Portfolio optimization for piecewise concave criteria functions Laurence Carassus

Huyˆen Pham

Laboratoire de Probabilit´es et

Laboratoire de Probabilit´es et

Mod`eles Al´eatoires, CNRS UMR 7599

Mod`eles Al´eatoires, CNRS UMR 7599

Universit´e Paris 7, Case 7012

Universit´e Paris 7, Case 7012

2 Place Jussieu

2 Place Jussieu

75251 Paris Cedex 05, France

75251 Paris Cedex 05, France

[email protected]

and Institut Universitaire de France [email protected]

Abstract In the context of a complete financial model, we study the portfolio optimization problem when the objective function may have a change of concavity at a given positive constant level. This typically includes utility maximization of terminal wealth when the agent modifies her preferences structure from a certain level of wealth. This also allows to consider the portfolio management problem of an investor willing to achieve a given level of performance by penalizing net loss and maximizing net gain. We finally compare some of our results with the classical portfolio choice problem of Merton by doing some numerical experiments.

Key words: Utility maximization, gain/loss criterion, nonconvex optimization, conjugate duality approach, Malliavin calculus. MSC 1991 subject classifications : Primary : 90A09, 93E20; secondary : 60H07. JEL classification : G11, C61.

1

1

Introduction

Portfolio optimization problems in finance usually assume concave (or convex) objective functions which are specified independently from the wealth process. The main examples are utility maximization from consumption and terminal wealth and hedging problems (meanvariance, shortfall risk, ...), see Karatzas and Shreve (1998) and references therein. In this paper, we consider a continuous objective function U on (0, ∞) of the form : ( U1 (x), 0 h, where h > 0, and U1 , U2 are C 1 , and strictly concave functions. Function U is in general only piecewise C 1 and piecewise concave. We study the problem of maximizing the expected objective function U of terminal wealth in the context of complete Itˆo processes model. Such a problem typically arises in utility maximization when the preferences structure of the agent changes from U1 to U2 at h. The constant h is then interpreted as a level of wealth at which the risk-aversion of the agent is modified. For example, the agent may be less risk-averse when her wealth is large enough and may be more risk-averse when her wealth decreases a lot. Hence, such a piecewise concave utility function allows to take into account an effect of the agent’s wealth on her risk behavior. On the other hand, consider an investor who wants to achieve a given level of performance h by adopting the following criterion : She penalizes net loss, i.e. when terminal wealth is below h, and maximizes net gain, i.e. when terminal wealth is above h. We call this criterion the portfolio gain/loss management. This is embedded in our optimization problem by choosing U1 (x) = −l(h − x) and U2 (x) = g(x − h), where l is a convex C 1 loss function and g is a concave C 1 gain function. We solve our piecewise concave optimization problem by using a martingale duality approach. We consider the conjugate function U˜ of U , i.e. U˜ (y) = supx>0 (U (x) − xy) and we provide an explicit expression of a function χ that attains the supremum in the definition of U˜ . In general, function χ is not continuous. In a second step, we prove continuity of the function H(y) = E[ZT0 χ(yZT0 )] under a certain condition, namely that the drift process of the asset is nonzero. Actually, by means of Malliavin calculus, this last condition ensures that the density of the unique martingale measure, ZT0 , is absolutely continuous with respect to the Lebesgue measure. This continuity result is essential to state the budget constraint and then to adapt the standard martingale approach of Cox and Huang (1989) or Karatzas et al. (1987). In the case where the drift process of the asset is zero, we derive directly, by the dynamic programming methods, the value function of our optimization problem. It appears that when U is not concave, the control problem is singular and there is no optimal portfolio. We also show that duality relation between the value functions of the primal optimization problem and of the dual problem holds, even in the case where the objective function U is not 2

concave. Finally, we analyze the qualitative behavior of the optimal strategy and compare with the classical Merton’s portfolio strategy by doing some numerical experiments. The paper is organized as follows. Section 2 describes the financial model and Section 3 formulates the portfolio optimization problem. In Section 4, we solve the problem by using a martingale duality approach. We also derive the solution by a dynamic programming approach when the drift process of the asset is zero. Section 5 presents some examples and we derive in Section 6 closed-form expressions for the optimal portfolio in the Black-Scholes model. Finally, Section 7 presents some numerical results.

2

The financial model

We consider the standard setup of a complete Itˆo processes model for a financial market, as described for example in Karatzas and Shreve (1998). There are one bank account, with constant price process S 0 , normalized to unity, and d risky assets of price process S = (S 1 , . . . , S d )0 governed by : dSt = µt dt + σt dWt ,

S0 = s ∈ Rd .

Here W = (W 1 , . . . , W d )0 is a standard d-dimensional Brownian motion on a complete probability space (Ω, F, P ) equipped with a filtration F = {Ft , 0 ≤ t ≤ T }; this is the P -augmentation of the filtration generated by W . The Rd -valued process µ and the Rd×d valued process σ are assumed to be progressively measurable with respect to F. We shall also assume that the matrix σt is invertible for all t ∈ [0, T ], P a.s. We define then the ‘market price of risk’ process : λt = σt−1 µt , 0 ≤ t ≤ T, which is assumed to satisfy martingale : Zt0

RT 0

|λt |2 dt < ∞. We consider then the exponential P -local

  Z t Z 1 t 2 0 |λu | du , 0 ≤ t ≤ T. = exp − λu dWu − 2 0 0

(2.1)

We shall assume Z 0 is a P -martingale, i.e. E[ZT0 ] = 1, so that one can define a probability measure P 0 equivalent to P on (Ω, FT ) by : P 0 (A) = E[ZT0 1A ],

A ∈ FT .

Recall that sufficient condition ensuring that E[ZT0 ] = 1 is the Novikov criteh a well-known i R T rion : E exp 21 0 |λt |2 dt < ∞, which is obviously satisfied if λ is bounded in (t, ω).

