8Efficient hedging

In an incomplete financial market model, a contingent claim typically will not admit a perfect hedge. Superhedging provides a method for staying on the safe side, but the required cost is usually too high both from a theoretical and from a practical point of view. It is thus natural to relax the requirements.

As a first preliminary step, we consider strategies of quantile hedging which stay on the safe side with high probability. In other words, we maximize the probability for staying on the safe side under a given cost constraint. The main idea consists in reducing the construction of such strategies for a given claim H to a problem of superhedging for a modified claim , which is the solution to a static optimization problem of Neyman–Pearson type. Typically, will have the form of a knock-out option, that is, = H · A . At this stage, we only focus on the probability that a shortfall occurs; we do not take into account the size of the shortfall if it does occur.

In Sections 8.2 and 8.3 we take a more comprehensive view of the downside risk. Our discussion of risk measures in Section 4.8 suggests to quantify the downside risk in terms of an acceptance set for suitably hedged positions. If acceptability is defined in terms of utility-based shortfall risk as in Section 4.9, we are led to the problem of constructing efficient strategies which minimize the utility-based shortfall risk under a given cost constraint. As in the case of quantile hedging, this problem can be decomposed into a static optimization problem and the construction of a superhedging strategy for a modified payoff profile . In Section 8.3 we go even one step further and assess the shortfall risk in terms of a general convex risk measure. For a complete market model and the case of AV@R, we discuss the structure of the modified payoff profile and point out the relation to the problem of robust utility maximization discussed in Section 3.5.

8.1Quantile hedging

Let H be a discounted European claim in an arbitrage-free market model such that P[ H = 0 ] < 1 and

We saw in Corollary 7.15 that there exists a self-financing trading strategy whose value process V↑ satisfies

By using such a superhedging strategy, the seller of H can cover almost any possible obligation which may arise from the sale of H and thus eliminate completely the corresponding risk. The smallest amount for which such a superhedging strategy is available is given by πsup(H). This cost will often be too high from a practical point of view, as illustrated by Example 7.21. Furthermore, if H is not attainable then πsup(H), viewed as a price for H, is too high from a theoretical point of view since it permits arbitrage. Even if H is attainable, a complete elimination of risk by using a replicating strategy for H would consume the entire proceeds from the sale of H, and any opportunity of making a profit would be lost along with the risk.

Let us therefore suppose that the seller is unwilling to put up the initial amount of capital required by a superhedge and is ready to accept some risk. What is the optimal partial hedge which can be achieved with a given smaller amount of capital? In order to make this question precise, we need a criterion expressing the seller’s attitude towards risk. Several of such criteria will be studied in the following sections. In this section, our aim is to construct a strategy which maximizes the probability of a successful hedge given a constraint on the initial cost.

More precisely, let us fix an initial amount

We are looking for a self-financing trading strategy whose value process maximizes the probability

over all those strategies whose initial investment V0 is bounded by v and which respect the bounds Vt ≥ 0 for t = 0, . . . , T. In view of Theorem 5.25, the second restriction amounts to admissibility in the following sense:

Definition 8.1. A self-financing trading strategy is called an admissible strategy if its value process satisfies VT ≥ 0.

◊

The problem of quantile hedging consists in constructing an admissible strategy such that its value process V∗ satisfies

where the maximum is taken over all value processes V of admissible strategies subject to the constraint

Note that this problem would not be well posed if considered without the constraint of admissibility.

Let us emphasize that the idea of quantile hedging corresponds to a Value at Risk criterion, and that it invites the same criticism: Only the probability of a shortfall is taken into account, not the size of the loss if a shortfall occurs. This exclusive focus on the shortfall probability may be reasonable in cases where a loss is to be avoided by any means. But for most applications, other optimality criteria as considered in the next section will usually be more appropriate from an economic point of view. In view of the mathematical techniques, however, some key ideas already appear quite clearly in our present context.

Let us first consider the particularly transparent situation of a complete market model before passing to the general incomplete case. The set

will be called the success set associated with the value process V of an admissible strategy. As a first step, we reduce our problem to the construction of a success set of maximal probability.

Proposition 8.2. Let P∗ denote the unique equivalent martingale measure in a complete market model, and assume that A∗ ∈ FT maximizes the probability P[A] over all sets A ∈ FT satisfying the constraint

Then the replicating strategy ∗ of the knock-out option

solves the optimization problem defined by (8.1) and (8.2), and A∗ coincides up to P-null sets with the success set of ∗.

