Optimal portfolios with regime switching and value-at-risk constraint

Automatica 46 (2010) 979–989

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Optimal portfolios with regime switching and value-at-risk constraintI Ka-Fai Cedric Yiu a,∗ , Jingzhen Liu a , Tak Kuen Siu b , Wai-Ki Ching c a

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hunghom, Kowloon, Hong Kong, China

b

Department of Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney, NSW 2109, Australia

c

Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, China

article

info

Article history: Received 5 June 2007 Received in revised form 30 June 2009 Accepted 25 February 2010 Available online 18 April 2010 Keywords: Optimal portfolio selection Regime-switching Maximum value-at-risk constraints Dynamic programming Regime-switching HJB equations Utility maximization

abstract We consider the optimal portfolio selection problem subject to a maximum value-at-Risk (MVaR) constraint when the price dynamics of the risky asset are governed by a Markov-modulated geometric Brownian motion (GBM). Here, the market parameters including the market interest rate of a bank account, the appreciation rate and the volatility of the risky asset switch over time according to a continuous-time Markov chain, whose states are interpreted as the states of an economy. The MVaR is defined as the maximum value of the VaRs of the portfolio in a short time duration over different states of the chain. We formulate the problem as a constrained utility maximization problem over a finite time horizon. By utilizing the dynamic programming principle, we shall first derive a regime-switching Hamilton–Jacobi–Bellman (HJB) equation and then a system of coupled HJB equations. We shall employ an efficient numerical method to solve the system of coupled HJB equations for the optimal constrained portfolio. We shall provide numerical results for the sensitivity analysis of the optimal portfolio, the optimal consumption and the VaR level with respect to model parameters. These results are also used to investigating the effect of the switching regimes. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction The optimal portfolio allocation problem is of great importance in finance from both theoretical and practical perspectives. The pioneering work of Markowitz (1952) first provided a mathematically elegant way to formulate the optimal portfolio allocation problem and developed the celebrated mean-variance approach for optimal portfolio allocation. He considers a singleperiod model and adopts variance (or standard deviation) as a measure of risk from the portfolio. The novelty of his meanvariance approach is that it reduces the optimal portfolio allocation problem to the one in which only the mean and the variance of the rate of return are involved under the normality assumption for the rates of return of the risky assets. This greatly simplifies

I The first and second authors are supported by RGC Grant PolyU. 5321/07E and the Research Committee of The Hong Kong Polytechnic University. The third author is supported by the Discovery Grant from the Australian Research Council (ARC), (Project No.: DP1096243). The last author is supported in part by HKRGC Grant No. 7017/07P, HKUCRGC Grants, HKU Strategy Research Theme fund on Computational Sciences, Hung Hing Ying Physical Research Sciences Research Grant. The material in this paper was not presented at any conference. This paper was recommended for publication under the direction of Editor Berç Rüstem. ∗ Corresponding author. Tel.: +852 27666923; fax: +852 23629045. E-mail addresses: [email protected] (K.F.C. Yiu), [email protected] (J. Liu), [email protected] (T.K. Siu), [email protected] (W.-K. Ching).

0005-1098/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2010.02.027

the problem of optimal portfolio allocation and makes a great leap forward to the development of the field. The mean-variance approach by Markowitz has also laid down solid theoretical foundation to the optimal portfolio allocation problem and has opened up an important field, namely, the modern portfolio theory, which, together with risk management and asset pricing, are coined as the three pillars in modern finance. It also provides the theoretical background and motivation for the development of Capital Asset Pricing Model (CAPM) by the seminal work of Sharpe (1964), which laid down the theory of equilibrium asset pricing. Merton (1969, 1971) pioneered the development of the optimal portfolio allocation problem in a continuous-time framework, which provides a more realistic setting to deal with the problem. He explores the state of art of the stochastic optimal control techniques to provide an elegant solution to the optimal portfolio allocation problem. His work has opened up an important field in modern finance, namely, the continuous-time finance. Under the assumption that the returns from the risky assets are stationary (i.e. the coefficients of the dynamics of the returns are constant) and some specific forms of the utility function, Merton derives closed-form solutions to the optimal portfolio allocation in a continuous-time setting. In reality, the returns from the risky assets might not be stationary. So, it would be of practical relevance and importance to consider asset pricing models with non-constant coefficients, which can incorporate the feature of non-stationary returns. Boyle and Yang (1997) considered the

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optimal asset allocation problem in the presence of nonstationary asset returns and transaction costs. They considered the Duffie and Kan (1996) multi-factor stochastic interest model and adopt a viscosity solution approach to deal with the problem. Recently, regime-switching, or Markov-modulated, models have received much attention among both researchers and market practitioners. Hamilton (1989) pioneered the econometric applications of regime-switching models by considering a discrete-time Markov-switching autoregressive time series model. Since then, regime-switching models, both discrete-time and continuoustime, have found a wide range of applications in economics and finance. Some papers on the use of regime-switching models in finance include Elliott and van der Hoek (1997) for asset allocation, Pliska (1997) and Elliott, Hunter, and Jamieson (2001), Elliott and Kopp (2004) for short rate models, Elliott and Hinz (2002) for portfolio analysis and chart analysis, Elliott, Chan, and Siu (2005) and Guo (2001) for option pricing under incomplete markets, Buffington and Elliott (2002a,b) for pricing European and American options, Elliott, Malcolm, and Tsoi (2003) for volatility estimation, Elliott, Siu, and Chan (2006) for valuing options under Markovswitching GARCH models and Elliott, Siu, and Chan (2007) for pricing and hedging variance and volatility swaps, and others. Regime-switching models provide a natural and convenient way to describe the impact of the structural changes in (macro)-economic conditions and business cycles on the price dynamics. They provided a pertinent way to describe the non-stationary feature of returns of risky assets. More recently, Yin and Zhou (2003) and Zhou and Yin (2004) established a mean-variance portfolio selection problem under Markovian regime-switching models in a continuous-time economy. They introduced the stochastic linearquadratic control to deal with the problem and established closedform solutions to mean-variance efficient portfolios and efficient frontiers. Maximizing profits by choosing the optimal portfolio allocation rule is an important issue. Another equally important issue is to control the risk we are taking. This raises the issues of risk measurement and management. Some recent financial crises, such as the Asian financial crisis, the collapse of Long-Term Capital Management (LTCM), and the turmoil at Barings and Orange Country, point out the importance of appropriate practice of risk measurement and management. Various methods and techniques for measuring, managing and controlling risk have been proposed in the literature that cater for the practical needs of regulators, central bankers and market traders. Value at Risk (VaR) has emerged as an important and popular tool for risk measurement. It has widely been adopted in the finance industries. VaR describes the extreme loss from a portfolio at a certain (small) probability level over a fixed time horizon, say 1 day or 7 days. Technically, it can be defined as a quantile of the profit/loss distribution of a portfolio. For example, if the VaR of the portfolio at a 1% probability level over the next trading day is £2 millions, it is expected that the actual loss from the portfolio over the next trading day is at least £2 millions with 1% probability. Duffie and Pan (1997) and Jorion (2001) provided excellent introduction and survey to VaR. In the original work of Merton, the expected utility of wealth or consumption was maximized over a fixed time horizon without imposing any risk limits or constraints. However, maximizing profits is not the only objective that needs to be taken into account in the optimal portfolio allocation problem. It is also important to control or limit the amount of risk being taken. Several approaches have been introduced in the literature to investigate the mean-VaR optimization, in which VaR is used as a measure of risk and a VaR constraint is imposed in the problem of maximizing the mean rate of return of the portfolio. Some of the works along this direction include Alexander and Baptista (1999), Kast, Luciano, and Peccati (1999) and Kluppelberg

