9Hedging under constraints

So far, we have focussed on frictionless market models, where asset transactions can be carried out with no limitation. In this chapter, we study the impact of market imperfections generated by convex trading constraints. Thus, we develop the theory of dynamic hedging under the condition that only trading strategies from a given convex class S may be used. In Section 9.1 we characterize those market models for which S does not contain arbitrage opportunities. Then we take a direct approach to the superhedging duality for American options. To this end, we first derive a uniform Doob decomposition under constraints in Section 9.2. The appropriate upper Snell envelopes are analyzed in Section 9.3. In Section 9.4 we derive a superhedging duality under constraints, and we explain its role in the analysis of convex risk measures in a financial market model.

9.1Absence of arbitrage opportunities

In practice, it may be reasonable to restrict the class of trading strategies which are admissible for hedging purposes. As discussed in Section 4.8, there may be upper bounds on the capital invested into risky assets, or upper and lower bounds on the number of shares of an asset. Here we model such portfolio constraints by a set S of d-dimensional predictable processes, viewed as admissible investment strategies into risky assets. Throughout this chapter, we will assume that S satisfies the following conditions:

(a) 0 ∈ S.

(b) S is predictably convex: If ξ , η ∈ S and h is a predictable process with 0 ≤ h ≤ 1, then the process

belongs to S.

(c) For each t ∈ {1, . . . , T}, the set

is closed in L0(Ω,Ft−1, P;ℝd).

(d) For all t, ξt ∈ St implies

In order to explain condition (d), let us recall from Lemma 1.66 that each ξt ∈ L0(Ω,Ft−1, P;ℝd) can be uniquely decomposed as

and where

Remark 9.1. Under condition (d), we may replace ξt · (Xt − Xt−1) by and P-a.s. implies ξ⊥t = 0. Note that condition (d) holds if the price increments satisfy the following nonredundance condition: For all t ∈ {1, . . . , T} and ξt ∈ L0(Ω,Ft−1, P;ℝd),

Indeed, under (9.1) we have Nt = {0} and Nt ⊥= L0(Ω,Ft−1, P;ℝd).

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Example 9.2. For each t let Ct be a closed convex subset of ℝd such that 0 ∈ Ct. Take S as the class of all d-dimensional predictable processes ξ such that ξt ∈ Ct P-a.s. for all t. If the nonredundance condition (9.1) holds, then S satisfies conditions (a) through (d). This case includes short sales constraints and restrictions on the size of a long position.

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Example 9.3. Let a, b be two constants such that −∞ ≤ a < 0 < b ≤ ∞, and take S as the set of all d-dimensional predictable processes such that

This class S corresponds to constraints on the capital invested into risky assets. If we assume that the nonredundance condition (9.1) holds, then S satisfies conditions (a) through (d). More generally, instead of the two constants a and b, one can take dynamic margins defined via two predictable processes (at) and (bt).

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Let S denote the set of all self-financing trading strategies = (ξ0, ξ) which arise from an investment strategy ξ ∈ S, i.e.,

In this section, our goal is to characterize the absence of arbitrage opportunities in S. The existence of an equivalent martingale measure P∗ ∈ P is clearly sufficient. Under an additional technical assumption, a condition which is both necessary and sufficient will involve a larger class PS ⊃ P. In order to introduce these conditions, we need some preparation. Recall that T denotes the set of all finite stopping times.

Definition 9.4. An adapted stochastic process Z on (Ω,F, (Ft), Q) is called a local Q-martingale if there exists a sequence of stopping times (τn)n∈N ⊂ T such that τn ↗ T Q-a.s., and such that the stopped processes are Q-martingales. The sequence (τn)n∈N is called a localizing sequence for Z. In the same way, we define local supermartingales and local submartingales.

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Remark 9.5. If Q is a martingale measure for the discounted price process X, then the value process V of each self-financing trading strategy = (ξ0, ξ) is a local Q-martingale. To prove this, one can take the sequence

as a localizing sequence. With this choice, |ξt| ≤ n on {τn ≥ t}, and the increments

of the stopped process are Q-integrable and satisfy

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The following proposition is a generalization of an argument which we have already used in the proof of Theorem 5.25. We emphasize that throughout this chapter, we will assume that F0 = {∅, Ω} and FT = F.

