Contingent Claims with Random Expiration and Time-optimal Portfolio Selection Thomas Balzer (e-mail: [email protected]) Abstract. In a complete financial market model where the prices of the assets are modeled as Ito diffusion processes, the problem of hedging a contingent claim whose expiration is a stopping time is considered. As in models with a deterministic time horizon, each sufficiently integrable claim can be replicated by a dynamic strategy. The analysis of such claims is motivated by problems of timeoptimal portfolio selection. Facing such a problem, an investor is trying to find an optimal stochastic time horizon such that an originally horizon-independent investment goal can be achieved. As examples of such goals the problems of reaching a predetermined level of wealth and of reaching a specified level of expected utility are discussed. Key words: contingent claims, stopping times, reaching a target wealth, expected utility, risk aversion JEL Classification: G11 Mathematics Subject Classification (1991): 90A09, 60G44

1

Introduction

The mathematical analysis of contingent claims is not only of interest within the context of option pricing, it also plays a central role in the martingale approach to portfolio optimization (cf. [16]). For the problem of finding a trading strategy to maximize the expected utility from terminal wealth this approach consists in decomposing the dynamic problem of choosing the optimal portfolio into a static problem of determining the optimal terminal wealth and a representation problem namely deriving a strategy that yields this predetermined wealth (for an introduction cf. [21] pp. 59ff.). The optimal portfolio turns out to be the replicating or hedging strategy – which always exists in a complete market – for the optimal terminal wealth interpreted as a contingent claim.

1

Since the majority of portfolio optimization problems analyzed in the literature deal with a time horizon fixed in advance (e.g. maximizing utility from terminal wealth at some specified date, cf. [16], [21] or [24]; maximizing the probability to reach a target wealth at a latter date, cf. [6] or [9]), the relevant contingent claims arising from these problems are options with a deterministic expiration date. If one studies the problem of structuring a portfolio such that a target wealth is reached in minimal expected time, the theory of contingent claims with a fixed expiration fails to be applicable since there is no time horizon specified in advance (for a study of such problems cf. [11], [12]). The problem of reaching a target wealth can be interpreted as a special problem of time-optimal portfolio selection, an optimization problem which consists in finding a portfolio such that an investment goal which is independent of some specified time horizon is reached within an optimal stochastic period. The aim of this article is to show that such a problem indeed allows for a martingale approach if a special type of option is considered: a contingent claim whose expiration date is a stopping time. Related contingent claims have lately received growing attention in optimization problems with an uncertain timehorizon: In the context of portfolio optimization with discretionary stopping as studied in [17], options with a random but bounded expiration arise. In [4] and [8] options with a random expiration not necessarily being a stopping time are analyzed. In this case the market model becomes incomplete due to additional timing risk. In the context of optimization problems where the time horizon serves as objective function however, it seems reasonable to restrict to stopping times: the investor should only be able to choose a random time horizon based on available market information. The structure of the article is as follows: In Section 2 the standard mathematical model for a complete financial market (as in [19] or [22]) is set up. Since there is no time horizon fixed in advance, a model with an infinite time horizon has to be considered. In Section 3 the concept of contingent claims with random expiration is introduced. As Proposition 1 implies, the “randomization” of the expiration date does not imply new risk, the market model remains complete which also has been mentioned in [8]. This result extends well-known facts from the literature on European options with a deterministic expiration date (cf. [19]). Section 4 serves as an introduction into the principle of time-optimal portfolio selection where the stochastic time horizon is the objective function. Different time intervals are compared by means of the expected return of one growthoptimally invested unit of wealth. This turns out to be both economically reasonable and convenient from a mathematical point of view. The problem of reaching a target wealth and of reaching a level of expected utility are laid out

2

as prime examples of typical problems where the time horizon can be optimized. In Section 5 the investment problem of reaching a target wealth is studied in detail. The relevant optimization problem involves minimization over all portfolio-stopping time pairs which allow for reaching the aspired wealth. By means of options with random expiration this problem can be equivalently restated in terms of a minimization over such options. In this form the investment problem allows for an immediate solution which is already known from the literature (cf. [11] and [12]): the log-optimal portfolio is the appropriate strategy. In Section 6 the investment problem of reaching a level of expected utility is analyzed. Again, the relevant optimization problem can be restated as a problem of minimization over options with random expiration. For utility functions which inhibit a global risk aversion greater than the one of a logarithmic utility function, the log-optimal strategy is again optimal. This is however not true in the converse case of a smaller risk aversion which is shown by means of a counterexample. In Section 7 it is seen by a simple Lagrange multiplier argument that the log-optimal strategy is as well optimal if the investor is more risk averse than a logarithmic investor locally at the certainty equivalent of the aspired utility. In Section 8 we analyze a non-standard utility function which arises when the investor is aiming at a predetermined probability to exceed a target wealth. This analysis is mainly based on tools from sequential testing. Conclusions from the derived results are drawn in Section 9, the proof of Proposition 1 is deferred to an appendix. It has to be mentioned that it is not the aim of this article to derive an entire duality theory for problems of time-optimal portfolio optimization but only to state some principles and immediate results. A duality theory for these problems relies heavily on methods of stopping and deciding under constraints (cf. [3]) and can be found in [1]. For many problems however, as shown in this contribution, a duality approach is not necessary to state optimality of the trading strategies.

2

The market model

Let there be given a complete probability space (Ω, F, P), endowed with a standard d-dimensional Brownian motion W = (W (1) , . . . , W (d) )∗ . The information structure is described by the filtration F := {F(t)}0≤t<∞ , the usual Paugmentation of F W (t) := σ(W (s), 0 ≤ s ≤ t) for t ∈ R+ . The financial market consists of d+1 financial assets, one riskless bond, whose price is modeled by the equation dP0 (t) = P0 (t)r(t)dt ,

3

P0 (0) = 1

(1)

and d risky stocks whose prices are governed by the equations (i = 1, . . . , d) dPi (t) = Pi (t)[bi (t)dt +

d X

σij (t)dW (j) (t)] , Pi (0) = pi > 0.

