A Variational Approach for Mean-Variance-Optimal Deterministic Consumption and Investment

Transcription

1 A Variational Approach for Mean-Variance-Optimal Deterministic Consumption and Investment Marcus C. Christiansen Abstract A significant number of life insurance contracts are based on deterministic investment strategies this justifies to restrict the set of admissible controls to deterministic controls. Optimal deterministic controls can be identified by Hamilton-Jacobi-Bellman techniques, but for the corresponding partial differential equations only numerical solutions are available and so the general existence of optimal controls is unclear. We present a non-constructive existence result and derive necessary characterizations for optimal controls by using a Pontryagin maximum principle. Furthermore, based on the variational idea of the Pontryagin maximum principle, we derive a numerical optimization algorithm for the calculation of optimal controls. 1 Introduction Among many other applications, individual investment strategies arise in pension saving contracts, see for example Cairns [6]. While in dynamic optimal consumptioninvestment problems one typically aims to find an optimal control from the set of adapted processes, in insurance practice quite a number of contracts rely on deterministic investment strategies. Deterministic investment and consumption strategies have the advantage that they are easier to organize in asset management, that they make future consumption predictable, and that they are easier to communicate. From a mathematical point of view, deterministic control avoids unwanted features of stochastic control such as diffusive consumption, satisfaction points and consistency problems. For further arguments and a detailed comparison of stochastic versus deterministic control see also Menkens [17]. The present paper is motivated by Christiansen and Steffensen [9], where meanvariance-optimal deterministic consumption and investment is discussed in a Black- Scholes market. Sufficient conditions for optimal strategies are derived from a Hamilton-Jacobi-Bellman approach, but only numerical solutions and no analytical M.C. Christiansen B University of Ulm, 8969 Ulm, Germany marcus.christiansen@uni-ulm.de The Authors 215 K. Glau et al. eds., Innovations in Quantitative Risk Management, Springer Proceedings in Mathematics & Statistics 99, DOI 1.17/ _13 225

2 226 M.C. Christiansen solutions are given. That means that the general existence of solutions remains unclear. We fill that gap, allowing for a slightly more general model with non-constant Black-Scholes market parameters. By applying a Pontryagin maximum principle, we additionally verify that the sufficient conditions of Christiansen and Steffensen [9] for optimal controls are actually necessary. Furthermore, we present an alternative numerical algorithm for the calculation of optimal controls. Therefore, we make use of the variational idea behind the Pontryagin maximum principle. In a first step, we define generalized gradients for our objective function, which, in a second step, allows us to construct a gradient ascent method. Mean-variance investment is a true classic since the seminal work by Markowitz [16]. Since then various authors have improved and extended the results, see for example Korn and Trautmann [12], Korn [13], Zhou and Li [18], Basak and Chabakauri [3], Kryger and Steffensen [15], Kronborg and Steffensen [14], Alp and Korn [1], Björk, Murgoci and Zhou [5] and others. Deterministic optimal control is fundamental in Herzog et al. [11] and Geering et al. [1]. But apart from other differences, they disregard income and consumption and focus on the pure portfolio problem without cash flows. Bäuerle and Rieder [2] study optimal investment for both, adapted stochastic strategies and deterministic strategies. They discuss various objectives including mean-variance objectives under constraints. In the present paper, we discuss an unconstrained mean-varianceobjective and we also control for consumption. The paper is structured as follows. In Sect. 2, we set up a basic model framework and specify the optimal consumption and investment problem that we discuss here. In Sect. 3, we present an existence result for the optimal control. Section 4 derives necessary conditions for optimal controls by applying a Pontryagin maximum principle. Section 5 defines and calculates generalized gradients for the objective, which helps to set up a numerical optimization algorithm in Sect. 6. In Sect. 7 we illustrate the numerical algorithm. 2 The Mean-Variance-Optimal Deterministic Consumption and Investment Problem Let B[, T ] denote the space of bounded Borel-measurable functions, equipped with the uniform norm. On some finite time interval [, T ], we assume that we have a continuous income with nonnegative rate a B[, T ] and a continuous consumption with nonnegative rate c B[, T ]. LetC[, T ] bet the set of continuous functions on [, T ]. The positive initial wealth x and the stochastic wealth X t at t > is distributed between a bank account with risk-free interest rate r C[, T ] and a stock or stock fund with price process dst = Stαtdt + Stσ tdw t, S = 1,