Proposition 9.6. For a local Q-supermartingale Z, the following conditions are equivalent.

(a) Z is a Q-supermartingale.

(b)

If, moreover, ZT ≥ 0 Q-a.s., then Zt ≥ 0 Q-a.s. for all t.

Proof. The implication (a)⇒(b) follows immediately from the definition of a Q-supermartingale, which includes the integrability of Z t for all t.

Now we prove the implication (b)⇒(a). We show first that the negative parts are integrable for all t. To this end, we proceed by backward induction on t. If t = T, our claim holds by assumption. So let us assume that is integrable for some t ∈ {0, . . . , T − 1}. Let (τn) be a localizing sequence for Z. Since τn ↗ T Q-a.s., we have

Since the random variables are uniformly bounded from below by the integrable random variable we may apply Fatou’s lemma for conditional expectations as follows, and subsequently use the fact that {τn ≥ t +1} = {τn ≤ t}c ∈ Ft and the supermartingale property of

Next, the convexity of x x− and Jensen’s inequality for conditional expectations imply that

Since the right-hand side is integrable according to the induction hypothesis, the integrability of Z − t now follows from (9.3). Note that (9.3) also yields the supermartingale inequality

Hence, Z will be a supermartingale if we can show that Zt ∈ L1 for all t. To this end, we now apply (9.3) successively to get

In view of this yields and in turn the integrability of Zt. If, moreover, ZT ≥ 0 Q-a.s., then backward induction and (9.4) yield Zt ≥ 0 Q-a.s. for all t.

Exercise 9.1.1. Use Proposition 9.6 to show that the following conditions are equivalent for a local Q-martingale Z.

(a) Z is a Q-martingale.

(b) Zt ∈ L1 for all t.

(c) ZT ∈ L1.

(d)

(e)

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Exercise 9.1.2. Let Y1, Y2 ∈ L1 be two independent, centered, and nonconstant random variables on (Ω,F, P). Suppose furthermore that f is a measurable function onℝ such that ξ := f (Y1) is real-valued but not integrable. Provide a concrete example for this situation. Then show that M0 := 0, M1 := Y1, and M 2 := ξY2 is a local martingale, but not a martingale, with respect to the filtration F0 = {∅, Ω},F1 = σ(Y1), and F2 = F.

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Definition 9.7. By PS we denote the class of all probability measures ≈ P such that

and such that the value process of any trading strategy in S is a local -supermartingale.

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Remark 9.8. If S contains all self-financing trading strategies = (ξ0, ξ) with bounded ξ, then PS coincides with the class P of all equivalent martingale measures. To prove this, let ∈ PS , and note that ξt · (Xt − Xt−1) ∈ L1() by (9.5) and the fact that ξt is bounded. Therefore, the value process V of any such is a -supermartingale by Proposition 9.6. The same applies to the strategy −, so V is in fact a -martingale, and Theorem 5.14 shows that is a martingale measure for X.

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Our first goal is to extend the “fundamental theorem of asset pricing” to our present setting; see Theorem 5.16. Let us introduce the positive cone

generated by S. Accordingly, we define the cones R and Rt. Clearly, R contains no arbitrage opportunities if and only if S is arbitrage-free. We will need the following condition on the L0-closure Rt of Rt:

This condition clearly holds if Rt itself is closed in L0 and in particular if St = Rt for all t.

Theorem 9.9. Under condition (9.6), there are no arbitrage opportunities in S if and only if PS is nonempty. In this case, there exists a measure ∈ PS which has a bounded density d/dP.

Example 9.10. Consider the situation of Example 9.2, where S consists of all predictable processes ξ such that ξt ∈ Ct P-a.s. for closed convex sets Ct ⊂ ℝd with 0 ∈ Ct.

