model, generated using ARSS method, is trained on all TS and MTM classes simultaneously. Whereas, the MCAR model, generated using ARSS method, is trained only on MTM classes simultaneously; and the M4S4 model, generated using static training-subset selection method, is trained separately on each subset of TS.

As the adaptive-random training subset selection method is considered the best one to generate implied volatility model for call options, it is applied to put options. The decoding of volatility forecasting formulas generated for call and put options as well as their forecasting errors are reported in Table 7.

A detailed examination of the formulas in Table 7 shows that the implied volatilities generated by genetic programming are function of all the inputs used, namely the option price divided by strike price ( *<sup>C</sup> K* for calls and *<sup>P</sup> <sup>K</sup>* for puts), the index price divided by strike

price *<sup>S</sup> K* and time to maturity . The implied volatilities generated for calls and puts cannot be negative since they are computed using the square root and the normal cumulative distribution functions as the root nodes. Furthermore, the performance of models is uniform as they present near MSE on the enlarged sample.

### **4.2. Dynamic hedging results:**

162 Genetic Programming – New Approaches and Successful Applications

**Models MSE Total Global** 

MSR 0.002367 (0.003934) MCR 0.002427 (0.003777) MGR 0.002034 (0.003501) **MSS 0.002076 (0.004044)** MCS 0.007315 (0.025811) MGS 0.002492 (0.003013) MSAS 0.002594 (0.003796) MCAS 0.002831 (0.004662) MGAS 0.001999 (0.003587) MSAR 0.002232 (0.003782) **MCAR 0.001424 (0.003527) MGAR 0.001599 (0.003590) Table 5.** Performance of the generated genetic programming volatility models, using dynamic trainingsubset selection method, according to MSE total for the TS samples, the MTM classes and both TS and

Based on the MSE total as performance criterion, the generated genetic programming volatility models MSS, MCAR and MGAR are selected. They seem to be more accurate in forecasting implied volatility than the other models because they have the smallest MSE in enlarged sample. However, the MTM model MCAR and the global model MGAR outperform the TS model MSS. Figure 13 describes the evolution's pattern of the squared

Figure 13 shows that almost all models relative to each data's group are performing on the enlarged sample and present forecasting errors which are small and very closed. Forecasting errors are higher for the MTM classes than for the TS samples and both TS and MTM samples. Comparison between models generated using static training-subset selection method (Figure 12) and dynamic training-subset selection methods (Figure 13) respectively, reveals that the amplitude of forecasting errors relative to TS and MTM classes respectively is lower for the models generated using dynamic training-subset selection methods than for the models generated using static training-subset selection method. Actually, the quality of the generated genetic programming forecasting models has been improved with the

**Figure 13.** Evolution of the squared errors for total sample of the best generated GP volatility models, using dynamic training-subset selection methods, relative to TS samples (a), MTM classes (b) and both

(b) MSE pattern for MTM classes **Models MSE Total** 

(c) MSE pattern for TS+MTM

**Models MSE Total MTM** 

errors for these generated volatility models.

dynamic training, in particular for MTM classes.

TS and MTM samples (c).

(a) MSE pattern for TS samples

**TS** 

MTM samples

The performance of the best genetic programming forecasting models is compared to the Black-Scholes model in delta, gamma and vega hedging strategies. Table 8 reports the

average hedging errors for call options using Black-Scholes (BS) and genetic programming (GP) models, at the 1-day and 7-days rebalancing frequencies. Values in bold correspond to the GP hedging errors which are less than the BS ones.

Dynamic Hedging Using Generated Genetic Programming Implied Volatility Models 165

**Model <60 60-180 >=180 <60 60-180 >=180** 

GP **0,000028** 0,000057 0,000036 **0,000227** 0,000429 0,000175

GP **0,000067 0,000057 0,00005 0,000831 0,000864 0,000186** 

**Vega hedging** BS 0,000362 0,000060 0,000052 0,001757 0,002015 0,000247

**Table 8.** Average hedge errors of dynamic hedging strategies relative to BS and GP models for call

Results in Table 9 show that the delta-gamma hedging performance improves for all moneyness classes of put options (except in-the-money put options) at longer maturities, regarding BS model at daily hedge frequency. However, the delta-gamma hedging performance improves for in-the money put options and at-the-money put options at medium maturities and for out-of-the money put options at longer maturities, regarding genetic programming model at daily hedge revision frequency. The best delta-gamma hedging performance is achieved, for BS model, using at-the-money long term put options

programming model is better than the BS model on in-the-money call options class. Further, the total of hedging errors relative to genetic programming model is about 21 percent slightly lower than 19 percent relative to BS model. Table 9 displays the average hedge errors for put options using BS and genetic programming models, at the 1-day and 7-days rebalancing frequencies. Values in bold correspond to the genetic programming hedging

