Comparison Between Mean-Variance and Monotone Mean-Variance Preferences Under Jump Diffusion and Stochastic Factor Model

Since Markowits (1952) proposed the mean-variance preference, it became a popular criterion in portfolio selection problems. Previous works have widely considered both multi-period discrete and continuous-time portfolio selection problems under mean-variance preferences. Li and Ng (2000) and Zhou and Li (2000) use embedding techniques to overcome the time-inconsistency of mean-variance preferences, leading a great number of following works, see e.g., Černỳ and Kallsen (2007), Li et al. (2002), Lim and Zhou (2002) and Lim (2004). However, when we consider the optimal investment problems, there is still another drawback of mean-variance preferences: non-monotonicity.

For a preference $\rho$ with monotonicity, if two random variables satisfy $X\leqslant Y$ almost surely, we expect $\rho(X)\leqslant\rho(Y)$. However, mean-variance preferences do not have such property. A strictly wealthier investor may have a lower utility than others, which violates the basic assumption of economic rationality. Maccheroni et al. (2009) provide an example to illustrate the non-monotonicity of mean-variance preferences. To conquer such a disadvantage, they propose a new criterion named the monotone mean-variance preference. It is the best possible monotone approximation of the classical mean-variance preference without loss of good tractability. In this paper, we use the abbreviations MMV and MV to denote monotone mean-variances and mean-variances.

There are several works related to MMV preferences. Trybuła and Zawisza (2019) study the optimal investment problem in a continuous-time financial market model with a stochastic factor. Using the method of dynamic programming and solving the Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation, they propose a verification theorem and obtain an explicit optimal strategy of the investment under MMV preferences. Trybuła and Zawisza (2019) use the result from Zawisza (2012) and prove that the optimal strategies under MMV and MV preferences are the same, which is different from the conclusion of the single-period case considered in Maccheroni et al. (2009). The reason of this difference may be either the discreteness or the discontinuity of the wealth process. Using the analytic approach rather than comparing the two optimal strategies directly, Strub and Li (2020) generalize the conclusion of Trybuła and Zawisza (2019) to all continuous semimartingale financial market models. In the continuous market scenario, Li and Guo (2021) solve the optimal reinsurance and investment problem in a multidimensional diffusion model, using the same HJBI approach as Mataramvura and Øksendal (2008) and Trybuła and Zawisza (2019). Recently, there are also works studying MMV investment problems when the trading strategy is constrained, see e.g., Shen and Zou (2022), Hu et al. (2023) and Du and Strub (2023). They all conclude that the solutions to MMV and MV problems coincide when the asset is continuous.

As the method and result of Strub and Li (2020) can only be applied to continuous markets, a crucial question is whether the conclusion still holds true in any Lévy market. It is also an open question raised by Trybuła and Zawisza (2019) for further study. Thus, to find the relationship between the two optimal strategies under MMV and MV preferences in the Lévy market, we consider a jump diffusion financial market. Besides, an untradable stochastic factor is also included in our model for two reasons. First, if we follow the model setup of Trybuła and Zawisza (2019), then we can compare our result with the former work and the difference is the pure effect of the jump risk. Second, markets with stochastic factors are incomplete. Bäuerle and Grether (2015) propose a detailed definition of the free cash flow stream and prove that there is no free cash flow stream in any complete and arbitrage-free market. That is why the optimal strategy of an MV investor would never escape the monotone domain of MV preferences in such market. Therefore, it is necessary to consider an incomplete market in our research. It is important to note that the jump diffusion part of our model can also lead to the incomplete property of the market, depending on the specific coefficient setting of the jump strength.

From another perspective, we can regard the MMV preference as a robust utility, which is considered in many robust optimization problems, e.g., Gilboa and Schmeidler (1989), Zawisza (2010) and Øksendal and Sulem (2014). In a jump diffusion model, Mataramvura and Øksendal (2008) consider such kind of max-min problem as a stochastic differential game. An interesting characterization on the set of ambiguity probability measures is proved in Li and Guo (2021), which helps us understand the range of ambiguity probability measures of MMV preferences even in the jump diffusion market. We will discuss more about this in Remark 1. In this paper, we mainly refer to Mataramvura and Øksendal (2008) and take a similar dynamic programming method to solve the MMV optimal investment problem in our market model.