Here, condition (9.6) will be satisfied as soon as the cones generated by the convex sets C t are closed in ℝd. This case includes short sales constraints and constraints on the size of a long position, which are modeled by taking for certain numbers such that

Example 9.11. Consider now the situation of Example 9.3, where S consists of all predictable processes ξ with a ≤ ξt · Xt−1 ≤ b P-a.s. for two constants a, b with −∞ ≤ a < 0 < b ≤ +∞.We claim that S does not contain arbitrage opportunities if and only if the unconstrained market is arbitrage-free. To prove this, note that the existence of an arbitrage opportunity in the unconstrained market is equivalent to the existence of some t and some Ft−1-measurable ξt such that ξt · (Xt − Xt−1) ≥ 0 P-a.s. and P[ ξt · (Xt − Xt−1) > 0 ] > 0 (see Proposition 5.11). In this case, there exists a constant c > 0 such that these properties are shared byand in turn by εt, where ε > 0. But εt ∈ St if ε is small enough. This argument shows moreover that PS = P.

As to the proof of Theorem 9.9, we will first show that the condition PS ≠ ∅ implies the absence of arbitrage opportunities in S:

Proof of sufficiency in Theorem 9.9. Suppose is a measure in PS , and V is the value process of a trading strategy in S such that VT ≥ 0 P-almost surely. Proposition 9.6 shows that V is a -supermartingale. Hence V0 ≥ [ VT ], so V cannot be the value process of an arbitrage opportunity.

Let us now prepare for the proof that the condition PS ≠ ∅ is necessary. First we argue that the absence of arbitrage opportunities in S is equivalent to the absence of arbitrage opportunities in each of the embedded one-period models, i.e., to the nonexistence of ξt ∈ St such that ξt · (Xt − Xt−1) amounts to a nontrivial positive gain. This observation will allow us to apply the techniques of Section 1.6. Let us denote

Similarly, we define

and the class S∞ ⊂ S. The following two lemmas show that for many purposes it is sufficient to consider strategies in S∞. The first lemma concerns arbitrage opportunities, the second probability measures in the class PS .

Lemma 9.12. The following conditions are equivalent:

(a) There exists an arbitrage opportunity in S.

(b) There exist t ∈ {1, . . . , T} and ξt ∈ St such that

(c) There exist t ∈ {1, . . . , T} and which satisfies (9.7).

(d) There exists an arbitrage opportunity in

Proof. The proof is essentially the same as the one of Proposition 5.11.

Lemma 9.13. For a probability measure ≈ P, the following two conditions are equivalent.

(a) ∈ PS .

(b) The value process of every strategy in S∞ is a -supermartingale.

Proof. The implication (a)⇒(b) follows just as in Remark 9.8. To prove the converse implication, (b)⇒(a), we need to show that the value process V of any given ξ = (ξ0, ξ) ∈ S is a local -supermartingale. To this end, we define τn := inf{t | |ξt| ≥ n} ∧ T. Clearly, τn ↗ T as n ↑ ∞. Since 0 ∈ S and is predictable, the predictable process belongs to S and in fact to S∞. Hence, condition (b) implies that the value process corresponding to ξ(n) and initial investment V0 is a -supermartingale. But this value process coincides with and so we conclude that V is a local -supermartingale.

In order to apply the results of Section 1.6, we introduce the convex sets

for t ∈ {1, . . . , T}. Lemma 9.12 shows that S contains no arbitrage opportunities if and only if the condition

holds for all t ∈ {1, . . . , T}.

Lemma 9.14. Condition (9.8) implies thatis a closed convex subset of L0(Ω,Ft , P).

Proof. The proof is essentially the same as the one of Lemma 1.68. Only the following additional observation is required: If (ξ n) is sequence in St, and if α and σ are two Ft−1-measurable random variables such that 0 ≤ α ≤ 1 and σ is integer-valued, then ζ := αξσ ∈ St. Indeed, predictable convexity of S and our assumption that 0 ∈ S imply that

for each n, and the closedness of St in L0(Ω,Ft−1, P;ℝd) yields

From now on, we will assume that

For the purpose of proving Theorem 9.9, this can be assumed without loss of generality: If (9.9) does not hold, then we replace P by an equivalent measure Pʹ which has a bounded density dPʹ/dP and for which the price process X is integrable. For instance, we can take

where c denotes the normalizing constant. If there exist a measure ≈ Pʹ such that each value process for a strategy in S is a local -supermartingale and such that the density d/dPʹ is bounded, then ∈ PS , and the density d/dP is bounded as well.