Proof. As a first step, let V be the value process of any admissible strategy such that V0 ≤ v. We denote by A := {VT ≥ H} the corresponding success set. Admissibility yields that VT ≥ H · A . Moreover, the results of Section 5.3 imply that V is a P∗-martingale. Hence, we obtain that

Therefore, A fulfills the constraint (8.3) and it follows that

As a second step, we consider the trading strategy ∗ and its value process V∗. Clearly, ∗ is admissible, and its success set satisfies

On the other hand, the first part of the proof yields that

It follows that the two sets A∗ and {V∗≥ H} coincide up to P-null sets. In particular, T ∗ is an optimal strategy.

Our next goal is the construction of the optimal success set A∗, whose existence was assumed in Proposition 8.2. This problem is solved by using the Neyman–Pearson lemma. To this end, we introduce the measure Q∗ given by

The constraint (8.3) can be written as

Thus, an optimal success set must maximize the probability P[ A ] under the constraint Q∗[ A ]≤ α. We denote by dP/dQ∗ the generalized density of P with respect to Q∗ in the sense of the Lebesgue decomposition as constructed in Theorem A.17. Thus, we may define the level

and the set

Proposition 8.3. If the set A∗ in (8.7) satisfies

then A∗ maximizes the probability P[ A ] over all A ∈ FT satisfying the constraint

Proof. The condition E∗[ H · ] ≤ v is equivalent to Q∗[ A ]≤ α = Q∗[ A∗ ]. Thus, the A particular form of the set A∗ in (8.7) and the Neyman–Pearson lemma in the form of Proposition A.33 imply that P[ A ]≤ P[ A∗ ].

By combining the two Propositions 8.2 and 8.3, we obtain the following result.

Corollary 8.4. Denote by P∗ the unique equivalent martingale measure in a complete market model, and assume that the set A∗ of (8.7) satisfies

Then the optimal strategy solving (8.1) and (8.2) is given by the replicating strategy of the knock-out option

Our solution to the optimization problem (8.1) and (8.2) still relies on the assumption that the set A∗ of (8.7) satisfies Q∗[ A∗ ] = α. This condition is clearly satisfied if

However, it may not in general be possible to find any set A whose Q∗-probability is exactly α. In such a situation, the Neyman–Pearson theory suggests replacing the indicator function A∗ of the “critical region” A∗ by a randomized test, i.e., by an FT-measurable [0, 1]-valued function ψ. Let R denote the class of all randomized tests, and consider the following optimization problem:

where Q∗ is the measure defined in (8.4) and α = v/E∗[ H ] as in (8.5). The generalized Neyman–Pearson lemma in the form of Theorem A.35 states that the solution is given by

where c∗ is defined through (8.6) and γ is chosen such that EQ∗ [ ψ∗ ] = α, i.e.,

Definition 8.5. Let V be the value process of an admissible strategy . The success ratio of is defined as the randomized test

Note that the set {ψV = 1} coincides with the success set {VT ≥ H} of V. In the extended version of our original problem, we are now looking for a strategy which maximizes the expected success ratio E [ ψV ] under the measure P under the cost constraint V0 ≤ v:

Theorem 8.6. Suppose that P∗ is the unique equivalent martingale measure in a complete market model. Let ψ∗ be given by (8.8), and denote by a replicating strategy for the discounted claim H ∗ = H · ψ∗. Then the success ratio ψV∗ of ∗ maximizes the expected success ratio E[ ψV ] among all admissible strategies with initial investment V0 ≤ v. Moreover, the optimal success ratio ψV∗ is P-a.s. equal to ψ∗.

We do not prove this theorem here, as it is a special case of Theorem 8.7 below and its proof is similar to the one of Corollary 8.4, once the optimal randomized test ψ∗ has been determined by the generalized Neyman–Pearson lemma. Note that the condition

implies that ψ∗ = ∗ with A∗ as in (8.7), so in this case the strategy ∗ reduces to the Aone described in Corollary 8.4.

Now we turn to the general case of an arbitrage-free but possibly incomplete market model, i.e., we no longer assume that the set P of equivalent martingale measures consists of a single element, but we assume only that

In this setting, our aim is to find an admissible strategy whose success ratio ψV∗ satisfies

where the maximum on the right-hand side is taken over all admissible strategies whose initial investment satisfies the constraint

Theorem 8.7. In any arbitrage-free market model there exists a randomized test ψ∗ such that

and which maximizes E[ ψ ] over all ψ ∈ R subject to the constraints

Moreover, the superhedging strategy for the modified claim

with initial investment πsup(H∗) solves the problem (8.9) and (8.10).

