RISK AVERSE ASYMPTOTICS AND THE OPTIONAL DECOMPOSITION PETER GRANDITS, CHRISTOPHER SUMMER* Abstract. We consider the problem of maximizing expected utility for a general utility function on R when the agent becomes increasingly risk averse. The limiting strategy will be shown to be a special, unique superhedging strategy, the so called balanced strategy. The connections to the optional decomposition and the class of minimal hedging strategies described in [5] are examined.

1. Introduction In this paper we consider a financial market with an agent who has to satisfy a contingent claim, e.g. an option, at final time T . The agent, who has the possibility of investing in an asset of the underlying financial market, has to decide on a trading strategy and there are, at least, two natural, but different possibilities: utility maximization and superhedging. We will investigate the relationship between these two concepts and show how superhedging can be interpreted as the limit of utility maximization. This interpretation will give some new insight into the notion of superhedging. For example the notion of superhedging in general does not yield a unique outcome, i.e., there are different superhedging strategies that lead to different terminal wealth at time T . But by looking at superhedging as a limit of utility maximization one is naturally led to a special, unique outcome. Let us assume we are given a finite probability space Ω = {ω1 , . . . , ωN }, a finite time horizon T ∈ N, a filtration (Ft )t=0,1,...,T , and a probability measure P. For finite Ω we always assume that P [ωi ] > 0 for all i = 1, . . . N . Furthermore we assume, as is typically done in arbitrage pricing theory, that F0 is trivial. The market consists of a stock S = (St )t=0,1,...,T , i.e., an Rvalued adapted process, and a riskless bond B. We assume without loss of generality that the stock S represents the discounted price process, i.e., that Date: Janury,13st, 2005. 2000 Mathematics Subject Classification. 91B28, 60G42, 46E30. Key words and phrases. hedging, exponential utility, risk aversion, optional decomposition. The authors would like to thank an anonymous referee for valuable suggestions that helped presenting the results in a more convincing, distinct way. * Financial support by the Austrian Nationalbank, Jubil¨ aumsfond 8699 and the Austrian Science Fund (FWF) under grant SFB010 is gratefully acknowledged. 1

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it is denoted in terms of units of the riskless bond, and that the bond is constant equal to 1. The stock process S can be represented by an event tree. Each node in the tree corresponds to an atom F ∈ Ft , t = 0, 1, . . . , T . By an atom F ∈ Ft we mean an element F ∈ Ft , F 6= ∅ such that G ∈ Ft with G ( F implies G = ∅. By a slight abuse of notation we say node F at time t for an atom F ∈ Ft . G is an successor of F ∈ Ft , if G ∈ Ft+1 and G ⊂ F , G 6= ∅. In this finite setting a trading strategy ϑ, i.e., an Ft−1 -measurable process, ~ ∈ RM , where M is the number of nodes in the can be viewed as a vector ϑ tree from time 0 to time T − 1. These are the nodes at which one can make a decision on how to trade. By a slight abuse of notation we will write ϑ also for the vector, not just for the process. We denote by ϑF ∈ R the value of the strategy ϑ at the node F ∈ Ft , t = 0, . . . , T . This means that ϑF denotes the number of shares the agent holds in the risky asset from time t up to time t + 1, if the state of the world ω satisfies ω ∈ F , which is, since F ∈ Ft , known at time t. The final R T gain of a trading strategy ϑ, in general the stochastic integral GT (ϑ) = 0 ϑdS, is in the finite setting just a finite sum. Note that GT (ϑ)(ωi ) is linear in ϑ. We assume that our market is arbitrage free (see, e.g., [4]), this is a standing assumption which we always make. In the current finite setting this simply means that at each time step the stock is not allowed to move only up or only down. In mathematical terms this means that for each atom F ∈ Ft , t = 0, 1, . . . , T − 1 either P [4St+1 = 0; F ] = P [St+1 = St ; F ] = 1 (this corresponds to the case when the stock does not move at all), or P [4St+1 > 0; F ] > 0 and P [4St+1 < 0; F ] > 0. Let us now introduce an economic agent who has a short position in an FT measurable contingent claim C, Ci := C(ωi ), and starts with an initial endowment c ∈ R. So by employing strategy ϑ the agent achieves an output Zi (ϑ) := Z(ϑ)(ωi ) := c + GT (ϑ)(ωi ) − Ci . We define the set  Z := Z(ϑ)| ϑ ∈ RM of all possible outcomes. 2. Balanced strategies ¯ ∈ Z, introducing a We will now define a special outcome Z¯ = Z(ϑ) balanced strategy and the corresponding balanced outcome. To our knowledge this is a new concept. One nice feature of the definition will be that it is very easy and straightforward to determine, given the stock price process and the contingent claim, the balanced strategy and the balanced outcome. But the definition will also turn out to be useful from a theoretical point of view. Before giving the formal definition, let us describe the idea behind a balanced strategy. At each time when the agent has to make a decision about