3

By Girsanov’s theorem, the process Wt0

Z

t

λu du, 0 ≤ t ≤ T,

= Wt + 0

is a P 0 -Brownian motion, and the dynamics of S under the so-called risk-neutral equivalent martingale measure P 0 is : dSt = σt dWt0 , 0 ≤ t ≤ T.

(2.2)

A portfolio is an Rd -valued F-adapted process θ = (θ1 , . . . , θd )0 such that : Z

T

|σt0 θt |2 dt < ∞,

a.s.

0

Here, θti describes the number of shares invested in the i-th risky asset at time t. The (selffinanced) wealth process X x,θ corresponding to an initial capital x ≥ 0 and a portfolio θ is defined by : dXtx,θ = θt0 dSt = θt0 σt dWt0 ,

X0x,θ = x.

(2.3)

A portfolio θ is called admissible for the initial capital x ≥ 0, and we write θ ∈ A(x), if : Xtx,θ ≥ 0,

a.s., 0 ≤ t ≤ T.

(2.4)

Remark 2.1 From (2.3), the process X x,θ is a P 0 -local martingale, and from (2.4), it is nonnegative, thus also a P 0 -supermartingale. We deduce that : h i x,θ 0 E Z T XT ≤ x, ∀θ ∈ A(x). (2.5)

3

The portfolio optimization problem

We consider a continuous function U : (0, ∞) → R defined by : ( U1 (x), 0 h, where h > 0, U1 is strictly concave, of class C 1 on (0, h], and U2 is strictly concave, of class C 1 on (h, ∞). Notice that continuity of U means that U2 (h) := limx↓h U2 (x) = U1 (h). We shall assume U20 (∞) := lim U20 (x) = 0. x→∞

4

(3.1)

This last condition is an Inada type condition on the behaviour at infinity of U 0 . Notice that we do not impose an Inada type condition on the behaviour at zero of U 0 . In particular, we can choose an exponential utility function U1 (x) = −e−x . We denote U20 (h) := limx↓h U20 (x). Notice that when U10 (h) 6= U20 (h), the function U is not differentiable. Moreover, function U is concave on (0, ∞) iff U10 (h) ≥ U20 (h). In the general case, the function U is only piecewise C 1 and piecewise concave. In the limiting case, h = 0, we recover the usual case of concave and C 1 utility function U = U2 . Our interest is on the optimization problem : h i J(x) = sup E U (XTx,θ ) , x > 0. (3.2) θ∈A(x)

Application 1 When U1 and U2 are standard utility functions, problem (3.2) is an utility maximization problem from terminal wealth. The constant h is interpreted as a level of wealth at which risk aversion of the agent may change. Application 2 Consider the case where U1 (x) = −l(h − x) and U2 (x) = g(x − h), with l, a C 1 strictly convex function on [0, ∞) and g, a C 1 strictly concave function on [0, ∞), such that l(0) = g(0) = 0. Then problem (3.2) is written equivalently as : h i J(x) = sup E −l(h − XTx,θ )+ + g(XTx,θ − h)+ , x > 0. θ∈A(x)

This is a portfolio management problem for an investor who wishes to achieve a level of performance h, by penalizing net loss and maximizing net gain.

4

Solution to the optimization problem

We define the conjugate function of U : U˜ (y) = sup (U (x) − xy) ,

y > 0,

x>0

which is a nonincreasing and convex function from (0, ∞) into R∪{∞}. Notice that function U˜ is not necessarily smooth C 1 . We also define the dual value function :
˜ J(y) = E U˜ yZT0 , y > 0. In a first step, we provide an explicit characterization of a function χ that attains the supremum in definition of U˜ . We need to introduce some notations. We denote by Ii 5

the inverse of the derivative of Ui , i = 1, 2; I1 is a continuous strictly decreasing function from [U10 (h), U10 (0)) into (0, h] and is extended by continuity on [U10 (h), U10 (0)] when U10 (0) := limx→0 U10 (x) < ∞, by setting I1 (U10 (0)) = 0. Notice that when U10 (0) < ∞, U1 is also extended by continuity in 0 by setting U1 (0) = 0; I2 is a continuous strictly decreasing function from (0, U20 (h)) into (h, ∞) and is extended by continuity by h on [U20 (h), ∞). In the case where U10 (h) < U20 (h), we define the function φ : [U10 (h), U20 (h)] → R by 1 : ( φ(y) =

U1 ◦ I1 (y) − U2 ◦ I2 (y) − y(I1 − I2 )(y), U1 (0) − U2 ◦ I2 (y) + yI2 (y),

U10 (h) ≤ y ≤ U10 (0) ∧ U20 (h) (4.1) U10 (0) ∧ U20 (h) < y ≤ U20 (h).

Proposition 4.1 There exists a nonnegative function χ defined on (0, ∞) such that : U˜ (y) = U (χ(y)) − yχ(y),

y > 0.

Function χ is explicitly characterized as follows : When U10 (h) ≥ U20 (h), we get :  I2 (y), 0 < y < U20 (h)     h, U20 (h) ≤ y ≤ U10 (h) χ(y) =  I1 (y), U10 (h) < y < U10 (0)    0, y ≥ U10 (0). When U10 (h) < U20 (h), we have :    I2 (y), χ(y) = I1 (y),   0,

0 < y < y(h) y(h) ≤ y < y(h) ∨ U10 (0) y ≥ y(h) ∨ U10 (0).

(4.2)

(4.3)

where y(h) is the unique element in (U10 (h), U20 (h)) such that φ(y(h)) = 0. 2

Proof. See Appendix.

Remark 4.1 In the case where U10 (h) < U20 (h), function χ of Proposition 4.1 may be not continuous on (0, ∞); from (4.3), there is a discontinuity at point y = y(h) whenever I1 (y(h)) 6= I2 (y(h)) or y(h) > U10 (0). Remark 4.2 By definition of U˜ , it is clear that :   e e (x) := inf U˜ (y) + xy , U (x) ≤ U y>0

1

x > 0.

For any real numbers a and b, we denote by a ∧ b (resp. a ∨ b) , the minimum (resp. maximum) of a and b.

