# A Comparison of Artificial Neural Networks and Bootstrap Aggregating Ensembles in a Modern Financial Derivative Pricing Framework

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. The Piterbarg Framework

#### 2.2. The Piterbarg PDE

#### 2.3. Solutions for European Call Options in the Piterbarg Framework

#### 2.4. Neural Network Modeling Approaches

#### 2.4.1. Artificial Neural Networks

#### 2.4.2. Ensemble Methods

## 3. Data Generation and Base Learner Configuration

#### 3.1. Training Data

#### 3.2. Testing Data

- If zero collateral trades were considered, it was assumed that the implied volatility surface was constructed using zero collateral trades;
- If fully collateralized trades were considered, it was assumed that the implied volatility surface was constructed using fully collateralized trades.

#### 3.3. Base Learner Configuration

## 4. Results

#### 4.1. Zero Collateral Numerical Results

#### 4.2. Fully Collateralized Numerical Results

#### 4.3. Numerical Results: Bagging Ensemble vs. Monte Carlo Simulation

- The stochastic process of the underlying asset follows a geometric Brownian motion;
- Constant interest rates were assumed;
- The underlying asset does not pay any dividends;
- Trades are European in nature;
- Trades are devoid of any friction costs;
- An Actual/365 day-count convention is used;
- The implied volatility parameters obtained from the volatility skew dated 9 April 2019 were assumed to be constructed using either zero collateral or a fully collateralized trades, depending on which type of trade was considered;
- We assumed vanilla options only.

- Given the relation in Equation (2), the price of a zero collateral European call option must be less than that of a fully collateralized European call option for any trade;
- For the bagging ensemble to be a viable alternative to a Monte Carlo simulation, it must be shown that the numerical accuracy of the bagging ensemble is within the three standard deviation error bounds of Monte Carlo simulation estimates for a reasonable number of simulations.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Parameter | Range |
---|---|

Spot price of the underlying asset (${S}_{t}$) ^{1} | (R10,000, R150,000) |

Strike price (K) | (60.00% to 140.00% of ${S}_{t}$) |

Time-to-maturity ($\tau $) | (7/365, 3.5) |

Repurchase agreement rate (${r}_{R}^{S}$) | (3.00%, 35.00%) |

Collateral rate (${r}_{C}$) | (60.00% to 80.00% of ${r}_{R}^{S}$ ) |

Funding rate (${r}_{F}$) | (120.00% to 140.00% of ${r}_{R}^{S}$) |

Implied volatility (${\sigma}_{P}$) | (2.00%, 80.00%) |

^{1}Prices are denoted in Rand (R), which is the domestic currency for South African markets.

Configuration | ANN | Bagging Ensemble |
---|---|---|

Training Sample Size | 1,500,000 | 1,500,000 |

Number of Members | 1 | 25 |

Sampling with Replacement | N/A | Yes |

Training Split | 85% | 85% |

Validation Split | 15% | 15% |

Parameter | Configuration |
---|---|

Number of hidden layers | 2 |

Neurons in first hidden layer | 512 |

Neurons in second hidden layer | 512 |

Neurons in output layer | 1 |

Hidden layer activation function | ReLU |

Output layer activation function | Softplus |

Optimizer | Adam |

Batch size | 64 |

Epochs | 20 |

Network Type | MSE | RMSE | ${\mathit{R}}^{2}$ |
---|---|---|---|

ANN | 1.67 × 10^{−6} | 0.001291 | 0.999948 |

Bagging Ensemble | 1.23 × 10^{−7} | 0.000350 | 0.999996 |

Metric | Analytical | ANN | Bagging Ensemble |
---|---|---|---|

Min price | R1.32 | R3.17 | R1.61 |

Max price | R22,219.81 | R22,221.17 | R22,216.90 |

Min absolute error | N/A | R0.00 | R0.00 |

Max absolute error | N/A | R148.30 | R47.62 |

Network Type | MSE | RMSE | ${\mathit{R}}^{2}$ |
---|---|---|---|

ANN | 1.63 × 10^{−6} | 0.001276 | 0.999952 |

Bagging Ensemble | 1.99 × 10^{−7} | 0.000446 | 0.999994 |

Metric | Analytical | ANN | Bagging Ensemble |
---|---|---|---|

Min price | R1.32 | R6.65 | R3.40 |

Max price | R23,553.12 | R23,479.49 | R23,543.68 |

Min absolute error | N/A | R0.02 | R0.00 |

Max absolute error | N/A | R194.81 | R63.68 |

Parameter | Trade 1 | Trade 2 | Trade 3 |
---|---|---|---|

Trade type | Call | Call | Call |

Spot price of the underlying asset $\left({S}_{t}\right)$ | R51,564.09 | R51,564.09 | R51,564.09 |

Strike $\left(K\right)$ | R42,716.80 | R53,396.00 | R64,075.20 |

Time to maturity | 345 days | 345 days | 345 days |

Implied volatility $\left({\sigma}_{P}\right)$ | 22.32% | 18.50% | 15.17% |

Repurchase agreement rate $\left({r}_{R}^{S}\right)$ | 7.00% | 7.00% | 7.00% |

Funding rate $\left({r}_{F}\right)$ | 8.50% | 8.50% | 8.50% |

Parameter | Trade 1 | Trade 2 | Trade 3 |
---|---|---|---|

Trade type | Call | Call | Call |

Spot price of the underlying asset $\left({S}_{t}\right)$ | R51,564.09 | R51,564.09 | R51,564.09 |

Strike $\left(K\right)$ | R42,716.80 | R53,396.00 | R64,075.20 |

Time to maturity | 345 days | 345 days | 345 days |

Implied volatility $\left({\sigma}_{P}\right)$ | 22.32% | 18.50% | 15.17% |

Repurchase agreement rate $\left({r}_{R}^{S}\right)$ | 7.00% | 7.00% | 7.00% |

Collateral rate $\left({r}_{C}\right)$ | 5.50% | 5.50% | 5.50% |

Option | Monte Carlo | Bagging Ensemble |
---|---|---|

Zero Collateral: Trade 2 | 0.035036 | 2.421635 |

Fully Collateralized: Trade 2 | 0.060940 | 1.670318 |

Option | Monte Carlo | Bagging Ensemble |
---|---|---|

Zero Collateral | 353.856228 | 18.082844 |

Fully Collateralized | 356.446953 | 17.612219 |

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**MDPI and ACS Style**

du Plooy, R.; Venter, P.J.
A Comparison of Artificial Neural Networks and Bootstrap Aggregating Ensembles in a Modern Financial Derivative Pricing Framework. *J. Risk Financial Manag.* **2021**, *14*, 254.
https://doi.org/10.3390/jrfm14060254

**AMA Style**

du Plooy R, Venter PJ.
A Comparison of Artificial Neural Networks and Bootstrap Aggregating Ensembles in a Modern Financial Derivative Pricing Framework. *Journal of Risk and Financial Management*. 2021; 14(6):254.
https://doi.org/10.3390/jrfm14060254

**Chicago/Turabian Style**

du Plooy, Ryno, and Pierre J. Venter.
2021. "A Comparison of Artificial Neural Networks and Bootstrap Aggregating Ensembles in a Modern Financial Derivative Pricing Framework" *Journal of Risk and Financial Management* 14, no. 6: 254.
https://doi.org/10.3390/jrfm14060254