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her strategy, she considers two cases: the stock going up or down. Within each of these two cases, the agent looks at the worst possible outcome for her, taking into account her short position in the contingent claim C. She then chooses a strategy to balance those two worst case scenarios, i.e., she takes the strategy that makes the worst outcome in the case of a rising stock equal the worst outcome in the case of a falling stock. Definition 2.1. Given a finite probability space Ω, |Ω| = N , a finite time horizon T ∈ N, a filtration (Ft )t=0,...,T , an adapted, R-valued stock process (St )t=0,...,T , and a probability measure P, we define for an atom F ∈ Ft , t = 0, 1, . . . , T − 1, F+ := {ω ∈ F : 4St+1 (ω) > 0} ∈ Ft+1 and F− := {ω ∈ F : 4St+1 (ω) < 0} ∈ Ft+1 . Assume we are further given an initial endowment c ∈ R and a claim C ∈ FT . ¯ = c+GT (ϑ)−C ¯ We then call a strategy ϑ¯ respectively an outcome Z¯ := Z(ϑ) balanced, if ¯ := min Z(ϑ)(ω) ¯ ¯ ¯ zF+ (ϑ) = min Z(ϑ)(ω) =: zF− (ϑ) ω∈F+

ω∈F−

for all atoms F ∈ Ft , t = 0, 1, . . . , T − 1. Remark 2.2. The probability measure P does not enter into the definition. The only important fact about the probability measure is that P [ω] > 0 for all ω ∈ Ω, a condition we always impose on a probability measure on a finite probability space Ω. Also the initial endowment c ∈ R has no influence on the definition of a balanced strategy. The first thing we have to check is whether a balanced strategy ϑ¯ really exists. For that we go backwards through the tree described by S. At an atom F ∈ FT −1 , i.e., a node at time T − 1, one has to find a ϑ¯F ∈ R such that

min ϑ¯F 4ST (ω) − C(ω) = min ϑ¯F 4ST (ω) − C(ω) . (1) ω∈F+

ω∈F−

If F+ = ∅, i.e., the stock does not move up, then the no-arbitrage condition implies that F− = ∅, and the above equation simplifies to 0 = 0, holding true for all ϑ¯F ∈ R. This is a special case in which the strategy ϑ¯F is irrelevant. We want to exclude this trivial case and impose thus the standing assumption that the tree generated by the process S is non-degenerate, see, e.g., [8]. This means intuitively that at each node the stock moves to at least two values with positive probability, mathematically we impose that for all atoms F ∈ Ft , t = 0, 1, . . . , T − 1   E (4St+1 )2 ; F − E[4St+1 ]2 6= 0.

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Assumption 2.3. From now on we will always impose the standing assumption that the tree generated by the stock process S is non-degenerate. Thus we assume, together with the standing assumption of no-arbitrage, P [4St+1 > 0; F ] > 0

and

P [4St+1 < 0; F ] > 0

for all atoms F in Ft , t = 0, . . . , T − 1. A moment’s reflection reveals that in our setting the assumption of nondegeneracy is equivalent to a one-to-one correspondence between strategies ϑ ∈ RM and outcomes Z ∈ Z. Since we assume that the tree is non-degenerate, ϑF 7→ min ϑF 4ST (ω) ω∈F+

is the minimum over a strictly positive, finite number of affine, strictly increasing functions and therefore continuous, strictly increasing and the limits for ϑF to ∞ respectively −∞ are lim min ϑF 4ST (ω) = ∞

ϑF →∞ ω∈F+

and

lim

min ϑF 4ST (ω) = −∞.

ϑF →−∞ ω∈F+

Analogously, by no-arbitrage also F− 6= ∅ a.s., ϑF 7→ minω∈F− ϑF 4ST (ω) is continuous, strictly decreasing in ϑ, with limits lim min ϑF 4ST (ω) = −∞ and

ϑF →∞ ω∈F−

lim

min ϑF 4ST (ω) = ∞.