and Korn (1997). These studies were conducted in a static setting. Recently, the formulation of the problem in a continuoustime was introduced by Basak and Shapiro (2001) and Luciano (1998). Both of them considered the optimal portfolio allocation problem by maximizing the utility function of an economic agent with the VaR constraint. Luciano (1998) provided analysis on derivations from the VaR constraint instead of explicitly applying the constraint to the optimal portfolio allocation problem while Basak and Shapiro (2001) imposed the VaR constraint at one point in time to investigate trading between recalculated VaRs. Cuoco, He, and Issaenko (2001) employed the martingale approach to study optimal dynamic trading strategies with VaR constraints. They considered the case that the price dynamics of the risky asset are governed by a geometric Brownian motion. Yiu (2004) imposed the VaR as a dynamic constraint. To make the calculations tractable, he calculated the constraints abstracting from withininterval trading and from considerations of backtesting. His approach applied the VaR constraint over time and stresses the repeated recalculations of the VaRs. It also described how the VaR affects the investment decision dynamically. However, in his framework, the returns from the risky assets are assumed to be stationary. The empirical studies show that the traditional log normal model can not catch the extreme stock movement and stock variability in the variance model. The switching behavior of the economic states can be due to the structural changes in economic conditions and business cycles. There can be substantial fluctuations in economic variables, which affect the dynamics of the market values of the assets, over a long period of time. Hence, it is of practical importance and relevance to incorporate the switching behavior of the economic states in modelling the dynamics of the market values of the assets for investment and risk management. Gabih, Sass, and Wunderlich (2005) considered the utility maximization problem with shortfall risk constraints when the dynamics of the stock returns are modulated by a continuoustime, finite-state hidden Markov chain. They employed the separation principle to separate the control problem or the utility maximization problem and the filtering problem of the hidden Markov chain. Gundel and Weber (2008) obtained closedform solution to an utility maximization problem under a joint budget and downside risk constraint, where the risk constraint is specified by a class of convex risk measures proposed in Föllmer and Schied (2002) and Frittelli and Rosazza Gianin (2002). They considered a general semi-martingale framework for the asset price dynamics and developed the closed-form solution based on the martingale approach for constrained maximization problems. Sotomayor and Cadenillas (2009) considered an optimal consumption and investment problem with the bankruptcy constraint under a Markovian regime-switching model for the asset price dynamics. They determined a consumption-investment policy so as to maximize the expected total discounted utility of consumption until bankruptcy and employed techniques of classical stochastic optimal control to derive the regime-switching Hamilton–Jacobi–Bellman equation. They were able to obtain explicit solutions to the problem for some HARA utility functions. In this paper, we consider the optimal portfolio selection problem subject to the MVaR constraint when the price dynamics of the risky asset are governed by a Markov-modulated geometric Brownian motion (GBM). In particular, the market parameters including the market interest rate of a bank account, the appreciation rate and the volatility of the risky asset switch over time according to a continuous-time Markov chain, whose states are interpreted as the states of an economy. The MVaR is defined as the maximum value of the VaRs of the portfolio in a short time duration over different states of the chain. We formulate the problem as a constrained utility maximization problem over a finite time horizon. By utilizing the optimal control theory (see, for

K.F.C. Yiu et al. / Automatica 46 (2010) 979–989

example, Ahmed and Teo (1981) and Teo, Reid, and Boyd (1980)) and the dynamic programming principle, we shall derive a regimeswitching Hamilton–Jacobi–Bellman (HJB) equation. Our work is different from those in the existing literature in four major aspects. Firstly, we consider the use of the dynamic programming approach together with the method of static constrained optimization approach to deal with the constrained optimal portfolio problem while most of the literature considered the use of the martingale approach for the constrained optimization problem. Secondly, we consider the case that both the regime-switching effect and the VaR constraint are present while most of the aforementioned literature, except Gabih et al. (2005), considered the case that the regime-switching effect is absent. Thirdly, most of the literature mentioned above did not consider the consumption process while we consider the optimal consumption and investment problem. Indeed, the proposed approach seems to be a convenient method to deal with the optimal consumption and investment problem with risk constraints. Lastly, the constraint we considered here is the maximum VaR over different states of an economy. This means that the optimal consumption and investment results developed here are uniformly optimal over different states of the economy described by the chain. In other words, our method here can provide a conservative and prudent approach to determine the optimal consumption and investment with risk constraints. Most of the literature considered the case that the VaR does not depend on the states of the economy. Our work is different from Gabih et al. (2005). Firstly, we consider an observable Markov chain while Gabih et al. (2005) assumed that the Markov chain is hidden. Secondly, the approach to deal with the risk constraint here is different from that employed in Gabih et al. (2005). Thirdly, we consider the maximum VaR constraint here, which is different from the constraint considered in Gabih et al. (2005). The work in this paper is also different from that of Sotomayor and Cadenillas (2009). The methodology and risk constraints used in the two papers are different. We shall propose an efficient numerical method to solve the regime-switching HJB equation and the optimal constrained portfolio. We shall provide numerical results for the sensitivity analysis of the optimal portfolio, the optimal consumption and the VaR level with respect to model parameters. These results are also used to investigate the effect of the switching regimes. The introduction of the regime-switching effect to the optimal consumption and investment problem with VaR constraints is important for at least three reasons. From a statistical perspective, the Markovian regime-switching model can describe and explain some important ‘‘stylized’’ features of financial time series, namely, time-varying conditional volatility, the heavy-tailedness of unconditional distribution of returns, regime switchings and nonlinearity. So, it provides a more realistic description to the behavior of asset returns than its constantcoefficient counterpart. Neglecting the regime-switching effect may lead to underestimation of the risk of a portfolio and suboptimal results for consumption and investment. It is especially important to incorporate the regime-switching effect when the decision horizon of the optimal consumption and investment problem is long, say 30–40 years, since there could be substantial changes in the economic condition over a long period of time. For example, one may consider the asset allocation problem of a pension fund. From an economic perspective, the Markovian regime-switching model can describe the stochastic evolution of investment opportunity sets due to structural changes in the state of the economy. This important economic feature cannot be captured by a constant-coefficient model. This paper is structured as follows. In Section 2, we shall describe the price dynamics of the model and formulate the constrained optimal portfolio problem. We shall derive the regimeswitching HJB equation using the dynamic programming principle