(2)

j=1

The market coefficients r (interest rate), b (vector of stock appreciation rates) and σ (volatility matrix) are all assumed to be progressively measurable with respect to F. Moreover, σ is assumed to be invertible and the processes r, b, σ as well as σ −1 are bounded uniformly in (t, ω) ∈ [0, T ] × Ω for each T ∈ R+ . Furthermore, r is assumed to be non-negative. Due to the invertibility of σ one is able to define the d-dimensional process θ – the market price of risk – by θ(t) := (σ(t))−1 (b(t) − r(t)1)

(3)

for t ∈ R+ with 1 := (1, . . . , 1)∗ ∈ Rd . As θ is bounded and F-progressively measurable as well, the process {Z0 (t)}0≤t<∞ defined as Z t Z 1 t Z0 (t) := exp{− θ∗ (s)dW (s) − kθ(s)k2 ds} (4) 2 0 0 with Z0 (0) = 1 is a non-negative exponential martingale (cf. [18] p. 199) An agent who starts out with an initial capital x0 and who cannot influence the asset prices by his actions is able to decide at each instant t ∈ R+ which proportion πi (t) of his wealth X(t) to invest in the i-th stock. With the vector Pd π(t) = (π1 (t), . . . , πd (t))∗ chosen, the proportion 1 − i=1 πi (t) of the agent’s wealth is invested risklessly. The assumption of a self-financing condition leads to the wealth equation dX(t)

= X(t){(r(t) + π ∗ (t)σ(t)θ(t))dt + π ∗ (t)σ(t)dW (t)}

(5)

with X(0) = x0 > 0. Subsequently, a portfolio π is assumed to be a F-progressively measurable process π = {π(t)}0≤t<∞ with values in Rd such that for the associated wealth, RT i.e. the solution X = X x,π of (5), the equation 0 X 2 (s)kπ(s)k2 ds < ∞ holds a.s. for each T ∈ R+ . A portfolio such that X x,π is a non-negative process is called admissible. The set of all portfolios admissible for x is denoted A0 (x). The state-price-density process H0 , defined as Z t  Z t  1 Z0 (t) H0 (t) := = exp − (r(s) + kθ(s)k2 )ds − θ∗ (s)dW (s) (6) P0 (t) 2 0 0 for t ∈ R+ , plays a central role subsequently. It can be easily seen that for each π ∈ A0 (x0 ) the equation d(H0 (t)X x0 ,π (t))

= H0 (t)X x0 ,π (t){π ∗ (t)σ(t) − θ∗ (t)}dW (t) 4

(7)

holds true, such that the process H0 X x0 ,π is a non-negative, local P-martingale, hence a P-supermartingale. In particular, for two F-stopping times τ1 and τ2 satisfying P(τ1 ≤ τ2 ) = 1 E[H0 (τ2 )X x0 ,π (τ2 )|F(τ1 )] ≤ H0 (τ1 )X x0 ,π (τ1 )

(8)

holds with H0 (∞)X x0 ,π (∞) := 0. The choice τ1 = 0 leads to E(H0 (τ )X x0 ,π (τ )) ≤ x0

(9)

for each F-stopping time τ (cf. [25], Theorem 3.3, p. 70).

3

Contingent claims with random expiration

In the subsequent paragraph, the problem of hedging a contingent claim whose expiration is a stopping time is considered. Such a claim can be interpreted as a special case of a Bermudan option (cf. [13] p. 459) where the set of possible exercise (stopping) times is a singleton. T := {τ | τ is F-stopping time with P(τ < ∞) = 1}

(10)

Definition 1 (Contingent claim with random expiration) Let τ ∈ T be a F-stopping time. 1. A contingent claim with expiration τ or shortly a τ -option is a F(τ )-measurable random variable B : Ω → [0, ∞) with 0 < E(H0 (τ )B) < ∞.

(11)

2. The hedging price of a τ -option B is defined as uB := inf{x0 > 0| ∃ π ∈ A0 (x0 ) with X x0 ,π (τ ) ≥ B a.s.}.

(12)

As common, we set inf ∅ = ∞. 3. A τ -option B is called attainable, if there exists an initial capital x0 ∈ (0, ∞) and a portfolio process π ∈ A0 (x0 ) such that for the associated wealth process X x0 ,π X x0 ,π (τ ) = B (13) holds almost surely. For deterministic expiration dates, the following proposition on the completeness of the market model is a well-known fact (cf. [15] p. 16 or [21] pp. 25ff.). The proof that this result extends to random expiration dates as well is deferred to the appendix. 5

Proposition 1 The infimum in (12) is attained and its value is uB = E(H0 (τ )B).

(14)

Furthermore there exists a portfolio process πB ∈ A(uB ) such that XB := X uB ,πB is defined by XB (t) =

1 E[H0 (τ )B|F(t)]; 0 ≤ t < ∞. H0 (t)

(15)

This portfolio is called the hedging portfolio for the τ -option B. In particular X uB ,πB (τ ) = B holds a.s. Hence, the contingent claim B is attainable with initial capital uB and portfolio πB .

4

Time-optimal portfolio selection

4.1

The time horizon as objective function

The basic idea of time-optimal portfolio selection consists in regarding the time horizon as the objective function that is to be minimized: the planning horizon is not specified in advance. The investor’s aim is to reach an investment goal which is independent of a specific date within an optimal time interval (for a more detailed analysis of these problems cf. [1], a related study can be found in [7]). Admissible periods are stochastic intervals of the form [0, [ τ]] := {(t, ω) ∈ R+ × Ω | 0 ≤ t ≤ τ (ω)}

(16)

for F-stopping times τ . To determine an optimal time interval which allows for reaching the investment goal, a criterion is needed to compare different intervals. Although it is possible to compare intervals by means of their expected length, we define that a stopping time τ1 ∈ T is preferred to some τ2 ∈ T if E(ζ(τ1 )) ≤ E(ζ(τ2 )) holds with the process ζ defined as Z t 1 ζ(t) := (17) (r(s) + kθ(s)k2 )ds 2 0 for t ∈ R+ . Subsequently, only stopping times belonging to the set T ∗ := {τ ∈ T | E(ζ(τ )) < ∞}

(18)

are considered for which the objective function ζ has an inherent financial interpretation. Lemma 1 (Version of Wald’s equality) For each stopping time τ ∈ T ∗ Z τ 1 (19) E(− ln(H0 (τ ))) = E( (r(s) + kθ(s)k2 )ds) = E(ζ(τ )) 2 0 holds. 6

Proof: Due to the definition of H0 in (6) the equality Z τ Z τ 1 E(− ln(H0 (τ ))) = E( (r(s) + kθ(s)k2 )ds + θ∗ (s)dW (s)) 2 0 0 Rτ follows. Since r is non-negative, E(ζ(τ )) < ∞ implies E( 0 kθ(s)k2 ds) < ∞. Rτ ∗ With E( 0 θ (s)dW (s)) = 0 (cf. [10], p. 62) the assertion follows.  Remark 1 (on the choice of the objective function) The portfolio π K , defined for t ∈ R+ as π K (t) = (σ ∗ (t))−1 θ(t) has the property – besides other optimality properties proven later on – that π K maximizes the long-term growth rate of capital (cf. [15], pp. 42ff.). Furthermore, inserting this special portfolio K in (7) proves that the wealth process X 1,π coincides with H0−1 . Consequentially, for τ ∈ T ∗ the quantity E(ζ(τ )) = E(− ln(H0 (τ ))) is simply the expected logarithmic return of one growth-optimally invested unit of wealth up to time τ . If time is measured by means of the objective function ζ, an interval [0, [ τ1]] is preferred to another interval [0, [ τ2]] if the lost profit associated with the first time period is less than the one associated with the latter period. If the market coefficients r, b σ are constant with −r 6= kθk2 /2 and τ ∈ T ∗ is a stopping time 1 E(τ ) = E(ζ(τ ))/(r + kθk2 ) 2

(20)

follows. Hence, in a constant coefficient framework comparing intervals by means of the objective function ζ is essentially equivalent to comparing intervals by their expected length.

