Mean-variance portfolio selection under Volterra Heston model
MMean-variance portfolio selection under Volterra Heston model
Bingyan Han ∗ Hoi Ying Wong † January 24, 2020
Abstract
Motivated by empirical evidence for rough volatility models, this paper investigatescontinuous-time mean-variance (MV) portfolio selection under the Volterra Heston model.Due to the non-Markovian and non-semimartingale nature of the model, classic stochas-tic optimal control frameworks are not directly applicable to the associated optimizationproblem. By constructing an auxiliary stochastic process, we obtain the optimal investmentstrategy, which depends on the solution to a Riccati-Volterra equation. The MV efficientfrontier is shown to maintain a quadratic curve. Numerical studies show that both roughnessand volatility of volatility materially affect the optimal strategy.
Keywords:
Mean-variance portfolio, Volterra Heston model, Riccati–Volterra equations,rough volatility.
Mathematics Subject Classification:
There has been a growing interest in studying rough volatility models [15, 11, 20]. Roughvolatility models are stochastic volatility models whose trajectories are rougher than the pathsof a standard Brownian motion in terms of the H¨older regularity. Specifically, when the H¨olderregularity is less than 1/2, the stochastic path is regarded as rough. The roughness is closelyrelated to the Hurst parameter H . This paper focuses on the Volterra Heston model, whoseprobabilistic characterization does not involve the rough paths theory [11].Rough volatility models are attractive because they capture the dynamics of historical andimplied volatilities remarkably well with only a few additional parameters. Investigations of thetime series of the realized volatility from high frequency data estimate the Hurst parameter H to be near 0 .
1, which is much smaller than the 0 . H , the rougher the timeseries model. Therefore, the empirical finding suggests a rougher realized path of volatility thanthe standard Brownian motion. Although previous studies have found a long memory propertywithin realized volatility series, it is shown in [15] that rough volatility models can generate theillusion of a long memory. However, the simulated paths with a small Hurst parameter resemblethe realized ones.Rough volatility models also better capture the term structure of an implied volatility sur-face, especially for the explosion of at-the-money (ATM) skew when maturity goes to zero.More precisely, let σ BS ( k, τ ) be the implied volatility of an option where k is the log-moneynessand τ is the time to expiration. The ATM skew at maturity τ is defined by φ ( τ ) (cid:44) (cid:12)(cid:12)(cid:12) ∂σ BS ( k, τ ) ∂k (cid:12)(cid:12)(cid:12) k =0 . (1.1) ∗ Department of Statistics, The Chinese University of Hong Kong, Hong Kong, [email protected] † Department of Statistics, The Chinese University of Hong Kong, Hong Kong, [email protected] See, for example, Oxford-Man Institute’s realized library at https://realized.oxford-man.ox.ac.uk/data a r X i v : . [ q -f i n . P M ] J a n mpirical evidence shows that the ATM skew explodes when τ ↓
0. However, conventionalvolatility models such as the Heston model [21] generate a constant ATM skew for a small τ . If the volatility is modeled by a fractional Brownian motion, then the ATM skew has anasymptotic property [14], φ ( τ ) ≈ τ H − / , when τ ↓ , (1.2)where H is the Hurst parameter. Rough volatility models can fit the explosion remarkably wellby simply adjusting the H .Recent advances offer elegant theoretical foundations for rough volatility models. We notethe martingale expansion formula for implied volatility [14], asymptotic analysis of fBM [14,Section 3.3], the microstructural foundation of rough Heston models by scaling the limit ofproper Hawkes processes [9], the closed-form characteristic function of rough Heston models upto the solution of a fractional Riccati equation [11], and the hedging strategy for options underrough Heston models [10]. In this paper, we are particularly interested in the affine Volterraprocesses [3] because these models embrace rough Heston model [11] as a special case. Thecharacteristic function in [11] is extended to the exponential-affine transform formula in termsof Riccati-Volterra equations [3]. Affine Volterra processes are applied to finance problems in[23]. In addition, an alternative rough version of the Heston model is introduced in [20], wheresome asymptotic results are derived.While the rough volatility literature focuses on option pricing, only a few works contribute toportfolio optimization such as [12, 13, 4]. All of them consider utility maximization. To the bestof our knowledge, this is the first paper to consider the mean-variance (MV) portfolio selectionunder a rough stochastic environment. The MV criterion in portfolio selection pioneered byMarkowitz’s seminal work is the cornerstone of the modern portfolio theory. We cannot give afull list of research outputs related to this Nobel Prize winning work, but mention contributionsin continuous-time settings [36, 27, 26, 6, 22, 31] as important references. We formulate the MV portfolio selection under the Volterra Heston models in a reasonablyrigorous manner. As pointed out by [3, 23], the Volterra Heston model (2.6)-(2.7) has a unique inlaw weak solution, but its pathwise uniqueness is still an open question in general. This enforcesus to consider the MV problem under a general filtration F that satisfies the usual conditionsbut may not be the augmented filtration generated by the Brownian motion. A similar generalsetting also appears in [22]. We emphasize that the probability basis and Brownian motions arealways fixed for the problem in Section 3. Therefore, our formulation is still considered to be a strong formulation , because the filtered probability space and Brownian motions are not partsof the control.Under such a problem formulation, we construct in Section 4 an auxiliary stochastic process M t to solve the MV portfolio selection by completion of squares. Several properties of M t arederived in Theorem 4.1, which is a main result of this paper. Like [11, 10, 3], we encounterdifficulties due to the non-Markovian and non-semimartingale structure of the Volterra Hestonmodel (2.6)-(2.7). Inspired by the exponential-affine formulas in [3, 11], the process M t isconstructed upon the forward variance under a proper alternative measure. The explicit solutionfor the optimal investment strategy is obtained in Theorem 4.3.Under the rough Heston model, we investigate the impact of roughness on the optimal invest-ment strategy u ∗ . Recently, a trading strategy has been proposed to leverage the information ofroughness [18]. The strategy longs the roughest stocks and shorts the smoothest stocks. Excessreturns from this strategy are not fully explained by standard factor models like the CAPMmodel and Fama-French model. We examine this trading signal under the MV setting. Ourtheory predicts that the effect of roughness on investment strategy is opposite under differentvolatility of volatility (vol-of-vol). We also discuss the roughness effect on the efficient frontier.2he rest of the paper is organized as follows. Section 2 presents the Volterra Heston modeland some useful properties. We discuss a related Riccati-Volterra equation. We then formulatethe MV portfolio selection problem in Section 3 and solve it explicitly in Section 4. Numericalillustrations are given in Section 5. Section 6 concludes the paper. The existence and uniquenessof the solution to Riccati-Volterra equations are summarized in Appendix A. An auxiliary resultused in Theorem 4.1 is proved in Appendix B. Our problem is defined under a given complete probability space (Ω , F , P ), with a filtration F = {F t } ≤ t ≤ T satisfying the usual conditions, supporting a two-dimensional Brownian motion W = ( W , W ). The filtration F is not necessarily the augmented filtration generated by W ;thus, it can be a strictly larger filtration. This consideration is different from some previousstudies like [27, 26, 31] but is consistent with [22] for the MV hedging problem under a generalfiltration. This consideration is important because the stochastic Volterra equation (2.6)-(2.7)only has a unique in law weak solution but its strong uniqueness is still an open question ingeneral. Recall that for stochastic differential equations, X is referred to as a strong solution ifit is adapted to the augmented filtration generated by W , and a weak solution otherwise. For aweak solution, the driving Brownian motion W is also a part of the solution [30, Chapter IX].Therefore, F cannot be simply chosen as the augmented filtration generated by W , as extrainformation may be needed to construct a solution to (2.6)-(2.7).To proceed, we introduce a kernel K ( · ) ∈ L loc ( R + , R ), where R + = { x ∈ R | x ≥ } , andmake the following standing assumption throughout the paper, in line with [3, 23]. A func-tion f is called completely monotone on (0 , ∞ ), if it is infinitely differentiable on (0 , ∞ ) and( − k f ( k ) ( t ) ≥ t >
