# Gaussian Process Regression for Swaption Cube Construction under No-Arbitrage Constraints

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## Abstract

**:**

## 1. Introduction

- The constrained GP regression approach has been extended to an input space of dimension 3. Numerical implementation of the 3D ‘constrained’ kriging problem is much more involved than in the 2D case considered in Chataigner et al. (2021). We show that considering an anisotropic stationary kernel allows us to benefit from the tensorization of the GP covariance matrix, and it significantly reduces numerical complexity.
- Compared to the equity price surface, the no-arbitrage constraints on swaption prices are no more directional. Indeed, the in-plane triangular inequality involves prices of swaptions with different maturities and tenors. This constraint is by far more complex to handle than the no calendar spreads arbitrage constraint in the equity option case. It can be viewed as a weaker form of the no calendar spreads arbitrage condition, and it requires an infinite number of checks in order to be imposed everywhere on the domain. In our methodology, we consider a weaker version of the in-plane inequality constraint, and we empirically check in Section 4 that the weaker form is enough to ensure no arbitrage almost everywhere.
- Computation of the GP prior hyperparameters by Maximum Likelihood Estimation (MLE) is much longer in the swaption case than in the equity case (see Section 3.3). Search for the maximal likelihood has been considerably improved by explicitly computing the Jacobian of the log-likelihood function.

## 2. Absence of Static Arbitrages for the Swaption Cube

**Proposition**

**1.**

- (i)
- For all $T,t,K\ge 0$, ${S}_{w}(T,T+t,K)\ge 0$;
- (ii)
- $K\to {S}_{w}(T,T+t,K)$ is a convex, decreasing (increasing) function;
- (iii)
- The in-plane triangular condition holds:For all $T,t,h,K\ge 0$, ${S}_{w}(T,T+t,K)+{S}_{w}(T+t,T+t+h,K)\ge {S}_{w}(T,T+t+h,K)$.

## 3. GP Regression to Construct an Arbitrage Free Swaption Cube

#### 3.1. Classical 3-Dimensional Gaussian Regression

#### 3.2. Imposing No Arbitrage Constraints to GP Regression

**Proposition**

**2.**

- (i)
- The finite-dimensional process ${S}_{w}^{N}$ uniformly converges to ${S}_{w}$ on $\mathcal{D}$ as ${N}_{T}\to \infty $, ${N}_{t}\to \infty $ and ${N}_{K}\to \infty $, almost surely;
- (ii)
- ${S}_{w}^{N}$ is a decreasing function of K on $\mathcal{D}$ if and only if ${\xi}_{i,j,k}\ge {\xi}_{i,j,k+1}$;
- (iii)
- ${S}_{w}^{N}$ is a convex function of K on $\mathcal{D}$ if and only if ${\xi}_{i,j,k+2}-{\xi}_{i,j,k+1}\ge {\xi}_{i,j,k+1}-{\xi}_{i,j,k}$.

**Remark**

**1.**

#### 3.3. Maximum Likelihood Estimation

**Remark**

**2.**

#### 3.4. The Most Probable Response Cube and Measurement Noises

#### 3.5. Sampling

## 4. Empirical Results

#### 4.1. Data

#### Absence of Arbitrages in the Data Set

#### 4.2. SABR Model Benchmark

#### 4.2.1. SABR Model

#### 4.2.2. Model Calibration and Pricing

#### 4.2.3. Unobserved Parameter Estimation

#### 4.3. Absence of Arbitrage, Calibration and Prediction Performance

#### 4.3.1. Implementation of the GP Regression Approach

#### 4.3.2. Calibration and Testing Methodology

#### 4.3.3. Absence of Arbitrage in the Predicted Swaption Cube

#### 4.3.4. Calibration Performance

#### 4.3.5. Prediction Performance

#### 4.3.6. SABR Model Calibrated on the GP Predicted Cube

#### 4.4. Uncertainty Quantification

## 5. Conclusions

## 6. Further Research

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Calibration of the SABR Model

- Calibration of the parameters of the SABR model for the pairs $(5,5)$, $(5,10)$, $(10,5)$ and $(10,10)$.
- For all the other pairs, take as a starting value for the parameters $\alpha $, $\rho $ and $\sigma $ the value of the calibrated parameters of the nearest neighbor among the pairs $(5,5)$, $(5,10)$, $(10,5)$ and $(10,10)$.
- Then, for the pairs $(T,t),$ calibrate:
- (a)
- $\alpha $, $\rho $ and $\sigma $ if there is at least three observations.
- (b)
- $\alpha $ only if there is one or two observations of payer swaptions OTM (or receiver swaptions OTM).
- (c)
- $\sigma $ only if the only observation is the price of a swaption ATM.
- (d)
- $\rho $ and $\alpha $ if there is an observation of a payer swaption OTM and an observation of a receiver swaption OTM.
- (e)
- $\alpha $ and $\sigma $ if we observed one swaption ATM and one payer or receiver swaption OTM.

