Power Utility Maximization in Exponential Lévy Models: Convergence of Discrete-Time to Continuous-Time Maximizers

In exponential Lévy models the stock is given – just as its name suggests – as the (stochastic) exponential of a Lévy process. Continuous-time exponential Lévy models and their discrete-time counterparts share the defining property of independent and identically distributed logarithmic returns. Exponential Lévy models have become widely popular in the last decade since they are analytically tractable and provide a reasonable approximation to financial data.

We consider maximization of expected utility from terminal wealth with finite time horizon for an investor equipped with a power utility function $U(x)=(1-p)^{-1}x^{1-p}$, $p>0$, in the framework of one bond (normalized to 1) and one stock which follows an exponential Lévy process. The problem of maximizing expected power utility from terminal wealth was first studied by Merton ([15]) for Brownian motion with drift and by Samuelson ([20]) for the discrete-time analogon in an $N$-period model. The optimal portfolio selection for general time-continuous exponential Lévy processes was investigated in [5], [10] and [2]. Nutz ([16], for $p\neq 1$), as well as Kardaras ([12], for $p=1$) have recently shown under minimal assumptions that the optimal fraction¹¹1Following a usual practice for power utiltiy functions, we note that we interpret in this exposition trading strategies as the fraction of current wealth invested in the stock, rather than the amount of shares the agent holds. Thus, for a given trading strategy $\pi_{t}$ the agent invests the fraction $\pi_{t}$ of her current wealth in the stock, and the fraction $1-\pi_{t}$ of current wealth in the bond. $\pi^{*}$ of wealth invested in the continuous-time exponential Lévy model is constant and given as the maximizer of a deterministic concave function $g$, which is defined in terms of the Lévy triplet (also cf., e.g., [6] for sufficient conditions in a general setting, and [4] for sufficient conditions on the optimality of constant trading strategies). Similarly, the optimal trading strategy for an $N$-period exponential Lévy model is to invest a constant fraction $\pi^{*}_{N}$ of current wealth in each step. This stems from the fact that the logarithmic return of the stock in each period is i.i.d. and consequently the $N$-period model becomes an iterated 1-period model. The optimal fraction $\pi_{N}^{*}$ is given as the maximizer of a deterministic concave function $g^{N}$, where $g^{N}$ only depends on the evolution of the Lévy process in the first step.

In this exposition we assume that a continuous-time exponential Lévy model is given. We define the discrete-time $N$-period exponential Lévy model as the restriction of the continuous-time model to discrete time instants of distance $\frac{T}{N}$, where $T$ is the finite time horizon. In particular, the discrete logarithmic returns are given by i.i.d. random variables. Hence, trading in this discrete-time model amounts to trading in the original continuous-time model with the restriction that the portfolio can only be adjusted at given time instant, i.e., if $t=\frac{kT}{N}$, $k=0,\ldots,N$. This restriction also involves a discretization gap in the sense that the set of admissible trading strategies for the $N$-period model is in general strictly smaller than the respective set of the continuous-time model. For this reason the optimal fraction $\pi^{*}$ invested in the stock of the continuous-time model might in general be not admissible for the discrete-time models. This somewhat surprising consequence was already mentioned by Rogers ([19]) in the case of the classical Merton problem, which we consider in more detail below.

If the Lévy process has a non-zero Brownian motion part or allows both positive and negative jumps, it is easily seen that the interval of admissible constant trading strategies of all $N$-period models is given by $[0,1]$. This means that the agent is allowed to invest in each period the fraction of 0% to 100% of current wealth in the stock. Short-selling and investing more than 100 % of current wealth is prohibited since negative wealth is not allowed by power utility investors. In the other (less important) case of a pure jump process allowing either only positive or only negative jumps, the interval of admissible trading strategies for the $N$-period model is more complicated and depends on the number of discretization points $N$. Although not technically necessary, in order to simplify the further discussion we will assume throughout this exposition that the Lévy process has a non-zero Brownian motion part or allows both positive and negative jumps. Hence, the interval of admissible constant $N$-period trading strategies is in particular given by $[0,1]$. We comment separately on the case of a pure jump process with either only positive or only negative jumps.