ϑF →−∞ ω∈F−

Therefore there exists a unique ϑ¯F ∈ R that gives us equality in (1). So we get a unique ϑ¯F for each atom F ∈ FT −1 . Having fixed the strategy at time T − 1, we can, following the same argument as above, determine ϑ¯F for each atom F ∈ FT −2 . Continuing this way using backward induction ¯ The corresponding we get the existence of a unique balanced strategy ϑ. ¯ balanced outcome Z(ϑ) is also unique. Summing up the above considerations, we get the following proposition. Proposition 2.4. Given a market as in Definition 2.1 that satisfies in addition Assumption 2.3. Then there exists a unique balanced strategy ϑ¯ and ¯ a unique balanced outcome Z¯ = Z(ϑ). Example 2.5. Let us consider a one time-step model with 4 states, i.e., Ω = {ω1 , ω2 , ω3 , ω4 }. The stock starts at S0 = 3 and 4S1 = (3, 0, −1, −2). The contingent claim is given by C = (3, 3, 2, 0) and the initial endowment is assumed to be c = 3. So we get Z(ϑ)(ω1 ) = 3ϑ, Z(ϑ)(ω2 ) = 0, Z(ϑ)(ω3 ) = −ϑ + 1, Z(ϑ)(ω4 ) = −2ϑ + 3. The only atom F ∈ F0 is F = Ω and we have F+ = {ω1 }, F− = {ω3 , ω4 }. So zF+ (ϑ) = 3ϑ

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and

( −ϑ + 1 ϑ ≤ 2, zF− (ϑ) = −2ϑ + 3 2 < ϑ. Therefore the unique balanced strategy is ϑ¯ = 1/4. Note however that any strategy ϑ ∈ [0, 1] is a superhedging strategy. 3. General utility functions As indicated in the introduction, we are interested in utility maximization, thus we are going to look at the underlying utility function. We will also introduce the concept of increasing risk aversion. Definition 3.1. Let U (x) denote a general utility function on R, which is twice differentiable. By this we mean a function U : R → R which is strictly increasing and strictly concave, i.e., U 0 (x) > 0 and U 00 (x) < 0, and which satisfies the Inada condition1 U 0 (−∞) = ∞. Then we denote by r(x) := −

U 00 (x) >0 U 0 (x)

the (absolute) risk aversion. This notion goes back to Arrow [1] and Pratt [6]. We consider a family of utility functions (Uαn )n≥1 , whose risk aversion tends to infinity uniformly on R as n → ∞. By that we mean that for all K > 0 there exists n0 such that U 00 (x) >K rαn (x) = − α0 n Uαn (x) holds for all x and for all n ≥ n0 . As an example one can think of the exponential utility function, since for Uαn (x) = −e−nx the risk aversion equals rαn (x) = αn = n. Using the setting described in Section 1 und being interested in utility maximization, we look at E[Uα (Z(ϑ))] = E[Uα (c + GT (ϑ) − C)] 7→ max . ϑ∈RM

Let us denote by

ϑα

the optimal strategy, i.e.,

E[Uα (Z α )] = E[Uα (Z(ϑα ))] = sup E[Uα (Z(ϑ))] , ϑ∈RM

where

Zα

:=

Z(ϑα )

is the optimal outcome.

Remark 3.2. How do we know that the optimal strategy ϑα really exists? One could deduce this fact from general results, e.g., Theorem 2.1 in [3] for the exponential utility function. But for the finite setting we can use some tools from convex analysis to easily deduce the existence of a unique 1Normally the Inada conditions also imply the assumption U 0 (∞) = 0, but we do not need this additional assumption in the following.

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maximizing strategy ϑ? , since the function ϑ 7→ E[U (Z(ϑ)] is a continuous, concave function on RM . Proposition 3.3. In the setting described in Section 1 there exists a unique utility maximizing ϑ? ∈ RM , i.e., E[U (Z(ϑ? ))] = sup E[U (Z(ϑ))] ϑ∈RM

for a utility function U (x). Proof. −E[U (Z(ϑ))] is a continuous, thus closed, proper convex function. We can therefore use [7, Theorem VI.27.1(d)], which guarantees the existence of a minimizer provided −E[U (Z(ϑ))] has no direction of recession. Remember that a vector ρ is called a direction of recession, if for every ϑ the function t 7→ −E[U (Z(ϑ + tρ))] is nonincreasing in t ∈ R. To show that no direction of recession exists, we use [7, Proposition II.8.6 together with the definition following Corollary 8.6.2] and show that lim sup E[U (Z(λϑ))] = −∞ λ→∞