3 A Variational Approach for Mean-Variance-Optimal Deterministic Consumption 227 where αt >rt, σt > and α, σ C[, T ]. We write πt for the proportion of the total value invested in stocks and call it the investment strategy. The wealth process X t is assumed to be self-financing. Thus, it satisfies dx t = X trt + αt rtπtdt + at ctdt + X tσ tπtdw t 1 with initial value X = x and has the explicit representation where X t = x e t du + t t as cse s du ds, 2 dut = rt + αt rtπt 1 2 σt2 πt 2 dt + σtπtdw t, U =. 3 It is important to note that the process X t t depends on the choice of the investment strategy πt t and the consumption rate ct t. In order to make that dependence more visible, we will also write X = X π,c. For some arbitrary but fixed risk aversion parameter γ> of the investor, we define the risk measure We aim to maximize the functional Gπ, c := MV γ MV γ [ ]:= E[ ] γ Var[ ]. T e ρs csds + e ρt X π,c T 4 with respect to the investment strategy π and the consumption rate c. The parameter ρ describes the preference for consuming today instead of tomorrow. 3 Existence of Optimal Deterministic Control Functions In Christiansen and Steffensen [9], where a Hamilton-Jacobi-Bellman approach is used, the existence of optimal control functions is related to the existence of solutions for the Hamilton-Jacobi-Bellman partial differential equation. However, only numerical solutions are available, so the general existence of solutions is unclear. Here, we fill that gap by giving an existence result for optimal deterministic control functions. The proof needs rather weak assumptions, but it is not constructive.

4 228 M.C. Christiansen Theorem 1 Let G : D, be defined by 4 for D := { π, c B[, T ] B[, T ] : ct ct ct, t [, T ] } with lower and upper consumption bounds c, c B[, T ]. Then, the functional G is continuous and has a finite upper bound. Proof We first show that MV γ [X T ] =MV γ [X π,c T ] has a finite upper bound that does not depend on π, c. Defining the stochastic process Y t := X t γ X t 2 + γ E[X t] 2, t [, T ], 5 we have MV γ [X T ] =E [Y T ]. So it suffices to show that E [Y T ] has a finite upper bound that does not depend on π, c. Since the quadratic variation process of X satisfies d[x]t = X t 2 σt 2 πt 2 dt, from Ito s Lemma we get that de[x t] =E[X t] { rt + αt rtπt } dt + at ctdt, 6 de[x t 2 ]=2E[Xt 2 ] { rt + αt rtπt } dt + E[X t] at ctdt + E[X t 2 ]σt 2 πt 2 dt. 7 Hence, the expectation function of Y solves the differential equation de[y t] =E[X t] { rt + αt rtπt } dt + at ctdt γ E[X t 2 ]σt 2 πt 2 dt 2γ E[X t 2 ] E[X t] 2 { rt + αt rtπt } dt. 8 The right hand side of 8 is maximal with respect to πt for π t = 1 αt rt 2γ σt 2 Plugging 9into8 and rearranging terms yields E[X t] 2γ Var[X t] E[X t 2. 9 ] de[y t] rt E[X t] 2γVar[Xt] dt + at ctdt + 1 αt rt 2 2 E[X t] 2γ Var[X t] 4γ σt 2 E[X t 2 dt. ] 1 Recall that we assumed γ> and σt >, so the first and second denominator are never zero. If the third denominator E[X t 2 ] is zero, we implicitly get E[X t] =, and 1 is still true by defining / :=. The first line on the right hand side of 1 has an upper bound of rt E[Y t]dt + at ctdt.

