Optimal Portfolio under Fractional Stochastic Environment Jean-Pierre Fouque∗
Ruimeng Hu†
March 22, 2017
arXiv:1703.06969v1 [q-fin.MF] 20 Mar 2017
Abstract Rough stochastic volatility models have attracted a lot of attentions recently, in particular for the linear option pricing problem. In this paper, starting with power utilities, we propose to use a martingale distortion representation of the optimal value function for the nonlinear asset allocation problem in a (non-Markovian) fractional stochastic environment (for all Hurst index H ∈ (0, 1)). We rigorously establish a first order approximation of the optimal value, where the return and volatility of the underlying asset are functions of a stationary slowly varying fractional Ornstein-Uhlenbeck process. We prove that this approximation can be also generated by a fixed zeroth order trading strategy providing an explicit strategy which is asymptotically optimal in all admissible controls. Furthermore, we extend the discussion to general utility functions, and obtain the asymptotic optimality of this fixed strategy in a specific family of admissible strategies.
Keywords: Optimal portfolio, Fractional stochastic processes, Martingale distortion, Asymptotic optimality.
1
Introduction
In this paper, we study the Merton problem under a non-Markovian fractional stochastic environment, and we are able to provide an explicit trading strategy which is asymptotically optimal in the case of power utilities and asymptotically optimal in a specific family of general utilities. The portfolio optimization problem was first studied in the continuous-time framework by Merton Merton [1969, 1971], where risky assets are considered following the Black-Scholes-Merton model with constant returns and constant volatilities. Under this setup, Merton provided explicit solutions on how to trade stocks and/or how to consume so as to maximize one’s utility, when the utility function is of specific types, for instance, Constant Relative Risk Aversion (CRRA). After these seminal papers, the optimal portfolio and consumption problem has been extensively studied in the financial market subject to imperfections. For instance, Cox and Huang [1989] and Karatzas et al. [1987] studied the case of incomplete market; transaction cost has been considered by Magill and Constantinides [1976] and a user’s guide by Guasoni and Muhle-Karbe [2013]; investment under portfolio constraint are studied by Grossman and Zhou [1993], Cvitanic and Karatzas [1995] and Elie and Touzi [2008], just to name a few. A key factor in Merton problem is the modeling of underlying assets, and empirical studies suggest that volatility is stochastic. In this direction, we refer the readers to Zariphopoulou [1999] for the case of non-linear local volatility models, Chacko and Viceira [2005] for the case of a particular Heston-like stochastic volatility model, Lorig and Sircar [2016] for the case of local-stochastic volatility, and Kramkov and Schachermayer [2003] for the case of general analysis for semimartingale models, to list a few. Most of the work have focused on the Markovian models of the volatility. However, in a recent series of papers, non-Markovian structured models seem to better describe the data, especially short-range dependence. In Gatheral et al. [2014], it is beautifully demonstrated that stochastic volatility driven by a fractional Brownian motion (fBm) with Hurst coefficient H < 21 , so-called rough fractional stochastic volatility (RFSV), is essentially relevant to observed data. Jaisson and Rosenbaum [2016] and Omar et al. [2016] showed that RFSV is a natural scaling limit of a general model of Limit Order Book (LOB) based on Hawkes processes. ∗ Department of Statistics & Applied Probability, University of California,
[email protected]. Work supported by NSF grant DMS-1409434. † Department of Statistics & Applied Probability, University of California,
[email protected].
1
Santa Barbara,
CA 93106-3110,
Santa Barbara,
CA 93106-3110,
Meanwhile, multi-scale factor models for risky assets were considered in the portfolio optimization problem in Fouque et al. [2016] and Fouque and Hu [2016b], where return and volatility are driven by a fast mean-reverting factor and a slowly varying factor. Specifically, Fouque et al. [2016] heuristically provided the asymptotic approximation to the value function and the optimal strategy for general utility functions, by analyzing a non-linear Hamilton-Jacobi-Bellman partial differential equation (HJB PDE). In this paper, we shall consider both the scales and non-Markovian structure for modeling the underlying assets. As in Fouque and Hu [2016a], and in particular because of the relevance for long-term investments, we only consider one slowly varying fractional stochastic factor denoted by Ztδ,H for 0 < H < 1. The cases with fast mean-reverting as well as multi-scale models, limited to H > 1/2, are studied in the paper in preparation Fouque and Hu [2017]. As in Garnier and Solna [2015], we model Ztδ,H by a fractional Ornstein-Uhlenbeck (fOU) process, which satisfies the following stochastic differential equation (SDE) (H)
dZtδ,H = −δaZtδ,H dt + δ H dWt
,
(H)
where δ is a small parameter, and Wt is a fractional Brownian motion with Hurst index H. We refer to Section 3.1 for a brief introduction to fBm and fOU, and to Mandelbrot and Van Ness [1968], Cheridito et al. [2003], Coutin [2007], Biagini et al. [2008], Kaarakka and Salminen [2011] for more details. Pricing options under such RFSV models is indeed a challenge since the model is non-Markovian and PDE tools are no longer available. However, when the fractional stochastic volatility factor is slowly varying (small δ), one can obtain a practical approximation using the so-called “epsilon-martingale decomposition” method designed in Fouque et al. [2000] and Fouque et al. [2001]. This has been recently carried out for slowly varying RFSV models in Garnier and Solna [2015] where a correction to Black-Scholes formula for fractional SV is obtained. Note that the problem is non-Markovian but remains linear in the case of option pricing. Main results. In this paper, we study the nonlinear terminal utility maximization problem under the RFSV model (3.10). For power utilities, by a martingale distortion representation, we rigorously obtain an expression for the value process at any time and for all H ∈ (0, 1), as well as an expression for the corresponding optimal portfolio. In the regime of small δ, these expressions take the form of a leading order term plus a first order correction of order δ H . This is done by expanding the martingale distortion representation around a “frozen” volatility at the observed value Z0δ,H at time t = 0. For H relatively small, close to 0.1 as demonstrated in Gatheral et al. [2014], the first order correction of the value process is relatively large, and should also be generated by any good practical strategy. Our result nicely shows that the leading order of the optimal strategy, which is explicit in terms of the underlying asset and the current factor level, therefore easily implemented, will generate the value function up to order δ H , that is including the first correction. In other words, the δ H term in the expression of the optimal strategy is not needed to give such correction to the value process. However, it is given explicitly and can be easily implemented to improve the strategy by taking into account inter-temporal hedging. For general utility functions, using the epsilon-martingale decomposition method and the properties of the risk tolerance function for the Merton problem with constant coefficient, we obtain an approximation for the portfolio value corresponding to a given strategy, and, as in Fouque and Hu [2016a] in the Markovian case, we show that this strategy is asymptotically optimal in a specific class of admissible strategies. Organization of the paper. The rest of the paper is organized as follows. In Section 2, we present the martingale distortion transformation under general stochastic volatility models first derived in the Markovian case in Zariphopoulou [1999], and in non-Markovian settings in Tehranchi [2004]. Here the drift and volatility of the underlying asset are driven by a stochastic process which is not required to be Markovian nor a semimartingale. We also present a generalization to the multi-asset case. In Section 3, we derive the asymptotic results when the stochastic factor is fractional and slowly varying. The approximation to the value process and optimal portfolio are given in Section 3.3 and Section 3.4 respectively. It is also shown that the leading order of the optimal portfolio is optimal in the full class of admissible strategies up to δ H . Merton problem with a general utility function is discussed and asymptotic optimality results are presented in Section 4. We conclude in Section 5.
2
2
Merton Problem with Power Utilities and Stochastic Environment
Denote by St the underlying asset price whose return and volatility are driven by a stochastic factor Yt , dSt = µ(Yt )St dt + σ(Yt )St dWt ,
(2.1)
with assumptions on µ(y) and σ(y) to be specified later. Here Yt is a general stochastic process that is adapted to Gt := σ WuY : 0 ≤ u ≤ t , where WtY is a Brownian motion generally correlated with the Brownian motion Wt driving the price St :
d Wt , WtY = ρ dt, |ρ| < 1.
Also define Ft as the natural filtration generated by (Wt , WtY ). Denote by π the investor’s strategy and by Xtπ the corresponding wealth process. The quantity πt ∈ Ft represents the amount of money invested in the risky asset at time t, while the rest Xtπ − πt earns a risk-free interest rate r (constant). Assuming that the strategy π is self-financing, and, without loss of generality, that the risk-free interest rate is zero, r = 0, the dynamics of the wealth process Xtπ is given by: dXtπ = πt µ(Yt ) dt + πt σ(Yt ) dWt .
(2.2)
The investor’s goal is to find the optimal strategy so as to maximize her expected utility of terminal wealth. Mathematically, she aims at identifying the optimal value Vt := sup E [U (XTπ )|Ft ] ,
(2.3)
π∈At
and the optimal strategy π ∗ . The set At is the class of all admissible strategies: At := {πt ∈ Ft : Xsπ in (2.2) stays nonnegative ∀s ≥ t, given Ft } ,
(2.4)
where zero is an absorbing state for Xtπ (bankruptcy), and U (·) is a utility function. Additional assumptions on At will be described later. Specifically, U (·) will be of power type in this section and Section 3, and be a general utility function with additional assumptions in Section 4. In order to motivate the martingale distortion transformation that we will introduce in Section 2.2, we first recall in the next subsection the distortion transformation obtained by Zariphopoulou [1999] in the Markovian case with power utility U (x) =
x1−γ , 1−γ
γ > 0,
γ 6= 1.
(2.5)
In Section 2.3, we generalize to the multi-asset case.
2.1
The Distortion Transformation
In the Markovian setup, Yt is a diffusion process following the stochastic differential equation of the form dYt = k(Yt ) dt + h(Yt ) dWtY , and the value function V (t, x, y) := supπ∈At E [U (XTπ )|Xt = x, Yt = y] is a solution to the Hamilton-JacobiBellman (HJB) equation given in Fouque et al. [2016]. The distortion transformation is given by V (t, x, y) = with q=
x1−γ Ψ(t, y)q , 1−γ
γ γ + (1 − γ)ρ2 3
(2.6)
(2.7)
which results in canceling (Ψy )2 terms in the HJB equation. Consequently, Ψ solves the linear PDE 1 2 1−γ 2 1−γ Ψt + h (y)∂yy + k(y)∂y + λ(y)ρh(y)∂y Ψ + λ (y)Ψ = 0, Ψ(T, y) = 1, 2 γ 2qγ where λ(y) is the Sharpe ratio λ(y) := µ(y)/σ(y). By Feynman-Kac formula, we observe that Ψ can be expressed as h 1−γ R T i e e 2qγ t λ2 (Ys ) ds Yt = y , Ψ(t, y) = E
e W fY = W Y − where under P, t t
(2.8)
R t 1−γ ρ γ λ(Ys ) ds is a standard Brownian motion. 0
The formula in the next subsection generalizes (2.8) without using any PDE argument.
2.2
Martingale Distortion Transformation
The martingale distortion transformation is motived by the formulas (2.6) and (2.8). It has been derived in Tehranchi [2004] with a slightly different utility function. For the sake of clarity, we restate it here, and we propose a short proof based on verification using stochastic calculus. Note that in the following Proposition 2.3, Yt is a general stochastic process adapted to Gt , and it does not need to be Markovian, nor a semimartingale. In particular, in the Section 3, we will be able to apply it to the case that Yt is a fractional process. Assumption 2.1 (Power Utility). The admissible strategies π ∈ At in (2.4) satisfy # "Z " # T π 2p(1−γ) π −2γ 2 2 < +∞, f or some p > 1, and E E sup (Xt ) πt σ (Yt ) dt < ∞. (Xt ) t∈[0,T ]
0
Before introducing the assumptions on the processes (St , Yt ), we need to define a new probability e by measure P ( Z ) Z T T e dP 1 = exp − as dWsY − a2 ds , (2.9) dP 2 0 s 0
where at is Gt -adapted given by
at = −ρ e W f Y := W Y + so that under P, t t at is bounded.
