*2.1. Model Description*

Consider a filtered probability space (<sup>Ω</sup>, F, Q) with information filtration {F*t*}0≤*t*≤*T* satisfying the usual conditions (increase, right-continuous, and augmented), where Q is a risk neutral measure. We model an equity market consisting of *N* firms with a single market factor, *It* (usually approximated by a market index in practice). The individual stock prices are denoted by *Sit*, for *i* = 1, 2, ... , *N*. For the sake of convenience, we ignore the superscript *i*, and denote (*St*)*t*≥0 the pricing process of an individual stock. Investors also have access to a risk free bond that pays a return rate of *r*. To start, the market factor *It* evolves under a risk neutral measure Q as:

$$\frac{dI\_{\rm I}}{I\_{\rm I-}} = rdt + \sqrt{V\_{\rm I,t}}dW\_{\rm I,t}^{\rm I} + \int\_{\rm R} (c^y - 1)\tilde{N}\_{\rm Y}(dt, dy),\tag{1}$$

$$dV\_{I,t} = \kappa\_I (\theta\_I - V\_{I,t})dt + \sigma\_I \sqrt{V\_{I,t}} d\mathcal{W}^I\_{2,t} \tag{2}$$

where *It*− stands for the value of *It* before a possible jump occurs, *y* ∈ *R* = R \ {0}, *VI*,*<sup>t</sup>* is the variance of market factor, *θI* denotes the long run variance, *κI* captures the mean reversion speed of *VI*,*<sup>t</sup>* to *θI*, *σI* measures the volatility of volatility, 2*κIθI* ≥ *σ*2*I* to ensure that the process *VI*,*<sup>t</sup>* remains strictly positive2, *<sup>W</sup>I*1,*t* and *<sup>W</sup>I*2,*t* are correlated standard Brownian motions, i.e., the innovations to the market return and volatility are correlated with correlation coefficient *ρ<sup>I</sup>*, Cov *dWI*1,*t*, *dWI*2,*t* = *ρIdt*, and *N* ˜ *<sup>y</sup>*(*dt*, *dy*) = *Ny*(*dt*, *dy*) − *<sup>ν</sup>y*(*dy*)*dt* is a compensated jump measure, where *Ny*(*dt*, *dy*) is the jump measure and the Lévy kernel (or density) *<sup>ν</sup>y*(*dy*) satisfies *R* min(1, *<sup>y</sup>*<sup>2</sup>)*<sup>ν</sup>y*(*dy*) < ∞.

Furthermore, we separate the effects of the market factor on individual equities' returns into two types of risks: the systematic diffusive volatility and jump. More specifically, the diffusive random variation of individual equities' returns is dependent on the Brownian motion that drives market returns through the coefficient *βdi f f* . In addition, the discontinuous movements in the market return can also trigger jumps in individual equities' returns through the coefficient *βjump*. Therefore, the individual equity prices are driven by the market factor, as well as an idiosyncratic component that also has stochastic volatility and jump, whose process under a risk neutral measure Q follows:<sup>3</sup>

$$\begin{aligned} \frac{dS\_{t}}{S\_{t-}} &= rdt + \underbrace{\beta\_{diff}\sqrt{V\_{l,t}}dW^{l}\_{1,t}}\_{\text{Systematic diffusivity}} + \underbrace{\int\_{R} (e^{\beta\_{jump}y} - 1)\bar{N}\_{y}(dt, dy)}\_{\text{Systematic jump}} + \underbrace{\sqrt{V\_{S,t}}dW^{S}\_{1,t}}\_{\text{Idivisors\text{recific diffusivity}}} \\ &+ \underbrace{\int\_{R} (e^{\xi} - 1)\bar{N}\_{\xi}(dt, d\xi)}\_{\text{Idivisors\text{recific jump}}}, \end{aligned} \tag{3}$$

$$dV\_{S,t} = \kappa\_S(\theta\_S - V\_{S,t})dt + \sigma\_S \sqrt{V\_{S,t}} dW\_{2,t}^S \tag{4}$$

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

<sup>3</sup> 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.

where *St*− stands for the value of *St* before a possible jump occurs, *ξ* ∈ *R* = R \ {0}, *VS*,*<sup>t</sup>* is the idiosyncratic variance of individual equity, *θS* denotes the long run idiosyncratic variance, *κS* captures the mean reversion speed of *VS*,*<sup>t</sup>* to *θS*, *σS* measures the volatility of idiosyncratic variance, 2*κSθS* ≥ *σ*<sup>2</sup> *S* to ensure that the process *VS*,*<sup>t</sup>* remains strictly positive4, *<sup>W</sup>S*1,*t* and *<sup>W</sup>S*2,*t* are correlated standard Brownian motions, i.e., the innovations to the idiosyncratic return and volatility are correlated with correlation coefficient *ρS*, Cov *dWS*1,*t* , *dWS*2,*t* = *ρSdt*, but they are independent of Brownian motions in the market factor, i.e., Cov *dW<sup>S</sup> i*,*t* , *dW<sup>I</sup> j*,*t* = 0 for *i*, *j* = 1, 2, and *N*˜ *ξ* (*dt*, *dy*) = *<sup>N</sup>ξ* (*dt*, *dξ*) − *νξ* (*dξ*)*dt* is a compensated jump measure, where *<sup>N</sup>ξ* (*dt*, *dξ*) is the jump measure and the Lévy kernel (or density) *νξ* (*dξ*) satisfies *R* min(1, *ξ*<sup>2</sup>)*νξ* (*dξ*) < ∞.
