# Calibrating FBSDEs Driven Models in Finance via NNs

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## Abstract

**:**

## 1. Introduction

## 2. Backward Stochastic Differential Equations

- $\zeta $$\in {L}_{T}^{2}\left({\mathbb{R}}^{d}\right)$, where ${L}_{T}^{2}\left({\mathbb{R}}^{d}\right)$ is the space of ${\mathbb{F}}_{t}$-measurable random variables $\xi $ s.t. $\mathbb{E}\left(\right|\xi {|}^{2})<\infty $
- $f(t,0,0)\in {H}_{T}^{2}\left({\mathbb{R}}^{d}\right)\phantom{\rule{1.em}{0ex}}\forall t\in [0,T]$, where ${H}_{T}^{2}\left({R}^{d}\right)$ is the space of predictable process Y s.t. ${\left|\right|Y\left|\right|}^{2}=\mathbb{E}({\int}_{0}^{T}|{Y}_{t}{|}^{2}dt)<\infty )$
- f is uniformly Lipschitz: there exists L s.t.$$|f(t,{y}_{1},{z}_{1})-f(t,{y}_{2},{z}_{2})|\le L(|{y}_{1}-{y}_{2}|+|{z}_{1}-{z}_{2}\left|\right)$$

**Theorem 1.**

#### Forward–Backward Stochastic Differential Equation

**Definition 1.**

## 3. Neural Networks

**Theorem 2.**

- input layer composed by n units, where n is the number of network entrances (n is the dimension of the features’ vector);
- hidden layer with $L-1,\phantom{\rule{0.166667em}{0ex}}L\ge 2$ neurons and outputs connected with the inputs of the following layer;
- output layer composed by $K\ge 1$ neurons that describe the network outputs;
- A set of directed and weighted edges (called w) that represent all possible connections among layers.

## 4. Solution of FBSDEs via NNs

## 5. Model Calibration via NN

#### 5.1. Calibration Objective

**Proposition 1**

#### 5.2. Deep Calibration Procedure

Algorithm 1: Deep calibration algorithm |

## 6. Empirical Results

#### 6.1. Black–Scholes–Barenblatt Equation

**Lemma 1.**

**Proof.**

#### 6.2. Heston Calibration via Neural Network

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Notation

W | Standard Brownian motion |

${L}_{n}^{p}\left([0,T]\right)$ | $p\in [1,\infty )$, set of ${\mathbb{R}}^{n}$-valued progressively measurable |

processes Y s.t. $\mathbb{E}{\int}_{0}^{T}{\left|{Y}_{s}\right|}^{p}ds<\infty $ | |

${S}^{2}\left([0,T]\right)$ | space of continuous $\mathbb{F}$-semimartingale Y s.t. |

$\mathbb{E}\left(su{p}_{t\in [0,T]}\right|{Y}_{t}{|}^{2})<\infty $ | |

${L}_{ad}^{2}([0,T]\times \Omega ;{\mathbb{R}}^{k})$ | space of $\mathbb{F}$-adapted process in ${L}^{2}([0,t]\times \Omega ;{\mathbb{R}}^{k})$ |

${L}_{T}^{2}\left({\mathbb{R}}^{d}\right)$ | space of ${\mathbb{F}}_{t}$-measurable random variables $\xi $ s.t. $\mathbb{E}\left(\right|\xi {|}^{2})<\infty $ |

${H}_{T}^{2}\left({R}^{d}\right)$ | space of predictable process Y s.t. ${\left|\right|Y\left|\right|}^{2}=\mathbb{E}({\int}_{0}^{T}|{Y}_{t}{|}^{2}dt)<\infty )$ |

## Appendix B. Definition

**Definition A1**

**Definition A2**

**Definition A3**

**Definition A4**

**Definition A5**

**Definition A6**

- ${\mathbb{E}}^{\mathbb{P}}\left[\right|{X}_{t}\left|\right]<\infty ,\forall t\ge 0$;
- ${\mathbb{E}}^{\mathbb{P}}\left[{X}_{t+s}\right|{\mathcal{F}}_{t}]={X}_{t}$, $\forall t,s\ge 0$

**Definition A7**

**Definition A8**

- 1.
- X has independent increments, i.e., ${X}_{t}-{X}_{s}$ is independent of ${\mathcal{F}}_{s}$, $0\le s<t<\infty $
- 2.
- $X\left(0\right)=0$ a.s;
- 3.
- X has stationary increments, i.e., $\forall t,h\phantom{\rule{1.em}{0ex}}{X}_{t+h}-{X}_{t}$ has a distribution that is independent of t;
- 4.
- it is continuous in probability, i.e., $\forall t$, $\forall \u03f5\phantom{\rule{1.em}{0ex}}{lim}_{s\to t}\mathbb{P}\left(\right|{X}_{t}-{X}_{s}|>\u03f5)=0$;
- 5.
- ${X}_{t}$ has cadlag trajectories, i.e., right continuous and with left limit defined everywhere.

**Definition A9**

**Definition A10**

**Definition A11**

## Note

1 | https://quantlib-python-docs.readthedocs.io/en/latest/, (accessed on 13 September 2022). |

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**Figure 1.**Example of Deep Neural Network Schema. In green the neurons of input layer, in blue the neurons of hidden layers and in red the neurons of output layer.

**Figure 2.**Evaluations of the learned solution $Yt=u(t,{X}_{t})$ at representative realizations of the underlying high-dimensional process ${X}_{t}$.

**Figure 5.**Train loss and validation loss of training. (

**a**) Loss of FCNN; (

**b**) Loss of Multi layer LSTM.

**Table 1.**Marginal priors of the model parameters $\mu $ for synthetically generating $\mathcal{D}$. The interest rate r is fixed.

Parameter | Marginal |
---|---|

$\eta $ | $\mathcal{U}[0,5]$ |

$\rho $ | $\mathcal{U}[-1,0]$ |

$\lambda $ | $\mathcal{U}[0,10]$ |

$\overline{\upsilon}$ | $\mathcal{U}[0,1]$ |

${\upsilon}_{0}$ | $\mathcal{U}[0,1]$ |

Parameter | Theoretical | FCNN | LSTM |
---|---|---|---|

$\eta $ | $0.3877$ | $0.3889$ | $0.4124$ |

$\rho $ | $-0.7165$ | $-0.6871$ | $-0.7345$ |

$\lambda $ | $1.3253$ | $1.3421$ | $1.105$ |

$\overline{\nu}$ | $0.0354$ | $0.0312$ | $0.0401$ |

${\nu}_{0}$ | $0.0174$ | $0.0169$ | $0.0171$ |

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**MDPI and ACS Style**

Di Persio, L.; Lavagnoli, E.; Patacca, M.
Calibrating FBSDEs Driven Models in Finance via NNs. *Risks* **2022**, *10*, 227.
https://doi.org/10.3390/risks10120227

**AMA Style**

Di Persio L, Lavagnoli E, Patacca M.
Calibrating FBSDEs Driven Models in Finance via NNs. *Risks*. 2022; 10(12):227.
https://doi.org/10.3390/risks10120227

**Chicago/Turabian Style**

Di Persio, Luca, Emanuele Lavagnoli, and Marco Patacca.
2022. "Calibrating FBSDEs Driven Models in Finance via NNs" *Risks* 10, no. 12: 227.
https://doi.org/10.3390/risks10120227