6

e e , see e.g Ekeland and Temam It is well-known that when U is concave, we have equality U = U (1976). In our context, in the case where U10 (h) < U20 (h), so that U is not concave, one can e e . For example, when U 0 (0) ≤ U 0 (h) and y(h) ≥ U 0 (0), i.e. φ(U 0 (0)) ≤ 0, check that U 6= U 1

2

1

1

a straightforward calculation shows that for x < I2 (y(h)) : e e (x) = min[U2 ◦ I2 (y(h)) − y(h)I2 (y(h)), U1 (0)] + xy(h), U e e differs from U . and so U Remark 4.3 By definition of χ and U˜ , we have for all y, z > 0 : U˜ (y) − χ(y)(z − y) = U (χ(y)) − χ(y)z ≤ U˜ (z). This shows that for all y > 0, −χ(y) ∈ ∂ U˜ (y), the subgradient of the convex function U˜ . When U is concave, the converse is true : any element xˆ ∈ −∂ U˜ (y) attains the supremum in U˜ (y), i.e. U˜ (y) = U (ˆ x) − xˆy (see e.g Ekeland and Temam (1976)). This property is crucial in the dual formulation when the set of martingale measures is not a singleton, see Cvitani´c (2000) or Deelstra, Pham and Touzi (2001). This last property is no more valid in our context. We shall give some examples in Section 5. Remark 4.4 In the case of Example 2, when U1 and U2 are on the form U1 (x) = −l(h − x) and U2 (x) = g(x − h), with l(0) = g(0) = 0, function φ defined in (4.1) reduces to : ( ˜l(y) − g˜(y), l0 (0) ≤ y ≤ g 0 (0) ∧ l0 (h) φ(y) = −l(h) − g˜(y) + yh, g 0 (0) ∧ l0 (h) < y ≤ g 0 (0), where, ˜l(y) = maxx>0 [−l(x) + xy] and g˜(y) = maxx>0 [g(x) − xy]. We shall make the following assumption : Assumption 4.1   E ZT0 I2 (yZT0 ) < ∞, ∀y ∈ (0, ∞). Remark 4.5 Suppose that there exists α ∈ (0, 1) and γ ∈ (1, ∞) such that αU20 (x) ≥ U20 (γx), ∀x ∈ (h, ∞). Then, by similar arguments as in Remark 6.9, p. 107 of Karatzas and Shreve (1998), Assumption 4.1 holds whenever E [ZT0 I2 (y0 ZT0 )] < ∞ for some y0 in (0, U20 (h)).

7

Remark 4.6 Suppose that there exist C ≥ 0, m, n > 0 such that : I2 (y) ≤ C(1 + y m + y −n ), ∀y > 0. Then the boundedness of the process λ in (t, ω) is a sufficient condition for Assumption 4.1 to hold.

From expression of function χ in Proposition 4.1 and recalling the nonincreasing feature of I1 and I2 , we easily see that : χ(y) ≤ I2 (y), ∀y > 0.

(4.4)

Under Assumption 4.1, one can then define the real-valued function on (0, ∞) by :   H(y) = E ZT0 χ(yZT0 ) ,

y > 0.

The second step is to state continuity of function H and then to prove that the budget ˆ = χ(ˆ constraint is satisfied : Given x > 0, one can find yˆ(x) > 0, such that X y (x)ZT0 ) is a 0 ˆ = H(ˆ terminal wealth satisfying E P [X] y (x)) = x. We need to make some assumptions on the market price of risk λ. We denote by H the Rt ˙ Cameron-Martin space formed by the functions of the form ψ(t) = 0 ψ(s)ds, t ∈ [0, T ], R  21 T ˙ with ψ˙ ∈ L2 ([0, T ], Rd ) equipped with the norm k ψ kH = |ψ(s)|2 ds . We denote 0 by D the Malliavin derivative operator defined on the domain ID1,2 of L2 (Ω); D : ID1,2 → L2 (Ω, H). We refer to Nualart (1995) for all unexplained notations. Given a random variable F ∈ ID1,2 , DF (ω) = (D1 F (ω), . . . , Dd F (ω))0 is valued in H for ω ∈ Ω, and Dt F (ω) = (Dt1 F (ω), . . . , Dtd F (ω))0 , 0 ≤ t ≤ T , is defined by : Z . DF = Dt F dt, a.s. 0

(CL) For all t ∈ [0, T ], λit ∈ ID1,2 , i = 1, . . . , d, and satisfy : E

P0

Z

T

 21 < ∞, |Dt λs | ds 2

(4.5)

0

where Dt λs = (Dt λ1s , . . . , Dt λds ). Moreover, λt (ω) 6= 0,

dt × dP a.s.

(4.6)

Remark 4.7 Notice that when λ is a bounded deterministic process, we have Dt λs = 0, and the last condition reduces to λt 6= 0 dt a.e. 8

Lemma 4.1 Suppose that condition (CL) holds. Then, for all z ≥ 0, P 0 [ZT0 = z] = 0. Proof. The case z = 0 is obvious since P 0 and P are equivalent. Fix now z> 0. The distribution law of ln ZT0 under P is given by : Z T Z 1 T 0 0 λs dWs − |λs |2 ds. ln ZT = − 2 0 0 From standard calculations on Malliavin derivative (see e.g. Proposition 2.3 in Ocone and Karatzas 1991), we then have for all 0 ≤ t ≤ T : Z T Z T 0 Dt λs λs ds Dt λs dWs − Dt ln ZT = −λt − t t Z T = −λt − Dt λs dWs0 , (4.7) t

The integrability condition (4.5) ensures that for all t ∈ [0, T ], the Itˆo stochastic integral Rt process { 0 Dt λs dWs0 , 0 ≤ t ≤ T } is a P 0 -martingale, see e.g. Jacod (1979). We deduce from (4.7) that : 0

E P [Dt ln ZT0 |Ft ] = −λt , 0 ≤ t ≤ T. Under condition (4.6) and recalling that P 0 is equivalent to P , this implies that : Z T 0 2 k D ln ZT kH = |Dt ln ZT0 |2 dt > 0, P 0 a.s. 0

From Theorem 3.1.1 in Nualart (1995), we deduce that the distribution law of ln ZT0 under P 0 admits a density with respect to the Lebesgue measure on R, and so the required result. 2 Proposition 4.2 Let Assumption 4.1 hold and suppose that one of the two following conditions holds : (i) U10 (h) ≥ U20 (h), (ii) U10 (h) < U20 (h) and condition (CL) holds. Then the function H is continuous on (0, ∞) and for all x > 0, there exists yˆ(x) > 0 (not necessarily unique) such that H(ˆ y (x)) = x. Proof. First, notice that from (4.4), we have : ZT0 χ(yZT0 ) ≤ ZT0 I2 (yZT0 ), ∀y > 0.

