Robust Portfolio Choice under the Modified Constant Elasticity of Variance

Abstract

This study investigates ambiguity aversion within the framework of a utility-maximizing investor under a modified constant-elasticity-of-volatility (M-CEV) model for the underlying asset. We derive closed-form solutions of a non-affine type for the optimal allocation and value function via a Cauchy problem. This work generalizes previous results in non-ambiguous settings by extending existing work to Hyperbolic Absolute Risk Aversion utility (HARA), correcting some typos in the literature for Constant Relative Risk Aversion utility (CRRA). Helpful details and derivations are also included in the manuscript.

1. Introduction

Portfolio optimization in continuous time is a constantly evolving line of research, with both practitioners and academics actively seeking more realistic objective functions, improved models, and refined stylized facts to enhance decision-making. All of these efforts are essential to keep pace with the growing complexity of financial markets and social interactions.

In this study, we embrace the widely utilized framework of expected utility theory (EUT) as the objective function for investors. EUT provides a fruitful ground for analytical solutions and is, therefore, more easily interpretable, as pioneered in the seminal work by Merton [1]. This early achievement relied on simple geometric Brownian motion (GBM) to describe the underlying asset price with considerations only for risk aversion. Although many extensions have been considered in the literature, the vast majority involve more than one source of risk in explaining asset prices. Examples include stochastic volatility models and the addition of jumps, as evidenced in references [2,3]. However, incorporating multiple risk sources often leads to inaccuracies in the estimation and evaluation methodologies, thereby partially affecting their benefits. To preserve the simplicity of a single source of risk while generalizing the GBM, this study uses a new member of the family of CEV models (see the seminal work of Cox in [4]); the so-called modified constant elasticity of variance (M-CEV). This model, explored for pricing purposes in the reference [5], has been adapted to expected utility optimization in the recent work of Muravei in [6]. Additionally, refer to the references [7,8] for portfolio optimization results on two other types of CEV models: CEV and LVO-CEV.

The primary innovation in our work lies in considering ambiguity in the model, thereby introducing ambiguity aversion in the decision-making of the investor. The assumption of model ambiguity can be traced back to the experimental studies of Ellsberg [9] and more recently the work in [10], who demonstrate that individuals are averse to ambiguity (unknown probability) in addition to their well-known aversion to risk. A crucial framework for analytical solutions was presented in Maenhout [11], leading to a Hamilton-Jacobi-Bellman (HJBI) representation of the solution in the GBM case. Most extensions of this work consider multiple sources of risk (e.g., the reference [12] for stochastic volatility and jumps). This study is the first to extend the analysis to a member of the CEV family of models.

For clarity, we list the main contributions of our work as follows:

• We solve the expected utility portfolio problem for a HARA (Hyperbolic Absolute Risk Aversion) investor in the absence of ambiguity aversion. This solution can be reinterpreted as a constant proportion portfolio insurance (CPPI), a strategy widely employed in the financial industry (see the reference [13]).
• As a by-product of solving HARA, we identified and rectified a few typos in the literature regarding the solution for the embedded Constant Relative Risk Aversion (CRRA) case.
• We find closed-form solutions for the optimal allocation, optimal reference model, value function, and optimal wealth process for an investor exhibiting both risk and ambiguity aversion within HARA utilities. Numerical and empirical studies with our findings will be conducted in a follow-up paper.

The remainder of this paper is organized as follows. Section 2 outlines the mathematical and financial settings and formulates the problem. In Section 3, we present the derived solutions and delve into details of various embedded cases that hold significance for readers. The appendix provides additional details that are organized into subsections related to the main proposition.

2. Problem Formulation

We consider the general price process $S_t$ given by the following stochastic differential equation (SDE):

$$\frac{d{S}_t}{S_t}=\left(r+\overline{\lambda }\sigma S_t^{\psi }\right)\mathrm{d}t+\sigma S_t^{\beta }\mathrm{d}{Z}_t,s_0>0,$$

(1)

where r and $\overline{\lambda }$ are positive real numbers and $s_0$ is the initial asset price value. In this general setting, $S_t^{\beta }$ represents the volatility of the risky asset return, and $\beta$ is the elasticity of variance with respect to the stock price. We assume $\psi =2\beta$ and $\beta \ne 0$, hence Equation (1) becomes the M-CEV model (see the manuscripts [6,14]).

We assume a market consisting of money market account B and risky asset $S_t$ (such as a stock). These prices follow the following dynamics:

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{B}_t}{B_t}& =r\mathrm{d}t,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{S}_t}{S_t}& =\left(r+\overline{\lambda }\sigma S_t^{2\beta }\right)\mathrm{d}t+\sigma S_t^{\beta }\mathrm{d}{Z}_t,s_0>0,\hfill \end{array}$$

(3)

where r is a constant risk-free interest rate. As explained in the references [5,6,8], the M-CEV allows for a non-zero probability of the underlying touching zero (default) if $\beta >1$, which makes it realistic for pricing and portfolio problems.

