Dating back to the 1980s, in the seminal work of [1] (equations II.1–II.8), the excess return of a security, also known as its risk premium, has been prescribed as proportional to powers of the volatility. Specifically, three models were proposed, all presented in terms of the market price of risk (MPR) —that is, technically the ratio of the excess return and the volatility of the security. The first model, type I, assumed a MPR proportional to volatility (i.e., power $1/2$). This model implies that each risk factor earns a risk premium that is proportional to the variance of the factor’s return. The second and the third models (types II and III) postulate constant MPR (i.e., power 0 on variance) and constant excess return (i.e., power $-1/2$ on variance, inversely proportional to volatility), respectively. These models have been widely used in the literature; see [2,3,4] for examples involving stochastic volatility (SV), stochastic interest, and jumps.

The specification of MPRs play a very important role in expected utility portfolio optimization. In this context, [5] solved the portfolio optimization problem for MPR of types I and II, in a setting of CRRA (power) utility, in an incomplete market with finite horizon for the Heston model (aka the 1/2 model). [6] considered the optimal investment and consumption problem in an incomplete market for the 3/2 model of [7] with Epstein-Zin-Weil recursive utility and an infinite horizon, which implies a value function independent of time. In particular, the authors considered two forms of excess return —constant and linear in the variance (i.e., the MPRs of types I and III). Nevertheless, the exact solution is only available when the agent’s elasticity of intertemporal substitution is one with constant excess return (type III). For all other cases, the solutions are approximations.

Our paper presents the very first closed-form analysis for type I MPR on the recently proposed 4/2 model (see [8]) with a finite horizon, consumption, and complete markets. This leads, as a by-product, to the first analysis of type I MPR on the 3/2 model. We incorporate several ingredients of interest to practitioners in an EUT setting: complete markets1, consumption and terminal wealth, and ambiguity aversion.

Two recent studies have been conducted on the 4/2 model under MPRs outside of the settings in [1], while excluding consumption in their analyses. First, [9] explored the optimal investment problem for a risk-averse investor in both incomplete and complete market in the absence of consumption. The authors employed the same MPR for the 1/2 and 3/2 components —proportional to $\sqrt{v_t}$, the driver of variance. This means the Heston component follows the type II MPR in [5], whereas the 3/2 component follows type I MPR in [6]. Second, the work of [10] considered an investor that is not only risk-averse, but also ambiguity-averse.

Solving the optimal consumption and asset allocation with the advanced 4/2 model for a type I MPR is challenging. The fact that our closed-form solutions are non-affine is proof of this challenge and an important departure from the exiting literature. When the MPR is proportional to a 4/2-structured volatility, the risk premium/excess return is proportional to the variance, ${\left(a\sqrt{v_t}+\frac{b}{\sqrt{v_t}}\right)}^2$. This means there are nonlinear elements in the drift of the equity, which jeopardizes affine solutions and the solvability of the implied partial differential equations (PDEs) in the corresponding Hamilton-Jacobi-Bellman (HJB) and HJB Isaacs (HJBI) equations.

The contributions of our work are as follows:

The paper is structured as follows. Section 2 describes the 4/2 model under consideration and the derivatives needed in the portfolio. Section 3 presents and solves the consumption and terminal wealth expected utility problem for a risk-averse investor. Section 3.1 focuses on the complete market case. Section 4 then extends the problematic to an ambiguity-averse investor, with a section on complete markets (Section 4.1). Section 5 studies and implements the top three main cases numerically. First, Section 5.1 analyses a risk-averse investor in a complete market with consumption. Second, Section 5.2 studies an ambiguity-averse complete market investor with no consumption. Finally, Section 6 concludes the paper. All proofs are provided with details in a complementary Appendix A.

We assume the stochastic processes describing the financial market are defined on a complete probability space $(\Omega ,\mathcal{F},\mathbb{P})$ with a right-continuous filtration ${\left\{{\mathcal{F}}_t\right\}}_{t\in [0,T]}$. The price process $S_t$ of the risky asset follows the so-called 4/2 model: where $v_t$ is the variance driver, which follows a CIR with mean-reversion rate $\kappa >0$, long-run mean $\theta >0$, and volatility of volatility $\sigma >0$. The Feller condition (i.e., $2\kappa \theta \ge {\sigma }^2$) is also imposed to keep the process $v_t$ strictly positive. The standard Brownian motions (BM) $W_t$ in dynamic of risky asset $S_t$ and $Z_{1t}$ in the dynamic of variance driver $v_t$ are correlated with parameter $\rho \in (-1,1)$. Thus, we will write $d{W}_t=\rho d{Z}_{1t}+\sqrt{1-{\rho }^2}d{Z}_{2t}$, where $Z_{2t}$ is another standard BM, independent of $Z_{1t}$. The variance, denoted by $z_t=z\left(v_t\right)$ is given as follows:This setting implies market prices of risk with the following representation: where $\overline{\lambda }=\left(\rho {\overline{\lambda }}_1+\sqrt{1-{\rho }^2}{\overline{\lambda }}_2\right)$, a and b are positive constants, ${\overline{\lambda }}_1$ and ${\overline{\lambda }}_2$ are constant. ${\lambda }_1\left(v_t\right)$ is the market price of variance risk, and $\lambda \left(v_t\right)$ is the market price of stock risk. Moreover, ${\lambda }_2\left(v_t\right)$ can be interpreted as the market price of stock idiosyncratic risk (i.e., with respect to $Z_{2t}$). Note that in this form of market price of risk, the excess return of the risky asset is proportional to its variance as recommended in the economics literature; see [1] equation (II.6), type I. As for the market price of variance risk ${\lambda }_1\left(v_t\right)$, we use Ito’s lemma to create the process of the variance:Hence our choice of market price of variance risk is: ${\overline{\lambda }}_1\left(a\sqrt{v_t}+\frac{b}{\sqrt{v_t}}\right)={\overline{\lambda }}_1\sqrt{z_t}$. That is, it is proportional to the volatility of the asset. This is similar to the proposal in [2].