The objective of this exposition is to answer the following two questions:

1. 1.)

   Do the optimal discrete-time strategies $\pi_{N}^{*}$ converge to the optimal continuous-time strategy $\pi^{*}$ as $N\to\infty$, i.e., as the number of discretization points increases?

2. 2.)

   How does the sign of the drift of the Lévy process and the risk aversion of the power utility funcion affect the optimal continuous-time strategy $\pi^{*}$ and the discrete-time strategies $\pi_{N}^{*}$?

Clearly, a necessary assumption for the convergence of $\pi_{N}^{*}\in[0,1]$ to $\pi^{*}$ is $\pi^{*}\in[0,1]$. But $\pi^{*}\in[0,1]$ generally fails to hold already in the Black-Scholes model due to the previously mentioned discretization gap: in the classical Merton problem the Lévy process is given by $L_{t}=\mu t+\sigma B_{t}$, where $B$ denotes Brownian motion. Let $U(x)=(1-p)^{-1}x^{1-p}$ with $p>0$, then the optimal continuous-time strategy is given by the constant $\pi^{*}=\frac{\mu}{\sigma^{2}p}$, cf. [15]. Thus, $\pi^{*}\notin[0,1]$ if e.g. $\mu$ is chosen large enough and $\pi_{N}^{*}$ will not converge to $\pi^{*}$.

To cope with this problem we introduce in the continuous-time exponential Lévy model the exogenous portfolio constraint $\mathscr{C}=[0,1]$. I.e., in this constrained continuous-time model the agent may only invest between $0$% and $100$% of her current wealth in the stock, which is the same restriction as in the $N$-period models. The optimal continuous-time strategy $\pi_{\mathscr{C}}^{*}$ of the constrained model can be found similarly as in the unconstrained case as the maximium of the same function $g$ on $\mathscr{C}=[0,1]$. In particular, $\pi^{*}\in[0,1]$ implies $\pi^{*}=\pi^{*}_{\mathscr{C}}$.

Coming back to our two questions we see that we have to refine the first one due to the discretization gap and rather ask

1. 1.)’

   Do the optimal discrete-time strategies $\pi_{N}^{*}$ converge to the optimal strategy $\pi^{*}_{\mathscr{C}}$ of the constrained continuous-time model as $N\to\infty$, i.e., as the number of discretization points increases?

In order to respond to both questions we observe that we can access $\pi^{*}_{\mathscr{C}}$ and $\pi_{N}^{*}$ as the maximizer of the previously mentioned deterministic and concave functions $g|_{[0,1]}$ and $g^{N}$, respectively. A major part in answering the questions is done in Theorem 4.1 below, which shows that $g^{N}$ converges uniformly on $[0,1]$ to $g$ as $N\to\infty$. Our main results can then be summarized as follows:

In the light of our results, we see that the exogenous portfolio constraint $\mathscr{C}=[0,1]$ is a natural assumption to make the continuous-time model resemble the discrete-time models. As we have already mentioned, $\pi^{*}\in[0,1]$ implies $\pi^{*}=\pi^{*}_{\mathscr{C}}$. On the other hand, if $\pi^{*}\notin[0,1]$ then $\pi_{\mathscr{C}}^{*}$ and $\pi^{*}_{N}$ are easily found: if $\pi^{*}<0$ then $\pi_{N}^{*}=\pi_{\mathscr{C}}^{*}=0$, and if $\pi^{*}>1$ then $\pi_{N}^{*}=\pi_{\mathscr{C}}^{*}=1$.

The approximation of a continuous-time Lévy model by discretizations has been covered under different aspects in the literature. Sufficient conditions for the convergence of optimal trading strategies in certain diffusion models and their discrete approximations were found in [7]. Rogers ([19]) compared optimal investors of a diffusion model and its discretization in terms of efficiency and his work was extended in [1] to the case of partially available information. Another aspect of discretization in terms of time-lagged trading and its impact on utility maximzation was discussed in [18] for exponential Lévy models.