Rt 0
1−γ γ
λ(Yt ),
(2.10)
as ds is a standard Brownian motion. Under the following assumption,
Assumption 2.2. (i) The SDE (2.1) for St has a unique strong solution. The function λ(·) is assumed to be bounded and C 2 (R), and functions λ′ (·), and λ′′ (·) are at most polynomially growing. e (ii) Define the P-martingale
i h 1−γ R e e 2qγ 0T λ2 (Ys ) ds Gt , Mt = E
then, one has
fY , dMt = Mt ξt dW t
′
by the Martingale Representation Theorem. We require ξ· ∈ L2p (Ω × [0, T ]) for some p′ ≥ p is the constant in Assumption 2.1.
(2.11) (2.12) p p−1 ,
where
Proposition 2.3. Let St follow the dynamics (2.1), and suppose the objective is (2.3) with power utility function (2.5). Under Assumption 2.1 and 2.2, the value process Vt is given by Vt =
iq RT 2 Xt1−γ h e 1−γ E e 2qγ t λ (Ys ) ds Gt . 1−γ 4
(2.13)
e is computed with respect to P e introduced in (2.9). The parameter q is given in term The expectation E[·] of γ and ρ by (2.7). The optimal strategy π ∗ is λ(Yt ) ρqξt πt∗ = Xt , (2.14) + γσ(Yt ) γσ(Yt ) where ξt is given in (2.12). The conditioning with respect to Gt corresponds to the separation of variable in the Markovian case presented in Section 2.1. Remark 2.4. (i) Note that γ = 1 in (2.5) is the log utility case, which can be treated separately. (ii) For the degenerate case λ(y) ≡ λ0 , the value process Vt is reduced to Vt =
2 Xt1−γ 1−γ e 2γ λ0 (T −t) . 1−γ
The quantity at = −ρ 1−γ λ0 is a constant and direct computation from (2.11) yields ξt = 0. γ Consequently, the optimal control π ∗ becomes πt∗ =
λ0 Xt . γσ(Yt )
In this case, both Vt and πt∗ do not depend on at and q as expected. (iii) In the uncorrelated case ρ = 0, the problem is already “linear”, since q = 1. The value process Vt and the optimal control π ∗ are simplified as Vt =
i RT 2 Xt1−γ h 1−γ E e 2γ t λ (Ys ) ds Gt , 1−γ
πt∗ =
λ(Yt ) Xt . γσ(Yt )
Proof of Proposition 2.3. The proof follows a verification argument, that is, in order to prove that Vt is indeed the value process and π ∗ given in (2.14) is optimal, one needs to prove (i) for any control πt ∈ At satisfying Assumption 2.1, the process (2.13) is a supermartingale, and (ii) Vt is a martingale under the control (2.14) which needs to be admissible. Let αt be the proportion of the wealth invested in St at time t, namely, πt = αt Xt , then the wealth process (2.2) can be rewritten as: dXt = Xt [αt µ(Yt ) dt + αt σ(Yt ) dWt ] .
(2.15)
In the following proof, we shall first derive the drift part of dVt , then obtain α∗t by maximizing the drift over α, and finally show that the drift part corresponding to α∗t is zero with the right choice of at and q. e Recall the P-martingale Mt defined in (2.11), and rewrite Vt using Mt as Vt =
where Nt = − 1−γ 2γ
Rt 0
Xt1−γ Nt q e Mt , 1−γ
(2.16)
λ2 (Ys ) ds. In the following derivation, we use the short notation λ = λ(Yt ), µ =
5
µ(Yt ), σ = σ(Yt ). By Itˆ o’s formula applied to Vt in (2.16), we deduce γ X 1−γ X 1−γ dVt = Xt−γ dXt − Xt−γ−1 d hXit eNt Mtq + t eNt Mtq dNt + t eNt qMtq−1 dMt 2 1−γ 1−γ 1−γ 1−γ 1 Xt X eNt q(q − 1)Mtq−2 d hM it + d eN , M q + 2 1−γ 1−γ t 1−γ X γ 1−γ 2 q −γ q −γ−1 2 2 2 t Nt Nt = Xt Xt αt µ − Xt dt Xt αt σ e Mt dt + e Mt − λ 2 1−γ 2γ
1 Xt1−γ Nt Xt1−γ Nt e qMtq−1 Mt ξt at dt + e q(q − 1)Mtq−2 Mt2 ξt2 dt + Xt−γ eNt qMtq−1 ρXt αt σMt ξt dt 1−γ 2 1−γ X 1−γ + t eNt Mtq (1 − γ)αt σ dWt + qξt dWtY . 1−γ
+
Under Assumptions 2.1 and 2.2, the last term is a true martingale. This follows from the boundedness of eNt Mtq guaranteed by the boundedness of λ(·), and square integrability of Xt1−γ αt σ and Xt1−γ ξt implied by: "Z # "Z # T
E
0
E
"Z
0
T
(Xtπ )2−2γ a2t σ 2 (Yt ) dt = E 2−2γ (Xtπ )
ξt2
#
" Z dt ≤ E
0
T
T
0
(Xtπ )−2γ πt2 σ 2 (Yt ) dt < ∞,
2p(1−γ) (Xtπ )
# p1 " Z E dt
T
0
2p/(p−1) ξt
dt
# p−1 p
< ∞.
By rewriting dVt = Xt1−γ eNt Mtq Dt (αt ) dt + d Martingale, the drift factor Dt (αt ) takes the form: Dt (αt ) := αt µ −
q q(q − 1) 2 γ 2 2 λ2 αt σ − + at ξt + ξ + ρqαt σξt . 2 2γ 1−γ 2(1 − γ) t
Differentiating Dt (αt ) with respect to α and checking the second order condition, one obtains the maximizer α∗t =
ρqξt λ ρqξt µ + = + . γσ 2 γσ γσ γσ
Evaluating the drift factor Dt at α∗t produces at qξt2 ρ2 q λρ q−1 ∗ Dt (αt ) = qξt + . + + 1−γ γ 2 γ 1−γ
(2.17)
(2.18)
Then, the drift factor Dt (α∗t ) vanishes under the choices (2.7) for q and (2.10) for at . In addition, using the relation πt = αt Xt and equation (2.17) for α∗t , the wealth process following πt∗ solves the SDE 2 ∗ ∗ λ (Yt ) + ρqλ(Yt )ξt λ(Yt ) + ρqξt dXtπ = Xtπ dt + dWt , γ γ thus stays non-negative, which implies the admissibility of πt∗ = α∗t Xt .
Remark 2.5. The choice of at and q in (2.10) and (2.7) is consistent with the distortion transformation in the Markovian case reviewed in Section 2.1. In fact, other choices only lead to the degenerate case mentioned in Remark 2.4(ii) by the following argument. From (2.18), one can factorize the expression and obtain 2 at ρ q λρ q−1 Dt (a∗t ) = qξt , + + ξt + 1−γ γ 2γ 2(1 − γ) 2 ρ q q−1 at + λρ +ξ + except the choice (2.10) and (2.7), one could consider: (a) ξt = 0, and (b) 1−γ t γ 2γ 2(1−γ) = 0, γ given q 6= γ+(1−γ)ρ2 . 6
(a) Suppose ξt = 0, by solving the martingale representation (2.12) for Mt , one has h 1−γ R T i e e 2qγ 0 λ2 (Ys ) ds , ∀t ∈ [0, T ], Mt = M0 = E
and Mt is deterministic, which is only valid under the situation in Remark 2.4(ii), where λ(y) is a constant. (b) The process ξt given in (2.12) does not depend on the choice of at , since by varyingat , one only varies ρ2 q q−1 at + λρ + ξ + the drift of Mt in (2.12), but not the diffusion part. Therefore the relation 1−γ t γ 2γ 2(1−γ) = 0 γ cannot hold if q 6= γ+(1−γ)ρ 2. Note that with choice (2.7) for q, the term ξt2 is canceled which corresponds to the cancellation of the nonlinear term (∂y Φ)2 in the PDE argument reviewed in Section 2.1.
2.3
Martingale Distortion Transformation with Multiple Assets
Proposition 2.3 can be generalized to the case of multiple risky assets modeled by dSti = µi (Yti )Sti dt +
k X
σij (Yti )Sti dWtj ,
i = 1, 2, . . . n.
(2.19)
j=1
Here, each Sti is driven by its own stochastic factor Yti , but all factors are adapted to the same single Brownian motion WtY with the correlation structure:
d W i , W j t = 0, d W i , W Y t = ρ dt, ∀ i, j = 1, 2, . . . , n. † Denote by π = π 1 , π 2 , · · · , π n ∈ Ft the trading vector such that πti represents the amount of money invested into Sti at time t († denotes the matrix transpose). In this multi-asset setup, under self-financing assumption and r = 0, the wealth process Xt satisfies dXt =
n X
πti µi (Yti ) dt +
n X i=1
i=1
πti
k X j=1
σij (Yti ) dWtj = πt · µ(Yt ) dt + πt · σ(Yt ) dWt ,
with vector notations Yt := [Yt1 , Yt2 , . . . , Ytn ]† , µ(Yt ) := [µ1 (Yt1 ), µ2 (Yt2 ), · · · , µn (Ytn )]† , σ(Yt ) := σi,j (Yti ) as an n × k matrix, and Wt := [Wt1 , Wt2 , · · · , Wtk ]† . The assumptions of the multi-assets case are similar to Assumption 2.1 and 2.2, which are summarized as follows. Assumption 2.6 (Power Utility with Multi-assets). The admissible trading vectors π ∈ At satisfy # "Z " # T 2 π 2p(1−γ) π −2γ < +∞, f or some p > 1, and E E sup (Xt ) (Xt ) kπt · σ(Yt )k dt < ∞. t∈[0,T ]
0
Assumption 2.7. The SDEs (2.19) for Sti have unique strong solutions for all i = 1, 2, . . . , n. Assume that the matrix σ(·) is of rank n, Σ(·) := σ(·)σ(·)† is invertible and positive definite. The function Λ(·) := µ(·)† Σ(·)−1 µ(·) is bounded and C 2 (R), and its derivatives Λ′ (·) and Λ′′ (·) are at most polynomially growing. e Moreover, define the P-martingale Mt by 1−γ R T e e 2qγ 0 µ(Ys )† Σ(Ys )−1 µ(Ys ) ds Gt , Mt = E e is the expectation under the probability measure (2.9) P e with where E 1−γ at = −ρ 1†k σ(Yt )† Σ−1 (Yt )µ(Yt ). γ
Then, one has and we require that ξ· ∈ L tion 2.6.
