*3.2. Characteristic Function*

This section aims at obtaining an analytical representation for the c.f. If *Z*(*t*) = *eβtY*(*t*) is defined such that *eβ<sup>t</sup>* is a matrix exponential, then *Zi*(*t*) is represented as:

$$\begin{split} dZ\_{i}(t) &= \sum\_{j=1}^{n} \left(\epsilon^{\delta t}\right)\_{ij} \left\{ \mathcal{L}\_{i} + \left(\varepsilon\_{j} - \frac{1}{2}\right) \sum\_{k=1}^{n} a\_{jk}^{2} \left(\sqrt{v\_{k}(t)} + \frac{b\_{k}}{\sqrt{v\_{k}(t)}}\right)^{2} + \left(\mathcal{E}\_{j} - \frac{1}{2}\right) \left(\sqrt{\vartheta\_{j}(t)} + \frac{\mathcal{E}\_{j}}{\sqrt{v\_{j}(t)}}\right)^{2} \right\} dt \\ &+ \sum\_{j=1}^{n} \left(\epsilon^{\delta t}\right)\_{ij} \left\{ \sum\_{k=1}^{n} a\_{jk} \left(\sqrt{v\_{k}(t)} + \frac{b\_{k}}{\sqrt{v\_{k}(t)}}\right) dW\_{k}(t) + \left(\sqrt{\vartheta\_{j}(t)} + \frac{\mathcal{E}\_{j}}{\sqrt{v\_{j}(t)}}\right) d\tilde{W}\_{j}(t) \right\} \end{split} \tag{6}$$

For convenience, we use *eβ<sup>t</sup>ij* as the *ij* component of the matrix *<sup>e</sup>β<sup>t</sup>*. Note that *Zi*(*t*) is no longer a mean-reverting process although it accounts for time dependent coefficients.

Next, we find the conditional c.f. for the increments of *Z*, defined as

$$\Phi\_{Z(t),v(t)}(T,\omega) = E\left[\exp\left\{i\omega'(Z(T)-Z(t))\right\} \mid Z(t) = z\_t, v(t) = v\_t\right] \tag{7}$$

Under a risk neutral measure, this c.f. can be used for pricing some financial products, given the integrability conditions (a discussion of the generalized c.f. as per Grasselli 2017 is beyond the scope of this paper.). For convenience, we formulate it as *<sup>v</sup>*(*t*)=(*<sup>v</sup>*1(*t*),..., *vn*(*t*), *<sup>v</sup>*#1(*t*),..., *<sup>v</sup>*#*n*(*t*)).

**Proposition 2.** *Let* (*Z*(*t*))*t*≥0 *evolve according to the model in Equation* (6)*. The c.f.* <sup>Φ</sup>*Z*(*t*),*<sup>v</sup>*(*t*) *is then given as follows:*

$$\begin{split} \Phi\_{Z(t),v(t)}(T,\omega) &= E\left[\exp i\omega^{\prime}(Z(T)-Z(t)) \mid Z(t) = y\_{t\prime}v(t) = v\_{t}\right] \\ &= \prod\_{k=1}^{n} \Phi\_{GG}\left(T, 1; L\_{k}(\omega), h\_{k}(\omega), g\_{k}(\omega), \kappa\_{k\prime}\theta\_{k\prime}\mathfrak{z}\_{k\prime}\rho\_{k\prime}b\_{k\prime}c\_{k\prime}v\_{k,t\prime}S\_{k,t}^{\*}\right) \\ &\times \prod\_{j=1}^{n} \Phi\_{GG}\left(T, 1; 0, L\_{j}(\omega), h\_{j}(\omega), g\_{j}(\omega), \tilde{\kappa}\_{j\prime}\tilde{\theta}\_{j\prime}\tilde{\xi}\_{j\prime}\tilde{\rho}\_{j\prime}\tilde{b}\_{j\prime}\tilde{c}\_{k\prime}\tilde{\upsilon}\_{k\prime}\tilde{\upsilon}\_{j,t\prime}S\_{t}^{\*\dagger}\right). \end{split}$$

*where* Φ*GG is a one-dimensional generalization of the c.f. from Grasselli (2017) provided in Lemma A1.*

Proof is provided in Appendix A. The c.f. above involves single expected values with respect to Brownian motion *<sup>B</sup>*(*t*). In each term, Φ*GG* (i.e., the second set of Brownian *W*(*t*)) is eliminated, hence this is a drastic simplification compared to the original 2*n* dimensional joint expectation.