4.2

Problems of time-optimal portfolio selection

It is possible to consider several different investment goals the investor can pursue within the framework of time-optimal portfolio selection. These goals can generally be classified by their variability: A static investment goal does not vary over time whereas a dynamic investment goal contains an inherent variability. A prime example for a dynamic goal is the aim to outperform a stochastic benchmark. In the present contribution only static goals are considered, dynamic goals are for example studied in [2]. Within this article two different kinds of static investment goals are considered: First the goal of reaching a target wealth and second, the aim of reaching a target level of expected utility. The initial capital the investor is endowed with at time t = 0 is always denoted x0 > 0.

7

Reaching a target wealth. The investment goal consists in reaching a target wealth x ˆ ∈ R for which 0 < x0 < x ˆ is assumed to avoid trivialities. The problem is to derive a stopping time τ and an admissible portfolio π ∈ A0 (x0 ), such that the associated wealth at the end of the stochastic interval [0, [ τ]] exceeds x ˆ. This problem can be stated as follows:

Problem 1 Minimize E(ζ(τ )) subject to (τ, π) ∈ G (1) , with

G (1) := {(τ, π) ∈ T ∗ × A0 (x0 ) | X x0 ,π (τ ) ≥ x ˆ a.s.}.

(21)

In this context, it is interesting under which conditions on the market coefficients G (1) 6= ∅ holds. The idea to derive a portfolio to reach a target wealth in minimal expected time goes back to Heath and Sudderth (cf. [11] and [12]). Although the optimal strategy for this investment goal is already known, this simplest problem of timeoptimal portfolio selection will subsequently serve as an example for the derived methods. Reaching a target level of expected utility. The preference structure of the investor is modelled by a utility function U which at this point only is assumed to be non-decreasing. The investment goal is to reach a level of expected utility u ˆ fulfilling U (x0 ) = U (X(0)) < u ˆ. Again, the time horizon and the portfolio have to be chosen such that the expected utility of the wealth at the end of the stochastic planning period exceeds the target level u ˆ. With the aim to find an optimal time horizon, the following optimization problem arises:

Problem 2 Let U be a utility function and let U (x0 ) < u ˆ. Minimize E(ζ(τ )) subject to (τ, π) ∈ G (2) , with

G (2) := {(τ, π) ∈ T ∗ × A0 (x0 ) | E(U (X x0 ,π (τ ))) ≥ u ˆ}.

(22)

Several choices of utility functions U can be thought of at this point. Of course, standard utility functions are those of the form U (x) = xα /α for α ∈ (−∞, 1)\{0} or U (x) = ln(x). With the choice U (x) = 1[ˆx,∞) (x), Problem 2 has the interpretation that the investor wants to acheive a predetermined probability of surpassing a target wealth x ˆ.

8

5

Investment goal: Reaching a target wealth

The optimization problems 1 and 2 involve the minimization of an objective function over a class of pairs of stopping times and admissible portfolios. Due to the completeness of the market model, each pair consisting of a stopping time and a portfolio can be identified with a pair consisting of a stopping time and a random expiration option. This allows to state the relevant problems in a more tractable way which is done for Problem 1 in the following section. The identification of an option with its replicating strategy is of course the core of the martingale approach to traditional portfolio optimization (cf. [19] pp. 101ff. or [16]). For E ⊆ R define L(E) := {Y : [0, ∞) × Ω → E | Y is progressively measurable}.

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Lemma 2 The equation inf

(τ,ξ)∈G (10 )

E(ζ(τ )) =

inf

E(ζ(τ ))

(24)

(τ,π)∈G (1)

0

holds with the set G (1 ) defined as 0

G (1 ) := {(τ, ξ) ∈ T × L({ˆ x}) | E(H0 (τ )ξ(τ )) ≤ x0 }.

(25)

Proof: For each pair (τ, π) ∈ G (1) the constant process ξ ≡ x ˆ fulfils (τ, ξ) ∈ (10 ) G , since (21) and (9) imply E(H0 (τ )ξ(τ )) = E(H0 (τ )ˆ x) ≤ E(H0 (τ )X x0 ,π (τ )) ≤ x0 . Hence E(ζ(τ )) ≥ inf (τ 0 ,ξ)∈G (10 ) E(ζ(τ 0 )) follows for each (τ, π) ∈ G (1) which implies inf

E(ζ(τ )) ≥

(τ,π)∈G (1)

inf

(τ,ξ)∈G (10 )

E(ζ(τ )).

This inequality holds trivially if G (1) = ∅. 0 Conversely, let (τ, ξ) ∈ G (1 ) 6= ∅ be given. The hedging price of the τ -option ξ(τ ) = x ˆ does not exceed x0 since E(H0 (τ )ξ(τ )) ≤ x0 . Thus, there exists an admissible portfolio π with X x0 ,π (τ ) ≥ x ˆ and hence (τ, π) ∈ G (1) . For each 0 (τ, ξ) ∈ G (1 ) one obtains E(ζ(τ )) ≥ inf (τ 0 ,π)∈G (1) E(ζ(τ 0 )) and consequentially inf

(τ,ξ)∈G (10 )

E(ζ(τ )) ≥

inf

E(ζ(τ )).

(τ,π)∈G (1)

0

The special case G (1 ) = ∅ is again obvious.  Taking into account the result of the preceding Lemma 2, Problem 1 can be transformed into a problem that involves the minimization over a class of stopping times and random expiration options. Problem 10 9

0

Minimize E(ζ(τ )) subject to (τ, ξ) ∈ G (1 ) . Before Problem 10 can solved, an auxiliary result is needed. Lemma 3 If ζ(∞) = ∞ holds a.s. then for each x > x0 the stopping time τx

:=

inf{t ∈ R+ |

1 x = } H0 (t) x0

(26)

fulfils τx ∈ T ∗ . Proof: Obviously, Z τx

=

t

inf{t ∈ R+ | 0

1 (r(s) + kθ(s)k2 )ds + 2

Z

t

θ∗ (s)dW (s) = ln(

0

x )} x0

holds. Since for k ∈ N the stopping time τx ∧ k is bounded, clearly τx ∧ k ∈ T ∗ follows and hence (cf. Lemma 1) E(ζ(τx ∧ k)) = E(− ln(H0 (τx ∧ k))) ≤ ln(

x ). x0

This implies E(ζ(τx )) ≤ ln(x/x0 ) < ∞, so only P(τx < ∞) = 1 remains to be proven. Since with c := E(1{τx <∞} ζ(τx )) > −∞ E(1{τx =∞} ζ(∞)) = E(ζ(τx )) − E(1{τx <∞} ζ(τx )) ≤ ln(

x )−c<∞ x0

holds, P(τx = ∞) = 0 and τx ∈ T ∗ follow.  The transformation of Problem 1 into the Problem 10 is justified by the fact that the latter problem admits an immediate solution. Proposition 2 Under the assumption ζ(∞) = ∞ a.s. the stopping time τxˆ := inf{t ∈ R+ |

x ˆ 1 = } H0 (t) x0

(27)

fulfils inf

(τ,ξ)∈G (10 )