## Notes

1 | This choice has been made for the sake of clarity. It is also possible to consider a grid with heterogeneous steps. |

2 | For the observed pairs $(T,t)$ that do not respect the convexity constraint, the violation errors range from 0.4% to 4% of the average swaption price involved in the convexity constraint. |

3 | A 1-dimensional Matern 5/2 kernel is given as $c(\mathbf{x},{\mathbf{x}}^{\prime})={\sigma}^{2}(1+\frac{\sqrt{5}|\mathbf{x}-{\mathbf{x}}^{\prime}|}{\theta}+\frac{5(\mathbf{x}-{\mathbf{x}}^{\prime})}{3{\theta}^{2}})\mathrm{exp}(-\frac{\sqrt{5}|\mathbf{x}-{\mathbf{x}}^{\prime}|}{\theta})$. |

4 | To give an idea, for $p=90\%$, the computation time for both methods takes less than 30 s on a laptop. |

## References

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**Figure 1.**Constrained GP (

**left**) and unconstrained GP (

**right**) price curves for $p=10\%$ (

**top**), $p=50\%$ (

**middle**) and $p=70\%$ (

**bottom**).

**Figure 2.**Constrained GP (

**left**) and unconstrained GP (

**right**) price curves with uncertainties for $p=10\%$ (

**top**), $p=40\%$ (

**middle**) and $p=80\%$ (

**bottom**).

**Table 1.**Percent of pairs (T, t) respecting the absence of arbitrage in strike constraints defined in Proposition 1.

Pairs (T, t) Respecting the AOA Constraints Input Data | Monotonicity Constraint in K | Convexity Constraint in K |
---|---|---|

Payer | 100% | 67.86% |

Receiver | 100% | 100% |

**Table 2.**Respect of the no-arbitrage constraints for the predicted OTM payer swaptions priced by constrained and unconstrained GP regression, for increasing sizes of training set.

Respect of the AOA Constraints GP Payer | Decrease in Strike (Constrained) | Convexity in Strike (Constrained) | In-Plane Triangular Inequality (Constrained) | Decrease in Strike (Unconstrained) | Convexity in Strike (Unconstrained) | In-Plane Triangular Inequality (Unconstrained) |
---|---|---|---|---|---|---|

p = 15% | 100% | 100% | 100% | 87.84% | 8.18% | 99.58% |

p = 30% | 100% | 100% | 100% | 93.79% | 13.22% | 99.97% |

p = 50% | 100% | 100% | 100% | 94.37% | 10.24% | 99.98% |

p = 70% | 100% | 100% | 100% | 99.55% | 5.98% | 99.97% |

p = 90% | 100% | 100% | 100% | 92.17% | 1.51% | 99.92% |

**Table 3.**Respect of the no-arbitrage constraints for the predicted OTM payer swaptions priced by the chosen SABR model with barycentric mean parameter interpolation, for increasing sizes of training set.

Respect of the AOA Constraints SABR Payer | Decrease in Strike | Convexity in Strike | In-Plane Triangular Inequality |
---|---|---|---|

p = 15% | 99.00% | 99.70% | 98.09% |

p = 30% | 99.49% | 99.82% | 97.96% |

p = 50% | 99.82% | 99.91% | 99.186% |

p = 70% | 99.96% | 99.99% | 99.90% |

p = 90% | 100% | 100% | 100% |

**Table 4.**Respect of the no-arbitrage constraints for the predicted OTM receiver swaptions priced by constrained and unconstrained GP regression, for increasing sizes of training set.

Respect of the AOA Constraints GP Payer | Increase in Strike (Constrained) | Convexity in Strike (Constrained) | In-Plane Triangular Inequality (Constrained) | Increase in Strike (Unconstrained) | Convexity in Strike (Unconstrained) | In-Plane Triangular Inequality (Unconstrained) |
---|---|---|---|---|---|---|

p = 15% | 100% | 100% | 99.51% | 31.46% | 82.67% | 97.10% |

p = 30% | 100% | 100% | 99.08% | 49.29% | 82.08% | 97.37% |

p = 50% | 100% | 100% | 99.53% | 56.49% | 86.41% | 98.00% |

p = 70% | 100% | 100% | 99.78% | 60.03% | 93.69% | 97.95% |

p = 90% | 100% | 100% | 99.94% | 56.36% | 91.74% | 97.87% |

**Table 5.**Respect of the no-arbitrage constraints for the predicted OTM receiver swaptions priced by the SABR model with barycentric mean parameter interpolation, for increasing sizes of training set.

