Optimal Design of Structured Products and the Role of Capital Protection Carole Bernard, Phelim Boyle, and Weidong Tian

∗

Abstract Many variants of structured products exist and these contracts have become very popular in recent years. This paper discusses the optimal design of structured products from the seller’s perspective. We use a simple model to find the optimal payoff for a class of structured products. We find that when there is a deterministic guaranteed floor in the contract the optimal design is robust and does not depend on the issuer’s preferences. This is no longer the the case when the contract has a stochastic floor. In this case we show how how the design depends explicitly on the issuer’s preferences. Our results provide a framework to study structured products known as Index Linked Products. An empirical analysis of structured products traded on the American Stock Exchange indicates that our theoretical framework provides some justification for the features found in existing contracts.

Keywords: Index-Linked Products, Optimal Design, Structured Products. JEL Classification: G10, G20.

Monday, November 19, 2007

Carole Bernard is with the University of Waterloo. Email: [email protected] Phelim Boyle is with Wilfrid Laurier University. Email: [email protected]. Weidong Tian is with the University of Waterloo. Email:[email protected]. ∗

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1

Introduction

In years financial innovation has had a profound impact on the market for structured products. There are some common features in many of these structured products sold by different issuers and at the same time there is also considerable heterogeneity. We assume that there is a consumer demand for contracts with certain generic features. Against this background the paper asks if there is an optimal contract design for such contracts with these features. To what extent, if any, will the optimal design depend on the characteristics of the issuer when we take as given certain1 contract features that are desired by the consumer. Many investors want both downside protection in bear markets and upside participation in bull markets and there is a wide variety of structured investment products that cater to such investors. These contracts can be equity-linked products sold by insurance companies or structured retail products marketed by banks. They often include a minimum guaranteed amount and a return linked to the performance of an index or a portfolio in a specific way 2 . A recent study by Breuer and Perst (2007) investigates the demand for reverse convertibles and reverse convertible bonds as examples of structured products. In particular, they show in the context of prospect theory (Tversky and Kahneman (1992)) that investors who underestimate the volatility of the expected return of the underlying stock, will be interested in these products. Most papers on structured products deal with the pricing issue. Stoimenov and Wilkens (2005) explain that these securities are designed for retail investors and facilitate complexpositions in options. They highlight a lack of transparency in the German market. More precisely, their study reveals that premiums charged to customers are not fair compared to the prices of exchange-traded options. Structured products appear to be overpriced and thus favor issuing institutions. They also provide a classification of structured exchangetraded products in Germany and conclude that the more complex the product the more it is overpriced. This is consistent with the theoretical model of Carlin (2007) which predicts that companies can increase their profits by increasing product complexity. However Carlin does not discuss how to measure complexity. Wilkens and Stoimenov (2007) focus on leverage products in the German retail market. They also reveal irregularities in the pricing of these products by analyzing differences between bid and ask quotes. They find that these products provide almost risk-free products for their issuers. Benet, Gianetti and Pissaris (2005) analyze the pricing of a particular type of structured products: reverse-exchangeable securities, which 1

to be described later Indexed Linked Products constitute a major class of structured products. Risk Magazine (Jan 2007,) estimates the size of the global retail products market to be 230 billion (in pounds sterling ) in 2005. 2

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are traded on the AMEX (American Stock Exchange). They also shows significant pricing bias in favor of issuers. The present paper differs from these paper in that we focus on the (optimal) design of structured products rather than their pricing. We first present some theoretical results on the optimal design in a simplified framework and then compare our theoretical predictions with the actual design of exchange-traded structured products. Thus, our study supplements the classification proposed by Stoimenov and Wilkens (2005). More precisely, we discuss a generic Index-Linked Product with a minimum guarantee (floor) and which meets (or beats) an index with a certain confidence level. We note that this generic index-linked contract contains standard index-linked products currently issued on the market as special cases3 . Several recent papers analyze the design of financial contracts that beat a benchmark with a confidence level, see for example: Jorion (2003), Alexander and Baptista (2004), Basak, Shapiro and Tepla (2006), Boyle and Tian (2007). Our aim is to find the optimal contract design (from the viewpoint of the issuer) that provides a guaranteed floor (to the investor) and beats a benchmark with a given confidence level. We assume that the market is incomplete. The incomplete4 market assumption is consistent with the security design5 literature. We solve the optimal design problem, from the issuer’s perspective with the relatively weak assumption that the issuer’s utility function is non decreasing with respect to the wealth. The insurer receives the single premium at time zero and the promised payment under the contract becomes a liability to the insurer. We assume the insurers objective is to minimize the expected utility of this liability. In order to concentrate on the design issue from the seller’s perspective we have adopted a particular focus. We assume the seller desires to design a product that meets certain specifications that are popular with consumers. The issuer’s objective is to minimize the expected utility of the liability under weak assumptions concerning the issuer’s preferences. 3