As pointed out by [8], a risk-neutral measure may not exist in the 4/2 model, which is a feature inherited from having the 3/2 model [11]. This failure causes the discounted asset price process may be a strict $\mathcal{Q}$-local martingale, and not a true $\mathcal{Q}$-martingale that equivalent to the historical measure $\mathcal{P}$. Thus, we explore the feasibility of changing measure under the market price of risk introduced in Equation (4).

In the next proposition we identify the parametric conditions needed for the existence of a valid risk-neutral measure $\mathcal{Q}$, which follows [9,12]. These conditions will be assumed throughout the paper.

See Appendix A.1 for the complete proof.

Furthermore, we assume the investor can also allocate on a financial derivative on the underlying. Let $O_t=m(S_t,v_t,t)$ denote the price of the option. It can be shown that the option price evolves with the stochastic differential equation (SDE): where $m_S=\frac{dm}{dS}$, and $m_v=\frac{dm}{dv}$ denote the partial derivatives of option price function m with respect to $S_t$ and $v_t$. Equations (1), (2), and (7) are considered as the reference model.

We consider the investor exhibits CRRA utility for both intermediate consumption and terminal wealth with the same risk-aversion level $\gamma$. That is, we define the utility functions for consumption and for terminal wealth (abusing notation slightly): where coefficients ${ϵ}_1$ and ${ϵ}_2$ are non-negative. The ratio ${ϵ}_1/{ϵ}_2$ indicates the relative importance of intermediate consumption and terminal wealth, and it thus affects decision-making (optimal strategy). Without loss of generality, we can set ${ϵ}_2=1$, and let ${ϵ}_1$ determine the relative importance ratio.

The objective of the investor is to maximize their utility from intermediate consumption $c_t$ and terminal wealth $X_T$; therefore the reward functional for the investor is defined as follows: where $\delta$ is a discount rate, $\Theta$ is a control variable to be clarified in the next section, and the goal is where $\overline{J}(x,v,t)$ is the value function and the space $\mathcal{U}$ of admissible controls ${\{{\Theta }_t,c_t\}}_{t\in [0,T]}$ with ${\Theta }_t\in {\mathbb{R}}^2$, $c_t\in {\mathbb{R}}^{+}$, is the set of feedback strategies that satisfy standard conditions (see [13]).

Assume the investor is uncertain about the probability distribution for the reference model and consider a set of plausible, alternative models when making investment decisions. Specifically, the investor is uncertain about the distribution of $Z_{1t}$ and $Z_{2t}$.

Let $e:=(e_t^v,e_t^S)$ be an ${\mathbb{R}}^2$-valued $\mathcal{F}$-progressively measurable process and define the Radon-Nikodym derivative process by According to Girsanov’s theorem, the process is a Wiener process under probability measure ${\mathbb{P}}^e$. Let $\epsilon [0,T]$ denote the set of all ${\mathcal{F}}_t$-progressively measurable processes such that the process (25) is a well-defined Radon-Nikodym derivative process. This formulation of incorporating investor’s model uncertainty is actually allowing uncertainty on the drift of diffusion risk factors of the stock and its variance’s driver (i.e., $Z_{2t}$ and $Z_{1t}$, respectively).

The alternative model then follows: The reward functional can be defined as in the previous section for a given probability measure ${\mathbb{P}}^e$:In the presence of a preference for robustness, the investor’s objective is to minimize the penalty term and maximize his utility from intermediate consumption $c_t$ and terminal wealth $X_T$: where $\overline{J}(x,v,t)$ is the value function, and the last two terms serves as the penalty term for deviating too far from the reference model. The space $\mathcal{V}$ of ${\mathcal{F}}_t$-adapted process ${\mathbf{e}}_t=\{e^S,e^v\}\in {\mathbb{R}}^2$, is the set of perturbations; the space $\mathcal{U}$ of admissible controls ${\{{\Theta }_t,c_t\}}_{t\in [0,T]}$ (i.e., ${\Theta }_t\in \mathbb{R}$, $c_t\in {\mathbb{R}}^{+}$) is the set of feedback admissible strategies. The perturbations $e_t^v$ and $e_t^S$ in the penalty term are scaled by ${\Phi }_1$ and ${\Phi }_2$, respectively. That is, the larger the values of ${\Phi }_1$ and ${\Phi }_2$, the smaller the penalties for deviating from the reference model, which implies that the investor is more uncertain about the model. Following [15], we assume where ${\varphi }_i>0$ denotes the ambiguity-aversion parameters. In this specification, the optimal strategy would be independent of the current wealth level for a power utility investor, namely homothetic robustness in [15]. Further, ${\varphi }_1$ can be interpreted as ambiguity aversion regarding the volatility driver, while ${\varphi }_2$ represents ambiguity about the stock process.