Asymptotic Optimal Portfolio in Fast Mean-reverting Stochastic Environments

# Asymptotic Optimal Portfolio in Fast Mean-reverting Stochastic Environments

## Abstract

This paper studies the portfolio optimization problem when the investor’s utility is general and the return and volatility of the risky asset are fast mean-reverting, which are important to capture the fast-time scale in the modeling of stock price volatility. Motivated by the heuristic derivation in [J.-P. Fouque, R. Sircar and T. Zariphopoulou, Mathematical Finance, 2016], we propose a zeroth order strategy, and show its asymptotic optimality within a specific (smaller) family of admissible strategies under proper assumptions. This optimality result is achieved by establishing a first order approximation of the problem value associated to this proposed strategy using singular perturbation method, and estimating the risk-tolerance functions. The results are natural extensions of our previous work on portfolio optimization in a slowly varying stochastic environment [J.-P. Fouque and R. Hu, SIAM Journal on Control and Optimization, 2017], and together they form a whole picture of analyzing portfolio optimization in both fast and slow environments.

{keywords}

Stochastic optimal control, asset allocation, stochastic volatility, singular perturbation, asymptotic optimality.

## I Introduction

The portfolio optimization problem in continuous time, also known as the Merton problem, was firstly studied in [16, 17]. In his original work, explicit solutions on how to allocate money between risky and risk-less assets and/or how to consume wealth are provided so that the investor’s expected utility is maximized, when the risky assets follows the Black–Scholes (BS) model and the utility is of Constant Relative Risk Aversion (CRRA) type. Since these seminal works, lots of research has been done to relax the original model assumptions, for example, to allow transaction cost [15], [10], drawdown constraints [9], [4], [5], price impact [3], and stochastic volatility [18], [2], [8] and [14].

Our work extends Merton’s model by allowing more general utility, and by modeling the return and volatility of the risky asset by a fast mean-reverting process :

 dSt=μ(Yt)Stdt+σ(Yt)StdWt, (1) dYt=1ϵb(Yt)dt+1√ϵa(Yt)dWYt. (2)

The two standard Brownian motion (Bm) are imperfectly correlated: . We are interested in the terminal utility maximization problem

 Vϵ(t,x,y)≡supπ∈AϵE[U(XπT)|Xπt=x,Yt=y], (3)

where is the wealth associated to self-financing :

 dXπt=π(t,Xπt,Yt)μ(Yt)dt+π(t,Xπt,Yt)σ(Yt)dWt, (4)

(assume the risk-free interest rate varnishes ) and is the set of strategies that stays nonnegative. Using singular perturbation technique, our work provide an asymptotic optimal strategy within a specific class of admissible strategies that satisfies certain assumptions:

 (5)

### I-a Motivation and Related Literature

The reason to study the proposed problem is threefold. Firstly, in the direction of asset modeling (1)-(2), the well-known implied volatility smile/smirk phenomenon leads us to employ a BS-like stochastic volatility model. Empirical studies have identified scales in stock price volatility: both fast-time scale on the order of days and slow-scale on the order of months [7]. This results in putting a parameter in (2). The slow-scale case (corresponding to large in (2)), which is particularly important in long-term investments, has been studied in our previous work [6]. An asymptotic optimality strategy is proposed therein using regular perturbation techniques. This makes it natural to extend the study to fast-varying regime, where one needs to use singular perturbation techniques. Secondly, in the direction of utility modeling, apparently not everyone’s utility is of CRRA type [1], therefore it is important to consider to work under more general utility functions. Thirdly, although it is natural to consider multiscale factor models for risky assets, with a slow factor and a fast factor as in [8], more involved technical calculation and proof are required in combining them, and thus, we leave it to another paper in preparation [11].

Our proposed strategy is motivated by the heuristic derivation in [8], where a singular perturbation is performed to the PDE satisfied by . This gave a formal approximation . They then conjectured that the zeroth order strategy

 π(0)(t,x,y)=−λ(y)σ(y)v(0)x(t,x,y)v(0)xx(t,x,y), (6)

reproduces the optimal value up to the first order ; see Section II-B for the formulation of and .