(4.8)

1) We suppose that condition (i) holds. Then by Proposition 4.1 (4.2), function χ is continuous on (0, ∞). From Assumption 4.1, (4.8) and the dominated convergence theorem, we obtain the continuity of H on (0, ∞). 9

2) We now suppose that condition (ii) holds. From Proposition 4.1 (4.3), function χ is right continuous. The right-continuity of H is then stated from Assumption 4.1, (4.8), the nondecreasing feature of function I2 and dominated convergence theorem. To prove the left-continuity of H in y > 0, take a nondecreasing sequence of positive real (yn )n such that yn % y. We then see from (4.3) that :

χ(yn ZT0 ) → χ(yZT0 ) + (I2 − I1 1y(h) 0. ZT χ(yn ZT ) ≤ ZT I2 2 We deduce by the dominated convergence theorem that : H(yn ) → H(y) + (I2 − I1 1y(h)
(4.9)

Using Lemma 4.1, this proves the left-continuity and then the continuity of H in y. 3) In all cases, by noting that χ(y) → ∞ when y → 0, we see, by Fatou’s Lemma, that H(y) → ∞ when y → 0. By noting that χ(y) → 0 when y → ∞, we obtain, by the dominated convergence theorem that H(y) → 0 when y → 0. This property combined with the continuity of H proves the existence of yˆ(x) > 0 such that H(ˆ y (x)) = x, for all x > 0. 2 Remark 4.8 The nonrandomness of the critical value h is only required in this last proposition, see (4.9). Indeed, in this case, y(h) is nonrandom and by Lemma 4.1, P 0 [yZT0 = y(h)] = 0, for all y > 0, which implies the continuity of function H. Adapting arguments of conjugate duality in complete markets, we characterize the solu˜ tion to problem (3.2) and prove the duality relation between value functions J and J. Theorem 4.1 Suppose that conditions of Proposition 4.2 hold. Then for all x > 0, there exists an optimal portfolio θˆ for problem (3.2) whose terminal wealth is given by :
ˆ = χ yˆ(x)Z 0 , X T

(4.10)

where χ is defined in Proposition 4.1 and yˆ(x) given by Proposition 4.2. The associated optimal wealth is given by : h i x,θˆ P0 ˆ Ft , 0 ≤ t ≤ T. Xt = E X (4.11) 10

Moreover, we have the duality relation :   ˜ + xy , J(x) = min J(y) y>0

x > 0.

ˆ given Proof. From definition of yˆ(x), the nonnegative FT -measurable random variable X in (4.10) lies in L1 (P 0 ) and 0

ˆ = x. E P [X]

(4.12)

0 ˆ t ], 0 ≤ t ≤ T . By the Consider then the nonnegative (P 0 , F)-martingale Mt = E P [X|F martingale representation property under P 0 (see e.g. Lemma 6.7 p.25 in Karatzas and Shreve 1998) and relation (2.2), we obtain the existence of a portfolio θˆ ∈ A(x) such that : Z t ˆ Mt = x + θˆu0 dSu = Xtx,θ , 0 ≤ t ≤ T. (4.13)

0

Now, by definition of χ in Proposition 4.1, we have for all θ ∈ A(x) : ˆ − yˆ(x)ZT0 X. ˆ U (XTx,θ ) − yˆ(x)ZT0 XTx,θ ≤ U˜ (ˆ y (x)ZT0 ) = U (X) Taking expectation and using (2.5), (4.12), we obtain that : ˆ EU (XTx,θ ) ≤ EU (X), ∀θ ∈ A(x). ˆ = X x,θˆ by (4.13), this proves that θˆ is solution to (3.2) and J(x) = EU (X). ˆ Relation Since X T (4.11) is simply relation (4.13). By definition of U˜ and from (2.5), we have for all x > 0, θ ∈ A(x), y > 0 :   h i
EU XTx,θ ≤ E U˜ yZT0 + yE ZT0 XTx,θ ˜ + xy, ≤ J(y) ˜ + xy). On the other hand, given x > 0, we have by definition of and so J(x) ≤ inf y>0 (J(y) χ and by (4.10), (4.12) : h i  
0 0 ˆ ˆ ˜ J(x) = EU X = E U yˆ(x)ZT + yˆ(x)E ZT X ˜ y (x)) + xˆ = J(ˆ y (x), which proves the last assertion of the theorem.

2

Conditions of the previous theorem does not include the case where λ ≡ 0 and U10 (h) < U20 (h). In such a context, recall that U is not concave, and function H is equal to χ, which may be discontinuous. One can then not apply the martingale approach. However, it is possible to derive directly the value function of problem (3.2). 11

Theorem 4.2 Suppose that λ ≡ 0 and U is bounded from below. Then the value function J of problem (3.2) is equal to U con , the concave envelope of U (i.e. the least concave majorant function of U ) and we have the duality relation :   ˜ J(x) = inf J(y) + xy , x > 0. (4.14) y>0

Proof. In the case λ ≡ 0, the dynamics of S is governed by dSt = σt dWt . It is convenient to change of control variable by defining πt = σt0 θt . We introduce then the dynamic value function associated to problem (3.2) by :   J (t, x) = sup E U (XTt,x,π ) , t ∈ [0, T ], x > 0, (4.15) π∈Π(t,x)

RT where Π(t, x) is the set of adapted processes (πs )t≤s≤T satisfying t |πs |2 ds < ∞ and such that : Z s t,x,π Xs := x + πu0 dWu ≥ 0, t ≤ s ≤ T, t

Xtt,x,π

= x.