We refer to model (3) as the reference model. Our investors face uncertainty regarding the probability distribution associated with the reference model and consider a set of plausible alternative models when making investment decisions. Specifically, the investors are uncertain about the distribution of S. This formulation enables us to capture the uncertainty regarding the drift in stock prices and the prices of risk in the stock market.

Let $e:=e_t^S$ denote an $\mathbb{R}$-valued ${\mathcal{F}}_t$-progressively measurable process. The Radon–Nikodym derivative process is defined as follows:

$$\begin{array}{c}\hfill {\zeta }_t^e=\mathbb{E}\left[\frac{\mathrm{d}{\mathbb{P}}^e}{\mathrm{d}\mathbb{P}}|{\mathcal{F}}_t\right]=exp\left(-{\int }_0^t\frac{{\left(e_{\tau }^S\right)}^2}{2}\mathrm{d}\tau +e_{\tau }\mathrm{d}Z\tau \right).\end{array}$$

(4)

According to Girsanov’s theorem, the process can be expressed as

$$\begin{array}{c}\hfill {\tilde{Z}}_t={\int }_0^t{e}_{\tau }^S\mathrm{d}\tau +Z_t.\end{array}$$

(5)

We denote the set of all ${\mathcal{F}}_t$-progressively measurable processes such that the process given by (4) is a well-defined Radon–Nikodym derivative process as $\mathcal{E}[0,T]$. This process, denoted by ${\tilde{Z}}_t$, represents the Wiener process under probability measure ${\mathbb{P}}^e$. The reference model is generally regarded as the most accurate representation of available data for an agent. However, viable alternative models pose a challenge in terms of statistical differentiation from reference models. Alternative models were generated via the perturbation process $e_t^S$ as follows:

$$\begin{array}{c}\hfill \frac{d{S}_t}{S_t}=\left(r+\overline{\lambda }\sigma S_t^{2\beta }-e_t^S\sigma S_t^{\beta }\right)\mathrm{d}t+\sigma S_t^{\beta }\mathrm{d}{Z}_t.\end{array}$$

(6)

In the model setup, it is important to acknowledge that this ambiguity stems from an investor’s inability to capture the expected returns precisely in the probability laws governing the stock price process. This assumption aligns with the perspectives presented in seminar studies, such as the work by Merton [15], and more recently shown in Tables 1 and 2 in reference [8] on the large standard errors in estimating the parameter $\overline{\lambda }$.

Let $\pi$ represent the investment strategy applied from time t to time T in model (3). We define space $\mathcal{U}[0,T]$ as the set of admissible strategies $\pi$ that satisfy the following conditions:

• $\pi$ is an ${\mathcal{F}}_t$-progressively measurable process.
• Under $\pi$, the wealth $X_t$ of the investor remains non-negative for $t\in [0,T]$.
• The integrability conditions necessary for the expectation operator in (7) to be well-defined are satisfied.

We consider an investor with a preference for Hyperbolic Absolute Risk Aversion (HARA) utility on terminal wealth with risk-aversion level $\gamma$. The utility function for terminal wealth $X_T$ is defined as:

$$u\left(X_T\right)=\frac{{\left(X_T-F\right)}^{\gamma }}{\gamma }.$$

The goal is to examine an ambiguous agent with HARA utility who aims to construct an investment strategy for the time interval $[0,T]$ that maximizes the expected utility for terminal wealth $X_T$. In line with this objective, we define the reward functional realized by the investor when selecting an alternative model specified by e as follows:

$$\begin{array}{c}\hfill w^e(x,t;\pi ,c)={\mathbb{E}}_{x,t}^{{\mathbb{P}}^e}\left[\frac{{\left(X_T-F\right)}^{\gamma }}{\gamma }\right],\end{array}$$

(7)

where $-\infty <\gamma <1$, $\gamma \ne 0$, and denotes the constant relative aversion parameter. Then, the indirect utility function is given by:

$$J(x,t)=\underset{\pi \in \mathcal{U}}{sup}\underset{e\in \mathcal{E}[t,T]}{inf}\left(w^e(x,t;\pi )+{\mathbb{E}}^{{\mathbb{P}}^e}\left[{\int }_t^T\frac{{\left(e_{\tau }^S\right)}^2}{2\Psi (\tau ,X_{\tau })}\mathrm{d}\tau \right]\right),$$

(8)

where space $\mathcal{U}$ consists of admissible controls ${\left\{{\pi }_t\right\}}_{t\in [0,T]}$ with ${\pi }_t\in \mathbb{R}$ and satisfies the standard conditions.