Proof. Denote by R0 the set of all ψ ∈ R which satisfy the constraints (8.12), and take a sequence ψn ∈ R0 such that

Lemma 1.70 yields a sequence of convex combinations n ∈ conv{ψn , ψn+1, . . . } converging P-a.s. to a function ∈ R. Clearly, n ∈ R0 for each n. Hence, Fatou’s lemma yields that

and it follows that ∈ R0. Moreover,

so ψ∗ := is the desired maximizer.

We must also show that (8.11) holds. To this end, we assume by way of contradiction that supP∗∈E∗[ H · ψ∗ ] < v. Then there exists ε ∈ (0, 1) such that P ψε := ε + (1 − ε)ψ∗ ∈ R0. Since P[ ψ∗ = 1 ] = 1 is impossible due to our assumption v < πsup(H), we have E [ ψε ] > E[ ψ∗ ], in contradiction to the maximality of E [ ψ∗ ]. This proves (8.11).

Now let be any admissible strategy whose value process V satisfies V0 ≤ v. If ψV denotes the corresponding success ratio, then

The P-martingale property of V yields that for all P∗ ∈ P,

Therefore, ψV is contained in R0 and it follows that

Consider the superhedging strategy ∗ of H ∗ = H · ψ∗ and denote by V∗ its value ∗ process. Clearly, is an admissible strategy. Moreover,

Thus, (8.14) yields that ψV∗ satisfies

On the other hand, dominates H ∗, and so

Therefore, ψV∗ dominates ψ∗ on the set {H > 0}. Moreover, any success ratio is equal to one on {H = 0}, and we obtain that ψV∗ ≥ ψ∗ P-almost surely. According to (8.15), this can only happen if the two randomized tests ψV∗ and ψ∗ coincide P-almost everywhere. This proves that solves the hedging problem (8.9) and (8.10).

8.2Hedging with minimal shortfall risk

Our starting point in this section is the same as in the previous one: At time T, an investor must pay the discounted random amount H ≥ 0. A complete elimination of the corresponding risk would involve the cost

of superhedging H, but the investor is only willing to put up a smaller amount

This means that the investor is ready to take some risk: Any “partial” hedging strategy whose value process V satisfies the capital constraint V0 ≤ v will generate a nontrivial shortfall

In the previous section, we constructed trading strategies which minimize the shortfall probability

among the class of trading strategies whose initial investment is bounded by v, and which are admissible in the sense of Definition 8.1, i.e., their terminal value VT is nonnegative. In this section, we assess the shortfall in terms of a loss function, i.e., an increasing function : ℝ → ℝ which is not identically constant. We assume furthermore that

A particular role will be played by convex loss functions, which correspond to risk aversion in view of the shortfall; compare the discussion in Section 4.9.

Definition 8.8. Given a loss function satisfying the above assumptions, the shortfall risk of an admissible strategy with value process V is defined as the expectation

of the shortfall weighted by the loss function .

◊

Our aim is to minimize the shortfall risk over all admissible strategies satisfying the capital constraint V0 ≤ v. Alternatively, we could minimize the cost under a given bound on the shortfall risk. In other words, the problem consists in constructing strategies which are efficient with respect to the trade-off between cost and shortfall risk. This generalizes our discussion of quantile hedging in the previous Section 8.1, which corresponds to a minimization of the shortfall risk with respect to the nonconvex loss function

Remark 8.9. Recall our discussion of risk measures in Chapter 4. From this point of view, it is natural to quantify the downside risk in terms of an acceptance set A for hedged positions. As in Section 4.8, we denote by A¯ the class of all positions X such that there exists an admissible strategy with value process V such that

for some A ∈ A . Thus, the downside risk of the position −H takes the form

Suppose that the acceptance set A is defined in terms of shortfall risk, i.e.,

where is a convex loss function and x0 is a given threshold. Then ρ(−H) is the smallest amount m such that there exists an admissible strategy whose value process V satisfies V0 = m and

For a given m, we are thus led to the problem of finding a strategy which minimizes the shortfall risk under the cost constraint V0 ≤ m. In this way, the problem of quantifying the downside risk of a contingent claim is reduced to the construction of efficient hedging strategies as discussed in this section.

