# A Solution to the Time-Scale Fractional Puzzle in the Implied Volatility

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## Abstract

**:**

## 1. Introduction

## 2. The Setup

#### 2.1. Integral Representation

#### 2.2. Some Special Cases

## 3. Approximation Formula

**Lemma**

**1.**

**Proposition**

**1.**

## 4. Numerical Examples

#### 4.1. Effect of ${H}_{2}$

#### 4.2. Effect of ${H}_{1}$

#### 4.3. Effect of ${\rho}_{1,2}$ and ${H}_{1}$

#### 4.4. Effect of Correlations

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Proof of Lemma 1

## Appendix B. Derivation of the Approximated Density Function

**Lemma**

**A1.**

## Appendix C. Formulas for Conditional Expectations

## References

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**Figure 1.**ATM implied volatility (

**a**) and ATM skew (

**b**) with respect to the option maturity. The model parameters are listed in Table 1.

**Figure 2.**ATM skew with respect to ${H}_{2}$, where ${H}_{2}$ denotes the Hurst index of the rough volatility. The model parameters are listed in Table 1.

**Figure 3.**ATM skew with respect to ${H}_{1}$, the Hurst index of volatility persistence. The model parameters are listed in Table 1.

**Figure 4.**Volatility smile with respect to strike and maturity. (

**a**–

**d**) show the volatility smile of $T=0.04$, $T=0.08$, $T=0.12$, and $T=0.2$, respectively. The model parameters are listed in Table 1.

**Figure 5.**Volatility smile with respect to correlation ${\rho}_{1,2}$. (

**a**–

**c**) show the volatility smile of ${\rho}_{1,2}=0.5$, 0, and $-0.5$, respectively. The maturity is $T=0.16$ and the other parameters are listed in Table 1.

**Figure 6.**Volatility smile with respect to the correlations. (

**a**–

**c**) show the volatility smile of ${\rho}_{1}$, ${\rho}_{2}$, and ${\rho}_{3}$, respectively. The maturity is $T=0.16$ and the other parameters are listed in Table 1.

**Figure 7.**ATM skew with respect to the correlations. (

**a**–

**c**) show the ATM skew of ${\rho}_{1}$, ${\rho}_{2}$, and ${\rho}_{3}$, respectively. The other parameters are listed in Table 1.

$({\mathit{H}}_{1},{\mathit{H}}_{2})$ | ${\mathit{S}}_{0}$ | r | q | ${\mathit{X}}_{0}^{1}$ | ${\mathit{X}}_{0}^{2}$ | ${\mathit{\theta}}_{1}$ | ${\mathit{\theta}}_{2}$ | ${\mathit{\kappa}}_{1}$ | ${\mathit{\kappa}}_{2}$ | ${\mathit{\gamma}}_{1}$ | ${\mathit{\gamma}}_{2}$ | ${\mathit{\rho}}_{1,2}$ | ${\mathit{\rho}}_{1}$ | ${\mathit{\rho}}_{2}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$(0.9,0.1)$ | 100 | 0.0131 | 0.017 | 0.2 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.12 | 0.12 | 0.8 | $-0.05$ | 0.1 |

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## Share and Cite

**MDPI and ACS Style**

Funahashi, H.; Kijima, M.
A Solution to the Time-Scale Fractional Puzzle in the Implied Volatility. *Fractal Fract.* **2017**, *1*, 14.
https://doi.org/10.3390/fractalfract1010014

**AMA Style**

Funahashi H, Kijima M.
A Solution to the Time-Scale Fractional Puzzle in the Implied Volatility. *Fractal and Fractional*. 2017; 1(1):14.
https://doi.org/10.3390/fractalfract1010014

**Chicago/Turabian Style**

Funahashi, Hideharu, and Masaaki Kijima.
2017. "A Solution to the Time-Scale Fractional Puzzle in the Implied Volatility" *Fractal and Fractional* 1, no. 1: 14.
https://doi.org/10.3390/fractalfract1010014