Effect of variance swap in hedging volatility risk

Shen, Yang Article Effect of variance swap in hedging volatility risk Risks Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Shen, Yang (2020) : Effect of variance swap in hedging volatility risk, Risks, ISSN 2227-9091, MDPI, Basel, Vol. 8, Iss. 3, pp. 1-34, https://doi.org/10.3390/risks8030070 This Version is available at: https://hdl.handle.net/10419/258023 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ risks Article Effect of Variance Swap in Hedging Volatility Risk Yang Shen School of Risk and Actuarial Studies, University of New South Wales, Sydney, NSW 2052, Australia; [email protected] Received: 20 May 2020; Accepted: 23 June 2020; Published: 1 July 2020   Abstract: This paper studies the effect of variance swap in hedging volatility risk under the mean-variance criterion. We consider two mean-variance portfolio selection problems under Heston’s stochastic volatility model. In the first problem, the financial market is complete and contains three primitive assets: A bank account, a stock and a variance swap, where the variance swap can be used to hedge against the volatility risk. In the second problem, only the bank account and the stock can be traded in the market, which is incomplete since the idiosyncratic volatility risk is unhedgeable. Under an exponential integrability assumption, we use a linear-quadratic control approach in conjunction with backward stochastic differential equations to solve the two problems. Efficient portfolio strategies and efficient frontiers are derived in closed-form and represented in terms of the unique solutions to backward stochastic differential equations. Numerical examples are provided to compare the solutions to the two problems. It is found that adding the variance swap in the portfolio can remarkably reduce the portfolio risk. Keywords: backward stochastic differential equation; efficient frontier; heston’s model; mean-variance portfolio selection; variance swap 1. Introduction It is widely recognized that the volatility of stocks evolves in a stochastic fashion rather than being constant or deterministic over time. Many empirical studies on stock options reveal that the implied volatility in option price data exhibit the so-called volatility smile/skew, which cannot be explained by the constant-volatility stock price models, say, the geometric Brownian motion model adopted in the Black and Scholes (1973) formula. Tremendous effort has been made to articulate this issue, and various stochastic volatility models have been developed to capture the volatility smile/skew as observed in the market. See, for example, French et al. (1987), Hull and White (1987), Wiggins (1987), Stein and Stein (1991), and Heston (1993). Among these stochastic volatility models, the most commonly used is certainly Heston’s model (Heston 1993). Indeed, the variance process described by Heston’s model follows a single-factor square-root process with mean-reversion, i.e., a Cox-Ingersoll-Ross (CIR) process, which is mathematically tractable even when the instantaneous volatility/variance is assumed to be correlated with the stock price. The early research of Heston’s model focuses on option pricing under this model. Recently, there has been emerging interest in portfolio selection problems under Heston’s model as well as other stochastic volatility models. One direction of research is on maximizing the expected utility from investment and consumption. Kraft (2005) investigates a utility maximization problem and provides an explicit solution of the problem under specific conditions on model parameters. With the help of a martingale criterion, Kallsen and Muhle-Karbe (2010) derive explicit solutions for a power utility maximization problem in a number of affine-form stochastic volatility models. Zeng and Taksar (2013) study an optimal portfolio selection problem under a general stochastic volatility model and obtain closed-form solutions for Heston’s model under more relaxed assumptions. Other relevant works Risks 2020,8, 70; doi:10.3390/risks8030070 www.mdpi.com/journal/risks Risks 2020,8, 70 2 of 34 along this direction include, for example, Zariphopoulou (2001), Fleming and Hernández-Hernández (2003), Liu et al. (2003), and Chacko and Viceira (2005). Another direction of research explores the mean-variance hedging, that is, the problem of approximating a given final payoff by a self-financing trading strategy so as to minimize the mean-squared error. Preceding works include Laurent and Pham (1999), Biagini et al. (2000), Hobson (2004), ˇ Cerný and Kallsen (2008), to name but only a few. Although portfolio selection problems under Heston’s model have been extensively studied, most existing works concentrate on portfolio selection under the utility-maximization or the mean-variance hedging criteria and few attention has been paid to that under Markowitz (1952)’s mean-variance paradigm. In fact, Markowitz (1952)’s mean-variance criterion and Merton (1969,1971 )’s utility-maximization criterion are both considered precursors of modern portfolio optimization theory, and, to some degree, they are of equal importance in the field. On the other hand, the history of the mean-variance portfolio selection is much longer than that of the mean-variance hedging. In Markowitz (1952) , the mean-variance portfolio selection problem is proposed in a single-period discrete-time setting. Using some delicate embedding techniques, Zhou and Li (2000) apply the linear-quadratic (LQ) control theory to solve a continuous-time mean-variance portfolio selection problem analytically. Subsequently, Lim and Zhou (2002), Lim (2004) and Lim (2005) study mean-variance portfolio selection problems with random parameters by using the LQ control and backward stochastic differential equations (BSDEs). At first glance, one may consider nesting the mean-variance portfolio selection problem under Heston’s model in Lim and Zhou’s framework. However, this is impossible since the variance process in Heston’s model is unbounded, which violates the boundedness assumption of model parameters in Lim and Zhou’s framework. In fact, a class of nonlinear BSDEs is used to solve mean-variance problems, and this class is termed backward stochastic differential Riccati equations (BSREs). Because there are no general results for BSREs with unbounded coefficients, it is challenging to use BSREs to solve mean-variance problems under Heston’s model. This attracts recent attention to studying mean-variance portfolio selection problems with unbounded random coefficients. See Chiu and Wong (2011,2014), Shen et al. (2014), Shen and Zeng (2015) , Shen (2015), Li et al. (2018), etc. The focus of the the current paper is on demonstrating that volatility derivatives (e.g., variance swap) is effective tool to manage volatility risk when the stock price dynamics is only partially correlated with the stochastic volatility dynamics. Particularly, the advantage of using variance swap contracts is that there is no cost of entering into these contracts since swaps are worth zero at issuance. Compared with other volatility derivatives, e.g., variance and volatility options, variance swaps provide non-directional exposure to volatility risk, which then reduces the need for delta-hedging residual volatility risk. To investigate the effect of variance swaps in hedging volatility risk, in this paper we consider two dynamic mean-variance portfolio selection problems under Heston’s stochastic volatility model in a complete market and an incomplete market, respectively. More specifically, we assume that a bank account, a stock and a variance swap are traded in the complete market and that only the bank account and the stock are traded in the incomplete market. The variance process of the stock is described by Heston’s model and is correlated with the stock price process. Throughout this paper, we make a standing assumption that the variance process/the market price of risk process is exponentially integrable. We employ a combined LQ control and BSDE approach to solve the two problems. To address the issue in the solvability of BSREs with unbounded coefficients, we use measure change techniques and study several transformed BSDEs under equivalent probability measures first. Based on the exponential-integrability assumption, Girsanov’s theorem, Hölder’s inequality and the one-to-one correspondence relationships between the solutions to the original BSREs and the transformed BSDEs, the uniqueness and existence of solutions to the original BSREs are proved under the real-world probability measure. Due to the Markovian structure, the unique solutions to these BSREs can be represented by the solutions to some Riccati-type ordinary differential equations (ODEs). With the unique solutions to related BSREs, a straightforward application of the Risks 2020,8, 70 3 of 34 LQ control theory leads to the explicit expressions of the efficient portfolio strategies and the efficient frontiers immediately. To examine the differences of the two problems and the effect of adding the variance swap in the portfolio, we provide numerical examples of the efficient frontiers with different parameter values in the complete and the incomplete markets. It is shown that the variance swap can reduce the overall risk of the terminal wealth through hedging against the volatility risk. In addition, we verify that if the stock price and variance processes are perfectly correlated, the complete market and the incomplete one are indistinguishable. Therefore, the variance swap is an effective tool to hedge idiosyncratic volatility risk. On technical side, this paper somehow extends Shen (2015) and Shen and Zeng (2015) to cater for the current setting. Shen et al. (2014)) consider a mean-variance problem under a constant elasticity of variance (CEV) model. The efficient strategy found in Shen et al. (2014) indeed is in a space smaller than the square-integrable space since the BSDE therein is proved to admit a unique solution in a space accommodating stochastic Lipschitz coefficients, which is smaller than the usually used square-integrable solution space for BSDEs. Shen and Zeng (2015) study an optimal investment-reinsurance problem for insurers under the mean-variance optimization criterion. They impose an exponential integrability condition of order 2 and solve the problem for a modified admissible control set. A similarly modified definition of the admissible set is adopted by Li et al. (2018) to investigate a mean-variance asset-liability management under with stochastic volatility. Though Shen and Zeng (2015) have considered Heston’s model in their framework, they only find the almost surely square-integrable efficient strategy. We should note that in most preceding works as well as the current one, admissible strategies are required to be square-integrable (in an expected sense). By extending some techniques in Shen (2015) developed exclusively for a complete market environment and increasing the order of exponential integrability of the market price of risk, in this paper we manage to find square-integrable efficient portfolio strategies under Heston’s model, where the market may be incomplete. The rest of this paper is structured as follows. Section 2introduces the basic notation, model dynamics and standing assumption. In Section 3, we formulate two mean-variance portfolio selection problems, one in the complete market with the variance swap and the other in the incomplete market without the variance swap. Using the combined LQ control and BSDE approach, we derive the explicit expressions of the efficient portfolio strategies and the efficient frontiers of the two problems in Sections 4and 5, respectively. Section 6provides numerical examples to illustrate the differences of the two problems and the effectiveness of the variance swap in hedging volatility risk. Finally, Section 7 concludes the paper. The Appendix contains proofs that can be adapted from the literature. 