◊

As in the preceding section, the construction of the optimal hedging strategy is carried out in two steps. The first one is to solve the “static” problem of minimizing

over all FT-measurable random variables Y ≥ 0 which satisfy the constraints

If Y∗ solves this problem, then so does := H ∧ Y∗. Hence, we may assume that 0 ≤ Y∗ ≤ H or, equivalently, that Y∗ = H ψ∗ for some randomized test ψ∗, which belongs to the set R of all FT-measurable random variables with values in [0, 1]. Thus, the static problem can be reformulated as follows: Find a randomized test ψ∗ ∈ R which minimizes the “shortfall risk”

over all ψ ∈ R subject to the constraints

The next step is to fit the terminal value VT of an admissible strategy to the optimal profile H ψ∗. It turns out that this step can be carried out without any further assumptions on our loss function . Thus, we assume at this point that the optimal ψ∗ of step one is granted, and we construct the corresponding optimal strategy.

Theorem 8.10. Given a randomized test ψ∗ which minimizes (8.16) subject to (8.17), a superhedging strategy for the modified discounted claim

with initial investment πsup(H∗) has minimal shortfall risk among all admissible strategies which satisfy the capital constraint 1 · X0 ≤ v.

Proof. The proof extends the last argument in the proof of Theorem 8.7. As a first step, we take any admissible strategy such that the corresponding value process V satisfies the capital constraint V0 ≤ v. Denote by

the corresponding success ratio. It follows as in (8.13) that ψV satisfies the constraints

Thus, the optimality of ψ∗ implies the following lower bound on the shortfall risk of :

In the second step, we consider the admissible strategy ∗ and its value process V∗. On the one hand,

so ∗ satisfies the capital constraint. Hence, the first part of the proof yields

On the other hand, and therefore

Hence, the inequality in (8.18) is in fact an equality, and the assertion follows.

Let us now return to the static problem defined by (8.16) and (8.17). We start by considering the special case of risk aversion in view of the shortfall.

Proposition 8.11. If the loss function is convex, then there exists a randomized test ψ∗ ∈ R which minimizes the shortfall risk

over all ψ ∈ R subject to the constraints

If is strictly convex on [0,∞), then ψ∗ is uniquely determined on {H > 0}.

Proof. The proof is similar to the one of Proposition 3.36. Let R0 denote the set of all randomized tests which satisfy the constraints (8.19). Take ψn ∈ R0 such that E [( H (1 − ψn) )] converges to the infimum of the shortfall risk, and use Lemma 1.70 to select convex combinations n ∈ conv{ψn , ψn+1, . . . } which converge P-a.s. to some ∈ R. Since is continuous and nonnegative, Fatou’s lemma implies that

where we have used the convexity of in the second step. Fatou’s lemma also yields that for all P∗ ∈ P

Hence ∈ R0, and we conclude that ψ∗ := is the desired minimizer. The uniqueness part is obvious.

Remark 8.12. The proof shows that the analogous existence result holds if we use a robust version of the shortfall risk defined as

where Q is a class of equivalent probability measures; see also Remark 3.37 and Sections 8.2 and 8.3.

◊

Combining Proposition 8.11 and Theorem 8.10 yields existence and uniqueness of an optimal hedging strategy under risk aversion in a general arbitrage-free market model.

Corollary 8.13. Assume that the loss function is strictly convex on [0,∞). Then there exists an admissible strategy which is optimal in the sense that it minimizes the shortfall risk over all admissible strategies subject to the capital constraint 1·X0 ≤ v. Moreover, any optimal strategy requires the exact initial investment v, and its success ratio is P-a.s. equal to

where ψ∗ denotes the solution of the static problem constructed in Proposition 8.11.

Proof. The existence of an optimal strategy follows by combining Proposition 8.11 and Theorem 8.10. Strict convexity of implies that ψ∗ is P-a.s. unique on {H > 0}. In particular, ψ∗ and the success ratio ψV∗ of any optimal strategy ∗ must coincide P-a.s. on {H > 0}. On {H = 0}, the success ratio ψV∗ is equal to 1 by definition.

Since is strictly increasing on [0, ∞), we must have that

for otherwise we could find some ε > 0 such that ψε := ε + (1 − ε)ψ∗ also satisfies the constraints (8.17). Since we have assumed that v < πsup(H), the constraints (8.17) imply that ψ∗ is not P-a.s. equal to 1 on {H > 0} and hence that

This, however, contradicts the optimality of ψ∗.

