*3.1. Change of Measure*

Here, we explore the topic of creating a risk-neutral measure Q for pricing purposes. As noted by Grasselli (2017); Platen and Heath (2010) and Baldeaux et al. (2015) among others, a risk-neutral measure may not be supported by data in the presence of a 3/2 model (e.g., <sup>√</sup><sup>1</sup>*v*(*t*)), as the parametric constraints needed for the discounted asset price process to be a Q- martingale are violated with real data; hence, we can only produce a strict Q-local martingale (i.e., Q would be absolute continuous but not equivalent to P). In such situation, the standard risk-neutral pricing methodology would fail (biased prices), and we have to turn to the benchmark approach for pricing (see Baldeaux et al. 2015).

The next proposition entertains the following changes of measure with constant *<sup>λ</sup>j*, *λ*⊥*j* , *λi* and # *λ*⊥*i* (see Escobar and Gong 2019 for other types of changes of measures) then identifies the parametric conditions needed for the existence of a valid risk-neutral measure Q.

$$d\boldsymbol{B}\_{j}^{\mathcal{Q}}(t) \quad = \ \lambda\_{j} \left(\sqrt{\boldsymbol{v}\_{j}(t)} + \frac{\boldsymbol{b}\_{j}}{\sqrt{\boldsymbol{v}\_{j}(t)}}\right) dt + d\boldsymbol{B}\_{j}^{\mathcal{P}}(t),\\d\boldsymbol{B}\_{i}^{\mathcal{Q}}(t) = \widetilde{\lambda}\_{i} \left(\sqrt{\boldsymbol{v}\_{i}(t)} + \frac{\widetilde{\boldsymbol{b}}\_{i}}{\sqrt{\boldsymbol{v}\_{i}(t)}}\right) dt + d\boldsymbol{B}\_{i}^{\mathcal{P}}(t)$$
 
$$d\boldsymbol{B}\_{j}^{\mathcal{Q}}(t)^{\perp} \quad = \ \lambda\_{j}^{\perp} \left(\sqrt{\boldsymbol{v}\_{j}(t)} + \frac{\boldsymbol{b}\_{j}}{\sqrt{\boldsymbol{v}\_{j}(t)}}\right) dt + d\boldsymbol{B}\_{j}^{\mathcal{P}}(t)^{\perp},\\d\boldsymbol{B}\_{i}^{\mathcal{Q}}(t)^{\perp} = \widetilde{\lambda}\_{i}^{\perp} \left(\sqrt{\boldsymbol{v}\_{i}(t)} + \frac{\widetilde{\boldsymbol{b}}\_{i}}{\sqrt{\boldsymbol{v}\_{i}(t)}}\right) dt + d\boldsymbol{B}\_{i}^{\mathcal{P}}(t)^{\perp}$$

**Proposition 1.** *The change of measure is well-defined for pricing purposes under the following four conditions:*

$$\begin{array}{rcl} \mathfrak{J}\_{\mathbb{J}}^{2} & \leq & 2a\_{\overline{\mathsf{I}}}\theta\_{\overline{\mathsf{I}}} - 2\mathfrak{J}\_{\overline{\mathsf{I}}} \max\left\{ \left| b\_{\overline{\mathsf{I}}}\lambda\_{\mathsf{I}} \right|, \left| \lambda\_{\overline{\mathsf{I}}}^{\bot}b\_{\overline{\mathsf{I}}} \right|, \left| b\_{\overline{\mathsf{I}}}a\_{\overline{\mathsf{I}}}\rho\_{\overline{\mathsf{I}}} \right|, \dots, \left| b\_{\overline{\mathsf{I}}}a\_{\overline{\mathsf{I}}}\rho\_{\overline{\mathsf{I}}} \right| \right\} \end{array} \tag{1}$$

$$
\hat{\xi}\_i^2 \le \quad 2\hat{\kappa}\_i \check{\theta}\_i - 2\hat{\xi}\_i \max\left\{ \left| \hat{b}\_i \check{\lambda}\_i \right|, \left| \hat{\lambda}\_i^\perp \check{b}\_i \right|, \left| \hat{b}\_i \check{\rho}\_i \right| \right\} \tag{2}
$$

$$\max \left\{ |\lambda\_j|, |\lambda\_j^\perp| \right\} \quad < \begin{array}{c} \alpha\_j \\ \frac{\mathcal{I}\_j}{\mathcal{J}\_j} \end{array} \tag{3}$$

#

$$\max \left\{ \left| \left. \widetilde{\lambda}\_{i} \right| , \left. \left| \widetilde{\lambda}\_{i}^{\perp} \right| \right| \right\} \right\} \\ \quad < \begin{array}{ll} \widetilde{\alpha}\_{i} \\ \frac{\widetilde{\chi}^{\prime}}{\widetilde{\xi}^{i}} \end{array} \tag{4}$$

*Moreover, if βij* = 0 *for i*, *j* = 1, . . . , *n, then the following must also be satisfied:*

$$L\_i = r,\\
\mathbf{c}\_i = \sum\_{j=1}^n a\_{ij} \left( \rho\_j \lambda\_j + \sqrt{1 - \rho\_j^2} \lambda\_j^\perp \right),\\
\tilde{c}\_i = \tilde{\rho}\_i \tilde{\lambda}\_i + \sqrt{1 - \tilde{\rho}\_i^2} \tilde{\lambda}\_i^\perp \tag{5}$$

Proof is included in Appendix A.
