*2.2. Characteristic Function*

In order to be able to derive the pricing formulas for the European call and put equity options, we are particularly interested in the characteristic function of the logarithm asset price. Given the dynamics of the underlying asset price under the Q measure, we consider the conditional characteristic function of log-asset price *XT* = ln *ST* given the market information up to time *t*, which is denoted by *ϕ*(*<sup>x</sup>*, *υ*1, *υ*2, *t*, *T*; *φ*):

$$\begin{aligned} \varphi(\mathbf{x}, \boldsymbol{\upsilon}\_{1}, \boldsymbol{\upsilon}\_{2}, t, T; \boldsymbol{\phi}) &= \mathrm{E}^{\mathbb{Q}} \left[ e^{i\boldsymbol{\phi}X\_{T}} \,|\, \mathbf{X}\_{t} = \mathbf{x}, V\_{I,t} = \boldsymbol{\upsilon}\_{1}, V\_{S,t} = \boldsymbol{\upsilon}\_{2} \right] \\ &\triangleq \mathrm{E}^{\mathbb{Q}}\_{t} \left[ e^{i\boldsymbol{\phi}X\_{T}} \right], \end{aligned} \tag{5}$$

where E Q *t* [·] denotes the condition expectation under the Q measure, *t* ≤ *T*, and i = √ −1.

**Lemma 1.** *Suppose that the market factor It and individual equity price St are driven by Equations (1) and (3), respectively. Then, the conditional characteristic function of log-asset price XT* = ln *ST is given by:*

$$\varrho(\mathbf{x}, \upsilon\_1, \upsilon\_2, t, T; \phi) = \exp\left\{A(\tau)\mathbf{x} + B(\tau)\upsilon\_1 + C(\tau)\upsilon\_2 + D(\tau)\right\},\tag{6}$$

*where:*

*<sup>A</sup>*(*τ*) = i*φ*, *<sup>B</sup>*(*τ*) = *κI* − <sup>i</sup>*φβdi f f <sup>σ</sup>IρI* − *d*1 *σ*<sup>2</sup> *I* 1 − *e*<sup>−</sup>*d*1*<sup>τ</sup>* 1 − *<sup>g</sup>*1*e*<sup>−</sup>*d*1*<sup>τ</sup>* , *<sup>C</sup>*(*τ*) = *κS* − i*φσSρS* − *d*2 *σ*<sup>2</sup> *S* 1 − *e*<sup>−</sup>*d*2*<sup>τ</sup>* 1 − *<sup>g</sup>*2*e*<sup>−</sup>*d*2*<sup>τ</sup>* , *<sup>D</sup>*(*τ*) = ⎡ ⎢ ⎢ ⎣i*φr* + *R e* <sup>i</sup>*φβjumpy* − 1 − i*φ e βjumpy* − 1 *<sup>ν</sup>y*(*dy*) I1 + *R e* i*φξ* − 1 − i*φ e ξ* − 1 *νξ* (*dξ*) I2 ⎤ ⎥ ⎥ ⎦ *τ* + *<sup>κ</sup>IθI σ*<sup>2</sup> *I κI* − <sup>i</sup>*φβdi f f <sup>σ</sup>IρI* − *d*1 *τ* − 2 ln 1 − *<sup>g</sup>*1*e*<sup>−</sup>*d*1*<sup>τ</sup>* 1 − *g*1 + *<sup>κ</sup>SθS σ*<sup>2</sup> *S* (*<sup>κ</sup>S* − i*φσSρS* − *d*2) *τ* − 2 ln 1 − *<sup>g</sup>*2*e*<sup>−</sup>*d*2*<sup>τ</sup>* 1 − *g*2 , *g*1 = *κI* − <sup>i</sup>*φβdi f f <sup>σ</sup>IρI* − *d*1 *κI* − <sup>i</sup>*φβdi f f <sup>σ</sup>IρI* + *d*1 ,