0, and k = 0 , , ... . Assumption 2.1. K is strictly positive and completely monotone on (0 , ∞ ) . There is γ ∈ (0 , ,such that (cid:82) h K ( t ) dt = O ( h γ ) and (cid:82) T ( K ( t + h ) − K ( t )) dt = O ( h γ ) for every T < ∞ . The convolutions K ∗ L and L ∗ K for a measurable kernel K on R + and a measure L on R + of locally bounded variation are defined by( K ∗ L )( t ) = (cid:90) [0 ,t ] K ( t − s ) L ( ds ) and ( L ∗ K )( t ) = (cid:90) [0 ,t ] L ( ds ) K ( t − s ) (2.1)for t > t = 0 by right-continuity ifpossible. If F is a function on R + , let( K ∗ F )( t ) = (cid:90) t K ( t − s ) F ( s ) ds. (2.2)Let W be a 1-dimensional continuous local martingale. The convolution between K and W is defined as ( K ∗ dW ) t = (cid:90) t K ( t − s ) dW s . (2.3)A measure L on R + is called resolvent of the first kind to K , if K ∗ L = L ∗ K ≡ id . (2.4)The existence of a resolvent of the first kind is shown in [19, Theorem 5.5.4] under the completemonotonicity assumption, imposed in Assumption 2.1. Alternative conditions for the existenceare given in [19, Theorem 5.5.5].A kernel R is called the resolvent or resolvent of the second kind to K if K ∗ R = R ∗ K = K − R. (2.5)3he resolvent always exists and is unique by [19, Theorem 2.3.1].Further properties of these definitions can be found in [19, 3]. Although the same notioncan be defined for higher dimensions and in matrix form, it suffices for us to consider the scalarcase. Commonly used kernels [3] summarized in Table 1 satisfy Assumption 2.1 once c > α ∈ (1 / , β ≥ K ( t ) R ( t ) L ( dt )Constant c ce − ct c − δ ( dt )Fractional (Power-law) c t α − Γ( α ) ct α − E α,α ( − ct α ) c − t − α Γ(1 − α ) dt Exponential ce − βt ce − βt e − ct c − ( δ ( dt ) + β dt )Table 1: Examples of kernels K and their resolvents R and L of the second and first kind. E α,β ( z ) = (cid:80) ∞ n =0 z n Γ( αn + β ) is the Mittag–Leffler function. See [11, Appendix A1] for its properties.The constant c (cid:54) = 0.The variance process within the Volterra Heston model is defined as V t = V + κ (cid:90) t K ( t − s ) ( φ − V s ) ds + (cid:90) t K ( t − s ) σ (cid:112) V s dB s , (2.6)where dB s = ρdW s + (cid:112) − ρ dW s and V , κ, φ , and σ are positive constants. The correlation ρ between stock price and variance is also constant. As documented in [15], the general overallshape of the implied volatility surface does not change significantly, indicating that it is stillacceptable to consider a variance process whose parameters are independent of stock price andtime. The rough Heston model in [11, 10] becomes a special case of (2.6) once K ( t ) = t α − Γ( α ) .Another rough version of the Heston model studied in [20] is adopted to investigate the powerutility maximization [4].Following [3] and [24, 6, 35, 32], the risky asset (stock) price S t is assumed to follow dS t = S t ( r t + θV t ) dt + S t (cid:112) V t dW t , S > , (2.7)with a deterministic bounded risk-free rate r t > θ (cid:54) = 0. The market price ofrisk, or risk premium, is then given by θ √ V t . The risk-free rate r t > Theorem 2.2. ([3, Theorem 7.1]) Under Assumption 2.1, the stochastic Volterra equation(2.6)-(2.7) has a unique in law R + × R + -valued continuous weak solution for any initial condition ( S , V ) ∈ R + × R + . Remark 2.3.
Our model (2.6)-(2.7) is defined under the physical measure, whereas the optionpricing model of [3, Equations (7.1)-(7.2)] is under a risk-neutral measure with a zero risk-freerate. However, the proofs are almost identical because the affine structure is maintained and S is determined by V . Remark 2.4.
For strong uniqueness, we mention [2, Proposition B.3] as a related result withkernel K ∈ C ([0 , T ] , R ) and [29, Proposition 8.1] for certain Volterra integral equations withsmooth kernels. However, the strong uniqueness of (2.6)-(2.7) is left open for singular kernels.For weak solutions, it is free to construct the Brownian motion as needed. However, the MV bjective only depends on the mathematical expectation for the distribution of the processes. Inthe sequel, we will only work with a version of the solution to (2.6)-(2.7) and fix the solution ( S, V, W , W ) , as other solutions have the same law. The following condition enables us to verify the admissibility of the optimal strategy. To bemore precise about the constant a , (4.23) gives an explicit sufficient large value needed. Assumption 2.5. E (cid:104) exp (cid:0) a (cid:82) T V s ds (cid:1)(cid:105) < ∞ for a large enough constant a > . To verify that Assumption 2.5 holds under reasonable conditions, we consider the Riccati-Volterra equation (2.8) for g ( a, t ) as follows: g ( a, t ) = (cid:90) t K ( t − s ) (cid:2) a − κg ( a, s ) + σ g ( a, s ) (cid:3) ds. (2.8)The existence and uniqueness of the solution to (2.8) are given in Lemmas A.2 and A.3. Theorem 2.6.
Suppose Assumption 2.1 holds and the Riccati-Volterra equation (2.8) has aunique continuous solution on [0 , T ] , then E (cid:104) exp (cid:0) a (cid:90) T V s ds (cid:1)(cid:105) = exp (cid:104) κφ (cid:90) T g ( a, s ) ds + V (cid:90) T (cid:2) a − κg ( a, s ) + σ g ( a, s ) (cid:3) ds (cid:105) < ∞ . (2.9) Moreover, denote L as the resolvent of the first kind to K , then E (cid:104) exp (cid:0) a (cid:90) T V s ds (cid:1)(cid:105) = exp (cid:104) κφ (cid:90) T g ( a, s ) ds + V (cid:90) T g ( a, T − s ) L ( ds ) (cid:105) . (2.10) Proof.
Note g ( a, t ) in (2.8) corresponds to [3, Equation (4.3)] with u = 0 and f = a . [3,Theorem 4.3] shows the equivalence between [3, Equation (4.4)] and [3, Equation (4.6)]. For t = T , the expressions in [3, Equation (4.4)-(4.6)] indicate that a (cid:90) T V s ds = Y − σ (cid:90) T g ( a, T − s ) V s ds + σ (cid:90) T g ( a, T − s ) (cid:112) V s dB s , (2.11)with Y = κφ (cid:90) T g ( a, s ) ds + V (cid:90) T (cid:2) a − κg ( a, s ) + σ g ( a, s ) (cid:3) ds. (2.12)As g ( a, · ) is continuous on [0 , T ] and therefore bounded, exp (cid:0) − σ (cid:82) t g ( a, T − s ) V s ds + σ (cid:82) t g ( a, T − s ) √ V s dB s (cid:1) is a martingale by [3, Lemma 7.3]. Therefore, E (cid:104) exp (cid:0) a (cid:90) T V s ds (cid:1)(cid:105) = exp( Y ) = exp (cid:104) κφ (cid:90) T g ( a, s ) ds + V (cid:90) T (cid:2) a − κg ( a, s ) + σ g ( a, s ) (cid:3) ds (cid:105) . (2.13)Note that K ∗ L = id implies (cid:90) T (cid:2) a − κg ( a, s ) + σ g ( a, s ) (cid:3) ds = (cid:90) T g ( a, T − s ) L ( ds ) . (2.14)The result follows.Theorem 2.6 recovers the same expression for E (cid:104) exp (cid:0) a (cid:82) T V s ds (cid:1)(cid:105) in [10, Theorem 3.2]. Westress that the proof circumvents the use of the Hawkes processes. In addition, we mention [17],which examines the moment explosions in the rough Heston model, as a related reference.5 Mean-variance portfolio selection
Let u t (cid:44) √ V t π t be the investment strategy, where π t is the amount of wealth invested in thestock. Then wealth process X t satisfies dX t = (cid:0) r t X t + θ (cid:112) V t u t (cid:1) dt + u t dW t , X = x > . (3.1) Definition 3.1.