2. Model Dynamics This section introduces the complete market and the incomplete market, and sets up the model dynamics of primitive assets, including a bank account, a stock and a variance swap. To begin with, we fix a complete probability space (Ω , F , P) , carrying two one-dimensional, independent standard Brownian motions {W1(t)}t≥0 and {W2(t)}t≥0 . We further equip (Ω , F , P) with a right-continuous, P -complete filtration F:={F(t)}t≥0 generated by W1(·) and W2(·) . Here P is a real-world probability measure. We denote by E[·] the expectation taken under P , Et[·] the conditional expectation under P given F(t) , |·| the Euclidean norm of Rn , and A> the transpose of any vector or matrix A . Let [ 0, T] be a finite horizon, where T<∞. For later use, we introduce several spaces of random variables and stochastic processes on (Ω,F,P). For any p∈[1, ∞), we define • Lp F(T),P(Ω ; Rn) : the space of Rn -valued, F(T) -measurable random variables ξ such that kξkLp F(T),P(Ω;Rn):={E[|ξ|p]}1 p<∞; Risks 2020,8, 70 4 of 34 • Lp F,P( 0, T ; Rn) : the space of Rn -valued, F -adapted processes f(·):={f(t)}t∈[0,T] such that kf(·)kLp F,P(0,T;Rn):={E[(RT 0|f(t)|2dt)p 2]}1 p<∞; • Sp F,P( 0, T ; Rn) : the space of Rn -valued, F -adapted, càdlàg processes f(·) such that kf(·)kSp F,P(0,T;Rn):={E[sup t∈[0,T]|f(t)|p]}1 p<∞; • S∞ F,P( 0, T ; Rn) : the space of Rn -valued, F -adapted, essentially bounded, càdlàg processes f(·) over [0, T]under P; • EF,P( 0, T ; Rn) : the space of Rn -valued, F -adapted, càdlàg processes f(·) such that the random variable f∗:=sup t∈[0,T]|f(t)|has exponential moments of all orders. Replacing the expectation E[·] by EQ[·] , where EQ[·] is the expectation under some probability measure Q equivalent to P , we can define similar spaces of random variables and stochastic processes on (Ω , F , Q) , i.e., Lp F,Q( 0, T ; Rn) , Sp F,Q( 0, T ; Rn) , Lp F(T),Q(Ω ; Rn) , L∞ F,Q( 0, T ; Rn) and EF,Q( 0, T ; Rn) . Furthermore, we define the following two spaces of deterministic functions: • C(0, T;Rn): the space of continuous functions φ:[0, T]→Rn; • Cb(0, T;Rn): the space of continuous, uniformly bounded functions φ:[0, T]→Rn. Throughout this paper, we will take Rnto be either Ror R2in different circumstances. We let ˜ P be the risk-neutral probability measure, which will be specified after we introduce our standing assumption. Suppose that the market prices of risks of W1(·) and W2(·) are given by two F -adapted processes {ξ1(t)}t≥0 and {ξ2(t)}t≥0 . We denote by {ξ(t)}t≥0 the vector process of the market prices of risks, where ξ(t):= (ξ1(t) , ξ2(t))> . We will specify the structure of the market prices of risks later. Indeed, ˜ P is a probability measure equivalent to P and the processes {˜ W1(t)}t≥0 and {˜ W2(t)}t≥0defined by ˜ W1(t):=W1(t) + Zt 0ξ1(u)du, and ˜ W2(t):=W2(t) + Zt 0ξ2(u)du, are two one-dimensional, (F , ˜ P) -standard Brownian motions. We denote by ˜ E[·] the expectation taken under ˜ P, and ˜ Et[·]the conditional expectation under ˜ Pgiven F(t). The price process of the bank account {P(t)}t≥0is given by dP(t) = rP(t)dt,P(0) = 1, (1) where r> 0 represents the risk-free, instantaneous interest rate. In what follows, we introduce the dynamics of the stock and variance processes under the risk-neutral measure ˜ P and the real-world measure Psequentially. Under ˜ P, the stock price process {S(t)}t≥0is governed by dS(t) = S(t)rdt +qv(t)d˜ W1(t),S(0) = s>0, (2) where pv(t) is the instantaneous volatility of the stock at time t ; the variance process {v(t)}t≥0 evolves according to Heston’s model dv(t) = ˜ κ˜ θ−v(t)dt +σqv(t)ρd˜ W1(t) + q1−ρ2d˜ W2(t),v(0) = v>0. (3) Risks 2020,8, 70 5 of 34 Here ˜ κ> 0 and ˜ θ> 0 are the speed of mean-reversion and the long-run average of v(t) under ˜ P ; σ> 0 is the volatility of volatility; the correlation coefficient satisfies ρ∈[− 1, 1 ] . We require that the Feller condition is satisfied, i.e., 2 ˜ κ˜ θ>σ2 , so that v(·) is positive, ˜ P -a.s. (refer to Chapter 9 in Elliott and Kopp 2005 ). Though the Feller condition may not be satisfied in practice, market volatility seldom becomes zero. Hence, having the Feller condition in place is meaningful in our model, which guarantees a strictly positive volatility. As in other literature on Heston’s model (see, for example, Zeng and Taksar 2013), we assume that the market prices of risks at time tare given by ξ1(t):=ξ1qv(t),ξ1∈R, and ξ2(t):=ξ2qv(t),ξ2∈R. The above specification of the market prices of risks ensures that the evolution of the variance process under the real-world probability measure P has a similar structure of affine drift and square-root volatility of volatility as that under the risk-neutral probability measure ˜ P (see Equation (5)). Indeed, this specification is closely related to the completely affine and the essentially affine specifications proposed by Duffee (2002). Furthermore, we require ξ2 1+ξ2 26= 0, which rules out the case that the real-world probability measure