The paper is organized as follows. In Section 1 we specify the continuous-time and discrete-time exponential Lévy models, as well as the optimization problem of maximizing expected utility from terminal wealth. We state the necessary assumptions (Assumption 2.1) for our results and recall certain findings of [16], as well as some properties of stochastic exponentials of Lévy processes that we shall frequently use. Section 3 summarizes important analytic properties of the deterministic functions $g$ and $g^{N}$. In Section 4 we state and prove our main results. We also comment on the extension of Theorem 4.1 to higher dimensional exponential Lévy processes. Certain technical results needed for the convergence of the wealth processes (Corollary 4.3) are presented in the Appendix.

It is important to mention that our convergence theorem (Theorem 4.1) relies on the work of [8] and [3] on moment asymptotics for Lévy processes. We also use results on the finiteness of $g$, that are proved in [16]. The convergence results in the Appendix are extensions of the results in [13] on Euler discretizations.

We fix a finite time horizon $T>0$ and a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},{\mathbb{P}})$ satisfying the usual assumptions of right-continuity and completeness. The exponential Lévy model of one bond normalized to $1$ and one stock $S$ with initial value $S_{0}>0$ is given as the solution of the SDE

where $(L_{t})_{t}$ is a Lévy process with Lévy-Khintchine triplet $(b(h),c,F)$ relative to a continuously differentiable truncation function $h$, with drift $b(h)\in\mathbb{R}$, diffusion $c>0$ and Lévy measure $F$ on $\mathbb{R}$. We refer to [9, II.4] for more information on Lévy processes.

By (2.1), $S$ is given by the stochastic exponential of $L$, i.e., $S_{t}=S_{0}\mathcal{E}\left(L\right)_{t}$. The agent is endowed with initial capital $x_{0}>0$ and her preferences are given by a power utility function $U(x)=\frac{x^{1-p}}{1-p}$ with $p>0$, where we set throughout this exposition $\frac{x^{0}}{0}:=\log(x)$, to cover also the case of logarithmic utility. Let $\mathcal{C}\subseteq\mathbb{R}$ denote an interval of portfolio constraints. Throughout this exposition we are only interested in the unconstrained case $\mathcal{C}=\mathbb{R}$ and in the exogenous constraint $\mathcal{C}=\mathscr{C}:=[0,1]$.

We define the set of all admissible constant trading strategies by the convex set

where $\pi$ denotes the fraction of wealth invested in the stock. Since we fix the utility function $U$, we denote the corresponding set of admissible constant trading strategies by $\mathcal{A}$ to simplify further notation, while we still have in mind that we mean either of (2.2), (2.3). The requirements in (2.2) and (2.3), respectively, ensure that $x_{0}\mathcal{E}\left(\pi L\right)_{t}\geq 0\;(>0)$ holds for all $\pi\in\mathcal{A}_{0}\;(\pi\in\mathcal{A}_{0,*})$ and $U(x_{0}\mathcal{E}\left(\pi L\right)_{T})>-\infty$, cf. [17, Lemma 2.5]. Here, $x_{0}\mathcal{E}\left(\pi L\right)_{t}$ is the wealth process corresponding to the constant trading strategy $\pi\in\mathcal{A}$, i.e., it is the solution of

We also consider non-constant admissible trading strategies $\pi_{t}(\omega)$ given by

where $\pi\;\begin{picture}(1.0,1.0)(0.0,-3.0)\circle*{3.0}\end{picture}\;L$ denotes the stochastic integral of $\pi_{t}$ w.r.t. $L$.