Lemma 9.15. If S contains no arbitrage opportunities and condition (9.6) holds, then for each t ∈ {1, . . . , T} there exists some Zt ∈ L∞(Ω,Ft , P) such that Zt > 0 P-a.s. and such that

Proof. Recall that R does not contain arbitrage opportunities if and only if S is arbitrage-free. Hence, for each t, by Lemma 9.12. By the equivalence of conditions (a) and (c) of same lemma and condition (9.6), we even get

where Rt denotes again the L0-closure of Rt. The cone Rt satisfies all conditions required from St, and hence Lemma 9.14 implies that each

is a closed convex cone in L1 which contains Furthermore, it follows from (9.11) and the argument in the proof of “(a)⇔(b)öf Theorem 1.55 that {0}, so satisfies the assumptions of the Kreps–Yan theorem, which is stated in Theorem 1.62. We conclude that there exist Zt ∈ L∞(Ω,, P) such that P[ Zt > 0] = 1, Ft and such that E[ Zt W ]≤ 0 for eachAs for each ξ ∈ S∞, Zt has property (9.10).

Now we can complete the proof of Theorem 9.9 by showing that the absence of arbitrage opportunities in S implies the existence of a measure that belongs to the class PS and has a bounded density d/dP.

Proof of necessity in Theorem 9.9. Suppose that S does not contain arbitrage opportunities. We are going to construct the desired measure via backward recursion. First we consider the case t = T. Take a bounded random variable ZT > 0 as constructed in Lemma 9.15, and define a probability measure T by

Clearly, T is equivalent to P, and Xt ∈ L1(T) for all t. We claim that

To prove this claim, consider the family

For ξ, ∈ S∞, let

and define ξʹ by for t < T and

The predictable convexity of S implies that ξʹ ∈ S∞. Furthermore, we have

Hence, the family Φ is directed upwards in the sense of Theorem A.37. By virtue of that theorem, ess sup Φ is the increasing limit of a sequence in Φ. By monotone convergence, we get

where we have used (9.10) in the last step. Since S contains 0, it follows that

which yields our claim (9.12).

Now we apply the previous argument inductively: Suppose we already have a probability measure t+1 ≈ P with a bounded density dt+1/dP such that

and such that

Then we may apply Lemma 9.15 with P replaced by t+1, and we get some strictly positive t ∈ L∞(Ω,Ft ,t+1) satisfying (9.10) with t+1 in place of P. We now proceed as in the first step by defining a probability measure t ≈ t+1 ≈ P as

Then t has bounded densities with respect to both t+1 and P. In particular, t[ |Xs| ] < ∞ for all s. Moreover, the Ft-measurability of dt/dt+1 implies that (9.14) is satisfied for t replacing t+1. Repeating the arguments that led to (9.12) yields

After T steps, we arrive at a measure := 1. Under this measure , the value process of every strategy in S∞ is a supermartingale, and so ∈ PS by Lemma 9.13.

9.2Uniform Doob decomposition

The goal of this section is to characterize those nonnegative adapted processes U which can be decomposed as

where the predictable d-dimensional process ξ belongs to S, and where B is an adapted and increasing process such that B0 = 0. In the unconstrained case where S consists of all strategies, we have seen in Section 7.2 that such a decomposition exists if and only if U is a supermartingale under each equivalent martingale measure P∗ ∈ P. In our present context, a first guess might be that the role of P is now played by PS . Since each value process of a strategy in S is a local -supermartingale for each ∈ PS , any process U which has a decomposition (9.15) is also a local -supermartingale for ∈ PS . Thus, one might suspect that the latter property would also be sufficient for the existence of a decomposition (9.15). This, however, is not the case, as is illustrated by the following simple example.

Example 9.16. Consider a one-period market model with the riskless bond and with one risky asset S101.0100We 11assume that and that takes the values and on 11Ω 11:= {ω−, ω+}. We choose any measure P on Ω which assigns positive mass to both ω+ and ω−. If we let S = [0, 1], then a measure belongs to PS if and only ifThus, for any positive initial value U0, the process defined by U1(ω−) := 0 and U1(ω+) := 2U0 is a PS -supermartingale. If U can be decomposed according to (9.15), then we must be able to write

for some B1(ω+)≥ 0. This requirement is equivalent to U0 ≤ ξ /2. Hence the decomposition (9.15) fails for U0 > 1/2.