For instance, Lehman Brothers Holdings Inc. currently sells Index-Plus Notes linked to the S&P500 Index on the American Stock Exchange. This product gives the opportunity to the investor to realize a multiple (greater than 1) of an increase in the value of the S&P 500 index when the final index is greater that the initial index level but otherwise the investor might incur some loss in his initial investment. The investor thus beats the index with a given probability (the probability that the initial index level is greater than the initial one). Another example is ABN AMRO’s “Constant Proportion Debt Obligation” which won the “Innovation of the Year 2006” an award by the IFR (International Financing Review) and is extensively studied by Benet et al. (2006). The basic idea behind this contract is to generate a targeted return at a given confidence level. 4 Strictly speaking, if a new non redundant traded security is introduced in an incomplete market the other asset prices will change (Detemple and Selden (1991), Boyle and Wang (2001)). Therefore the impact on the underlying asset price should be incorporated in the optimal design of a contract, which can lead to substantial difficulties. 5 Allen and Gale (1992), Boot and Thakor (1993), Madan and Soubra (1991), DeMarzo and Duffie (1999).

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Thus we do not discuss how the liabilities are to be funded. It is also beyond the scope of this paper to discuss how equilibrium is reached in the market6 for structured products. The main results of this paper are as follows. We use the term capital protection to denote the product feature corresponding to the guaranteed return of the investor’s initial investment. 1. When the contract includes capital protection, the optimal design of the generic ILP does not depend on the utility function of the issuer. 2. If the contract does not include capital protection, it is not possible to find an optimal ILP for some particular utility functions. This provides a theoretical motivation for capital protection. 3. If there is a stochastic guaranteed floor and the issuer is risk-neutral, then the optimal design is the same as in the case when the contract includes capital protection. 4. In general, for a given stochastic guaranteed floor, the optimal design for a generic ILP depends on the preferences of the issuer. 5. Optimal generic ILPs can generate discontinuous payoff structures which can explain why some exchange-traded ILPs exhibit discontinuities in their payoff structures. Our results can be used to provide a perspective on the design of existing contracts. First, the theoretical designs we obtain are consistent with existing structured products that often include a capital protection even though issuers have different preferences and might have different pricing models. Section 4 gives an overview of the structured products currently traded on the American Stock Exchange. Second, a product design without capital protection may not be optimal for the issuer. Third, if the guaranteed is random, issuers with different risk preferences will have different optimal designs. Our results are also related to the maximum-probability strategy, which was discussed by Browne (2000), and Spivak and Cvitani´c (1999). In the maximum-probability strategy the aim is to beat a benchmark with the highest possible probability. Under certain conditions on the benchmark, we show that the optimal contract design in this paper coincides with the optimal design based on the maximum-probability strategy, but in most cases, they differ. There are also certain similarities between the optimal design of a generic ILP with the optimal design of an insurance contract. The literature7 on the optimal design of insurance contracts usually takes the policyholder’s perspective. Even though we concentrate on index-linked products instead of insurance, the results in this paper should be useful for the optimal design 6 7

Carlin (2007) discusses this issue. See for instance Arrow (1963), Raviv (1979).

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of insurance contracts as well. Finally, our study complements some previous literature the optimal design of equity indexed annuities from the investor’s perspective (Boyle and Tian (2007)) or the work by Barrieu and El Karoui (2002) who look at Pareto financial contracts between two agents. The rest of this paper is as follows. Section 2 sets out the framework. We use a singleperiod setting. We show that the optimal design of a generic ILP with capital protection does not depend on the issuer’s preferences. In section 3, we first provide optimal designs in the absence of capital protection and show that there might be no optimal design or that they might depend on issuers’ risk aversion. We then extend the results to a dynamic setting, where early redemption is possible. In section 4, we present our empirical analysis of the exchange-traded structured products on the American Stock Exchange and compare the empirical analysis with our theoretical design structured. We show that an ILP with a discontinuous payoff structure can be constructed from a portfolio of a standard ILP and an optimal ILP. All proofs are given in Appendix.