**<0.98 Delta hedging** BS 0,013119 0,001279 0,000678 0,057546 0,010187 0,005607 GP **0,009669 0,001081 0,000662 0,053777 0,009585 0,005594 Gamma hedging** BS 0,000596 0,000732 0,000061 0,003026 0,007357 0,000429 GP 0,000892 0,002040 0,000075 0,003855 **0,001359 0,000153 Vega hedging** BS 0,000575 0,000050 0,000039 0,000525 0,000226 0,000099 GP **0,000473** 0,002035 0,004518 0,000617 0,004642 0,040071 **0.98-1.03 Delta hedging** BS 0,002508 0,000717 0,000730 0,019623 0,005416 0,002283 GP **0,002506 0,0007** 0,001725 0,020 **0,0054 0,0022 Gamma hedging** BS 0,000069 0,000018 0,000006 0,000329 0,000169 0,000027 GP 0,000377 0,000040 0,000029 0,000727 **0,000155** 0,000059  **Vega hedging** BS 0,000066 0,000373 0,003294 0,000527 0,023500 0,031375 GP 0,000281 **0,000013 0,000207** 0,001102 **0,000147 0,000134 >=1.03 Delta hedging** BS 0,000185 0,000906 0,001004 0,001602 0,006340 0,006401 GP **0,000184 0,000905 0,001 0,000840 0,005789 0,0064 Gamma hedging** BS 0,000323 0,000047 0,000028 0,001546 0,000386 0,000157

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

errors which are less than the BS ones.

**strategy** 

**S/K Hedging** 

options


**Table 7.** Performance of the best generated genetic programming volatility models for call and put options and their decoding formulas 0 12 , , *CP S X or X X KK K* 

Results in Table 8 show that the delta hedging performance improves for out-of-the money call options at longer maturities, for at-the-money call options at medium maturities and for in-the money call options at shorter maturities, regardless of the model used at daily hedge revision frequency. The best delta hedging performance is achieved using in-the-money short term call options for all MTM classes, regardless of the option model used.

The delta-gamma hedging performance improves for all moneyness classes of call options at longer maturities, regardless of the model used at daily hedge frequency (except in-themoney call options using the genetic programming model). The best delta-gamma hedging performance is achieved, for BS model, using at-the-money long term call options for all MTM classes. However, the best delta-gamma hedging performance is achieved, for genetic programming model, using in-the-money short term call options for all MTM classes.

The delta-vega hedging performance improves for out-of-the money and in-the-money call options at longer maturities and for at-the-money call options at shorter maturities, regarding BS model at daily hedge revision frequency. However, the delta-vega hedging performance improves for out-of-the money call options at shorter maturities, for at-themoney call options at medium maturities and for in-the money call options at longer maturities, regarding genetic programming model at daily hedge revision frequency. The best delta-vega hedging performance is achieved, for BS model, using out-of-the-money long term call options for all moneyness and time to maturity classes. However, the best delta-gamma hedging performance is achieved, for genetic programming model, using atthe-money medium term call options for all MTM classes.

The percentage of cases where the hedging error of the genetic programming model is less than the BS hedging error is around 59%. In particular, the performance of genetic programming model is better than the BS model on in-the-money call options class. Further, the total of hedging errors relative to genetic programming model is about 21 percent slightly lower than 19 percent relative to BS model. Table 9 displays the average hedge errors for put options using BS and genetic programming models, at the 1-day and 7-days rebalancing frequencies. Values in bold correspond to the genetic programming hedging errors which are less than the BS ones.

164 Genetic Programming – New Approaches and Successful Applications

the GP hedging errors which are less than the BS ones.

))\*X1)\*X1))\*X1))) 6 5

*GP*

options and their decoding formulas 0 12 , , *CP S X or X X*

the-money medium term call options for all MTM classes.

Call sqrt((X0/(multiply(X,((

Put ncdf (sin ((cos (sin

multiply(X1,plus(X1,X2

(minus (minus (-(cos (sin(X2))), ln(X0)), ln(X0))))-exp(X1))))

average hedging errors for call options using Black-Scholes (BS) and genetic programming (GP) models, at the 1-day and 7-days rebalancing frequencies. Values in bold correspond to

**Option LISP Expression Formula MSE** 

*C K S S K K*

 

**Table 7.** Performance of the best generated genetic programming volatility models for call and put

*KK K*

Results in Table 8 show that the delta hedging performance improves for out-of-the money call options at longer maturities, for at-the-money call options at medium maturities and for in-the money call options at shorter maturities, regardless of the model used at daily hedge revision frequency. The best delta hedging performance is achieved using in-the-money

The delta-gamma hedging performance improves for all moneyness classes of call options at longer maturities, regardless of the model used at daily hedge frequency (except in-themoney call options using the genetic programming model). The best delta-gamma hedging performance is achieved, for BS model, using at-the-money long term call options for all MTM classes. However, the best delta-gamma hedging performance is achieved, for genetic programming model, using in-the-money short term call options for all MTM classes.