Since the value process V∗ of an optimal strategy is a P-martingale, and since

we conclude from the above that

Thus, is equal to v.

Beyond the general existence statement of Proposition 8.11, it is possible to obtain an explicit formula for the optimal solution of the static problem if the market model is complete. Recall that we assume that the loss function (x) vanishes for x ≤ 0. In addition, we will also assume that

is strictly convex and continuously differentiable on (0,∞).

Then the derivative ʹ of is strictly increasing on (0, ∞). Let J denote the inverse function of ʹ defined on the range of ʹ, i.e., on the interval (a, b) where a := limx↓0 ʹ(x) and b := limx↑∞ ʹ(x). We extend J to a function J+ : [0,∞] → [0,∞] by setting

From now on, we assume also that

i.e., P∗ is the unique equivalent martingale measure in a complete market model. Its density will be denoted by

Theorem 8.14. Under the above assumptions, the solution of the static optimization problem of Proposition 8.11 is given by

where the constant c is determined by the condition E∗[H ψ∗ ] = v.

Proof. The problem is of the same type as those considered in Section 3.3. It can in fact be reduced to Corollary 3.43 by considering the random utility function

Just note that the shortfall risk E [ (H−Y) ] coincides with the negative expected utility −E[u(Y, ·) ] for any profile Y such that 0 ≤ Y ≤ H. Moreover, since our market model is complete, it has a finite structure by Theorem 5.37, and so all integrability conditions are automatically satisfied. Thus, Corollary 3.43 states that the optimal profile H ∗ := Y∗ which maximizes the expected utility E[ u(Y, ·) ] under the constraints 0 ≤ Y ≤ H and E∗[ Y ]≤ v is given by

Dividing by H yields the formula for the optimal randomized test ψ∗.

Corollary 8.15. In the situation of Theorem 8.14, suppose that the objective probability measure P is equal to the martingale measure P∗. Then the modified discounted claim takes the simple form

Example 8.16. Consider the discounted payoff H of a European call option with strike K under the assumption that the numéraire S0 is a riskless bond, i.e., that for a certain constant r ≥ 0. If the assumptions of Corollary 8.15 hold, then the modified profile H ∗ is the discounted value of the European call option struck at := K + J+(c∗) · (1 + r)T, i.e.,

Example 8.17. Consider an exponential loss function (x) = (eαx − 1)+ for some α > 0. In this case,

and the optimal profile is given by

Example 8.18. If is the particular loss function

for some p > 1, then the problem is to minimize a lower partial moment of the difference VT − H. Theorem 8.14 implies that it is optimal to hedge the modified claim

where the constant cp is determined by

◊

Let us now consider the limit p ↑ ∞ in (8.20), corresponding to ever increasing risk aversion with respect to large losses.

Proposition 8.19. Let us consider the loss functions

for p > 1. As p ↑ ∞, the modified claims H ψ∗pof (8.20) converge P-a.s. and in L1(P∗) to the discounted claim

where the constant c∞ is determined by

Proof. Let γ(p) be shorthand for 1 /(p − 1) and note that

Hence, if (pn) is a sequence for which converges to some ∈ [0,∞], then

Hence,

Since each term on the left-hand side equals v, we must have

which determines uniquely as the constant c∞ of (8.21).

Example 8.20. If the discounted claim H in Proposition 8.19 is the discounted payoff of a call option with strike K, and the numéraire is a riskless bond as in Example 8.16, then the limiting profile is equal to the discounted call with the higher strike price

◊

In the remainder of this section, we consider loss functions corresponding to risk neutrality and to risk-seeking preferences. In the first case, the loss function will be convex but not strictly convex. In the second case, it will not be convex. Let us first consider the risk-neutral case.

Example 8.21. In the case of risk neutrality, the loss function is given by

Thus, the task is to minimize the expected shortfall

under the capital constraint V0 ≤ v. Let P∗ be the unique equivalent martingale measure in a complete market model. Then the static problem corresponding to Proposition 8.11 is to maximize the expectation

under the constraint that ψ ∈ R satisfies

We can define two equivalent measures Q and Q∗ by

The problem then becomes the hypothesis testing problem of maximizing EQ[ ψ ] under the side condition

Since the density dQ/dQ∗ is proportional to the inverse of the density φ∗ = dP∗/dP, Theorem A.35 implies that the optimal test takes the form

where the constant c1 is given by

and where the constant γ∈ [0, 1] is chosen such that can be arbitrary.