Respect of the AOA Constraints SABR Receiver | Increase in Strike | Convexity in Strike | In-Plane Triangular Inequality |
---|---|---|---|

p = 15% | 99.77% | 98.73% | 97.67% |

p = 30% | 99.82% | 99.22% | 97.62% |

p = 50% | 99.94% | 99.67% | 99.05% |

p = 70% | 100% | 99.94% | 99.88% |

p = 90% | 100% | 100% | 100% |

**Table 6.**Mean relative calibration error and calibration RMSE of the constrained GP regression and SABR model with barycentric mean parameter interpolation, for increasing sizes of training set.

Average Calibration Error | Constrained GP Mean Relative Error | SABR Mean Relative Error | Constrained GP RMSE | SABR RMSE |
---|---|---|---|---|

p = 15% | 1.77% | 1.61% | 2.68 | 12.65 |

p = 30% | 2.29% | 1.47% | 2.78 | 10.56 |

p = 50% | 2.67% | 2.21% | 3.43 | 11.67 |

p = 70% | 3.02% | 2.87% | 3.88 | 13.85 |

p = 90% | 3.25% | 3.22% | 4.11 | 14.59 |

**Table 7.**RMSE of the constrained GP regression and SABR model with barycentric mean parameter interpolation, for increasing sizes of training set.

Average Prediction Error | Constrained GP RMSE | SABR RMSE |
---|---|---|

p = 15% | 29.01 | 172.42 |

p = 30% | 27.83 | 58.55 |

p = 50% | 9.98 | 33.25 |

p = 70% | 6.30 | 20.57 |

p = 90% | 4.71 | 16.54 |

**Table 8.**Average gain in RMSE when calibrating the SABR model with the output of the GP regression model.

Percentage of Data Used as Input | p = 15% | p = 30% | p = 50% | p = 70% | p = 90% |
---|---|---|---|---|---|

RMSE relative gain | 63% | 68% | 59% | 35% | 19% |

**Table 9.**Respect of the no-arbitrage constraints for the predicted OTM payer swaptions priced by the SABR model calibrated on observations or on the output of the GP model, for increasing sizes of training set.

Respect of the AOA Constraints Payer | Decrease in Strike (SABR) | Convexity in Strike (SABR) | In-Plane Triangular Inequality (SABR) | Decrease in Strike (GP-SABR) | Convexity in Strike (GP-SABR) | In-Plane Triangular Inequality (GP-SABR) |
---|---|---|---|---|---|---|

p = 15% | 99.45% | 99.80% | 98.51% | 99.90% | 99.98% | 100% |

p = 30% | 99.68% | 99.98% | 96.63% | 100% | 100% | 100% |

p = 50% | 99.75% | 99.90% | 99.37% | 100% | 100% | 100% |

p = 70% | 100% | 100% | 100% | 100% | 100% | 100% |

p = 90% | 100% | 100% | 100% | 100% | 100% | 100% |

**Table 10.**Respect of the no-arbitrage constraints for the predicted OTM receiver swaptions priced by the SABR model calibrated on observations or on the output of the GP model, for increasing sizes of training set.

Respect of the AOA Constraints Receiver | Increase in Strike (SABR) | Convexity in Strike (SABR) | In-Plane Triangular Inequality (SABR) | Increase in Strike (GP-SABR) | Convexity in Strike (GP-SABR) | In-Plane Triangular Inequality (GP-SABR) |
---|---|---|---|---|---|---|

p = 15% | 99.87% | 99.20% | 98.22% | 99.99% | 99.53% | 99.52% |

p = 30% | 99.88% | 99.40% | 96.43% | 100% | 99.76% | 99.97% |

p = 50% | 99.88% | 99.70% | 99.10% | 100% | 99.80% | 100% |

p = 70% | 100% | 100% | 100% | 100% | 99.80% | 100% |

p = 90% | 100% | 100% | 100% | 100% | 99.80% | 100% |

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**MDPI and ACS Style**

Cousin, A.; Deleplace, A.; Misko, A.
Gaussian Process Regression for Swaption Cube Construction under No-Arbitrage Constraints. *Risks* **2022**, *10*, 232.
https://doi.org/10.3390/risks10120232

**AMA Style**

Cousin A, Deleplace A, Misko A.
Gaussian Process Regression for Swaption Cube Construction under No-Arbitrage Constraints. *Risks*. 2022; 10(12):232.
https://doi.org/10.3390/risks10120232

**Chicago/Turabian Style**

Cousin, Areski, Adrien Deleplace, and Adrien Misko.
2022. "Gaussian Process Regression for Swaption Cube Construction under No-Arbitrage Constraints" *Risks* 10, no. 12: 232.
https://doi.org/10.3390/risks10120232