The delta-vega hedging performance improves for out-of-the money and in-the-money call options at longer maturities and for at-the-money call options at shorter maturities, regarding BS model at daily hedge revision frequency. However, the delta-vega hedging performance improves for out-of-the money call options at shorter maturities, for at-themoney call options at medium maturities and for in-the money call options at longer maturities, regarding genetic programming model at daily hedge revision frequency. The best delta-vega hedging performance is achieved, for BS model, using out-of-the-money long term call options for all moneyness and time to maturity classes. However, the best delta-gamma hedging performance is achieved, for genetic programming model, using at-

The percentage of cases where the hedging error of the genetic programming model is less than the BS hedging error is around 59%. In particular, the performance of genetic

short term call options for all MTM classes, regardless of the option model used.

 

\*

sin cos sin exp 2 \* ln *GP*

cos sin

*K*

**Total** 

*S*

*<sup>P</sup> <sup>K</sup>*

0.001599

0.001539


**Table 8.** Average hedge errors of dynamic hedging strategies relative to BS and GP models for call options

Results in Table 9 show that the delta-gamma hedging performance improves for all moneyness classes of put options (except in-the-money put options) at longer maturities, regarding BS model at daily hedge frequency. However, the delta-gamma hedging performance improves for in-the money put options and at-the-money put options at medium maturities and for out-of-the money put options at longer maturities, regarding genetic programming model at daily hedge revision frequency. The best delta-gamma hedging performance is achieved, for BS model, using at-the-money long term put 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 classes.

Dynamic Hedging Using Generated Genetic Programming Implied Volatility Models 167

programming model is better than the BS model on at-the-money put options class. But, the total of hedging errors relative to genetic programming model is about 50 percent slightly

In summary, the genetic programming model is more accurate in all hedging strategies than the BS model, for in-the-money call options and at-the-money put options. 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 BS 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 BS delta-vega hedging error is 100% for in-the-money call options as well as for at-themoney put options. The percentage of cases where the delta-gamma hedging error of the genetic programming model is less than the BS delta-gamma hedging error is 100% for at-

Furthermore, results exhibit that as the rebalancing frequency changes from 1-day to 7-days revision, as the hedging errors increase and vice versa. The option value is a nonlinear function of the underlying, therefore, hedging is instantaneous and hedging with discrete rebalancing gives rise to error. Frequent rebalancing can be impractical due to transactions costs. In the literature, consequences of discrete time hedging have been considered usually in conjunction with the existence of transaction costs, that's why hedgers would like to trade at least frequently as possible. Pioneered by Leland [43], asymptotic approaches are used as well [44-46]. For most MTM classes, delta-gamma and delta-vega hedging strategies are shown to perform better in dynamic hedging when compared with delta hedging strategy, regardless of the model used. The delta-gamma strategy enables the performance of a discrete rebalanced hedging to be improved. The delta-vega strategy corrects partly for the

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

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

higher than 46 percent relative to BS model.

the-money put options.

risk of a randomly changing volatility.

**5. Conclusion** 

vega hedging.


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

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

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 programming model is better than the BS model on at-the-money put options class. But, the total of hedging errors relative to genetic programming model is about 50 percent slightly higher than 46 percent relative to BS model.

In summary, the genetic programming model is more accurate in all hedging strategies than the BS model, for in-the-money call options and at-the-money put options. 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 BS 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 BS delta-vega hedging error is 100% for in-the-money call options as well as for at-themoney put options. The percentage of cases where the delta-gamma hedging error of the genetic programming model is less than the BS delta-gamma hedging error is 100% for atthe-money put options.

Furthermore, results exhibit that as the rebalancing frequency changes from 1-day to 7-days revision, as the hedging errors increase and vice versa. The option value is a nonlinear function of the underlying, therefore, hedging is instantaneous and hedging with discrete rebalancing gives rise to error. Frequent rebalancing can be impractical due to transactions costs. In the literature, consequences of discrete time hedging have been considered usually in conjunction with the existence of transaction costs, that's why hedgers would like to trade at least frequently as possible. Pioneered by Leland [43], asymptotic approaches are used as well [44-46]. For most MTM classes, delta-gamma and delta-vega hedging strategies are shown to perform better in dynamic hedging when compared with delta hedging strategy, regardless of the model used. The delta-gamma strategy enables the performance of a discrete rebalanced hedging to be improved. The delta-vega strategy corrects partly for the risk of a randomly changing volatility.