′
2p
ftY , dMt = Mt ξt dW ′
(Ω × [0, T ]) for some constant p > 7
(2.20)
(2.21) p p−1 ,
where p is the constant in Assump-
Proposition 2.8. Under Assumption 2.6 and 2.7, the portfolio value Vt can be expressed as Vt =
iq RT † −1 Xt1−γ h e 1−γ E e 2qγ t µ(Ys ) Σ(Ys ) µ(Ys ) ds Gt , 1−γ
e is calculated under P e with at in (2.20), the constant q is chosen to be: where E q=
γ
γ + (1 −
γ)ρ2 1†k σ(Yt )† Σ−1 (Yt )σ(Yt )1k
,
and 1k is a k-vector of ones. The optimal control π ∗ is given by Σ(Yt )−1 µ(Yt ) ρqξt Σ(Yt )−1 σ(Yt )1k πt∗ = Xt , + γ γ with ξ given in (2.21). Proof. The proof is a straightforward generalization of the one given in the single asset case, thus we omit the details here. Note that the case n = k = 1 is reduced to Proposition 2.3.
3
Application to Fractional Stochastic Environment
In this section, we first briefly review the fractional Brownian motion (fBm) and fractional OrnsteinUhlenbeck (fOU) processes, and then introduce the slowly varying fOU process. Under such a model, we will derive an approximation of portfolio value Vt based on results in Proposition 2.3. More importantly, note that the optimal trading strategy π ∗ given by (2.14) is not explicit due to the presence of ξt given by the martingale representation theorem, and we will obtain an explicit approximation to this optimal strategy.
3.1
Fractional Brownian Motion and Fractional Ornstein-Uhlenbeck Processes (H)
A fractional Brownian motion is a continuous Gaussian process (Wt ) with zero mean and the covariance structure: i σ2 h (H) 2H 2H 2H E Wt Ws(H) = H |t| + |s| − |t − s| , (3.1) 2 where σH is a positive constant and H ∈ (0, 1) is called Hurst index. According to Mandelbrot and Van Ness (H) [1968], Wt has the following moving-average stochastic integral representation: Z 1 H− 21 H− 21 (H) dWs , (3.2) Wt = − (−s) (t − s) + + Γ(H + 12 ) R where (Wt )t∈R+ is the usual Brownian motion and (Wt )t∈R− := (B−t )t∈R− is another Brownian motion 2 independent of (Wt )t∈R+ . Therefore, σH is calculated as: Z ∞ 2 1 1 1 2 H− 12 H− 21 = 2 σH (1 + s) − s ds + = . (3.3) 2H Γ(2H + 1) sin(πH) Γ (H + 21 ) 0 Now we consider the Langevin equation with fractional Brownian motion (H)
dZtH = −aZtH dt + dWt
,
with the initial condition Z0H = η. In Cheridito et al. [2003], it is proved that Z t H,η −at au (H) Zt := e η+ e dWu 0
8
(3.4)
Rt (H) is the unique almost surely continuous process that solves equation (3.4), where 0 eau dWu exists as a path-wise Riemann-Stieltjes integral (by integration by parts) and is almost surely continuous in t. Particularly, for t ∈ R+ , Z t Z t (H) e−a(t−s) Ws(H) ds, (3.5) e−a(t−s) dWs(H) = Wt − a ZtH := −∞
−∞
is a stationary solution with initial condition η = Z0H , and every other stationary solution has the same distribution as ZtH . In the sequel, we shall only consider this stationary solution and call it the stationary fractional Ornstein-Uhlenbeck process. It has zero mean and (co)variance structure: 1 −2H 2 a Γ(2H + 1)σH , 2 Z ∞ H H x1−2H 2 2 sin(πH) 2 E Zt Zt+s = σou dx := σou CZ (s). cos(asx) π 1 + x2 0
2 σou =
(H)
Using the moving-average representation (3.2) for Wt ZtH =
Z
(3.6) (3.7)
, the stationary solution (3.5) can be expressed as:
t
−∞
K(t − s) dWsZ ,
(3.8)
where WtZ t∈R is a standard BM on R as described in (3.2), with the superscript Z indicating that it drives the process ZtH . The kernel K is defined by Z t 1 H− 21 H− 21 −as K(t) = t − a e ds . (3.9) (t − s) Γ(H + 12 ) 0 We refer to [Garnier and Solna, 2015, Section 2.2] for asymptotic properties of K(t) when t ≪ 1 and t ≫ 1, for short-range correlation properties when H ∈ (0, 12 ), and for long-range correlation properties when H ∈ ( 21 , 1). In what follows, we will be mainly interested in the case H < 12 as explained in the introduction, but our asymptotic results are also valid for H > 12 . As noted in [Garnier and Solna, 2015, Appendix B], a more general class of Gaussian volatility factors can be considered. But for the sake of simplicity and length ,we restrict ourselves to the case of fOU process.
3.2
The Slowly Varying fOU Process
As explained in the introduction, we consider the slowly varying fractional factor denoted by Ztδ,H . In the regime of small δ, Ztδ,H is defined as a rescaled stationary fOU process, Ztδ,H
=δ
H
Z
t
−∞
e
−δa(t−s)
dWs(H)
=
Z
t
−∞
Kδ (t − s) dWsZ ,
Kδ (t) =
√ δK(δt),
(3.10)
(H)
where Wt is a fBm driven by the Brownian motion WtZ via (3.2), and K(t) is given in (3.9). According to Section 3.1, Ztδ,H is a stationary solution to the SDE (H)
dZtδ,H = −δaZtδ,H dt + δ H dWt
.
2 It is a zero-mean, stationary Gaussian process with variance σou and covariance Z ∞ h i x1−2H δ,H 2 2 2 sin(πH) E Ztδ,H Zt+s = σou CZ (δs) = σou dx. cos(δasx) π 1 + x2 0
(3.11)
(3.12)
The covariance function depends on δs only, which indicates that 1/δ is the natural scale of Ztδ,H as desired. More properties and estimates regarding Ztδ,H are stated in Lemma A.1. 9
As δ goes to zero, by dominated convergence theorem, the covariance becomes Z ∞ 1−2H i h x δ,H 2 2 sin(πH) 2 2 sin(πH) π 2 = σou dx = σou csc(πH) = σou , (3.13) lim E Ztδ,H Zt+s 2 δ→0 π 1+x π 2 0 D = (σou Z)t∈R , where Z is a standard normal and the process Ztδ,H converges in distribution to Z0δ,H t∈R
random variable.
3.3
First order Approximation to the Value Process
In this section, we study the problem discussed in Section 2 with Yt = Ztδ,H and WtY = WtZ . To be precise, the underlying asset St is driven by the slowly varying fractional stochastic factor Ztδ,H defined in (3.10), dSt = µ(Ztδ,H )St dt + σ(Ztδ,H )St dWt . Still, we denote by Xtπ the wealth process, and it follows dXtπ = πt µ(Ztδ,H ) dt + πt σ(Ztδ,H ) dWt . The value process is denoted by Vtδ to indicate its dependence of δ introduced by the slowly varying process Z·δ,H : Vtδ := sup E [U (XTπ )|Ft ] . π∈At
Z·δ,H
Note that, by definition, the process is neither Markovian nor a semimartingale when H 6= 12 , therefore the HJB equation is not available. However, it is adapted to Gt . In order to use Proposition 2.3, we need to check that Ztδ,H satisfies Assumption 2.2(ii). Lemma 3.1. The slowly varying fractional factor Ztδ,H defined in (3.10) satisfies the Assumption 2.2(ii). Proof. To obtain the process (ξt )t∈[0,T ] in (2.12), we shall use Malliavin calculus. By the Clark-Ocone Formula under change of measure (see Di Nunno et al. [2009]), we obtain # " Z T Z e Dt MT − MT e MT F[t,T ] Gt , f Gt := E Mt ξt = E Dt as dW s t where Dt denotes the Malliavian derivative, and F[t,T ] is F[t,T ] =
1−γ qγ
Z
T
t
1−γ γ
Z
t
T
fZ λ′ (Zsδ,H )Kδ (s − t) dW s
Z 1 − γ T ′ δ,H δ δ,H ′ δ,H δ λ(Zs )λ (Zs )K (s − t) ds + ρ λ (Zs )K (s − t) dWsZ . γ t t e MT M −1 F[t,T ] |Gt . Then Assumption 2.2(i) and Lemma A.1(i) imply that ξ· ∈ Lk (Ω×[0, T ]), Thus, ξt = E t for any k ≥ 2: "Z # Z Z T h T T i RT 2 k R T k 1 e k −1 E E F[t,T ] e− t as dWs − 2 t as ds dt |ξt | dt = E E MT Mt F[t,T ] |Gt dt ≤ C 1−γ = γ
0
Z
λ(Zsδ,H )λ′ (Zsδ,H )Kδ (s − t) ds + ρ
T
0
0
≤ C′
Z
0
T
i h 2k E F[t,T ] dt < ∞.
e In the derivation, we have used the facts that Mt is a bounded positive P-martingale and the (2k)th moments of F[t,T ] exist for any k. The second fact follows from the polynomial growth property of λ′ (·), existence of moments of Ztδ,H and uniformly integrability of Kδ (t) on [0, T ]. 10
Theorem 3.2. Under Assumption 2.1 and 2.2, for fixed t ∈ [0, T ), Xt = x and the observed value Z0δ,H , Vtδ takes the form Vtδ = Qδt (Xt , Z0δ,H ) + O(δ 2H ), (3.14) where
" !# H+ 23 1−γ 1−γ 2 (T − t) x 1 − γ 1 − γ λ (z)(T −t) e 2γ λ(z)λ′ (z) φδt + δ H ρλ(z) Qδt (x, z) = 1+ . 1−γ γ γ Γ(H + 52 )
(3.15)
Here φδt is defined by φδt = E
"Z
t
T
Zsδ,H −
Z0δ,H
# # "Z T δ,H ds Gt , Zsδ,H − Z0 ds Ft = E t
(3.16)
and φδt is of order δ H as proved in Lemma A.1 in the sense that its variance is of order δ 2H . Note that O(δ 2H ) denotes a Ft -adapted random variable and it is of order δ 2H in L2 . Proof. A straightforward application of Proposition 2.3 with Yt = Ztδ,H and WtY = WtZ gives the following representation of the value process Vtδ iq R T 2 δ,H Xt1−γ h e 1−γ E e 2qγ t λ (Zs ) ds Gt . 1−γ i h 1−γ R T e e 2qγ t λ2 (Zsδ,H ) ds Gt , and then apply Taylor formula to the function xq . We start by expanding Ψδt := E The formula for the conditional expectation under an absolute continuous change of measure, together with the value of at given by (2.10) and Taylor expansion in z at the point Z0δ,H yields, i h 1−γ R T 2 δ,H RT 2 RT Z 1 Ψδt = E e 2qγ t λ (Zs ) ds e− t as dWs − 2 t as ds Gt i h 1−γ R T 2 δ,H RT R T 2 1−γ 2 2 δ,H 1−γ δ,H Z 1 = E e 2qγ t λ (Zs ) ds e t ρ( γ )λ(Zs ) dWs − 2 t ρ ( γ ) λ (Zs ) ds Gt i h R T 1−γ R T 2 1−γ 2 2 δ,H δ,H δ,H 1−γ 2 Z 1 = e 2qγ λ (Z0 )(T −t) E e t ρ( γ )λ(Z0 ) dWs − 2 t ρ ( γ ) λ (Z0 ) ds+A[t,T ] +B[t,T ] Gt , Vtδ =
where A[t,T ] and B[t,T ] are given by A[t,T ]
B[t,T ]
Z T Z T 1−γ 1−γ δ,H δ,H δ,H δ,H ′ ′ δ,H = ds + ρ λ (Z0 ) Zsδ,H − Z0δ,H dWsZ λ(Z0 )λ (Z0 ) Zs − Z0 qγ γ t t 2 Z T 1 − γ Zsδ,H − Z0δ,H ds, λ(Z0δ,H )λ′ (Z0δ,H ) − ρ2 γ t Z T Z 2 2 1−γ T 1−γ = λλ′′ + λ′2 (χs ) Zsδ,H − Z0δ,H ds + ρ λ′′ (ηs ) Zsδ,H − Z0δ,H dWsZ qγ γ t t 2 Z T 2 1 − γ − ρ2 λλ′′ + λ′2 (χs ) Zsδ,H − Z0δ,H ds, γ t
with χs and ηs being the Lagrange remainders: χs , ηs ∈ [Z0δ,H ∨ Zsδ,H , Z0δ,H ∧ Zsδ,H ]. Since λ(·) is bounded, one can expand eA[t,T ] +B[t,T ] and deduce h R T 1−γ R T 2 1−γ 2 2 δ,H i δ,H δ,H 1−γ 2 Z 1 Ψδt =e 2qγ λ (Z0 )(T −t) E e t ρ( γ )λ(Z0 ) dWs − 2 t ρ ( γ ) λ (Z0 ) ds 1 + A[t,T ] + R[t,T ] Gt h R T 1−γ R T 2 1−γ 2 2 δ,H i δ,H δ,H 1−γ 2 Z 1 =e 2qγ λ (Z0 )(T −t) E e t ρ( γ )λ(Z0 ) dWs − 2 t ρ ( γ ) λ (Z0 ) ds 1 + A[t,T ] Gt + O(δ 2H ),
where R[t,T ] is given by
R[t,T ] = B[t,T ] +
∞ X k 1 A[t,T ] + B[t,T ] , k!