5 A Variational Approach for Mean-Variance-Optimal Deterministic Consumption 229 With the help of the equality 2 E[X t] 2γ Var[X t] = E[X t] 2 4γ Var[X t] E[Y t] and the inequalities E[X t] 2 E[X t 2 ] and Var[X t] E[X t 2 ], we can show that the second line on the right hand side of 1 has an upper bound of All in all, we obtain 1 αt rt 2 4γ σt γ E[Y t] dt. de[y t] C 1 E[Y t] + C 2 dt, t [, T ] 11 for some finite positive constants C 1 and C 2, since the functions rt, at, αt are uniformly bounded on [, T ], since ct ct for a uniformly bounded function c, and since the positive and continuous function σt has a uniform lower bound greater than zero. Thus, we have E[Y t] gt for gt defined by the differential equation dgt = C 1 gt +C 2 dt, g = Y = x >. 12 This differential equation for gt has a unique solution, which is bounded on [, T ] and does not depend on the choice of π, c. Hence, also MV γ [X T ] =E[Y T ] has a finite upper bound that does not depend on the choice of π, c. Thesameis true for the functional 4, since Gπ, c T e ρs csds + e ρt MV γ e ρt [X T ]. Now we show the continuity of the functional G. Suppose that π n, c n n 1 is an arbitrary but fixed sequence in D that converges to π, c with respect to the supremum norm. Since D is a Banach space, the limit π, c is also an element of D.LetX n t := X π n,c n t for all t. As the sequence π n, c n n 1 is convergent and within D, the absolutes π n t and c n t have finite upper bounds, uniformly in n and uniformly in t. Therefore, analogously to inequality 11, from Eq. 6 we get that de[x n t] C 3 E[X n t] + C 4 dt, t [, T ], n =, 1, 2,... for some positive finite constants C 3 and C 4. Arguing analogously to 12, we obtain that E[X n t] f t for some bounded function f t. Using similar arguments for E[X n t], we get that also the absolute E[X n t] is uniformly bounded in n and

6 23 M.C. Christiansen in t. Applying Eq. 7, we obtain de[x n t 2 ]=2E[X n t 2 ] { rt + αt rtπ n t } dt + E[X n t] at c n tdt + E[X n t 2 ]σt 2 π n t 2 dt. Using the uniform boundedness of E[X n t], π n t and c n t, we can conclude that de[x n t 2 ] C 5 E[X n t 2 ]+C 6 dt, t [, T ], n =, 1, 2,... for some positive finite constants C 5 and C 6. Hence, arguing analogously to above, the value E[X n t 2 ] is uniformly bounded in n and in t. LetY n t be the process according to definition 5 but with X n instead of X. Using8 and the uniform boundedness of E[X n t], E[X n t 2 ], π n t and c n t, we can show that de[y t Y n t] C 7 sup π t π n t + sup c t c n t dt, t [, T ] t [,T ] t [,T ] for some positive finite constant C 7. Thus, we get E[Y T Y n T ] TC 7 sup π t π n t +T sup c t c n t, t [,T ] t [,T ] whereweusedthaty Y n = x x =. Arguing similarly for E[Y t Y n t], we can conclude that Gπ, c Gπ n, c n T = e ρs c s c n sds + e ρt MV γ e ρt [X T ] e ρt MV γ e ρt [X n T ] T TC 8 sup c t c n t +e ρt E[Ỹ T Ỹ n T ] t [,T ] sup π t π n t +2T sup c t c n t t [,T ] t [,T ] for some finite constant C 8, where the processes Ỹ t and Ỹ n t are defined as above but with γ replaced by γ e ρt. Since we assumed that π n, c n n 1 converges in supremum norm, we obtain that Gπ n, c converges to Gπ, c, i.e. the functional G is continuous.

7 A Variational Approach for Mean-Variance-Optimal Deterministic Consumption 231 As G has a finite upper bound on the domain D, the supremum sup Gπ, c π,c D indeed exists. Since G is continuous and D is a Banach space, we can conclude that on each compact subset K of D there exists a pair π, c for which Gπ, c = sup Gπ, c. 13 π,c K 4 A Pontryagin Maximum Principle Christiansen and Steffensen [9] identify characterizing equations for optimal investment and consumption rate by using a Hamilton-Jacobi-Bellman approach. Here, we show that those characterizing equations are indeed necessary by using a Pontryagin maximum principle cf. Bertsekas [4]. Defining the moment functions m i t = E[X t i ], i = 1, 2, p i t = E[ T t as cs e T s du ds i ], i = 1, 2, n i t = E[e i T t du ], i = 1, 2, kt = E[e T t du T t as cse T s du ds], 14 as in Christiansen and Steffensen [9], we can represent the objective function Gπ, c by T Gπ, c = e ρs csds + e ρt m 1 tn 1 t + p 1 t γ e 2ρT m 2 tn 2 t + 2m 1 tkt + p 2 t m 1 tn 1 t + p 1 t 2 15 for any t in [, T ]. Simple calculations give us that d dt m 1t = rt + αt rtπt m 1 t + at ct, d dt m 2t = 2rt + 2 αt rt πt + πt 2 σt 2 m 2 t + 2 at c t m 1 t. Similarly to m 1 and m 2,alson 1, n 2, p 1, p 2, and k solve a system of ordinary differential equations but with terminal instead of initial conditions, see Christiansen and Steffensen [9]. 16