An investment strategy u ( · ) is said to be admissible if(1). u ( · ) is F -adapted;(2). E (cid:104)(cid:16) (cid:82) T |√ V t u t | dt (cid:17) (cid:105) < ∞ and E (cid:104) (cid:82) T | u t | dt (cid:105) < ∞ ; and(3). the wealth process (3.1) has a unique solution in the sense of [34, Chapter 1, Definition6.15], with P - a.s. continuous paths.The set of all of the admissible investment strategies is denoted as U . Remark 3.2.
In Condition (1), F is possibly strictly larger than the Brownian filtration of W = ( W , W ) , which means that extra information in addition to W can be used to construct anadmissible strategy. In general, u can rely on a local P -martingale that is strongly P -orthogonalto W . See hedging strategy (3.6) in [22, Theorem 3.1] for such examples. However, our optimalstrategy u ∗ turns out to only depend on the variance V and Brownian motion W , as shown inTheorem 4.3. Remark 3.3.
We emphasize once again that the underlying probability space and Brownianmotions are not parts of our control. Therefore, our formulation should still be referred to asa strong formulation. Readers may refer to [34, Chapter 2, Section 4] for discussions of thedifference between strong and weak formulations of stochastic control problems.
The MV portfolio selection in continuous-time is the following problem . min u ( · ) ∈U J ( x ; u ( · )) = E (cid:2) ( X T − c ) (cid:3) , subject to E [ X T ] = c, ( X ( · ) , u ( · )) satisfy (3.1) . (3.2)The constant c is the target wealth level at the terminal time T . We assume c ≥ x e (cid:82) T r s ds following [27, 26, 31]. Otherwise, a trivial strategy that puts all of the wealth into the risk-freeasset can dominate any other admissible strategy. The MV problem is said to be feasible for c ≥ x e (cid:82) T r s ds if there exists a u ( · ) ∈ U that satisfies E [ X T ] = c . Note that r t > E [ (cid:82) T V t dt ] >
0. It is then clear that the feasibility of our problem is guaranteed for any c ≥ x e (cid:82) T r s ds by a slight modification to the proof in [26, Propsition 6.1].As Problem (3.2) has a constraint, it is equivalent to the following max-min problem [28]. (cid:26) max η ∈ R min u ( · ) ∈U J ( x ; u ( · )) = E (cid:2) ( X T − ( c − η )) (cid:3) − η , ( X ( · ) , u ( · )) satisfy (3.1) . (3.3)Let ζ = c − η and consider the inner Problem (3.4) of (3.3) first. (cid:26) min u ( · ) ∈U J ( x ; u ( · )) = E (cid:2) ( X T − ζ ) (cid:3) − η , ( X ( · ) , u ( · )) satisfy (3.1) . (3.4) There are several equivalent formulations. Optimal investment strategy
To solve Problem (3.4), we introduce a new probability measure ˜ P by d ˜ P d P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t = exp (cid:16) − θ (cid:90) t V s ds − θ (cid:90) t (cid:112) V s dW s (cid:17) , (4.1)where the stochastic exponential is a true martingale [3, Lemma 7.3]. Then ˜ W t (cid:44) W t +2 θ (cid:82) t √ V s ds is a new Brownian motion under ˜ P . Hence, V t = V + (cid:90) t K ( t − s ) ( κφ − λV s ) ds + (cid:90) t K ( t − s ) σ (cid:112) V s d ˜ B s , (4.2)where λ = κ + 2 θρσ and d ˜ B s = ρd ˜ W s + (cid:112) − ρ dW s .Denote ˜ E [ · ] and ˜ E [ ·|F t ] as the ˜ P -expectation and conditional ˜ P -expectation, respectively.The forward variance under ˜ P is the conditional ˜ P -expected variance: ˜ E [ V s |F t ] (cid:44) ξ t ( s ). Thefollowing identity is proven in [23, Propsition 3.2] by an application of [3, Lemma 4.2]. ξ t ( s ) = ˜ E [ V s |F t ] = ξ ( s ) + (cid:90) t λ R λ ( s − u ) σ (cid:112) V u d ˜ B u , (4.3)where ξ ( s ) = (cid:18) − (cid:90) s R λ ( u ) du (cid:19) V + κφλ (cid:90) s R λ ( u ) du, (4.4)and R λ is the resolvent of λK such that λK ∗ R λ = R λ ∗ ( λK ) = λK − R λ . (4.5)If λ = 0, interpret R λ /λ = K and R λ = 0.Consider the stochastic process, M t = 2 exp (cid:104) (cid:90) Tt (cid:0) r s − θ ξ t ( s ) + (1 − ρ ) σ ψ ( T − s ) ξ t ( s ) (cid:1) ds (cid:105) , (4.6)where ψ ( t ) = (cid:90) t K ( t − s ) (cid:2) (1 − ρ ) σ ψ ( s ) − λψ ( s ) − θ (cid:3) ds. (4.7)The existence and uniqueness of the solution to (4.7) are established in Lemma A.4.The process M is the key to applying the completion of squares technique in Theorem 4.3,inspired by [27, 26, 31]. Heuristically speaking, the non-Markovian and non-semimartingalecharacteristics of the Volterra Heston model are overcome by considering M . The construction of M is based on the following observations. To make a completion of squares, we need an auxiliaryprocess M as an additional stochastic factor in a place consistent with previous studies of MVportfolios under semimartingales. The completion of squares procedure for proving Theorem4.3 indicates that M should satisfy (4.8). We then link M with the conditional expectation in(4.13) via a proper transformation. The exponential-affine transform formula in [3, Equation(4.7)] is applied to obtain (4.6). Theorem 4.1.