k=2
11
and O(δ 2H ) is a random variable of order δ 2H in L2 sense as mentioned before. b such that under P, b W ctZ = WtZ − ρ 1−γ λ(Z δ,H )t is a We introduce a new probability measure P, 0 γ
standard Brownian motion. Then Ψδt can be rewritten as 1−γ
2
Ψδt =e 2qγ λ
(Z0δ,H )(T −t) b
E
1 + A[t,T ] Gt + O(δ 2H )
# Z T (1 − γ) δ,H δ,H δ,H δ,H ′ ds Gt =e Zs − Z0 λ(Z0 )λ (Z0 ) E 1+ qγ t # " Z T δ,H 1−γ 2 b ρ 1 − γ λ′ (Z δ,H ) Zsδ,H − Z0δ,H dWsZ Gt + e 2qγ λ (Z0 )(T −t) E 0 γ t # " 2 Z T δ,H 1−γ 2 1 − γ δ,H δ,H δ,H )(T −t) λ (Z δ,H ′ 2 b ρ 0 − e 2qγ ds Gt + O(δ 2H ), Zs − Z0 λ(Z0 )λ (Z0 ) E γ t 1−γ 2qγ
λ2 (Z0δ,H )(T −t) b
"
and the second term is canceled with the third one, since # " Z T 1 − γ δ,H δ,H ′ δ,H Z b ρ E λ (Z0 ) Zs − Z0 dWs Gt γ t # " Z T 1−γ δ,H δ,H Z ′ δ,H b c dWs Gt λ (Z0 ) Zs − Z0 =E ρ γ t # " Z T 1 − γ 1 − γ δ,H δ,H δ,H δ,H ′ b ρ +E ρ Zs − Z0 λ (Z0 ) λ(Z0 ) ds Gt γ γ t # " 2 Z T b ρ2 1 − γ Zsδ,H − Z0δ,H ds Gt . λ(Z0δ,H )λ′ (Z0δ,H ) =E γ t Thus, the term Ψδt is simplified to Ψδt
=e
1−γ 2qγ
λ2 (Z0δ,H )(T −t)
(1 − γ) δ,H δ,H ′ δ λ(Z0 )λ (Z0 )Φt + O(δ 2H ), 1+ qγ
(3.17)
with b Φδt = E
"Z
t
T
Zsδ,H − Z0δ,H
# # "Z T b ds Gt = E Zsδ,H ds Gt − Z0δ,H (T − t). t
To further simplify Φδt , we use the moving average representation (3.10) for Zsδ,H and deduce # # "Z "Z Z T T s δ,H δ δ,H δ Z b b Φt = E Zs ds Gt − Z0 (T − t) = E K (s − u) dWu ds Gt − Z0δ,H (T − t) t t −∞ # # "Z "Z Z Z t T T T b b =E Kδ (s − u) ds dWuZ Gt + E Kδ (s − u) ds dWuZ Gt − Z0δ,H (T − t) t u −∞ t # " Z t Z T Z TZ T δ,H δ Z δ Z b K (s − u) ds dWu − Z0 (T − t) + E = K (s − u) ds dWu Gt −∞ t t u # "Z Z Z Z T T T T b c Z Gt + ρ 1 − γ λ(Z δ,H ) Kδ (s − u) ds du = φδt + E Kδ (s − u) ds dW u 0 γ t u t u δ H (T − t)H+3/2 1−γ δ λ(Z0δ,H ) = φt + ρ + O(δ H+1 ). (3.18) γ Γ(H + 25 ) 12
cZ = W Z − In the derivation, we have changed the order of ds and dWuZ , and use the relation W t t δ,H 1−γ γ λ(Z0 )ρt. Now combining (3.17) and (3.18), we obtain Vtδ =
q Xt1−γ Ψδt 1−γ
Xt1−γ 1−γ 1−γ δ,H δ,H λ2 (Z0δ,H )(T −t) ′ δ 2γ = e λ(Z0 )λ (Z0 )Φt + O(δ 2H ) 1+ 1−γ γ !) ( H+ 23 δ,H 2 Xt1−γ 1−γ (T − t) 1 − γ 1 − γ δ,H δ,H δ,H )(T −t) λ (Z ′ δ H 0 = e 2γ λ(Z0 )λ (Z0 ) φt + δ ρλ(Z0 ) 1+ 1−γ γ γ Γ(H + 25 ) + O(δ 2H ).
Observe that there are two corrections to the leading term: a random component φδt , and a deterministic function of (t, Xt , Z0δ,H ), both being of order δ H . δ Remark 3.3 (Discussion of the i assumptions on λ(·)). In order to expand Ψt , we need a uniform bound h 1−γ R T 2 δ,H λ (Z ) ds s . Notice that if γ > 1, this is automatically satisfied, since the exponential (in δ) of E e 2qγ t function is bounded by 1. For 0 < γ < 1, it is also satisfied under the assumption λ(·) bounded as stated in Assumption 2.2(i). Moreover, the assumption can be relaxed to have uniform bounds for exponential moments of the function λ2 (·).
3.4
Optimal Strategy
We now turn to the expansion to the optimal portfolio given in (2.14) # " ρqξt λ(Ztδ,H ) ∗ Xt , + πt = γσ(Ztδ,H ) γσ(Ztδ,H ) where the process ξt given by the representation theorem (2.12) is usually not known explicitly. In this section, we approximate ξt using the results derived in Theorem 3.2, and we obtain the following asymptotic result for πt∗ . Theorem 3.4. Under Assumption 2.1 and 2.2, the optimal strategy πt∗ is approximated by " # H+1/2 λ(Ztδ,H ) δ,H δ,H H ρ(1 − γ) (T − t) ∗ ′ +δ πt = λ(Z0 )λ (Z0 ) Xt + O(δ 2H ) 3 γσ(Ztδ,H ) γ 2 σ(Ztδ,H ) Γ(H + 2 ) (0)
:= πt
(1)
+ δ H πt
(3.19)
+ O(δ 2H ).
Remark 3.5. (i) For the case H = 21 , Ztδ,H becomes the Markovian OU process, and (3.19) coincides with the approximation of feedback form derived in [Fouque et al., 2016, Section 3.2.2 and 6.3.2]. (0)
(ii) In the approximation (3.19) to πt∗ , the leading order strategy πt follows the process Ztδ,H , the first (1) order correction πt is partially frozen at Z0δ,H , and the random correction φδt appearing in Vt disap(0) (1) pears here. This makes the approximated strategy πt + δ H πt easier to implement. Moreover, under additional smoothness assumption on σ(·), typically σ(·) is C 1 and (1/σ(·))′ is (1) bounded, then the correction term πt can be fully frozen at Z0δ,H without changing the order of accuracy, namely, (1)
πt
=
ρ(1 − γ) (T − t)H+1/2 λ(Z0δ,H )λ′ (Z0δ,H )Xt + O(δ H ). 3 δ,H 2 ) Γ(H + γ σ(Z0 ) 2 13
(iii) Denote by Xtπ
(0)
(0)
the wealth process following the zeroth order strategy πt dXtπ
and V·π
(0)
,δ
(0)
(0)
=
λ(Ztδ,H ) Xt γσ(Ztδ,H )
(0)
= µ(Ztδ,H )πt dt + σ(Ztδ,H )πt dWt ,
the corresponding value process Vtπ
(0)
h (0) i := E U XTπ Ft .
,δ
(0)
In Section 4.3 Proposition 4.5, we derive the expansion to Vtπ ,δ for general utility function. When (0) applied to the case of power utility (2.5), one can deduce that V π ,δ − Qδt is of order δ 2H with Qδt given in (3.15). Therefore, by Theorem 3.2, Vtπ
that
(0) πt
λ(Ztδ,H ) γσ(Ztδ,H )
=
(0)
,δ
− Vtδ is of order δ 2H , and we conclude
generates the approximated value process given by (3.14), and is asymptotically
optimal within all admissible strategy At up to order δ H . Proof. It suffices to derive the expansion of hξt determined by (2.12). In the previous section, we have i 1−γ R T 2 λ (Zsδ,H ) ds δ e 2qγ t obtained a rigorous expansion for Ψt := E e Gt ; see (3.17) and (3.18). Rewrite Mt defined in (2.11) using Ψδt as where It =
1−γ 2qγ
Rt 0
Mt = eIt Ψδt ,
λ2 (Zsδ,H ) ds. Applying Itˆ o’s formula to Mt yields,
dMt =eIt Ψδt dIt + eIt dΨδt =
1 − γ 2 δ,H λ (Zt )Mt dt + eIt 2qγ + O(δ 2H ) 1−γ
2
=δ H eIt e 2qγ λ
(Z0δ,H )(T −t) 1
−
δ,H 1−γ 2 1−γ 1 − γ 2 δ,H δ λ (Z0 )Ψt dt + e 2qγ λ (Z0 )(T −t) λ(Z0δ,H )λ′ (Z0δ,H ) dΦδt 2qγ qγ
−γ f Z + O(δ 2H ). λ(Z0δ,H )λ′ (Z0δ,H )θt,T dW t qγ
Here in the derivation, we have successively used the relation (3.17) and (3.18), dψtδ = dφδt + (Ztδ,H − Z0δ,H ) dt, where ψtδ is given by # "Z T ψtδ = E Zsδ,H − Z0δ,H ds Ft (3.20) 0
and dψtδ = δ H θt,T dWtZ + δ H+1 θet,T dWtZ with θt,T and θet,T specified in Lemma A.1. Noticing that from (3.17), one can deduce 1−γ
2
Ψδt = e 2qγ λ
(Z0δ,H )(T −t)
+ O(δ H ),
then dMt becomes 1−γ ftZ + O(δ 2H ) λ(Z0δ,H )λ′ (Z0δ,H )θt,T dW dMt = δ H eIt Ψδt qγ " # H+ 12 1 − γ (T − t) δ,H δ,H ftZ + O(δ 2H ), = δH λ(Z0 )λ′ (Z0 ) Mt dW qγ Γ(H + 23 )
and the approximation of ξt is given by
1
ξt = δ H
(T − t)H+ 2 1−γ + O(δ 2H ). λ(Z0δ,H )λ′ (Z0δ,H ) qγ Γ(H + 32 )
Plugging the above expression into (2.14) yields the desired result (3.19). 14
4
General Utilities and Fractional Stochastic Environment
In this section, we study the nonlinear portfolio optimization through asymptotics with general utility U (x), and when the drift µ and volatility σ of the underlying asset St are driven slowly varying fractional stochastic factor Ztδ,H defined in (3.10) . This is motivated by two recent works: in Fouque and Hu [2016a], we developed asymptotic results for the value function following a given strategy in the slow varying Markovian environment, and proved the optimality of such a strategy up to a certain order; on the other hand, asymptotics of linear pricing problem has been done and implied volatility is provided in Garnier and Solna [2015] when the volatility is driven by Ztδ,H . Using the notation M (t, x; λ) for the classical Merton value with constant Sharpe-ratio λ, we denote by v (0) the value function at frozen Sharpe-ratio λ(z), v (0) (t, x, z) = M (t, x, λ(z)).