Beside the non-monotonicity, another huge drawback of MV preferences is the time-inconsistency. The main reason for the time-inconsistency is the variance term in MV preferences, which yields the in-applicability of the dynamic programming method. Bjork and Murgoci (2010) propose a general theory of Markovian time inconsistent stochastic control problems, in which three approaches are raised to fix the time-inconsistency and find the optimal control. The first is to choose whatever seems to be optimal based on the investor’s current preference, without knowing the fact that his preference may change over time. The second is to find a pre-commitment solution, which means that the investor only considers the optimal problem once at the initial time and keeps taking that strategy all the time. All of the literature on MV preferences mentioned above actually uses the pre-commitment approach. The third is to define a new equilibrium solution and use game theory terms to formulate the problem. For example, Bjork and Murgoci (2010) find an equilibrium solution to the general utility form that includes MV preferences. Zeng et al. (2013) derive the optimal strategy of reinsurance and investment that holds time-consistency for MV insurers. All three approaches can be reasonable in different circumstances, depending on the goal of the study. In our paper, similar to Trybuła and Zawisza (2019), we use the pre-commitment approach, which is the best situation for us to analyze the relationship between MMV and MV problems. As we find that the optimal strategy under MMV preferences depend on the investor’s wealth and the market state, MMV preferences also have the time-inconsistency property as MV preferences.

The main contributions of our work are as follows. We refine the optimal investment problems with respect to MMV and MV preferences in a jump diffusion and stochastic factor model. To the best of our knowledge, this is the first time that these two preferences are considered in such a financial market model with jumps. We obtain the explicit optimal strategies of MMV and MV problems under certain assumptions (Conditions of Proposition 1 and Assumption 1). We prove that the two optimal strategies and value functions coincide if and only if Assumption 1 holds. Thus, the discontinuity of the market model may cause investors with MMV preferences to choose a different strategy from those with MV preferences. We answer the open question proposed by Trybuła and Zawisza (2019) and extend the conclusion of Strub and Li (2020). This phenomenon is interesting because it contradicts the consistency in continuous markets, where the two preferences are shown to yield the same result. Besides, under Assumption 1, we prove that the achieved maximum utilities of MMV and MV preferences are also the same, which is somehow ignored by Trybuła and Zawisza (2019). Theoretically, Assumption 1 may not hold, leading to the inconsistency of MMV and MV problems. Nevertheless, we also provide some empirical evidences to illustrate the validity of Assumption 1 in real financial market.

More specifically, we introduce the Doléans-Dade exponentials to replace the target probability measures of MMV preferences. In the MMV problem, we transform the problem into a stochastic differential game and use the dynamic programming method to obtain the HJBI equation. In the MV problem, we deal with the nonlinearity of the variance term first, and then obtain the HJB equation by combining the Lagrange multiplier and dynamic programming methods. To solve the HJBI and HJB equations, we obtain a new type of highly nonlinear parabolic partial differential equation. The equation cannot be simply transformed to a linear parabolic equation by the traditional Hopf-Cole transformation method used in Zariphopoulou (2001), Zawisza (2010) and Trybuła and Zawisza (2019). Instead, we refer to a new type existence and uniqueness theorem on parabolic equations from Ladyzhenskaya et al. (1968) to prove the existence and uniqueness of the solutions to HJBI and HJB equations. Then, we propose a jump diffusion type Feynman-Kac formula to prove the consistency of the two optimal strategies of MMV and MV problems under Assumption 1. Finally, we prove that the optimal strategies and value functions of MMV and MV problem are the same if and only if Assumption 1 holds. We also derive the efficient frontier of the investment and analyze the economic impact caused by the jump diffusion part of the risky asset.