A particular, fully solvable case is the FG model (*βij* = 0, *i*, *j* = 1, . . . , *n*).

**Corollary 1.** *Let* (*Z*(*t*))*t*≥0 *evolve according to the FG model (βij* = 0*, i*, *j* = 1, ... , *n). The c.f.* <sup>Φ</sup>*Z*(*t*),*<sup>v</sup>*(*t*) *is subsequently presented as follows:*

$$\begin{split} \Phi\_{Z(t),\boldsymbol{v}(t)}(T,\omega) &= E\left[\exp i\omega^{\prime}(Z(T)-Z(t)) \mid Z(t) = y\_{t\prime}\boldsymbol{v}(t) = \boldsymbol{v}\_{t}\right] \\ &= \prod\_{k=1}^{n} \Phi\_{\boldsymbol{G}}\left(T, 1; L\_{k}(\omega), h\_{k}(\omega), \boldsymbol{g}\_{k}(\omega), \kappa\_{k\prime}\theta\_{k\prime}\boldsymbol{\upgamma}\_{k\prime}\rho\_{k\prime}b\_{k\prime}c\_{k\prime}, \boldsymbol{v}\_{k,t}, \boldsymbol{S}\_{k,t}^{\*}\right) \\ &\times \prod\_{j=1}^{n} \Phi\_{\boldsymbol{G}}\left(T, 1; 0, L\_{j}(\omega), h\_{j}(\omega), \boldsymbol{g}\_{j}(\omega), \tilde{\kappa}\_{j\prime}\tilde{\theta}\_{j\prime}\tilde{\zeta}\_{j\prime}\tilde{\rho}\_{j\prime}\tilde{b}\_{j\prime}\tilde{\kappa}\_{k\prime}\tilde{\nu}\_{k\prime}\boldsymbol{\upgamma}\_{k\prime}\boldsymbol{S}\_{t}^{\*j}\right) \end{split}$$

*where* Φ*G is the one-dimensional c.f provided by Grasselli (2017) in Proposition 3.1 and given in the Appendix B for completeness.*

See Appendix A for proof. Next, we turn to the conditional c.f. of the increments of *Z* given the terminal value of the CIR processes. This is defined as follows:

$$\Phi\_{Z(t)x(T)}(\tau,\omega) = E\left[\exp\left[\omega'(Z(T) - Z(t))\right] \mid Z(t) = z\_{t\prime}v(T) = v\_T\right] \tag{8}$$

The above is useful when we need to work with the joint distribution of (*Z*(*T*), *v*(*T*)) given (*Z*(*t*), *<sup>v</sup>*(*t*)). For such cases, we can try to rely on a convenient simulation scheme combining the distribution of *Z*(*T*) given (*Z*(*t*), *v*(*T*)) (via Equation (8)) with that of *v*(*T*) given *<sup>v</sup>*(*t*), the latter is known to be non-centered chi-squared. In this way, we can avoid usual discretization algorithms such as the Euler–Maruyama or Milstein schemes, which are generally not suitable for the CIR process (due to failure of the Lipschitz condition at 0).