Notice that with these notations, we have J(x) = J (0, x). From dynamic programming principle (see e.g. Fleming and Soner 1993), the value function J is a lower-semicontinuous viscosity supersolution of :   1 2 ∂ 2w ∂w + inf − p = 0. − ∂t p∈R 2 ∂x2

(4.16)

By using similar arguments as in Lemma 5.1 in Cvitanic, Pham and Touzi (1999), we deduce from this last relation that function J is concave in x and nonincreasing in t (this is formally proved by sending p respectively to infinity and zero in (4.16)). Moreover, we clearly have from (4.15) and Fatou’s lemma (recall that U is bounded from below) that J (T − , x) ≥ U (x). By definition of the concave envelope, this implies that :

J (t, x) ≥ U con (x),

t ∈ [0, T ), x > 0.

On the other hand, since U (∞) > −∞, it is clear that U con is nondecreasing. We then have for all t ∈ [0, T ), x > 0, π ∈ Π(t, x) :     E U (XTt,x,π ) ≤ E U con (XTt,x,π )
≤ U con E[XTt,x,π ] ≤ U con (x), 12

where the second relation follows from Jensen’s inequality and the third from the fact that E[XTt,x,π ] ≤ x and the nondecreasing feature of U con . This proves that J (t, x) ≤ U con (x) and ˜ so the required equality. Finally, by noting that J(y) = U˜ (y) (since ZT0 = 1), the duality relation (4.14) follows from Proposition I.4.1. in Ekeland and Temam (1976). 2 Remark 4.9 In the case where U is concave and so J ≡ U , the optimal control is given by π ≡ 0. This means that the optimal portfolio is to invest nothing in the stocks. When U is not concave, the control problem (4.15) is singular : there is no optimal control in the class A(x) (an optimal one would be obtained for a process θ taking only values 0 and infinity). Remark 4.10 Theorems 4.1 and 4.2 show that, although duality relation between U and U˜ does not hold (see Remark 4.2), we have duality relation between value functions J and ˜ J.

5 5.1

Examples Power-Power utility function

We consider the example where Ui , i = 1, 2, are power utility function with constant relative risk aversion 1 − αi , αi ∈ (0, 1) : xα 1 , α1 xα 2 U2 (x) = + C, α2 U1 (x) =

α

α

where C = hα11 − hα22 is a constant added in order to ensure continuity of the utility function U , i.e. U1 (h) = U2 (h). Notice that U10 (0) = ∞ and U10 (h) ≥ U20 (h) iff hα1 ≥ hα2 . Case : hα1 ≥ hα2 We have :  − 1 1−α2  ,  y χ(y) = h,  1  − 1−α 1, y

0 < y < hα2 −1 hα2 −1 ≤ y ≤ hα1 −1 y > hα1 −1

Case : hα1 < hα2 We have :

φ(y) =

y −β1 y −β2 − − C, β1 β2 13

y ∈ [hα1 −1 , hα2 −1 ],

where βi = αi /(1 − αi ). Then y(h) is the unique solution in (hα1 −1 , hα2 −1 ) of hα1 y −β1 − β1 α1

=

y −β2 hα2 − . β2 α2

Function χ is explicitly expressed in : ( χ(y) =

y y

1 − 1−α

2

,

0 < y < y(h)

1 − 1−α 1

,

y ≥ y(h).

Notice that χ is discontinuous in y(h). A straightforward computation leads to : ( U˜ (y) =

y −β2 + β2 −β 1 y , β1

C,

0 < y < y(h) y ≥ y(h).

Function U˜ is differentiable on (0, ∞) except in y(h). For y 6= y(h), we have U˜ 0 (y) = −χ(y). For y = y(h), the subgradient of U˜ is given by h i − 1 − 1 −∂ U˜ (y(h)) = y(h) 1−α1 , y(h) 1−α2 . We easily check that any element xˆ in the interior of −∂ U˜ (y(h)) does not attain the maximum in U˜ (y(h)), i.e. U (ˆ x) − xˆy(h) < U˜ (y(h)).

5.2

Exponential-Logarithm utility function

We consider the example where U1 is an exponential utility function with absolute risk aversion η and U2 is a logarithm utility function :

U1 (x) = − exp(−ηx), U2 (x) = ln x + C, where C = − exp(−ηh) − ln h is a constant added in order to ensure continuity of the utility function U , i.e. U1 (h) = U2 (h). We see that U10 (h) = ηe−ηh < U20 (h) = h1 and function U is non concave. We have U10 (0) = η and ( − ηy + ln y − ηy ln ηy + 1 − C, ηe−ηh < y ≤ η ∧ h1 φ(y) = ln y − C, η ∧ h1 < y ≤ h1 . We have to distinguish two cases depending on the sign of φ(U10 (0)) = ln ηh + e−ηh . 14

Case : ln ηh + e−ηh ≥ 0 Then y(h) is the unique solution in (ηe−ηh , η ∧ h1 ) of − ηy + ln y − ηy ln ηy = C − 1 and  1   y, χ(y) = − η1 ln ηy ,   0,

0 < y < y(h) y(h) ≤ y < η y ≥ η.

Case : ln ηh + e−ηh < 0 Notice that this implies ηh < 1. We then have y(h) = eC and ( 1 , 0 < y < eC y χ(y) = 0, y ≥ eC Notice that χ is discontinuous in eC . A straightforward computation leads to : ( U˜ (y) =

− ln y + C − 1, −1,

0 < y < eC y ≥ eC .

Function U˜ is differentiable on (0, ∞) except in eC . For y 6= eC , we have U˜ 0 (y) = −χ(y). For y = eC , the subgradient of U˜ is given by −∂ U˜ (eC ) = [0, e−C ]. We easily see that any element xˆ in the interior of −∂ U˜ (eC ) does not attain the maximum in U˜ (eC ).