### I-B Main Theorem and Organization of the Paper

Let (resp. ) be the expected utility of terminal wealth associated to (resp. )

 Vπ(0),ϵ:=E[U(Xπ(0)t)|Xπ(0)t=x,Yt=y],

and be the wealth process given by (4) with (resp. in ). By comparing and , we claim that performs asymptotically better up to order than the family . Mathematically, this is formulated as:

###### Theorem I.1

Under paper assumptions, for any family of trading strategies , the following limit exists and satisfies

 ℓ:=limϵ→0˜Vϵ(t,x,y)−Vπ(0),ϵ(t,x,y)√ϵ≤0. (7)

Proof will be given in Section IV as well as the interpretation of this inequality according to different ’s.

The rest of the paper is organized as follows. Section II introduces some preliminaries of the Merton problem and lists the assumptions needed in this paper. Section III rigorously gives ’s first order approximation . Section IV is dedicated to the proof of Theorem I.1, where the expansion of is analyzed first. The precise derivations are provided, but the detailed technical assumption is referred to our recent work [6].

## Ii Preliminaries and Assumptions

In this section, we firstly review the classical Merton problem, and the notation of risk tolerance function . Then heuristic expansion results of in [8] are summarized. Standing assumptions of this paper are listed, as well as some estimation regarding and . We refer to our recent work [6, Section 2, 3] for proofs of all these results.

### Ii-a Merton problem with constant coefficients

We shall first consider the case of constant and in (1). This is the classical Merton problem, which plays a crucial role in interpreting the leading order term and analyzing the singular perturbation. This problem has been widely studied extensively, for instance, see [13].

Let be the wealth process in this case. Using the notation in [8], we denote by the problem value. In Merton’s original work, closed-form was obtained when the utility is of power type. In general, one has the following results, with all proofs found in [6, Section 2.1] or the references therein.

###### Proposition II.1

Assume that the utility function is , strictly increasing, strictly concave, such that is finite, and satisfies the Inada and Asymptotic Elasticity conditions: , , , then, the Merton value function is strictly increasing, strictly concave in the wealth variable , and decreasing in the time variable . It is and is the unique solution to the HJB equation

 Mt+supπ{σ22π2Mxx+μπMx}=0, (8) M(T,x;λ)=U(x),

where is the constant Sharpe ratio. It is with respect to , and the optimal strategy is given by

 π⋆(t,x;λ)=−λσMx(t,x;λ)Mxx(t,x;λ).

We next define the risk-tolerance function

 R(t,x;λ)=−Mx(t,x;λ)/Mxx(t,x,;λ),

and two operators, following the notations in [8],

 Dk =R(t,x;λ)k∂kx,k=1,2,⋯, (9) Lt,x(λ) =∂t+12λ2D2+λ2D1. (10)

By the regularity and concavity of , is continuous and strictly positive. Further properties are presented in Section II-D. Using the relation , the nonlinear Merton PDE (8) can be re-written in a “linear” way: . We now mention an uniqueness result to this PDE, which will be used repeatedly in Sections III.

###### Proposition II.2

Let be the operator defined in (10), and assume that the utility function satisfies the conditions in Proposition II.1, then

 Lt,x(λ)u(t,x;λ)=0,u(T,x;λ)=U(x), (11)

has a unique nonnegative solution.

### Ii-B Existing Results in [8]

In this subsection, we review the formal expansion results of derived in [8]. To apply singular perturbation technique, we assume that the process is ergodic and equipped with a unique invariant distribution . We use the notation for averaging with respect to , namely, . Let be the infinitesimal generator of :

 L0=12a2(y)∂2y+b(y)∂y.

Then, by dynamic programming principle, the value function formally solves the Hamilton-Jacobi-Bellman (HJB) equation:

 Vϵt+1ϵL0Vϵ+maxπ∈Aϵ(σ(y)2π2Vϵxx/2+π(μ(y)Vϵx+ρa(y)σ(y)Vϵxy/√ϵ))=0.

In general, the value function’s regularity is not clear, and is the solution to this HJB equation only in the viscosity sense. In [8], a unique classical solution is assumed, so that heuristic derivations can be performed. However, in this paper, such assumption is not needed, as we focus on the quantity defined in (18). It corresponds to a linear PDE, which classical solutions exist. Under the assumptions in [8], the optimizer to this nonlinear PDE is of feedback form:

 π∗=−λ(y)Vϵxσ(y)Vϵxx−ρa(y)Vϵxy√ϵσ(y)Vϵxx,

and the simplified HJB equation reads:

 Vϵt+1ϵL0Vϵ−(λ(y)Vϵx+1√ϵρa(y)Vϵxy)2/(2Vϵxx)=0,

for .

