# Sound Deposit Insurance Pricing Using a Machine Learning Approach

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## Abstract

**:**

## 1. Introduction

## 2. Sound Deposit Insurance

- The initial capital at time 0 i.e., $exp\left(-rT\right)b$;
- The global loss, $\mathcal{L}$;
- The insurance policy, $-I$;
- The premium payed for the insurance policies, at time T, $exp\left(rT\right)\pi \left(I\right)$.

**Theorem**

**1.**

- If $exp\left(rT\right)\le l$ then $I=0$
- If $exp\left(rT\right)>l$$$f\circ L\left(x\right)=\left\{\begin{array}{cc}0\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}exp\left(rT\right){S}_{0}-l\le x\hfill \\ exp\left(rT\right){S}_{0}-x-l\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}exp\left(rT\right){S}_{0}-u\le x\le exp\left(rT\right){S}_{0}-l\hfill \\ u-l\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}x\le exp\left(rT\right){S}_{0}-u\hfill \end{array}\right..$$$$f\circ L\left(x\right)={\left(x-exp\left(rT\right){S}_{0}+l\right)}_{+}-{\left(x-exp\left(rT\right){S}_{0}+u\right)}_{+}+u-l.$$

**Corollary**

**1.**

#### 2.1. Black-Scholes Model

#### 2.2. Implied Volatility

#### 2.3. Static Arbitrage

**Definition**

**1.**

**Theorem**

**2.**

- 1.
- ${\partial}_{\tau}C>0$
- 2.
- $\underset{k\to \infty}{lim}C(\tau ,\phantom{\rule{0.166667em}{0ex}}k)=0$
- 3.
- $\underset{k\to -\infty}{lim}C(\tau ,\phantom{\rule{0.166667em}{0ex}}k)+k=a\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}a\in R$
- 4.
- $C(\tau ,\phantom{\rule{0.166667em}{0ex}}k)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathit{is}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathit{convex}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathit{in}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}k$
- 5.
- $C(\tau ,\phantom{\rule{0.166667em}{0ex}}k)\ge 0$

**Theorem**

**3.**

- 1.
- ${\partial}_{\tau}{w}_{imp}>0;$
- 2.
- $\underset{k\to \infty}{lim}\phantom{\rule{0.166667em}{0ex}}{d}_{1}\left(k\right)=-\infty ;$
- 3.
- $\tau {\sigma}_{imp}\ge 0;$
- 4.
- ${\left(1-\frac{x}{2{w}_{imp}}{\partial}_{x}\left({w}_{imp}\right)\right)}^{2}-\frac{1}{4}\left(\frac{1}{{w}_{imp}}-\frac{1}{4}\right){\left({\partial}_{x}\left({w}_{imp}\right)\right)}^{2}+\frac{1}{2}{\partial}_{xx}\left({w}_{imp}\right)\ge 0.$

**Definition**

**2.**

- 1.
- It is free of calendar spread arbitrage;
- 2.
- The volatility slice is free of butterfly arbitrage for any fixed time to maturity.

#### 2.4. Parameterization of the Implied Volatility

#### 2.5. Machine Learning Approach

## 3. The Quadratic Parametrization

#### 3.1. The Raw Quadratic Model

#### 3.2. Elimination of Static Arbitrage

**Definition**

**3.**

**Proposition**

**1.**

- 1.
- ${\psi}_{\tau}\left({\partial}_{\tau}{\psi}_{\tau}\right)>0$
- 2.
- ${\partial}_{\tau}\left[ln\left(\frac{{v}_{\tau}}{{v}_{\tau}-{\mu}_{\tau}}\right)\right]>\frac{\left({v}_{\tau}-{\mu}_{\tau}\right)}{4{v}_{\tau}^{3}}$
- 3.
- ${\partial}_{\tau}\left[ln{\psi}_{\tau}\right]<\frac{2{v}_{\tau}}{{v}_{\tau}-{\mu}_{\tau}}-\frac{1}{{v}_{\tau}}$

**Proof.**

**Proposition**

**2.**

- 1.
- ${\theta}_{22}-{\theta}_{21}>0;$
- 2.
- ${\theta}_{22}{\theta}_{01}+{\theta}_{21}{\theta}_{02}<\frac{{\theta}_{12}{\theta}_{11}}{2}.$

