# Transformations of Telegraph Processes and Their Financial Applications

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Transformations of Classical Telegraph Process

**Remark**

**1.**

**Remark**

**2.**

## 3. Financial Applications of Transformed Telegraph Process

#### 3.1. Application of Classical Telegraph Process in Finance: Balance Case

**Example**

**1**

**Remark**

**3.**

#### 3.2. Application of Classical Telegraph Process in Finance: Dis-Balance Case

#### 3.2.1. Asymptotic Results for Scaled Telegraph Process

**Remark**

**4.**

#### 3.2.2. Application in Finance: Black-Scholes Formula for Geometric Limiting Telegraph Process

**Example**

**2**

**Remark**

**5.**

#### 3.3. Asymmetric Telegraph Process and Its Financial Application

**Remark**

**6.**

**Example**

**3**

**Remark**

**7.**

## 4. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Time evolution of European call option price $C\left(t\right)$ as a function of v (

**a**) and $\lambda $ (

**b**).

**Figure 2.**Time evolution of European call option price $C\left(t\right)$ as a function of v and $\lambda $ for $t=0$ (

**a**) and $t=0.6$ (

**b**).

**Figure 3.**Time evolution of European call option price $C\left(t\right)$ as a function of $\sigma $.

**Figure 4.**Time evolution of European put option price $P\left(t\right)$ and dependent on v (

**a**) and $\lambda $ (

**b**).

**Figure 5.**Time evolution of European put option price $P\left(t\right)$ and dependent on v and $\lambda $ for $t=0$ (

**a**) and $t=0.6$ (

**b**).

**Figure 7.**Time evolution of European call option price $C\left(t\right)$ by fix $\lambda $, $t=0$ (

**a**) and $t=0.6$ (

**b**) in the asymmetric case.

**Figure 8.**Time evolution of European call option price $C\left(t\right)$ for fix v, $t=0$ (

**a**) and $t=0.6$ (

**b**) in the asymmetric case.

**Figure 9.**Time evolution of European call option price $C\left(t\right)$ as a function of $\sigma $.

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**MDPI and ACS Style**

Pogorui, A.A.; Swishchuk, A.; Rodríguez-Dagnino, R.M.
Transformations of Telegraph Processes and Their Financial Applications. *Risks* **2021**, *9*, 147.
https://doi.org/10.3390/risks9080147

**AMA Style**

Pogorui AA, Swishchuk A, Rodríguez-Dagnino RM.
Transformations of Telegraph Processes and Their Financial Applications. *Risks*. 2021; 9(8):147.
https://doi.org/10.3390/risks9080147

**Chicago/Turabian Style**

Pogorui, Anatoliy A., Anatoliy Swishchuk, and Ramón M. Rodríguez-Dagnino.
2021. "Transformations of Telegraph Processes and Their Financial Applications" *Risks* 9, no. 8: 147.
https://doi.org/10.3390/risks9080147