# Financial Applications on Fractional Lévy Stochastic Processes

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## Abstract

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## 1. Introduction and Notations

#### An Example of a Lévy Process: Lévy Stable Processes (LSP)

## 2. The Stochastic Model and Deterministic Representation

## 3. Numerical Discretization of the Fractional PDE

- For a call option, the payoff is defined by ${\mathrm{\Pi}}_{T}=\mathrm{max}({S}_{T}-K,0)$. Since we must truncate the spatial domain in order to make it workable from a numerical point of view, we will use the following boundary conditions $u({x}_{min},t)=0$ and $u({x}_{max},t)={e}^{{x}_{max}}-K{e}^{-r(T-t)}$ where ${x}_{min}=-log\left(4K\right)$ and ${x}_{max}=log\left(4K\right)$.
- For a put option, the payoff is defined by ${\mathrm{\Pi}}_{T}=max(K-{S}_{T},0)$ and we will use the following conditions $u({x}_{max},t)=0$ and $u({x}_{min},t)=K{e}^{-r(T-t)}-{e}^{{x}_{min}}$ where ${x}_{min}=-log\left(4K\right)$ and ${x}_{max}=log\left(4K\right)$.

## 4. Numerical Simulations and Discussions

#### Approximation of Lévy Model Greeks

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Comparison between numerical solutions (SDE and FPDE). Upon the comparison, using the same parameters, of the left hand side of Equation (17) obtained by a FPDE approach and the right hand side of Equation (17) computed using Monte Carlo simulation, we observe that both solutions are close to each other.

**Figure 2.**This figure shows the evolution of the call option price with respect to the time to maturity using the FPDE representation. In comparison with the Black–Scholes call price when the underlying is driven by the Brownian motion, the two prices have the same shape.

**Figure 3.**This figure shows the evolution of the call put option price with respect to the time to maturity using the FPDE representation. In comparison with the Black–Scholes put price when the underlying is driven by the Brownian motion, the two prices have the same shape.

**Figure 4.**Lévy model Greeks for the call option: By analyzing the Greeks plots, we observe that the call option’s price have much higher Delta values than out of the call option’s price of Black–Scholes model, and this value oscillates around $2.5$, which ranges between $2.49$ and $2.51$. Gamma reaches its maximum when the underlying price is a little bit smaller, exactly equal to the strike of the call option, and the chart shows that Gamma is significantly constant for the Lévy model. We display also Vega as a function of the asset price and time to maturity for a call options with strike at 80 and we highlight the fact that Vega is much higher for the call option’s price than for the call option’s price for Black–Scholes. At last, Rho reaches its maximum when the underlying price is a little smaller, not exactly equal to the strike of the call option’s price, and it is significantly higher with hyper-volatility than for the call option’s price of Black–Scholes.

**Figure 5.**Lévy model Greeks for the put option: in the figures above, we plotted the Greeks for the put option in 3D. We observe that the Delta is constant with value equal to $1.51$ for put option’s price for Levy model, but it oscillates for Black–Scholes. We can see that the put–call parity is maintained for the Lévy model: Vega(Call) = Vega(Put) and Gamma(Call) = Gamma(Put). We have a negative Rho which ranges between $-1000$ and 0, and the figure displays its fluctuation with respect to the underlying asset.

${\mathit{S}}_{0}$ | Strike K | r | q | $\mathit{\lambda}$ | $\mathit{\eta}$ | m | $\mathit{\sigma}$ | $\mathit{\beta}$ | $\mathit{\alpha}$ |
---|---|---|---|---|---|---|---|---|---|

100 | 80 | $0.02$ | 1 | 10 | $0.01$ | $0.02$ | $0.15$ | $-1$ | $1.76$ |

**Table 2.**Call price obtained with Euler scheme and finite difference. Upon comparison, the two call prices are very closed to each other, with small error ${E}_{max}$ which is defined by ${E}_{max}=max|u({x}_{i},{t}_{n})-{U}_{i}^{n}|$.

Strike K | Call Price with SDE | Call Price with FPDE | ${\mathit{E}}_{max}$ |
---|---|---|---|

80 | $43.9820$ | $43.9789$ | $0.0040$ |

100 | $40.9454$ | $40.8985$ | $0.0068$ |

110 | $39.8865$ | $39.8783$ | $0.0082$ |

**Table 3.**Put price obtained with Euler scheme and finite difference. Upon comparison, the two put prices are very close to each other, with small error ${E}_{max}$ which defined by ${E}_{max}=max|u({x}_{i},{t}_{n})-{U}_{i}^{n}|$.

Strike K | Put Price with SDE | Put Price with FPDE | ${\mathit{E}}_{max}$ |
---|---|---|---|

80 | $26.0659$ | $26.1043$ | $0.0384$ |

100 | $5.5968$ | $5.5481$ | $0.0487$ |

110 | $2.7987$ | $2.7683$ | $0.0304$ |

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**MDPI and ACS Style**

Aljethi, R.A.; Kılıçman, A.
Financial Applications on Fractional Lévy Stochastic Processes. *Fractal Fract.* **2022**, *6*, 278.
https://doi.org/10.3390/fractalfract6050278

**AMA Style**

Aljethi RA, Kılıçman A.
Financial Applications on Fractional Lévy Stochastic Processes. *Fractal and Fractional*. 2022; 6(5):278.
https://doi.org/10.3390/fractalfract6050278

**Chicago/Turabian Style**

Aljethi, Reem Abdullah, and Adem Kılıçman.
2022. "Financial Applications on Fractional Lévy Stochastic Processes" *Fractal and Fractional* 6, no. 5: 278.
https://doi.org/10.3390/fractalfract6050278