2

Optimal Design with Capital Protection

This section describes the set up, explains the assumptions and develops the model. We also present the first main result of the paper in Proposition 1. This proposition gives the explicit form of the optimal payoff of a contract which satisfies two basic constraints. This result is robust in the sense that the payoff does not depended on the issuer’s preferences. We consider a one-period setting; the two dates are zero and T . There are many investors who invest in financial products and there are many firms issueing these products. We assume that firms are facing the same group of investors but that the firms can have different risk preferences and might have different business constraints. A probability space (Ω, F, P ) characterizes uncertainty, where Ω denotes the states of the real world, F the events, and P the probability measure. Any structured product is defined as an element of L2 (P ). We assume the existence of a risk-free asset: a zero-coupon bond with maturity T and a basic financial commodity: a traded portfolio or an index. For simplicity we will refer to it as the index. Each firm has a general utility function ua (.) (a stands for a particular firm). We assume that ua (.) is increasing but not necessarily concave or convex8 . Note that the risk neutral assumption is a special of our preference assumption. Prices in the financial market are endogenous and the index price is known to all firms. Each firm has a pricing operator 8

It is documented in the behavioral finance literature that u(.) might be partly convex and partly concave. See Kahneman and Tversky (1979). A recent application of this type of preferences to the pricing of structured products has been developed by Breuer and Perst (2007).

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pa : L2 (P ) → R and this pricing operator can differ across firms. Let A and B be two claims in L2 (P ). By a pricing operator we mean a linear functional from L2 (P ) to R satisfying the following additive property, and semi-positivity property9 : (1). pa (A) + pa (B) = pa (A + B), ∀A, B ∈ L2 (P ) (2). pa (A) ≥ 0, ∀A ∈ L2 (P )+ , (3). pa (A) > 0, ∀A ∈ L2 (P )++ . where L2 (P )+ = {X ∈ L2 (P ) : X ≥ 0, a.s.}, and L2 (P )++ = {X ∈ L2 (P )+ : P (X > 0) > 0}. Later, we will consider structured products with a guaranteed floor, a cap, or with a benchmark. The guaranteed floor, the cap and/or the benchmark are all assumed to be positive almost surely. Assume Γ ∈ L2 (P )++ is the benchmark that the investor wants to match or to beat with a confidence level α. The cases α = 0 or α = 1 are special because in the first case no constraint is imposed and in the second case a guaranteed floor is imposed. We consider the general case when α ∈ (0, 1). Let Γ0 be the minimum guaranteed floor (Γ0 ≤ Γ a.s.). The issuer receives an upfront payment to fund the contract and agrees to make a payoff to the investor at the future time, T . The payoff X satisfies: X ≥ Γ0 , a.s.,

P (X ≥ Γ) ≥ α

(1)

We now discuss the constraints in (1). Since X ≥ Γ0 , a.s., by the no-arbitrage principle, the present value of X must be at least as large as the present value of Γ0 : pa (X) ≥ pa (Γ0 ). Assume that pa (Γ) < ∞ and that P (Γ > Γ0 ) > 1 − α. We first consider the capital protection case, in which Γ0 = f is a positive constant.

2.1

Capital Protection

We now describe the payoff in more detail and formulate the issuer’s design problem. Assume X is the terminal payoff at time T of the structured product, Γ is the given benchmark, which is linked to the market performance, f is the fixed guaranteed amount. The issuer’s design 9

By Riesz representation theorem, the pricing operator is constructed as the expectation under a riskneutral measure. See Harrison-Kreps (1979). In an incomplete market model, since there are many equivalent martingale measures, the pricing operator might be different across the firms.

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problem is to solve the optimization problem, min E[ua (X)] s.t.

(

X ≥ f, a.s., P (X ≥ Γ) ≥ α

(2)

where ua (.) represents the preferences of the issuer (firm). Recall that we only assume that ua (.) is increasing. The solution of the optimal design problem (2) is given in the next Proposition. We denote the optimal payoff at time T by X ∗ . Proposition 1 Given a firm with utility function ua (.) and a pricing operator pa (.). Assume that ua (f ) > −∞, f > 0, Γ ∈ L2 (P )++ , α ∈ [0, 1], P (Γ > f ) > 1 − α, and the distribution function of Γ is continuous, then there exists a constant d > 0 such that the following payoff X ∗ is the optimal payoff subject to constraints (1), where

for all ω ∈ Ω.

   f ∗ X (ω) = Γ(ω)   f

if Γ(ω) ≤ f if f < Γ(ω) < f + d if Γ(ω) ≥ f + d

(3)

The proof of proposition 1 is given in Appendix A. The result is surprisingly simple. First, the payoff does not depend on the preference function ua (.). Moreover the payoff does not depend on the pricing operator pa (.) as well since d is determined by the distribution of the benchmark Γ only. Precisely, P (Γ − f ≥ d) = 1 − α. (4) Therefore, the optimal design X ∗ is the same for different issuers. Second, the proof does not depend on the market completeness. 10 The payoff of X is quite straightforward. It can be decomposed into two positions. The first component is a zero-coupon bond with face value f and maturity T . The second component is a knockout digital option with payoff at maturity (Γ − f )1{f 0 such that the payoff X ∗ is the optimal payoff, where    Γ0 (ω) ∗ X (ω) = Γ(ω)   Γ0 (ω)

if ua (Γ(ω)) ≤ ua (Γ0 (ω)) if 0 < ua (Γ(ω)) − ua (Γ0 (ω)) < d if ua (Γ(ω)) − ua (Γ0 (ω)) ≥ d

