# Equity Option Pricing with Systematic and Idiosyncratic Volatility and Jump Risks

## Abstract

**:**

## 1. Introduction

## 2. Equity Option Valuation

#### 2.1. Model Description

#### 2.2. Characteristic Function

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

#### 2.3. Valuation of the European Index and Equity Options

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

## 3. Empirical Studies

#### 3.1. Data Description

#### 3.2. Parameter Estimation

#### 3.3. Pricing Performance

## 4. Conclusions

## Funding

## Conflicts of Interest

## References

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1. | In fact, the work of Xiao and Zhou (2018) is a complement to the recent studies that disentangle the four types of risks in equity premiums, such as Bégin et al. (2020), who developed a GARCH-jump model in which an individual firm’s systematic and idiosyncratic risk have both a Gaussian diffusive and a jump component. Their empirical results showed that normal diffusive and jump risks have drastically different effects on the expected return of individual stocks by using 20 years of returns and options on the S&P 500 and 260 stocks. |

2. | One can refer to Assumption 2.1 of Cheang et al. (2013) and Cheang and Garces (2019) for a more detailed explanation. |

3. | Obviously, our proposed model for the dynamics of the market factor and individual equity prices is an extension of Christoffersen et al. (2018). In fact, our model also can be regarded as a further generalization of Cheang et al. (2013) and Cheang and Garces (2019) by taking into account the factor structure. |

4. | One can refer to the Assumption 2.1 of Cheang et al. (2013) and Cheang and Garces (2019) for a more detailed explanation. |

5. | The relative error is defined by $\frac{|{C}_{model}-{C}_{market}|}{{C}_{market}}\times 100\%$, where ${C}_{model}$ and ${C}_{market}$ denote the theoretical model option prices and the real market prices, respectively. |

**Figure 1.**The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity T = 24 May 2019.

**Figure 2.**The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity T = 31 May 2019.

**Figure 3.**The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity T = 7 June 2019.

**Figure 4.**The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity T = 14 June 2019.

**Figure 5.**The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity T = 21 June 2019.

**Figure 6.**The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity T = 19 July 2019.

**Figure 7.**The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity T = 16 August 2019.

**Figure 8.**The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity T = 20 September 2019.

**Figure 9.**The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity T = 18 October 2019.

**Figure 10.**The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity T = 17 January 2019.

**Table 1.**Estimated parameters. Note: This table shows the average of the estimated parameters obtained by minimizing the root mean squared pricing errors between the market price and the model price for each option on 8 May 2019. Standard errors are reported in parentheses.

Parameters | Our | 2-FSV | 2-SV | 2-SVJ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