◊

Assume now that the shortfall risk is assessed by an investor who, instead of being risk-averse, is in fact inclined to take risk. In our context, this corresponds to a loss function which is concave on [0,∞) rather than convex. It is not difficult to generalize Theorem 8.14 so that it covers this situation. Here we limit ourselves to the following explicit case study.

Example 8.22. Consider the loss function

for some q ∈ (0, 1). In order to solve our static optimization problem, one could apply the results and techniques of Section 3.3. Here we will use an approach based on the Neyman–Pearson lemma. Note first that for ψ ∈ R

Hence, we get a lower bound on the “shortfall risk” of ψ:

The problem of finding a minimizer of the right-hand side is equivalent to maximizing the expectation EQ[ ψ ] under the constraint that EQ∗ [ ψ ] ≤ v /E ∗[ H ] for the measures Q and Q∗ defined via

if we assume again P[ H > 0 ] = 1. As in Example 8.21, we then conclude that the optimal test must be of the form

for certain constants and γ. Under the simplifying assumption that

the formula (8.23) reduces to

By taking we obtain an identity in (8.22), and so must be a minimizer for E[ (H(1 − ψ)) ] under the constraint that E∗[H ψ] ≤ v.

◊

In our last result of this section, we recover the knock-out option

which was obtained as the solution to the problem of quantile hedging by taking the limit q ↓ 0 in (8.25). Intuitively, decreasing q corresponds to an increasing appetite for risk in view of the shortfall.

Proposition 8.23. Let us assume for simplicity that (8.24) holds for all q ∈ (0, 1), that P[ H > 0] = 1, and that there exists a unique constant such that

Then the solutions of (8.25) converge P-a.s. to the solution

of the corresponding problem of quantile hedging as constructed in Proposition 8.3.

Proof. Take any sequence qn ↓ 0 such that converges to some ∈ [0,∞]. Then

Hence,

Since we assumed (8.24) for all q ∈ (0, 1), the left-hand terms are all equal to v, and it follows from (8.26) that This establishes the desired convergence.

8.3Efficient hedging with convex risk measures

As in the previous sections of this chapter, we consider the shortfall

arising from hedging the discounted claim H with a self-financing trading strategy with initial capital

In this section, our aim is to minimize the shortfall risk

where ρ is a given convex risk measure as discussed in Chapter 4. Here we assume that ρ is defined on a suitable function space, such as Lp(Ω,F, P), so that the shortfall risk is well-defined and finite; cf. Remark 4.44. In particular, we assume that ρ(Y) = ρ() whenever Y = P-a.s.

As in the preceding two sections, the construction of the optimal hedging strategy can be carried out in two steps. The first step is to solve the static problem of minimizing

over all FT-measurable random variables Y ≥ 0 that satisfy the constraint

If Y∗ solves this problem, then so does H ∧Y∗.Hence,we may assume that 0 ≤ Y∗ ≤ H, and we can reformulate the problem as

The next step is to fit the terminal value VT of an admissible strategy to the optimal profile Y∗. It turns out that this step can be carried out without any further assumptions on our risk measure ρ. Thus, we assume at this point that the optimal Y∗ of step one is granted, and we construct the corresponding optimal strategy.

Proposition 8.24. A superhedging strategy for a solution Y∗ of (8.27) with initial investment πsup(Y∗) has minimal shortfall risk among all admissible strategies whose value process satisfies the capital constraint V0 ≤ v.

Proof. Let V be the value process of any admissible strategy such that V0 ≤ v. Due to Doob’s systems theorem in the form of Theorem 5.14, V is a martingale under any P∗ ∈ P, and so

Thus, Y := H ∧ VT satisfies the constraints in (8.27), and we get

Next let V∗ be the value process of a superhedging strategy for Y∗ with initial investment

Then we have and Moreover, P-a.s., and thus

This concludes the proof.

Let us now return to the static problem defined by (8.27).

Proposition 8.25. If the convex risk measure ρ is lower semicontinuous with respect to P-a.s. convergence of random variables in the class {−Y |0 ≤ Y ≤ H} and ρ(−Y) < ∞ for one such Y, then there exists a solution of the static optimization problem (8.27). In particular, there exists a solution if H is bounded and ρ is continuous from above.