E(ζ(τ )) = E(− ln(H0 (τxˆ ))) = ln(ˆ x) − ln(x0 ). 0

(28)

Proof: Let τ ∈ T ∗ be a stopping time such that (τ, x ˆ) ∈ G (1 ) holds. Jensen’s inequality implies E(ζ(τ )) = E(− ln(H0 (τ ))) ≥ − ln(E(H0 (τ ))) ≥ ln(ˆ x) − ln(x0 ) 0

due to the definition of G (1 ) . Therefore inf

(τ,ξ)∈G (10 )

E(ζ(τ )) ≥ ln(ˆ x) − ln(x0 ).

10

0

The assumption ζ(∞) = ∞ ensures that (τxˆ , x ˆ) ∈ G (1 ) holds (cf. Lemma 3). Since the sample paths of H0 are a.s. continuous, H0 (τxˆ ) = x0 /ˆ x and E(ζ(τxˆ )) = E(− ln(H0 (τxˆ ))) = ln(ˆ x) − ln(x0 ) follow.  Although by Lemma 2 the Problems 1 and 10 have the same value, the solution of Problem 10 does not shed light on the optimal trading strategy that has to be pursued to reach the investment goal. In this case however, the optimal strategy can be readily stated. Proposition 3 Under the assumption ζ(∞) = ∞ a.s. the pair (τxˆ , π K ) ∈ T ∗ × A0 (x0 ) with τxˆ defined as in (27) and the portfolio π K = (σ ∗ )−1 θ fulfils (τxˆ , π K ) ∈ G (1) and is optimal for Problem 1. Proof: Due to Lemma 3, τxˆ ∈ T ∗ holds. Furthermore, the wealth process K according to the initial capital x0 and the portfolio π K is given as X x0 ,π = K x0 (H0 )−1 . Hence, X x0 ,π (τxˆ ) = x0 (H0 (τxˆ ))−1 = x ˆ. Consequentially, (τxˆ , π K ) ∈ (1) G and ln(

x ˆ Prop. ) = E(ζ(τxˆ )) = x0

2

inf

(τ,ξ)∈G (10 )

E(ζ(τ ))

Lemma 2

=

inf

E(ζ(τ ))

(τ,π)∈G (1)

which implies that (τxˆ , π K ) attains the infimum in Problem 1.  Remark 2 (Kelly strategy) The portfolio π K which is optimal for reaching a target wealth x ˆ within the time interval [0, [ τxˆ]] is called Kelly strategy, due to [20], where a similar strategy is proposed within a different setting. Breiman (cf. [5]) proved that for favorable discrete-time games the Kelly system minimizes the expected time to reach a target wealth. In a continuous-time framework, this property of the Kelly strategy, which is furthermore the log-optimal strategy for traditional portfolio optimization (cf. [21], p. 71f. or [22] pp. 212-213), is known due to [12]. Nevertheless, the method of deriving this result used here differs from all above mentioned approaches. Example 1 The market coefficients r, b σ are all assumed to be constant. Let σ be invertible and r > − 21 kθk2 . With these assumptions ζ(∞) = ∞ holds such that Propositions 2 and 3 are applicable and the Kelly strategy is the optimal portfolio for Problem 1. The minimal expected time to reach the target wealth x ˆ is (cf. (20)) 1 1 E(τxˆ ) = E(− ln(H0 (τxˆ )))/(r + kθk2 ) = (ln(ˆ x) − ln(x0 ))/(r + kθk2 ). 2 2

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This minimal expected time to reach x ˆ increases with a higher target wealth and decreases with a higher interest rate and/or a higher market price of risk.

11

6

Investment goal: Reaching a level of expected utility

Problem 2 which results from the investment goal of aiming at a target level of expected utility can be as well transformed into a problem of minimization over stopping times and random expiration options. Lemma 4 The equation inf

(τ,ξ)∈G (20 )

E(ζ(τ )) =

inf

E(ζ(τ ))

(30)

(τ,π)∈G (2)

0

holds with the set G (2 ) defined as 0

G (2 ) := {(τ, ξ) ∈ T ∗ × L(R+ )| E(H0 (τ )ξ(τ )) ≤ x0 , E(U (ξ(τ ))) ≥ u ˆ }.

(31)

Proof: Similar as in the proof of Lemma 2 it is sufficient to show the unique 0 relation between G (2 ) and G (2) , i.e. for each (τ, π) ∈ G (2) there exists ξ ∈ L(R+ ) 0 with (τ, ξ) ∈ G (2 ) and vice versa. For (τ, π) ∈ G (2) define ξ(t) := X x0 ,π (t) for t ∈ R+ . Since π is admissible ξ ∈ L(R+ ) follows. Furthermore E(U (ξ(τ ))) = E(U (X x0 ,π (τ ))) ≥ u ˆ holds. Equation (9) implies E(H0 (τ )ξ(τ )) = E(H0 (τ )X x0 ,π (τ )) ≤ x0 and hence (τ, ξ) ∈ 0 G (2 ) . 0 Conversely, let (τ, ξ) ∈ G (2 ) be given. The random variable ξ(τ ) can be regarded as a τ -option with a hedging price E(H0 (τ )ξ(τ )) ≤ x0 . Hence, Proposition 1 yields the existence of a portfolio π such that X x0 ,π (τ ) ≥ ξ(τ ) a.s. holds. Since U is assumed to be non-decreasing, E(U (X x0 ,π (τ ))) ≥ E(U (ξ(τ ))) ≥ u ˆ and (τ, π) ∈ G (2) follow.  Again, the Problem 2 can be transformed into a minimization problem with identical value. Problem 20 0

Minimize E(ζ(τ )) subject to (τ, ξ) ∈ G (2 ) . In fact, for a special class of utility functions, Problem 20 allows again for an immediate solution. Proposition 4 Let U be a strictly increasing utility function such that the mapping x 7→ ln(U −1 (x)) is convex. Furthermore, ζ(∞) = ∞ a.s. is assumed. Then inf

(τ,ξ)∈G (20 )

E(ζ(τ )) = E(− ln(H0 (τU ))) = ln(U −1 (ˆ u)) − ln(x0 )

(32)

holds with τU := inf{t ∈ R+ |

1 U −1 (ˆ u) = }. H0 (t) x0 12

(33)

With these assumptions, the Kelly strategy π K is optimal for reaching the utility level u ˆ, i.e. the pair (τU , π K ) ∈ T ∗ × A0 (x0 ) is optimal for Problem 2. 0