for any ϑ 6= 0. Since ϑ 6= 0 we know by the no-arbitrage condition together with the non-degeneracy condition, see Assumption 2.3, that there has to exist an ω1 ∈ Ω such that GT (ϑ)(ω1 ) < 0. We also get that maxω∈Ω GT (ϑ)(ω) > 0. So E[U (Z(λϑ))] ≤ P [ω1 ] U (c + λGT (ϑ)(ω1 ) − C(ω1 ))   + (1 − P [ω1 ])U c + λ max GT (ϑ)(ω) − min C(ω) . ω∈Ω

ω∈Ω

and therefore there exist constants c1 , c2 , d1 , d2 , where d1 , d2 > 0 such that E[U (Z(λϑ))] ≤ P [ω1 ] U (c1 − λd1 ) + (1 − P [ω1 ])U (c2 + λd2 )   1 − P [ω1 ] U (c2 + λd2 ) . = P [ω1 ] U (c1 − λd1 ) 1 + P [ω1 ] U (c1 − λd1 )

(2)

We know from concavity that U (−∞) = −∞. If U (∞) < ∞ it follows directly, otherwise we deduce from de L’Hospital and the Inada condition U 0 (−∞) = ∞ that the expression in the square bracket tends to 1 for λ → ∞. Therefore the right hand side in equation (2) goes to −∞. This proves that −E[U (Z(ϑ))] has no direction of recession. To show uniqueness, we just assume that there are two different maximizing strategies and consider a linear combination of those with coefficients 1/2. Due to non-degeneracy this provides us with an outcome that differs from the two maximizing ones on a set with probability greater than zero (cf. Remark 2.2). The utility function is by assumption strictly concave and thus the expected utility of the new strategy would be greater than the one of the maximizing strategies - the desired contradiction. 

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Our goal is to show that Z α converges to the balanced outcome Z¯ and determine the speed of convergence. First we will show that the optimal outcome Z α becomes ‘almost balanced’ as the risk aversion rα (x) tends to infinity. Proposition 3.4. Let Z αn = c + GT (ϑαn ) − C be the optimal outcome for the utility maximization problem E[Uαn (c + GT (ϑ) − C)] 7→ max . ϑ

Let further rαn (x) denote the risk aversion for the utility function Uαn (x), n ≥ 1. Then for any atom F ∈ Ft , t = 0, . . . , T − 1, there exists a constant K(F ) such that | min Z αn (ω) − min Z αn (ω)| ≤ ω∈F+

ω∈F−

K(F ) inf x∈I rαn (x)

holds true for all n ≥ 1, where I := (−∞, min Z αn (ω)] ∪ (−∞, min Z αn (ω)]. ω∈F−

ω∈F+

Proof. Fix an atom F ∈ Ft , t = 0, . . . , T − 1 and let us denote by ϑαFn the entry of the optimal strategy ϑαn that corresponds to the node given by F . Since ϑαn is optimal, we get, if we differentiate E[Uαn (Z(ϑ))] with respect to the entry ϑF ,   0 = E Uα0 n (Z αn (ω))4St+1 (ω); F , where E[. . . ; F ] denotes the expected value with respect to the atom F . We can split this sum into a sum over ω ∈ F+ and ω ∈ F− , since ω ∈ F \ {F+ ∪ F− }, i.e. ω for which 4St+1 (ω) = 0, do not contribute to the sum. Noting that Uα0 n (x) > 0 is strictly decreasing we get      0 0 αn αn ω )) 4St+1 (ω); F− 0 ≥ E Uαn (Z (ω))4St+1 (ω); F+ + E max Uαn (Z (˜ ω ˜ ∈F− 0 αn = Uαn ( min Z (˜ ω )) ω ˜ ∈F−    Uα0 n (Z αn (ω)) E 0 4St+1 (ω); F+ + Uαn (minω˜ ∈F− Z αn (˜ ω ))

 E[4St+1 ; F− ] .