**Proof.**

**Proposition**

**3.**

- 1.
- ${\theta}_{1}^{2}-4{\theta}_{0}{\theta}_{2}+{\theta}_{2}<0;$
- 2.
- $\frac{1}{4}<{\theta}_{0}<1.$

**Proof.**

**Theorem**

**4.**

- 1.
- ${\psi}_{\tau}\left({\partial}_{\tau}{\psi}_{\tau}\right)>0$
- 2.
- ${\partial}_{\tau}\left[ln\left(\frac{{v}_{\tau}}{{v}_{\tau}-{\mu}_{\tau}}\right)\right]>\frac{\left({v}_{\tau}-{\mu}_{\tau}\right)}{4{v}_{\tau}^{3}}$
- 3.
- ${\partial}_{\tau}\left[ln{\psi}_{\tau}\right]<\frac{2{v}_{\tau}}{{v}_{\tau}-{\mu}_{\tau}}-\frac{1}{{v}_{\tau}}$
- 4.
- $0<\tau {v}_{\tau}<\frac{1}{4}$
- 5.
- $\frac{{v}_{\tau}{\psi}_{\tau}^{2}}{{v}_{\tau}-{\mu}_{\tau}}\left(4\tau {\mu}_{\tau}-1\right)>0$

## 4. Numerical Implementation

#### 4.1. The Cost Function

#### 4.2. The Algorithm, Step by Step

- Start by a volatility data $({x}^{\left(i\right)},{w}^{\left(i\right)})$ for any fixed time to maturity.
- Using the training set data and the conditions in Propositions 2 and 3, estimate parameters by minimizing the cost function for a fixed value of $\lambda $ (For the first implementation let $\lambda =0$).
- Using the estimated parameters, compute training error and cross-validation error for different values of m.
- Plot learning curve which is the training error and the cross-validation error versus m.
- (a) If the learning curve shows no drawback of overfitting and underfitting, plot Durrleman’s function based on the estimated parameters.(b) Otherwise, plot the validation curve which is the cross-validation error versus the regularization parameter $\lambda $, and choose the value of $\lambda $ which minimizes the cross-validation error, then move on to step 2.

#### 4.3. Ruling Out Calendar Spread Arbitrage

- ${\theta}_{2\left(2\right)}>a$
- $c{\theta}_{2\left(2\right)}+a{\theta}_{0\left(2\right)}<\frac{b{\theta}_{1\left(2\right)}}{2}$

- ${\theta}_{2\left(n\right)}>{\theta}_{2(n-1)}$
- ${\theta}_{2\left(n\right)}{\theta}_{0(n-1)}+{\theta}_{2(n-1)}{\theta}_{0\left(n\right)}<\frac{{\theta}_{1\left(n\right)}{\theta}_{1(n-1)}}{2}$

#### 4.4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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1 | For technical reasons we assume the value of b at time T and discount it to make it comparable to today’s value. |

**Figure 1.**Plots of the total implied variance for six different times to maturity following the forward slice-by-slice method of Section 4.3.

**Table 1.**Times to maturity and the optimum values of the regularization parameter for each volatility slice.

Expiry Date | Time to Maturity | $\mathit{\lambda}$ |
---|---|---|

20 December 2014 | 0.0136 | 0.3 |

2 January 2015 | 0.0465 | 3 |

17 January 2015 | 0.0876 | 1.2 |

23 January 2015 | 0.1041 | 2.8 |

20 February 2015 | 0.178 | 0.9 |

20 March 2015 | 0.232 | 1.3 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Assa, H.; Pouralizadeh, M.; Badamchizadeh, A.
Sound Deposit Insurance Pricing Using a Machine Learning Approach. *Risks* **2019**, *7*, 45.
https://doi.org/10.3390/risks7020045

**AMA Style**

Assa H, Pouralizadeh M, Badamchizadeh A.
Sound Deposit Insurance Pricing Using a Machine Learning Approach. *Risks*. 2019; 7(2):45.
https://doi.org/10.3390/risks7020045

**Chicago/Turabian Style**

Assa, Hirbod, Mostafa Pouralizadeh, and Abdolrahim Badamchizadeh.
2019. "Sound Deposit Insurance Pricing Using a Machine Learning Approach" *Risks* 7, no. 2: 45.
https://doi.org/10.3390/risks7020045