The remainder of this paper is organized as follows. Section 2 introduces the market model and monotone mean-variance preferences. Section 3 solves the optimal investment problem with respect to MMV preferences. We formulate the HJBI equation and find the corresponding solution. Section 4 solves the optimal investment problem with respect to MV preferences. In Section 5, we compare the two optimal strategies and value functions and provide the sufficient and necessary condition for them to be consistent. Besides, we derive the efficient frontier and make some economic explanations. We also provide empirical evidences to demonstrate the validity of the assumption in real financial market. Finally, Section 6 concludes the paper.

We consider a filtered probability space $\left(\Omega;\mathcal{F};\{\mathcal{F}_{t}\}_{t\geqslant 0};\mathbb{P}\right)$ satisfying the usual conditions. $W^{1}=\{W^{1}_{t},t\in[0,T]\}$ and $W^{2}=\{W^{2}_{t},t\in[0,T]\}$ are Brownian motions. ${N}$ is a Poisson random measure on $\left([0,T]\times\mathbb{R},\mathcal{B}([0,T]\times\mathbb{R})\right)$ with respect to a Lévy process $L$. Filtration $\{\mathcal{F}_{t}\}_{t\geqslant 0}$ is generated by $W^{1}$, $W^{2}$ and $N$. We assume that $W^{1}$, $W^{2}$ and $N$ are independent. Further more, we denote $\mathrm{d}t\nu(\mathrm{d}p)$ as the compensator (or intensity measure), and $\tilde{N}$ as the corresponding compensated martingale measure of $N$, i.e., $\tilde{N}\left(\mathrm{d}t,\mathrm{d}p\right)=N\left(\mathrm{d}t,\mathrm{d}p\right)-\mathrm{d}t\nu(\mathrm{d}p)$. We assume that $\nu(\mathbb{R})<+\infty$. For more information about the Poisson random measure, see Protter (2005) and Tankov (2003).

Suppose that there are two investment assets. The price process of the risk-free asset $B=\{B_{t},t\in[0,T]\}$ is given by

And the risky asset $S=\{S_{t},t\in[0,T]\}$ follows a Lévy-Itö process, satisfying the following stochastic differential equation (SDE):

where the stochastic factor $Z=\{Z_{t},t\in[0,T]\}$ follows SDE:

Besides, we assume $\gamma>-1$ with $\int_{\mathbb{R}}\gamma(z,p)^{2}\nu(\mathrm{d}p)<\infty$, coefficients $\mu,\sigma,a$ and $b$ are continuous functions, $\gamma$ is continuous with respect to $z$, and they satisfy all the required regularity conditions to ensure that SDEs (2) and (3) have unique strong solutions. Let $\rho_{W}$ represent the correlation of Brownian motion parts of $S$ and $Z$ and $\rho_{W}^{2}+\bar{\rho}_{W}^{2}=1$.

In this paper, we compare the optimal investment strategies of investors under MMV preferences and MV preferences. Now, we introduce the MMV preferences first. Based on Maccheroni et al. (2009), the class of MMV preferences is defined by

where

Moreover, MMV preferences can be rewritten as

where

For simplicity, we denote $\Lambda_{\theta}$ as the objective function used in the MMV investment problem.

Naturally, we consider the set of all the Doléans-Dade exponential measures

to replace $\mathcal{Q}_{0}$, where the range of $(\eta,\phi)=(\eta_{1,t},\eta_{2,t},\phi_{t})_{0\leqslant t\leqslant T}$ is given by

For $\mathbb{Q}\in\mathcal{Q}$, we just restrict it to satisfy $\mathbb{Q}\ll\mathbb{P}$ and $\frac{\mathrm{d}\mathbb{Q}}{\mathrm{d}\mathbb{P}}\geqslant 0$ rather than $\mathbb{Q}\sim\mathbb{P}$ as in Trybuła and Zawisza (2019). It is a notable difference, because we cannot ensure the target $\mathbb{Q}$ to be equivalent to $\mathbb{P}$ when the market faces jump risks. For the aim of clarity, we state the definition of the Doléans-Dade exponential and stress one of its important properties here.