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

$$\begin{split} g\_2 &= \frac{\kappa\_S - \mathrm{i}\phi\sigma\_S\rho\_S - d\_2}{\kappa\_S - \mathrm{i}\phi\sigma\_S\rho\_S + d\_2}, \\ d\_1 &= \sqrt{\left(\mathrm{i}\phi\beta\_{diff}\sigma\_I\rho\_I - \kappa\_I\right)^2 + \beta\_{diff}^2\sigma\_I^2(\mathrm{i}\phi + \phi^2)}, \\ d\_2 &= \sqrt{\left(\mathrm{i}\phi\sigma\_S\rho\_S - \kappa\_S\right)^2 + \sigma\_S^2(\mathrm{i}\phi + \phi^2)}, \end{split}$$

*and τ* = *T* − *t.*

**Proof.** To obtain the conditional characteristic function of log-asset price *XT* = ln *ST*, we first take the following transformation by using the Itô lemma for Equation (3):

$$\begin{split} d\ln S\_{t} &= \left(\tau - \frac{1}{2}\beta\_{diff}^{2}V\_{I,t} - \frac{1}{2}V\_{S,t} - \int\_{R}\left(\boldsymbol{\epsilon}^{\mathcal{S}\_{j\text{imp}}}\boldsymbol{y}^{\mathcal{Y}} - 1\right)\boldsymbol{\nu}\_{\mathcal{Y}}(d\boldsymbol{y}) - \int\_{R}\left(\boldsymbol{\epsilon}^{\mathcal{S}} - 1\right)\boldsymbol{\nu}\_{\mathcal{Y}}(d\boldsymbol{\xi})\right)dt \\ &+ \beta\_{diff}\sqrt{V\_{I,t}}d\mathcal{W}\_{1,t}^{I} + \beta\_{jump}\int\_{R}\boldsymbol{y}\mathcal{N}\_{\mathcal{Y}}(d\boldsymbol{t},d\boldsymbol{y}) + \sqrt{V\_{\mathcal{S},t}}d\mathcal{W}\_{1,t}^{\mathcal{S}} + \int\_{R}\boldsymbol{\mathcal{S}}\_{\mathcal{Y}}^{\mathcal{N}}(d\boldsymbol{t},d\boldsymbol{\xi}). \end{split}$$

The Feynman–Kac formula states that *ϕ*(*<sup>x</sup>*, *υ*1, *υ*2, *t*, *T*; *φ*) is governed by the following partial integro-differential equation (PIDE):

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩*∂ϕ∂τ* = *r* − 12 *β*2*di f f VI*,*<sup>t</sup>* − 12*VS*,*<sup>t</sup>* − *R eβjumpy* − 1 *<sup>ν</sup>y*(*dy*) − *R eξ* − 1 *νξ* (*dξ*) *∂ϕ∂x* + 12 *β*2*di f f υ*1 + *υ*2 *<sup>∂</sup>*<sup>2</sup>*ϕ ∂x*<sup>2</sup> + *<sup>κ</sup>I*(*<sup>θ</sup>I* − *<sup>υ</sup>*1) *∂ϕ∂υ*1 + 12*σ*2*I υ*1 *<sup>∂</sup>*<sup>2</sup>*ϕ ∂υ*21 + *<sup>κ</sup>S*(*<sup>θ</sup>S* − *<sup>υ</sup>*2) *∂ϕ∂υ*2 + 12*σ*2*Sυ*<sup>2</sup> *<sup>∂</sup>*<sup>2</sup>*ϕ ∂υ*22 + *βdi f f <sup>σ</sup>IρIυ*1 *<sup>∂</sup>*<sup>2</sup>*ϕ ∂x∂υ*1 + *<sup>σ</sup>SρSυ*2 *<sup>∂</sup>*<sup>2</sup>*ϕ ∂x∂υ*2 + *R* !*ϕ*(*x* + *βjumpy*, *υ*1, *υ*2, *t*, *T*; *φ*) − *ϕ*(*<sup>x</sup>*, *υ*1, *υ*2, *t*, *T*; *φ*)" *<sup>ν</sup>y*(*dy*) + *R* [*ϕ*(*x* + *ξ*, *υ*1, *υ*2, *t*, *T*; *φ*) − *ϕ*(*<sup>x</sup>*, *υ*1, *υ*2, *t*, *T*; *φ*)] *νξ* (*dξ*), *ϕ*(*<sup>x</sup>*, *υ*1, *υ*2, *t*, *T*; *φ*)|*<sup>t</sup>*=*<sup>T</sup>* = *e*i*φXT* . (7)