Proof: For each (τ, ξ) ∈ G (2 ) one obtains in application of Jensen’s inequality E(− ln(H0 (τ )))

= E(− ln(H0 (τ )ξ(τ ))) + E(ln(U −1 [U (ξ(τ ))])) ≥

− ln(E(H0 (τ )ξ(τ ))) + ln(U −1 [E(U (ξ(τ )))])

since x 7→ ln(U −1 (x)) and x 7→ − ln(x) are convex. Furthermore, the mappings − ln and x 7→ ln(U −1 (x)) are decreasing resp. increasing such that E(− ln(H0 (τ ))) ≥

− ln(x0 ) + ln(U −1 (ˆ u))

follows from E(H0 (τ )ξ(τ )) ≤ x0 and E(U (ξ(τ ))) ≥ u ˆ. Hence, inf

(τ,ξ)∈G (20 )

E(ζ(τ )) ≥ ln(

U −1 (ˆ u) ). x0

This infimum is attained by the pair (τU , ξU ) with ξU ≡ U −1 (ˆ u) which can be 0 shown to belong to the set G (2 ) (cf. proof of Proposition 2). The optimality of the Kelly strategy follows as in Proposition 3.  It is interesting to notice that under the assumptions of Proposition 4, the wealth according to the optimal portfolio takes at the stopping time τU a.s. the K value X x0 ,π (τU ) = x0 (H0 (τU ))−1 = U −1 (ˆ u). Apparently, this wealth generates K an expected utility of E(U (X x0 ,π (τU ))) = u ˆ, but this utility level is in fact not only reached in the mean but with probability 1. Although the investment goal consists of reaching an average utility of u ˆ, the optimal trading strategy acheives this goal pathwise. This feature of the solution of Problem 2 in the above special case can be understood with a closer look on the assumptions imposed on U . Remark 3 The convexity of the mapping x 7→ ln(U −1 (x)) can be interpreted in terms of the investor’s risk aversion. Obviously, x 7→ ln(U −1 (x)) is convex if and only if x 7→ U (ln−1 (x)) is concave. The concavity of x 7→ U (ln−1 (x)) means that the investor whose preference structure is modelled by the function U is globally more risk averse than an investor with a logarithmic utility function. (cf. [14], Theorem 5, p. 40). This is for example the case for utility functions of the form U (x) = xα /α with α < 0. An investor who is globally less inclined to take risk than a “logarithmic investor” is not willing to reduce the time span to reach his investment goal if the price he has to pay for this is the acceptance of an average utility of u ˆ rather than a safe utility of this extent. Proposition 4 can not be applied for utility functions of the form U (x) = xα /α with α ∈ (0, 1). However, it can be shown here that – in contrast to the case 13

α < 0 – the time to reach the target utility can be strictly reduced by refraining from reaching u ˆ almost surely. Example 2 The market coefficients r, b and σ are assumed constant, σ is invertible and r > 0. Let U be defined as U (x) = xα /α with α ∈ (−∞, 1) \ {0}. First, consider a deterministic time horizon [0, T ]. Within the context of traditional portfolio optimization, the aim is to maximize the expected utility E(U (X x0 ,π (T ))) at time T . For this special choice of utility function, the optimal portfolio for this objective is well-known to be the portfolio π1 = (σ ∗ )−1 θ/(1 − α) (cf. [15], p. 39). The maximal utility that can be reached is then given as E(U (X x0 ,π1 (T )))

= U (x0 exp((r +

1 1 kθk2 )T )). 2 (1 − α)

(34)

As an easy consequence, E(U (X x0 ,π1 (T ))) ≥ u ˆ

⇔ T ≥

ln(U −1 (ˆ u)) − ln(x0 ) 1 1 kθk2 r + 2 (1−α)

1 follows. Hence, the time horizon T 0 := (ln(U −1 (ˆ u)) − ln(x0 ))/(r + 21 (1−α) kθk2 ) is the shortest deterministic time horizon that enables the investor to reach an expected utility equal to u ˆ. Comparing the values of the objective function ζ at time T 0 and at the stopping time τU as defined in (33) yields

E(ζ(τU )) < E(ζ(T 0 )) ⇔ α < 0. For α < 0 the stochastic time interval [0, [ τU],] that allows for reaching u ˆ has to be preferred to the deterministic interval [0, T 0 ]. This is no surprise since for α < 0 Proposition 4 yields the optimality of τU . For α ∈ (0, 1) however, there exists a deterministic time horizon [0, T 0 ] which is more preferable than [0, [ τU]] such that the target utility level can be reached. 0 The pair (T 0 , ξ) with ξ(t) := X x0 ,π1 (t) (t ∈ R+ ) is in G (2 ) such that for α ∈ (0, 1) inf

(τ,ξ)∈G (20 )

E(ζ(τ )) ≤ E(ζ(T 0 )) < E(ζ(τU ))

(35)

follows. In particular, the Kelly strategy can not be optimal for reaching an expected utility level in the case U (x) = xα /α with α ∈ (0, 1). For this choice of a preference structure where the investor is globally less risk averse than an investor with a logarithmic utility function, refraining from almost surely reaching the aspired utility does in fact result in a shorter expected time to reach the goal. The utility level can be reached in the mean faster than it can be reached with probability 1. 14

The case where the investor’s preference structure is modelled by the utility function U (x) = ln(x) deserves further attention. Example 3 (Logarithmic Utility) Since the mapping x 7→ ln(ln−1 (x)) is convex, Proposition 4 implies that for the logarithmic utility function U (x) = ln(x), the Kelly strategy π K enables the investor to reach the target utility u ˆ over the time horizon [0, [ τU].] Assume there exists some T ∈ R+ with E(− ln(H0 (T ))) = u ˆ − ln(x0 ). The expected utility of an investor pursueing the Kelly strategy at time T is given as E(ln(x0 H0 (T )−1 )) = ln(x0 ) − E(ln(H0 (T ))) = u ˆ. With the prerequisites of Proposition 4 one obtains inf

(τ,ξ)∈G (20 )

E(ζ(τ )) = E(− ln(H0 (τU ))) = u ˆ − ln(x0 ) = E(− ln(H0 (T ))).

(36)

Consequentially, in this case the optimal time horizon can also be chosen to be deterministic. Both [0, T ] and the stochastic interval [0, [ τU]] allow for reaching the target utility u ˆ when the Kelly strategy is pursued. Although both intervals are equivalent in terms of the objective function ζ and the optimal trading strategy is in both cases identical, there are profound differences: The interval [0, [ τU]] is bounded by a stopping time, the wealth according to π K evaluated at time τU however has as distribution the unit mass in U −1 (ˆ u). The interval [0, T ] has the advantage of having a bounded length, the wealth at the end of this interval however is not distributed according to a Dirac measure. If a bounded time horizon is preferred this has to be paid with a variance on the wealth side. If the wealth should not inhibit variance, insecurity has to be tolerated on the time axis.