P coincides with the risk-neutral probability measure ˜ P . Otherwise, the portfolio selection problems do not have non-trivial solutions. Under P, the stock price process {S(t)}t≥0follows dS(t) = S(t)r+ξ1v(t)dt +qv(t)dW1(t),S(0) = s>0. (4) Here r+ξ1v(t) can be considered as the appreciation rate of the stock at time t . The variance process {v(t)}t≥0under Psatisfies dv(t) = κθ−v(t)dt +σqv(t)ρdW1(t) + q1−ρ2dW2(t),v(0) = v>0, (5) where κ:=˜ κ−σξρ,θ:=˜ κ˜ θ ˜ κ−σξρ,ξρ:=ρξ1+q1−ρ2ξ2. Even when ˜ κ−σξρ= 0, we still have κθ =˜ κ˜ θ in view of the convention ˜ κ−σξρ ˜ κ−σξρ=0 0= 1. Then Equation (5) is well-defined even if ˜ κ−σξρ= 0. Hence, we do not require that ˜ κ−σξρ6= 0 in this paper. The equivalence between ˜ P and P implies that v(·) is also positive, P -a.s.. In fact, this is guaranteed by the Feller condition under P , i.e., 2 κθ = 2 ˜ κ˜ θ>σ2 . In Heston’s model (5), W1(·) can be considered as the common shock of the stock price and the variance, while W2(·) as the idiosyncratic shock of the variance. Therefore, ξ1(·) and ξ2(·) represent the market price of the common risk of the stock and the variance and that of the idiosyncratic risk of the variance, respectively. We now introduce our assumption: Standing Assumption. For any p∈[1, ∞), EZp(T)<∞, where Z(T):=exp ZT 0|ξ(u)|2du=exp (ξ2 1+ξ2 2)ZT 0v(u)du. Risks 2020,8, 70 6 of 34 The Standing Assumption states that the variance process/the market price of risk is exponentially integrable of all orders under P . This assumption allows us to define a family of probability measures equivalent to P through a family of Radon-Nikodým derivatives. Indeed, under the Standing Assumption, it holds that Ee−m2 1ξ2 1+m2 2ξ2 2 2RT 0v(u)du−RT 0√v(u)[m1ξ1dW1(u)+m2ξ2dW2(u)]=1, ∀m1,m2∈R. (6) The above equation will be used frequently throughout this paper. Particularly, if we take m1=m2= 1 in (6) or p=1 2 in the Standing Assumption, then we can specify the risk-neutral probability measure ˜ P as follows d˜ P dPF(T) =Γ(T):=exp −1 2ZT 0|ξ(u)|2du −ZT 0ξ(u)>dW(u). (7) Moreover, it will turn out that the Standing Assumption ensures that a set of BSDEs admits unique solutions in proper spaces and mean-variance portfolio selection problems have optimal solutions. We now introduce the dynamics of a variance swap under Heston’s model. The research on pricing variance swaps can be dated back to the early works of Neuberger (1990,1994) and Dupire (1992,1993 ). Simply speaking, a variance swap contract is a financial contract with two legs on the future realized variance of the price changes of the underlying asset. One leg of the variance swap is floating and pays a variable amount based upon the annualized realized variance over a specified period, while the other leg is fixed and pays a fixed amount. This fixed amount is called the strike, which is usually chosen such that there is no cost of entering the contract at the issue time. The terminal payoff of the variance swap is equal to the realized variance minus the strike multiplying a notational amount. Under the continuous sampling scheme, we consider a variance swap written on the realized variance of the stock S over [ 0, TV] with a one-unit notational amount. Mathematically, the timet value of this variance swap is given by V(t) = e−r(TV−t)˜ Et1 TVZTV 0v(u)du −KV, (8) where the strike KVis KV:=˜ E1 TVZTV 0v(u)du. From (3), we have v(u) = v(0)e−˜ κu+˜ θ(1−e−˜ κu) + e−˜ κuZu 0e˜ κµσqv(µ)ρd˜ W1(µ) + q1−ρ2d˜ W2(µ). (9) Substituting (9) into (8) gives V(t) = e−r(TV−t)˜ Et1 TVZTV 0e−˜ κuZu 0e˜ κµσqv(µ)ρd˜ W1(µ) + q1−ρ2d˜ W2(µ)du =e−r(TV−t)˜ Et1 TVZTV 0ZTV µe−˜ κudue˜ κµσqv(µ)ρd˜ W1(µ) + q1−ρ2d˜ W2(µ) (10) =e−r(TV−t)Zt 0 σ ˜ κTV1−e−˜ κ(TV−µ)qv(µ)ρd˜ W1(µ) + q1−ρ2d˜ W2(µ). Here the first equality is obtained via canceling out the first two deterministic terms on the right hand side of (9) and noting that the third term on the right hand side of (9) is a ˜ P -martingale and hence has a zero expectation under ˜ P; the second equality comes from interchanging the order of integration; the Risks 2020,8, 70 7 of 34 third equality is due to the martingale property of the Itô integral inside the conditional expectation ˜ Et[·]. Therefore, differentiating both sides of (10) with respect to t , we obtain the dynamics of the variance swap dV(t) = rV(t)dt +φ(t)qv(t)ρd˜ W1(t) + q1−ρ2d˜ W2(t) =rV(t) + ξρφ(t)v(t)dt +φ(t)qv(t)ρdW1(t) + q1−ρ2dW2(t), (11) where φ(t):=σ ˜ κTVe−r(TV−t)−e−(r+˜ κ)(TV−t). 3. Problem Formulation In this section, we formulate two mean-variance portfolio selection problems for an economic agent. In the first problem, the agent can invest in the bank account, the stock and the variance swap; in the second problem, the agent can only invest in the bank account and the stock. Since the variance process is driven by the common shock W1(·) and the idiosyncratic shock W2(·) , the market in the first problem is complete, while that in the second problem is incomplete in general. Note when ρ=− 1 or ρ= 1, the idiosyncratic shock W2(·) disappears in the dynamics of the variance process and the variance swap. The randomness of the financial market is entirely driven by the common shock W1(·) . In this case, the variance swap becomes a redundant asset, and the market in the second problem becomes complete. The special case of ρ=− 1 or ρ= 1 will be discussed in detail in the rest of the paper. In what follows, we introduce the mean-variance portfolio selection problem in the complete market. In this circumstance, the agent can allocate his wealth to the bank account, the stock and the variance swap over the finite-horizon [ 0, T] . Here we assume that the investment horizon is not longer than the term