(12)

The constant d is such that P (ua (Γ) − ua (Γ0 ) ≥ d) = 1 − α. This proposition is proved in the Appendix. Proposition 2 shows that the optimal design now depends on the issuer’s preferences. For a stochastic guaranteed amount, the optimal design depends on the firm’s preferences. This stands in sharp contrast to the first case with capital protection where the optimal design is independent of the issuer’s preferences. We next illustrate this proposition with three examples: Example 3.1 ua (x) = x. In this case we assume the issuing firm is risk-neutral. Here the optimal design is formally the same as the optimal design in the case when there is capital protection.    Γ0 (ω) ∗ X (ω) = Γ(ω)   Γ0 (ω)

if Γ(ω) ≤ Γ0 (ω) if 0 < Γ(ω) − Γ0 (ω) < d if Γ(ω) − Γ0 (ω) ≥ d

where the constant d is such that P (Γ − Γ0 ≥ d) = 1 − α (which is the probability constraint that has to be satisfied in the deterministic case (4)). If the company is risk-neutral, the optimal design looks similar to the case when the guarantee is deterministic but could be could be very different in shape from the the nose profile if we use a pathological example. Let us illustrate this by a simple example. Assume an initial investment of x0 , and then a guarantee given for instance by: Γ0 (ω) =

x0 1S ≤S (ω) + x0 egT 1ST >S0 (ω) 2 T 0

which means that in case of a negative return of the index the investor gets at least x20 and in case of a positive return in the index the investor gets its investment x0 with a minimum guaranteed rate g. Let Γ = x0 SST0 . Then Γ < Γ0 is equivalent to ST < S20 or ST ∈ (S0 , S0 egT ).

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So that the optimal contract becomes:

or

  Γ0 (ω) = x20      Γ(ω)     Γ (ω) = x0 0 2 X ∗ (ω) =  Γ0 (ω) = x0 egT      Γ(ω)     Γ (ω) = x egT 0 0

if if if if if if

ST (ω) ≤ S20 S0 < ST (ω) < S20 + d˜ 2 S0 + d˜ ≤ ST (ω) ≤ S0 2 S0 ≤ ST (ω) ≤ S0 egT S0 egT < ST (ω) < S0 egT + d˜ S0 egT + d˜ ≤ ST (ω)

 x0 gT   Γ0 (ω) = 2 1ST ≤S0 + x0 e 1ST >S0 X ∗ (ω) = Γ(ω)   Γ0 (ω) = x0 egT

if ST (ω) ≤ S0 egT if S0 egT < ST (ω) < S0 egT + d˜ if S0 egT + d˜ ≤ ST (ω)

It will depend on the probability α. For instance, for a large α, the second design is more possible. For very small alpha, the first design is optimal. It shows that the optimal design in absence of capital protection can now be more complicated than a nose shape and corresponds more or less as a “double nose” shape. Example 3.2 ua (x) = log(x). In this case, the optimal design is different from the optimal design with capital protection. When there is capital protection, Γ − Γ0 plays a key role since the optimal contract is finally determined by the probability constraint (4), that is P (Γ − Γ0 ≥ d) = 1 − α. If the issuer has a log-utility function, that is ua (x) = log(x), then the ratio of the benchmark over the guaranteed one ΓΓ0 plays a similar role. The optimal design X ∗ given in proposition 2 becomes: X ∗ = Γ0 + (Γ − Γ0 )+ 1n where d is such that: P

µ

Γ ≥ ed Γ0

¶

Γ 0. E := Γ0 < Γ ≤ (Γ0 + c) 14

so that the optimal contract becomes: X ∗ = Γ0 + (Γ − Γ0 )1E . From these three examples, we see that the optimal contract’s payoff depends on the issuer’s preferences when the guaranteed amount is stochastic. Again, the continuous assumption in Proposition 2 is crucial to guarantee the existence of an optimal solution X ∗ described above.