7

From global to local risk aversion

In the last section it has been possible to derive time-optimal trading strategies in the case where the investor is globally more risk averse than an investor with a logarithmic utility function. In fact, with refined methods it is possible to analyze the impact of local risk aversion as well. For this purpose, we need to distinguish among utility functions. Definition 2 A strong utility function is a mapping U : R → R∪{−∞} which is strictly increasing, strictly concave, continuous differentiable in dom(U ) := {x ∈ R | U (x) > −∞} and which satisfies limx→∞ U 0 (x) = 0 and limx↓a U 0 (x) = ∞ with a := inf{x ∈ R | x ∈ dom(U )}. 15

For a strong utility function U we denote I := (U 0 )−1 and the convex ˜ (y) := supx∈R (U (x) − xy) for y > 0. Furthermore, conjugate function U ˜ U (0) = limx→∞ U (x) is set. The following simple result on Lagrange multipliers for constrained optimization serves as the key lemma. ˜ ∈ G (20 ) such that Lemma 5 If there exists µ = (µ1 , µ2 )∗ ∈ R2+ and a pair (˜ τ , ξ) one of the following conditions holds: ˜ τ )) = x0 , E(U (ξ(˜ ˜ τ ))) = u 1. E(H0 (˜ τ )ξ(˜ ˆ and ˜ τ ) − x0 ) + µ2 (ˆ ˜ τ )))] E[ζ(˜ τ ) + µ1 (H0 (˜ τ )ξ(˜ u − U (ξ(˜ =

inf

(τ,ξ)∈T ∗ ×L(R+ )

E[ζ(τ ) + µ1 (H0 (τ )ξ(τ ) − x0 ) + µ2 (ˆ u − U (ξ(τ )))] (37)

if U is an arbitrary increasing utility function or 2. ˜ ( µ1 H0 (τ ))] ˆ − µ2 U E[ζ(˜ τ )] = inf ∗ E[ζ(τ ) − µ1 x0 + µ2 u τ ∈T µ2

(38)

for a strong utility function U , we have inf

(τ,ξ)∈G (20 )

E(ζ(τ )) = E(ζ(˜ τ )).

(39)

0

Proof: Let (τ, ξ) ∈ G (2 ) be arbitrary. Under condition (37) we have E(ζ(τ )) ≥ E[ζ(τ ) + µ1 (H0 (τ )ξ(τ ) − x0 ) + µ2 (ˆ u − U (ξ(τ )))] ˜ τ ) − x0 ) + µ2 (ˆ ˜ τ )))] = E(ζ(˜ ≥ E[ζ(˜ τ ) + µ1 (H0 (˜ τ )ξ(˜ u − U (ξ(˜ τ )) which implies the assertion. If U is a strong utility function and (38) holds, we have with E(ζ(τ )) ≥ E[ζ(τ ) + µ1 (H0 (τ )ξ(τ ) − x0 ) + µ2 (ˆ u − U (ξ(τ )))] ˜( ≥ E[ζ(τ ) − µ1 x0 + µ2 u ˆ − µ2 U

µ1 H0 (τ ))] ≥ E(ζ(˜ τ )) µ2

the assertion.  For a twice differentiable, strong utility function U the Arrow-Pratt measure of relative risk aversion defined by RRAU (x) := −

xU 00 (x) x>0 U 0 (x) 16

(40)

(cf. [14] p. 39) will now play a central role. Utility functions of the type Uα (x) = xα /α for α ∈ (−∞, 1) \ {0} have constant relative risk aversion RRAUα ≡ 1 − α, for the logarithmic utility function RRAln (x) = 1 follows for all x. Proposition 5 Let U be a strong utility function which is furthermore assumed to be twice differentiable. It is assumed that ζ(∞) = ∞ holds a.s. and the target utility level u ˆ satisfies U (x0 ) < u ˆ as well as RRAU (U −1 (ˆ u)) > 1. Under these prerequisites (τU , ξU ) with τU as in (33) and ξU ≡ U −1 (ˆ u) is optimal for Problem 2’. The pair (τU , π K ) is optimal for Problem 2. Proof: The proof is an application of Lemma 5. As multipliers, we choose µ1 = x−1 0 and µ2 :=

1 U 0 (¯ u)I(U 0 (¯ u))

=

1 U 0 (¯ u)¯ u

with u ¯ := U −1 (ˆ u). Since U 0 (¯ u) > 0 and u ¯ > x0 > 0 hold, µ2 > 0 follows. Define ˜ (y) for κ := −µ1 x0 +µ2 u the function g(y) := κ−ln(y)−µ2 U ˆ+ln(µ1 )−ln(µ2 ). The 0 function g is differentiable with g (y) = −1/y + µ2 I(y). This derivative is zero at y = U 0 (¯ u) due to the choice of µ2 . The second derivative g 00 (y) = 1/y 2 + µ2 I 0 (y) at y0 = U 0 (¯ u) is strictly positive if 1 1 1 >0 ⇔ + 2 00 y0 y0 I(y0 ) U (I(y0 ))

−

I(y0 )U 00 (I(y0 )) >1 y0

due to the concavity of U . The last expression can be easily seen to be equivalent to RRAU (¯ u) > 1 which has been assumed. Hence, g takes a global minimum at y = U 0 (¯ u) with value g(U 0 (¯ u)) = ln(U −1 (ˆ u)) − ln(x0 ). (20 ) We now have (τU , ξU ) ∈ G (cf. Proposition 2) with E(H0 (τU )ξU (τU )) = x0 and E(U (ξU (τU ))) = u ˆ as well as E(ζ(τU ))

= ln(U −1 (ˆ u)) − ln(x0 ) ≤

≤

inf E(g(

τ ∈T ∗

µ1 H0 (τ ) )) µ2

˜( inf E[ζ(τ ) − µ1 x0 + µ2 u ˆ − µ2 U

τ ∈T ∗

µ1 H0 (τ ))] µ2

such that the optimality of the pair (τU , ξU ) follows from Lemma 5. The optimality of the Kelly strategy for Problem 2 is again immediate under the given prerequisites. 

Example 4 (Exponential Utility) Consider the exponential utility function U (x) = 1 − e−λx for λ > 0. The target utility level u ˆ is assumed to satisfy 17

U (x0 ) < u ˆ as well as U −1 (ˆ u) > λ−1 which is equivalent to RRAU (U −1 (ˆ u)) > 1 = RRAln (U −1 (ˆ u)). Then, (τU , ξU ) with τU as defined in (33) and ξU ≡ U −1 (ˆ u) is optimal for Problem 20 and the best possible trading strategy is again the Kelly one according to Proposition 5. Remark 4 The prerequisite of Proposition 5 means that the Kelly strategy is optimal for reaching an expected utility of u ˆ if the investor is locally more risk averse at the capital U −1 (ˆ u) than an investor with a logarithmic utility function. If this is case the investor is again reaching the utility level almost surely and not only in the mean. Consequentially, the results derived in Section 6 do not depend on a preference structure that is globally more risk averse than a logarithmic utility function. For utility functions with constant relative risk aversion, global and local risk aversion do in fact coincide. Hence, Proposition 5 proves again the – already known – optimality of the Kelly strategy for U (x) = xα /α and 1 − α > 1. The capital U −1 (ˆ u) at which the relative risk aversion has to exceed 1 can be interpreted as the certainty equivalent of a utility u ˆ, it is the safe amount of money that generates a utility u ˆ. If the investor is more risk averse than a logarithmic investor locally at the certainty equivalent of the aspired utility, the Kelly system turns out to be the appropriate trading strategy.