of the variance swap, i.e., 0 <T≤TV . Let πS(t) denote the amount of the agent’s wealth invested in the stock at time t , and πV(t) the notational amount invested in the variance swap at time t . We call π(·):={π(t)}t∈[0,T]={(πS(t) , πV(t))>}t∈[0,T] a portfolio strategy of the agent. Let X(t):=Xπ(t) be the total wealth of the agent at time t when the portfolio strategy π(·) is adopted. Suppose that the market is frictionless, short-selling is allowed and the portfolio strategy is self-financing. Then the amount of the agent’s wealth invested in the bank account at time t is equal to X(t)−πS(t)−πV(t)V(t) . Thus, the agent’s wealth process {X(t)}t∈[0,T] satisfies the following stochastic differential equation (SDE): dX(t) = r[X(t)−πS(t)−πV(t)V(t)]dt +πS(t)dS(t) S(t)+πV(t)dV(t) =rX(t) + π(t)>B(t)dt +π(t)>σ(t)dW(t),X(0) = x>0, (12) where B(t):=ξ1v(t),ξρφ(t)v(t)>∈R2, (13) and σ(t):= pv(t)0 ρφ(t)pv(t)p1−ρ2φ(t)pv(t)!∈R2×2, (14) Risks 2020,8, 70 8 of 34 represent the risk premium vector and the volatility matrix of the risky assets (i.e., the stock and the variance swap), respectively. Denote by Σ(t):=σ(t)σ(t)>∈R2×2, (15) the variance-covariance matrix of the risky assets. If ρ6=± 1, the market price of risk (vector) and its squared-norm satisfy B(t)>(σ(t)>)−1=ξ(t)>= (ξ1qv(t),ξ2qv(t)), and B(t)>Σ(t)−1B(t) = |ξ(t)|2= (ξ2 1+ξ2 2)v(t),∀t∈[0, T]. However, if ρ=− 1 or ρ= 1, the volatility matrix and the variance-covariance matrix are singular. To integrate this singular case into a unified framework, we define an auxiliary market price of risk (vector) and its squared-norm as ϑ(t):= (ξ1qv(t),1{ρ6=±1}ξ2qv(t))>, (16) and |ϑ(t)|2:= (ξ2 1+1{ρ6=±1}ξ2 2)v(t),∀t∈[0, T]. (17) It should be noted that the following equality holds B(t) = σ(t)ϑ(t),∀t∈[0, T], for any ρ∈[−1, 1]. Definition 1. In the complete market, a portfolio strategy π(·) is said to be admissible if (1) π(·) is F -adapted; (2) π(·)>B(·)∈ L2 F,P( 0, T ; R) and π(·)>σ(·)∈ L2 F,P( 0, T ; R2) . The set of admissible portfolio strategies is denoted by AC. From the standard theory of SDEs (refer to Chapter 1 in Yong and Zhou 1999), we know that for any π(·)∈ AC , the SDE (12) has a unique strong solution X(·) such that X(·)∈ Sp F,P( 0, T ; R) , for any p∈[ 1, ∞) . The agent’s objective is to find a portfolio π(·)∈ AC , such that the expected terminal wealth satisfies E[X(T)] = dfor a given d∈R, while the variance of the terminal wealth Var[X(T)] = E(X(T)−E[X(T)])2=E(X(T)−d)2 is minimized. We specify the mean-variance problem in the complete market as follows: Definition 2. In the complete market, the mean-variance portfolio selection is the following stochastic control problem with a terminal state constraint, parameterized by d ∈R:          min π(·)∈AC J(x,v;π(·)) :=E(X(T)−d)2, subject to (E[X(T)] = d, (X(·),v(·);π(·)) satisfy (5)and (12). (18) Here π∗(·) denotes an optimal portfolio strategy of the above problem, and X∗(·) denotes the wealth process associated with π∗(·) . The optimal portfolio strategy is called an efficient portfolio Risks 2020,8, 70 15 of 34 which can be also written as E[X∗(T)] = d=s1−h(0)e−2rT h(0)e−2rT qVar[X∗(T)] + xerT. (47) Proof. Denote by ˆ X(t):=X(t)−ce−r(T−t), (48) which solves the same SDE as X(t) in Equation (12) but has a different initial value ˆ X( 0 ) = x−ce−rT . Hence for any π(·)∈ AC , we know ˆ X(·)∈Sp F,P( 0, T ; R) , for p≥ 1. By Itô’s differentiation rule, we have dh(t)ˆ X2(t) = h(t)σ(t)>π(t) + ϑ(t) + η(t) h(t)ˆ X(t)>σ(t)>π(t) + ϑ(t) + η(t) h(t)ˆ X(t)dt +2h(t)ˆ X(t)π(t)>σ(t) + ˆ X2(t)η(t)>dW(t). (49) We employ a localization technique and define a sequence of stopping times as follows τn:=inf t≥0 : Zt 02h(u)ˆ X(u)π(u)>σ(u) + ˆ X2(u)η(u)> 2du ≥n, for n≥1. Here we adopt the convention inf ∅:= +∞ . Since ˆ X(·)∈S2 F,P( 0, T ; R) and π(·)>σ(·)∈ L2 F,P( 0, T ; R2) , for any π(·)∈ AC , and (h(·) , η(·)) ∈ L∞ F,P( 0, T ; R)×L2 F,P( 0, T ; R2) , the sequence of the stopping times τnis increasing and converges to +∞,P-a.s., when napproaches +∞. Moreover, we note h(T∧τn)ˆ X2(T∧τn)≤e2rTsup t∈[0,T] ˆ X2(t), where the right hand side is a P-integrable random variable. Integrating from 0 to T∧τnand taking expectations on both sides of (49) yield E[h(T∧τn)ˆ X2(T∧τn)] −h(0)ˆ X2(0) = ERT∧τn 0h(u)σ(u)>π(u) + ϑ(u) + η(u) h(u)ˆ X(u)> ×σ(u)>π(u) + ϑ(u) + η(u) h(u)ˆ X(u)du. (50) Sending n→∞ in the above equation and using the dominated convergence theorem and the monotone convergence theorem, we obtain E[ˆ X2(T)] −h(0)ˆ X2(0) = EZT 0h(u)σ(u)>π(u) + ϑ(u) + η(u) h(u)ˆ X(u)> ×σ(t)>π(u) + ϑ(u) + η(u) h(u)ˆ X(u)du. (51) Depending on the value of correlation coefficient, the optimal strategy of the quadratic-loss minimization problem (20) is given by the following two cases π(t) = −Σ(t)−1B(t) + σ(t)η(t) h(t)X(t)−ce−r(T−t), if ρ6=±1, (52) or πS(t) + ρφ(t)πV(t) = −ξ1−ρσK(t)X(t)−ce−r(T−t), if ρ=−1 or 1. (53) Risks 2020,8, 70 16 of 34 The optimal cost functional is given by J0(x,v;π(·)) = h(0)[x−ce−rT]2. (54) From the Lagrangian duality theorem, solving the original problem (18) is reduced to maximizing the following cost functional J(x,v;π(·),λ) = J0(x,v;π(·)) −λ2 =h(0)x−(d−λ)e−rT2−λ2, (55) over λ∈R. From Lemma 1, we know ∂2J ∂λ2(x,v;π(·),λ) = 2e−2rTh(0)−2<0. Using the first-order condition to (55) with respect to λ, we obtain the following optimal value of λ: λ∗=−h(0)e−rT(x−de−rT) h(0)e−2rT −1. Substituting λ∗ into (52) or (53) and (54) leads to the efficient portfolio strategy (43) or (44) and the efficient frontier (46) and (47). Next we claim that the efficient portfolio strategy (43) or (44) is admissible. Evidently, π∗(·) is F-adapted, i.e., Condition (1) in Definition 1is satisfied. For both (43) and (44), we can see