Optimal Consumption-Investment Problems in Incomplete Markets

Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, December 12-15, 2005

ThA11.1

Optimal Consumption-Investment Problems in Incomplete Markets with Random Coe!cients Netzahualcóyotl Castañeda-LeyvaW Universidad Autónoma de Aguascalientes Departamento de Estadística Av. Universidad 940, Ciudad Universitaria 20100 Aguascalientes, Ags., MÉXICO [email protected]

Daniel Henández-Hernández Centro de Investigación en Matemáticas Callejón de Jalisco S/N, Valenciana Guanajuato, Gto. 36240 MÉXICO [email protected]

of the hyperbolic absolute risk aversion (HARA) and In this work we present the explicit solution of an op- the logarithmic utility functions. timal investment problem in an incomplete financial market, for HARA and logarithmic utility functions. 2 The model The market follows a generalization of the Black and In this section we shall introduce a factor model for Scholes diusion model, which consists of a bank the financial market based on the model presented account, a risky asset, and an economic external by Bielecki and Pliska [1]; we also state the problem factor. The coe!cients of the underlying diusion we want to solve. The financial market is governed by a stanprocesses are random and depend on the economic dard two-dimensional Brownian motion (BM) external factor. This market includes more realistic financial scenarios where the martingale methodol- {(W1t , W2t ) , Ft }0$t$T , defined in a complete proba{Ft }0$t$T is the augogy and stochastic control techniques, established in bility space ( , FT , P ) , where o n Castañeda-Leyva and Hernández-Hernández [2], are mentation of the filtration F (W1 ,W2 ) . This t 0$t$T applied. market is composed by a bank account, a risky asset, Keywords: Optimal investment and consumption, and a correlated external factor, such that: incomplete markets, stochastic volatility, martingale 1. The bank account process is given by method, optimal control, Black-Scholes model. µZ t ¶ 0 St $ exp r (Yu ) du ; t 5 [0, T ] , 1 Introduction Abstract

We consider the problem of maximizing the expected utility of final wealth and consumption in a finite horizon T > 0, when the financial market is incomplete. This problem is solved using the martingale method, consisting in translating the original investor’s problem, called primal problem into a convex optimization problem, and then solve the associated dual problem; see Karatzas and Shreve [9]. In Section 2 the model is presented, whereas the martingale method is developed in Section 3. Finally, an explicit solution is given for the particular examples W The

research of this author was supported by grants PROMEPUAAGS-EXB-69 and UAA-PIEST-04-01.

0-7803-9568-9/05/$20.00 ©2005 IEEE

0

where r (·) is the interest rate function. 2. The asset price process S satisfies the stochastic dierential equation (1)

dSt = St [µ (Yt ) dt +  (Yt ) dW1t ] ,

with S0 = 1. Here µ (·) and  (·) are the rate return and the volatility functions, respectively. 3. The dynamics of the external factor Y is give by the stochastic dierential equation (2)

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dYt = g (Yt ) dt +  (Yt ) dW1t + % (Yt ) dW2t ,

with Y0 = y 5 R. All the coe!cients may be random and depend on the external economic factor Y . In particular, when  and % are constants with 2 + %2 = 1, then  is the correlation coe!cient between the underlying BM’s of the asset price and the external factor. This is the case studied in Castañeda-Leyva and Hernández-Hernández [2]. In particular, when  = ±1 the market is complete. Otherwise, it becomes incomplete, since the external factor cannot be traded.

Given U1 , U2 : R+ $ R utility functions, the investor’s problem consists in to ¾ ½ Z T ¡ Z,c ¢ U2 (ct ) dt , (3) maximize E U1 XT + 0