Proof. Take a sequence Yn with 0 ≤ Yn ≤ H and supP∗∈P E∗[ Yn ] ≤ v such that ρ(Yn − H) converges to the infimum A of the shortfall risk. We can use Lemma 1.70 to select convex combinations Z n ∈ conv{Yn , Yn+1, . . . } which converge P-a.s. to some random variable Z. Then 0 ≤ Z ≤ H and Fatou’s lemma yields that

for all P∗ ∈ P. Lower semicontinuity of ρ implies that

Moreover, the right-hand side is equal to A, due to the convexity of ρ. Hence, Z is the desired minimizer.

Combining Proposition 8.25 and Proposition 8.24 yields the existence of a risk-minimizing hedging strategy in a general arbitrage-free market model. So far, all arguments were practically the same as in the preceding two sections.

Beyond the general existence statement of Proposition 8.25, it is sometimes possible to obtain an explicit formula for the optimal solution of the static problem (8.27) if the market model is complete and so P = {P∗}. In this case, the static optimization problem simplifies to

By substituting Z for H − Y, this is equivalent to the problem

where := E∗[ H ] − v. We will now discuss this problem in the case ρ = AV@Rλ. Our approach relies on the general idea that a minimax problem can be transformed into a standard minimization problem by using a duality result for the expression involving the maximum. In the case of AV@Rλ, we can use the following representation of AV@Rλ from Proposition 4.51,

for Z ≥ 0. Our discussion of problem (8.28) will be valid also beyond the setting of a complete discrete-time market model, whose underlying probability space has necessarily a discrete structure by Theorem 5.37. In fact it applies whenever P and P∗ are two equivalent probability measures on a given measurable space (Ω,F). This is important in view of the application of the next theorem in Example 3.50.

Theorem 8.26. Suppose that H ∈ L1(P) and denote by φ := dP∗/dP the price density of P∗ with respect to P. Then the problem (8.28) admits a solution for ρ = AV@Rλ which is of the form

for certain constants c > 0, r∗ ≥ 0, and γ ∈ [0, 1].

Proof. Recall from Section 4.4 that

where Qλ is the set of all probability measures Q P whose density dQ/dP is P-a.s. bounded by 1/λ. Thus it follows from Fatou’s lemma that AV@Rλ is lower semicontinuous with respect to P-a.s. convergence of random variables in the class {−Y |0 ≤ Y ≤ H }⊂ L1 (the same argument also gives upper semicontinuity and hence continuity, but this fact is not needed here). Proposition 8.25 hence yields the existence of a solution Z ∗ of the minimization problem (8.28). By (8.29), Z ∗ must solve

where r∗ ≥ 0 is such that

Proposition 4.51 states that r∗ is a λ-quantile for Z ∗.

Let us now solve (8.31). To this end, we consider first the case in which r∗ = 0. Then we are in the situation of Example 8.21 and obtain

for constants c > 0 and γ ∈ [0, 1]. This representation coincides with (8.30) for r∗ = 0.

Now we consider the case r∗ > 0. Note first that we must have Z∗ ≥ H ∧ r∗. Indeed, let us assume P[ Z ∗ < H ∧ r∗ ] > 0. Then we could obtain a strictly lower risk AV@Rλ(−Z∗) either by decreasing the level r∗ in case P[ Z∗ ≤ H ∧ r∗ ] = 1 or, in case P[ Z∗ > H∧r∗ ] > 0, by shifting mass of Z∗ from {Z∗ > H∧r∗} to the set {Z∗ < H∧r∗}.

Thus, we can solve our problem by minimizing E[ (+ H ∧ r∗ − r∗)+ ] subject to

Any satisfying these constraints must be concentrated on {H > r∗}, so that the problem is equivalent to

But this problem is equivalent to the one for r∗ = 0 if we replace H by H −H∧r∗. Hence, it is solved by

for some constants c > 0 and γ ∈ [0, 1]. It follows that

We now solve (8.28) in the more specific situation in which H = 1. In this case, one sees that r∗ and c are functions of the capital . The next result shows that these functions behave as follows. As long as is below some critical threshold v∗, we have r∗ = 0, and c = c() is determined by the requirement E∗[ Z∗ ] = . Above the critical threshold v∗, the value of c is always equal to c(v∗), and now r∗ > 0 is determined by the requirement E ∗[ Z ∗ ] = .