Due to the affine structure of our model, we postulate *ϕ*(*<sup>x</sup>*, *υ*1, *υ*2, *t*, *T*; *φ*) admitting the form of (6). Substituting Equation (6) into the above PIDE (7) gives the following system of ordinary differential equations (ODEs) for *<sup>A</sup>*(*τ*), *<sup>B</sup>*(*τ*), *<sup>C</sup>*(*τ*), and *<sup>D</sup>*(*τ*):

$$\begin{cases} \frac{\partial A(\tau)}{\partial \tau} = 0, \\ \frac{\partial B(\tau)}{\partial \tau} = \frac{1}{2} \sigma\_I^2 B^2(\tau) + \left[ \beta\_{diff} \sigma\_I \rho\_I A(\tau) - \kappa\_I \right] B(\tau) - \frac{1}{2} \beta\_{diff}^2 \left[ A(\tau) - A^2(\tau) \right], \\ \frac{\partial C(\tau)}{\partial \tau} = \frac{1}{2} \sigma\_S^2 C^2(\tau) + \left[ \sigma\_S \rho\_S A(\tau) - \kappa\_S \right] C(\tau) - \frac{1}{2} \left[ A(\tau) - A^2(\tau) \right], \\ \frac{\partial D(\tau)}{\partial \tau} = r A(\tau) + \kappa\_I \theta\_I B(\tau) + \kappa\_S \theta\_S C(\tau) + \int\_R \left[ e^{A(\tau) \beta\_{imp} y} - 1 - A(\tau) \left( e^{\theta\_{imp} y} - 1 \right) \right] v\_\theta(dy) \\ \qquad + \int\_R \left[ e^{A(\tau) \tilde{\xi}} - 1 - A(\tau) \left( e^{\tilde{\xi}} - 1 \right) \right] v\_\xi(d\xi), \end{cases}$$

where the boundary conditions are given as *A*(0) = i*φ* and *B*(0) = *C*(0) = *D*(0) = 0.

By solving the above ODEs, we can obtain the characteristic function (6).

**Lemma 2.** *Suppose that the market factor It is driven by Equation (1). Then, the conditional characteristic function of log-market factor ZT* = ln *IT is given by:*

$$\begin{split} \psi(\mathbf{z}, \boldsymbol{\upsilon}\_{1}, \mathbf{t}, T; \boldsymbol{\Phi}) &= \mathbf{E}^{\mathbb{Q}} \left[ e^{i\boldsymbol{\phi} \mathbf{Z}\_{\mathrm{T}}} \middle| \mathbf{Z}\_{\mathrm{I}} = \mathbf{z}, V\_{\mathrm{I}, \mathrm{t}} = \boldsymbol{\upsilon}\_{1} \right] \\ &= \exp \left\{ \bar{A}(\boldsymbol{\tau}) \boldsymbol{z} + \bar{\boldsymbol{\mathcal{B}}}(\boldsymbol{\tau}) \boldsymbol{\upsilon}\_{1} + \bar{\boldsymbol{D}}(\boldsymbol{\tau}) \right\}, \end{split} \tag{8}$$

*where:*

$$\begin{split} \tilde{A}(\tau) &= \text{i}\phi, \\ B(\tau) &= \frac{\kappa\_{I} - \text{i}\phi\tau\_{I}\rho\_{I} - d}{\sigma\_{I}^{2}} \left[ \frac{1 - e^{-d\tau}}{1 - g e^{-d\tau}} \right], \\ D(\tau) &= \left[ \text{i}\phi\tau + \underbrace{\int\_{R} \left( e^{\text{i}\phi y} - 1 - \text{i}\phi \left( e^{y} - 1 \right) \right) \nu\_{y}(dy)} \right] \tau + \frac{\kappa\_{I}\theta\_{I}}{\sigma\_{I}^{2}} \left[ \left( \kappa\_{I} - \text{i}\phi\tau\_{I}\rho\_{I} - d \right) \tau - 2\ln\frac{1 - g e^{-d\tau}}{1 - g} \right], \\ g &= \frac{\kappa\_{I} - \text{i}\phi\sigma\_{I}\rho\_{I} - d}{\kappa\_{I} - \text{i}\phi\sigma\_{I}\rho\_{I} + d}, \\ d &= \sqrt{\left( \text{i}\phi\sigma\_{I}\rho\_{I} - \kappa\_{I} \right)^{2} + \sigma\_{I}^{2} \left( \text{i}\phi + \phi^{2} \right)}, \\ \end{split}$$