The equation is fully nonlinear and is only explicitly solvable in some cases; see [2] for instance. The heuristic expansion results overcome this by providing approximations to . This is done by the so-called singular perturbation method, as often seen in homogenization theory. To be specific, one substitutes the expansion into the above equation, establishes equations about by collecting terms of different orders. In [8, Section 2], this is performed for and we list their results as follows:

1. The leading order term is defined as the solution to the Merton PDE associated with the averaged Sharpe ratio :

 v(0)t−12¯¯¯λ2(v(0)x)2v(0)xx=0,v(0)(T,x)=U(x), (12)

and by the uniqueness discussed in Proposition II.1, is identified as:

 v(0)(t,x)=M(t,x;¯¯¯λ). (13)
2. The first order correction is identified as the solution to the linear PDE:

 v(1)t+12¯¯¯λ2(v(0)xv(0)xx)2v(1)xx−¯¯¯λ2v(0)xv(0)xxv(1)x=12ρBD21v(0), (14)

with . The constant is , and solves Rewrite equation (14) in terms of the operators in (9)-(10), solves the following PDE which admits a unique solution:

 Lt,x(¯¯¯λ)v(1)=12ρBD21v(0),v(1)(T,x)=0. (15)
3. is explicitly given in term of by

 v(1)(t,x)=−12(T−t)ρBD21v(0)(t,x). (16)

### Ii-C Assumptions

Basically, we work under the same set of assumptions on the utility and on the state processes as in [6]. We restate them here for readers’ convenience. Further discussion and remarks are referred to [6, Section 2]

###### Assumption II.3

Throughout the paper, we make the following assumptions on the utility :

1. is , strictly increasing, strictly concave and satisfying the following conditions (Inada and Asymptotic Elasticity):

2. is finite. Without loss of generality, we assume .

3. Denote by the risk tolerance, Assume that , is strictly increasing and on , and there exists , such that for , and ,

 ∣∣∂ixRi(x)∣∣≤K. (17)
4. Define the inverse function of the marginal utility as , , and assume that, for some positive , satisfies the polynomial growth condition:

Note that Assumption II.3(ii) is a sufficient condition, and rules out the case , for , and . However, all theorems in this paper still hold under minor modifications to the proof. Next are the model assumptions.

###### Assumption II.4

We make the following assumptions on the state processes :

1. For any starting points and fixed , the system of SDEs (1)–(2) has a unique strong solution . Moreover, the functions and are at most polynomially growing.

2. The process with infinitesimal generator is ergodic with a unique invariant distribution, and admits moments of any order uniformly in : . The solution of the Poisson equation is assumed to be polynomial for polynomial functions .

3. The wealth process is in uniformly in , i.e., , where is independent of and .

### Ii-D Existing estimates on R(t,x;λ) and v(0)

In this subsection, we state several estimations of the risk tolerance function and the zeroth order value function , which are crucial in the proof of Theorem III.1.

By Proposition II.1 and the relation (13), is concave in the wealth variable , and decreasing in the time variable , therefore has a linear upper bound, for : , for some constant . Combining it with Assumption II.4(iii), we deduce:

###### Lemma II.5

Under Assumption II.3 and II.4, the process is in uniformly in , i.e. : where is defined in Section II-B and satisfies equation (12).

###### Proposition II.6

Suppose the risk tolerance is strictly increasing for all in (this is part of Assumption II.3 (iii)), then, for each , is strictly increasing in the wealth variable .

###### Proposition II.7

Under Assumption II.3, the risk tolerance function satisfies: , , such that , Or equivalently, , there exists , such that Moreover, one has

## Iii Portfolio performance of a given strategy

Recall the strategy defined in (6), and assume is admissible. In this section, we are interested in study its performance. That is, to give approximation results of the value function associated to , which we denote by :

 Vπ(0),ϵ(t,x,y)=E{U(Xπ(0)T)|Xπ(0)t=x,Yt=y}, (18)

where is a general utility function satisfying Assumption II.3, is the wealth process associated to the strategy and is the fast factor. Our main result of this section is the following, with the proof delayed in Section III-B.

###### Theorem III.1

Under assumptions II.3 and II.4, the residual function defined by is of order . In other words, , there exists a constant , such that , where may depend on but not on .

We recall that the usual “big O” notation: means that the function is of order , that is, for all , there exists such that , where may depned on , but not on . Similarly, we denote , if .