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The reason for the failure of the decomposition (9.15) for certain PS-supermartingales is that PS does not reflect the full structure of S; the definition of PS depends only on the cone

generated by S. In the approach we are going to present here, the structure of S will be reflected by a stochastic process which we associate to any measure Q P:

Definition 9.17. For a measure Q P, the upper variation process for S is the increasing process AQ defined by

for t = 0, . . . , T − 1. By QS we denote the set of all Q ≈ P such that

and such that

◊

Clearly, the upper variation process of any measure Q ≈ P satisfies

where S∞ are the bounded processes in S. Hence for Q ∈ QS and ξ ∈ S∞, the condition EQ[ |Xt+1 − Xt| | Ft ] < ∞guarantees that

and it follows that

Moreover, we have

which implies the inclusion

Proposition 9.18. If Q ∈ QS , and V is the value process of a trading strategy in S, then V − AQ is a local Q-supermartingale. If, moreover, then V − AQ is a Q-supermartingale.

Proof. Let V be the value process of = (ξ0, ξ) ∈ S. Denote by τn(ω) the first time t at which

If such a t does not exist, let τn(ω) := T. Then τn is a stopping time. Since

belongs to L1(Q), and

This proves that is a Q-supermartingale, and so V − AQ is a local Q-supermartingale. Finally, since and by definition of QS , Proposition 9.6 yields that V − AQ is a Q-supermartingale if

Let us identify the class QS in some special cases.

Remark 9.19. If S∞ consists of all bounded predictable processes ξ with nonnegative components, then QS = PS . To prove this, take Q ∈ QS , and note first that AQ ≡ 0, due to (9.16) and the fact that S is a cone. Thus, value processes of strategies in S are local Q-supermartingales by Proposition 9.18. By taking ξ ∈ S such that and supermartingale, and Proposition 9.6 implies for j ≠ i, we get that X i is a local Q-that X i is a Q-supermartingale. In particular, is Q-integrable, and we conclude Q ∈ PS.

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Remark 9.20. If S∞ consists of all bounded predictable processes ξ, then QS = P. This follows by combining Remarks 9.8 and 9.19.

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Example 9.21. Suppose our market model contains just one risky asset, and S consists of all predictable processes ξ such that

where a and b are two given predictable processes with

If we assume in addition that E [ |Xt+1 − X t| | Ft ] > 0 P-a.s., then the nonredundance condition (9.1) holds, and S satisfies the assumptions (a) through (d) stated at the beginning of this chapter. If Q ≈ P is any probability measure such that

then

Hence, QS consists of all measures Q ≈ P with (9.19), EQ[ bt+EQ[ Xt+1−Xt | ]+ ] < 1 Ft ∞, and EQ[ at+1 EQ[ Xt+1 − Xt | Ft ]− ] > −∞ for all t.

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We now state the uniform Doob decomposition under constraints, which is the main result of this section.

Theorem 9.22. Suppose that PS is nonempty. Then for any adapted process U with UT ≥ 0 P-a.s., the following conditions are equivalent.

(a) U − AQ is a Q-supermartingale for every Q ∈ QS .

(b) There exists ξ ∈ S and an adapted increasing process B such that B0 = 0 and

Proof. (b)⇒(a): Fix Q ∈ QS and let

Since P-a.s. and Proposition 9.18 yields that MQ is a Q-supermartingale. Moreover, since

we have BT ∈ L1(Q). The fact that 0 ≤ Bt ≤ BT furthermore yields Bt ∈ L1(Q) for all t. Therefore, M − B = U − AQ is a Q-supermartingale.

(a)⇒(b):We must show that for any given t ∈ {1, . . . , T} there exist some ξ ∈ S and a nonnegative random variable Rt playing the role of Bt−Bt−1 such that Ut−Ut−1 = ξt · (Xt − Xt−1) − Rt, i.e.,

where

The formulation of this problem does not change if we switch from P to any equivalent probability measure, so we can assume without loss of generality that P ∈ PS . In this case AP ≡ 0, and U is a P-supermartingale. In particular, Us ∈ L1(P) for all s.