3.2

Stochastic Floor with Γ0 > 0 and a random maturity

In this section we consider the generic ILP in a continuous-time setting. We assume there is a stochastic guaranteed floor which is positive almost surely. In addition, we assume the product can be redeemed before maturity. This is a realistic feature since some structured products can be redeemed earlier by the issuer (with callable options) or have surrender options for the investor. Let τ denotes the early redemption time, it thus represents the termination date of the product. We assume that P (Γ0 (τ ) > 0) = 1 which means that the guarantee amount (at the random time τ ) is still assumed to be almost surely postive. The optimal design problem writes in this case as: ( X ≥ Γ0 (τ ), a.s., min E[ua (X)] s.t. (13) X P (X ≥ Γ(τ )) ≥ α The optimal design Xτ is given in the next proposition: Proposition 3 Given a firm with utility function ua (.) which is defined over (0, ∞) and a pricing operator15 pa (.; τ ) for a future cash flow at random time τ . Assume that the distribution function of Γ(τ ) − Γ0 (τ ) is continuous, then there exists a constant d > 0 such that the payoff Xτ is the optimal payoff, where

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   Γ0 (τ (ω)) Xτ (ω) = Γ(τ (ω))   Γ0 (τ (ω))

if ua (Γ(τ (ω))) ≤ ua (Γ0 (τ (ω))) if 0 < ua (Γ(τ (ω))) − ua (Γ0 (τ (ω))) < d if ua (Γ(τ (ω))) − ua (Γ0 (τ (ω))) ≥ d

(14)

See Blanchet-Scalliet, El Karoui and Martellini (2005) for asset pricing for a general random maturity cash flow.

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In particular, if the guaranteed floor Γ0 is constant, or equivalently, there is capital protection, then there exists d > 0 such that: Xτ = Γ0 + (Γ(τ ) − Γ0 )+ 1{Γ(τ )−Γ0 0 such that the optimal payoff X ∗ writes as: ( Γ(ω) if Γ(ω) < d (15) X ∗ (ω) = 0 if Γ(ω) ≥ d where d is such that P (Γ ≥ d) = 1 − α. 2. If ua (0) = −∞, then there is no optimal design for a future cash flow X such that X ≥ 0, a.s., P (X ≥ Γ) ≥ α, and E[ua (X)] > −∞. It is still possible for some issuers (from Proposition 4) to have an optimal design even when there is no guaranteed floor. For instance for a risk-neutral issuer, ua (x) = x, it is always possible to design an optimal contract to beat a benchmark up to a confidence level, even though there is no capital protection. Often issuers are assumed to be risk-neutral and in this case the result is identical: the absence of capital protection does not modify the optimal design. Even though there is no guaranteed floor, Proposition 4 provides an optimal design for some risk-averse issuers. However, for an aggressive issuer with a logutility function ua (x) = log(x), Proposition 4 implies that there exists no optimal design. 16

Some previous literature supports capital protection products. Bodie and Crane (1998) show that a rolling capital protected investment outperforms many other strategies and undershoots the benchmark on very few occasions.

3.4

Capped Index-Linked Products

Index-linked products often have an upper limit on the payoff under the contract. We now assume the final payoff is capped by Γ2 (τ ) if τ is the maturity of the contract. A capped index-linked product clearly appeals to some issuers since from the issuer’s perspective, the capped amount limits the liability under the contract. From proposition 3, Xτ is the optimal design for the issuer without a cap. Intuitively, min{Xτ , Γ2 (τ )} should be the optimal design for the same issuer with the cap Γ2 . The next proposition shows that this is almost the case. Proposition 5 Assume that the distribution function of ua (Γ(τ )) − ua (Γ0 (τ )) is continuous, Γ0 ≤ γ2 a.s. and P (Γ(τ ) > Γ0 (τ )) ≥ 1 − α, then there exists a positive constant d such that P (Γ(τ ) > Γ2 (τ )) + P (G) = 1 − α

(16)

where G = { Γ0 (τ ) < Γ(τ ) ≤ Γ2 (τ ), ua (Γ(τ )) − ua (Γ0 (τ )) ≥ d } and the payoff X cap = Γ0 (τ ) + (Γ(τ ) − Γ0 (τ ))1G

(17)

is the optimal design of the capped generic ILP. From Proposition 5, the capped feature does not change the optimal contract payoff much since the liability is either the guaranteed amount Γ0 , or the benchmark Γ. The constant d might be different from the constant we determined in Proposition 3. However, the shape of X cap is the same as the shape of the contract min{Xτ , Γ2 (τ )}.

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Exchange-traded Index-Linked Products

This section presents our empirical study of exchange-traded structured products. Structured products are available on exchanges and we restrict ourselves to the American Stock Exchange (AMEX). On the AMEX there are five categories of structured products: Index-Linked Notes, Equity-Linked Term Note, Currency Notes, Index-Linked Term Notes and others. We focus on Index-Linked Notes since represent the largest share (at least 62%) among all structured

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products. Moreover the other products show strong similarities in their payoff structure with Index-Linked Notes16 . We first consider the index-linked notes on AMEX in general and then we focus on two specific index-linked notes with discontinuous payoffs.