According to reference [16], investors consider alternative models that are statistically challenging to distinguish from reference models. To address this issue, the value function incorporates a penalty term designed to discourage significant deviations from the reference model. This penalty term, which appears as the last expectation term in the problem (8), is computed based on relative entropy. The perturbations $e_t^S$ in the penalty term were scaled by $\Psi (\tau ,X_{\tau })$. Function $\Psi$ reflects the level of ambiguity or uncertainty associated with the models and is used to adjust the penalties accordingly. Higher values of $\Psi$ indicate smaller penalties for deviating from the reference model, signifying greater uncertainty on the part of the investor.

By incorporating the penalty term, the investor considers both model ambiguity and the associated diffusion risk. This approach allows investors to make decisions that weigh the potential benefits of alternative models while accounting for the risks and uncertainties involved.

For the sake of analytical tractability, we maintain the assumption proposed in reference [11], that the ambiguity aversion parameter $\varphi$ is related to the value function $J(x,t)$ and the risk aversion level $\gamma$ by the expression

$$\begin{array}{c}\hfill \Psi =\frac{\varphi }{\gamma J(x,t)},\end{array}$$

(9)

where $\varphi >0$ denotes the level of ambiguity aversion. This assumption allows us to incorporate the degree of ambiguity into the model and analyze its impact on investors’ decision-making processes.

In summary, the value function considers investors’ preferences for terminal wealth while also considering the uncertainty associated with alternative models. The penalty term within the value function discourages significant deviation from the reference model. Scaling factor $\Psi$ is a key component that reflects investors’ level of ambiguity aversion. Higher values of $\Psi$ indicate greater uncertainty and ambiguity regarding the alternative models, which corresponds to smaller penalties for deviations from the reference model. By incorporating this penalty term and adjusting it based on the level of ambiguity, investors can make decisions that balance their preferences with the associated uncertainties in the alternative models.

3. Optimal Investment Strategies

In this section, we address the problem described in (8) by employing the stochastic control approach for the M-CEV model with ambiguity. Our objective is to derive closed-form solutions to the investment problem. The solution will provide insights into the impact of ambiguity aversion levels on investors’ decision-making processes.

3.1. The Hamilton-Jacobi-Bellman-Isaacs (HJBI) Equation

Assuming that ${\pi }_t$ represents the fraction of wealth invested in stock $S_t$, the remaining portion of wealth $(1-{\pi }_t)$ is invested in the risk-free money account with a constant interest rate r. For mathematical benefits, our focus is on the investor’s position in the stock, denoted by ${\psi }_t$ (i.e., the number of units of the asset held). Then, investor wealth $X_t$ is governed by the following stochastic differential equation (SDE) derived from a self-financing condition:

$$\begin{array}{cc}\hfill d{X}_t& =X_t\left[{\pi }_t\frac{\mathrm{d}{S}_t}{S_t}+(1-{\pi }_t)r\mathrm{d}t\right]\hfill \\ \hfill & =X_t\left(r+{\pi }_t\overline{\lambda }\sigma S_t^{2\beta }-{\pi }_t{e}_t^S\sigma S_t^{\beta }\right)\mathrm{d}t+{\pi }_t\sigma S_t^{\beta }{X}_t\mathrm{d}{Z}_t.\hfill \end{array}$$

(10)

Since ${\psi }_t={\pi }_t\frac{X_t}{S_t}$, we can express Equation (10) as follows:

$$\begin{array}{cc}\hfill d{X}_t& ={\psi }_t\mathrm{d}{S}_t+r\left(X_t-{\psi }_t{S}_t\right)\mathrm{d}t\hfill \\ \hfill & =\left[{\psi }_{t}r{S}_t+{\psi }_t\overline{\lambda }\sigma S_t^{2\beta +1}-{\psi }_t{e}_t^S\sigma S_t^{\beta +1}+r\left(X_t-{\psi }_t{S}_t\right)\right]\mathrm{d}t+{\psi }_t\sigma S_t^{\beta +1}\mathrm{d}{Z}_t\hfill \\ \hfill & =\left(r{X}_t+{\psi }_t\overline{\lambda }\sigma S_t^{2\beta +1}-{\psi }_t{e}_t^S\sigma S_t^{\beta +1}\right)\mathrm{d}t+{\psi }_t\sigma S_t^{\beta +1}\mathrm{d}{Z}_t.\hfill \end{array}$$

(11)

Consequently, the expression for the value function (8) should be modified to:

$$J(X,S,t)=\underset{\psi \in \mathcal{U}}{sup}\underset{e\in \mathcal{E}[t,T]}{inf}\left(w^e(X,S,t;\psi )+{\mathbb{E}}^{{\mathbb{P}}^e}\left[{\int }_t^T\frac{{\left(e_{\tau }^S\right)}^2}{2\Psi (\tau ,X_{\tau },S_{\tau })}\mathrm{d}\tau \right]\right),$$

(12)

where $J(X,S,t)$ satisfies a HJBI equation.