Assume Assumption 2.1 holds and (4.7) has a unique continuous solution on [0 , T ] , then M satisfies the following properties.(1). M t is essentially bounded and < M t < e (cid:82) Tt r s ds , P - a.s. , ∀ t ∈ [0 , T ) . M T = 2 . M on t , then dM t = (cid:2) − r t + θ V t (cid:3) M t dt + (cid:2) θ (cid:112) V t U t + U t M t (cid:3) dt + U t dW t + U t dW t , (4.8) where U t = ρσM t (cid:112) V t ψ ( T − t ) , (4.9) U t = (cid:112) − ρ σM t (cid:112) V t ψ ( T − t ) . (4.10) (3). M = 2 exp (cid:104) (cid:90) T r s ds + κφ (cid:90) T ψ ( s ) ds + V (cid:90) T (cid:2) (1 − ρ ) σ ψ ( s ) − λψ ( s ) − θ (cid:3) ds (cid:105) . (4.11) Furthermore, for fractional kernel K ( t ) = t α − Γ( α ) , denote the fractional integral as I α ψ ( t ) = K ∗ ψ ( t ) . Then M = 2 exp (cid:104) (cid:90) T r s ds + κφI ψ ( T ) + V I − α ψ ( T ) (cid:105) . (4.12) (4). E (cid:104)(cid:0) (cid:82) T U it dt (cid:1) p/ (cid:105) < ∞ for p ≥ , i = 1 , .Proof. Property (1) .It is straightforward to see that M t > − ρ = 0,note (cid:82) Tt ξ t ( s ) ds > P -a.s. by Lemma B.1, then M t < e (cid:82) Tt r s ds , P -a.s.. If 1 − ρ (cid:54) = 0, we claim M − ρ t = 2 − ρ exp (cid:2) − ρ ) (cid:90) Tt r s ds (cid:3) ˜ E (cid:104) exp (cid:0) − θ (1 − ρ ) (cid:90) Tt V s ds (cid:1)(cid:12)(cid:12)(cid:12) F t (cid:105) . (4.13)It is equivalent to show that˜ E (cid:104) exp (cid:0) − θ (1 − ρ ) (cid:90) Tt V s ds (cid:1)(cid:12)(cid:12)(cid:12) F t (cid:105) (4.14)= exp (cid:104) (cid:90) Tt (cid:0) − (1 − ρ ) θ ξ t ( s ) + (1 − ρ ) σ ψ ( T − s ) ξ t ( s ) (cid:1) ds (cid:105) . Denote ˜ ψ = (1 − ρ ) ψ . Then ˜ ψ satisfies˜ ψ = K ∗ (cid:0) σ ψ − λ ˜ ψ − (1 − ρ ) θ (cid:1) . (4.15)Therefore, (4.14) holds for all t ∈ [0 , T ] by [3, Theorem 4.3] applying to ˜ ψ . The martingaleassumption in [3, Theorem 4.3] is verified by [3, Lemma 7.3].If 1 − ρ >
0, then ˜ E (cid:104) exp (cid:0) − θ (1 − ρ ) (cid:82) Tt V s ds (cid:1)(cid:12)(cid:12)(cid:12) F t (cid:105) < P -a.s., which implies M t < e (cid:82) Tt r s ds , P -a.s.. 1 − ρ < Property (2).
Denote M t = 2 e Z t in (4.6) with proper Z t . We first derive the equation for dZ t . From (4.3),apply Itˆo’s lemma to ξ t ( s ) on time t and get dξ t ( s ) = 1 λ R λ ( s − t ) σ (cid:112) V t d ˜ B t . (4.16)8hen dZ t = (cid:2) − r t + θ V t − (1 − ρ ) σ ψ ( T − t ) V t (cid:3) dt − θ (cid:90) Tt λ R λ ( s − t ) σ (cid:112) V t d ˜ B t ds + (1 − ρ ) σ (cid:90) Tt ψ ( T − s ) 1 λ R λ ( s − t ) σ (cid:112) V t d ˜ B t ds = (cid:2) − r t + θ V t − (1 − ρ ) σ ψ ( T − t ) V t (cid:3) dt − θ (cid:90) Tt σ λ R λ ( s − t ) ds (cid:112) V t d ˜ B t + (1 − ρ ) σ (cid:90) Tt σψ ( T − s ) 1 λ R λ ( s − t ) ds (cid:112) V t d ˜ B t = (cid:2) − r t + θ V t − (1 − ρ ) σ ψ ( T − t ) V t (cid:3) dt + d ˜ B t · σ (cid:112) V t (cid:90) Tt (cid:104) (1 − ρ ) σ ψ ( T − s ) − θ (cid:105) λ R λ ( s − t ) ds. The second equality is guaranteed by the stochastic Fubini theorem [33].We claim the following representation for (4.9)-(4.10). U t = σρM t (cid:112) V t (cid:90) Tt (cid:104) (1 − ρ ) σ ψ ( T − s ) − θ (cid:105) λ R λ ( s − t ) ds, (4.17) U t = σ (cid:112) − ρ M t (cid:112) V t (cid:90) Tt (cid:104) (1 − ρ ) σ ψ ( T − s ) − θ (cid:105) λ R λ ( s − t ) ds. (4.18)Indeed, we only have to show (cid:90) Tt (cid:104) (1 − ρ ) σ ψ ( T − s ) − θ (cid:105) λ R λ ( s − t ) ds = ψ ( T − t ) . (4.19)Although one can verify (4.19) in the same fashion as [3, Lemma 4.4], we still detail the deriva-tion here for a self-contained paper. As (cid:90) Tt (cid:104) (1 − ρ ) σ ψ ( T − s ) − θ (cid:105) λ R λ ( s − t ) ds = (cid:90) T − t (cid:104) (1 − ρ ) σ ψ ( T − t − s ) − θ (cid:105) λ R λ ( s ) ds = (cid:2) (1 − ρ ) σ ψ − θ (cid:3) ∗ λ R λ ( T − t ) , we have (cid:90) Tt (cid:104) (1 − ρ ) σ ψ ( T − s ) − θ (cid:105) λ R λ ( s − t ) ds − ψ ( T − t )= (cid:2) (1 − ρ ) σ ψ − θ (cid:3) ∗ λ R λ ( T − t ) − K ∗ (cid:2) (1 − ρ ) σ ψ − λψ − θ (cid:3) ( T − t )= (cid:2) (1 − ρ ) σ ψ − θ (cid:3) ∗ (cid:2) λ R λ − K (cid:3) ( T − t ) + λK ∗ ψ ( T − t )= − R λ ∗ K ∗ (cid:2) (1 − ρ ) σ ψ − θ (cid:3) ( T − t ) + λK ∗ ψ ( T − t ) . The application of (4.7) leads to R λ ∗ ψ = R λ ∗ K ∗ (cid:2) (1 − ρ ) σ ψ − λψ − θ (cid:3) . (4.20)9onsequently, − R λ ∗ K ∗ (cid:2) (1 − ρ ) σ ψ − θ (cid:3) ( T − t ) + λK ∗ ψ ( T − t )= (cid:2) λK − R λ − λK ∗ R λ (cid:3) ∗ ψ ( T − t ) = 0 . This shows that dZ t = (cid:2) − r t + θ V t − (1 − ρ ) σ ψ ( T − t ) V t (cid:3) dt + U t M t d ˜ W t + U t M t dW t . (4.21)Applying Itˆo’s lemma to M t = 2 e Z t with function f ( z ) = 2 e z yields dM t = M t dZ t + 12 M t dZ t dZ t = M t (cid:2) − r t + θ V t − (1 − ρ ) σ ψ ( T − t ) V t (cid:3) dt + U t + U t M t dt + U t d ˜ W t + U t dW t = (cid:2) − r t + θ V t (cid:3) M t dt + (cid:2) θ (cid:112) V t U t + U t M t (cid:3) dt + U t dW t + U t dW t . Property (3) .The proof for the property of Y t in [3, Theorem 4.3] indicates (cid:90) T (cid:2) − θ ξ ( s ) + (1 − ρ ) σ ψ ( T − s ) ξ ( s ) (cid:3) ds = (cid:90) T (cid:2) − θ V + ( κφ − λV ) ψ ( s ) + (1 − ρ ) σ ψ ( s ) V (cid:3) ds. Under the fractional kernel, we show by integration by parts that (cid:90) T (cid:2) − θ − λψ ( s ) + (1 − ρ ) σ ψ ( s ) (cid:3) ds = I − α ψ ( T ) . (4.22)This gives the desired result. Property (4) .It is sufficient to consider the case with p >
2. As ψ ( t ) is continuous on [0 , T ] and M t isessentially bounded, E (cid:104)(cid:0) (cid:90) T U it dt (cid:1) p/ (cid:105) ≤ C E (cid:104)(cid:0) (cid:90) T V t dt (cid:1) p/ (cid:105) ≤ C (cid:90) T E (cid:2) V p/ t (cid:3) dt ≤ C sup t ∈ [0 ,T ] E (cid:2) V p/ t (cid:3) < ∞ . The last term is finite by [3, Lemma 3.1].We first propose a candidate optimal control u ∗ . In the following theorem, we prove theadmissibility of u ∗ and the integrability of the corresponding X ∗ . Theorem 4.2 is in the spiritof [27, 26, 31]. Finally, we prove the optimality of u ∗ in (4.24) by Theorem 4.3. Theorem 4.2.