4.1

Index-Linked Products on the American Stock Exchange

Most ILPs on AMEX have payoffs of the following form ¶¾ ¾ ½ ½ µ S T − S0 ,c X = X0 min max f, 1 + κ S0

(18)

where X0 is the initial investment (principal of the index-linked notes). If there exists a floor parameter f , capital is protected, and we will see later than more than half of ILPs are capital protected. For non capital protected ILP, the participation rate κ should be greater than the participation rate in capital protected ILP. A cap c is often imposed to limit the liability of issuers. About 95% of traded index-linked notes have roughly a payoff given by (18) where we include the cases when c = +∞ (no cap) and f = 0 (no protection). These empirical payoffs are generally consistent with those obtained in Proposition 1 and Proposition 5. However, the theoretical result is more general since in addition to capital protection (that is the minimum guaranteed amount Γ0 ), we include a benchmark Γ to be matched or beaten with a confidence level α. When α = 1, the optimal design in Proposition 5 is exactly the same as the payoff in (18). About 40% of the issued products have no full capital protection. The theory suggests then that the optimal design depends on the issuer except for risk-neutral issuers. We now examine more additional features of these ILPs. For index-linked notes on the AMEX, we first classify them into four types in terms of the payoff structures (see Table 1). This classification is different from the one proposed by Stoimenov and Wilkens (2005) (see Figure 1 of their paper) but it is consistent with theirs. In particular, we just analyze the terminal payoffs. Thus, we do not distinguish between European style and path-dependent style structured products. In keeping with our theoretical approach, we focus on the existence of a guarantee (including for instance the “guarantee products” proposed by Stoimenov and Wilkens (2005)). Without guarantees, Type 3 include products called classic products and turbo products in their study. To simplify, classic products never give a higher return than the market (that is κ = 1 whereas Turbo products can have a strictly higher return than the 16

As of Oct 3, 2006, there were 208 Index-Linked Notes, and 320 structured products in total.

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Table 1: Continuous ILPs on the American Stock Exchange This table presents the continuous payoff structure of those index-linked product traded on American Stock Exchange, date Oct 3, 2006. Since those products have no early redemption and the coupons (if any ) are very small we ignore these coupon structures and focus on the terminal payoff structure. Denote RT = SST as 0 the index gross return from time 0 to time T , X0 as the initial premium. Source: www.amex.com. The amount C1 represents the capital protection, if any (that is the amount guaranteed at maturity). The constant C2 represents the maximum benefit that the investor can receive if the contract includes an upper-limit on benefits.

Type

Feature

Payoff Structure

Type 1 Capital Protected & Unlimited Benefits

max {C1 , X0 (1 + κ(RT − 1))}

Type 2

Capital Protected & Capped Benefits

max {C1 , min {C2 , X0 (1 + κ(RT − 1))}}

Type 3

No Guarantee & Capped Benefits

X0 SST0 1ST ≤S0 + min {C2 , X0 (1 + κ(RT − 1))} 1ST >S0

Type 4

No Guarantee & Unlimited Benefits

X0 SST0 1ST ≤S0 + X0 (1 + κ(RT − 1))1ST >S0

market (κ > 1) when the underlying is between a specified range of prices. These structured products provide good illustrations of contracts with a probabilistic constraint. The payoff under a turbo product beats the market with a given confidence level (which is equal to the probability the underlying is between the specified range at the maturity of the product). Moreover, following Figure 2. of Stoimenov and Wilkens (2005), all products with a fixed maturity on the German market seem to have limited benefits (that is type 2 and type 3 of Table 1). In contrast, some American17 style structured products do not have an upper limit on the benefits (no cap). Table 1 provides a classification of continuous ILPs. Here continuous means that the payoff of this index-linked product is continuous with respect to the underlying. Most (almost all) final payoffs of index-linked notes on the American Stock Exchange are continuous as can be seen from Table 2. Percentages in Table 2 are established by counting the number of contracts issued by each company divided by the total number of contracts available in the market. We do not look at the volume. The discontinuous products appear in the column others which is the category of products that are not included in the four types described 17

We use American here to denote the early exercise feature. The term has nothing to do with geography in this context.