(4.1)
Then we define the strategy π (0) by (0)
π (0) (t, x, z) = − and the associate value process V π
(0)
,δ
(0)
(4.2)
is
Vπ where Xtπ
λ(z) vx (t, x, z) , (0) σ(z) vxx (t, x, z)
(0)
,δ
h (0) i := E U XTπ Ft ,
(4.3)
is the wealth process following strategy π (0) : dXtπ
(0)
= µ(Ztδ,H )π (0) (t, Xtπ
(0)
, Ztδ,H ) dt + σ(Ztδ,H )π (0) (t, Xtπ
(0)
, Ztδ,H ) dWt .
(4.4)
(0)
We first derive the expansion for V π ,δ , and then we show that π (0) is optimal up to order δ H among the strategies of the form (4.5) Aeδt [e π0 , π e1 , α] := π = π e0 + δ α π e1 : π ∈ Aδt , α > 0, 0 < δ ≤ 1 . (0)
As a byproduct, by applying the expansion results for V π ,δ to power utility, π (0) obtained in Theorem 3.4 is optimal up to order δ H within the full class of strategies Aδt . In the next subsection, we first review the classical Merton problem when µ and σ are constants in (0) (2.1), which plays a crucial role in deriving the expansion (4.21) to V π ,δ . Then we define some notations for later use.
4.1
Merton Problem with Constant Coefficients
This problem has been extensively studied, for example, in Karatzas and Shreve [1998]. Here we summarize the results about the classical Merton value function M (t, x; λ). Assume that the utility function U (x) is C 2 (0, ∞), strictly increasing, strictly concave, and satisfies the Inada and Asymptotic Elasticity conditions: U ′ (0+) = ∞,
U ′ (∞) = 0,
AE[U ] := lim x x→∞
U ′ (x) < 1, U (x)
then, the Merton value function M (t, x; λ) is strictly increasing, strictly concave in the wealth variable x, and decreasing in the time variable t, which is C 1,2 ([0, T ] × R+ ) and solves the HJB equation 1 M2 1 2 2 (4.6) Mt + sup σ π Mxx + µπMx = Mt − λ2 x = 0, M (T, x; λ) = U (x), 2 2 Mxx π where λ = µ/σ is the constant Sharpe ratio. It is C 1 with respect to λ, and the optimal strategy is π ∗ (t, x; λ) = −
λ Mx (t, x; λ) . σ Mxx (t, x; λ)
15
(4.7)
Given the Merton value function M (t, x; λ), one can define the risk-tolerance function by R(t, x; λ) = −
Mx (t, x; λ) . Mxx (t, x; λ)
(4.8)
It is clear that R(t, x; λ) is continuous and strictly positive due to the regularity, concavity and monotonicity of M (t, x; λ). For further properties, we refer to K¨allblad and Zariphopoulou [2014, 2017] and Fouque and Hu [2016a]. We use the notation from Fouque et al. [2016]: Dk = R(t, x; λ)k ∂xk , k = 1, 2, · · · , 1 Lt,x (λ) = ∂t + λ2 D2 + λ2 D1 . 2
(4.9) (4.10)
Note that the coefficients of Lt,x (λ) depend on R(t, x; λ), and therefore on M (t, x; λ). The Merton PDE (4.6) can be re-written as Lt,x (λ)M (t, x; λ) = 0.
(4.11)
Next, we summarize all assumptions needed in the rest of this section. This will include properties of (0) the utility function U (x), the state processes (Xtπ , St , Ztδ,H ) as well as v (0) (t, x, z).
4.2
Assumptions
Basically, we work under the same set of assumptions as in Fouque and Hu [2016a], and we restate them here for readers’ convenience. Detailed discussion about general utility functions can be found there in Section 2.3. Assumption 4.1. Throughout the paper, we make the following assumptions on the utility U (x): (i) U(x) is C 6 (0, ∞), strictly increasing, strictly concave and satisfying the following conditions (Inada and Asymptotic Elasticity): U ′ (0+) = ∞,
U ′ (∞) = 0,
AE[U ] := lim x x→∞
U ′ (x) < 1. U (x)
(4.12)
(ii) U(0+) is finite. Without loss of generality, we assume U(0+) = 0. (iii) Denote by R(x) the risk tolerance, R(x) := −
U ′ (x) . U ′′ (x)
(4.13)
Assume that R(0) = 0, R(x) is strictly increasing and R′ (x) < ∞ on [0, ∞), and there exists K ∈ R+ , such that for x ≥ 0, and 2 ≤ i ≤ 4, (i) i (4.14) ∂x R (x) ≤ K.
(iv) Define the inverse function of the marginal utility U ′ (x) as I : R+ → R+ , I(y) = U ′(−1) (y), and assume that, for some positive α, κ, I(y) satisfies the polynomial growth condition: I(y) ≤ α + κy −α ,
(4.15)
as well as for positive constants cn , Cn , n = 1, 2, 3, with c2 > 1, c1 I(x) ≤ |xI ′ (x)| ≤ C1 I(x),
c2 |I ′ (x)| ≤ xI ′′ (x) ≤ C2 |I ′ (x)| and |xI ′′′ (x)| ≤ C3 I ′′ (x), 1−γ
(4.16)
Remark 4.2. The item (ii) excludes the case of power utility U (x) = x1−γ when γ > 1. However, all results in this section still hold for the case γ > 1, with a slight modification in the proofs. The conditions (4.16) which were introduced in K¨ allblad and Zariphopoulou [2017], are crucial assumptions in their Proposition 4, which will be used in our derivation. They also give a mixture of inverse of the marginal utilities as an example that satisfies this conditions. 16
Below are the additional assumptions needed on the state processes (Xtπ
(0)
, St , Zt ) and on v (0) (t, x, z).
Assumption 4.3. (i) The function λ(z) = µ(z)/σ(z) is C 2 (R). Moreover, λ(z), λ′ (z) and λ′′ (z) are at most polynomially growing. (ii) The value function v (0) (t, x, z) = M (t, x; λ(z)) satisfies the relation: 2 (0) x vxx (t, x, z) ≤ d(z)v (0) (t, x, z),
(4.17)
with d(z) being polynomial growth. Note that this is automatically satisfied by the power utility (2.5).
(iii) The process v (0) (t, Xtπ
(0)
, Z0δ,H ) is in L4 ([0, T ] × Ω) uniformly in δ, i.e., "Z # T 4 δ,H (0) π (0) E v (s, Xs , Z0 ) ds ≤ C1
(4.18)
0
where C1 is independent of δ and Z0δ,H is given in (3.10) with t = 0. Remark 4.4. Notice that condition (4.17) is actually a hidden assumption on the general utility, and it is automatically satisfied by power utility. In order to guarantee (4.18), there is a list of assumptions discussed in [Fouque and Hu, 2016a, Section 2.4].
The Epsilon-Martingale Decomposition with a Given Strategy π (0)
4.3
As introduced in Fouque et al. [2000] in the context of linear pricing problem and further developed in Garnier and Solna [2015], the idea of epsilon-martingale decomposition is to find a process which is in the form of a martingale plus something small with the right terminal condition. Specifically, we aim (0) (0) to find Qπ ,δ such that its terminal condition coincides with the quantity of interest Vtπ ,δ , namely, QTπ
(0)
,δ
= VTπ
(0)
,δ
= U (XTπ
(0)
), and that can be decomposed as Qtπ
Mtδ
Rtδ
(0)
,δ
= Mtδ + Rtδ ,
(4.19)
2H
H
where is a martingale and is of order δ . Note that the term of order δ will be absorbed in the martingale Mtδ . Suppose we obtain such a decomposition (4.19), and then taking conditional expectation with respect (0)
to Ft on both sides of the equation QTπ ,δ = MTδ + RTδ gives h (0) i (0) (0) Vtπ ,δ = E QTπ ,δ |Ft = Mtδ + E RTδ |Ft = Qtπ ,δ + E RTδ |Ft − Rtδ . (0)
(4.20)
(0)
Since Rtδ is of order δ 2H , Qtπ ,δ is the approximation to Vtπ ,δ up to δ H . Therefore the above argument (0) leads to the desired approximation result. Now it remains to find Qtπ ,δ so that the decomposition holds, and we have the following proposition. Proposition 4.5. Under Assumption 4.1 and 4.3, for fixed t ∈ [0, T ), Xtπ
Z0δ,H ,
the Ft -measurable value process Vtπ
where Qtπ
(0)
,δ
,δ
= Qπt
= x, and the observed value
defined in (4.3) is of the form
(0)
,δ
(Xtπ
(0)
, Z0δ,H ) + O(δ 2H ),
(4.21)
(x, z) is given by:
Qtπ
v (0)
(0)
(0) Vtπ ,δ
(0)
(0)
,δ
(x, z) = v (0) (t, x, z) + λ(z)λ′ (z)D1 v (0) (t, x, z)φδt + δ H ρλ2 (z)λ′ (z)v (1) (t, x, z), (4.22) δ and D1 are defined in (4.1) and (4.9) respectively, φt t∈[0,T ] is the Ft -measurable process of order δ H
given in (3.16) and v (1) (t, x, z) is defined as
v (1) (t, x, z) = D12 v (0) (t, x, z)Dt,T , 17
Dt,T =
(T − t)H+3/2 . Γ(H + 25 )
(4.23)
The proof of Proposition 4.5 will be given after Corollary 4.6 and Proposition 4.7. As explained in Remark 3.5, we have the following corollary. 1−γ
Corollary 4.6. In the case of power utility U (x) = x1−γ , with γ > 0, γ 6= 1, and under Assumption 2.2 and 4.3, π (0) given by (3.19) is asymptotically optimal in the full class of admissible strategies Aδt up to order δ H . Proof. Straightforward computations give, under power utilities, v (0) , D1 v (0) and v (1) as v (0) (t, x, z) = v (1) (t, x, z) =