Assume Assumption 2.1 holds and (4.7) has a unique continuous solution on [0 , T ] . Denote A t (cid:44) θ + ρσψ ( T − t ) . Suppose Assumption 2.5 holds with constant a given thefollowing: a = max (cid:110) p | θ | sup t ∈ [0 ,T ] | A t | , (8 p − p ) sup t ∈ [0 ,T ] A t (cid:111) , for certain p > . (4.23)10 onsider u ∗ ( t ) = ( θ + ρσψ ( T − t )) (cid:112) V t ( ζ ∗ e − (cid:82) Tt r s ds − X ∗ t ) , (4.24) where X ∗ t is the wealth process under u ∗ and ζ ∗ = c − η ∗ with η ∗ = e − (cid:82) T r s ds M x − e − (cid:82) T r s ds M c − e − (cid:82) T r s ds M . (4.25) u ∗ ( · ) in (4.24) is admissible and X ∗ under u ∗ ( · ) satisfies E (cid:104) sup t ∈ [0 ,T ] | X ∗ t | p (cid:105) < ∞ , (4.26) for p ≥ . Moreover, ζ ∗ e − (cid:82) Tt r s ds − X ∗ t ≥ , P - a.s. , ∀ t ∈ [0 , T ] . (4.27) Proof.
The wealth process under u ∗ is given by (cid:40) dX ∗ t = (cid:2) r t X ∗ t + θA t V t ( ζ ∗ e − (cid:82) Tt r s ds − X ∗ t ) (cid:3) dt + A t √ V t ( ζ ∗ e − (cid:82) Tt r s ds − X ∗ t ) dW t ,X ∗ = x . (4.28)To find a solution to X ∗ , define Y t satisfying (cid:40) dY t = − r t Y t dt − θ √ V t Y t dW t + Y t (cid:112) − ρ σψ ( T − t ) √ V t dW t ,Y = M ( ζ ∗ e − (cid:82) T r s ds − x ) . (4.29)The unique solution of Y t is given by Y t = Y exp (cid:104) − (cid:90) t (cid:0) r s + θ V s + (1 − ρ ) σ ψ ( T − s ) V s (cid:1) ds − (cid:90) t θ (cid:112) V s dW s + (cid:90) t (cid:112) − ρ σψ ( T − s ) (cid:112) V s dW s (cid:105) . Itˆo’s lemma yields X ∗ t = ζ ∗ e − (cid:82) Tt r s ds − Y t M t (4.30)as the unique solution of the wealth process. Indeed, d Y t M t = (cid:104) r t Y t M t − θA t V t Y t M t (cid:105) dt − A t (cid:112) V t Y t M t dW t . (4.31)The existence of u ∗ is also guaranteed by the existence of the solution X ∗ . Furthermore, Y t M t = Y M Φ( t ), whereΦ( t ) (cid:44) exp (cid:104) (cid:90) t (cid:2) r s − (cid:0) θA s + A s (cid:1) V s (cid:3) ds − (cid:90) t A s (cid:112) V s dW s (cid:105) . As Y t /M t ≥
0, (4.27) follows from (4.30).For (4.26), note that by Doob’s maximal inequality and [3, Lemma 7.3], E (cid:104) sup t ∈ [0 ,T ] | Φ( t ) | p (cid:105) ≤ C E (cid:104) sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12) e − (cid:82) t θA s V s ds (cid:12)(cid:12)(cid:12) p (cid:105) + C E (cid:104) sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12) exp (cid:16) − (cid:90) t A s V s ds − (cid:90) t A s (cid:112) V s dW s (cid:17)(cid:12)(cid:12)(cid:12) p (cid:105) ≤ C E (cid:104) e p (cid:82) T | θA s | V s ds (cid:105) + C E (cid:104) exp (cid:16) − (cid:90) T pA s V s ds − (cid:90) T pA s (cid:112) V s dW s (cid:17)(cid:105) . a = 2 p | θ | sup t ∈ [0 ,T ] | A t | . The secondterm is also finite. In fact, by H¨older’s inequality and Assumption 2.5 with a constant a =(8 p − p ) sup t ∈ [0 ,T ] A t , E (cid:104) exp (cid:16) − (cid:90) T pA s V s ds − (cid:90) T pA s (cid:112) V s dW s (cid:17)(cid:105) ≤ (cid:110) E (cid:104) e (8 p − p ) (cid:82) T A s V s ds (cid:105)(cid:111) / (cid:110) E (cid:104) exp (cid:16) − p (cid:90) T A s V s ds − p (cid:90) T A s (cid:112) V s dW s (cid:17)(cid:105)(cid:111) / < ∞ . E (cid:104) sup t ∈ [0 ,T ] | X ∗ t | p (cid:105) < ∞ is proved. As for admissibility of u ∗ , u ∗ is F -adapted at first. Forintegrability, let 1 / ˆ p + 1 / ˆ q = 1, ˆ p, ˆ q >
1, we have E (cid:104)(cid:16) (cid:90) T | (cid:112) V t u ∗ t | dt (cid:17) (cid:105) ≤ C E (cid:104)(cid:16) (cid:90) T | A t V t Φ( t ) | dt (cid:17) (cid:105) ≤ C E (cid:104) sup t ∈ [0 ,T ] Φ ( t ) (cid:16) (cid:90) T V t dt (cid:17) (cid:105) ≤ C (cid:110) E (cid:104) sup t ∈ [0 ,T ] Φ p ( t ) (cid:105)(cid:111) / ˆ p (cid:110) E (cid:104)(cid:16) (cid:90) T V t dt (cid:17) q (cid:105)(cid:111) / ˆ q ≤ C (cid:110) E (cid:104) sup t ∈ [0 ,T ] Φ p ( t ) (cid:105)(cid:111) / ˆ p (cid:16) sup t ∈ [0 ,T ] E (cid:2) V qt (cid:3)(cid:17) / ˆ q < ∞ and E (cid:104) (cid:90) T | u ∗ t | dt (cid:105) ≤ C E (cid:104) (cid:90) T A t V t Φ ( t ) dt (cid:105) ≤ C E (cid:104) sup t ∈ [0 ,T ] Φ ( t ) (cid:90) T V t dt (cid:105) ≤ C (cid:110) E (cid:104) sup t ∈ [0 ,T ] Φ p ( t ) (cid:105)(cid:111) / ˆ p (cid:110) E (cid:104)(cid:16) (cid:90) T V t dt (cid:17) ˆ q (cid:105)(cid:111) / ˆ q ≤ C (cid:110) E (cid:104) sup t ∈ [0 ,T ] Φ p ( t ) (cid:105)(cid:111) / ˆ p (cid:16) sup t ∈ [0 ,T ] E (cid:2) V ˆ qt (cid:3)(cid:17) / ˆ q < ∞ . The last terms in the two inequalities above are finite by [3, Lemma 3.1] and take p = 2ˆ p .We are now ready to prove u ∗ in (4.24) is optimal and to derive the efficient frontier. Theorem 4.3.