over (, c) 5 A (x, y) , as well as to provide an optimal trading strategy (ˆ  , cˆ) . Here U (·) = U1 (·) , U2 (·) is strictly increasing, strictly concave, and dierentiable, such that U 0 (4) $ limb" U 0 (b) = 0 and U 0 (0+) $ limb0 U 0 (b) = 4. To obtain the solution of this optimization problem, we will use the martingale approach and stoCondition 2.1. 1. µ (·) ,  (·) , and r (·) belong to chastic control techniques; see Castañeda-Leyva and Cb2 (R). Hernández-Hernández [2] and Harrison and Pliska 2. r (·)  0 and  (·) >  0 , for some  0 > 0. [7]. 3. g (·) 5 C 1 (R) such that g 0 (·) 5 Cb (R) . Define the equivalent local martingale measures 4.  (·) , % (·) belong to Cb1 (R) such that  (·)  0 family as and % (·)  %0 , for some 0 , %0 > 0. Without loss of © generality assume that  0  2 (·) + %2 (·)  1. P $ Q | P ! Q ! P, S/S 0 is a Q-local martingale, for all y 5 R} , An important example is when the external factor Y is modeled as an Ornstein-Uhlenbeck process, and M as the set of all progressively measurable RT where in this case, g (y) =  (y  y¯) and  (·) and processes { t , Ft } , with E 0  2u du < 4, such tM[0,T ] % (·) are constant functions with 2 +%2 = 1; for some that the local martingale fixed parameters , y¯ > 0. R Rt 1 t 2 2 Now, consider a single investor with initial capZtD $ e3 0 [w(Yu )dW1u +D u dW2u ]3 2 0 [w (Yu )+D u ]du ital x, who generates a wealth process is a martingale, for all y 5 R. Here XtZ,c $ Xtx,y,Z,c ; t 5 [0, T ] , µ (·)  r (·) .  (·) $  (·) following the integral equation Z t cu du XtZ,c + 0 Z t (r (Yu ) XuZ,c + [µ (Yu )  r (Yu )]  u ) du $ x+ 0 Z t  u  (Yu ) dW1u . + 0

Note that  (·) 5 Cb2 (R) , since  (·) >  0 > 0. The process  in M plays the role of control process in the associate dual problem. Moreover, for each  5 M, a probability measure P D 5 P can be defined in ( , FT ) as Z D ZTD dP ; A 5 FT . P (A) $ A

Here { t , Ft }0$t$T and {ct , Ft }0$t$T are the trading portfolio process and consumption process respec- In this sense, M is a subset of P. Under the measure D D D tively, such that are progressively measurable with P , {(W1t , W2t ) , Ft }0$t$T is a BM, where Z t Z T Z T D 2 W1t $ W1t + (4)  (Yu ) du,  u du, ct dt < 4, 0 0 0 Z t D $ W +  u du. W 2t 2t and c  0 a.s. The trading strategy (, c) is 0

admissible if X Z,c  0 a.s. and the set of such 0 is a continuous P D -martingale and trading strategies is denoted as A (x, y). Finally, In addition,R S/S ¢ ·¡ Z,c 0 X· /S· + 0 ct /St0 dt is a nonnegative continuous assume that x, T 5 R+ $ (0, 4) , and y 5 R.

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P D -local martingale and a P D -supermartingale; see From elementary calculus, it follows that Karatzas and Shreve [8]. ˜ (z) = U (I (z))  zI (z) ; z > 0, (8) U Finally, note that our market is incomplete and is 0 free of arbitrage opportunities, since P is non empty where I (·) is the inverse function of U (·) . The associated dual functional to the primal and non trivial; see Delbaen and Schachermayer [4]. problem (6) is defined as 3 Martingale method L (, ) ½ · ¸ Z T In this section the investor’s problem is formulated E U1 (B) + U2 (ct ) dt $ x + sup as a convex optimization problem and, using the BD0,cD0 0 martingale method, the associated dual problem ¸¾ · Z T ct is obtained. Furthermore, a relevant relationship D B + ; E 0 dt ST0 between the optimal solutions of both problems is 0 St presented. for  5 M and   0. The dual problem is to The following lemma allows us to obtain a useful minimize L (, ) , over  5 M,  > 0. characterization of the set of admissible trading (9) strategies A (x, y). It is analogous to Theorem 1 Note that the dual functional L (, ) is given by · µ D¶ in Cuoco [3] and Theorem 5.6.2 in Karatzas and ˜ 1  ZT Shreve [9]; see also Lemma 2.2 in Castañeda-Leyva (10) L (, ) = E U ST0 and Hernández-Hernández [2]. µ D¶ ¸ Z T ˜2  Zt dt + x; + U Lemma 3.1. Let B  0 be a FT -measurable random St0 0 variable and c a consumption process such that for  5 M,  > 0. It is easy to see that ½ ¾ Z T ½ ¾ Z T ct B D (11) E U1 (B) + U2 (ct ) dt sup + (5) sup E 0 dt  x. ST0 0 (B,c)MB(x,y) DMM 0 St  inf L (, ) . Then, there exists a trading portfolio  such that DMM,b>0 (, c) 5 A (x, y) and XTZ,c  B a.s. Conversely, if When the equality holds in (11), we say that there (, c) 5 A (x, y) , then B $ XTZ,c satisfies the budget is no duality gap. This is true for logarithmic and constraint (5). HARA utility functions. The next proposition shows the relationship beTherefore, the investor’s problem (3) can be tween the optimal solutions of the problems (6) and written as (9); compare with Proposition 6.3.8 in Karatzas and ¾ ½ Z T Shreve [9]. U2 (ct ) dt , (6) maximize E U1 (B) + ³ ´ 0 ˆ 5 Proposition 3.1. Assume that for some ˆ,  ³ ´ over (B, c) 5 B (x, y) , where ˆ cˆ , defined as M × R the pair B, +

B (x, y) $ {(B, c) | B  0 and c  0

(12)

such that satisfy (5)} .