SPX | AAPL | SPX | AAPL | AAPL | AAPL | |||||||

${V}_{I,0}$/${V}_{1,0}$ | 0.0133 | 0.0119 | 0.0239 | 0.0181 | ||||||||

(0.0000) | (0.0000) | (0.0002) | (0.0001) | |||||||||

${V}_{S,0}$/${V}_{2,0}$ | 0.0470 | 0.0514 | 0.0197 | 0.0176 | ||||||||

(0.0000) | (0.0000) | (0.0002) | (0.0002) | |||||||||

${\kappa}_{I}$/${\kappa}_{1}$ | 0.2496 | 0.2929 | 0.3489 | 0.4064 | ||||||||

(0.0212) | (0.0148) | (0.0118) | (0.0311) | |||||||||

${\kappa}_{S}$/${\kappa}_{2}$ | 0.2454 | 0.1504 | 0.4131 | 0.4108 | ||||||||

(0.0288) | (0.0797) | (0.0729) | (0.0171) | |||||||||

${\theta}_{I}$/${\theta}_{1}$ | 0.2820 | 0.3066 | 0.3314 | 0.2817 | ||||||||

(0.0181) | (0.0317) | (0.0534) | (0.0348) | |||||||||

${\theta}_{S}$/${\theta}_{2}$ | 0.2303 | 0.3683 | 0.2447 | 0.3415 | ||||||||

(0.0190) | (0.0590) | (0.0365) | (0.0423) | |||||||||

${\sigma}_{I}$/${\sigma}_{1}$ | 0.3472 | 0.3932 | 0.1615 | 0.1898 | ||||||||

(0.0127) | (0.0137) | (0.0081) | (0.0106) | |||||||||

${\sigma}_{S}$/${\sigma}_{2}$ | 0.1496 | 0.1640 | 0.2206 | 0.1970 | ||||||||

(0.0056) | (0.0135) | (0.0386) | (0.0059) | |||||||||

${\lambda}_{I}$ | 0.0450 | |||||||||||

(0.0017) | ||||||||||||

${\lambda}_{S}$ | 0.3413 | 0.3065 | ||||||||||

(0.2463) | (0.1194) | |||||||||||

${\mu}_{I}$ | 0.1657 | |||||||||||

(0.0599) | ||||||||||||

${\mu}_{S}$ | 0.0889 | 0.0333 | ||||||||||

(0.0391) | (0.0042) | |||||||||||

${\delta}_{I}$ | 0.0850 | |||||||||||

(0.0113) | ||||||||||||

${\delta}_{S}$ | 0.0679 | 0.0534 | ||||||||||

(0.0078) | (0.0013) | |||||||||||

${\beta}_{diff}$ | 0.3891 | 0.2457 | ||||||||||

(0.0381) | (0.0983) | |||||||||||

${\beta}_{jump}$ | 0.8429 | |||||||||||

(0.8091) | ||||||||||||

${\rho}_{I}$/${\rho}_{1}$ | −0.9290 | −0.8498 | −0.9222 | −0.7445 | ||||||||

(0.0063) | (0.0080) | (0.0096) | (0.0297) | |||||||||

${\rho}_{S}$/${\rho}_{2}$ | −0.9926 | −0.8938 | −0.7673 | −0.7817 | ||||||||

(0.0001) | (0.0469) | (0.1632) | (0.0549) |

**Table 2.**Out-of-sample pricing errors. Note: This table shows the out-of-sample pricing errors across different maturities. Pricing errors are reported as the root mean squared errors (RMSE) of option prices for four models.

RMSE | Our | 2-FSV | 2-SV | 2-SVJ | Improvement Rate | ||
---|---|---|---|---|---|---|---|

Maturity | Our vs. 2-FSV | Our vs. 2-SV | Our vs. 2-SVJ | ||||

24 May 2019 | 0.2573 | 0.2574 | 0.2596 | 0.2707 | 0.0373% | 0.8803% | 4.9568% |

31 May 2019 | 0.2507 | 0.2508 | 0.2564 | 0.2652 | 0.0392% | 2.2499% | 5.4846% |

7 June 2019 | 0.2343 | 0.2347 | 0.2527 | 0.2474 | 0.1764% | 7.2947% | 5.3044% |

14 June 2019 | 0.1992 | 0.2041 | 0.2261 | 0.2099 | 2.4278% | 11.9155% | 5.0858% |

21 June 2019 | 0.1824 | 0.1827 | 0.1873 | 0.1916 | 0.1399% | 2.5963% | 4.7934% |

19 July 2019 | 0.3256 | 0.3301 | 0.3326 | 0.3383 | 1.3434% | 2.0948% | 3.7368% |

16 August 2019 | 0.2856 | 0.2835 | 0.2879 | 0.2922 | −0.7573% | 0.7946% | 2.2384% |

20 September 2019 | 0.3177 | 0.3159 | 0.3162 | 0.3222 | −0.5932% | -0.4851% | 1.4002% |

18 October 2019 | 0.1185 | 0.1180 | 0.1215 | 0.1272 | −0.4458% | 2.4886% | 6.8593% |

17 January 2020 | 0.4882 | 0.4882 | 0.4893 | 0.4943 | −0.0071% | 0.2182% | 1.2201% |

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**MDPI and ACS Style**

Li, Z.
Equity Option Pricing with Systematic and Idiosyncratic Volatility and Jump Risks. *J. Risk Financial Manag.* **2020**, *13*, 16.
https://doi.org/10.3390/jrfm13010016

**AMA Style**

Li Z.
Equity Option Pricing with Systematic and Idiosyncratic Volatility and Jump Risks. *Journal of Risk and Financial Management*. 2020; 13(1):16.
https://doi.org/10.3390/jrfm13010016

**Chicago/Turabian Style**

Li, Zhe.
2020. "Equity Option Pricing with Systematic and Idiosyncratic Volatility and Jump Risks" *Journal of Risk and Financial Management* 13, no. 1: 16.
https://doi.org/10.3390/jrfm13010016