We assume by way of contradiction that

Recall that we have proved in Lemma 9.14 that is a closed convex subset of L1(Ω,Ft , P). The Hahn–Banach separation theorem, Theorem A.60, now implies the existence of a random variable Z ∈ L∞(Ω,Ft , P) such that

Note that the function −λ{Z<0} belongs to for all λ ≥ 0. Thus

for every λ ≥ 0, and it follows that Z ≥ 0 P-almost surely.

In fact, we can always modify Z such that it is bounded from below by some ε > 0 and still satisfies (9.21). To see this, note first that every W ∈ C St is dominated by some with integrable negative part. Therefore

where we have used our assumption that P ∈ PS . If we let Zε := ε 1 + (1 − ε)Z, then Zε still satisfies E[ Zε W ]≤ 0 for all and for ε small enough, the expectation E [ Zε (Ut − Ut−1) ] is still larger than α. So Zε also satisfies (9.21). Therefore, we may assume from now on that our Z with (9.21) is bounded from below by some constant ε > 0.

For the next step, let Zt−1 := E[ Z | Ft−1 ] and

Since this density is bounded and since P ∈ PS, we get

Moreover, it is not difficult to check that

see the proof of Theorem 7.5 for details. We now consider the case s = t. As we have seen in the proof of Theorem 9.9, the family

is directed upwards. Therefore, we may conclude as in (9.13) that

Since Z t−1 ≥ ε, (9.24) implies that

By (9.23) we may conclude that and so (9.22) yields Q ∈ QS .

As a final step, we show that U − AQ cannot be a Q-supermartingale, thus leading our assumption (9.20) to a contradiction with our hypothesis (a). To this end, we use again (9.24):

Thus, we cannot have

so U − AQ cannot be a Q-supermartingale, in contradiction to our hypothesis (a).

9.3Upper Snell envelopes

From now on, we assume the condition PS ≠ ∅. Let H be a discounted American claim. Our goal is to construct a superhedging strategy for H that belongs to our class S of admissible strategies. The uniform Doob decomposition suggests that we should find an adapted process U ≥ H such that U−AQ is a Q-supermartingale for each Q ∈ QS . If we consider only one such Q, then the minimal process U which satisfies these requirements is given by Q + AQ, where

is the Snell envelope of H − AQ with respect to Q. Thus, one may guess that

is the minimal process U which dominates H and for which U − AQ is a Q-supermartingale for each Q ∈ QS . Let us assume that

Note that this condition holds if H is bounded.

Definition 9.23. The process

will be called the upper QS -Snell envelope of H.

◊

The main result of this section confirms our guess that ↑ is the process we are looking for.

Theorem 9.24. The upper QS -Snell envelope of H is the smallest process U ≥ H such that U − AQ is a Q-supermartingale for each Q ∈ QS .

For a European claim, we have the following additional result.

Proposition 9.25. For a discounted European claim HE with

the upper QS -Snell envelope takes the form

Proposition 9.25 will follow from Lemma 9.30 below. The next result provides a scheme for the recursive calculation of ↑. It will be used in the proof of Theorem 9.24.

Proposition 9.26. For fixed Q0 ∈ QS let Qt(Q0) denote the set of all Q ∈ QS which coincide with Q0 on Ft. Then ↑ satisfies the following recursion formula:

The proofs of this proposition and of Theorem 9.24 will be given at the end of this section. Let us recall the following concepts from Section 6.4. The pasting of two probability measures Q1 ≈ Q2 in a stopping time τ ∈ T = {σ | σ is a stopping time ≤ T} is the probability measure

It was shown in Lemma 6.41 that, for all stopping times σ and FT-measurable Y ≥ 0,

Recall also that a set Q of equivalent probability measures on (Ω,F) is called stable if for any pair Q1, Q2 ∈ Q and all τ ∈ T the corresponding pasting also belongs to Q. A technical inconvenience arises from the fact that our set QS may not be stable. We must introduce a further condition on τ which guarantees that the pasting of Q1, Q2 ∈ QS in τ also belongs to QS .