In this vein, when working with the non mean-reverting factor model (*βij* = 0, *i*, *j* = 1, ... , *n*), we can easily adapt the procedures in Grasselli (2017) to provide an exact simulation scheme for the model given the vector of the independent CIR process at maturity *T* (i.e., *v*(*T*)). This requires only the c.f. provided next:

**Corollary 2.** *Let* (*Z*(*t*))*t*≥0 *evolve according to the FG model (βij* = 0*, i*, *j* = 1, ... , *n). Then, the c.f.* <sup>Φ</sup>*Z*(*t*),*<sup>v</sup>*(*T*) *is then given as follows:*

$$\begin{split} \Phi\_{Z(t),v(T)}(T,\omega) &= \prod\_{j=1}^{n} \Phi\_{\mathcal{G},T} \left( T, \Phi; L, h\_{\flat}, \mathcal{g}\_{j}, \kappa\_{j}, \theta\_{j}, \tilde{\xi}\_{j}, \tilde{\nu}\_{j}, \rho\_{j}, b\_{j}, c\_{j}, \upsilon\_{j,T}, S\_{j,t}^{\*} \right) \\ &\times \prod\_{i=1}^{n} \Phi\_{\mathcal{G},T} \left( T, \mathbf{1}; \mathbf{0}, h\_{\mathcal{i}i}, \tilde{\xi}\_{i}, \tilde{\nu}\_{i}, \tilde{\theta}\_{i}, \tilde{\xi}\_{i}, \tilde{\rho}\_{i}, \tilde{\nu}\_{i}, \tilde{\upsilon}\_{i}, \tilde{\upsilon}\_{i,T}, S\_{t}^{\*i} \right) \end{split}$$

*where* <sup>Φ</sup>*G*,*<sup>T</sup> is the one-dimensional c.f provided by Grasselli (2017) in Proposition 4.1 and given in the Appendix B for completeness.*

Proof of this result is provided in Appendix A. Unsurprisingly, the previous result cannot be extended to the mean-reverting case, due to the absence of closed formulas for the object:

$$\mathbb{E}\left[\exp\left\{\mu\left(\int\_t^T B(s)\nu(s)ds + \int\_t^T \mathbb{C}(s)\frac{1}{\nu(s)}ds + \int\_t^T D(s)\ln(\nu(s))ds\right)\right\} \mid \nu(T)\right]$$

which is not solvable even when two of the three deterministic functions *<sup>B</sup>*(*s*), *<sup>C</sup>*(*s*) and *<sup>D</sup>*(*s*) are zero.

### **4. Discussion: One Common Factor in Two Dimensions**

We assume two assets, i.e., *<sup>X</sup>*1(*t*) and *<sup>X</sup>*2(*t*), with one common volatility component, and one intrinsic factor each. The asset prices thereby follow the system of SDE for *i* = 1, 2:

$$\begin{split}dY\_{i}(t) &= \left(L\_{i} - \beta\_{i}Y\_{i}(t)\right)dt \\ &+ \left( (c\_{i} - \frac{1}{2})[a\_{i}^{2}(\sqrt{v\_{1}(t)} + \frac{b\_{1}}{\sqrt{v\_{1}(t)}})^{2}] + (\tilde{c}\_{i} - \frac{1}{2})(\sqrt{\tilde{v}\_{i}(t)} + \frac{\tilde{b}\_{i}}{\sqrt{\tilde{v}\_{i}(t)}})^{2} \right)dt \\ &+ a\_{i}\left(\sqrt{v\_{1}(t)} + \frac{b\_{1}}{\sqrt{v\_{1}(t)}}\right)dW\_{1}(t) + \left(\sqrt{\tilde{v}\_{i}(t)} + \frac{\tilde{b}\_{i}}{\sqrt{\tilde{v}\_{i}(t)}}\right)d\tilde{W}\_{i}(t) \end{split}$$

$$\begin{array}{rcl} d\boldsymbol{v}\_{1}(t) &=& \boldsymbol{\alpha}\_{1}(\boldsymbol{\theta}\_{1} - \boldsymbol{\upsilon}\_{1}(t))dt + \boldsymbol{\xi}\_{1}\sqrt{\boldsymbol{\upsilon}\_{1}(t)}dB\_{1}(t) \\ d\boldsymbol{\overline{\boldsymbol{\upsilon}}\_{i}(t)} &=& \boldsymbol{\overline{\boldsymbol{\alpha}}\_{i}(\boldsymbol{\theta}\_{i} - \boldsymbol{\upsilon}\_{i}(t))}dt + \boldsymbol{\xi}\_{i}\sqrt{\boldsymbol{\upsilon}\_{i}(t)}dB\_{i}(t) \end{array}$$

with +*dBj*(*t*), *dWj*(*t*) , = *<sup>ρ</sup>jdt*, \$*dB*#*i*(*t*), *dW*#*i*(*t*) %= *ρ*#*idt* for *j* = 1; *i* = 1, 2.

