BP neural network model is a three-layer feed forward network composed of input layer, hidden layer and output layer. Each layer contains several disconnected neuron nodes, and the adjacent nodes are connected according to a certain weight. The direction of information transmission is from input layer to hidden layer to output layer. There is transfer matrix between input layer and hidden layer, and transfer matrix exists between hidden layer and output layer. If the difference between the actual output and the expected output can not meet the required error, the error value is fed back layer by layer along the network path, and the connection weights and thresholds of each layer are corrected.
Artificial bee colony (ABC) algorithm is inspired by the intelligent behavior of bees. In ABC algorithm, the location of food represents a possible solution of the optimization problem, while the amount of nectar represents the quality or fitness of the corresponding solution.
Firstly, the algorithm generates the initial population randomly, and then sets the limit and the maximum number of cycles. After initialization, the bees begin to search circularly. They use greedy mechanism to search the neighborhood of the old solution. If the fitness of the new solution is greater than that of the old solution, the bee will forget the old solution and remember the new one. Then the possible values (Pi) of these new solutions are calculated, and the onlookers begin to use the greedy mechanism to search for new solutions and solutions remembered by bees near the possible values (Pi). If the latest solutions obtained can no longer be updated, the Scout will abandon these solutions and replace them with new ones. This cycle reaches the maximum number of cycles. There is only one scout per cycle.
The principle of Monte Carlo simulation option pricing can be understood in this way. Firstly, according to the given price movement process of the underlying asset, the price change of the underlying asset is simulated. When the number of simulations reaches a certain number, the average value is taken to obtain the expected value. According to the law of large numbers, the Monte Carlo simulation results finally meet the convergence. Compared with the BS model, this method has the following advantages: First, it is flexible, easy to implement and improve; Second, the error and convergence speed of the simulation estimation have nothing to do with the dimensionality of the problem, and it can better solve the multi-asset option pricing and The path depends on the issue of option pricing. However, in order to obtain high accuracy, thousands of simulations are usually required, so the Monte Carlo method usually takes more time than the BS model.
The correlation among three kinds of volatilities can be seen in the Table below.
| IV | Realized Volatility | VIX | |
|---|---|---|---|
| IV | 1 | 0.5028 | 0.7220 |
| Realized Volatility | 0.5028 | 1 | 0.6988 |
| VIX | 0.7220 | 0.6899 | 1 |
The figure below shows the comparisons between predicted and sample true values based on ABC-BP neural network model.This result suggests that the model have predictability.
The figure below shows the comparisons based on BP and ABC-BP neural network model optimal and true value. We find that the ABC-BP neural network model is closer to the volatility index, which indicates that the volatility prediction under this method is more optimized.
The figure below shows comparison of MSE based on BP neural network model and ABC-BP neural network model.we can conclude that MSE decreases with the increase of iterations, and the MSE of ABC-BP neural network model is obviously smaller than that of BP neural network model under the optimal fitting condition. Under the best fit, the MSE of the ABC-BP neural network model is significantly smaller than the BP neural network model. So, the validity of model prediction is illustrated.
The error comparison under the three models can be seen in the Table below.In this table, we show the mean square error MSE, mean absolute error MAE, mean deviation error MBE, and the calculation results of the BS model of the BP and ABC-BP neural network models from January 2017 to May 2018. It can be seen from Table 3 that the MSE, MAE and MBE of the ABC-BP neural network model are the smallest among the three models. Therefore, the overall performance of the ABC-BP neural network model proposed in this paper is better than the BP neural network model and the BS model.
| MSE | MAE | MBE | |
|---|---|---|---|
| BS | 0.001 | 0.024 | 0.001 |
| BP | 0.000** | 0.010** | -0.000** |
| ABC-BP | 0.000*** | 0.010** | 0.000*** |
Finally, the volatility data is introduced into the montecarlo simulation to calculate the predicted option price and yield (including delta and vega). The prediction results are shown in Table below.
| Maximum | Minimum | Mean | Median | Std.Dev | |
|---|---|---|---|---|---|
| Underlying price | 2872.820 | 2257.830 | 2530.086 | 2648.980 | 234.174 |
| Call | 1611.265 | 108.0463 | 1088.706 | 1334.268 | 395.1518 |
| Put | 1121.573 | 110.7903 | 633.1015 | 593.3589 | 292.2569 |
| delta | 0.970963 | 0.536306 | 0.757982 | 0.812611 | 0.126825 |
| vega | 1234.165 | 171.0937 | 864.9403 | 876.3806 | 199.771 |
In this section, we apply the model's predictable results to major options trading strategies: long, butterfly, and calendar spreads to verify the validity of the model's predictions. In addition, we apply our model to fluctuation momentum and mean recovery. Momentum refers to short-term volatility behavior, while mean reversion refers to long-term behavior. We chose three-day volatility momentum cycles and used 10% and 90% volatility movements as benchmarks for mean reregression.
the figure below shows the comparison of volatility based on different four models (Robustness Test).We find that the decline in returns is due to the increasing value of volatility prediction error, especially for the GARCH model, but not for the ABC-BP neural network model. Cash options are still the best performers. These results show that ABC-BP neural network model has better performance than GARCH model.
In this paper, we examine the IV predictability of the artificial bee colony improved model (ABC-BP neural network model) and apply the model to three popular options trading strategies. We documented two main findings.
- the BP neural network model is inherently slow in convergence. The model is easy to fall into local optimum and easy to overfit. The accuracy of implied volatility is low while the training time is prolonged. The experimental results show that the ABC-BP neural network model is superior to the BP neural network model in terms of speed and predictability, and also superior to the traditional GARCH model.
- we demonstrate that the ABC-BP neural network model is applicable to options trading strategies, such as cross-seat, butterfly, and calendar spreads. The performance of this model is better than that of the traditional GARCH model. We believe that this conclusion is useful for option traders to select trading strategies and specific trading products.