5.3

Power Loss function-Power Utility function

We consider the case where : (h − x)p , 0 < x ≤ h, p (x − h)α U2 (x) = , x > h, α

U1 (x) = −

where p > 1 and 0 < α < 1. Then, U10 (h) = 0 and U20 (h) = ∞, and so U10 (h) ≤ U10 (0) < U20 (h). q −β p α We easily see that ˜l(y) = yq , where q = p−1 and g˜(y) = y β , where β = 1−α . Moreover, ( −β yq − yβ , 0 ≤ y ≤ hp−1 q φ(y) = −β p − hp − y β + yh, y > hp−1 . 15

p

Then φ(U10 (0)) = hq − of φ(U10 (0)). Case : hp+(p−1)β ≤ βq

h−(p−1)β β

and we have to distinguish two cases depending on the sign

p

Then y(h) is the solution of − hp −

y −β β

( χ(y) = Case : hp+(p−1)β >

+ hy = 0 in (hp−1 , ∞), and we get :

h + y −1−β , 0,

0 < y < y(h) y ≥ y(h)

q β

Then y(h) ∈ (0, hp−1 ) and is equal to y(h) =

χ(y) =

1   q+β

  −1−β  ,   h+y

q β

, and we have :

0
h − y q−1 ,     0,

1   q+β

q β

≤ y < hp−1

y ≥ hp−1

A straightforward computation leads to :  y −β     β − yh, U˜ (y) = yq − yh,  q    hp −p,

0
1   q+β

q β

≤ y < hp−1

y ≥ hp−1

1 1   q+β   q+β Function U˜ is differentiable on (0, ∞) except in βq . For y 6= βq , we have U˜ 0 (y) 1   q+β q = −χ(y). For y = β , the subgradient of U˜ is given by

−∂ U˜

! " # q−1 1   q+β   α−1   q+β q q q q+β = h− ,h − . β β β

We easily check that any element xˆ in the interior of −∂ U˜   1  q q+β maximum in U˜ . β Notice that in both cases, hp+(p−1)β > y(h).

6

q β

  1  q+β q β

does not attain the

and hp+(p−1)β ≤ βq , function χ is discontinuous in

Case of constant market price of risk

In this section, we consider the case where the market price of risk λ is a nonzero constant; this is essentially the Black-Scholes model. The density of the risk-neutral martingale 16

measure is then given by : Zt0

  1 2 0 0 = exp −λ Wt + |λ| t , 0 ≤ t ≤ T. 2

(6.1)

It follows that ZT0 /Zt0 is independent of Ft and has same distribution law under P 0 as ZT0 −t . We deduce from (4.11) that the optimal wealth process is given by : ˆ

Xtx,θ = H(t, yˆ(x)Zt0 ), 0 ≤ t ≤ T,

(6.2)

where H(t, y) = E P

0



 χ(yZT0 −t ) , (t, y) ∈ [0, T ] × (0, ∞),

and yˆ(x) > 0 is solution of H(0, yˆ(x)) = H(ˆ y (x)) = x. Moreover, when function H is 1,2 smooth C , the optimal portfolio is simply obtained by applying Itˆo’s formula on (6.2) and identifying diffusion terms : ∂H θˆt = − (t, yˆ(x)Zt0 )ˆ y (x)Zt0 (σt0 )−1 λ, 0 ≤ t ≤ T. ∂y

(6.3)

In the sequel, we provide some explicit examples where we compute function H. We introduce the following notations : for all τ ∈ (0, T ], c ∈ (0, ∞), γ ∈ R, we denote ln c − |λ|2 τ (γ + 21 ) √ . d(τ, c, γ) = |λ| τ We also denote by Φ the distribution function of the standard normal law : Z d Φ(d) = ϕ(z)dz, d ∈ R, −∞

where ϕ(z) =

√1 2π

exp(−z 2 /2).

Lemma 6.1 For all τ ∈ (0, T ], c ∈ (0, ∞), γ ∈ R, we have :    0 γ  γ(γ + 1)|λ|2 τ P0 E (Zτ ) 1Zτ0 ≤c = exp Φ(d(τ, c, γ)). 2    
 |λ|2 τ 1 P0 0 Φ(d(τ, c, 0)) − ϕ d τ, c, E ln Zτ 1Zτ0 ≤c = 2 2 2

Proof. See Appendix.

We now provide explicit expressions of function H for the examples of the previous section. Power-power utility function 17

We consider the example of paragraph 5.1. By using Lemma 6.1, a straightforward calculation leads to : • For hα1 ≥ hα2 :      1 β2 |λ|2 (T − t) hα2 −1 1 − 1−α 2 exp H(t, y) = y Φ d T − t, ,− 1 − α2 2 y 1 − α2        α2 −1 α1 −1 h h ,0 − Φ d T − t, ,0 + h Φ d T − t, y y      1 hα1 −1 1 β1 |λ|2 (T − t) − 1−α 1 exp 1 − Φ d T − t, ,− . +y 1 − α1 2 y 1 − α1 • For hα1 < hα2 :     y(h) 1 β2 |λ|2 (T − t) 2 exp H(t, y) = y Φ d T − t, ,− 1 − α2 2 y 1 − α2      2 1 β1 |λ| (T − t) y(h) 1 − 1−α 1 exp +y 1 − Φ d T − t, ,− . 1 − α1 2 y 1 − α1 1 − 1−α



Exponential-logarithm utility function We consider the example of paragraph 5.2. By using Lemma 6.1, we get : • For ln ηh + e−ηh ≥ 0 :    1 y(h) H(t, y) = Φ d T − t, , −1 y y         1 y(h) y |λ|2 (T − t) η − − Φ d T − t, ,0 ln + Φ d T − t, , 0 η η 2 y y        1 η 1 y(h) 1 + ϕ d T − t, , − ϕ d T − t, , . η y 2 y 2 • For ln ηh + e−ηh < 0 :    eC 1 H(t, y) = Φ d T − t, , −1 . y y Power loss function-power utility function We consider the example of paragraph 5.3. Again, by using Lemma 6.1, we obtain : • For hp+(p−1)β ≤ βq :    y(h) H(t, y) = hΦ d T − t, ,0 y      y(h) |λ|2 (T − t) −1−β Φ d T − t, +y exp β(1 + β) , −1 − β . 2 y