8

Example: bability

Reaching an above-target pro-

Within this section we study Problem 2 for the utility function U (x) = 1[ˆx,∞) (x). This choice implies that the investor wants to reach a fixed probability u ˆ ∈ (0, 1) to exceed the target wealth x ˆ > x0 which is obviously a weaker goal than the one pursued in Problem 1. For the rest of this section, we assume zero interest rate r ≡ 0. For this special utility function the equivalent problem 2’ takes a rather special form. Lemma 6 For the choice U (x) = 1[ˆx,∞) (x) the equation inf

(τ,ξ)∈G (2∗ )

holds with the set G (2 G (2

∗

)

∗

)

E(ζ(τ )) =

inf

E(ζ(τ ))

(41)

(τ,π)∈G (2)

defined as

:= {(τ, ξ) ∈ T ∗ × L({0, 1})| E(H0 (τ )ξ(τ )) ≤

x0 , E(1 − ξ(τ )) ≤ 1 − u ˆ}. (42) x ˆ

Proof: The proof is similar to the one of Lemma 4.

18

For (τ, π) ∈ G (2) define ξ(t) := U (X x0 ,π (t)) for t ∈ R+ . The choice of U implies ξ ∈ L({0, 1}). Furthermore E(ξ(τ )) = E(U (X x0 ,π (τ ))) ≥ u ˆ or equivalently E(1 − ξ(τ )) ≤ 1 − u ˆ. The admissibility of π and the definition of ξ yields x ˆξ(t) ≤ X x0 ,π (t), thus E(H0 (τ )ˆ xξ(τ )) ≤ E(H0 (τ )X x0 ,π (τ )) ≤ x0 due to (2∗ ) Equation (9). Hence (τ, ξ) ∈ G . ∗ Let (τ, ξ) ∈ G (2 ) be given. The τ -option B := x ˆξ(τ ) fulfils E(H0 (τ )ξ(τ )) ≤ x0 /ˆ x. Thus (cf. Proposition 1) there exists a portfolio π with X x0 ,π (τ ) ≥ x ˆξ(τ ) a.s. Consequentially, E(U (X x0 ,π (τ ))) ≥ E(U (ˆ xξ(τ ))) = E(ξ(τ )) ≥ u ˆ which implies (τ, π) ∈ G (2) .  In this equivalent form, Problem 2’ allows for an immediate solution based on the theory of sequential analysis as presented in [23]. Since the process H0 is a martingale when interest rates are neglected, Problem 2’ coincides with the problem of sequentially testing for the drift of a Brownian Motion. For the proof of the following result we therefore refer to [23], Section 17.6. Proposition 6 Assume P(ζ(∞) = ∞) = 1 and x0 /ˆ x
(τ,ξ)∈G (2∗ )

:=

inf{t ∈ R+ |

1 1−u ˆ u ˆx ˆ ∈ /( , )}, H0 (t) 1 − x0 /ˆ x x0

(43)

0≤t<∞

(44)

:= 1{H0 (t)−1 ≥ˆuxˆ/x0 } , ∗

)

and satisfies

E(ζ(τ )) = E(ζ(τP )) = u ˆ ln(

1−u ˆ u ˆx ˆ ). ) + (1 − u ˆ) ln( x0 1 − x0 /ˆ x

(45)

The optimal solution for Problem 2 is the pair (τP , πP ) where πP is the hedging strategy for the τP -option B := x ˆξP (τP ). Although the optimal time horizon is known, it is not possible in this case to derive the optimal trading strategy explicitly. However, it is clear, that the stratK egy π K cannot coincide with the optimal πP , since X x0 ,π (τP ) = x0 H0 (τP )−1 = u ˆx ˆ
9

Conclusion

Comparing different time intervals by means of the expected return of one growth-optimally invested unit of wealth contains an implicit risk aversion. This 19

risk aversion is exactly equivalent to a preference structure modelled by a logarithmic utility function. In particular, in a constant coefficient framework, comparing time intervals by means of their expected length inhibits risk aversion comparable to logarithmic utility. This explains as well why the log-optimal trading strategy is optimal for minimizing the time until a target wealth is reached. When the problem of reaching an expected utility level is analyzed, two preference structures are considered: One on the time axis which has been seen to correspond to logarithmic utility, a second preference structure is considered on the wealth side, modelled by a function U . The results obtained in Section 6 and 7 can now be interpreted in the following way: A problem of time-optimal portfolio selection deals with the interplay of time and wealth. The investor tends to tolerate risk (and some insecurity always has to be tolerated) on the axis where he is less risk averse. If the function U represents a risk aversion globally or locally greater than the one of a logarithmic utility function (i.e. the risk aversion on the wealth axis is greater than the one on the time axis) insecurity is tolerated on the time axis: The optimal time horizon is a stochastic period whereas the final wealth is deterministic. If the risk aversion on the time axis surpasses the one on the wealth axis (which is the case for U (x) = xα /α and α ∈ (0, 1)) a deterministic time horizon is preferred to a deterministic wealth. Insecurity is tolerated on the wealth side by means of non-degenerate distributed final wealth. If both preference structures on the time and on the wealth axis coincide, insecurity can be tolerated in each component: it is however not possible to completely eliminate any risk.