19

above in Table 1. Most of the time, the capital is fully protected, that is the amount C1 is equal to f X0 , where f = (1 + g), g is a guaranteed interest rate and X0 stands for the initial investment. Sometimes it is partly protected, for example only 50% of the initial investment is guaranteed. Some products have no cap, but an embedded callable option. That is, under specified conditions, the issuer may elect to call the products. In that case, that means investors receive a cash payment per unit of index-linked note and will not receive any supplemental redemption amount at maturity of the notes. In our analysis, we include the presence of the callable option as a cap on the product since losses for the issuer are then capped (note that this option also induces discontinuities in the payoff). We now discuss how the features of the Index-Linked Notes on AMEX (see table 2) vary across different issuers. We see that Merrill Lynch and Morgan Stanley are the two major issuers in this market. Together they account for 41.2% of the entire market in number of notes. Together with the third ranking firm (Bank of America 13.9%), the fourth ranking one (Citigroup 11.6%), these four firms issue over two thirds of the notes of the whole market. We note that that these four main issuers do not issue similar products. For instance, 100% of products sold by Bank of America provide capital protection while only 24% of Merrill Lynch’s ILPs include a minimum guarantee. Almost half of the products sold by Merrill Lynch have no capital protection and no cap on the benefits. Among the total of 208 index-linked notes, we find that 44% are of Type 1, 22% are of type 2, 12% are of type 3 and 17% are of type 4, 5% are of some other type. Note that Morgan Stanley is a typical representative of the average issuer of index-linked notes because it issues approximately these percentages of each type. In particular, that means 66% provide a protection of the initial investment (at least partly). The term other types includes discontinuous payoffs and/or decreasing payoffs. These are mainly issued by Lehman Brothers Holdings, Inc. and we will study two examples in the next section. Note that both Merrill Lynch and Morgan Stanley issue an ILP with a decreasing payoff, linked to the performance of the Philadelphia Stock exchange Housing Sector IndexSM . In these products an increase in this index implies a decrease in the investment’s return. Both have a capital protection feature of 50% of the principal and capped benefits18 . Some ILPs have path-dependent features that we do not analyze here. For example, some payoffs are based on the average (capped) return instead of the final (capped) index return. These products contain features of Asian-style options. In practice, most of products with no 18

Descriptions of these products can be found on www.amex.com. The Morgan Stanley product is called the Bear Market PLUS Linked to the PHLX Housing Sector Index, referred to as MFP. The Merrill Lynch product is called Accelerated Return Bear Market Notes Linked to the Performance of the PHLX Housing Sector Index referred to as MPL.

20

Table 2: Major Issuers of ILPs on the Exchange This table illustrate the percentage of those market major issuers on ILPs on American Stock Exchange, dated on Oct 3 2006. There are 17 issuers of index-linked notes. We ignore those who only issue one or two index-linked notes. These issuers include ABN AMRO Bank, Credit Suisse (USA), Goldman Sachs, International Finance Group. The first column of percentages is obtained by dividing the number of contracts issued by each company over the total number of contracts in the market. Then for a given company, we analyse and classify the issued contracts among the different types defined previously in Table 1. Source: www.amex.com.

Firm Percentage Bank of America 13.9% CIBC 6.7% Citigroup 11.1% J.P.Morgan 2.9% Lehman Brothers 6.7% Merrill Lynch 20.2% Morgan Stanley 21.2% Structured Products Corp 2.9% Bear Stearns 1.4% UBS AG 2.4% Wachovia 3.4% Wells Fargo 2.9%

Type 1 Type 2 Type 3 Type 4 Others 59% 41% 0% 0% 0% 93% 7% 0% 0% 0% 35% 26% 22% 17% 0% 17% 83% 0% 0% 0% 43% 7% 0% 0% 50% 19% 5% 26% 48% 2% 50% 16% 11% 21% 2% 0% 100% 0% 0% 0% 33% 67% 0% 0% 0% 20% 60% 0% 20% 0% 57% 0% 29% 14% 0% 100% 0% 0% 0% 0%

cap are based on the average index return instead of directly on the index. Some also include American options because of the existence (for the investor) of an American redemption right (during a specified period), or a Bermudan redemption right (at fixed dates) or a callable right for the issuer. In our analysis, we focus on the shape of the terminal payoff of the structured product and not on these exotic aspects.

4.2

Examples of discontinuous ILPs

In general, ILPs have increasing payoffs (see Table 2). However, approximately 5% of traded ILPs exhibit discontinuities, decreasing payoffs or both. We now illustrate two examples of ILPs on AMEX with discontinuous payoffs. These examples illustrate how our theoretical findings could provide a partial explanation of the design of discontinuous products in the market. The optimal design in Proposition 1 involves a digital call option.