Suppose the assumptions in Theorem 4.2 hold, then the optimal investmentstrategy for Problem (3.2) is given by (4.24) . Moreover, (4.24) is unique under a given solution ( S, V, W , W ) to (2.6)-(2.7). The variance of X ∗ T isVar [ X ∗ T ] = M − e − (cid:82) T r s ds M (cid:0) ce − (cid:82) T r s ds − x (cid:1) . (4.32) Proof.
First, we consider the inner Problem (3.4) with an arbitrary ζ ∈ R . Denote h t = ζe − (cid:82) Tt r s ds . By Itˆo’s lemma with the property of M and completing the square, for any admis-sible strategy u , d M t ( X t − h t ) = 12 (cid:2) ( X t − h t ) M t θ V t + 2( X t − h t ) θ (cid:112) V t U t + ( X t − h t ) U t M t + 2 M t ( X t − h t ) θ (cid:112) V t u t + 2( X t − h t ) u t U t + M t u t (cid:3) dt + 12 (cid:2) ( X t − h t ) U t + 2 M t ( X t − h t ) u t (cid:3) dW t + 12 ( X t − h t ) U t dW t = 12 M t (cid:104) u t + (cid:0) θ (cid:112) V t + U t M t (cid:1) ( X t − h t ) (cid:105) dt + 12 (cid:2) ( X t − h t ) U t + 2 M t ( X t − h t ) u t (cid:3) dW t + 12 ( X t − h t ) U t dW t . M t and h t are bounded, E (cid:104) (cid:82) T U it dt (cid:105) < ∞ for i = 1 , u t is admissible, and X t has P -a.s.continuous paths, then stochastic integrals (cid:90) t (cid:2) ( X s − h s ) U s + 2 M s ( X s − h s ) u s (cid:3) dW s and (cid:90) t ( X s − h s ) U s dW s are ( F , P )-local martingales. There is an increasing localizing sequence of stopping times { τ k } k =1 , ,... such that τ k ↑ T when k → ∞ . The local martingales stopped by { τ k } k =1 , ,... are true martingales. Consequently,12 E [ M τ k ( X τ k − h τ k ) ] = 12 M ( x − h ) + 12 E (cid:104) (cid:90) τ k M t (cid:16) u t + (cid:0) θ (cid:112) V t + U t M t (cid:1) ( X t − h t ) (cid:17) dt (cid:105) . (4.33)From (3.1), by Doob’s maximal inequality and the admissibility of u ( · ), E [ X τ k ] ≤ C (cid:104) x + E (cid:104)(cid:0) (cid:90) T | u t (cid:112) V t | dt (cid:1) (cid:105) + E (cid:104) (cid:90) T u t dt (cid:105)(cid:105) < ∞ . (4.34)Then M τ k ( X τ k − h τ k ) is dominated by a non-negative integrable random variable for all k .Sending k to infinity, by the dominated convergence theorem and the monotone convergencetheorem, we derive E [( X T − ζ ) ] = 12 M ( x − h ) + 12 E (cid:104) (cid:90) T M t (cid:16) u t + (cid:0) θ (cid:112) V t + U t M t (cid:1) ( X t − h t ) (cid:17) dt (cid:105) . (4.35)Therefore, the cost functional E [( X T − ζ ) ] is minimized when u t = − (cid:0) θ (cid:112) V t + U t M t (cid:1) ( X t − h t ) . (4.36)Then E [( X T − ζ ) ] = M ( x − h ) . The uniqueness of u ∗ follows directly from (4.35) and M t > P -a.s., ∀ t ∈ [0 , T ]. To solve the outer maximization problem in (3.3), consider J ( x ; u ( · )) = 12 M (cid:2) x − ( c − η ) e − (cid:82) T r s ds (cid:3) − η . (4.37)The first and second order derivatives are ∂J∂η = M (cid:2) x − ( c − η ) e − (cid:82) T r s ds (cid:3) e − (cid:82) T r s ds − η,∂ J∂η = M e − (cid:82) T r s ds − < , where we have used the strict inequality M < e (cid:82) T r s ds , by Theorem 4.1.Then the optimal value for η is given by (4.25), solved from ∂J∂η = 0. Var[ X ∗ T ] is obtained bydirect simplification of J ( x ; u ( · )) with η ∗ .Although the Volterra Heston model is non-Markovian and non-semimartingale in nature,the optimal control u ∗ in (4.24) does not rely on the whole volatility path. Moreover, theoptimal amount of wealth in the stock, π ∗ t , does not depend on the volatility value directly, butrather on the roughness and dynamics of volatility through parameters and the Riccati-Volterraequation (4.7). If we let kernel K = id, it is then clear that the Volterra Heston model (2.6)reduces to the classic Heston model [21]. Our results in Theorem 4.2 and Theorem 4.3 indicatethat the u ∗ in (4.24) is optimal even under a general filtration F . It extends the correspondingresult in [6, 32] where the filtration is chosen as the Brownian filtration. As a sanity check, thefollowing corollary verifies that our solution reduces to the one under the Heston model.13 orollary 4.4. Consider the Heston model, that is, the kernel K = id . Suppose other as-sumptions in Theorem 4.2 hold, then the optimal strategy (4.24) is the same as the one in[6].Proof. Without loss of generality, suppose r t = 0, as in [6]. We first match M t / L t in [6, Equation (3.2)].Note the resolvent in (4.5) reduces to R λ ( t ) = λe − λt and the forward variance in (4.3) is ξ t ( s ) = e − λ ( s − t ) V t + κφλ (cid:16) − e − λ ( s − t ) (cid:17) . (4.38)Therefore, (cid:90) Tt ξ t ( s ) ds = 1 − e − λ ( T − t ) λ V t + κφλ (cid:32) T − t − − e − λ ( T − t ) λ (cid:33) and (cid:90) Tt ψ ( T − s ) ξ t ( s ) ds = V t (cid:90) Tt ψ ( T − s ) e − λ ( s − t ) ds + κφλ (cid:90) Tt (cid:2) − e − λ ( s − t ) (cid:3) ψ ( T − s ) ds. Then (cid:90) Tt (cid:2) − θ ξ t ( s ) + (1 − ρ ) σ ψ ( T − s ) ξ t ( s ) (cid:3) ds = w ( T − t ) V t + y ( T − t ) , (4.39)where w ( T − t ) (cid:44) (1 − ρ ) σ (cid:90) Tt ψ ( T − s ) e − λ ( s − t ) ds − θ − e − λ ( T − t ) λ ,y ( T − t ) (cid:44) (1 − ρ ) σ κφλ (cid:90) Tt (cid:2) − e − λ ( s − t ) (cid:3) ψ ( T − s ) ds − θ κφλ (cid:32) T − t − − e − λ ( T − t ) λ (cid:33) . Replacing t with T − t and taking derivative on t give˙ w ( t ) = (1 − ρ ) σ ψ ( t ) − λ (1 − ρ ) σ (cid:90) TT − t ψ ( T − s ) e − λ ( s − T + t ) ds − θ e − λt = (1 − ρ ) σ ψ ( t ) − λw ( t ) − θ . Comparing with (4.7), we find w ( t ) = ψ ( t ). Moreover,˙ y ( t ) = (1 − ρ ) σ κφλ (cid:90) TT − t λe − λ ( s − T + t ) ψ ( T − s ) ds − θ κφλ (cid:16) − e − λt (cid:17) = κφw ( t ) .y ( t ) and w ( t ) satisfy the same ODEs as in [6, Equations (A.1)-(A.4)], with our notations.Therefore, M t / L t in [6, Equation (3.2)].Consider the inner Problem (3.4). With a constant H = ζ , terms in the optimal hedge ϕ ( x, H ) [6, p.476] are reduced to ξ = 0 , a = ( θ + ψ ( T − t ) ρσ ) /S t , V = ζ, and x + ϕ ( x, H ) · S = X ∗ t . (4.40)Then it is clear that the optimal strategies are the same.14 Numerical studies
In this section, we restrict ourself to the case with K ( t ) = t α − Γ( α ) , α ∈ (1 / , α = 1 recovers the classic Heston model. We examine the effect of α onthe optimal investment strategy and efficient frontier.The first step is to solve the Riccati-Volterra equation (4.7) numerically. Following [11], weuse the fractional Adams method in [7, 8]. The convergence of this numerical method is givenin [25]. Readers may refer to [11, Section 5.1] for more details about the procedure.In Figure (1a), ψ decreases when α becomes smaller under certain specific parameters, closeto the calibration result in [11] with one extra risk premium parameter θ . However, one cannotexpect ψ to be monotone in α in general (see Figure (1b)). Figures (1a)-(1b) also confirm theclaim that ψ ≤ − ρ > ψ α = 0.6α = 0.7α = 0.8α = 0.9α = 1.0 (a) ψ under parameters in [11] ψ α=0.6α=0.7α=0.8α=0.9α=1.0 (b) ψ under another setting Figure 1: Plot of ψ under different α . Other parameters are as follows. In Figure (1a), vol-of-vol σ = 0 .