Theorem 8.27. Consider the setting of Theorem 8.26. Assume in addition that H = 1 and that φ has a continuous and strictly increasing quantile function qφ with respect to P and satisfies φ∞ > λ−1. Then the solution Z∗ of problem (8.28) is P-a.s. unique. Moreover, there exists a critical capital level v∗ such that

where t0 is determined by the condition E ∗[ Z ∗ ] = , and

where and, for ds, the level tλ is the unique solution of the equation

Finally, the critical capital level v∗ is equal to 1 − Φ(tλ).

Proof. We fix ∈ (0, 1). For a constant c we then let

where r = r(c) ≥ 0 is such that E∗[ Zr ] = , i.e.,

This makes sense as long as c ≥ c0, where c0 is defined via = E[ φ; φ > c0 ]. Theorem 8.26 states that a solution of our problem can be found within the class {Zr(c) | c ≥ c0}. Thus, we have to minimize

over c ≥ c0. Here we have used Lemma A.27 in the second identity. This minimization problem can be simplified further by using the reparameterization c = qφ(t), which is one-to-one according to our assumptions. Indeed, by letting

we simply have to minimize the function

over t ≥ t0 := Fφ(c0). For t ≤ 1 −λ, we get R(t) = λ, which cannot be optimal. We show next that the function

has a unique maximizer tλ ∈ (1−λ, 1], which will define the solution as soon as tλ ≥ t0 and as long as t = t0 does not give a better result. To this end, we note first that

The numerator of this expression is strictly larger than zero for t ≤ 1 − λ and equal to Φ(1 − λ) > 0 at t = 1 − λ. Moreover, for t > 1 − λ,

which is easily seen to be strictly decreasing in t. For t ↑ 1 this expression converges to 1− λφ∞, which is strictly negative due to our assumption φ∞ > λ−1. Hence, the numerator in (8.35) has a unique zero tλ ∈ (1 − λ, 1), which is the unique solution of the equation

and this solution tλ is consequently the unique maximizer of Ψ.

If tλ ≤ t0, then R has no minimizer on (t0, 1], and it follows that t∗ = t0 is its minimizer. Let us compare R(tλ) with R(t0) in case tλ > t0. We have

and

Since tλ is the unique maximizer of the function t (t−1+λ)+/ Φ(t),we thus see that R(tλ) is strictly smaller than R(t0). Hence the solution is defined by

Clearly, tλ is independent of , while t0 decreases from 1 to 0 as increases from 0 to 1. Thus, by taking v∗ as the capital level for which tλ = t0, we see that the optimal solution has the form

where

Finally, if is equal to the critical capital level v∗, we must have that tλ = t0. But t0 was defined to be the solution of 1 − Φ(t) = , and so v∗ = 1 −Φ(tλ).

Let us now point out the connections of the preceding theorem with robust statistical test theory as explained in Section 3.5 and in particular in Remark 3.54. To this end, let R denote the set of all measurable functions ψ : Ω → [0, 1], which will be interpreted as randomized statistical tests; see Remark A.36. Problem (8.28) for H = 1 can then be rewritten as

where Qλ is the set of all probability measures Q P whose density dQ/dP is P-a.s. bounded by 1 /λ. The solution ψ∗ of this problem is described in Theorem 8.27. It is easy to see that ψ∗ must also solve the following dual problem:

where α = supQ∈EQ[ ψ∗ ]. Thus, ψ∗ is an optimal randomized test for testing the ∗ Qλ hypothesis P* against the composite null hypothesis Qλ; see Remark 3.54. If Qλ admits a least-favorable measure Q0 with respect to P∗ in the sense of Definition 3.48, then ψ∗ must also be a standard Neyman–Pearson test for testing the hypothesis P∗ against the null hypothesis Q0. By Theorem A.35, it must hence be of the form

where c > 0 and κ ∈ [0, 1] are constants and

Under the assumptions of Theorem 8.27, this is the case if π = c(φ ∨ qφ(tλ)), where c is a suitable constant. It follows that

We have by (8.34),

This yields that c = λ, and we see that

defines a probability measure Q0 ∈ Qλ with

The following result now follows immediately from Theorem 8.27:

Corollary 8.28. For a measure P∗≈ P satisfying the assumptions of Theorem 8.27, the measure Q0 in (8.36) is a least-favorable measure for

in the sense of Definition 3.48.