2 x1−γ 1−γ e 2γ λ (z)(T −t) , 1−γ
D1 v (0) (t, x, z) =
2 x1−γ 1−γ e 2γ λ (z)(T −t) , γ
3
2 (1 − γ) 1−γ 1−γ (T − t)H+ 2 2γ λ (z)(T −t) . x e γ2 Γ(H + 52 ) (0)
(0)
Then, one can deduce Qδt = Qtπ ,δ , where Qδt is given by (3.15) and Qtπ ,δ is given by (4.22). Combining (0) with Theorem 3.2, V δ and V π ,δ admits the same first order approximation. Therefore, we obtain the desired asymptotic optimality. For general utilities, we will derive a similar result in the smaller class Aeδt [e π0 , π e1 , α] in Section 4.4. (0)
Proposition 4.7. For the Markovian case H = 21 , the approximation Qtπ ,δ given in (4.21) coincides with the result derived in [Fouque and Hu, 2016a, Theorem 3.1], √ δ π (0) ,δ (0) V (t, x, z) = v (t, x, z) + (T − t)2 ρλ2 (z)λ′ (z)D12 v (0) (t, x, z) + O(δ). (4.24) 2 Proof. First observe that when H = 12 , Dt,T =
1 (T − t)2 , 2
√
(0)
and the third term in Qtπ ,δ becomes 2δ (T − t)2 ρλ2 (z)λ′ (z)D12 v (0) (t, x, z). Using the moving-average representation (3.10) for Zsδ,H with H = 1/2, φδt is explicitly computed as φδt =
1 − e−aδ(T −t) δ,H Zt − (T − t)Z0δ,H = (T − t) Ztδ,H − Z0δ,H + O(δ). aδ (0)
′ (0) Then using the p“Vega-Gamma” relation vz (t, x, z) = (T − t)λ(z)λ (z)D1 v (t, x, z) and the fact Ztδ,H − Z0δ,H ∼ O(δ pH ), one can deduce
Vtπ
(0)
,δ
(0) (0) =v (0) (t, Xtπ , Z0δ,H ) + vz(0) (t, Xtπ , Z0δ,H ) Ztδ,H − Z0δ,H √ (0) δ + (T − t)2 ρλ2 (Z0δ,H )λ′ (Z0δ,H )D12 v (0) (t, Xtπ , Z0δ,H ) 2 √ (0) (0) δ =v (0) (t, Xtπ , Ztδ,H ) + (T − t)2 ρλ2 (Ztδ,H )λ′ (Ztδ,H )D12 v (0) (t, Xtπ , Ztδ,H ) + O(δ), 2
which is consistent with the result derived in [Fouque and Hu, 2016a, Theorem 3.1]. We now turn to the proof of Proposition 4.5. Proof of Proposition 4.5. According to the epsilon-martingale decomposition strategy, our goal is to show (0) that Qtπ ,δ can be written as Mtδ + Rtδ , where Mtδ is a martingale, and Rtδ is of order δ 2H . We shall (0)
mainly focus on the derivation of Qtπ ,δ and delay the proofs of accuracy in the Appendix A for the sake of clarity and simplicity. The technique is very similar to the one presented in Garnier and Solna [2015] in the context of option pricing problem with fractional stochastic volatility. The main difference is that their case involves the linear Black-Scholes operator, as in our case, it involves the non-linear Merton operator 18
Lt,x (λ). Amazingly, the properties of risk-tolerance function R(t, x; λ) will enable us to carry the proof as follows. In order to avoid differentiating the fOU process Ztδ,H , we freeze it at Z0δ,H . The corresponding error will be compensated in the following calculation. This technique has also been use in the context of pricing when deriving hedging strategy with frozen volatility in [Fouque et al., 2011, Section 8.4]. By Itˆ o’s formula applied to v (0) defined in (4.1) and Taylor expansion in z at the point Z0δ,H , we deduce dv (0) (t, Xtπ
(0)
, Z0δ,H ) = Lt,x (λ(Ztδ,H ))v (0) (t, Xtπ + σ(Ztδ,H )π (0) (t, Xtπ
(0)
(0)
, Z0δ,H ) dt
, Ztδ,H )vx(0) (t, Xtπ
(0)
, Z0δ,H ) dWt
(0)
= Lt,x (λ(Z0δ,H ))v (0) (t, Xtπ , Z0δ,H ) dt i h (0) (1) + (Ztδ,H − Z0δ,H )(λ2 R)z z=Z δ,H + gt vx(0) (t, Xtπ , Z0δ,H ) dt 0 i h (0) (2) 1 (0) (1) δ,H δ,H 2 2 + (Zt − Z0 )(λ R )z z=Z δ,H + gt vxx (t, Xtπ , Z0δ,H ) dt + dMt 0 2 (0) (1) = (Ztδ,H − Z0δ,H )λ(Z0δ,H )λ′ (Z0δ,H )D1 v (0) (t, Xtπ , Z0δ,H ) dt + dMt (0) (0) 1 (2) (0) (1) + gt vx(0) (t, Xtπ , Z0δ,H ) dt + gt vxx (t, Xtπ , Z0δ,H ) dt, (4.25) 2 where in the derivation, we have used the relation Lt,x (λ(z))v (0) (t, x, z) = 0, (1)
Mt
D1 v (0) = −D2 v (0) , and π (0) (t, x, z) =
λ(z) R(t, x; λ(z)), σ(z)
(4.26)
is the martingale defined by (1)
dMt
= σ(Ztδ,H )π (0) (t, Xtπ
(0)
, Ztδ,H )vx(0) (t, Xtπ
(0)
, Z0δ,H ) dWt , (1)
(4.27) (2)
and the last two terms in (4.25) are of order δ 2H (see Appendix A), with gt and gt being the Lagrange remainders: 2 2 1 δ,H 1 δ,H (1) (2) δ,H 2 2 gt = , g = , (4.28) Zt − Z0δ,H Z − Z λ2 R zz λ R t t 0 (1) zz z=χ(2) 2 2 z=χt t i h (i) and χt ∈ Z0δ,H ∧ Ztδ,H , Z0δ,H ∨ Ztδ,H , i = 1, 2. Now it remains to find the epsilon-martingale decomposition for the term R δ,H (0) (Zs − Z0δ,H )D1 v (0) (s, Xsπ , Z0δ,H ) ds in (4.25). To this end, we recall φδt and ψtδ given in (3.16) and (3.20) respectively, which satisfy the relation Ztδ,H − Z0δ,H
dt = dψtδ − dφδt and consequently
(0) (0) Ztδ,H − Z0δ,H D1 v (0) (t, Xtπ , Z0δ,H ) dt = D1 v (0) (t, Xtπ , Z0δ,H ) dψtδ − dφδt .
(4.29)
On the right-hand side, the first term is proved to be a true martingale in Appendix A, while the second term need further analysis, namely, the differential of φδt D1 v (0) will be computed. In the sequel, without (0) any confusion, the arguments of v (0) (t, Xtπ , Z0δ,H ) shall be omitted for simplicity. (0) d φδt D1 v (0) =D1 v (0) dφδt + φδt Lt,x (λ(Ztδ,H ))D1 v (0) dt + φδt σ(Ztδ,H )π (0) (t, Xtπ , Ztδ,H )∂x D1 v (0) dWt
(0) + σ(Ztδ,H )π (0) (t, Xtπ , Ztδ,H )∂x D1 v (0) d W, φδ t
(0) =D1 v (0) dφδt + ρλ(Z0δ,H )D12 v (0) d W Z , ψ δ t + φδt σ(Ztδ,H )π (0) (t, Xtπ , Ztδ,H )∂x D1 v (0) dWt
(3) (4) 1 (5) + φδt gt ∂x D1 v (0) dt + φδt gt (4.30) ∂xx D1 v (0) dt + ρgt ∂x D1 v (0) d W Z , ψ δ t , 2
where in the above derivation, we have used
Lt,x (λ(Z0δ,H ))D1 v (0) = D1 Lt,x (λ(Z0δ,H ))v (0) = 0, and d W, φδ t = ρ d W Z , ψ δ t , 19
(4.31)
(3)
(4)
(5)
with the first one being proved in [Fouque et al., 2016, Lemma 2.5]. Again, gt , gt and gt are Lagrange remainders from Taylor series (4) (3) δ,H δ,H 2 2 , λ R , g = Z − Z gt = Ztδ,H − Z0δ,H (λ2 R)z t t 0 (3) z z=χ(4) z=χt t (5) , gt = Ztδ,H − Z0δ,H (λR)z (5) z=χt
h i (i) with χt ∈ Z0δ,H ∧ Ztδ,H , Z0δ,H ∨ Ztδ,H , i = 3, 4, 5. Now combining (4.29) and (4.30) yields:
(2) Ztδ,H − Z0δ,H D1 v (0) dt = − d φδt D1 v (0) + ρλ(Z0δ,H )D12 v (0) d W z , ψ δ t + dMt (4) 1
(3)
+ φδt gt ∂x D1 v (0) dt + φδt gt
(2) Mt
where
(2)
dMt
2
is the martingale given by (0)
= D1 v (0) (t, Xtπ , Z0δ,H ) dψtδ + φδt σ(Ztδ,H )π (0) (t, Xtπ
(5) ∂xx D1 v (0) dt + ρgt ∂x D1 v (0) d W Z , ψ δ t
(0)
, Ztδ,H )∂x D1 v (0) (t, Xtπ
(0)
, Z0δ,H ) dWt , (4.32)
(1)
following a similar proof as for Mt . δ Let d W Z , ψ δ t := θt,T dt, from Lemma A.1(iv), one has Z T −t δ Kδ (s) ds = δ H θt,T + δ H+1 θet,T , θt,T = 0
and a straightforward computation gives
∂t Dt,T = −θt,T ,
(4.33)
where Dt,T is defined in (4.23). Then applying Itˆ o’s formula to v (1) defined in (4.23) brings dv (1) (t, Xtπ
(0)
, Z0δ,H ) = Lt,x (λ(Ztδ,H ))v (1) dt + σ(Ztδ,H )π (0) (t, Xtπ
= Lt,x (λ(Z0δ,H ))v (1) dt + σ(Ztδ,H )π (0) (t, Xtπ (3) (4) 1 (1) + gt vx(1) dt + gt v dt 2 xx (3)
= −D12 v (0) θt,T dt + dMt (3)
with the last two terms of order O(δ H ), and Mt (3)
dMt
= σ(Ztδ,H )π (0) (t, Xtπ
(0)
, Ztδ,H )vx(1) dWt
(0)
, Ztδ,H )vx(1) dWt
(4) 1 (1) vxx
(3)
+ gt vx(1) dt + gt
2
dt,
(4.34)
as the martingale: (0)
, Ztδ,H )vx(1) (t, Xtπ
(0)
, Z0δ,H ) dWt .
(4.35)
Collecting equation (4.25), (4.30) and (4.34), we obtain (0) (0) (0) (0) dQtπ ,δ (Xtπ , Z0δ,H ) = d v (0) (t, Xtπ , Z0δ,H ) + λ(Z0δ,H )λ′ (Z0δ,H )φδt D1 v (0) (t, Xtπ , Z0δ,H ) (0) +δ H ρλ2 (Z0δ,H )λ′ (Z0δ,H )v (1) (t, Xtπ , Z0δ,H ) = dMtδ + dRtδ ,
where dMtδ and dRtδ are (1)
(2)
(3)
+ δ H ρλ2 (Z0δ,H )λ′ (Z0δ,H ) dMt , (4.36) 1 (2) (0) (1) (3) (4) 1 (1) dRtδ = gt vx(0) dt + gt vxx dt + δ H ρλ2 (Z0δ,H )λ′ (Z0δ,H ) gt vx(1) dt + gt v dt (4.37) 2 2 xx (3) (4) 1 (5) + λ(Z0δ,H )λ′ (Z0δ,H ) φδt gt ∂x D1 v (0) dt + φδt gt ∂xx D1 v (0) dt + ρgt ∂x D1 v (0) δ H θt,T + δ H+1 θet,T dt . 2 dMtδ = dMt
+ λ(Z0δ,H )λ′ (Z0δ,H ) dMt
20
(0)
Noticing that v (0) (T, XTπ , Z0δ,H ) = U (XTπ
(0)
(0)
), φδT D1 v (0) (T, XTπ , Z0δ,H ) = 0 since φδT = 0, and
(0)
(0)
(0)
v (1) (T, XTπ , Z0δ,H ) = 0 by definition, the terminal condition for Qπ ,δ indeed coincides with VTπ ,δ . Combining with the proof that Mtδ is a true martingale and Rtδ is of order δ 2H detailed in Appendix A, we obtain the desired result in Proposition 4.5.