03, mean-reversion speed κ = 0 .
1, risk premium parameter θ = 5, correlation ρ = − . T = 1. In Figure (1b), σ = 0 . κ = 2 . θ = 0 . ρ = − .
56, and T = 1 . u ∗ and α is not straightforward and may change with differentcombinations of parameters. We emphasize that the following analysis is based on the parametersetting detailed in the descriptions of the figures. Consider the setting in Figure (1a) first.Interestingly, the effect of α on u ∗ is significantly influenced by σ . This can be explained using(4.24). If the correlation ρ between stock and volatility is negative due to the leverage effectin the equity market, θ + ρσψ ( T − t ) will increase as α decreases, as shown in Figure (1a). Incontrast, ζ ∗ e − (cid:82) Tt r s ds − X ∗ t ≥ ζ ∗ = c − η ∗ = 2 c − e − (cid:82) T r s ds M x − e − (cid:82) T r s ds M . (5.1)The M in (4.12) is an increasing function on α because ψ is negative. Then ζ ∗ will be smallerif α is smaller, under certain parameters. Therefore, ζ ∗ e − (cid:82) Tt r s ds − X ∗ t and θ + ρσψ ( T − t )move in different directions when α is decreasing. If σ is small, ζ ∗ e − (cid:82) Tt r s ds − X ∗ t will dominate θ + ρσψ ( T − t ). Then u ∗ will decrease as α becomes smaller. If σ is relatively large, θ + ρσψ ( T − t )will dominate ζ ∗ e − (cid:82) Tt r s ds − X ∗ t . Then u ∗ increases when α becomes smaller. The above effect ofvol-of-vol σ also appears under the parameters setting in Figure (1b), where ψ is not monotonein α . Figures (2a)-(2b) display the optimal investment strategy u ∗ . We make use of the open-source Python package differint to calculate the fractional integrals I − α and I in (4.12).Assumption 2.5 is validated under the setting in Figures (2a)-(2b). Available at https://github.com/differint/differint .0 0.2 0.4 0.6 0.8 1.0 1.2 1.4time (in years)2.742.752.762.772.78 u * α=0.6α=0.7α=0.8α=0.9α=1.0 (a) u ∗ under σ = 0 . u * α=0.6α=0.7α=0.8α=0.9α=1.0 (b) u ∗ under σ = 3 Figure 2: Optimal strategy u ∗ with α = 0 . , . , . , .
9, and 1 .
0. In both subplots, we set initialwealth x = 1, risk-free rate r = 0 .
01, initial variance V = 0 .
5, long-term mean level φ = 0 . c = x e ( r +0 . T . For simplicity, we set V t = 0 . X ∗ t = 1 forall time t ∈ [0 , T ]. The other parameters are the same as in Figure (1b), namely, κ = 2 . θ = 0 . ρ = − .
56, and T = 1 .
35. Figures (2a)-(2b) only differ in the vol-of-vol σ .Figures (2a)-(2b) are a sensitivity analysis as we keep most of the parameters unchanged,and vary a few of them. Specifically, the use of constant V t and X ∗ t in Figures (2a)-(2b) hasthe following interpretation. We are interested in the sensitivity of the optimal control on theHurst parameter through α . As the other parameters being fixed, if we observe V t = 0 . X ∗ t = 1 at t ∈ [0 , T ], Figures (2a)-(2b) illustrate the marginal effect of the Hurst parameter onthe investment strategy. The constant values of V t and X ∗ t are not from a realized path.Figures (2a)-(2b) only provide a marginal effect of α ; thus, we conduct a further numericalanalysis under the settings in [1]. Consider a realistic situation in which the investor calibratestwo sets of parameters for the Heston model and rough Heston model for a given impliedvolatility surface. We contrast the two strategies induced from the calibrated parameters.Figure (3c) exhibits the optimal amount of wealth π ∗ with one simulation path of V t in Figure(3b) by the lifted Heston approach [1]. Assumption 2.5 holds true under the setting in Figure3. Figure (3a) plots the A t = θ + ρσψ ( T − t ). Furthermore, ζ ∗ = 30 . ζ ∗ = 21 . ζ ∗ reported. A rough Heston investor has a larger A t ζ ∗ but a smaller A t .Moreover, Figure (4c) illustrates that the rough Heston strategy has an average terminal wealthcloser to the target c = 1 . α . Figures (2a)-(2b) indicate that α is not the only factor determining the investment in a stock. The trading idea in [18] agreeswith Figure (2b), because the optimal investment position u ∗ is larger for a smaller α . However,an inconsistency occurs in Figure (2a). Indeed, if we use the VVIX index as a proxy for thevol-of-vol, then the vol-of-vol seems larger in 2007, 2008, 2010, and 2015. The buy-rough-sell-smooth strategy [18] performs better in 2005, 2007, 2008, 2010, and 2014 than in other years, asshown in [18, Figure 3]. This consistency suggests that vol-of-vol may also be important when16
50 100 150 200 250time (in days)0.4000.4050.4100.4150.4200.4250.4300.435 A t RoughHeston (a) A t V t (b) volatility π * RoughHeston (c) π ∗ Figure 3: Investment strategies under the Heston and rough Heston models. The varianceprocess is simulated with the lifted Heston model in [1]. The parameters for simulation arespecified in [1, Equations (23) and (26)] with α = 0 .