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and present the method of the Lagrange multiplier to deal with the MVaR constraint in Section 3. The results of the numerical experiments will be presented and discussed in Section 4. The final section summarizes the paper. 2. Price dynamics and optimization In this section, we shall consider a continuous-time financial model consisting of a bank account B and a risky asset S, that are tradable continuously over a finite time horizon [0, T ], where T ∈ (0, ∞). First, we fix a complete probability space (Ω , F , P ), where P is a real-world probability. Denote X := {X (t )}t ∈T as a continuous-time, finite-state Markov chain with state space X := (x1 , x2 , . . . , xN ). The states of X are interpreted as different states of an economy. Following Elliott, Aggoun, and Moore (1994), we shall represent the state X as a finite set of unit vectors E := {e1 , e2 , . . . , eN }, where ei = (0, . . . , 1, . . . , 0)0 ∈
t

Z

QX (s)ds + M (t ),

(2.1)

0

where {M (t )}t ∈T is an
(2.2)

where h·, ·i denote an inner product in 0 (i = 1, 2, . . . , N). Then, the price process of the bank account B is governed by B(t ) = exp

t

Z



r (s)ds ,

B(0) = 1.

(2.3)

0

Suppose µ(t ) and σ (t ) denote the appreciation rate and the volatility rate of the risky asset at time t, respectively. We suppose that

µ(t ) := µ(t , X (t )) = hµ, X (t )i , σ (t ) := σ (t , X (t )) = hσ , X (t )i ,

(2.4)

µ := (µ1 , µ2 , . . . , µN )0 ∈ ri and σi > 0, for each i = 1, 2, . . . , N. Let W (t ) denote a standard Brownian motion on (Ω , F , P ) with respect to {F (t )}t ∈T , the P -augmentation of the natural filtration generated by {W (t )}t ∈T . The price process {S (t )}t ∈T of the risky asset S is assumed to be dS (t ) = S (t ) (µ(t )dt + σ (t )dW (t )) ,

S (0) = s.

(2.5)

For each t ∈ T , write G(t ) := F (t ) ∨ F (t ), an enlarged information set generated by F X (t ) and F (t ). We assume that the market agent can invest into the money market account and the risky asset S. Let π := {π (t )}t ∈T denote a G-adapted portfolio process, where π (t ) := π (t , ω) denotes the amount of wealth allocated to the risky asset S at time t. Let {c (t , ω)}t ∈T denote a non-negative G-adapted process, where c (t ) := c (t , ω) represents the consumption rate of an economic agent at time t. Define u(t ) := (c (t ), π (t )) to be our control process. We suppose that {u(t )}t ∈T is a G(t )-measurable adapted process such that X

t

Z

(π (s)2 + c (s))ds < ∞, 0

t ≥ 0 P -a.s.

(2.6)

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Suppose {V (t )}t ∈T denotes the wealth process of the economic agent with initial wealth V (0) = v > 0 and initial state X (0) = ei ∈ E . Then, dV (t ) = [π (t )(µ(t ) − r (t )) + r (t )V (t ) − c (t )]dt

+ π (t )σ (t )dW (t ).

(2.7)

Let U (·, ·) : T × [0, ∞) → < denote a utility function such that for each t ∈ T , U (t , ·) is C 2 (0, ∞), strictly increasing and strictly concave. limc →∞ U 0 (t , c ) = 0 and limc →0 U 0 (t , c ) = ∞. Let ρ denote an impatient factor or a discount factor, where ρ is a positive constant. Suppose the economic agent needs continuous consumption over the time horizon [0, T ] and he/she uses the portfolio process π (t ) available at the initial wealth v0 and given that the initial state of the economy X (0) = x. Then, the expected discounted utility of the agent is defined as J (v0 , x; u(·)) := E

T

Z



e−ρ t U (t , c (t ))dt |V (0) = v0 , X (0) = x . 0

(2.8) We call the control process u(·) := (c (·), π (·)) an admissible control process if π (·) is a portfolio process and c (·) is a consumption rate process. The set of all admissible control processes is A. The unconstrained optimal portfolio/consumption problem of the economic agent with initial wealth v0 and given that X (0) = x is max J (v0 , x; u(·)),

(2.9)

u(·)∈A

and the conditional variance of V (t + h) given G(t ) and X (t ) = ei is Var[V (t + h)|G(t ), X (t ) = ei ] =

∆i V (t , h) := V (t ) − e−ri h V (t + h) ∀i = 1, 2, . . . , N , (2.16) which, conditional on G(t ), can be viewed as a function of the random variable V (t + h). Then, under P , the VaR of the portfolio VaR(t , h, i, β) with the probability level β over the time interval [t , t + h] given G(t ) and X (t ) = ei is defined as P (∆i V (t , h) ≥ VaR(t , h, i, β)|G(t ), X (t ) = ei ) = β. (2.17) Since ∆i V (t , h) is normally distributed conditional on G(t ) and X (t ) = ei , it can be shown that VaR(t , h, i, β) = −θi (t )(1 − eri h ) s σ 2 π 2 (t ) − Φ −1 (β) i (1 − e−2αi h ) (2.18) 2αi which depends on the portfolio decision π (t ) we made at time t. Let

s a1i := −Φ −1 (β)



dV (t ) = [π (t )(µ(t ) − r (t )) + r (t )V (t ) − c (t )]dt (2.10)

bi :=

In the sequel, we shall present the MVaR constraint of the problem. First, we assume that the portfolio is adjusted frequently so that the interval from t to t + h is small, where h > 0. It is reasonable to assume that there is no trading in between constraint reevaluation and that the consumption is approximately constant in the small time interval [t , t + h] since in reality, one can only adjust the portfolio discretely over time and the portfolio/consumption decision is made at the beginning of the time horizon. That is, π(τ ) = π (t ) and c (τ ) = c (t ), for all τ ∈ [t , t + h]. We also assume that there is no regime switching in the small time interval. In other words, X (τ ) = X (t ), for all τ ∈ [t , t + h]. First, we define the following three quantities:

Then,

+ π (t )σ (t )dW (t ).