**Proof.**Similartheproofof Lemma1,easilyverifythe

 we can

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

Once the characteristic function is found, it is straightforward to calculate the prices of European options by using Fourier inversion. Let *<sup>C</sup>*(*St*, *T*, *K*) and *<sup>P</sup>*(*St*, *T*, *K*) be the prices of the European equity call and put options at time *t* with strike price *K* and maturity *T* under the risk neutral measure Q, respectively. Then, these option prices are determined by:

 above results.

$$\mathcal{C}(S\_t, T, K) = \mathfrak{e}^{-r\tau} \mathcal{E}\_t^{\mathbb{Q}} \left[ \max(S\_T - K, 0) \right],$$

and:

 −

> to

$$P(S\_t, T, K) = e^{-r\tau} \mathcal{E}\_t^{\mathbb{Q}} \left[ \max(K - S\_{T}, 0) \right],$$

where *τ* = *T* − *t* is the time to maturity.

**Theorem 1.** *Suppose that the market factor It and the individual equity price St are driven by Equations (1) and (3), respectively. Then, the prices of the European equity call and put options with strike price K and maturity τ* = *T* − *t are given by:*

$$\mathcal{L}\left(\mathbb{S}\_{t\prime},T,\mathcal{K}\right) = \mathbb{S}\_{t}\Pi\_{1}\left(\mathbb{S}\_{t\prime}T,\mathbb{K};\mathbb{X}\_{\prime}\mathbb{R}\_{diff},\mathbb{A}\_{\mathrm{jump}}\right) - \mathrm{Ker}^{-r\tau}\Pi\_{2}\left(\mathbb{S}\_{t\prime}T,\mathbb{K};\mathbb{X}\_{\prime}\mathbb{R}\_{diff\prime\prime}\mathbb{A}\_{\mathrm{jump}}\right) \tag{9}$$

*and:*

$$P(\mathcal{S}\_{l\prime},T,K) = K\varepsilon^{-r\tau} \left[ 1 - \Pi \mathbf{1}\_2 \left( \mathcal{S}\_{l\prime}, T, K; \mathcal{S}\_{diff}, \mathcal{S}\_{\text{jump}} \right) \right] - \mathcal{S}\_l \left[ 1 - \Pi\_1 \left( \mathcal{S}\_{l\prime}, T, K; \mathcal{S}\_{\text{diff}}, \mathcal{S}\_{\text{jump}} \right) \right] \tag{10}$$

*where the risk neutral probability distribution functions* Π1 *and* Π2 *are defined by:*

$$\Pi\_1 \left( \mathbb{S}\_t, T, \mathbb{K}; \mathbb{X}\_t \mathbb{H}\_{diff}, \mathbb{A}\_{\text{jump}} \right) = \frac{1}{2} + \frac{e^{-r\tau}}{\pi \mathbb{S}\_t} \int\_0^{+\infty} \Re \left[ \frac{e^{-\mathrm{i}\phi \ln K} \varrho(\mathbf{x}, \boldsymbol{\upsilon}\_1, \boldsymbol{\upsilon}\_2, t, T; \phi - \mathbf{i})}{\mathbf{i}\phi} \right] d\phi$$

*JRFM* **2020**, *13*, 16

*and:*

$$\Pi\_2\left(\mathbb{S}\_t, T, \mathbb{K}; \mathfrak{f}\_{diff}, \mathfrak{f}\_{jump}\right) = \frac{1}{2} + \frac{1}{\pi} \int\_0^{+\infty} \mathfrak{R}\left[\frac{e^{-\mathrm{i}\boldsymbol{\phi}\ln K} \varrho(\mathbf{x}, \boldsymbol{\upsilon}\_1, \boldsymbol{\upsilon}\_2, t, T; \boldsymbol{\phi})}{\mathrm{i}\boldsymbol{\phi}}\right] d\boldsymbol{\phi}\_t$$