Define

$$\begin{array}{cc}\hfill {\theta }_t& :={\pi }_t\sigma S_t^{\beta },\hfill \end{array}$$

(13)

where ${\theta }_t$ represents the portfolio exposure to the fundamental risk factor $Z_t$, and ${\pi }_t$ is the portfolio weight that determines the allocation of wealth in the M-CEV model. Expression (13) illustrates how the investor’s desired exposure to the risk factor is calculated, depending on the portfolio weight ${\pi }_t$ and the volatility of the asset return $\sigma$, which is scaled by $S_t^{\beta }$. The portfolio weight determines the portion of investor wealth allocated to the risky asset, thereby influencing the overall exposure to the risk factor. As before, we can represent (13) as:

$$\begin{array}{cc}\hfill {\theta }_t& :=\frac{{\psi }_t{S}_t}{X_t}\sigma S_t^{\beta }\hfill \\ \hfill & ={\psi }_t{X}_t^{-1}\sigma S_t^{\beta +1}.\hfill \end{array}$$

(14)

This means that we can also state that the number of units of the asset dictates the allocation of the investor’s wealth to the risky asset, thus impacting overall exposure. To simplify our analysis, we shift our focus to these exposures rather than investor position $\psi$ in stock. As a result, the transformation of model (11) is given by:

$$\begin{array}{cc}\hfill d{X}_t& =\left(r{X}_t+{\psi }_t\overline{\lambda }\sigma S_t^{2\beta +1}-{\psi }_t{e}_t^S\sigma S_t^{\beta +1}\right)\mathrm{d}t+{\psi }_t\sigma S_t^{\beta +1}\mathrm{d}{Z}_t\hfill \\ \hfill & =X_t\left(r+\overline{\lambda }{\theta }_t{S}_t^{\beta }-e_t^S{\theta }_t\right)\mathrm{d}t+{\theta }_t{X}_t\mathrm{d}{Z}_t.\hfill \end{array}$$

(15)

Accordingly, the value function (12) satisfies the Hamilton (Jacobi)-Isaacs equation:

$$\begin{array}{cc}\hfill \underset{\theta }{sup}\underset{e^S}{inf}& \{J_t+x\left(r+\overline{\lambda }{S}^{\beta }\theta -e^S\theta \right)J_x+\frac{1}{2}{x}^2{\theta }^2{J}_{xx}\hfill \\ \hfill & +\left(rS+\overline{\lambda }\sigma S^{2\beta +1}-e_t^S\sigma S^{\beta +1}\right)J_s+\frac{1}{2}{\sigma }^2{S}^{2\beta +2}{J}_{ss}+x\sigma S^{\beta +1}\theta J_{xs}+\frac{{\left(e^S\right)}^2}{2\Psi }\}=0,\hfill \end{array}$$

(16)

where $J_t$, $J_x$, $J_s$, $J_{xx}$, $J_{ss}$, and $J_{xs}$ denote the partial derivatives of the first and second order with respect to time, stock and wealth, respectively. Moreover, $\Psi =\frac{\varphi }{\gamma J}$ (see (9)), where $\varphi$ is a positive ambiguity-aversion parameter.

We first find the infimum, denoted as $e^{S_{*}}$, and then differentiate (16) to obtain the optimal exposure as:

$$\left\{\begin{array}{c}{\theta }^{*}=\frac{e^{S_{*}}{J}_x-\overline{\lambda }{S}^{\beta }{J}_x-\sigma S^{\beta +1}{J}_{xs}}{x{J}_{xx}}.\hfill \end{array}\right.$$

The solution to (16) is provided in the next section.

3.2. Closed-Form Solutions for Hara Utility

To obtain the optimal strategy, we solve HJBI Equation (16) (for additional details, refer to Appendix A). The main proposition is presented below.

Proposition 1.

Assume that $\beta <0$, $\varphi >0$, and $\varphi \ne \gamma -1$. The solution to Problem (16) is as follows:

1.

The indirect utility function of an ambiguity and risk averse investor is given by

$$\begin{array}{c}\hfill J(x,S,t)=\frac{{\left(x-F{e}^{-r(T-t)}\right)}^{\gamma }}{\gamma }{g}^{1/\alpha }(S,T-t),\end{array}$$

(17)

where $\alpha =\frac{\gamma -\varphi }{\gamma (\varphi -\gamma +1)}$, and we have the following properties:

(a) 

Function $g(S,T-t)$ is determined by solving the Cauchy problem:

$$\left\{\begin{array}{c}\mathcal{L}g\equiv -g_t+\frac{1}{2}{\sigma }^2{S}^{2\beta +2}{g}_{ss}+\alpha S\left(r\frac{\gamma (\varphi -\gamma +1)}{\gamma -\varphi }+\frac{\overline{\lambda }\gamma \sigma S^{2\beta }}{\gamma -\varphi }\right)g_s+\frac{\alpha \gamma {\left(\overline{\lambda }{S}^{\beta }\right)}^2}{2(\varphi -\gamma +1)}g+\alpha \gamma rg=0,\hfill \\ g(S,0)=1\mathit{when}t=T.\hfill \end{array}\right.$$