αi := −ri π (t )(µi − ri ) − c (t ) , −ri Y (τ ) := eαi τ V (τ ), i = 1, 2, . . . , N , τ ∈ [t , t + h]. θi (τ ) := θi (t ) =

(2.11)

+

e

αi τ

π (t )σi dW (τ ).

2ri

µi − r i



ri

, (2.19)

(eri h − 1),

(eri h − 1).

VaR(t , h, i, β) = a1i |π (t )|σi + a2i π (t ) + bi c (t ), i = 1, . . . , N .

(2.20)

We define the MVaR of the portfolio with the probability level β over the time horizon [t , t + h] given G(t ) as MVaR(t , h, β) =

max

i=1,2,...,N

VaR(t , h, i, β).

(2.21)

Then, the constraint of restricting the MVaR at the level R is MVaR(t , h, β) ≤ R.

(2.22)

This is equivalent to the following N constraints: a1i |π (t )|σi + a2i π (t ) + bi c (t ) ≤ R,

i = 1, 2, . . . , N .

(2.23)

(a1i σi + a2i )π (t ) + bi c (t ) ≤ R,

i = 1, 2, . . . , N .

(2.24)

We shall have an upper bound to constrain the investment in the risky asset S. That is, π (t ) is bounded above. The constraint on the consumption will be imposed if R is small. The portfolio optimization problem with the MVaR constraint is then formulated as

Y (t + h) − Y (t ) = θi (t )(eαi (t +h) − eαi t ) t +h

1 ri

e2ri h − 1

For simplicity, assume there is no short selling. We have

Then, in the small time interval [t , t + h],

Z

(2.15)

Now, we define the discounted net loss of the portfolio over the time interval [t , t + h] as

a2i := −

subject to

σi2 π 2 (t ) (1 − e−2αi h ). 2αi

Z (2.12)

t

max E

π (t ),c (t )

T



U (t , c (t ))dt ,

(2.25)

0

This then implies that

subject to:

V (t + h) = e

dV (t ) = [π (t )(µ(t ) − r (t )) + r (t )V (t ) − c (t )]dt + π (t )σ (t )dW (t ),

(2.26)

(a1i σi + a2i )π (t ) + bi c (t ) ≤ R,

(2.27)

−αi h

(V (t ) − θi (t )) + θi (t ) t +h

Z +

e−αi (t +h−τ ) π (t )σi dW (τ ),

(2.13)

t

which is an Ornstein–Uhlenbeck process with a negative meanreverting parameter αi . The conditional mean of V (t + h) given G(t ) and X (t ) = ei under the measure P is E [V (t + h)|G(t ), X (t ) = ei ] = θ (t ) + e−αi h (V (t ) − θi (t )),

(2.14)

i = 1, 2, . . . , N .

3. Regime-switching HJB equation and the optimality conditions In this section, we shall derive a regime-switching HJB equation for the value function described in the last section. We shall also

K.F.C. Yiu et al. / Automatica 46 (2010) 979–989

derive a system of coupled HJB equations corresponding to the regime-switching HJB equation. We shall assume that the control process u is Markovian with respect to G. That is, u(t ) = u(t , V (t ), X (t )). For simplicity of notation, we use π (t ) to symbolize π (t , v, x) and c (t ) to symbolize c (t , v, x) in the following for some equations, with V (t ) = v and X (t ) = x. In many cases, it suffices to consider Markovian control processes. First, we consider the following optimal value function given V (0) = v and X (0) = x: V (t , v, x) := sup E π,c

T

Z



U (τ , c (τ ))dτ |V (t ) = v, X (t ) = x . (3.1) t

solve for the HJB equation even without the regime-switching effect and the risk constraint (see, for example, Huang, Wang, and Teo (2000), Huang, Wang, and Teo (2004), Wang, Jennings, and Teo (2003), Wang, Yang, and Teo (2006), Zhang, Yang, and Teo (2006)). In the sequel, we shall consider the situation when there are two states in the Markov chain and the agent has a power utility function. We shall illustrate how to simplify the above system of HJB equations in this situation. First, we assume that the rate matrix Q of the chain is

 Q =

−p p



p , −p

(3.6)

where p is a positive real constant. We then consider the following power utility function for consumption: U (t , c (t , v, x)) := e−δ t c (t , v, x)γ ,

Let F (t , v, x, π (t ), c (t )) = π (t )(µ(t ) − r (t )) + r (t )v − c (t ),

(3.2)

and G(t , v, x, π (t ), c (t )) = π 2 (t )σ 2 (t ).

(3.3)

Let Vi := V (t , v, ei ), for each i = 1, 2, . . . , N, and V := (V1 , V2 , . . . , VN ). Then, by the principle of dynamic programming in the stochastic optimal control, it can be shown that the value function V satisfies the following regime-switching HJB equation:

" ∂V ∂V + sup U (t , c (t )) + F (t , v, X (t ), π (t ), c (t )) ∂t ∂v π,c # ∂ 2V 1 + G(t , v, X (t ), π (t ), c (t )) 2 + hV, QX (t )i = 0, 2 ∂v

δ > 0, 0 < γ < 1.

(3.7)

Here, δ represents an impatient factor for consumption and it is assumed to be a positive constant. In this case, the value functions for the two economic states satisfy the following pair of coupled HJB equations:

∂ V1 γ + e−δt copt (t , v, e1 ) + [πopt (t , v, e1 )(µ1 − r1 ) ∂t ∂ V1 1 2 ∂ 2 V1 + r1 v − copt (t , v, e1 )] + πopt (t , v, e1 )σ12 2 ∂v 2 ∂v + p(V2 − V1 ) = 0,

(3.8)

and (3.4)

with terminal and boundary conditions: V (T , v, x) = 0,

∂ V2 γ + e−δt copt (t , v, e2 ) + [πopt (t , v, e2 )(µ2 − r2 ) ∂t ∂ V2 1 2 ∂ 2 V2 + r2 v − copt (t , v, e2 )] + πopt (t , v, e2 )σ22 2 ∂v 2 ∂v + p(V1 − V2 ) = 0.

(3.9)

Following the approach in Merton (1971), we assume that the value function is of the following form:

and V (t , 0, x) = 0,

Vi = V (t , v, ei ) = e−δ t hi (t , v)v γ ,

subject to a set of N VaR constraints:

(a1i σi + a2i )π (t ) + bi c (t ) ≤ R,

i = 1, 2, . . . , N .