18

• For hp+(p−1)β > βq :  

1   q+β

q β

  H(t, y) = hΦ d T − t,

  , 0

y

   1   q+β q   β |λ|2 (T − t)    Φ d T − t, , −1 − β  + y −1−β exp β(1 + β) 2 y    1   q+β q    β hp−1     + h Φ d T − t, ,0 − Φ d T − t, , 0 y y 

     |λ|2 (T − t) hp−1 ,q − 1 − y exp q(q − 1) Φ d T − t, 2 y      1 q−1

  − Φ d T − t,

q+β

q β

 , q − 1 .

y

In all those examples, function H is smooth C 1,2 and the optimal portfolio is given by (6.3).

7

Numerical results

In this section, we consider an agent in a Black-Scholes-Merton model :   σ2 St = S0 exp (µ − )t + σWt , 2

(7.1)

with power nonconcave utility functions, who modifies her risk aversion from 1−α1 to 1−α2 at a level of wealth h. This is the example of paragraph 5.1 with hα1 < hα2 . We provide numerical results for the optimal wealth-proportion invested in the risky asset S starting from initial wealth x, and which is given by : π ˆt :=

θt St

= pˆ(t, St ),

ˆ

Xtx,θ

where : µ pˆ(t, s) := σ2

−y ∂H ∂y H

!

 t, yˆ(x)

s S0

− µ2 σ

e

1 µ2 ( −µ)t 2 σ2

! ,

from (2.1), (6.3) and (7.1). We compare our results with the constant optimal wealthproportion in the Merton model for power utility function with risk aversion 1 − αi : µ 1 πmerton (αi ) = . σ 2 1 − αi 19

We focus first on an agent whose risk aversion decreases when she reaches a high level of wealth : Graphicss 1 illustrate this case for the values α1 = 0.2, α2 = 0.7 and h = 10x = 900. We have also set µ = 0.15, σ = 0.1 and S0 = 90. Given a trajectory of the asset price St (Figure 1b), Figure 1.a gives the evolution of the optimal portfolio t 7→ π ˆt = pˆ(t, St ). Graphics 1.c and 1.d. provide the graph of the optimal proportion function pˆ(t, .) for a long and short maturity. Graphics 1 show that the agent starts with a strategy close to the Merton’s optimal strategy πmerton (α2 ) corresponding to the lower risk aversion 1 − α2 . When the time to maturity decreases, she switchs to the Merton’s optimal strategy πmerton (α1 ) corresponding to the higher risk aversion 1 − α1 . Notice that the size of this switch of strategy is more important as the range between α1 and α2 is large. For large time to maturity, the agent adopts the behavior of the Merton’s agent with the lower risk aversion 1 − α2 since she expects a higher objective value function. However, for short time to maturity, she must take into account her actual wealth which shall remain with large probability under the level h, and so she adopts the behavior of the Merton’s agent with risk aversion 1 − α1 . Similarly, Graphics 2 illustrate the case of an agent whose risk aversion increases when her wealth decreases largely. We choose the values α1 = 0.7, α2 = 0.2 and h = x/100 = 0.9. We have also set µ = −0.15, σ = 0.1 and S0 = 90. Again, for large time to maturity, the agent follows the strategy of the Merton’s agent with lower risk aversion 1 − α1 . For short time to maturity, she must take into account her actual wealth which shall remain with large probability above the level h, and so she follows the strategy of the Merton’s agent wity risk aversion 1 − α2 .

20

Figure 1: Graphics 1ab

21

Figure 2: Graphics 1cd

22

Figure 3: Graphics 2ab

23

Figure 4: Graphics 2cd

8 8.1

Appendix Proof of Proposition 4.1

We introduce the Fenchel-Legendre transform U˜i of the convex function −Ui (−), for i = 1, 2 : U˜1 (y) = sup0 0, so that : U˜ (y) = max[U˜1 (y), U˜2 (y)],

y > 0.

The functions U˜i are convex and we easily see that U˜i (y) = Ui (χi (y)) − χi (y)y, i = 1, 2, where χ1 is a continuous nonincreasing function valued in (0, h], defined by :   0 < y < U10 (h)  h, χ1 (y) = (8.1) I1 (y), U10 (h) ≤ y < U10 (0)   0 0, y ≥ U1 (0) 24

(where the third domain is empty when U10 (0) = ∞) and χ2 is a continuous nonincreasing function valued in [h, ∞) and defined by ( I2 (y), 0 < y < U20 (h) χ2 (y) = (8.2) h, y ≥ U20 (h). We then have U˜ (y) = U (χ(y)) − χ(y)y, where : χ(y) = χ1 (y)1U˜1 (y)≥U˜2 (y) + χ2 (y)1U˜1 (y)
(8.3)

In order to compute explicitly χ, we have to characterize the domain {y > 0 : U˜1 (y) < U˜2 (y)}. Let us define the following functions : U¯1 (y) = U1 ◦ I1 (y) − y (I1 (y) − h) , for U10 (h) ≤ y < U10 (0), U¯2 (y) = U2 ◦ I2 (y) − y (I2 (y) − h) , for 0 < y ≤ U20 (h). These two functions are continuously differentiable and we have : U¯10 (y) = h − I1 (y) > 0, for U10 (h) < y < U10 (0) U¯20 (y) = h − I2 (y) < 0, for 0 < y < U20 (h). Therefore, function U¯1 is strictly increasing and function U¯2 is strictly decreasing. Noting that for i = 1, 2, we have U¯i (Ui0 (h)) = Ui (h), we deduce that : U¯1 (y) − U1 (h) > 0, for U10 (h) < y < U10 (0) U¯2 (y) − U2 (h) > 0, for 0 < y < U20 (h).