For more information about the Doléans-Dade exponential, see Protter (2005), Section 8 of Chapter II. Furthermore, similar to the well-known Kazamaki’s and Novikov’s criteria, we need the following criterion to determine whether a Doléans-Dade exponential is a martingale and then $\mathbb{Q}^{\eta,\phi}$ to be a true probability measure.

Now, we introduce a new process $Y^{\eta,\phi}=\{Y^{\eta,\phi}_{t},t\in[0,T]\}$ as follows:

Based on the uniqueness of SDE’s solution, we have $Y_{T}^{\eta,\phi}=y\frac{\mathrm{d}\mathbb{Q}^{\eta,\phi}}{\mathrm{d}\mathbb{P}}$ when starting with $t=0$ and $Y_{0}=y$. If we set $y=\frac{1}{2\theta}$, the objective function restricted by the maximization on $\mathcal{Q}$ becomes

Notably, when $y\neq 0$, $\frac{Y_{T}^{\eta,\phi}}{y}=\frac{\mathrm{d}\mathbb{Q}^{\eta,\phi}}{\mathrm{d}\mathbb{P}}$ should be greater than 0. Thus, not all the predictable $(\eta,\phi)$ meet the requirement. We consider a series of admissible sets of $(\eta,\phi)$:

Denote $\mathcal{Q}^{t}$ as the set of all the Doléans-Dade exponential measures with respect to $\mathcal{M}^{t}$. It is trivial to see that $\mathcal{M}^{0}=\mathcal{M}$ and $\mathcal{Q}^{0}=\mathcal{Q}$. Further, we denote $\mathbb{E}^{\mathbb{Q}^{\eta,\phi}}$ by $\mathbb{E}^{\eta,\phi}$ for simplicity. In the following discussion, we restrict the maximization (minimization) over $\mathbb{Q}$ on $\mathcal{Q}^{t}$ in the definition of $\Lambda_{\theta}$ ($V_{\theta}$), unless otherwise noted. See Remark 1 for more detailed explanations.

In this section, we solve the optimal investment problem under MMV preferences. We treat the max-min optimization problem as a stochastic differential game between an investor and a potential market player. We use the dynamic programming method to get the HJBI equation and propose a verification theorem. The optimal investment strategy is then obtained by solving the HJBI equation and verifying the verification theorem.

First, we introduce the investor’s wealth process. Let $\bar{\pi}=\{\bar{\pi}_{t},t\in[0,T]\}$ be the process of investment amount in the risky asset. Then, the wealth process $\bar{X}$ satisfies the following SDE:

Similar to Trybuła and Zawisza (2019), we consider $\pi=\{e^{r(T-t)}\bar{\pi}_{t},t\in[0,T]\}$ and $X^{{\pi}}=\{e^{r(T-t)}\bar{X}_{t}^{\bar{\pi}},t\in[0,T]\}$ as the forward value processes of the investment amount and wealth for simplicity. Then, the wealth process of the self-financed investor satisfies:

Next, we define the objective function as

Based on the Girsanov Theorem, $W^{\eta_{1}}_{t}=W^{1}_{t}-\int_{0}^{t}\eta_{1,s}\mathrm{d}s$ and $W^{\eta_{2}}_{t}=W^{2}_{t}-\int_{0}^{t}\eta_{2,s}\mathrm{d}s$ are Brownian motions under $\mathbb{Q}^{\eta,\phi}$. For the jump part, the compensator of $N\left(\mathrm{d}t,\mathrm{d}p\right)$ becomes $(1+\phi_{t}(p))\nu(\mathrm{d}p)\mathrm{d}t$, i.e., $\tilde{N}^{\phi}\left(\mathrm{d}t,\mathrm{d}p\right)=\tilde{N}\left(\mathrm{d}t,\mathrm{d}p\right)-\phi_{t}(p)\nu(\mathrm{d}p)dt$ is compensated Poisson random measure under $\mathbb{Q}^{\eta,\phi}$. Then, the dynamics under $\mathbb{Q}^{\eta,\phi}$ become:

Referring to Øksendal and Sulem (2007b), the generator $\mathscr{A}$ of $(X,Y,Z)$ is:

See Øksendal and Sulem (2007a) for more information about the Girsanov theorem and the generator.