Hence, the vector V of the value functions at different regimes satisfies the following system of coupled HJB equations:

" ∂ Vi ∂ Vi + sup U (t , c (t )) + F (t , v, ei , π (t ), c (t )) ∂t ∂v π,c # 1 ∂ 2 Vi + G(t , v, ei , π (t ), c (t )) 2 + hV, Qei i = 0, 2 ∂v

(3.5)

with terminal and boundary conditions: Vi (T , v, ei ) = 0,

i = 1, 2.

(3.10)

This form is in line with the form of the power utility function. As in Yiu (2004), we neglect the derivatives of hi (i = 1, 2) with respect to v and obtain

∂ Vi = γ e−δt hi (t , v)v γ −1 , ∂v ∂ 2 Vi = γ (γ − 1)e−δt hi (t , v)v γ −2 , ∂v 2

(3.11) (3.12)

and

∂ Vi = e−δt h0i (t , v)v γ − δ e−δt hi (t , v)v γ , ∂t

i = 1, 2

(3.13)

where h0i represents the derivative of hi with respect to t. For each i = 1, 2, we define

and Vi (t , 0, ei ) = 0,

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i = 1, 2, . . . , N ,

subject to a set of N VaR constraints:

(a1i σi + a2i )π (t ) + bi c (t ) ≤ R,

i = 1, 2, . . . , N .

Due to the non-linearity in copt (t , v, ei ) and πopt (t , v, ei ) and the presence of the regime-switching effect, the first-order conditions and the system of the coupled HJB equations are highly nonlinear. We shall employ some numerical methods to solve for πopt (t , v, ei ), copt (t , v, ei ) and Vi (i = 1, 2, . . . , N ). Indeed, except for some simple cases, numerical methods are often required to

 πopt (t , v, ei ) Ai (πopt (t , v, ei ), v)hi (t , v) = γ (µi − ri ) + ri v ! 2 1 πopt (t , v, ei ) + σi2 γ (γ − 1) − δ − p, (3.14) 2 v2 

and Bi (πopt (t , v, ei ), hi (t , v))

=

copt (t , v, ei )

vγ

−

γ hi (t , v)copt (t , v, ei ) . v

(3.15)

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K.F.C. Yiu et al. / Automatica 46 (2010) 979–989

Substituting (3.11)–(3.13) into (3.8) and (3.9), we obtain

4.1. The iterative algorithm

h01 (t , v) + A1 (πopt (t , v, e1 ), v)h1 (t , v)

+ B1 (copt (t , v, e1 ), h1 (t , v)) + ph2 (t , v) = 0,

(3.16)

and h02 (t , v) + A2 (πopt (t , v, e2 ), v)h2 (t , v)

+ B2 (copt (t , v, e2 ), h2 (t , v)) + ph1 (t , v) = 0,

(3.17)

with terminal conditions hi (T , v) = 0, for i = 1, 2. γ Let ψ := − 1−γ and gi (t , v) := (hi (t , v))1−ψ , for each i = 1, 2. Then, we obtain the following system of coupled ordinary differential equations (O.D.E.s) for gi (i = 1, 2): g10 (t , v) + A1 (πopt (t , v, e1 ), v)g1 (t , v)

+ (1 − ψ)pg2 (v, t ) g1 (v, t ) = 0, g2 (t , v) + A2 (πopt (t , v, e2 ), v)g2 (t , v)

(3.18)

0

+ (1 − ψ)B2 (copt (t , v, e2 ), g2 (t , v) + (1 − ψ)pg1 (t , v)

1−γ

−1/(1−γ )

(k)

γ

1−γ

(0)

Step I: For each i = 1, 2, the initial values πopt (t , v, ei ) = and copt (t , v, ei ) = v hi

+ (1 − ψ)B1 (copt (t , v, e1 ), g1 (t , v)1−γ ) 1−γ

In this subsection, we shall present the iterative algorithm and specify some specimen values of the model parameters in our implementation. In the iterative algorithm, we use the unconstrained solution as an initial guess. We divide the domain of the computation into a grid of Nt × Nv mesh points, where Nt and Nv represent the number of mesh points in the space and the time domains, respectively. The steps in the iterative algorithm are presented as follows: For each n = Nt − 1, . . . , 0, v = [0, ∆v, . . . , Nv ∆v] and t = [(Nt − 1)∆t , . . . , ∆t , 0]:

(0) g1,n

and

(0)

(0) g2,n

(0, v) (i = 1, 2).

are computed from the following two equations:

(0)

(0)

g1,n = g1,n+1 + ∆t (1 − ψ)A1 g1,n+1 + ∆t (1 − ψ)B1

)

γ

g2 (t , v) = 0,

+ (1 − ψ)p(g2(0,n)+1 )1−γ (g1(0,n)+1 )γ ,

(3.19)

with the terminal conditions gi (T , v) = 0, for each i = 1, 2. For the case when there is no MVaR constraint, gi (i = 1, 2) satisfy the following system of coupled ordinary differential equations: g10 (t ) + (1 − ψ)A1 g1 (t ) + (1 − ψ)B1

+ (1 − ψ)pg2 (t )1−γ g1 (t )γ = 0,

(3.20)

(4.1)

and (0)

(0)

(0)

g2,n = g2,n+1 + ∆t (1 − ψ)A2 g2,n+1 + ∆t (1 − ψ)B2

+ (1 − ψ)p(g1(0,n)+1 )1−γ (g2(0,n)+1 )γ .

(4.2)

Step II: With the risk constraint, according to (3.5), we look for π and c from

and

"

g2 (t ) + (1 − ψ)A2 g2 (t ) + (1 − ψ)B2 0

π ,c

(3.21) 1

where Ai = γ

∂ Vi ∂v #

max U (t , c (t , v, ei )) + F (t , v, ei , π (t , v, ei ), c (t , v, ei ))

+ (1 − ψ)pg1 (t )1−γ g2 (t )γ = 0, 

(µi − ri )2 1 (µi − ri )2 +r + 2 2 σi2 (1 − γ ) σi (1 − γ )

(µi −ri )v σi2 (1−γ )

+ G(t , v, ei , π (t , v, ei ), c (t , v, ei )) 

2

− δ − p,

(3.22)

and Bi = 1 − γ , for i = 1, 2.