(8.4) (8.5)

We first compute χ on the two following domains : a) For y ∈ (0, U10 (h) ∧ U20 (h)), we have χ1 (y) = h and χ2 (y) = I2 (y). Hence, U˜1 (y) − U˜2 (y) = U1 (h) − U¯2 (y) = U2 (h) − U¯2 (y) < 0, by (8.5). Therefore, by (8.3), χ(y) = I2 (y). d) For y ∈ [U10 (0) ∨ U20 (h), ∞), (notice that when U10 (h) ≥ U20 (h), U10 (0) ∨ U20 (h) = U10 (0)), we have χ1 (y) = 0 and χ2 (y) = h. If U10 (0) = ∞, this case is vacuous. Otherwise, U˜1 (y)− U˜2 (y) = U1 (0)−U2 (h)+hy = U1 (0)−U1 (h)+hy ≥ U1 (0)−U1 (h)+hU10 (0) > 0, since U1 is strictly concave on (0, h]. Therefore, by (8.3), χ(y) = 0. For the other domains, we now distinguish two cases : First case : U10 (h) ≥ U20 (h). b1) For y ∈ [U20 (h), U10 (h)], we have χ1 (y) = χ2 (y) = h. Hence, by (8.3), χ(y) = h. 25

c1) For y ∈ (U10 (h), U10 (0)), we have χ1 (y) = I1 (y) and χ2 (y) = h. Hence, U˜1 (y) − U˜2 (y) = U¯1 (y) − U2 (h) = U¯1 (y) − U1 (h) > 0, by (8.4). Therefore, by (8.3), χ(y) = I1 (y). Relation (4.2) is then stated by combining a), b1 ), c1 ) and d ). Second case : U10 (h) < U20 (h). We shall see below that U˜1 − U˜2 is actually equal to the function φ introduced in (4.1) on [U10 (h), U20 (h)]. We first prove that there exists a unique y(h) ∈ (U10 (h), U20 (h)) such that φ(y(h)) = 0. For all y ∈ [U10 (h), U20 (h)], we have : ( U¯1 (y) − U¯2 (y), for U10 (h) ≤ y ≤ U10 (0) ∧ U20 (h), φ(y) = U1 (0) − U¯2 (y) + hy, for U10 (0) ∧ U20 (h) < y ≤ U20 (h). Function φ is continuous on [U10 (h), U20 (h)]. Since φ is differentiable for y ∈ (U10 (h), U10 (0) ∧ U20 (h))∪(U10 (0)∧U20 (h), U20 (h)), and φ0 (y) > 0, we get that φ is a continuous strictly increasing function on [U10 (h), U20 (h)]. Recalling that U1 (h) = U2 (h), we have : φ(U10 (h)) = U2 (h) − U¯2 (U10 (h)) = U¯2 (U20 (h)) − U¯2 (U10 (h)), ( U1 (0) − U¯2 (U20 (h)) + hU20 (h) > U1 (0) − U1 (h) + hU10 (0), φ(U20 (h)) = U¯1 (U20 (h)) − U¯1 (U10 (h)),

for U10 (0) < U20 (h) for U10 (0) ≥ U20 (h).

From the strictly monotonicity of U¯2 and U¯1 , and the strict concavity of U1 , we obtain that, φ(U10 (h)) < 0 and φ(U20 (h)) > 0. We deduce the existence of an unique y(h) ∈ (U10 (h), U20 (h)) such that φ(y(h)) = 0, and : φ(y) < 0, for U10 (h) ≤ y < y(h)

(8.6)

U20 (h).

(8.7)

φ(y) > 0, for y(h) < y ≤ Notice that y(h) < U10 (0) iff φ(U10 (0) ∧ U20 (h)) > 0. To compute χ, we have to distinguish several cases.

b2) For y ∈ [U10 (h), U10 (0) ∧ U20 (h)), we have χ1 (y) = I1 (y) and χ2 (y) = I2 (y). Hence, U˜1 (y)− U˜2 (y) = φ(y), and from (8.6)-(8.7) and (8.3), χ(y) = I2 (y)1y U20 (h), we have χ1 (y) = I1 (y) and χ2 (y) = h. Hence, by same arguments as in c1 ), we have χ(y) = I1 (y).

26

Finally, we obtain the expression (4.3) of χ by noting the following points. When U10 (0) > U20 (h), we have φ(U20 (h) ∧ U10 (0)) > 0 and so y(h) < U10 (0). When U10 (0) ≤ U20 (h), we have either [y(h), y(h) ∨ U10 (0)) = ∅ and χ(y) = I2 (y) on [U10 (h), U10 (0)), whenever φ(U10 (0)) ≤ 0; or [y(h), y(h) ∨ U10 (0)) = [y(h), U10 (0)) and χ(y) = 0 on [U10 (0), U20 (h)), whenever φ(U10 (0)) > 0. The proof is ended by combining a), b2 ), c2 ) and d ).

8.2

Proof of Lemma 6.1

Consider the probability measure Qγ with density with respect to P 0 given by :   γ 2 |λ|2 dQγ 0 0 T . = exp −γλ WT − dP 0 2 Then, from (6.1) and Bayes formula, we have :    0 γ  γ(γ + 1)|λ|2 τ P0 E (Zτ ) 1Zτ0 ≤c = exp Qγ [Zτ0 ≤ c]. 2 By noting that Zτ0

  1 0 Qγ 2 = exp −λ Wτ + |λ| τ (γ + ) , 2

γ

where WtQ = Wt0 + γλt is a Qγ -brownian motion by Girsanov’s theorem, we obtain the first relation of the Lemma. On the other hand, from (6.1), we have : i |λ|2 τ h 
 0 P 0 [Zτ0 ≤ c], ln Zτ0 1Zτ0 ≤c = E P N 1N ≤d(c,τ, 1 ) + 2 2 √ where N = −λ0 Wτ0 /(|λ| τ ) is a standard normal random variable under P 0 . Finally, using the first relation of the Lemma for γ = 0, we obtain the required result. EP

0

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