Lemma 9.27. For τ ∈ T , the pasting of Q1, Q2 ∈ QS in τ satisfies E[ |Xt+1 − Xt| | Ft ] < ∞ P-a.s. for all t, and its upper variation process is given by

Moreover, we have ∈ QS under the condition that there exists ε > 0 such that

Proof. The identity (9.26) yields

and each of the two conditional expectations is finite almost surely.

Now we will compute the upper variation process A of . As above, (9.26) yields

Taking the essential supremum over gives

and from this our formula for follows.

In a final step, we show that belongs to QS under condition (9.27). We must show that Let Zt denote the density of Q2 with respect to Q1 on Ft. Then, by our formula for A,

which is finite for Q1, Q2 ∈ QS .

Lemma 9.28. Suppose we are given Q1, Q2 ∈ QS , a stopping time τ ∈ T , and a set B ∈ Fτ such that dQ2/dQ1|Fτ ≥ ε a.s. on B. Let be the pasting of Q1 and Q2 in the stopping time

Then ∈ QS , and the Snell envelopes associated with these three measures by (9.25) are related as follows:

Proof. We have dQ2/dQ1|Fσ ≥ ε, hence ∈ QS by Lemma 9.27. Let ρ be a stopping time in the set of all stopping times that are larger than or equal to τ. The formula Tτ for AQ in Lemma 9.27 yields

Moreover, (9.26) implies that

for all random variables Y such that all conditional expectations make sense. Hence,

Whenever ρ1, ρ2 are stopping times in Tτ, then is also a stopping time in Tτ. Conversely, every ρ ∈ Tτ can be written in that way for stopping times ρ1 and ρ2. Thus, taking the essential supremum over all ρ1 and ρ2 and applying Proposition 6.37 yields (9.28).

In fact, we have in (9.28), as we will have in the following lemma.

Lemma 9.29. For any Q0 ∈ QS , τ ∈ T , and δ > 0, there exist a set Λδ ∈ Fτ such that Q0[ Λδ ]≥ 1 − δ and measures Qk ∈ QS such that Qk = Q0 on Fτ and

Proof. By Theorem A.37 and its proof, there exists a sequence such that n n 00

We will recursively define measures Qk ∈ QS and sets such that and

By letting this will imply the first part of the assertion. We start this recursion in k = 0 by taking Q0 and

For Qk given, the equivalence of Qk and implies that there exists some ε > 0 such that the set

satisfies Q0[ D ]≥ 1−2−(k+1) δ. Thus, satisfies We now define a set

and consider the pasting Qk+1 of Qk and in the stopping time By Lemma 9.28, Qk+1 ∈ QS and

Now we can proceed to proving the main results in this section.

Proof of Proposition 9.26. For Q0 ∈ QS and t ∈ {t, . . . , T}, Qt(Q0) denotes the set of all Q ∈ QS which coincide with Q0 on Ft. By Lemma 9.29 and by the definition of Q as the Snell envelope of H − AQ,

Since twe Qget

For the proof of the converse inequality, let us fix an arbitrary Q ∈ Qt(Q0). For any δ > 0, Lemma 9.29 yields a set Λδ ∈ Ft+1 with measure Q[ Λδ ]≥ 1−δ and Qk ∈ Qt+1(Q) such thatP-a.s. on Λδ. Since Q k coincides with Q onFt+1, we have P-a.s. on Λδ

By taking δ ↓ 0 and by recalling we arrive at the converse of the inequality (9.29).

Proof of Theorem 9.24. Since Q0 ∈ QS is obviously contained in Qt(Q0), the recursion formula of Proposition 9.26 yields

i.e., is indeed a Q0-supermartingale for each Q0 ∈ QS . We also know that ↑ dominates H.

Let U be any process which dominates H and for which U − AQ is a Q-supermartingale for each Q ∈ QS . For fixed Q, the Q-supermartingale U − AQ dominates H − AQ and hence also Q, since Q is the smallest Q-supermartingale dominating H − AQ by Proposition 6.10. It follows that

Proposition 9.25 we will be implied by taking τ∗ ≡ T in the following lemma.