(k)

(k)

(k)

(k)

(k)

Un (c1 ) + Fn (v, π1 , c1 )

sup (k) (k)

In this section, we shall conduct numerical experiments to provide sensitivity analysis for the optimal portfolio, the optimal consumption and the VaR level arising from the Markovmodulated model when the model parameters vary. We shall identify the model parameters that have significant effects on the optimal portfolio, the optimal consumption and the VaR level. Here, we also make comparisons of the qualitative behaviors of the optimal portfolio, the optimal consumption and the VaR level obtained from our model (Model I) to those arising from the model without switching regimes (Model II). For each of the comparisons, we vary the specimen value of one parameter of Model I and keep the other parameters of Model I being fixed. We also assume that the specimen values of Model II are the same as those of Model I, except for the varying parameter in Model I. If the value of the varying parameter in Model I is identical to the value of its counterpart in Model II, Model I and Model II are identical to each other. In this fashion, we can perform both the sensitivity analysis and the comparison between Model I and Model II at the same time. For illustration, we consider the situation when there are two states in the Markov chain in Model I, and, so we have a pair of coupled HJB equations for the optimal investment and consumption problem under Model I. Here, we assume that State 1 and State 2 of the chain X represent a Economy 1 (E1 ) and Economy 2 (E2 ), respectively. We shall solve this pair of coupled HJB equations numerically by employing an iterative algorithm.

(4.3)

That is, for k > 1, πopt (t , v, ei ) and copt (t , v, ei ) (i = 1, 2) are solved from

" 4. Numerical experiments and discussions

∂ 2 Vi + hV, Qei i . ∂v 2

π 1 , c1

1

(k)

(k)

+ Gn (π1 , c1 )

∂ 2 V1(,kn−1) ∂v 2

2

"

(k)

(k)

−

(k−1) pV1,n

(k)

Un (c2 ) + Fn (v, π2 , c2 )

sup (k) (k)

π 2 , c2

1

(k)

(k)

+ Gn (π2 , c2 )

∂ 2 V2(,kn−1) ∂v 2

2

−

∂ V1(k−1) ∂v +

(k−1) pV2,n

# ,

(4.4)

∂ V2(k−1) ∂v

(k−1) pV2,n

+

(k−1) pV1,n

# .

(4.5)

Here (k−1)

V i ,n

= e−δt h(i,kn−1) (t , v)v γ = e−δt gi(,kn−1) (t , v)1−γ v γ (k)

i = 1, 2.

(4.6)

(k)

Also, we compute g1,n and g2,n from the following equations recursively: (k)

(k)

(k)

(k)

g1,n = g1,n+1 + ∆t (1 − ψ)A1 (πopt (t , v, e1 ), v)g1,n+1

  (k) 1) + ∆t (1 − ψ)B1 copt (t , v, e1 ), h1(k,− n+1  1−γ  γ (k−1) + ∆tp(1 − ψ) g2(k,n−+11) g1,n+1 ,

(4.7)

K.F.C. Yiu et al. / Automatica 46 (2010) 979–989

and

588.4

(k)

588. 2

(k)

(k)

(k)

g2,n = g2,n+1 + ∆t (1 − ψ)A1 (πopt (t , v, ei )g2,n+1 ) (k−1) copt (t , v, e2 ), h2,n+1

 1−γ  γ (k−1) + ∆tp(1 − ψ) g1(k,n−+11) g2,n+1 .

v=800,t=0.2

588



587. 8

(4.8)

Step III: Return to Step II with k = k + 1 until kV (k−1) − V (k) k1 <  . Here, we implement the above iterative algorithm by Matlab. We consider some hypothetical values for the model parameters and assume that T = 20 years, v = 1000, Nv = 500, Nt = 1000, δ = 0.2, γ = 0.5, r1 = 0.1, µ1 = 0.2 and σ1 = 0.5. The maximum loss is limited not to above R = 100 with probability k = 0.99. Indeed, we found that the optimal results are robust with respect to R. These hypothetical values are similar to those used in Yiu (2004). When the interest rate r2 of the bond, the appreciation rate µ2 and the volatility rate σ2 of the stock in E2 are identical to their corresponding values in E1 , Model I and Model II are identical to each other. In this case, the numerical results for the optimal investment, the optimal consumption and the VaR level obtained from Model I are the same as those arising from Model II no matter what the value of the parameter p in the rate matrix of the chain is. We find that the optimal results are robust with respect to the change in the value of p. By varying one parameter at one time, we can perform the sensitivity analysis for the optimal investment, the optimal consumption and the VaR level with respect to that particular parameter and also make comparison between Model I and Model II. Here, we further assume that the parameter p in the rate matrix is 0.5 and focus on how the optimal investment, the optimal consumption and the VaR level change when the market parameters r2 , µ2 and σ2 vary.

587. 6

π1

(k)

587. 4 587. 2 587 586. 8 586. 6 0.5

0.7

0.8

0.9

1

1.1

1.2

Fig. 4.2.1. The optimal investment (E1 ) against the volatility σ2 . for t=14 1200 σ

1000

σ

800 600 400 200 0 0

200

400

600

800

1000

v

Fig. 4.2.2. The optimal investment (E2 ) against the portfolio value v for different

σ2 .

for t=14

4.2. The effect of σ2

100 90 80 70 60 VaR 2

Here, we shall focus on the effect of the volatility σ2 of the stock in E2 on the optimal investment, the optimal consumption and the VaR level. When σ1 > (<)σ2 , E1 is said to be a ‘‘Bad’’ (‘‘Good’’) economy relative to E2 . In this case, Model I and Model II are different from each other. When σ1 = σ2 (E1 and E2 coincide), there is no switching regime and Model I is identical to model II. Fig. 4.2.1 plots π1 against the volatility in regime E2 . It can be seen that when the MVaR constraint is active, the current π1 decreases as σ2 increases. Figs. 4.2.2 and 4.2.3 depict the plots of the optimal proportion of investment in the stock, the VaR level, respectively, against the optimal portfolio value and t = 14 years. The plots compared when there is no switching (σ2 = 0.5) and when the state switches to E2 . From Fig. 4.2.2, we see that when the Markov chain is changed to E2 , the optimal investment in the stock under Model II decreases significantly as σ2 increases. The result here also reflects that the regime switching in volatility has a significant impact on the optimal investment. The VaR level decreases substantially when σ2 increases. This reflects that the agent does not prefer to take the risk by investing in the stock when the volatility σ2 is high. Figs. 4.2.4 and 4.2.5 plots the consumption level in E1 and E2 , respectively, when the volatility in E2 varies. Fig. 4.2.4 shows that the current consumption level c1 in E1 is affected by σ2 which is in the other regime. Fig. 4.2.5 shows how c2 is affected by σ2 in the same regime. From both Figs. 4.2.4 and 4.2.5 we see that the change in volatility in one regime has a common effect on both regimes. That is, the agent will consume more to increase his utility when the volatility in either regime increases.