The following table (Table 1) gives a baseline parameter set for the one-factor, two-dimensional 4/2 factor model used in the subsequent sections. The choice of parameters in Scenario **A** was made by combining the seminal works of Schwartz (1997) (see Oil and Copper in Tables IV and V) and Heston (1993). Scenario **B** combines Schwartz (1997) (see Oil and Copper, Tables IV and V) with Heston et al. (2009). In both cases, we assume a simple structure for the market price of risk (*<sup>c</sup>*1 = *c*2 = *c*˜ 1 = *c*˜ 2 = 0)2.

<sup>2</sup> Variations on *c* will be studied in future research as part of a calibration exercise (see Medvedev and Scaillet 2007 for viable approaches and Escobar and Gschnaidtner 2016 for some pitfalls).

The ˜ *θi*, *i* = 1, 2 in the table are set to match the long term volatilities as estimated in Schwartz (1997), which are 0.334 (Oil, Table IV) and 0.233 (Copper, Table V):

$$\begin{aligned} &\mathbb{E}\left[a\_1^2 \left(\sqrt{v\_1(t)} + \frac{b\_1}{\sqrt{v\_1(t)}}\right)^2 + \left(\sqrt{\vec{v}\_1(t)} + \frac{\tilde{b}\_1}{\sqrt{v\_1(t)}}\right)^2\right] \\ &= \quad a\_1^2 \left(\frac{2a\_1b\_1^2}{2a\_1\theta\_1 - \xi\_1^2} + 2b\_1 + \theta\_1\right) + \frac{2a\_1b\_1^2}{2a\_1\theta\_1 - \xi\_1^2} + 2\tilde{b}\_1 + \tilde{\theta}\_1 = (0.334)^2 \end{aligned} \tag{9}$$

This explains the values of *θi* in the table.

˜

**Table 1.** Toy parametric values.


The present section considers two independent cases. First, we study the impact of the parameters *b*1, # *b*1 and # *b*2 on implied volatility surfaces and on two risk measures for a portfolio of underlyings. We then assess the impact of the commonalities *a*1 and *a*2 on these same targets, i.e., implied volatilities and risk measures. To ensure that the cases lead to reasonable assets behavior, we report the expected return, variance of return for each asset, as well as the correlation between two assets and the leverage effects in Tables 2 and 3 under Scenarios **A** and **B**, respectively.

We simulated 500,000 paths with *dt* = 0.1 and considered the following scenarios for *b*: *b*1 = 0.008, ˜ *b*1 = ˜ *b*2 = 0; *b*1 = 0, ˜ *b*1 = ˜ *b*2 = 0.008; *b*1 = ˜ *b*1 = ˜ *b*2 = 0 and *b*1 = ˜ *b*1 = ˜ *b*2 = 0.008.

**Table 2.** First four moments for scenarios on 3/2 component (*b*), Scenario **A**.



**Table 3.** First four moments for scenarios on 3/2 component (*b*), Scenario **B**.

Similarly, we considered the following scenarios for *a*: *a*1 = *a*2 = 0; *a*1 = 0.75, *a*2 = 0; *a*1 = 0, *a*2 = 0.75 and *a*1 = *a*2 = 0.75. Tables 4 and 5 present key statistics for the returns under Scenarios **A** and **B**, respectively.

**Table 4.** First four moments for scenarios on commonalities (*a*), Scenario **A**.


**Table 5.** First four moments for scenarios on commonalities (*a*), Scenario **B**.