(3)

This inequality must hold true for all n ≥ 1, thus Uα0 n (Z αn (ω)) 4St+1 (ω) > 0 Uα0 n (minω˜ ∈F− Z αn (˜ ω )) must, for all n ≥ 1, be bounded from above for ω ∈ F+ . It follows from the definition of risk aversion, i.e., r(x) = −

U 00 (x) d = − log U 0 (x) 0 U (x) dx

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that

Rb U 0 (a) a r(x)dx . = e U 0 (b) Therefore we get, if minω˜ ∈F− Z αn (˜ ω ) ≥ Z αn (ω), ) (Z minω∈F Z αn (˜ ω) ˜ − Uα0 n (Z αn (ω)) = exp rαn (x)dx ω )) Uα0 n (minω˜ ∈F− Z αn (˜ Z αn (ω)     αn αn ≥ exp min Z (˜ ω ) − Z (ω) inf rαn (x) . ω ˜ ∈F−

x∈I

So we get from (3) the necessary condition that for each ω ∈ F+ (note that ω ) < Z αn (ω) the following holds trivially), there exists a for minω˜ ∈F− Z αn (˜ constant K1 (ω) such that for any n ≥ 1    αn αn min Z (˜ ω ) − Z (ω) inf rαn (x) ≤ K1 (ω). ω ˜ ∈F−

x∈I

So in particular, since |F+ | < ∞, there exists a constant K1 (F+ ) such that for any αn ≥ 1   αn αn ω ) inf rαn (x) ≤ K1 (F+ ), ω ) − min Z (˜ min Z (˜ ω ˜ ∈F−

ω ˜ ∈F+

x∈I

i.e., ω) ≤ ω ) − min Z αn (˜ min Z αn (˜ ω ˜ ∈F+

ω ˜ ∈F−

K1 (F+ ) , inf x∈I rαn (x)

∀αn ≥ 1.

If one uses      0 αn ω )) 4St+1 (ω); F+ + E Uα0 n (Z αn (ω))4St+1 ; F− 0 ≤ E max Uαn (Z (˜ ω ˜ ∈F+

instead of (3), one analogously gets ω) ≤ ω ) − min Z αn (˜ min Z αn (˜ ω ˜ ∈F−

ω ˜ ∈F+

K1 (F− ) , inf x∈I rαn (x)

∀αn ≥ 1,

thus completing the proof for the constant K(F ) = max {K1 (F+ ), K1 (F− )}.  The previous proposition showed that the optimal outcome Z αn gets, for large risk aversion, ‘almost balanced’. Now we have to investigate whether this property of the outcome also implies that the corresponding optimal trading strategy ϑαn becomes ‘almost balanced’. Actually we do not need the fact the strategy ϑαn is optimal. The following proposition can be formulated for any strategy ϑβn and outcome Z βn = Z(ϑβn ). Proposition 3.5. Let βn be a sequence in R and r : R 7→ R a function such that r(βn ) tends to infinity as n ≥ 1 tends to ∞. If for all atoms F ∈ Ft , t = 0, 1, . . . , T − 1 | min Z βn (˜ ω ) − min Z βn (˜ ω )| = O (1/r(βn )) ω ˜ ∈F+

ω ˜ ∈F−

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for n → ∞, then

βn ¯ ϑ − ϑ

∞

= O (1/r(βn )) ,

where k·k∞ denotes the supremum norm on RM . Proof. Suppose the proposition would not hold true. Then for n = 1, 2, . . . there exist nodes Fn and a function g : N 7→ N tending to infinity for n → ∞ such that |ϑβFnn − ϑ¯Fn | ≥ g(n)/r(βn ). Since there are only finitely many nodes Fn , there have to exist one fixed node F and a subsequence nk tending to ∞, such that βnk

|ϑF

− ϑ¯F | ≥ g(nk )/r(βnk ).

(4)

If there are more such atoms Fi , which correspond to times ti , we choose a node F = Fi such that t = ti is maximal. This way we get for nodes G βn following F , i.e., G ( F , |ϑG k − ϑ¯G | = O(1/r(βnk )). We will consider the subtree starting at this specially chosen node F at time t, and we will deduce a contradiction by showing that for any given K1 > 0 there exists an β0 = β0 (K1 ) such that β β 0 0 min Z (˜ ω ) > K1 /r(β0 ). ω ) − min Z (˜ ω˜ ∈F+ ω ˜ ∈F− To do this, we define the outcome Z˜ β = Z(ϑ˜β ), which we get if we follow the strategy ϑ¯ everywhere except on the subtree starting at F . There we use the balanced strategy ϑ¯ just for the first time step (t, t + 1] and after that we follow the strategy ϑβ , i.e., ϑ˜β = ϑ¯ everywhere except on nodes G ( F , where ϑ˜βG = ϑβG . We know that there exists K2 > 0 such that βnk ˜ < K2 /r(βn ) min Z˜ βnk (˜ (˜ ω ) ω ) − min Z k ω˜ ∈F+ ω ˜ ∈F− holds for all βnk . This follows since the node F was chosen such that t is maximal and since the outcome Z˜ βnk = c + GT (ϑ˜βnk ) − C depends in the affine way on the strategy. If we set m = min{|4St+1 (ω)| : |4St+1 (ω)| > 0 and ω ∈ F }, we know from the non-degeneracy condition that m > 0 and we can take 1 g(nk ) ≥ 2m (K1 + K2 ). βn Let us now suppose ϑ k > ϑ¯F , the other case work analogously, just the F