0.6

σ2

π2

+ ∆t (1 − ψ)B2



985

50 40 30 20 10 0

0

200

400

600

800

1000

v

Fig. 4.2.3. The VaR level (E2 ) against the portfolio value v for different σ2 .

4.3. The effect of µ2 In this subsection, we investigate the impact of the appreciation rate µ2 of the stock in State 2 on the optimal investment, the optimal consumption and the VaR level. When µ1 > (<)µ2 , E1 is said to be a ‘‘Good’’ (‘‘Bad’’) economy relative to E2 . In this case, Model I and Model II are different from each other. When µ1 = µ2 , E1 and E2 coincide, and Model I and Model II are identical. With the imposed MVaR constraint, Fig. 4.3.1 plots π1 against µ2 in regime E2 . It can be seen that when the MVaR constraint is active, π1 increases with µ2 . Figs. 4.3.2 and 4.3.3 depict the plots of the optimal proportion of investment in the stock and the VaR level, respectively, against the optimal portfolio value for different values of µ2 and t = 14 years. In these figures, we compared the cases when there is no switching regime and when the state

986

K.F.C. Yiu et al. / Automatica 46 (2010) 979–989 for t=14

for t=14

350

700 600

250

500

200

400 π

2

c1

300

150

300

100

200

50

100

0

0

200

400

600

800

0

1000

0

200

400

v

600

800

1000

v

Fig. 4.2.4. The optimal consumption (E1 ) against the portfolio value v for different σ2 .

Fig. 4.3.2. The optimal investment (E2 ) against the portfolio value v for different

µ2 .

for t=14 350

for t =14

120

300 100 250 80 VaR

c2

2

200 150

60

100

40

50

20

0

0

200

400

600

800

0

1000

0

v

200

400

600

800

1000

v

Fig. 4.2.5. The optimal consumption (E2 ) against the portfolio value v for different σ2 .

Fig. 4.3.3. The optimal VaR (E2 ) against the portfolio value v for different µ2 . for t=14

610 350 605

300 250

600

c

1

π

1

200 595

150 100

590

50 585 0.1

0.2

0.3

0.4

0.5

μ2

0.6

0.7

0.8

0 0

200

400

600

800

1000

v

Fig. 4.3.1. The optimal investment (E1 ) against µ2 .

switches to E2 . From Fig. 4.3.2, the effect of µ2 on the qualitative behavior of the optimal investment against the portfolio value v is significant. The optimal investment in the stock increases as µ2 increases. Also, for the cases when µ2 = 0.2 and µ2 = 0.5, there is a critical point at which there is a reversal of the optimal investment behavior from increasing to decreasing. From Fig. 4.3.3, the qualitative behavior of the VaR level changes significantly as µ2 vary. In particular, an increase in µ2 shifts the curve of the VaR level against the portfolio value v upwards. This reflects that when the appreciation rate µ2 of the stock is higher, the economic agent is more willing to take higher risk by investing more in the stock. Figs. 4.3.4 and 4.3.5 plots the consumption level in E1 and E2 , respectively, when µ2 in E2 varies. Both Figures show that the optimal consumptions against the portfolio value v shift

Fig. 4.3.4. The optimal consumption (E1 ) against the portfolio value v for different µ2 .

downward when µ2 increases. This reveals that the agent consumes less and invests more in the stock when the appreciation rate of the stock is higher. 4.4. The effect of r2 We shall consider the impact of the interest rate r2 of the bond on the optimal investment, the optimal consumption and the VaR level. When r1 > (<)r2 , E1 is said to be a ‘‘Good’’ (‘‘Bad’’) economy relative to E2 . In this case, Model I and Model II are different from each other. When r1 = r2 , E1 and E2 coincide, and Model I and Model II are identical. With the imposed MVaR constraint,

K.F.C. Yiu et al. / Automatica 46 (2010) 979–989

987

for t=14

350

for t=14 100 90

300

80 250

70 60

c2

VaR

2

200 150

50 40

100

30 20

50

10 0 0

200

400

600

800

0

1000

0

200

400

600

v

800

1000

v

Fig. 4.3.5. The optimal consumption (E2 ) against the portfolio value v for different µ2 .

Fig. 4.4.3. The VaR level (E2 ) against the portfolio value v for different r2 .

for t=14

592 350 300

590

250

589

200 c1

π

591

150

588

100 587 50 586 0

0.05

0.1

0.15

0

0.2

0

r2

200

400

600

800

1000

v

Fig. 4.4.1. The optimal investment (E1 ) against the interest rate r2 .

Fig. 4.4.4. The optimal consumption (E1 ) against the portfolio value v for different r2 .

for t=14

600

for t=14

350

500

300

400

π

2

250 300 200 c

200

150 100

100

0 0

200

400

600

800

1000

50

v 0

Fig. 4.4.2. The optimal investment (E2 ) against the portfolio value v for different r2 .

Fig. 4.4.1 plots π1 against µ2 . The figure shows that when the MVaR constraint is active, the current π1 is smaller if the interest r2 in E2 gets higher. Figs. 4.4.2 and 4.4.3 depict the plots of the optimal proportion of investment in the stock and the VaR level, respectively, against the optimal portfolio value for different values of r2 and t = 14 years. We compared the cases when there is no switching regime and when the state changes to E2 . In Fig. 4.4.2, the impact of r2 on the qualitative behavior of the optimal investment against the portfolio value v is significant. The curve of the optimal investment against the portfolio value shifts upwards as r2 decreases. Also, when r2 decreases to a certain level, say r2 = 0.1, the qualitative behavior of the optimal investment against the portfolio value changes. There is a critical point at which the optimal investment against the portfolio value

0

200

400

600

800

1000

v

Fig. 4.4.5. The optimal consumption (E2 ) against the portfolio value v for different r2 .

v changes from increasing to decreasing. From Fig. 4.4.3, the effect of r2 on the VaR level is significant. A decrease in r2 shifts the curve of the VaR level against the portfolio value upwards. It also results in the change in the qualitative behavior of the VaR level. For example, when r2 decreases to 0.1, there is a critical point or a threshold level for the portfolio value, below which the VaR level increases as v does and above which the VaR level stays constant at a saturated level no matter what the value of r2 is. This might be attributed to the presence of the VaR constraint, which limits the amount of risk taken by the agent. Figs. 4.4.4 and 4.4.5 plots the consumption level in E1 and E2 , respectively, against the