8 232 M.C. Christiansen Theorem 2 Let π, c be an optimal control in the sense of 13, and let mi t, pi t, n i t, i = 1, 2, and k t be the corresponding moment functions according to 14. Then, we have necessarily π αt rt e ρt m 1 tn 1 t 2γ m 1 t k t n 1 tm 1 T t = σt 2 2γ m 2 tn 2 t 1, 17 { c ct if e t = ρt t n 1 t + 2γ e ρt m 1 tn 2 t + k t n 1 tm 1 T > ct else. 18 Proof With π, c being an optimal control, we define local alternatives by π ε t, c ε t = { π t, c t + ht, lt for t t ε, t ] π t, c t else for continuous functions h and l. AsGπ, c is maximal, by applying 15 for t = t we obtain that t Gπ, c Gπ ε, c ε = e ρs lsds + e ρt m 1 t m ε 1 t n 1 t t ε { γ e 2ρT m 2 t m ε 2 t n 2 t + 2 m 1 t m ε 1 t k t m 1 t 2 m ε 1 t 2 n 1 t 2 2m 1 t } m ε 1 t n 1 t p1 t t = e ρs lsds + m 1 t m ε 1 t t ε {e ρt n 1 t 2γ e 2ρT k t n 1 t p1 t } m 1 t 2 m ε 1 t 2 γ e 2ρT n 1 t 2 m 2 t m ε 2 t γ e 2ρT n 2 t 19 must be nonnegative. Equation 16 impliesthat m 1 t m ε 1 t = = t t ε t t ε d ds m 1 s d ds mε 1 s ds {rs + αs rsπ s m 1 s

9 A Variational Approach for Mean-Variance-Optimal Deterministic Consumption 233 rs + αs rsπ ε s } m ε 1 s + ls ds, since m 1 mε 1 forε. Moreover, since we have m 1 t m 1 t, rt rt, αt αt, σt σt for t t, we get that m 1 t m ε 1 t = αt rt m 1 t t t ε t hsds + lsds + oε. t ε 2 For the squared functions we use m 1 t 2 m ε 1 t 2 = m 1 t m ε 1 t m 1 t + m ε 1 t = 2m 1 t m 1 t m ε 1 t m 1 t m ε 1 t 2 and then apply the asymptotic formula 2, which leads to m 1 t 2 m ε 1 t 2 = 2αt rt m 1 t 2 t t ε t + 2m 1 t lsds + oε. t ε hsds Similarly, we can show that { m 2 t m ε 2 t = 2αt rt m 2 t } 2π t σ t 2 m 2 t t hsds + 2m 1 t t t ε t ε lsds + oε. 21 Plugging Eq. 21 into Eq. 19 and rearranging, we get oε t t ε { lsds e ρt + e ρt n 1 t 2γ e 2ρT m 1 t n 2 t + k t n 1 t n 1 t m 1 t + p 1 t }

10 234 M.C. Christiansen + t t ε hs ds αt rt m 1 t n 1 t e ρt 2γ e 2ρT m 1 t { 2γ e 2ρT m 2 t n 2 t { k t + n 1 t n 1 t m 1 t + p1 t } 1 π σt 2 } t αt rt for all continuous functions l and h. Note that n 1 t m 1 t + p1 t = m 1 T. Consequently, we must have that the sign of lt equals the sign of e ρt + e ρt n 1 t 2γ e 2ρT m 1 t n 2 t + k t n 1 t m 1 T, which means that 18 holds, and we have necessarily that = m 1 t n 1 t e ρt 2γ e 2ρT m 2 t n 2 t 1 π t 2γ e 2ρT m 1 t k t + n 1 t m 1 T, which means that 17 is satisfied. σt 2 αt rt Recalling that n 1 t m 1 t + p1 t = m 1 T, we observe that Eqs. 17 and 18 are equal to Eqs. 19 and 2 in Christiansen and Steffensen [9], which means that the latter equations are not only sufficient but also necessary Generalized Gradients for the Objective For differentiable functions on the Euclidean space, a popular method to find maxima is to use the gradient ascent method. We want to follow that variational concept, however our objective is a mapping on a functional space. Therefore, we first need to discuss the definition and calculation of proper gradient functions. Theorem 3 Let π, c D for D as defined in Theorem 1. For each pair of continuous functions h, l on [, T ], we have Gπ + δh, c + δl Gπ, c T lim = hs π Gπ, csds δ δ + T ls c Gπ, csds