(18)

(b) 

Solution $g(S,T-t)$ can be represented as:

$$g(S,T-t)=e^{R\tilde{\tau }+zB\left(\tilde{\tau }\right)}{D}^{\delta }\left(\tilde{\tau }\right)\frac{\Gamma (\eta -\delta +1/2)}{\Gamma (1+2\eta )}{e}^{-\frac{z}{2}A\left(\tilde{\tau }\right)}{\left(zA\left(\tilde{\tau }\right)\right)}^{\delta }{M}_{\delta ,\eta }\left(zA\left(\tilde{\tau }\right)\right),$$

(19)

where $z=\omega S^{-2\beta }$, $\tilde{\tau }={\sigma }^2{\beta }^2\omega (T-t)$, $\Gamma \left(x\right)$ is the gamma function, $M_{\delta ,\eta }\left(x\right)$ is the Whittaker function with parameters

$$\delta =-\frac{1}{2}-\frac{1}{2\beta }\left(\frac{1}{2}-\frac{\alpha \overline{\lambda }\gamma }{\sigma (\gamma -\varphi )}\right),\eta =\sqrt{{\left(\delta +\frac{1}{2}\right)}^2+\frac{\alpha \gamma {\overline{\lambda }}^2}{4{\sigma }^2{\beta }^2(\gamma -\varphi -1)}}.$$

Meanwhile, the remaining constants and functions are given by

$$\omega =\frac{r}{{\sigma }^2\mid \beta \mid },Q=\frac{r}{{\sigma }^2\beta \omega },R=\frac{r\alpha \gamma }{{\sigma }^2{\beta }^2\omega }-2Q\delta ,$$

(20)

$$A\left(\tilde{\tau }\right)=\frac{1}{2{sinh}^2\tilde{\tau }(coth\tilde{\tau }+Q)},B\left(\tilde{\tau }\right)=\frac{Q^2-1}{2\left(coth\tilde{\tau }+Q\right)},D\left(\tilde{\tau }\right)=\frac{Q^2-1}{4A\left(\tilde{\tau }\right)B\left(\tilde{\tau }\right)}.$$

(21)

2.

The optimal exposure to the risk factor $Z_t$ is

$${\theta }^{*}=\frac{\left(x-F{e}^{-r(T-t)}\right)S^{\beta }}{x}\left[\frac{\overline{\lambda }}{\varphi -\gamma +1}+\sigma S\left(B\left(\tilde{\tau }\right)+\frac{\delta +\eta +\frac{1}{2}}{z}\frac{M_{\delta +1,\eta }\left(zA\left(\tilde{\tau }\right)\right)}{M_{\delta ,\eta }\left(zA\left(\tilde{\tau }\right)\right)}\right)\frac{\mathrm{d}z}{\mathrm{d}S}\right],$$

(22)

which is equivalent to writing

$${\theta }^{*}=\frac{(x-F{e}^{-r(T-t)})S^{\beta }}{x}\left[\frac{\overline{\lambda }}{\varphi -\gamma +1}-2\sigma \omega \beta S^{-2\beta }B\left(\tilde{\tau }\right)-2\sigma \beta \left(\delta +\eta +\frac{1}{2}\right)\frac{M_{\delta +1,\eta }\left(\omega A\left(\tilde{\tau }\right)S^{-2\beta }\right)}{M_{\delta ,\eta }\left(\omega A\left(\tilde{\tau }\right)S^{-2\beta }\right)}\right].$$

(23)

3.

The worst-case measure is determined by:

$$\begin{array}{c}\hfill e^{s_{*}}=\frac{\varphi \overline{\lambda }{S}^{\beta }}{\varphi -\gamma +1}+\frac{\varphi \sigma S^{\beta +1}}{\gamma -\varphi }\left[B\left(\tilde{\tau }\right)+\frac{\delta +\eta +\frac{1}{2}}{z}\frac{M_{\delta +1,\eta }\left(zA\left(\tilde{\tau }\right)\right)}{M_{\delta ,\eta }\left(zA\left(\tilde{\tau }\right)\right)}\right]\frac{\mathrm{d}z}{\mathrm{d}S},\end{array}$$

(24)

which can also be written as follows:

$$e^{s_{*}}=\frac{\varphi \overline{\lambda }{S}^{\beta }}{\varphi -\gamma +1}+\frac{\varphi \sigma S^{\beta }}{\varphi -\gamma }\left[-2\omega \beta S^{-2\beta }B\left(\tilde{\tau }\right)-2\beta (\delta +\eta +\frac{1}{2})\frac{M_{\delta +1,\eta }\left(\omega A\left(\tilde{\tau }\right)S^{-2\beta }\right)}{M_{\delta ,\eta }\left(\omega A\left(\tilde{\tau }\right)S^{-2\beta }\right)}\right].$$

(25)

Proof. 