988

K.F.C. Yiu et al. / Automatica 46 (2010) 979–989

optimal portfolio value for different values of r2 . Fig. 4.4.4 shows that current consumption is affected by µ2 in the other regime. From both Figs. 4.4.4 and 4.4.5, we see that the curve of the optimal consumptions against the portfolio value shifts downwards as r2 decreases in general. 5. Summary We considered the optimal portfolio selection problem under the MVaR constraint when the price dynamics of the risky asset are governed by a Markov-modulated GBM. The market parameters were assumed to switch over time according to a continuoustime Markov chain. The optimal portfolio selection problem was formulated as a constrained utility maximization problem over a finite time horizon. We derived a system of coupled HJB equations for the problem and adopted the static constrained optimization approach to deal with the MVaR constraint. An iterative method was used to solve numerically the system of coupled HJB equations and the optimal constrained portfolio. Numerical results for the sensitivity analysis of the optimal portfolio, the optimal consumption and the VaR level with respect to model parameters and for investigating the effect of the switching regimes were provided and discussed. Acknowledgements We would like to thank the referees for their valuable and helpful comments and suggestions. References Ahmed, N. U., & Teo, K. L. (1981). Optimal control of distributed parameter systems. North Holland. Alexander, G., & Baptista, A. (1999). Value at risk and mean-variance analysis. Working Paper. University of Minnesota. Basak, S., & Shapiro, A. (2001). Value-at-risk-based risk management: optimal policies and asset prices. The Review of Financial Studies, 14(2), 371–405. Boyle, P. P., & Yang, H. (1997). Asset allocation with time variation in expected returns. Insurance: Mathematics and Economics, 21, 201–218. Buffington, J., & Elliott, R. J. (2002a). Regime switching and European options. In Stochastic theory and control, proceedings of a workshop, Lawrence, K.S (pp. 73–81). Berlin: Springer. Buffington, J., & Elliott, R. J. (2002b). American options with regime switching. International Journal of Theoretical and Applied Finance, 5, 497–514. Cuoco, D., He, H., & Issaenko, S. (2001). Optimal dynamic trading strategies with risk limits. Preprint. The Wharton School—Yale School of Management—The Wharton School. Duffie, D., & Kan, R. (1996). A yield-factor model of interest rates. Mathematical Finance, 6, 379–406. Duffie, D., & Pan, J. (1997). An overview of value at risk. Journal of Derivatives, 7–49. Spring. Elliott, R. J., & Hinz, J. (2002). Portfolio analysis, hidden Markov models and chart analysis by PF-diagrams. International Journal of Theoretical and Applied Finance, 5, 385–399. Elliott, R. J., & Kopp, P. E. (2004). Mathematics of financial markets (2nd ed.). Springer. Elliott, R. J., & van der Hoek, J. (1997). An application of hidden Markov models to asset allocation problems. Finance and Stochastics, 3, 229–238. Elliott, R. J., Aggoun, L., & Moore, J. B. (1994). Hidden Markov models: estimation and control. Berlin, Heidelberg, New York: Springer. Elliott, R. J., Chan, L. L., & Siu, T. K. (2005). Option pricing and Esscher transform under regime switching. Annals of Finance, 1(4), 423–432. Elliott, R. J., Hunter, W. C., & Jamieson, B. M. (2001). Financial signal processing. International Journal of Theoretical and Applied Finance, 4, 567–584. Elliott, R. J., Malcolm, W. P., & Tsoi, A. H. (2003). Robust parameter estimation for asset price models with Markov modulated volatilities. Journal of Economics Dynamics and Control, 27(8), 1391–1409. Elliott, R. J., Siu, T. K., & Chan, L. L. (2006). Option pricing for GARCH models with Markov switching. International Journal of Theoretical and Applied Finance, 9(6), 825–841. Elliott, R. J., Siu, T. K., & Chan, L. L. (2007). Pricing volatility swaps under Heston’s stochastic volatility model with regime switching. Applied Mathematical Finance, 14(1), 41–62. Föllmer, H., & Schied, A. (2002). Convex measures of risk and trading constraints. Finance and Stochastics, 6, 429–447. Frittelli, M., & Rosazza Gianin, E. (2002). Putting order in risk measures. Journal of Banking and Finance, 26(7), 1473–1486.

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Ka-Fai Cedric Yiu received his M.Sc. from University of Dundee and University of London, and D.Phil. from University of Oxford. Over the decades, he had worked closely with the industry on different projects in University of Oxford and University College of London. He started his lecturing career in the University of Hong Kong. He is currently working in the Hong Kong Polytechnic University. He was awarded the Donald Julius Groen Prize in 2002. His current research interests include optimization and optimal control, financial engineering and signal processing. He has published over 40 journal publications and given over 30 conference presentations in these fields. He holds a US patent in signal processing.

Jingzhen Liu received a first class honours degree in Mathematics from the South China Normal University, in China in 2003, and received her Master Degree in the probability and stochastic process from Nankai University in 2006. She is currently a Ph.D. candidate in the Department of Applied Mathematics, The Hong Kong Polytechnic University. Her research interests include risk management, financial engineering and optimal control.

K.F.C. Yiu et al. / Automatica 46 (2010) 979–989 Tak Kuen Siu received his Bachelor degree from Hong Kong University of Science and Technology and his Ph.D. from University of Hong Kong. He is an Associate Professor in Actuarial Studies, Faculty of Business and Economics at Macquarie University, Australia. His research interests include mathematical finance, actuarial science and risk management. He has authored over 70 papers in these fields. He serves as a member in the editorial boards of several journals, including Stochastics and IMA Journal of Management Mathematics. He is a Research Affiliate of the Centre for Research into Industry, Enterprise, Finance and the Firm (CRIEFF) in the School of Economics and Finance at University of St. Andrews, United Kingdom and a Member of Advanced Modeling and Applied Computing Laboratory in the Department of Mathematics at the University of Hong Kong. He was a Honorary Fellow of the College of Science and Engineering, The University of Edinburgh from 2004 to 2008. He has been an Affiliate Member of the Financial Integrity Research Network (FIRN), an ARC Research Network, since 2009.

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Wai-Ki Ching is an Associate Professor in the Department of Mathematics at the University of Hong Kong. He obtained his B.Sc. and M.Phil. degrees in Mathematics and Applied Mathematics from the University of Hong Kong in 1991 and 1994 respectively. In 1998 he obtained his Ph.D. degree in Systems Engineering and Engineering Management from the Chinese University of Hong Kong. He was a visiting post-doc fellow at the Judge Business School of the Cambridge University (1999–2000). Ching was awarded the Best Student Paper Prize (2nd Prize) in the Copper Mountain Conference, US (1998), the Outstanding Ph.D. Thesis Prize in the Engineering Faculty, the Chinese University of Hong Kong (1998) and the Croucher Foundation Fellowship (1999). He is an author of more than 200 publications. His research interests are mathematical modelling and applied computing with focus in risk management, game theory and bioinformatics.