11 A Variational Approach for Mean-Variance-Optimal Deterministic Consumption 235 with π Gπ, cs = αs rs m 1 sn 1 se ρt 2γ e 2ρT m 2 sn 2 s σs πs 2γ e 2ρT m 1 s αs rs ks n 1 sm 1 T and c Gπ, cs = e ρs e ρt n 1 s + 2γ e 2ρT m 1 sn 2 s + ks n 1 sm 1 T. The limit Gπ + δh, c + δl Gπ, c lim = d Gπ + δh, c + δl δ δ dδ δ= is the so-called Gateaux derivative or directional derivative of the functional G at π, c in direction h, l. Following Christiansen [7], we interpret the twodimensional function π Gπ, c, π Gπ, c as the gradient of G at π, c. Proof Proof of Theorem 3 In the proof of Theorem 2 we already implicitly showed that Gπ, c Gπ + h1 t ε,t ], c + l1 t ε,t ] = π Gπ, ct c Gπ, ct t t ε t t ε hsds lsds + oε for all t [, T ], π, c D, and h, l C[, T ]. Defining an equidistant decomposition of the interval [, T ] by τ i := i n T, i =,...,n,

12 236 M.C. Christiansen we can rewrite the difference Gπ + δh, c + δl Gπ, c to Gπ + δh, c + δl Gπ, c n = Gπ + δh1 [,τi ], c + δl1 [,τi ] Gπ + δh1 [,τi 1 ], c + δl1 [,τi 1 ] = δ i=1 + δ n π Gπ + δh1 [,τi 1 ], c + δl1 [,τi 1 ]τ i i=1 n c Gπ + δh1 [,τi 1 ], c + δl1 [,τi 1 ]τ i i=1 τ i τ i 1 τ i hsds τ i 1 lsds + n ot/n for all <δ 1. The moments p 1, p 2, n 1, n 2, k, interpreted as mappings of π, c from the domain B[, T ] 2 with L 2 -norm into the codomain C[, T ] with supremum norm, are continuous. Hence, the gradient functions on the right hand side of the last equation are continuous with respect to the parameters τ i 1 and τ i. Thus, for n we obtain Gπ + δh, c + δl Gπ, c δ = T T + i=1 π Gπ + δh1 [,t], c + δl1 [,s] shsds c Gπ + δh1 [,s], c + δl1 [,s] slsds. Since the moment functions p 1, p 2, n 1, n 2, k interpreted as mappings of π, c from the domain B[, T ] 2 with supremum-norm into the codomain C[, T ] with supremum norm are even uniformly continuous, the above gradient functions are uniformly continuous with respect to parameter δ. Thus, for δ we end up with the statement of the theorem. 6 Numerical Optimization by a Gradient Ascent Method With the help of the gradient function π Gπ, c, π Gπ, c of the objective Gπ, c, we can construct a gradient ascent method. A similar approach is also used in Christiansen [8]. Algorithm 1. Choose a starting control π, c. 2. Calculate a new scenario by using the iteration

13 A Variational Approach for Mean-Variance-Optimal Deterministic Consumption 237 Fig. 1 Sequence of investment rates π i, i =,...,4 calculated by the gradient ascent method. The higher the number i the darker the color of the corresponding graph π i+1, c i+1 := π i, c i + K π Gπ i, c i, π Gπ i, c i where K > is some step size that has to be chosen. If c i+1 is above or below the bounds c and c, we cut it off at the bounds. 3. Repeat step 2 until Gπ i+1, c i+1 Gπ i, c i is below some error tolerance. 7 Numerical Example Here, we demonstrate the gradient ascent method of the previous section with a numerical example. For simplicity, we fix the consumption rate c and only control the investment rate π. We take the same parameters as in Christiansen and Steffensen [9] in order to have comparable results: For the Black-Scholes market we assume that r =.4, α =.6 and σ =.2. The time horizon is set to T = 2, the initial wealth is x = 2, and the savings rate is at ct = 1 8 = 2. The preference parameter of consuming today instead tomorrow is set to ρ =.1, and the risk aversion parameter is set to γ =.3. Starting from π =.5, Fig. 1 shows the converging series of investment rates π i, i =,...,4 for K =.2. The last iteration step π 4 perfectly fits the corresponding numerical result in Christiansen and Steffensen [9].

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