6. The path is rougher than that of theclassic Heston model. Moreover, we implement the Euler scheme for the stock process. Thesimulation is run with 250 time steps for one year, corresponding to the 250 trading days in ayear. The investor under the Heston model uses the calibrated parameters in [1, Table 6] toimplement the optimal strategy with α = 0 . x = 1, r = 0 . θ = 0 . T = 1,and c = x e ( r +0 . T = 1 . (a) u ∗ under the rough Hestonmodel (b) u ∗ under the Heston model (c) Wealth Figure 4: Statistics for strategies and wealth. Based on 3000 simulated paths, the solid line plotsthe mean and the shadow area is the 95% confidence interval estimated by bootstrapping. Therough Heston model suggests investing more and the terminal wealth is closer to the expectedvalue c = 1 . α and expectedwealth level c . Their relationship is clear, and the variance of the optimal wealth is reduced if α decreases, as M decreases when α decreases and Var[ X ∗ T ] in (4.32) is an increasing functionon M . We have also verified Assumption 2.5 under the setting in Figures (5a)-(5b). To the best of our knowledge, this is the first study of the continuous-time Markowitz’s mean-variance portfolio selection problem under a rough stochastic environment. We specifically focuson the Volterra Heston model. By deriving the optimal strategy and efficient frontier, we obtainfurther insights into the effect of roughness on them.There are many possible future research directions. Natural considerations are the utility17 E [ X * T ] V a r [ X * T ] (a) Efficient frontier V a r [ X * T ] c=1.2c=1.3c=1.4c=1.5 (b) Var[ X ∗ T ] under different expected value c Figure 5: Plots of the efficient frontier and variance. Roughness parameter α ∈ [0 . , r = 0 . V = 0 . x = 1, φ = 0 . σ = 0 . κ = 0 . θ = 0 . ρ = − . T = 1, and c ∈ [ x e ( r +0 . T , x e ( r +0 . T ].maximization and time-inconsistency of the MV criterion. In addition, we have already includedmodel ambiguity with rough volatility in our research agenda. Acknowledgements
The authors would like to thank two anonymous referees and the Editor for their careful readingand valuable comments, which have greatly improved the manuscript.
A Solutions of Riccati-Volterra equations
To demonstrate the existence and uniqueness of the solution to a Riccati-Volterra equation, wefirst rephrase the following result from a recent monograph [5] with more general assumptions.The underlying idea of the proof is the Picard iteration.
Theorem A.1.
Suppose kernel K ( · ) is bounded or is the fractional kernel with α ∈ (0 , . Let c , c , c be constant. Then there exsits δ > such that f ( t ) = (cid:90) t K ( t − s ) (cid:2) c + c f ( s ) + c f ( s ) (cid:3) ds (A.1) has a unique continuous solution f on [0 , δ ] .Proof. Note that quadratic function is locally Lipschitz; then according to Theorem 3.1.2 andTheorem 3.1.4 in [5], the claim holds.However, δ in Theorem A.1 is not explicit. Tighter results exist if more assumptions areimposed.We investigate g ( a, t ) in (2.8) first. Based on [16, Theorem A.5], we have Lemma A.2.
Suppose Assumption 2.1 holds and κ − aσ > . Then (2.8) has a uniqueglobal continuous solution. Moreover, < g ( a, t ) ≤ r ( t ) < w ∗ , ∀ t > , (A.2) where w ∗ (cid:44) κ −√ κ − aσ σ and r ( t ) (cid:44) Q − (cid:0) (cid:82) t K ( s ) ds (cid:1) ; that is, the inverse function of Q , givenby Q ( w ) = (cid:90) w dua − κu + σ u . (A.3)18 roof. To apply the result in [16, Theorem A.5], we define H ( w ) = a − κw + σ w . Then H ( w ) satisfies Assumption A.1 in [16] with w max (cid:44) κσ and w ∗ defined above. The claimfollows from [16, Theorem A.5 (c)] with a ( t ) ≡ K ( t ) = t α − Γ( α ) , [10, Theorem 3.2] obtains the following tighterresults and the proof is based on the scaling limits of the Hawkes processes. Lemma A.3. If K ( t ) = t α − Γ( α ) , α ∈ (1 / , , then g ( a, t ) in (2.8) satisfies g ( a, t ) ≤ cσ (cid:104) κ + t − α Γ(1 − α ) + σ (cid:112) a ( t ) − a (cid:105) , (A.4) with a ( t ) = σ (cid:2) κ + t − α Γ(1 − α ) (cid:3) and a constant c > . In other words, if a < a ( T ) , thenAssumption 2.5 is satisfied. Next, we study ψ ( · ) in (4.7). (4.7) has a unique continuous solution on some interval [0 , δ ] ifthe conditions in Theorem A.1 are satisfied. Without Theorem A.1, we also have the followingresult. Lemma A.4.
Suppose Assumption 2.1 holds.(1). If − ρ > , then (4.7) has a unique global continuous solution ψ ∈ L loc ( R + , R ) and ψ < for t > .(2). If − ρ = 0 , then (4.7) is linear and has a unique continuous solution on [0 , T ] .(3). If − ρ < , further assume λ > and λ + 2(1 − ρ ) θ σ > . Then (4.7) has aunique global continuous solution. Moreover, ¯ w ∗ − ρ < ¯ r ( t )1 − ρ ≤ ψ ( t ) < , ∀ t > , (A.5) with ¯ w ∗ = λ − √ λ +2(1 − ρ ) θ σ σ and ¯ r ( t ) (cid:44) ¯ Q − (cid:0) (cid:82) t K ( s ) ds (cid:1) , where ¯ Q ( w ) = (cid:90) w du σ u − λu − (1 − ρ ) θ . (A.6) Proof.
The claim in (1) follows from [3, Theorem 7.1]. The continuity follows from the unique-ness of the global solution and [19, Theorem 12.1.1]. The claim in (2) is classic and can befound in [5, Theorem 1.2.3]. For (3), we consider ˜ ψ = (1 − ρ ) ψ . Then ˜ ψ satisfies˜ ψ = K ∗ (cid:0) σ ψ − λ ˜ ψ − (1 − ρ ) θ (cid:1) . (A.7)Define H ( w ) = σ w − λw − (1 − ρ ) θ . (A.8)Then ¯ w ∗ is the unique root of H ( w ) = 0 on ( −∞ , ¯ w max ] with ¯ w max = λσ . H ( w ) satisfiesAssumption A.1 in [16]. Therefore, [16, Theorem A.5 (c)] with a ( t ) ≡ < ˜ ψ ( t ) ≤ ¯ r ( t ) < ¯ w ∗ , ∀ t > . (A.9)Note ˜ ψ = (1 − ρ ) ψ . This gives the result desired.19 Positivity of integrals with forward variance
Lemma B.1.
Suppose Assumption 2.1 holds. The forward variance ξ t ( s ) in (4.3) satisfies (cid:82) Tt ξ t ( s ) ds > , P - a.s. , for every t ∈ [0 , T ) .Proof. As (cid:82) Tt ξ t ( s ) ds = ˜ E [ (cid:82) Tt V s ds |F t ] and V s is non-negative by Theorem 2.2, it is sufficient toshow that (cid:82) Tt V s ds > P -a.s..Given t ∈ [0 , T ), for ω ∈ Ω such that V s ( ω ) is continuous in s , we suppose (cid:82) Tt V s ( ω ) ds = 0.By the continuity of V s ( ω ), V s ( ω ) = 0 for s ∈ [ t, T ]. Using the argument given in [3, Theorem3.5, Equation (3.8)], for 0 < h < T − t , we have V t + h ( ω ) = V + (cid:90) t K ( t + h − s ) ( κφ − λV s ( ω )) ds + (cid:90) t K ( t + h − s ) σ (cid:112) V s ( ω ) d ˜ B s ( ω )+ (cid:90) t + ht K ( t + h − s ) ( κφ − λV s ( ω )) ds + (cid:90) t + ht K ( t + h − s ) σ (cid:112) V s ( ω ) d ˜ B s ( ω ) ≥ (cid:90) t + ht K ( t + h − s ) ( κφ − λV s ( ω )) ds + (cid:90) t + ht K ( t + h − s ) σ (cid:112) V s ( ω ) d ˜ B s ( ω ) . 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