It is the agent’s aim to find an optimal trading strategy $\pi^{*}_{t}$ for the problem of maximizing expected utility problem from terminal wealth on $[0,T]$

In order to discuss the optimal trading strategy $\pi^{*}_{t}$ in more detail we now state the assumptions that we make throughout this exposition:

As already mentioned in the introduction, by (ii) we make the assumption that $L$ has a non-zero Brownian motion part or that $L$ has both positive and negative jumps. Although this is not technically necessary it simplifies further discussion on the admissibility of $N$-period trading strategies below. In particular, assumption (ii) assures a no arbitrage condition known as no unbounded increasing profit, which has been introduced in [12]. We comment on the less important case of a Lévy process with $c=0$ and either only positive or only negative jumps at the end of this section.

Assumption (iii) is a common assumption assuring existence and uniqueness of a maximizer of (2.5), i.e., the $\sup$ in (2.5) is in fact a $\max$.

Under Assumption 2.1, the optimal trading strategy $\pi^{*}_{t}$ of (2.5) becomes trackable and is of a simple structure: Nutz (cf. [16, Theorem 3.2 and Remark 3.3], for $p\neq 1$) and Kardaras (cf. [12, Lemma 5.1], for $p=1$) have shown that under our assumptions the optimal trading strategy $\pi^{*}$ is unique and constant, $\pi^{*}\in\mathcal{A}\cap\mathcal{C}$. In the following, we denote the optimal strategy of the unconstrained model ($\mathcal{C}=\mathbb{R}$) by $\pi^{*}$ and the optimal strategy under the exogenous portfolio constraint $\mathcal{C}=\mathscr{C}=[0,1]$ by $\pi^{*}_{\mathscr{C}}$, which we refer to as the optimal strategy of the constrained continuous-time model.

The optimal wealth processes are given by $x_{0}\mathcal{E}\left(\pi^{*}L\right)$, $x_{0}\mathcal{E}\left(\pi^{*}_{\mathscr{C}}L\right)$ and an investigation of [14] even shows that the Inada conditions upon $U$ ensure

i.e., $\pi^{*}\in\mathcal{A}_{0,*}$ for all $p>0$. $\mathcal{E}\left(\pi^{*}_{\mathscr{C}}L\right)_{t}>0$ holds true since $\mathcal{A}\cap[0,1]=[0,1]$ and $\mathcal{E}\left(\pi L\right)_{t}>0$ for all $\pi\in[0,1]$. [16, Theorem 3.2] (for $p\neq 1$) and [11, Lemma 5.1] (for $p=1$) show that $\pi^{*}$, $\pi^{*}_{\mathscr{C}}$ are given as the unique maximizer of the deterministic concave function $g:\mathcal{A}\cap\mathcal{C}\to\mathbb{R}$ defined by

where we have used $F(-\infty,-1]=0$. In particular, the function $g$ corresponding to the constrained case $\mathscr{C}=[0,1]$ is the restriction on $[0,1]$ of the function $g$ corresponding to unconstrained case $\mathcal{C}=\mathbb{R}$. This also shows that $\pi^{*}\in[0,1]$ implies $\pi^{*}=\pi^{*}_{\mathscr{C}}$. Moreover, the concavity of $g$ shows that $\pi^{*}>1$ ($\pi^{*}<0$) implies $\pi^{*}_{\mathscr{C}}=1$ ($\pi^{*}_{\mathscr{C}}=0$).

(2.10) implies that $\pi^{*}$, $\pi^{*}_{\mathscr{C}}$ are independent of the initial capital $x_{0}$, which is a well known property of power utility functions even in more general market models. By [16, Theorem 3.2] the optimal wealth processes $x_{0}\mathcal{E}\left(\pi^{*}L\right)_{t}$ and $x_{0}\mathcal{E}\left(\pi^{*}_{\mathscr{C}}L\right)_{t}$, respectively, lead to the terminal utility