The proof is divided into several steps, all of which are presented in Appendix A.

Appendix A.1 derives the solution to the HJBI equation up to the Cauchy representation of g.

Appendix A.2 details the solution to the Cauchy equation. In particular, Appendix A.2.1 provides the scaling transformations needed to obtain an equation for a new function h. Appendix A.2.2 finds the PDE for h. Appendix A.2.3 applies a Laplace transform to solve the PDE for h. Appendix A.2.4 combines all results to obtain g. Lastly, Appendix A.2.5 computes a ratio involving derivatives of g needed for the next step.

Appendix A.3 uses the previous results to derive the optimal exposure and worst-case measure, denoted as ${\theta }^{*}$ and $e^{s_{*}}$, respectively, which are dedicated to the Whittaker function. □

As indicated by reference [6], in the absence of ambiguity and for CRRA, it is crucial to emphasize that assuming a risk-free interest rate r of zero leads to the following simplifications:

$$\begin{array}{c}\omega =z=\tilde{\tau }=0,\\ Q=R=0.\end{array}$$

Meanwhile, the limit values from the expressions (21) are provided by:

$$\begin{array}{cc}\hfill & {lim}_{\omega ⟶0}D\left(\tilde{\tau }\right)=1,\hfill \\ \hfill & {lim}_{\omega ⟶0}zB\left(\tilde{\tau }\right)=0,\hfill \\ \hfill & {lim}_{\omega ⟶0}R\tilde{\tau }=0,\hfill \\ \phi (S,t)\hfill & :={lim}_{\omega ⟶0}zA\left(\tilde{\tau }\right)=\frac{1}{2}{\sigma }^{-2}{\beta }^{-2}{S}^{-2\beta }{(T-t)}^{-1}.\hfill \end{array}$$

Consequently, function $g(S,t)$ in Proposition 1 can be simplified, resulting in the following corollary.

Corollary 1.

Assume that the risk-free interest rate of $r=0$, $\beta <0$, $\varphi >0$, and $\varphi \ne \gamma -1$.

1.

The indirect utility function is given by:

$$\begin{array}{c}\hfill J(x,S,t)=\frac{{(x-F)}^{\gamma }}{\gamma }{g}^{1/\alpha }(S,T-t),\alpha =\frac{\gamma -\varphi }{\gamma (\varphi -\gamma +1)},\end{array}$$

(26)

and we have the following properties:

(a) 

Function $g(S,T-t)$ solves the Cauchy problem

$$\left\{\begin{array}{c}\mathcal{L}g\equiv -g_t+\frac{1}{2}{\sigma }^2{S}^{2\beta +2}{g}_{ss}+\frac{\alpha \overline{\lambda }\gamma \sigma S^{2\beta +1}}{\gamma -\varphi }{g}_s+\frac{\alpha \gamma {\left(\overline{\lambda }{S}^{\beta }\right)}^2}{2(\varphi -\gamma +1)}g=0,\hfill \\ g(S,0)=1.\hfill \end{array}\right.$$

(27)

(b) 

Solution $g(S,T-t)$ is given by

$$\begin{array}{c}\hfill g(S,T-t)=\frac{\Gamma (\eta -\delta +1/2)}{\Gamma (1+2\eta )}{e}^{-\frac{1}{2}\phi (S,t)}{\phi }^{\delta }(S,t)M_{\delta ,\eta }\left(\phi (S,t)\right),\end{array}$$

(28)

where $\Gamma \left(x\right)$ is the gamma function, $M_{\delta ,\eta }\left(x\right)$ is the Whittaker function with parameters

$$\delta =-\frac{1}{2}-\frac{1}{2\beta }\left(\frac{1}{2}-\frac{\alpha \overline{\lambda }\gamma }{\sigma (\gamma -\varphi )}\right),\eta =\sqrt{{\left(\delta +\frac{1}{2}\right)}^2+\frac{\alpha \gamma {\overline{\lambda }}^2}{4{\sigma }^2{\beta }^2(\gamma -\varphi -1)}},$$

and there is a limit value

$$\begin{array}{c}\hfill \phi (S,t)=\underset{\omega ⟶0}{lim}zA\left(\tilde{\tau }\right)=\frac{1}{2}{\sigma }^{-2}{\beta }^{-2}{S}^{-2\beta }{(T-t)}^{-1}.\end{array}$$

(29)

2.

The optimal exposure is determined by:

$$\begin{array}{c}\hfill {\theta }^{*}=\frac{\left(x-F\right)S^{\beta }}{x}\left[\frac{\overline{\lambda }}{\varphi -\gamma +1}-2\sigma \beta (\delta +\eta +\frac{1}{2})\frac{M_{\delta +1,\eta }\left(\phi (S,t)\right)}{M_{\delta ,\eta }\left(\phi (S,t)\right)}\right].\end{array}$$

(30)

3.