Here the argument ”B  0 and c  0” means that B is a nonnegative FT measurable random variable and c is a consumption process. This problem will be referred to as the primal problem; see Section 8.6 in Luenberger [10]. For a utility function U (·) , define the conjugate function (7)

˜ (z) $ sup {U (b)  zb} ; z > 0. U b>0

µ Dˆ ¶ ˆ ZT , ˆ $ I1  B ST0

µ Dˆ ¶ ˆ Zt ; cˆt $ I2  St0

belongs to B (x, y) and satisfies # " Z T ˆ c ˆ B t (13) E Dˆ + 0 dt = x. ST0 0 St ³ ´ ˆ cˆ is the optimal solution to the primal Then, B, ³ ´ ˆ is the optimal solution problem (6), whereas ˆ,  to the dual problem (9). In particular, there is no duality gap.

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for  5 M, where · ¸ In this section the solution to the consumption¢ ¡ 2 1 2 q (y, v) $  r (y) + (1  )  (y) + v ; investment problem shall be given when the utility 2 function is HARA and logarithmic. The martingale method allows us to obtain an explicit form for the for y, v 5 R. Note that q (y, v) is bounded, provided optimal process ˆ and the optimal trading strategy the control space is compact. Define the value function of (14) as (ˆ  , cˆ) . Here we assume that U (·) = U1 (·) = U2 (·) , (16) W (T, y) $ inf J (T, y, ) . DMM where b Now, temporarily, consider [M, M ] as the bounded U (b) $ ; b > 0,  control space, that is  5 MM ; for M > 0 given. The verification theorem below states that with 0 6=  < 1. From (9) and (10), we wish to 4

HARA and logarithmic utility functions

w (T, y) = W M (T, y) ;

minimize J (T, y, ) , over  5 M,

(14)

where ¡ w (T, y)¢ is the ¡ unique¢ smooth function in 1,2 ¯ ¯ R C + × R _ Cp R+ × R satisfying the associated Hamilton Jacobi Bellman equation ¢ 1¡ 2 wT = 1 +  + %2 wyy + (g  ) wy ¸ ·2 1 2 (17)  r + (1  )  w +  × 2 ½ ¾ 1 sup %wy v  (1  ) wv2 , 2 vM[3M,M]

where J (T, y, ) $ E

·µ

ZTD ST0

¶k

+

Z

T

0

µ

ZtD St0

¶k

¸ dt

and  $ / (1  ) . Note that 0 6=  < 1. We present only the solution to the case when  > 0, since the solution for  < 0 is similar. An advantage of the martingale method applied in this case is the reduction to just one control variable ( 5 M). On with w (0, y) = 1; see Theorem IV.4.3 and Remark the other hand, observe that, for t 5 [0, T ] IV.4.1 in Fleming and Soner [6]. The Markov policy Rt 2 1 2 induced by the previous equation is (Z D )k = Z k,D e3 2 k(13k) 0 [w (Ys )+D s ]ds , t

t

(18)

where Ztk,D

µ Z t $ exp  [ (Yu ) dW1u +  u dW2u ] 0 ¶ Z ¤ 2 t £ 2  (Yu ) +  2u du .  2 0

|% (y) wy (t, y)|  M (1  ) w (t, y) , w (t, y) 6= 0, and  W (t, y) = M sgn (% (y) wy (t, y)) , otherwise. We will prove below that W M (T, y) does not depend on M . Theorem 4.1. (Verification) For M > 0, let w (T, y) be the unique solution to (17). Then w (T, y)  J (T, y, ) ,

dYt = [g (Yt )   (Yt )  (Yt )  % (Yt )  t ] dt k,D k,D + (Yt ) dW1t + % (Yt ) dW2t .

Therefore

% (y) wy (t, y) , 1   w (t, y)

if

Analogous to the measure P D and the BM on (4), using the process Z k,D , define the measure P k,D in FT and the BM (W1k,D , W2k,D ) ; see Fleming and Hernández-Hernández [5]. Under P k,D , the dynamics of the process Y satisfies

(15)

 W (t, y) = 

w (T, y) = W M (T, y) = J (T, y, ˆ) , for  5 MM , where

ˆ t =  W (T  t, Yt ) 5 MM

J (T, y, ) · R Z T k,D q(Y ,D )dt t t e0 + = E

0

T

¸ q(Y ,D )ds s s e0 dt , Rt

and  W (t, y) is the Markov policy defined in (18). That is, ˆ is the optimal control process for the constrained auxiliary problem relative to (16).

