*3.2. Parameter Estimation*

Our proposed model allowed a general distribution for jump components of the market factor and individual equity price and thus could be easily introduced to the special cases such that the jump components follow the compound Poisson process of Merton (1976) and Kou (2002), etc. For different types of Lévy kernels, different forms of our model can be presented. In order to keep consistent with Bates (2000) for comparative analysis, in the following, we assumed that the jump components of the dynamics for the market factor and individual equity followed compound Poisson processes and the jump magnitude was drawn from the log-normal distribution of Merton (1976). Thus, the Lévy kernels for the market factor and individual equity, respectively, are given by:

$$\nu\_{\mathcal{Y}}(dy) = \lambda\_I \frac{1}{\sqrt{2\pi\delta\_I^2}} \exp\left\{-\frac{\left(y-\mu\_I\right)^2}{2\delta\_I^2}\right\} dy \tag{13}$$

and:

$$\nu\_{\vec{\xi}}(d\vec{\xi}) = \lambda\_S \frac{1}{\sqrt{2\pi\delta\_{\vec{S}}^2}} \exp\left\{-\frac{\left(\vec{\xi} - \mu\_S\right)^2}{2\delta\_{\vec{S}}^2}\right\} d\vec{\xi}\_{\prime} \tag{14}$$

where *<sup>λ</sup>j*, for *j* = *I*, *S*, denotes the jump intensity, *μj* is the mean of the jump size, and *δj* is the variance of the jump size. Then, the integrals I*i*, for *i* = 1, 2, 3, in Lemmas 1 and 2 can be calculated as follows:

$$\begin{split} \mathcal{I}\_{1} &= \lambda\_{I} \left[ \mathfrak{e}^{\mathsf{i}\mathfrak{p}\mathcal{G}\_{\text{jupup}}\mu\_{I} - \frac{1}{2}\mathfrak{e}^{\mathsf{i}\mathfrak{p}}\mathfrak{k}\_{\text{jupup}}^{2}\delta\_{I}^{2}} - 1 - \mathsf{i}\mathfrak{p} \left( \mathfrak{e}^{\mathcal{G}\_{\text{jupup}}\mu\_{I} + \frac{1}{2}\mathfrak{k}\_{\text{jupup}}^{2}\delta\_{I}^{2}} - 1 \right) \right], \\ \mathcal{I}\_{2} &= \lambda\_{S} \left[ \mathfrak{e}^{\mathsf{i}\mathfrak{p}\mu\_{S} - \frac{1}{2}\mathfrak{e}^{\mathsf{i}\mathfrak{d}\_{S}}\mathcal{S}\_{S}^{2}} - 1 - \mathsf{i}\mathfrak{p} \left( \mathfrak{e}^{\mu\_{S} + \frac{1}{2}\mathcal{S}\_{S}^{2}} - 1 \right) \right], \end{split}$$

and

$$\mathbf{I}\_3 = \lambda\_I \left[ \mathfrak{e}^{\mathbf{i}\mathfrak{\phi}\mu\_I - \frac{1}{2}\mathfrak{p}^2 \delta\_I^2} - 1 - \mathbf{i}\mathfrak{p} \left( \mathfrak{e}^{\mathfrak{u}\_I + \frac{1}{2}\delta\_I^2} - 1 \right) \right].$$

Based on Theorems 1 and 2, we employed a two step calibration procedure (see, for example, Wong et al. 2012; Christoffersen et al. 2018) to estimate the model parameters. First, we calibrated the market index dynamic Θ*I* based on the S&P 500 index option price alone. Second, we used the equity option price to calibrate the individual equity dynamic Θ*S* conditional on estimates of Θ*I*. Consider the situation in which an investor wants to hedge his or her equity position with index options and hedging horizon *T*. For brevity, we further suppose that the investor observes index option prices and equity option prices both with maturity *T*, the same as hedging horizon. Specifically, the dataset contains *Mt* index option prices *<sup>C</sup>*(*It*, *T*, *Ki*), for *i* = 1, 2, ... , *Mt*, and *Nt* equity option prices *<sup>C</sup>*(*St*, *T*, *Kj*), for *j* = 1, 2, . . . , *Nt*.

In the calibration process, the risk neutral model parameters were backed out by minimizing a loss function capturing the fit between the theoretical model and market prices. We employed the root mean squared errors (RMSE) as the objective function. The first step calibrated the risk neutral parameters for the index process, which are calibrated by:

$$\text{RMSE}(I) = \arg\min\_{\Theta\_l} \sqrt{\frac{1}{M\_l} \sum\_{i=1}^{M\_l} \left[ \mathbb{C}\_{i,\text{market}}(I\_{t\prime}, T, K\_i) - \mathbb{C}\_{i,\text{model}}^{\Theta\_l}(I\_{t\prime}, T, K\_i) \right]^2},\tag{15}$$

where *Ci*,*market*(*It*, *T*, *Ki*) is the market price of the index call option contract from the sample and *C*Θ*I <sup>i</sup>*,*model*(*It*, *T*, *Ki*) represents the model price calculated using Equation (15) and the vector of model input parameters Θ*I*.

The second calibrated the beta and the parameters for the idiosyncratic risk:

$$\text{RMSE}(S) = \arg\min\_{\Theta\_S} \sqrt{\frac{1}{N\_t} \sum\_{j=1}^{N\_t} \left[ C\_{j,\text{market}}(S\_{t\prime}, T\_{\prime}K\_j) - C\_{j,\text{model}}^{\Theta\_S}(S\_{t\prime}T\_{\prime}K\_j) \right]^2},\tag{16}$$

where *Cj*,*market*(*St*, *T*, *Kj*) is the market price of the equity call option contract from the sample and *C*Θ*<sup>S</sup> <sup>j</sup>*,*model*(*St*, *T*, *Kj*) represents the model price calculated using Equation (13) and the vector of model input parameters Θ*S*.

On the basis of the above calibration method, Table 1 presents the risk neutral parameter estimates across various model specifications. Note that the values of the diffusive beta *βdi f f* and jump beta *βjump* for our proposed model were 0.3891 and 0.8429, respectively. The corresponding value of *βdi f f* for the 2-FSV model was 0.2457. Obviously, both our proposed model and the 2-FSV model showed that AAPL tended to have a relatively low exposure to diffusive market movements. However, the jump exposure coefficient *βjump* = 0.8429 indicated that the AAPL had a strong exposure to market jumps, which meant that the factor structure of the jumps was much stronger than the one of the diffusive movements. The reason for this result may be related to the sample data we selected. If we can ge<sup>t</sup> more sample data in the future, we will do an in-depth analysis. Moreover, we also can see that the values of correlation *ρ* were strongly negative for four models, capturing the so-called leverage effect both in the index and individual equity.

**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 .