4.4
Asymptotic Optimality of π (0)
In this subsection, we first derive the approximation of Vtπ,δ Vtπ,δ := E [ U (XTπ )| Ft ] ,
(4.38)
for any admissible strategy π taking the form π e0 + δ α π e1 using epsilon-martingale decomposition technique π as demonstrated in Proposition 4.5, where Xt is the wealth process following the trading strategy π: dXtπ = µ(Ztδ,H )π(t, Xtπ , Ztδ,H ) dt + σ(Ztδ,H )π(t, Xtπ , Ztδ,H ) dWt .
Then, given the previously established results of Vtπ these approximations for
(0) Vtπ ,δ
and
Vtπ,δ ,
(0)
,δ
(4.39)
in Proposition 4.5, we asymptotically compare
and then prove Theorem 4.8.
Theorem 4.8. Under Assumptions 4.1, 4.3, 4.9 and B.1, for any family of trading strategies Aeδt [e π0 , π e1 , α], 2 the following limit exists in L and satisfies V π,δ − Vtπ ℓ := lim t δ→0 δH
where Vtπ
(0)
,δ
(0)
,δ
≤ 0, in L2 ,
(4.40)
and Vtπ,δ are defined in (4.3) and (4.38) respectively. (0)
That is, the strategy π (0) that generate Vtπ ,δ performs asymptotically better up to order δ H than any family Aeδt [e π0 , π e1 , α]. Moreover, the inequality can be written according to the following four cases: (i) π e0 = π (0) , α > H/2: ℓ = 0 and Vtπ,δ = Vtπ
(0)
,δ
+ o(δ H );
(ii) π e0 = π (0) , α = H/2: −∞ < ℓ < 0 and Vtπ,δ = Vtπ
(iii) π e0 = π (0) , α < H/2: ℓ = −∞ and Vtπ,δ = Vtπ (iv) π e0 6= π (0) : limδ→0 Vtπ,δ < limδ→0 Vtπ
where all relations between Vtπ,δ and Vtπ
(0)
(0)
,δ
,δ
(0)
,δ
(0)
,δ
+ O(δ H ) with O(δ H ) < 0;
+ O(δ 2α ) with O(δ 2α ) < 0;
,
hold under L2 sense.
Assumption 4.9. For a fixed choice of (e π0 , π e1 , α > 0), we require:
(i) The whole family (in δ) of strategies {e π0 + δα π e1 } is contained in Aδ (t, x, z);
(ii) The function µ(z) is C 1 (R).
(iii) Functions π e0 (t, x, z) and π e1 (t, x, z) are continuous on [0, T ] × R+ × R, and C 1 in z. (iv) The process v (0) (t, Xtπ , Z0δ,H ) is in L4 ([0, T ] × Ω) uniformly in δ, i.e., "Z # T 4 δ,H E v (0) (s, Xsπ , Z0 ) ds ≤ C2
(4.41)
0
where C2 is independent of δ, Z0δ,H follows (3.10) with t = 0, and Xtπ follows (4.39) with π = π e0 + δ α π e1 .
Remark 4.10. We have π e0 + δ 0 π e1 = π e0 + π e1 + δ α · 0, so it is enough to consider α > 0. 21
Proof. We first deal with the case π = π (0) + δ α π e1 . The derivation is similar to the one in Section 4.3. As usual, in order to condense the notation, we systematically omit the argument (s, Xsπ , Z0δ,H ) for v (0) in what follows. 1 (0) (0) dv (0) (t, Xtπ , Z0δ,H ) =vt dt + µ(Ztδ,H )π(t, Xtπ , Ztδ,H )vx(0) dt + σ 2 (Ztδ,H )π 2 (t, Xtπ , Ztδ,H )vxx dt 2 + σ(Ztδ,H )π(t, Xtπ , Ztδ,H )vx(0) dWt 1 (2) (0) (1) f(1) dt + dM =(Ztδ,H − Z0δ,H )λ(Z0δ,H )λ′ (Z0δ,H )D1 v (0) dt + gt vx(0) dt + gt vxx t 2 (0) 1 2 (1) (2) (0) (0) dt, e1 (t, Xtπ , Ztδ,H )vxx + δ α get vx(0) dt + δ α e gt vxx dt + δ 2α σ 2 (Ztδ,H ) π 2
f(1) , ge(1) and ge(2) are defined by where M t t t f(1) = σ(Z δ,H ) π (0) (t, X π , Z δ,H ) + δ α π dM e1 (t, Xtπ , Ztδ,H ) vx(0) (t, Xtπ , Z0δ,H ) dWt , t t t t (2) (1) δ,H δ,H 1 (µRe π ) π 1 )z , , g e = Z − Z gt = Ztδ,H − Z0δ,H (µe e z t t 0 (2) (1) z=χ et
z=χ et
h i (i) with χ et ∈ Z0δ,H ∧ Ztδ,H , Z0δ,H ∨ Ztδ,H , i = 1, 2. Then it suffices to find the epsilon-martingale decomposition for the term (Ztδ,H − Z0δ,H )D1 v (0) (t, Xtπ , Z0δ,H ) dt. Following a similar derivation as in Section 4.3, one can deduce (0) dQtπ ,δ (Xtπ , Z0δ,H ) = d v (0) (t, Xtπ , Z0δ,H ) + λ(Z0δ,H )λ′ (Z0δ,H )φδt D1 v (0) (t, Xtπ , Z0δ,H ) +δ H ρλ2 (Z0δ,H )λ′ (Z0δ,H )v (1) (t, Xtπ , Z0δ,H ) e δ + δ 2α dN δ , fδ + dR = dM t t t
where (1)
fδ = dM f dM t t
(2)
f + λ(Z0δ,H )λ′ (Z0δ,H ) dM t
(3)
f , + δ H ρλ2 (Z0δ,H )λ′ (Z0δ,H ) dM t
(2)
= D1 v (0) (t, Xtπ , Z0δ,H ) dψtδ + φδt σ(Ztδ,H )π(t, Xtπ , Ztδ,H )∂x D1 v (0) (t, Xtπ , Z0δ,H ) dWt ,
(3)
= σ(Ztδ,H )π(t, Xtπ , Ztδ,H )vx(1) (t, Xtπ , Z0δ,H ) dWt ,
f dM t
ft dM
1 (2) (0) (1) (0) (3) (4) 1 (1) δ,H δ,H δ ′ α (1) (0) α (2) (0) H 2 e dRt = gt vx dt + gt vxx dt + δ e gt vx dt + δ get vxx dt + δ ρλ (Z0 )λ (Z0 ) gt vx(1) + gt v 2 2 xx h 2 (1) 1 (3) (1) +δ α µe π 1 vx(1) + δ α σ 2 π (0) π e1 vxx + δ 2α σ 2 π dt + λ(Z0δ,H )λ′ (Z0δ,H )φδt gt ∂x D1 v (0) e1 vxx 2 1 2α 2 1 2 1 (4) (0) (0) α 1 (0) α 2 (0) 1 (0) dt ∂xx D1 v e π ∂x D1 v + δ σ π π e ∂xx D1 v + δ σ π + gt ∂xx D1 v + δ µe 2 2 i h (5) dt, π 1 ∂x D1 v (0) δ H θt,T + δ H+1 θet,T + ρλ(Z0δ,H )λ′ (Z0δ,H ) gt ∂x D1 v (0) δ H θt,T + δ H+1 θet,T + δ α σe 2 (0) e δ = 1 σ 2 (Z δ,H ) π dN e1 (t, Xtπ , Ztδ,H ) vxx (t, Xtπ , Z0δ,H ) dt. t t 2
To condense the expression for Rtδ , we omit the arguments for functions v (0) (t, Xtπ , Z0δ,H ), v (1) (t, Xtπ , Z0δ,H ), µ(Ztδ,H ), σ(Ztδ,H ), π (0) (t, Xtπ , Ztδ,H ) and π e1 (t, Xtπ , Ztδ,H ). Since the Merton value M (t, x; λ) is strictly concave, so does v (0) (t, x, z) = M (t, x; λ(z)), which implies ftδ is a true martingale that Nt is non-increasing. Moreover, under Assumption 4.9, B.1, one can prove M
22
eδ is of order δ H+H∧α , which yields and R t h (0) i i h eTδ + δ 2α NTδ |Ft ftδ + E R Vtπ,δ = E QTπ ,δ |Ft = M i h (0) eTδ − R etδ |Ft + δ 2α E NTδ − Ntδ |Ft = Qtπ ,δ (Xtπ , Z0δ,H ) + E R (0) (0) = Qtπ ,δ (Xtπ , Z0δ,H ) + δ 2α E NTδ − Ntδ |Ft + O(δ H+H∧α ) ≤ Qtπ ,δ (Xtπ , Z0δ,H ) + O(δ H+H∧α ), (4.42) ftδ + R etδ + Ntδ = Qtπ where in the derivation we have used M Nt .
(0)
,δ
(Xtπ , Z0δ,H ) and the decreasing property of
The second case is π = π e0 + δ α π e1 with π e0 6≡ π (0) . Here the wealth process Xtπ follows e0 + δ α π e1 (t, Xtπ , Ztδ,H ) dWt . e0 + δ α π e1 (t, Xtπ , Ztδ,H ) dt + σ(Ztδ,H ) π dXtπ = µ(Ztδ,H ) π
(4.43)
Under similar derivations, one can deduce
btδ ctδ + dR btδ + dN dv (0) (t, Xtπ , Z0δ,H ) = dM
where
ctδ = σ(Ztδ,H )π(t, Xtπ , Ztδ,H )vx(0) (t, Xtπ , Z δ,H ) dWt , dM 0 1 1 α 2 1 2 (0) (1) (2) δ (0) 1 (0) 2 0 1 (0) α (0) b dRt = b gt vx + b π vx + σ π e π e vxx + δ σ π dt + δ µe vxx dt, e g v 2 t xx 2 2 (0) b δ = 1 σ 2 (Z δ,H ) π (t, Xtπ , Z0δ,H ) dt, e0 − π (0) (t, Xtπ , Z0δ,H )vxx dN t 0 2
(1)
with gbt
(4.44)
(2)
and b gt
(1)
gbt
defined as π 0 )z = Ztδ,H − Z0δ,H (µe
(1) z=χ bt
i h (i) and χ bt ∈ Z0δ,H ∧ Ztδ,H , Z0δ,H ∨ Ztδ,H , i = 1, 2.