The worst-case measure can be obtained as:

$$\begin{array}{c}\hfill e^{s_{*}}=\frac{\varphi \overline{\lambda }{S}^{\beta }}{\varphi -\gamma +1}-\frac{2\varphi \sigma \beta S^{\beta }(\delta +\eta +\frac{1}{2})}{\gamma -\varphi }\frac{M_{\delta +1,\eta }\left(\phi (S,t)\right)}{M_{\delta ,\eta }\left(\phi (S,t)\right)}.\end{array}$$

(31)

Proof. 

The proof is available in Appendix A.4. □

For further simplicity, we set F equal to 0; this would effectively be a CRRA utility setting. Assuming a risk-free interest rate of $r=0$, we obtain an even simpler case.

Corollary 2.

Consider risk-free interest rates of $r=0$, $\beta <0$, $\varphi >0$, and $\varphi \ne \gamma -1$. We have the following properties.

1.

The indirect utility function of an ambiguity and risk averse investor is given by:

$$\begin{array}{c}\hfill J(x,S,t)=\frac{x^{\gamma }}{\gamma }{g}^{1/\alpha }(S,T-t),\alpha =\frac{\gamma -\varphi }{\gamma (\varphi -\gamma +1)},\end{array}$$

(32)

where the function $g(S,t)$ is identical to those in parts $\left(a\right)$ and $\left(b\right)$ of Corollary 1.

2.

The optimal exposure to the risk factor $Z_t$ is determined by:

$$\begin{array}{c}\hfill {\theta }^{*}=\frac{\overline{\lambda }{S}^{\beta }}{\varphi -\gamma +1}-2\sigma \beta S^{\beta }(\delta +\eta +\frac{1}{2})\frac{M_{\delta +1,\eta }\left(\phi (S,t)\right)}{M_{\delta ,\eta }\left(\phi (S,t)\right)}.\end{array}$$

(33)

Proof. 

The proofs are analogous to those presented in Corollary 1. □

Note that the Cauchy problem and worst-case scenario are the same as those in Corollary 1.

3.3. Corrections to the Crra Case

Our work extends the results of the previous study in presented in reference [14] and in the formal publications [6]. However, discrepancies were identified in the Cauchy problem (2.13), as well as in parameters (3.5) and (3.6). This is related to Theorems 1 and 2 in the manuscript [6]. In this section, we address and rectify these inaccuracies by amalgamating the corrections for the aforementioned issues.

Corollary 3.

(Correction) For the M-CEV model, the value function $J(x,S,t)$ is given by:

$$J(x,S,t)=\frac{x^{\gamma }}{\gamma }{f}^{\frac{1}{\delta }}(S,t),\delta =\frac{1}{1-\gamma }.$$

1.

The function f solves the Cauchy problem,

$$\left\{\begin{array}{c}\mathcal{L}f(S,t)\equiv f_t+\frac{a^2{S}^{2\beta +2}}{2}{f}_{ss}+\delta S\left(\alpha -\gamma r+c{a}^2{S}^{2\beta }\right)f_s+\frac{\delta (\delta -1)}{2a^2}{\left[(\alpha -r)S^{-\beta }+c{a}^2{S}^{\beta }\right]}^{2}f+r\gamma \delta f=0,\hfill \\ f(S,T)=1.\hfill \end{array}\right.$$

(34)

2.

The solution of boundary-value problem (34) is the function

$$\begin{array}{c}\hfill f(S,t)=e^{R\tau +zB\left(\tau \right)}{D}^{\lambda }\left(\tau \right)\frac{\Gamma (\eta -\lambda +\frac{1}{2})}{\Gamma (1+2\eta )}{e}^{-\frac{zA\left(\tau \right)}{2}}{\left(zA\left(\tau \right)\right)}^{\lambda }{M}_{\lambda ,\eta }\left(zA\left(\tau \right)\right),\end{array}$$

(35)

where $z=\Lambda S^{-2\beta }$, $\tau =a^2{\beta }^2\Lambda (T-t)$, $\Gamma \left(z\right)$ is the gamma function, $M_{\lambda ,\eta }\left(z\right)$ is the Whittaker function with parameters

$$\lambda =-\frac{1}{2}-\frac{1}{2\beta }\left(\frac{1}{2}-\delta c\right),\eta =\sqrt{{\left(\lambda +\frac{1}{2}\right)}^2+\frac{\delta (1-\delta )c^2}{4{\beta }^2}}.$$

The remaining constants and functions are given by:

$$\begin{array}{c}\Lambda =\frac{\sqrt{\delta }}{a^2\mid \beta \mid }\sqrt{{\alpha }^2-\gamma r^2},Q=\frac{\delta \left(\alpha -\gamma r\right)}{\Lambda a^2\beta },R=\frac{r\gamma \delta }{a^2{\beta }^2\Lambda }-2Q\lambda -\frac{\delta (1-\delta )(\alpha -r)c}{\Lambda a^2{\beta }^2},\hfill \\ A\left(\tau \right)=\frac{1}{2{sinh}^2\tau (coth\tau +Q)},B\left(\tau \right)=\frac{Q^2-1}{2\left(coth\tau +Q\right)},D\left(\tau \right)=\frac{Q^2-1}{4A\left(\tau \right)B\left(\tau \right)}.\hfill \end{array}$$

3.