π∗(t)>σ(t) = −ϑ(t) + η(t) h(t)>X∗(t)−(d−λ∗)e−r(T−t). Then substituting π∗(t)σ(t)into (12) gives dX∗(t) = a1(t)X∗(t) + a2(t)dt +a3(t)>X∗(t) + a4(t)>dW(t), (56) where a1(t):=r−|ϑ(t)|2+ϑ(t)>η(t) h(t)∈R, a2(t):= (d−λ∗)e−r(T−t)|ϑ(t)|2+ϑ(t)>η(t) h(t)∈R, and a3(t):=−ϑ(t) + η(t) h(t)∈R2, a4(t):= (d−λ∗)e−r(T−t)ϑ(t) + η(t) h(t)∈R2. By Itô’s differentiation rule, we can verify X∗(t) = (d−λ∗)e−r(T−t)−H(t)¯ h(t)e−Rt 0|ϑ(u)|2du, (57) where ¯ h(·)is the unique solution of the BSRE (29) and H(·)is defined by H(t) = h(0)(d−λ∗)e−rT −x0e−rtΠ(t),∀t∈[0, T]. Risks 2020,8, 70 17 of 34 Using Lemmas 1and 4, we deduce that Esup u∈[0,T]|X∗(u)|4≤K1+Esup u∈[0,T]Π(u)¯ h(u)4 ≤K1+Esup u∈[0,T]|Π(u)|8+Esup u∈[0,T]|¯ h(u)|8<∞. Furthermore, we derive that EZT 0π∗(u)>B(u) 2du≤CEZT 0v(u)X∗(u) 2du+1 ≤CEsup u∈[0,T]|X∗(u)|4+EeRT 0|v(u)|2du+1<∞. Hence, π∗(·)>B(·)∈ L2 F,P( 0, T ; R) . In the same vein, we can show π∗(·)>σ(·)∈ L2 F,P( 0, T ; R2) . This implies that Condition (2) in Definition 1is satisfied. Therefore, we can conclude π∗(·)∈ AC . This completes the proof. Remark 5. Although the efficient strategy has the same parametric form as that in the market with bounded coefficients, the admissibility of the efficient strategy needs to be carefully discussed. Simply speaking, this is because the product of two square-integrable variables is not necessarily square-integrable unless additional conditions (refer to Remark 3), i.e., Standing Assumption, are imposed. Remark 6. From Lemmas 2and 3, we can see that h( 0 ) is independent of the term of the variance swap, i.e., TV . Therefore, the efficient frontier (45) or (46) in the complete market does not depend on TV . This is interesting since investing in the variance swaps with different maturities does not make any difference to the efficient frontier. Remark 7. Depending on the correlation coefficient, the efficient portfolio strategies have different representations. Particularly, if ρ=− 1or 1, the optimal (notational) amounts allocated to the stock and the variance swap cannot be disentangled, since the volatility matrix now is singular and hence not invertible. Indeed, when ρ=− 1or 1, the stock and the variance swap play essentially the same role in the market and the existence of any one of them makes the market complete and the other a redundant asset. 5. Solution to the Incomplete Market Case In this section, we consider the mean-variance portfolio selection problem (22), where the variance swap is absent. Although the market with only the bank account and the stock in this section seems simpler than the market in the previous section, the derivations of the efficient portfolio strategy and the efficient frontier are more complicated due to market incompleteness. The derivations in this section also rely on the LQ control and BSDE approach. Consider a pair of processes g(·):={g(t)}t∈[0,T] and ζ(·):={ζ(t)}t∈[0,T]= {(ζ1(t),ζ2(t))>}t∈[0,T]satisfying the following BSDE      dg(t) = −2r+ξ2 1v(t)g(t) + 2ξ1qv(t)ζ1(t) + ζ2 1(t) g(t)dt +ζ(t)>dW(t), g(T) = 1. (58) As in the previous section, Equation (58) is a BSRE with random and unbounded coefficients, thereby the existing theory of BSREs cannot be used directly. Similarly, we apply a transformation method to prove the existence and uniqueness of a solution to BSRE (58). Indeed, in Section 4we use a reciprocal transformation to relate the BSRE (27) to the linear BSDE (29). Since the market is incomplete in this Risks 2020,8, 70 18 of 34 section, the structure of BSRE (58) is different from that of (27). We cannot use the same reciprocal transformation in this section. Instead, we use a logarithmic transformation to relate BSRE (58) to a quadratic BSDE. In that sense, this section is not a trivial repetition of the previous section. Under the Standing Assumption, we can define a new probability measure ˘ P equivalent to P as follows d˘ P dPF(T) =Γ(T):=exp −2ZT 0ξ2 1v(u)du −2ZT 0ξ1qv(u)dW1(u). By Girsanov’s theorem, the process ˘ W(·):= ( ˘ W1(·),˘ W2(·))>defined by ˘ W1(t):=W1(t) + 2Zt 0ξ1qv(u)du, and ˘ W2(t):=W2(t), is a two-dimensional, (F , ˘ P) -standard Brownian motion. We denote by ˘ E[·] the expectation taken under ˘ P, and ˘ Et[·]the conditional expectation under ˘ Pgiven F(t). Lemma 5. Suppose that the Standing Assumption holds. The BSRE (58) admits at least one solution (g(·),ζ(·)) ∈ S2 F,P(0, T;R)×L2 F,P(0, T;R2). Proof. We consider a pair of processes ˘ g(·):={˘ g(t)}t∈[0,T] and ˘ ζ(·):={˘ ζ(t)}t∈[0,T]= {(˘ ζ1(t),˘ ζ2(t))>}t∈[0,T]following a quadratic BSDE under ˘ P: d˘ g(t) = [−2r+ξ2 1v(t)] + 1 2[˘ ζ2 1(t)−˘ ζ2 2(t)]dt +˘ ζ(t)>d˘ W(t),˘ g(T) = 0. (59) From the Standing Assumption, we can see that the BSDE (59) belongs to a class of quadratic BSDEs satisfying the exponential integrability condition, as considered by Briand and Hu (2008). Therefore, under the Standing Assumption, there exists at least one solution (˘ g(·) , ˘ ζ(·)) ∈ EF,˘ P( 0, T ; R)× Lp F,˘ P(0, T;R2), for any p≥1 (see Corollary 4 in Briand and Hu 2008). We consider a pair of transformed processes g(·)and ζ(·)defined by g(t):=e˘ g(t), (60) and ζ(t):=˘ ζ(t)e˘ g(t). (61) Applying Itô’s differentiation rule to g(t)gives dg(t) = [−2r+ξ2 1v(t)]e˘ g(t)+˘ ζ2 1(t)e˘ g(t)dt +e˘ g(t)˘ ζ(t)>d˘ W(t) =−2r+ξ2 1v(t)g(t) + 2ξ1qv(t)ζ1(t) + ζ2 1(t) g(t)dt +ζ(t)>dW(t), Risks 2020,8, 70 19 of 34 which is exactly the BSRE (58). Therefore, by relationships (60) and (61), there also exists at least one solution (g(·),ζ(·)) to the BSRE (58). By (˘ g(·),˘ ζ(·)) ∈ EF,˘ P(0, T;R)×Lp F,˘ P(0, T;R2), where p=4, Hölder’s inequality and the Standing Assumption (see also Remark 3), we derive Esup u∈[0,T]|g(u)|2=˘ EΓ−1(T)sup u∈[0,T] e2˘ g(u) ≤EΓ−1(T)1 2˘ Ee4 supu∈[0,T]|˘ g(u)|1 2 ≤Ee−8ξ2 1RT 0v(u)du+4ξ1RT 0√v(u)dW1(u)1 4 ×Ee12ξ2 1RT 0v(u)du1 4˘ Ee4 supu∈[0,T]|˘ g(u)|1 2 ≤EZ12(T)1 4˘ Ee4 supu∈[0,T]|˘ g(u)|1 2 <∞, and EZT 0|ζ(t)|2dt=˘ EΓ−1(T)ZT 0e2˘ g(t)|˘ ζ(t)|2dt ≤C˘ EΓ−2(T)e4 supt∈[0,T]|˘ g(t)|1 2˘ EZT 0|˘ ζ(t)|2dt21 2 ≤CEΓ−3(T)1 4˘ Ee8 supt∈[0,T]|˘ g(t)|1 4||˘ ζ(·)||2 L4 F,˘ P(0,T;R2) ≤CEZ30(T)1 8˘ Ee8 supt∈[0,T]|˘ g(t)|1 4||˘ ζ(·)||2 L4 F,˘ P(0,T;R2)<∞. Therefore, (g(·),ζ(·)) ∈ S2 F,P(0, T;R)×L2 F,P(0, T;R2). Remark 8. As − 2 r+ξ2 1v(t) is not uniformly bounded in t , the classical theory of quadratic BSDEs established by Kobylanski (2000) is not working for (59). Furthermore, since the driver of (59) is neither convex nor concave in the second (control) component of the solution, the uniqueness result for quadratic BSDEs in Briand and Hu (2008) cannot be applied. Therefore, the BSDE (59) does not necessarily have a unique solution, and neither does the BSRE (58). Fortunately, we can find an explicit solution to the BRSE (58), thanks to its Markovian structure, and verify that this solution is exactly the unique solution. Lemma 6. Suppose that the Standing Assumption holds. A solution pair (g(·) , ζ(·)) to the BSRE (58) is given by g(t) = exp 2r(T−t)−M(t)v(t)−N(t), (62) and ζ1(t) = −σρqv(t)M(t)g(t), (63) ζ2(t) = −σq1−ρ2qv(t)M(t)g(t). (64) Risks 2020,8, 70 20 of 34 Here M ∈ C(0, T;R)and N ∈ C(0, T;R)are the solutions of the following Riccati and linear ODEs: dM(t) dt −(κ+2σξ1ρ)M(t) + ρ2−1 2σ2M2(t) + ξ2 1=0, M(T) = 0, (65) and dN(t) dt +κθM(t) = 0, N(T) = 0. (66) Proof. Refer to the Appendix A. Remark 9. Unlike the proof of Lemma 2, the solution of the BSRE (58) is found by trial and error in Lemma 6. This is because we cannot find a linear BSDE related to BSRE (58). In fact, it seems very difficult, if not impossible, to represent the solution in terms of an expectation expression. The difficulty is caused by market incompleteness. Lemma 7. The first component of the solution to the BSRE (58), i.e., g(t) given by (62), is in L∞ F,P( 0, T ; R) . More specifically, 0≤g(t)<e2rT, a.e. t ∈[0, T],P-a.s.. Proof. From the second line of Equation (A11), we see dg(t)e2rt−Rt 0[ξ1−ρσM(u)]2v(u)du=e2rt−Rt 0[ξ1−ρσM(u)]2v(u)duζ1(t)dW1(t) + ζ2(t)dW2(t). Since e2rt−Rt 0[ξ1−ρσM(u)]2v(u)du is bounded, a.e. t∈[ 0, T] , P -a.s. and ζ(·)∈ L2 F,P( 0, T ; R2) , we have that the process ng(t)e2rt−Rt 0[ξ1−ρσM(u)]2v(u)duot∈[0,T] is a square-integrable (F,P)-martingale. Then taking expectations gives 0≤g(t) = Etexp 2r(T−t)−ZT t[ξ1−ρσM(u)]2v(u)du≤e2r(T−t)≤e2rT, a.e. t∈[ 0, T] , P -a.s.. Furthermore, since v(t)> 0, a.e. t∈[ 0, T] , P -a.s., and M(t) = ξ1 ρσ is not the solution of (65), the upper bound is strict, for any t∈[0, T]. Remark 10. Although we have already obtained the solution space of g(·) in Lemma 5, that space is not delicate enough to be used in our following applications. Lemma 7gives a more accurate estimate for the first component of the solution g(·). The upper bound of g(·)guarantees that the first-order condition is satisfied in Theorem 2 for the outer maximization problem. As in the previous section, we provide the explicit representations of the solutions to (65) and (66). Risks 2020,8, 70 21 of 34 Lemma 8. The explicit solutions to (65) and (66) are given by M(t) =                            ξ2 1 sin(δIτ) δI cos(δIτ) + κ+2σξ1ρ 2 sin(δIτ) δI ,∆I<0, ξ2 1τ 1+κ+2σξ1ρ 2τ,∆I=0, ξ2 1 sinh(δIτ) δI cosh(δIτ) + κ+2σξ1ρ 2 sinh(δIτ) δI ,∆I>0, (67) and (i) if ρ26=1 2, then N(t) =                  −2κθ (2ρ2−1)σ2log  cos(δIτ) + κ+2σξ1ρ 2 sin(δIτ) δI +κθ(κ+2σξ1ρ) (2ρ2−1)σ2τ,∆I<0, −2κθ (2ρ2−1)σ2log  1+κ+2σξ1ρ 2τ +κθ(κ+2σξ1ρ) (2ρ2−1)σ2τ,∆I=0, −2κθ (2ρ2−1)σ2log  cosh(δIτ) + κ+2σξ1ρ 2 sinh(δIτ) δI +κθ(κ+2σξ1ρ) (2ρ2−1)σ2τ,∆I>0; (68) (ii) if ρ2=1 2and κ6=−2σξ1ρ, then N(t) = −κθξ2 1 κ+2σξ1ρsinh(δIτ) δI e−(κ+2σξ1ρ)τ 2−τ; (69) (iii) if ρ2=1 2and κ=−2σξ1ρ, then N(t) = 1 2κθξ2 1τ2. (70) where ∆I:= (κ+2σξ1ρ)2−2ξ2 1σ2(2ρ2−1),δI:=1 2p|∆I|and τ:=T−t. Proof. The proof is similar to that of Lemma 3. We omit it here. Lemma 9. Suppose that the Standing Assumption holds. The solution M(·) given by (67) is non-negative and non-explosive over [0, T]. Proof. Consider the following Riccati equation for M1(·)∈ C(0, T;R): dM1(t) dt −(κ+2σξ1ρ)M1(t) + ρ2−1 2σ2M2 1(t) = 0, M1(T) = 0. (71) Obviously, M1(t)≡0 is the solution to (71). Observe M1(T) = M(T)and 0−(κ+2σξ1ρ) 0(ρ2−1 2)σ2!≤ ξ2 1−(κ+2σξ1ρ) 0(ρ2−1 2)σ2!. Thus, using the comparison theorem for Riccati equations (see Theorem 2.1 in Freiling et al. 1996) gives 0≡M1(t)≤M(t),∀t∈[0, T]. Risks 2020,8, 70 22 of 34 Since v(·) is Markovian with respect to F , we can derive as in Lemma 2that there exists M2(·)∈ C(0, T;R)and N2(·)∈ C(0, T;R)such that ˘ Etexp ξ2 1ZT tv(u)du=exp M2(t)v(t) + N2(t). The dynamics of v(·)under ˘ Pis given by dv(t) = κθ −(κ+2σξ1ρ)v(t)dt +σqv(t)ρd˘ W1(t) + q1−ρ2d˘ W2(t),v(0) = v>0. As in Lemmas 2and 6, we have dM2(t) dt −(κ+2σξ1ρ)M2(t) + σ2M2 2(t) + ξ2 1=0, M2(T) = 0, and dN2(t) dt +κθM2(t) = 0, N2(T) = 0. Observe that M(T) = M2(T) = 0 and ξ2 1−(κ+2σξ1ρ) 0(ρ2−1 2)σ2!