βnk

roles of F+ and F− have to be interchanged. By (4) this implies ϑF ϑ¯F + g(nk )/r(βnk ), and we get min Z βnk (˜ ω ) > min Z˜ βnk (˜ ω ) + mg(nk )/r(βnk ), ω ˜ ∈F+

ω ˜ ∈F+

min Z βnk (˜ ω ) < min Z˜ βnk (˜ ω ) − mg(nk )/r(βnk ).

ω ˜ ∈F−

ω ˜ ∈F−

≥

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Therefore min Z βnk (˜ ω ) − min Z βnk (˜ ω) >

ω ˜ ∈F+

ω ˜ ∈F−

˜ βnk

> min Z ω ˜ ∈F+

(˜ ω ) − min Z˜ βnk (˜ ω ) + 2mg(nk )/r(βnk ) ω ˜ ∈F−

> −K2 /r(βnk ) + (K1 + K2 )/r(βnk ) = K1 /r(βnk ), the desired contradiction with β0 = βnk .



To get our main theorem, we now just have to apply Proposition 3.5 for the optimal outcome Z αn , where Proposition 3.4 guarantees that the necessary assumptions are fulfilled. Since ϑ 7→ Z(ϑ)(ω) is affine in ϑ, convergence of the strategy ϑαn implies convergence of the outcome Z αn at the same rate of convergence. Theorem 3.6. Assume we are given a finite probability space (Ω, (F)t=0,...,T , P) with a finite time horizon T ∈ N, an R-valued, adapted stochastic process (St )t=0,...,T representing the discounted stock price process, a riskless bond B = 1, the contingent claim C ∈ FT , and the initial endowment c for an agent holding a short position in the contingent claim C. Assume further that Assumption 2.3 is satisfied, i.e., S is non-degenerate. Given a family of utility functions (Uαn (x))n≥1 whose risk aversion tends to infinity uniformly as n tends to ∞, let ϑαn (respectively, Z αn ) denote the optimal strategy (respectively, outcome) for the optimization problem E[Uαn (c + GT (ϑ) − C)] 7→ max . ϑ

Then

 

α

ϑ n − ϑ¯ = O 1/ inf rαn (x) ∞ x∈R

and

  α n ¯ sup Z (ω) − Z(ω) = O 1/ inf rαn (x) . ω∈Ω

x∈R

4. Different notions of superhedging Now we are going to use the notion of a balanced strategy to show the connections to the concept of superhedging. As indicated in the introduction there are some intuitive connections between increasing risk aversion and superhedging. We saw however in the quite easy Example 2.5 that there might be more than one superhedging strategy, whereas the notation of a balanced strategy is, in the setting presented so far, unique. Therefore we look at a more restrictive notion of superhedging that is already known in the literature. In his paper [5] Kramkov considers two notions of superhedging (in fact he uses the word ‘hedging’, not ‘superhedging’), superhedging and minimal superhedging. The setting of the paper is a very general, continuous time model. But we are going to apply the results just to our finite Ω model,