A

Proof of Proposition 1

Proof: The proof is executed in two steps: Step 1: It is shown, that for z := E(H0 (τ )B) the inequality z ≤ uB holds true. For uB = ∞ this inequality holds trivially. In case uB < ∞ there exists x0 > 0 and a portfolio π ∈ A0 (x0 ) with X x0 ,π (τ ) ≥ B a.s. Since H0 X x0 ,π is a supermartingale, E(H0 (τ )X x0 ,π (τ )) ≤ x0 follows (cf. (9)). Hence, z = E(H0 (τ )B) ≤ E(H0 (τ )X x0 ,π (τ )) ≤ x0 . Taking the infimum over all x0 admitting a replication of B yields z ≤ uB . Step 2: It is shown that z ≥ uB holds. It suffices to prove the existence of a portfolio π ∈ A0 (z) such that X z,π (τ ) = B holds a.s. The random variable H0 (τ )B is integrable and F(τ )-measurable. For t ∈ R+ define the martingale M (t) := E[H0 (τ )B|F(t)] which can be considered in an a.s. right-continuous modification (cf. [18], Theorem 1.3.13, p. 16). Due to the martingale representation theorem (cf. ibd., pp. 182ff.) there exists a Rd -valued, progressively 20

measurable process m with for all t ∈ R+

RT 0

km(s)k2 ds < ∞ a.s. for all T ∈ R+ , such that t

Z

m∗ (s)dW (s)

M (t) = z + 0

holds. The process XB , defined by XB (t) := H0 (t)−1 M (t) for t ∈ R+ can be chosen to have a.s. continuous paths. Furthermore, it can be seen that XB coincides with the wealth process according to a capital z and the portfolio πB defined as πB (t) := (

(σ ∗ (t))−1 m(t) + (σ ∗ (t))−1 θ(t))1{H0 (t)XB (t)>0} , H0 (t)XB (t)

for t ∈ R+ (cf. [22], Lemma 2.65, pp. 69-71). RT It has to be proven that πB is admissible, i.e. 0 kXB (t)πB (t)k2 dt < ∞ holds for all T ∈ R+ . If T ∈ R+ is arbitrary and S denotes the stopping time S = inf{t | XB (t) = 0}, the integrability condition reduces to R T ∧S kXB (t)πB (t)k2 dt < ∞. On the set {(t, ω) | t < S(ω)} 0 πB (t)XB (t) =

(σ ∗ (t))−1 m(t) + (σ ∗ (t))−1 θ(t)XB (t) H0 (t)

holds. The process H0 restricted on [0, T ] is a.s. strictly positive with continuous paths. Consequentially there exists some set N1 ∈ F with P(N1 ) = 0 and 0 < H (T ) (ω) = mint∈[0,T ] H0 (t, ω) for all ω ∈ N1C . Furthermore there exists RT a P-zero set N2 ∈ F, such that 0 km(s, ω)k2 ds < ∞ for all ω ∈ N2C . The boundedness of (σ ∗ )−1 by a constant κ(σ, T ) implies Z

T ∧S

k 0

(σ ∗ (s))−1 m(s) 2 k ds ≤ H0 (s)

T ∧S

Z 0

κ(σ, T )2 km(s)k2 ds < ∞ (H (T ) )2

for all ω ∈ (N1 ∪ N2 )C . Due to the continuity of the sample paths of XB there exists a set N3 ∈ F with P(N3 ) = 0 and maxt∈[0,T ] XB (t, ω) =: X (T ) (ω) < ∞ for all ω ∈ N3C . Since θ is bounded by some κ(θ, T ) T ∧S

Z

k(σ ∗ (s))−1 θ(s)XB (s)k2 ds ≤

0

Z

T ∧S

κ(σ, T )2 κ(θ, T )2 (X (T ) )2 ds < ∞

0

holds for all ω ∈ N3C . Hence Z

T ∧S

kXB (s)πB (s)k2 ds

0

≤ 2(

κ(σ, T )2 (H (T ) )2

Z

T ∧S

km(s)k2 ds +

Z

0

0

21

T ∧S

(κ(σ, T )κ(θ, T )X (T ) )2 ds) < ∞

holds a.s. Since the process XB is non-negative, πB is an admissible strategy. It remains to be shown, that the wealth X z,πB = XB associated with πB fulfils X z,πB (τ ) ≥ B. For each t ∈ R+ XB (t) = (H0 (t))−1 E[H0 (τ )B|F(t)]. Therefore the assertion follows in application of Theorem 3.2 in [25], p. 69. 

22

References [1] Balzer, T.: Zeitorientierte Portfolio-Optimierung. Norderstedt: Books on Demand 2001 [2] Balzer, T.: Time-optimal Active Portfolio Selection. Preprint 2001 [3] Balzer, T., Janßen, K.: A Duality Approach to Problems of Combined Stopping and Deciding under Constraints. Submitted for publication (2001) [4] Blanchet-Scalliet, C., El Karoui, N., Jeanblanc, M., Martellini, L.: Optimal Investment and Consumption Decisions when Time-Horizon is Uncertain. Preprint 2001 [5] Breiman, L.: Optimal Gambling Systems for Favorable Games. In: Neyman, J. (ed): Fourth Berkeley Symposium on Probability and Statistics I. Berkeley: University of California Press 1961, pp 65-78 [6] Browne, S.: Reaching Goals by a Deadline: Digital Options and Continuous-Time Active Portfolio Management. Advances in Applied Probability 31, 551-577 (1999) [7] Burkhardt, T.: Wachstumsorientierte Portfolioselektion auf der Grundlage von Zielerreichungszeiten. OR Spektrum 22, 203-237 (2000) [8] El Karoui, N., Martellini, L.: Dynamic Asset Pricing Theory with Uncertain Time-Horizon. Preprint 2001 [9] F¨ ollmer, H., Leukert, P.: Quantile Hedging. Finance and Stochastics 3, 251-273 (1999) [10] Harrison, J.M.: Brownian Motion and Stochastic Flow Systems. New York: Wiley 1985 [11] Heath, D., Sudderth, W.: Continuous-Time Portfolio Management: Minimizing or maximizing the expected time to reach a goal. IMA Preprint Series 74 (1984) [12] Heath, D., Orey, S., Pestien, V. & Sudderth, W.: Minimizing or maximizing the expected time to reach zero. SIAM J. Control Optim. 25, 195-205 (1987) [13] Hull, J.C.: Options, Futures, and Other Derivatives, 4th edn. Upper Saddle River: Prentice-Hall International 2000 [14] Ingersoll, J.E.: Theory of financial decision making. Savage: Rowman & Littlefield 1987 [15] Karatzas, I.: Lectures on the Mathematics of Finance. Providence: American Mathematical Society 1997 23

[16] Karatzas, I., Lehoczky, J.P., Shreve, S.E.: Optimal portfolio and consumption decisions for a “small investor” on a finite horizon. SIAM J. Control Optim. 27, 1157-1186 (1987) [17] Karatzas, I., Wang, H.: Utility maximization with discretionary stopping. SIAM J. Control Optim. 39, 306-329 (2000) [18] Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. New York: Springer 1991 [19] Karatzas, I., Shreve, S.E.: Methods of Mathematical Finance, New York: Springer 1998 [20] Kelly Jr., J.: A New Interpretation of Information Rate. Bell System Tech. J. 35, 917-926 (1956) [21] Korn, R.: Optimal portfolios: stochastic models for optimal investment and risk management in continuous time. Singapore: World Scientific 1997 [22] Korn, R., Korn, E.: Option Pricing and Portfolio Optimization. Providence: American Mathematical Society 2001 [23] Liptser, R.S., Shiryaev, A.N.: Statistics of Random Processes II: Applications, 2nd edn. New York: Springer 2000 [24] Merton, R.C.: Continuous-time Finance, Malden: Blackwell 1990 [25] Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, Berlin: Springer 1999

24