≤ ξ2 1−(κ+2σξ1ρ) 0σ2!. Again, using the comparison theorem for Riccati equations gives 0≤M(t)≤M2(t),∀t∈[0, T]. It follows from Jensen’s inequality, the tower property, Hölder’s inequality and the Standing Assumption that exp M2(t)˘ E[v(t)] + N2(t)≤˘ Eexp M2(t)v(t) + N2(t) =˘ Eexp ξ2 1ZT tv(u)du ≤EΓ(T)exp ξ2 1ZT 0v(u)du ≤Ee−8ξ2 1RT 0v(u)du−4ξ1RT 0√v(u)dW(u)1 2Ee6ξ2 1RT 0v(u)du1 2 ≤EZ6(T)1 2<∞. Note that M2(t)≥0, N2(t)≥0, ∀t∈[0, T], and ˘ E[v(t)] = e−(κ+2σξ1ρ)tv+κθ Zt 0e(κ+2σξ1ρ)udu∈ Cb(0, T;R). Risks 2020,8, 70 23 of 34 Therefore, M2(·) does not explode over [ 0, T] , i.e., the investment horizon T is shorter than the first explosion time of M2(·), and 0=M1(t)≤M(t)≤M2(t)<∞,∀t∈[0, T]. This completes the proof. The next lemma states that (g(·) , ζ(·)) given in Lemma 6is, in fact, the unique solution to the BSRE (58). Lemma 10. Suppose that the Standing Assumption holds. The BSRE (58) admits a unique solution (g(·),ζ(·)) ∈ L∞ F,P(0, T;R)×L2 F,P(0, T;R2), which is given by (62)–(64). Proof. From relationships (60) and (61), we have ˘ g(t) = 2r(T−t)−M(t)v(t)−N(t), (72) and ˘ ζ(t)=(˘ ζ1(t),˘ ζ2(t))> = (−σρqv(t)M(t),−σq1−ρ2qv(t)M(t))>, (73) is a solution pair of the quadratic BSDE (59). Suppose that (˘ ˘ g(·) , ˘ ˘ ζ(·)) , where ˘ ˘ ζ(·):= ( ˘ ˘ ζ1(·) , ˘ ˘ ζ2(·))> , is another solution pair to the BSDE (59), i.e., d˘ ˘ g(t) = [−2r+ξ2 1v(t)] + 1 2[˘ ˘ ζ2 1(t)−˘ ˘ ζ2 2(t)]dt +˘ ˘ ζ(t)>d˘ W(t),˘ ˘ g(T) = 0. (74) By the Standing Assumption and Lemma 9, the following Novikov condition holds: ˘ Eexp 1 2RT 0|˘ ζ(u)|2du ≤EΓ(T)exp σ2supu∈[0,T]M2(u) 2RT 0v(u)du ≤Ee−8ξ2 1RT 0v(u)du−4ξ1RT 0√v(u)dW(u)1 2E(Z(T))4+σ2supu∈[0,T]M2(u) ξ2 1+ξ2 21 2 =E(Z(T))4+σ2supu∈[0,T]M2(u) ξ2 1+ξ2 21 2 <∞. Thus, we can define a new probability measure ˆ Pequivalent to ˘ Pas follows dˆ P d˘ PF(T) =Θ(T):=exp −1 2ZT 0|˘ ζ(u)|2du −ZT 0 ˘ ζ1(u)d˘ W1(u) + ZT 0 ˘ ζ2(u)d˘ W2(u). (75) By Girsanov’s theorem, the process ˆ W(·):= ( ˆ W1(·),ˆ W2(·))>defined by ˆ W1(t):=˘ W1(t) + Zt 0 ˘ ζ1(u)du, and ˆ W2(t):=˘ W2(t)−Zt 0 ˘ ζ2(u)du, Risks 2020,8, 70 24 of 34 is a two-dimensional, (F , ˆ P) -standard Brownian motion. Under ˆ P , (˘ g(·) , ˘ ζ(·)) and (˘ ˘ g(·) , ˘ ˘ ζ(·)) satisfy the following two equations      d˘ g(t) = [−2r+ξ2 1v(t)] −1 2[˘ ζ2 1(t)−˘ ζ2 2(t)]dt +˘ ζ(t)>dˆ W(t), ˘ g(T) = 0, (76) and      d˘ ˘ g(t) = [−2r+ξ2 1v(t)] + 1 2[˘ ˘ ζ2 1(t)−˘ ˘ ζ2 2(t)] −[˘ ζ1(t)˘ ˘ ζ1(t)−˘ ζ2(t)˘ ˘ ζ2(t)]dt +˘ ˘ ζ(t)>dˆ W(t), ˘ ˘ g(T) = 0. (77) Denote by (ˆ g(·),ˆ ζ(·)) := ( ˘ g(·),˘ ζ(·)) −(˘ ˘ g(·),˘ ˘ ζ(·)). Then subtracting (76) by (77) gives    dˆ g(t) = −1 2[ˆ ζ2 1(t)−ˆ ζ2 2(t)]dt +ˆ ζ(t)>dˆ W(t), ˆ g(T) = 0. (78) Indeed, Equation (78) is a BSDE with quadratic growth and bounded terminal value considered by Kobylanski (2000), hence admits a unique solution. Clearly, ˆ g(·)≡ 0 and ˆ ζ(·)≡( 0, 0 )> form the unique solution pair to (78). Therefore, (˘ g(·) , ˘ ζ(·)) is the unique solution to the quadratic BSDE (59). By definition, (g(·) , ζ(·)) ∈ S2 F,P( 0, T ; R)×L2 F,P( 0, T ; R2) is the unique solution to the BSRE (58). Furthermore, by Lemmas 5and 7, we conclude that (g(·),ζ(·)) ∈ L∞ F,P(0, T;R)×L2 F,P(0, T;R2). Next we derive the closed-form expressions of the efficient portfolio strategy and the efficient frontier of the mean-variance portfolio selection problem (22). Theorem 2. The efficient portfolio strategy of the mean-variance portfolio selection problem (22) is given by π∗ S(t) = −ξ1−ρσM(t)Y(t)−(d−β∗)e−r(T−t), (79) where β∗=−g(0)e−rT(y−de−rT) g(0)e−2rT −1, (80) and the efficient frontier is given by Var[Y∗(T)] = g(0)e−2rT 1−g(0)e−2rT d−yerT2, (81) which can be also written as E[Y∗(T)] = d=s1−g(0)e−2rT g(0)e−2rT qVar[Y∗(T)] + yerT. (82) Proof. As in Section 4, a two-step procedure is employed to derive the efficient portfolio strategy and the efficient frontier. Since the derivations are similar to those of Theorem 1, we only present key steps here. Risks 2020,8, 70 31 of 34 Using the Feynman-Kac formula gives ∂F ∂t+ [κθ −(κ+2σξρ)v]∂F ∂v+1 2σ2v∂2F ∂v2+ (ξ2 1+1{ρ6=±1}ξ2 2)vF =0, (A1) and ¯ η1(t) = e−2r(T−t)+Rt 0|ϑ(u)|2du ∂F ∂v(t,v(t))σqv(t)ρ, (A2) ¯ η2(t) = e−2r(T−t)+Rt 0|ϑ(u)|2du ∂F ∂v(t,v(t))σqv(t)q1−ρ2. (A3) Note that Equation (A1) is a PDE related to a square-root model. Then the solution of (A1) has an exponential affine-form (see Chapter 9 in Elliott and Kopp 2005). We try the following exponential affine-form solution for F F(t,v) = eK(t)v+L(t), (A4) where K(T) = 0, L(T) = 0. Substituting (A4) into (A1)–(A3) gives (38) and (39) and ¯ η1(t) = ¯ h(t)K(t)σqv(t)ρ, ¯ η2(t) = ¯ h(t)K(t)σqv(t)q1−ρ2. Furthermore, from the relationships (31) and (32) between (¯ h(·) , ¯ η(·)) and (h(·) , η(·)) , we obtain (35)–(37). Proof of Lemma 3.We try K(t) = Q2(τ) Q1(τ), (A5) where Q1(τ) , Q2(τ)∈ C( 0, T ; R) satisfy the initial conditions Q1( 0 ) = 1 and Q2( 0 ) = 0. Then using the product rule and Equation (38), we obtain d dτQ2(τ) = d dτ[K(t)Q1(τ)] =−(κ+2σξρ)Q2(τ) + 1 2σ2K(t)Q2(τ) + (ξ2 1+1{ρ6=±1}ξ2 2)Q1(τ) + K(t)d dτQ1(τ). Comparing the coefficients of K(t)on both sides results in a system of two-coupled ODEs: d dτQ1(τ) = −1 2σ2Q2(τ), (A6) and d dτQ2(τ) = (ξ2 1+1{ρ6=±1}ξ2 2)Q1(τ)−(κ+2σξρ)Q2(τ). (A7) Risks 2020,8, 70 32 of 34 The solution of this system is given by Q(τ) = exp " 0−1 2σ2 ξ2 1+1{ρ6=±1}ξ2 2−(κ+2σξρ)!τ# 1 0!. (A8) Furthermore, substituting (A5) and (A6) into (39) and integrating both sides, we obtain L(t) = −2κθ σ2log |Q1(τ)|. (A9) The matrix exponential in (A8) can be calculated as follows exp " 0−1 2σ2 ξ2 1+1{ρ6=±1}ξ2 2−(κ+2σξρ)!τ# =                          e−(κ+2σξρ)τ 2 cos(δCτ) + κ+2σξρ 2 sin(δCτ) δC−1 2σ2sin(δCτ) δC (ξ2 1+1{ρ6=±1}ξ2 2)sin(δCτ) δCcos(δCτ)−κ+2σξρ 2 sin(δCτ) δC!,∆C<0, e−(κ+2σξρ)τ 2 1+κ+2σξρ 2τ−1 2σ2τ (ξ2 1+1{ρ6=±1}ξ2 2)τ1−κ+2σξρ 2τ!,∆C=0, e−(κ+2σξρ)τ 2 cosh(δCτ) + κ+2σξρ 2 sinh(δCτ) δC−1 2σ2sinh(δCτ) δC (ξ2 1+1{ρ6=±1}ξ2 2)sinh(δCτ) δCcosh(δCτ)−κ+2σξρ 2 sinh(δCτ) δC!,∆C>0. (A10) Combining (A10) with (A5) and (A9) gives the closed-form solutions (40) and (41). Proof of Lemma 6.Applying Itô’s differentiation rule to (62) and substituting (63)–(66) yield dg(t) = ∂g ∂t+κ[θ−v(t)]∂g ∂v+1 2σ2v(t)∂2g ∂v2dt +∂g ∂vσpv(t)ρdW1(t) + p1−ρ2dW2(t) =−2r+ξ2 1v(t)g(t)−2σξ1ρv(t)M(t)g(t) + σ2ρ2v(t)M2(t)g(t)dt +ζ(t)>dW(t) =−2r+ξ2 1v(t)g(t) + 2ξ1pv(t)ζ1(t) + ζ2 1(t) g(t)dt +ζ(t)>dW(t). (A11) This coincides with the BSRE (58). 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