Lemma 9.30. Let H be a discounted American claim whose payoff is zero if it is not exercised at a given stopping time τ∗ ∈ T , i.e., Ht(ω) = 0 if t ≠ τ∗(ω). Then its upper QS -Snell envelope is given by

Proof. By definition,

Since each process AQ is increasing, it is clearly optimal to take τ = t on {τ∗ ≤ t}. Hence,

So we have to show that choosing τ ≡ τ∗ is optimal on {τ∗ > t}. If σ ∈ Tt is a stopping time with P[ σ > τ∗ ] > 0, then τ := σ ∧ τ∗ is as least as good as σ, since each process AQ is increasing. So it remains to exclude the case that there exists a stopping time σ ∈ Tt with σ ≤ τ∗ on {τ∗ > t} and P[ σ < τ∗ ] > 0, such that σ yields a strictly better result than τ∗. In this case, there exists some Q1 ∈ QS such that

with strictly positive probability on {τ∗ > t}. Take any ∈ PS and ε > 0, and define

Now let Qε be the pasting Q1 and in the stopping time According to Lemmas 9.27 and 9.28, Qε ∈ , and its upper variation process satisfies QS and as well as

By using our assumption that Hσ ≤ Hτ∗, we get

By letting ε ↓ 0, the P-measure of Bε becomes arbitrarily close to 1, and we arrive at a contradiction to (9.30).

9.4Superhedging and risk measures

Let H be a discounted American claim such that

Our aim in this section is to construct superhedging strategies for H which belong to our set S of admissible trading strategies. Recall that a superhedging strategy for H is any self-financing trading strategy whose process dominates H. If applied with t = 0, the following theorem shows that is the minimal amount for which a superhedging strategy is available.

Denote by (H) the set of all Ft-measurable random variables Ut ≥ 0 for which there exists some η ∈ S such that

Theorem 9.31. The upper QS -Snell envelopeof H is the minimal element of More precisely,

(a)

(b)

Proof. The uniform Doob decomposition in Section 9.2 combined with Theorem 9.24 yields an increasing adapted process B and some ξ ∈ S such that

So the fact that ↑ dominates H proves (a).

As to part (b), we first get ess inf (H) from (a). For the proof of the converse inequality, take (H) and choose a predictable process η ∈ S for which (9.31) holds. We must show that the set satisfies P[ B ] = 1. Let

Then and our claim will follow if we can show that Let ξ denote the predictable process obtained from the uniform Doob decomposition of the P-supermartingale and define

With this choice, ∈ S by predictable convexity, and Ût satisfies (9.31), i.e., Ût ∈ Let

Then ÛVs ≥ Hs ≥ 0 for all s, and so ÛV− AQ is a Q-supermartingale for each Q ∈ QS by Propositions 9.18 and 9.6. Hence,

This proves ess inf

For European claims, the upper QS -Snell envelope takes the form

By taking t = 0, it follows that

is the smallest initial investment which suffices for superhedging the claim H E. In fact, the formula above can be regarded as a special case of the representation theorem for convex risk measures in our financial market model. This will be explained next. Let us take L∞ := L∞(Ω,F, P) as the space of all financial positions. A position Y ∈ L∞ will be regarded as acceptable if it can be hedged with a strategy in S at no additional cost. Thus, we introduce the acceptance set

Due to the convexity of S, this set A S is convex, and under the mild condition

A S induces a convex risk measure via

see Section 4.1. Note that condition (9.33) holds in particular if S does not contain arbitrage opportunities. In this case, we have in fact

i.e., ρS is normalized. The main results of this chapter can be restated in terms of ρS :

Corollary 9.32. Under condition (9.6), the following conditions are equivalent:

(a) ρS is sensitive.

(b) S contains no arbitrage opportunities.

(c) PS ≠ ∅.

If these equivalent conditions hold, then

In other words, ρS can be represented in terms of the penalty function

Proof. That (a) implies (b) is obvious. The equivalence between (b) and (c) was shown in Theorem 9.9. Since both sides of (9.34) are cash invariant, it suffices to prove (9.34) for Y ≤ 0. But then the representation for ρS is just a special case of the superhedging duality (9.32). Finally, (9.34) and (c) imply that and the sensitivity of ρS follows.