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described by (Ω, (Ft )t=0,...,T , P), the stock process (St )t=0,...,T , the initial endowment c (which will, for consistency reasons, in this section be denoted by V0 ), the contingent claim C ∈ FT , and the riskless bond B = 1. Let us consider a trading strategy ϑ, which in our finite setting can be interpreted as a vector, together with the value process Vt = V0 + Gt (ϑ) − Kt , where Kt ≥ 0 is (in our finite setting) an adapted, non-decreasing process, which can be interpreted as cumulative consumption. Note that until R T now we only looked at the final gain of a trading strategy, i.e., GT (ϑ) = 0 ϑdS. Now we Rt also have to consider intermediate wealth, thus we use Gt (ϑ) = 0 ϑdS as the intermediate gain of the trading strategy ϑ. We call ϑt and Kt , together with the initial wealth V0 , a superhedging portfolio for the claim C, if C ≤ VT , i.e., if the strategy gives us, provided we start with initial endowment V0 , enough to satisfy the claim at final time T . We call a strategy ϑˆt , a nonb t ≥ 0 and an initial wealth Vb0 with value decreasing consumption process K ˆ −K b t minimal superhedging portfolio, if it satisfies process Vbt = Vb0 + Gt (ϑ) Vbt ≤ Vt ,

t ∈ [0, T ]

(5)

for all superhedging portfolios (V0 , ϑt , Kt ) with value processes Vt . Kramkov shows that such a minimal superhedging portfolio exists, that its value process Vbt is given by Vbt = supQ∈Me EQ[C|Ft ], and that it gives rise to the so called optional decomposition ˆ −K b t = Vbt . Vb0 + Gt (ϑ) Me denotes, see [5] for details, the set of martingale measures equivalent to P. At final time T this gives ˆ −K b T = VbT . Vb0 + GT (ϑ) Note that C = sup EQ[C|FT ] = VbT Q∈Me

holds and therefore the cumulative consumption at time T ˆ −C b T = Vb0 + GT (ϑ) K can be interpreted as the outcome of an agent starting with an initial endowˆ ment of Vb0 and a short position in the contingent claim C, using strategy ϑ. b T ≥ 0 this is a superhedging strategy. Since by assumption K What are the relations between the minimal superhedging portfolio introduced by Kramkov and the concept of balanced strategies? Note that the balanced strategy is independent of the initial endowment. It becomes however a superhedging strategy provided the initial endowment is big enough to be able to superhedge, i.e., if the endowment equals supQ∈M EQ[C]. We will actually show that it is a minimal superhedging strategy, i.e., the strategy, belonging to a minimal superhedging portfolio.

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The minimal superhedging portfolio seems to be more restrictive, since it imposes the condition (5), which is a condition concerning the whole timeperiod [0, T ], whereas the balanced strategy is only concerned with the outcome at the final time T . We will however show that every balanced outcome ¯ for some minimal superhedging stratZ¯ is given by the outcome Z¯ = Z(ϑ) ¯ egy ϑ. Before proving this, the following remark shows that an implication in the other direction, namely that every minimal superhedging strategy is a balanced strategy, does for blatant reasons not hold true. Remark 4.1. Note that the minimal superhedging strategy does not have to be unique. This can be seen by considering the following easy example: S0 = 2, 4S1 = (1, 0, −1), c = 1 = V0 , and C = (0, 1, 0). In this setting any ϑ ∈ [−1, 1] is a minimal superhedging strategy with V0 = 1, K0 = 0, V1 = (0, 1, 0) = C, and K1 = (1 + ϑ, 0, 1 − ϑ). So in this example the restriction to minimal superhedging strategies does not provide a superhedging agent with any help for deciding which strategy to choose, whereas the balanced strategy ϑ¯ = 0 is unique. Let us now show that the balanced outcome Z¯ corresponds, as claimed above, indeed to a minimal superhedging strategy. Theorem 4.2. Given a market as in Definition 2.1 that satisfies in addition ¯ defined in Section 2, is a Assumption 2.3. Then the balanced strategy ϑ, ¯ equals minimal superhedging strategy and the balanced outcome Z¯ = Z(ϑ) b the consumption KT for a minimal superhedging portfolio. ˆ a consumption process K b and an initial wealth Vb0 is a Proof. A strategy ϑ, minimal superhedging portfolio if and only if it satisfies Vbt = sup EQ[C|Ft ] Q∈Me

b t ≥ 0, determined by and K ˆ −K b t = Vbt , Vb0 + Gt (ϑ) is non-decreasing. To proof the theorem we will look at the balanced strategy ϑ¯ and choose the corresponding value process V¯t respectively the corresponding consump¯ in such a way, that the superhedging ¯ t = V¯0 − V¯t + Gt (ϑ) tion process K ¯ ¯ ¯ portfolio (V0 , ϑ, K) is a minimal superhedging portfolio. We start building up a strategy to get the balanced outcome Z¯ by working ¯T = backwards through the tree induced by the stock process St . We set K Z¯T and therefore V¯T = C = VbT . For each node F at time T − 1 we have to determine the corresponding strategy ϑ¯F to get the balanced outcome ¯ T ≥ 0. But this simply means that we have to take the unique ϑ¯F to Z¯ = K get ϑ¯F 4ST − V¯T balanced, i.e.,