###### Corollary III.2

In the case of power utility , is asymptotically optimal in up to order .

{proof}

This is obtained by straightly comparing the expansion of given in [8, Corollary 6.8], and the one of from the above Theorem. Since both quantities have the approximation at order , we have the desired result.

### Iii-a Formal expansion of Vπ(0),ϵ

In the following derivation, to condense the notation, we systematically use for the risk tolerance function , and for the zeroth order strategy given in (6).

By the martingality, solves the following linear PDE: , with terminal condition . Define two operators and by , and respectively, then this linear PDE can be rewritten as:

 (L2+1√ϵL1+1ϵL0)Vπ(0),ϵ=0. (19)

We look for an expansion of of the form

 Vπ(0),ϵ=vπ(0),(0)+√ϵvπ(0),(1)+ϵvπ(0),(2)+⋯,

while the terminal condition for each term is: Inserting the above expansion of into (19), and collecting terms of and give: Since and are operators taking derivatives in , we make the choice that and are free of . Next, collecting terms of yields whose solvability condition (Fredholm Alternative) requires that . This leads to a PDE satisfied by : which has a unique solution (see Proposition II.2). And since also solves this equation, we deduce that

 vπ(0),(0)(t,x)≡v(0)(t,x)=M(t,x;¯¯¯λ)

and admits a solution:

 vπ(0),(2)(t,x,y)=−12θ(y)D1v(0)+C1(t,x),

where is defined in Section II-B and is in (9).

Then, collecting terms of order yields and the solvability condition reads . This gives an equation satisfied by : , which is exactly equation (14). This equation is uniquely solved by (see (15)). Thus, we obtain

 vπ(0),(1)≡v(1)=−12(T−t)ρBD21v(0). (20)

Using the solution of and we just identified, one deduces an expression for : where is the solution to the Poisson equation:

### Iii-B First order accuracy: proof of Theorem iii.1

This section is dedicated to the proof of Theorem III.1, which is to show the residual function is of order . To this end, we define the auxiliary residual function by

 ˜E=Vπ(0),ϵ−(v(0)+ϵ1/2v(1)+ϵvπ(0),(2)+ϵ3/2vπ(0),(3)),

where we choose in the expression of and . Since, by definition, , it suffices to show that is of order .

According to the derivation in Section III-A, the auxiliary residual function solves with a terminal condition Note that is the infinitesimal generator of the processes , one applies Feynman-Kac formula and deduces:

 ˜E(t,x,y) =ϵE(t,x,y)[∫TtL1vπ(0),(3)(s,Xπ(0)s,Ys)ds] +ϵE(t,x,y)[∫TtL2vπ(0),(2)(s,Xπ(0)s,Ys)ds] +ϵ3/2E(t,x,y)[∫TtL2vπ(0),(3)(s,Xπ(0)s,Ys)ds] −ϵE(t,x,y)[vπ(0),(2)(T,Xπ(0)T,YT)] −ϵ3/2E(t,x,y)[vπ(0),(3)(T,Xπ(0)T,YT)]. (21)

The first three expectations come from the source terms while the last two come from the terminal condition. We shall prove that each expectation above is uniformly bounded in . The idea is to relate them to the leading order term and the risk-tolerance function , where some nice properties and estimates are already established (see Section II-D).

For the source terms, straightforward but tedious computations give:

 L2vπ(0),(2)=−14θ(y)(λ2(y)−¯¯¯λ2)D21v(0), (22) L1vπ(0),(3)=12ρ2a(y)λ(y)θ′1(y)D31v(0)+ (23) 12(T−t)ρ2Ba(y)λ(y)θ′(y)D1[12D2+D1]D21v(0), (24) L2vπ(0),(3)=14ρθ1(y)(λ2(y)−¯¯¯λ2)D31v(0) (25) +12θ(y)ρB{−[12D2+D1]D21v(0)+12(T−t)(λ2(y)−¯¯¯λ2)D41v(0)} (26) +14θ(y)ρB(T−t)[12(λ2(y)−¯¯¯λ2)D2D31v(0)−λ2(y)RRxx(D2+D1)D21v(0)], (27)

where in the computation of , we use the commutator between operators and : At terminal time , they becomes and .

Note that the quantity is bounded by a constant . This is proved for in Proposition II.7, and guaranteed by Assumption II.3(iii) for , since by definition . Therefore, the expectations related to the source terms in (21) are sum of terms of the following form:

 E(t,x,y)[∫T