We now construct the $N$-period model associated to (2.1): let $\sigma_{N}:=\{t_{0}=0,t_{1}=\frac{T}{N},t_{2}=\frac{2T}{N},\ldots,t_{N}=T\}$ for $N\in{\mathbb{N}}$. We define the discrete-time process $S^{N}$ by

i.e., $S^{N}$ is the restriction of $S$ to $\sigma_{N}$. Hence, investing in $S^{N}$ amounts to investing in $S$ with the restriction that the portfolio may only be adapted at time instants $t_{i}\in\sigma_{N}$. From now on we refer to $S^{N}$ as the $N$-period model approximation of (2.1). $S^{N}$ evolves in each period according to

where $Z^{N}_{i}=\frac{S(t_{i+1})-S(t_{i})}{S(t_{i})}=\frac{\mathcal{E}\left(L\right)_{t_{i+1}}-\mathcal{E}\left(L\right)_{t_{i}}}{\mathcal{E}\left(L\right)_{t_{i}}}=e^{\widetilde{L}_{t_{i+1}}-\widetilde{L}_{t_{i}}}-1$ are i.i.d. random variables. Investing in each period the fraction $\pi$ of current wealth in the stock $S^{N}$ amounts to the wealth process

For the same economic agent and utility function $U$ as above, the set of admissible constant trading strategies for $S^{N}$ is given by

$\mathcal{A}^{N}$ is given by $[0,1]$ and does not depend on $p$ and $N$ since we assume that $L$ satisfies $c\neq 0$ or allows both positive and negative jumps. We also find $\mathcal{A}^{N}=[0,1]\subseteq\mathcal{A}$.

The set of non-constant admissible trading strategies $(\pi_{i}(\omega))_{i}$ is given by

Similarly to (2.5) it is the agent’s aim to find for each $N$ a maximizer $(\pi_{N,i}^{*})_{i=0}^{N-1}$ of

Maximizing (2.13) leads to a constant maximizer $\pi^{*}_{N}\in\mathcal{A}^{N}=[0,1]$, i.e., $\pi_{N,i}^{*}=\pi_{N}^{*}\in[0,1]$. Moreover, $\pi_{N}^{*}$ is in fact the maximizer of the following 1-period model

and is independent of the initial wealth $x_{0}$, cf. [20]. We want to give a short sketch by backward induction why $\pi^{*}_{N}$ is constant: Given current wealth $X_{t_{N-1}}=x$, the Bellman principle and $Z^{N}$ being i.i.d. show that the optimal strategy $\pi^{*}_{N,N-1}(x)$ for the last step maximizes (2.14) with $x_{0}=x$. Moreover, since $U$ is a power utility function, $\pi^{*}_{N,N-1}$ is in fact independent of $x$. Hence, $\pi_{N,N-1}^{*}=\pi^{*}_{N}$. Similar arguments as well as $(Z^{N}_{i=n+1})^{N}$ being i.i.d. – and in particular independent of the current wealth $X_{t_{n}}$ – prove $\pi_{N,n}^{*}=\pi^{*}_{N}$ for the induction step.

Analogously to $g$ above, we define for each $N$ the deterministic concave function $g^{N}:[0,1]\to\mathbb{R}$ by

Clearly, $\pi^{*}_{N}$ is a maximizer of $g^{N}$. The reason why we have slightly modified $g^{N}$ in comparison to (2.14) will come apparent in the proof of Lemma 4.6 below.

This section summarizes some results on the continuity and differentiability of the functions $g$ and $g^{N}$, defined in (2.10) and (2.15), respectively.

For the proof of Proposition 3.1 we need the following elementary result on exchanging differentiation and integration for concave functions, which is a special case of [17, Lemma 5.14].

Our main convergence result is stated in Theorem 4.1 below, which also implies two important corollaries. Proposition 4.4 gives qualitative properties of the optimal strategies $\pi_{N}^{*}$ and $\pi^{*}$ related to their sign and the risk aversion of $U$. To improve readability, we move the proofs of Theorem 4.1 and Proposition 4.4 to the end of this section. We comment on the extension of Theorem 4.1 to higher dimensions at the end of its proof.

The proof of the next corollary of Theorem 4.1 requires some results on the convergence of Euler approximations of (2.4), which can be found in the Appendix.