min ϑ¯F 4ST (ω) − V¯T (ω) = min ϑ¯F 4ST (ω) − V¯T (ω) . ω∈F+

ω∈F−

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For a node F ∈ FT −1 we denote by V¯T −1 (F ) ∈ R the value of V¯T −1 (ω) for ω ∈ F , which is, since F ∈ FT −1 is an atom, equal for all ω ∈ F . Now we set for each such node F ∈ FT −1 the value V¯T −1 (F ) ∈ R minimal, such that V¯T −1 (F ) ≥ V¯T (ω) − ϑ¯F 4ST (ω) (6) for all ω ∈ F+ ∪F− . If we repeat this procedure for all the previous time steps T − 2, T − 3, . . . , 0, we get a superhedging strategy ϑ¯ and a non-decreasing ¯ t , since consumption process K
¯ t (ω) − K ¯ t−1 (F ) = V¯t−1 (F ) − V¯t (ω) − ϑ¯F 4St (ω) ≥ 0, K because of (6). Now it just remains to show that this superhedging strategy ϑ¯ is indeed a minimal superhedging strategy. We will do this by showing that, for some minimal superhedging portfolio Vb , V¯T −1 (F ) = VbT −1 (F ) for each node F at time T − 1 and then one just has to work again back in time. This gives V¯t = Vbt for all t ∈ [0, T ], thus proving the theorem. To show that V¯T −1 (F ) = VbT −1 (F ), first note that by the definition of a minimal superhedging strategy VbT −1 (F ) ≤ V¯T −1 (F ). If we assume that V¯T −1 (F ) > VbT −1 (F ), there exists by the choice of V¯T −1 (F ) an ω ∈ F+ ∪ F− such that V¯T (ω) − ϑ¯F 4ST (ω) > VbT −1 (F ), i.e.,
−VbT −1 (F ) > ϑ¯F 4ST (ω) − V¯T (ω) ≥ min ϑ¯F 4ST (˜ ω ) − V¯T (˜ ω) . ω ˜ ∈F+ ∪F−

Since ϑ¯ was chosen to give a balanced outcome, this implies that there exist ω+ ∈ F+ and ω− ∈ F− for which the minimum on the right hand side of the last inequality is attained. If we now assume that ϑˆF ≤ ϑ¯F take ω = ω+ , otherwise ω = ω− to get, using V¯T = VbT and the fact that the consumption b t is non-decreasing for the last inequality process K −VbT −1 (F ) > ϑ¯F 4ST (ω) − VbT (ω) ≥ ϑˆF 4ST (ω) − VbT (ω) ≥ −VbT −1 (F ), i.e., VbT −1 (F ) < VbT −1 (F ), the desired contradiction.



References [1] K. Arrow, Essays in the theory of risk-bearing, North-Holland Publishing Co., Amsterdam, 1970. [2] P. Cheridito, C. Summer, Utility-Maximizing Strategies under Increasing Risk Aversion., submitted [3] F. Delbaen, P. Grandits, T. Rheinl¨ ander, D. Samperi, M. Schweizer, and C. Stricker, Exponential hedging and entropic penalties, Mathematical Finance 12 (2002), no. 2, 99–123. [4] J.M. Harrison and S.R. Pliska, Martingales and stochastic integrals in the theory of continuous trading, Stochastic Process. Appl. 11 (1981), no. 3, 215–260. [5] D. O. Kramkov, Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets, Probab. Theory Related Fields 105 (1996), no. 4, 459–479. [6] J. Pratt, Risk aversion in the small and in the large, Econometrica 32 (1964), 122–136. [7] R.T. Rockafellar, Convex analysis, Princeton University Press, Princeton, N.J., 1970.

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[8] M. Schweizer, Variance-optimal hedging in discrete time, Math. Oper. Res. 20 (1995), no. 1, 1–32. [9] C. Summer, Utility Maximization and Increasing Risk Aversion. Ph.D. Thesis (2002), Vienna University of Technology. http://www.fam.tuwien.ac.at/~csummer/bin/phd.pdf ¨t Wien, Wirtschaftsuniviersita ¨t Wien Technische Universita E-mail address: [email protected], [email protected]