*where ϕ*(*<sup>x</sup>*, *υ*1, *υ*2, *t*, *T*; *φ*) *is the conditional characteristic function of* ln *ST, which can be seen in Equation (6), and* [·] *indicates the real part of a complex number.*

**Proof.** In order to ge<sup>t</sup> the pricing formulas of the European equity call and put options, let us first introduce a change of measure from Q to Q˜ by the following Radon–Nikodym derivative:

$$\frac{d\tilde{\mathbb{Q}}}{d\mathbb{Q}} = e^{-r(T-t)} \frac{\mathbb{S}\_T}{\mathbb{S}\_t}.$$

We denote the conditional characteristic function of *XT* = ln *ST* under the Q˜ measure by *ϕ*˜(*<sup>x</sup>*, *υ*1, *υ*2, *t*, *T*; *φ*). Then, *ϕ*˜(*<sup>x</sup>*, *υ*1, *υ*2, *t*, *T*; *φ*) can be expressed as:

$$\begin{aligned} \bar{\boldsymbol{\varrho}}(\boldsymbol{x}, \boldsymbol{\upsilon}\_{1}, \boldsymbol{\upsilon}\_{2}, t, \boldsymbol{T}; \boldsymbol{\phi}) &= \operatorname{\mathbf{E}}\_{t}^{\mathbb{Q}} \left[ \boldsymbol{e}^{\mathbf{i}\boldsymbol{\phi}\boldsymbol{X}\_{T}} \right] \\ &= \operatorname{\mathbf{E}}\_{t}^{\mathbb{Q}} \left[ \boldsymbol{e}^{-\boldsymbol{r}(T-t)} \frac{\boldsymbol{S}\_{T}}{\boldsymbol{S}\_{t}} \boldsymbol{e}^{\mathbf{i}\boldsymbol{\phi}\boldsymbol{X}\_{T}} \right] \\ &= \boldsymbol{e}^{-\boldsymbol{r}(T-t)-\boldsymbol{x}} \operatorname{\mathbf{E}}\_{t}^{\mathbb{Q}} \left[ \boldsymbol{e}^{\mathbf{i}(\boldsymbol{\phi}-\mathbf{i})\boldsymbol{X}\_{T}} \right] \\ &= \boldsymbol{e}^{-\boldsymbol{r}(T-t)-\boldsymbol{x}} \boldsymbol{q}(\boldsymbol{x}, \boldsymbol{\upsilon}\_{1}, \boldsymbol{\upsilon}\_{2}, t, T; \boldsymbol{\phi}-\mathbf{i}). \end{aligned}$$

Thus, the price of a European equity call option *<sup>C</sup>*(*St*, *T*, *K*) can be calculated by utilizing *ϕ*(*<sup>x</sup>*, *υ*1, *υ*2, *t*, *T*; *φ*) and *ϕ*˜(*<sup>x</sup>*, *υ*1, *υ*2, *t*, *T*; *φ*):

$$\begin{split} \mathbb{C}\left(S\_{t},T,K\right) &= e^{-r\tau} \mathbb{E}\_{t}^{\mathbb{Q}} \left[\max\{S\_{T}-K,0\}\right] \\ &= e^{-r\tau} \mathbb{E}\_{t}^{\mathbb{Q}} \left[S\_{T}\mathbbm{1}\_{\{S\_{T}\geq K\}}\right] - Ke^{-r\tau} \mathbb{E}\_{t}^{\mathbb{Q}} \left[\mathbbm{1}\_{\{S\_{T}\geq K\}}\right] \\ &= S\_{t} \mathbb{E}\_{t}^{\mathbb{Q}} \left[\mathbbm{1}\_{\{S\_{T}\geq K\}}\right] - Ke^{-r\tau} \mathbb{E}\_{t}^{\mathbb{Q}} \left[\mathbbm{1}\_{\{S\_{T}\geq K\}}\right] \\ &= S\_{t} \mathbb{E}\_{t}^{\mathbb{Q}} \left[\mathbbm{1}\_{\{X\_{T}\geq \ln K\}}\right] - Ke^{-r\tau} \mathbb{E}\_{t}^{\mathbb{Q}} \left[\mathbbm{1}\_{\{X\_{T}\geq \ln K\}}\right] \\ &= S\_{t} \Pi\_{1} \left(S\_{t},T,K; \beta\_{diff}, \beta\_{jump}\right) - Ke^{-r\tau} \Pi\_{2} \left(S\_{t},T,K; \beta\_{diff}, \beta\_{jump}\right) \end{split}$$