The optimal investor strategy is:

$${\pi }^{*}(X,S,t)=X\left[\delta \frac{\alpha -r+c{a}^2{S}^{2\beta }}{a^2{S}^{2\beta +1}}+\left(B\left(\tau \right)+\frac{\left(\lambda +\eta +\frac{1}{2}\right)M_{\lambda +1,\eta }\left(A\left(\tau \right)z\right)}{z{M}_{\lambda ,\eta }\left(A\left(\tau \right)z\right)}\right)\frac{\mathrm{d}z}{\mathrm{d}S}\right].$$

(36)

Based on the remark in reference [6], and assuming that $\alpha =r\sqrt{\gamma }$, it is easy to see that ${\alpha }^2-\gamma r^2=0$. Consequently, $\Lambda =\frac{\sqrt{\delta }}{a^2\mid \beta \mid }\sqrt{{\alpha }^2-\gamma r^2}=0$. Therefore, the corrected outcomes are as follows:

Corollary 4.

If $\alpha =r\sqrt{\gamma }$, Formulas (35) and (36) are simplified. In this case, the following properties exist:

1.

$$\Lambda =\tau =z=0,$$

and the limit values are

$$\begin{array}{c}{lim}_{\Lambda ⟶0}D\left(\tau \right)=1,\theta (S,t)={lim}_{\Lambda ⟶0}zA\left(\tau \right)=0.5a^{-2}{\beta }^{-2}{S}^{-2\beta }{(T-t)}^{-1},\hfill \\ \Omega (S,t)={lim}_{\Lambda ⟶0}zB\left(\tau \right)=\frac{1}{2S^{2\beta }}\frac{{\delta }^2{\left(\alpha -\gamma r\right)}^2(T-t)}{a^2\left[1+\beta \delta (\alpha -\gamma r)(T-t)\right]},\hfill \\ \Psi \left(t\right)={lim}_{\Lambda ⟶0}R\tau =r\gamma \delta (T-t)-2\delta \beta \lambda (\alpha -\gamma r)(T-t)-\frac{\delta (1-\delta )c(\alpha -r)(T-t)}{a^2}.\hfill \end{array}$$

2.

The solution $f(S,t)$ is given by:

$$\begin{array}{c}\hfill f(S,t)=\frac{\Gamma (\eta -\lambda +\frac{1}{2})}{\Gamma (1+2\eta )}{e}^{\Psi \left(t\right)+\Omega (S,t)-\frac{1}{2}\theta (S,t)}{\theta }^{\lambda }(S,t)M_{\lambda ,\eta }\left(\theta (S,t)\right).\end{array}$$

(37)

3.

The optimal policy is:

$${\pi }^{*}(X,S,t)=X\left[\delta \frac{\alpha -r+c{a}^2{S}^{2\beta }}{a^2{S}^{2\beta +1}}-\frac{2\beta \left(\lambda +\eta +\frac{1}{2}\right)}{S}\frac{M_{\lambda +1,\eta }\left(\theta (S,t)\right)}{M_{\lambda ,\eta }\left(\theta (S,t)\right)}\right].$$

4. Discussion

Our findings reveal closed-form solutions to all elements of interest to a financial investor: optimal investment allocation on the risky asset, optimal alternative model due to ambiguity, optimal wealth process evolution, and finally, the expression for the value function (i.e., the value of the objective function at the optimal). This is very rare outside exponentially linear structures, which are also known as affine solutions.

Our choice of model, the M-CEV, has the advantage of keeping only one source of risk, and therefore, a lower parametric space; this is ideal for practitioners, always searching for a combination of realism and simplicity.

Our methodology to find the solution involved many transformations of the original PDE problem (from the HJBI equation), resulting in analytical expressions for a non-trivial Cauchy problem. The details provided in our work can serve as a foundation for future research in dealing with complex PDEs and boundary conditions. Thanks to the detailed analysis, we could detect and correct problems in previously known solutions to embedded problems (M-CEV in the absence of ambiguity with a CRRA utility).

This work has the potential for many extensions and applications. First, other CEV models in the literature have provided solutions (in some cases, approximations) to the embedded problem of CRRA with no ambiguity. The methodology developed here can be used to solve such problems analytically, while extending to HARA and ambiguity. Once this is achieved, an empirical comparison would help shed light on the importance of the various models in the presence/absence of ambiguity as well as in the presence/absence of a floor on wealth (HARA versus CRRA).

5. Ethical Compliance

All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki Declaration and its later amendments or comparable ethical standards.