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Theorem 4.2. Let W (T, y) be the unconstrained with value function (16). There exists a constant K > 0 such that for M > K/ (1  ) , implies that w (T, y) = W (T, y) and, the optimal control process is give by

and cW (t, x, y) $

W

ˆt =  (T  t, Yt ) ; ¢ ¡ ¯ + × R _ C 0,1 (R ¯ + × R) Furthermore, W 5 C 1,2 R b and solves the dierential equation

(19)

WT (20)

1 = 1 + Wyy + (g  ) Wy 2 ¸ · %2 Wy2 1 2 ,  r + (1  )  W  2 2 W

 W (t, x, y) · ¸ Wy (t, y) x (1  )  (y) +  (y) $  (y) W (t, y) x . W (t, y)

Logarithmic example. In (3), suppose that U1 (b) = U2 (b) $ log b;

b > 0.

Then, from (9) and (10), the dual problem is to minimize E

½Z

0

with W (0, y) = 1.

T

 2t dt +

Z

0

T

Z

0

t

¾

 2s dsdt

,

over  5 M. Now we give the explicit optimal trading strategy The optimal values are given by for the investor’s problem. According to (12), we consider ˆ = 1+T. ˆ  0 and  µ ¶13k µ Dˆ ¶ x ZT x ST0 ˆ ˆ , B $ I  0 = Finally, similarly to the HARA case, the optimal Dˆ ZTDˆ ST trading strategy (ˆ  , cˆ) is given by the feedback form µ Dˆ ¶ µ 0 ¶13k Zt x St ˆ ³ ´ ; cˆt $ I  0 = Dˆ ZtDˆ St  ˆ t =  W t, XtZˆ ,ˆc , Yt , ³ ´ where cˆt = cW t, XtZˆ ,ˆc , Yt ; Dˆ = J (T, y, ˆ) = W (T, y) , µ ¶13k where x ˆ ,  = Dˆ µ (y)  r (y)  W (t, x, y) $ x  2 (y) and ˆ is given by (19). Now, define  (y) " # = x Z T ˆ ˆ  (y) cˆu Xt B $ E Dˆ + du | Ft ; t 5 [0, T ] . 0 0 0 St ST t Su and x ; cW (t, x, y) $ We get 1+T t £ Dˆ ¤k31 ˆt for (t, x, y) 5 [0, T ] × R+ × R. Z W (T  t, Yt ) X = x £ t 0 ¤k 0 W (T, y) St St References and, by Ito’s formula, d where

ˆ cˆ  ˆ X + 0 dt = 0 dW1Dˆ ; 0 S S S

[1] Bielecki, T.R. and S.R. Pliska. Risk Sensitive dynamic asset management, Appl. Math. Optim. 39 (1999), 337-360. [2] Castañeda-Leyva, N. and D. Hernández-Hernández. Optimal consumption-investment problems in incomplete financial markets with stochastic coe!cients, SIAM J. Control Optim. (2005).

³ ´ ˆ t , Yt ,  ˆ t $  W T  t, X ³ ´ ˆ t , Yt , cˆt = cW T  t, X

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[3] Cuoco, D. Optimal consumption and equilibrium prices with trading portfolio constraints and stochastic income, J. Econom. Theory. 72 (1997), 3373. [4] Delbaen, F. and W. Schachermayer. A general version of the fundamental theorem of asset pricing, Math. Ann. 300 (1994), 463-520. [5] Fleming, W.H. and D. Hernández-Hernández. An optimal consumption model with stochastic volatility, Finan. Stoch. 7 (2003), 245-262. [6] Fleming, W.H. and H.M. Soner. Controlled Markov processes and viscosity solutions, Springer-Verlag New York, 1993. [7] Harrison, J.M. and S.R. Pliska. Martingales and stochastic integrals in the theory of continuous trading, Stoch. Proc. and Appl. 11 (1981), 215-260. [8] Karatzas, I. and S.E. Shreve. Brownian Motion and Stochastic Calculus, Springer-Verlag New York, 1991. [9] Karatzas, I. and S.E. Shreve. Methods of Mathematical Finance, Springer-Verlag New York, 1998. [10] Luenberger, D.G. Optimization by Vector Space Methods, John Wiley & Sons New York, 1969.

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