Once the conditional characteristic function *ϕ*(*<sup>x</sup>*, *υ*1, *υ*2, *t*, *T*; *φ*) is obtained, we can easily calculate the probability distribution functions Π1 *St*, *T*, *K*; *βdi f f* , *βjump* and Π2 *St*, *T*, *K*; *βdi f f* , *βjump* according to the Lévy inversion formula:

.

$$\Pi\_1 \left( \mathbb{S}\_{t\prime} T\_\prime \mathcal{K}; \mathfrak{f}\_{diff\prime\prime} \mathfrak{f}\_{jump} \right) = \frac{1}{2} + \frac{1}{\pi} \int\_0^{+\infty} \mathfrak{R} \left[ \frac{e^{-\mathrm{i}\boldsymbol{\phi} \ln \boldsymbol{K}} \bar{\mathfrak{g}}(\mathbf{x}, \boldsymbol{\upsilon}\_1, \boldsymbol{\upsilon}\_2, \mathbf{t}, \boldsymbol{T}; \boldsymbol{\phi})}{\mathrm{i}\boldsymbol{\phi}} \right] d\boldsymbol{\phi}$$

and:

$$\Pi\_2\left(\mathbb{S}\_t, T, \mathbb{K}; \mathfrak{f}\_{diff}, \mathfrak{f}\_{jump}\right) = \frac{1}{2} + \frac{1}{\pi} \int\_0^{+\infty} \mathfrak{R}\left[\frac{e^{-\mathrm{i}\boldsymbol{\phi}\ln K} \mathfrak{g}(\mathbf{x}, \boldsymbol{\upsilon}\_1, \boldsymbol{\upsilon}\_2, \mathbf{t}, T; \boldsymbol{\phi})}{\mathrm{i}\boldsymbol{\phi}}\right] d\boldsymbol{\phi}\_t$$

A similar approach can be used to derive the pricing formula for the European equity put option.

In a similar way, we also can present the pricing formulas for the European index call and put options.

**Theorem 2.** *Suppose that the market factor It is driven by Equation (1). Then, the time t prices of the European index call and put options with strike price K and maturity τ* = *T* − *t are given by:*

$$\mathbb{C}\left(I\_{t\prime},T,K\right) = I\_t \hat{\Pi}\_1\left(I\_{t\prime},T,K\right) - Ke^{-r\tau} \hat{\Pi}\_2\left(I\_{t\prime},T,K\right) \tag{11}$$

*and:*

$$P(I\_{l},T,K) = K\varepsilon^{-r\tau} \left[1 - \tilde{\Pi}\_{2}\left(I\_{l},T,K\right)\right] - I\_{l}\left[1 - \tilde{\Pi}\_{1}\left(I\_{l},T,K\right)\right] \tag{12}$$

*where the risk neutral probability distribution functions* Π1 *and* Π2 *are defined by:*

$$\tilde{\Pi}\_1\left(I\_{\mathbf{f}},T,K\right) = \frac{1}{2} + \frac{e^{-r\tau}}{\pi I\_{\mathbf{f}}} \int\_0^{+\infty} \Re\left[\frac{e^{-i\boldsymbol{\phi}\ln K}\psi(z,\boldsymbol{\nu}\_1,\mathbf{t},T;\boldsymbol{\phi}-\mathbf{i})}{\mathbf{i}\boldsymbol{\phi}}\right] d\boldsymbol{\phi}$$

*and:*

$$\text{III}\_2\left(I\_{t\prime}, T, K\right) = \frac{1}{2} + \frac{1}{\pi} \int\_0^{+\infty} \Re\left[\frac{e^{-i\phi \ln K} \psi(z, \upsilon\_1, t, T; \phi)}{i\phi}\right] d\phi,$$

*where ψ*(*<sup>z</sup>*, *υ*1, *t*, *T*; *φ*) *is the conditional characteristic function of* ln *IT, which can be seen in Equation (8).*