,
(2)
gbt
2 e0 )z = Ztδ,H − Z0δ,H (σ 2 π
(2)
z=χ bt
,
(4.45) (4.46) (4.47)
(4.48)
btδ is strictly decreasing due to the strict concavity of v (0) . Under Assumption 4.9, B.1, M ctδ is a Here N δ H∧α b true martingale, and Rt is of order δ . Therefore we obtain i h bTδ − N btδ Ft + O(δ H∧α ) < v (0) (t, Xtπ , Z δ,H ) + O(δ H∧α ). Vtπ,δ = v (0) (t, Xtπ , Z0δ,H ) + E N (4.49) 0 Now comparing the approximation (4.21) with (4.42) (4.49), we obtain the desired result in Theorem 4.8.
5
Conclusion
In this paper, we have considered the portfolio allocation problem in the context of a slowly varying fractional stochastic environment driven by a fractional OU process with H ∈ (0, 1), and when the investor tries to maximize her terminal utility with, first, power utilities, and, then, in a general class of utility functions. In the power utility case, using a martingale distortion representation for the value process and the espsilon-martingale decomposition method, we are able to derive a first order asymptotic approximation for both the optimal portfolio value and the optimal strategy. The first order correction for the optimal portfolio value has both random and deterministic parts as in the linear option pricing problem studied in Garnier and Solna [2015]. However, the approximate optimal strategy does not involve a random part and can be easily implemented. We also show that the zeroth order of the optimal strategy generates the 23
portfolio value up the the first order and we observe that the first order correction is even more important in the case of H small as observed in volatility data (see Gatheral et al. [2014]). Finally, we extend our analysis to the case of general utilities where we can derive the first order asymptotic optimality within a specific subclass of strategies. The case of fast varying fractional stochastic environment with H ∈ ( 21 , 1) is the topic of the paper in preparation Fouque and Hu [2017].
A
Technical Lemmas
In this section, we present several lemmas which are used in Section 3 and 4. Lemma A.1. (i) The slowly varing fractional factor Ztδ,H defined in (3.10) is a stationary Gaussian process with zero mean and variance Z ∞ 2 Z t 2 δ,H δ 2 K (t − s) ds = K2 (s) ds = σou , (A.1) E Zt = 0
−∞
2 where σou is given in (3.6) and free of δ. Therefore Ztδ,H has finite moments of any order, and for + any p ∈ N , Z·δ,H ∈ Lp ([0, T ] × Ω) uniformly in δ. i h Any adapted process that χt ∈ Z0δ,H ∧ Ztδ,H , Z0δ,H ∨ Ztδ,H also satisfies that χ· ∈ Lp ([0, T ] × Ω) uniformly in δ.
(ii) The difference Ztδ,H − Z0δ,H is a Gaussian random variable with zero mean and variance 2 2 (δt)2H + o(δ 2H ), = σH E Ztδ,H − Z0δ,H
(A.2)
2 where σH is given in (3.3). Consequently, the k th moment of Ztδ,H − Z0δ,H is of order δ kH , uniformly in t ∈ [0, T ]. Moreover, Z·δ,H − Z0δ,H is of order δ H in Lp ([0, T ] × Ω) sense, for any p ∈ N+ .
(iii) The random correction φδt defined in (3.16) is a normal random variable of order δ H with zero mean and variance #2 H+ 21 Z " h 2 i δ 2H T 2+2H ∞ t t 1 t H− 21 δ H+ 21 = 2 dv E φt 1− +v −v − (1 − )(H + )(v − )+ T T 2 T Γ (H + 32 ) 0 + O(δ 2H+1 ),
(A.3)
where the integral is uniformly bounded in t ∈ [0, T ]. Therefore, the Lp ([0, T ] × Ω) norm φδ· is of order δ H , for any p ∈ N+ . (iv) The process ψtδ t∈[0,T ] defined in (3.20) is a square-integrable martingale satisfying dψtδ =
Z
T −t
0
with θt,T and θet,T given by θt,T
1 H+ 12 = , 3 (T − t) Γ(H + 2 )
Kδ (s − t) ds dWtZ := δ H θt,T + δ H+1 θet,T dWtZ , θet,T =
a Γ(H + 21 )
Z
0
T −t
Z
3
s 0
H− 12 −aδu
(s − u)
e
a(T − t)H+ 2 , du ds ≤ Γ(H + 52 )
and uniformly bounded in t ∈ [0, T ] and δ ≪ 1. Consequently, one has
d ψ, W
Z
t
=
Z
0
T −t
δ
K (s) ds
!
dt and d hψit =
24
(A.4)
Z
0
T −t
δ
K (s) ds
!2
dt.
(A.5)
Proof. All can be computed directly, and we refer to the statements in [Garnier and Solna, 2015, Section 6, Appendix A]. Lemma A.2. The processes M (i) t∈[0,T ] , i = 1, 2, 3 defined in (4.27), (4.32) and (4.35) are true martingales with respect to the filtration Ft , so does (M )t∈[0,T ] . i h h
1/2 i (1) < ∞, which is equivalent to E sups≤T Ms < ∞ Proof. We prove this result by showing E M (1) T
by Burkholder–Davis–Gundy inequality. This implies that M (1) is a martingle. To this end, we first bound its quadratic variation
E 2 D (0) (0) d M (1) = λ2 (Ztδ,H )R2 (t, Xtπ ; λ(Ztδ,H )) vx(0) (t, Xtπ , Z0δ,H ) dt t 2 (0) 2 (0) (0) ≤ λ2 (Ztδ,H )C 2 Xtπ vx(0) (t, Xtπ , Z0δ,H ) dt ≤ λ2 (Ztδ,H )C 2 v (0) (t, Xtπ , Z0δ,H ) dt
by using the estimate R(t, x; λ(z)) ≤ Cx and the concavity of v (0) , and then deduce !1/2 D Z T E1/2 2 (0) ≤ C2E E M (1) λ2 (Zsδ,H ) v (0) (s, Xsπ , Z0δ,H ) ds T
0
2
1/4
≤C E
"Z
0
T
4
λ
(Zsδ,H ) ds
#
1/4
·E
"Z
0
T
v
(0)
4
(0) (s, Xsπ , Z0δ,H )
#
ds < ∞,
where to conclude, we have used Assumption 4.3, and Lemma A.1(i) about Zsδ,H . The proofs for M (2) and M (3) are obtained in a similar way with estimates from [Fouque and Hu, 2016a, Proposition 3.5], which is of the form j (A.6) R (t, x; λ(z)) ∂x(j+1) R(t, x; λ(z)) ≤ Kj , ∀0 ≤ j ≤ 3, ∀(t, x, z) ∈ [0, T )t × R+ × R,
and Lemma A.1(iii)-(iv), and thus we omit the details here. Lemma A.3. The process Rtδ t∈[0,T ] defined in (4.37) is of order δ 2H .
(1) (0)
Proof. We shall prove that each term in Rtδ is of order δ 2H . The first term we deal with is gt vx (1) gt defined in (4.28):
with
1 2 (0) (0) (1) (0) 2 δ,H vx(0) (t, Xtπ , Z0δ,H ) Ztδ,H − Z0δ,H 2 (λ′ ) R + 2λλ′′ R + 4λλ′ Rz + λ2 Rzz gt vx (t, Xtπ , Z0 ) = (1) 2 z=χt 2 (0) (0) 1 δ,H (1) (1) δ,H ≤ d(χt )R(t, Xtπ ; λ(χt ))vx(0) (t, Xtπ , Z0δ,H ) Zt − Z0 2 2 (0) (0) 1 δ,H (1) Zt − Z0δ,H d(χt )CXtπ vx(0) (t, Xtπ , Z0δ,H ) ≤ 2 2 (0) (1) ≤ C Ztδ,H − Z0δ,H d(χt )v (0) (t, Xtπ , Z0δ,H ). Here the first inequality follows from [Fouque and Hu, 2016a, Propositon 3.7]: there exists non-negative functions de01 (z) and de02 (z) that have mostly polynomial growth and satisfy |Rz (t, x; λ(z))| ≤ de01 (z)R(t, x; λ(z)),
|Rzz (t, x; λ(z))| ≤ de02 (z)R(t, x; λ(z)),
and thus d(z) is also at most polynomially growing defined as 2 d(z) = 2 (λ′ (z)) + 2λ(z)λ′′ (z) + 4λ(z)λ′ (z)de01 (z) + λ2 (z)de02 (z) . 25
(A.7)
The second inequality is given by the estimate R(t, x; λ(z)) ≤ Cx and the concavity of v (0) . Therefore "Z # "Z # T 2 T δ,H δ,H δ,H (1) (0) π (0) δ,H (1) (0) π (0) E gs vx (s, Xs , Z0 ) ds ≤ CE Zs − Z0 d(χs )v (s, Xs , Z0 ) ds 0
0
" Z ≤ E
0
T
Zsδ,H
−
Z0δ,H
8
# 14 " Z ds E
T
0
4
d
(χ(1) s ) ds
# 41 " Z E
0
T
2 (0) v (0) (s, Xsπ , Z0δ,H ) ds
# 12
and is of order δ 2H . This is because, one has proved in Lemma A.1(ii) that the first expectation is of order δ 2H , the second expectation is uniformly bounded in δ due to the polynomial growth property of d(·) and Lemma A.1(i), while the third term is uniformly bounded by Assumption 4.3(iii). Other terms contained in Rtδ can be proved of order δ 2H in a similar way with additional Assumption 4.3(ii), estimates (A.6), Lemma A.1(iii)-(iv) and estimates from [K¨allblad and Zariphopoulou, 2017, Proposition 4].
B
Assumptions in Section 4.4
This set of assumptions is used in establishing the approximation accuracy (4.42) (resp. (4.49)) to Vtπ ftδ (resp. M ctδ ) is a true martingale and that defined in (4.38), namely, these assumptions will ensure that M etδ (resp. R btδ ) is of order δ H+H∧α (resp. δ H∧α ). R 0 1 Assumption B.1. Let A0 (t, x, z) π e ,π e , α be the family of trading strategies defined in (4.5). Recall that X π is the wealth generated by the strategy π = π e0 + δ α π e1 as defined in (4.39). In order to condense π the notation, we systematically omit the argument (s, Xs , Z0δ,H ) of v (0) and v (1) , the argument Zsδ,H of µ and σ, the argument Z0δ,H of λ, and (s, Xsπ , Zsδ,H ) of π e0 and π e1 in what follows. According to the different cases, we further require:
(i) If π e0 ≡ π (0) , the following quantities are uniformly bounded in δ: 2 2 RT RT R T 1 (0) 2 R T 1 (0) 2 (0) (0) E 0 (µe π 1 )z |z=ξe(1) vx ds, E 0 (µRe π 1 )z |z=ξe(2) vxx ds, E 0 µe π vx ds, E 0 σe π vx ds, s s h i h 2 2 2 (1) i RT RT RT (1) E 0 σ2 π e1 ∂xx D1 v (0) ds, E λ2 λ′ 0 µe π 1 vx ds , E λ2 λ′ 0 σ 2 π e1 vxx ds , # " # " 2 12 2 12 RT RT 2 ′ 1 (1) ′ 1 (0) δ ,E λ λ , σe π vx ds E λλ σe π vx φs ds 0 0
(ii) If π e0 6≡ π (0) , we require the uniformly boundedness (in δ) of the following: 2 2 2 RT RT RT RT (0) (0) (0) (0) π 0 )z |z=ξb(1) vx e0 )z |z=ξb(2) vxx e0 π e1 vxx ds, ds, E 0 (σ 2 π ds, E 0 µe π 1 vx ds, E 0 σ 2 π E 0 (µe s
E
RT 0
σ2
s
2 21 2 12 RT RT 1 2 (0) 0 (1) 1 (1) π e vxx ds, E 0 σe π vx ds , E 0 σe π vx ds .
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