21

4.2.1

Example of Discontinuous Index-Linked Note

In this section, we give an example of a discontinuous payoff which is similar to the optimal payoff we studied earlier. Lehman Brothers Holdings Inc. markets the Rebound RANGERSSM (which means Rebound Risk Adjusting Equity Range Securities). These index-linked notes provide investors with partial downside protection as well as fixed upside potential. The notes are senior unsecured debt securities linked to the performance of the Nasdaq-100 Index. The principal is $1, 000 for each unit19 . This contract is a four-year contract which has some standard features. It provides partial capital protection since 20% of the initial investment is guaranteed. It has also a probability of outperforming the index (in the event of an increase on the Nasdaq-100 Index). It gives no interest prior to maturity but includes the option of an early redemption. We ignore this feature (see Appendix, Table 3 for more details concerning this embedded option) and we focus on the terminal payoff. At maturity, assuming the note has not been redeemed earlier, the payoff XT on this ILP is given by:    $1, 310 XT = $1, 000   $1, 000 × (RT + 0.2)

if RT ≥ 1 if 0.8 ≤ RT < 1 if RT < 0.8

Figure 3 displays the payoff. These ILPs give investors the opportunity to realize a fixed return (of 31% over the life of the securities) upon an increase in the value of the Nasdaq100 Index while providing limited protection for a decline in the index (at least 20% of their principal). This product has an increasing payoff. As displayed in Figure 3, the payoff is discontinuous. This type of discontinuity is not frequently observed in practice and illustrates the heterogeneity of ILPs. However, it is a relevant product compared to our previous theoretical findings. Indeed the final payoff can also be interpreted as the following portfolio of derivatives written on the underlying index S. XT = 1000(RT + 0.2) − E1 − E2 19

(19)

Note also that the prospectus explains that “the calculation agent will calculate the amount payable to you by reference to the level of the Nasdaq-100 Index, which is calculated by reference to the prices of the common stocks included in the Nasdaq-100 Index without taking into consideration the value of dividends paid on those common stocks.” This explanation is very common and can be found in most prospectus. It is always explained that the return on the notes will not reflect the return that investors would realize if they actually own the underlying stocks.

22

1400 1200

Payoff

1000 800 600 400 200 0

0.2

0.4

0.6 0.8 1 1.2 Final Index Return R

1.4

1.6

1.8

T

Figure 3: Payoff w.r.t. Index total return RT =

ST S0

where E1 = max{1000RT − 800, 0}1{0.81}

(21)

and

The first component of (19) is a long position in a zero coupon bond (in an amount of 200) plus a long position in the index. The second one is a short position on a range digital call. It can easily be seen that E1 corresponds to an optimal design we obtained in proposition 1 by choosing a benchmark Γ = 1000RT − 800 and the probability: α = P (RT ≤ 1) = P (ST ≤ S0 ).

(22)

The third component (E2 ) in the above decomposition of XT is a short position in a digital call option. 4.2.2

Discontinuous and Non Increasing Payoff

We now consider another example of an ILP offered by Lehman Brothers Holdings Inc. where the payoff is neither increasing nor continuous. We then discuss to what extent this product can be optimal. These index-linked notes are called the Absolute Buffer Notes. They are three year notes with payment linked to the performance of the Dow Jones EURO STOXX 50SM Index. The

23

notes are senior unsecured debt securities maturing on July 29, 2008. The principal is equal to $1,000 (see Appendix Table 4 for a more detailed of the contract). At maturity T , we denote by RT the index total return, RT = SST0 . Then, the payoff at maturity T can be written as:

XT

   $1, 000 + $1, 000 × 130% × (RT − 1) = $1, 000 + $1, 000 × 100% × |RT − 1|   $1, 000 + $1, 000 × 100% × (RT − 1)

if RT ≥ 1 if 0.8 ≤ RT < 1 if RT < 0.8

Figure 4 displays this unusual payoff. Similar to most index-linked notes, no interest or other payments are made on the notes before maturity (85% of Index-Linked Notes have no coupon prior to maturity). Note also that this product is not capped and cannot be called by the issuer. 1600

1400

Payoff

1200

1000

800 Return1

600

400 0.5

0.6

0.7

0.8 0.9 1 Index Final Return

1.1

1.2

Figure 4: Payoff w.r.t. Index total return

1.3

ST S0

It is also possible to decompose the payoff XT of an Absolute Buffer Note in terms of a portfolio of derivatives: XT = 1000RT + F1 + F2 (23) where F1 = max{1000 × 30% × (RT − 1), 0}

(24)

F2 = max{2000 × (1 − RT ), 0}1{0.8≤RT 0 such that for each ω ∈ Ω, X ∗ (ω) solves the (static) optimization problem: (Eω ) :

min {ua (x) − λ1{x≥Γ(ω)} }

x≥Γ0 (ω)

(A1)

Then X ∗ solves the optimization problem (11). Proof. First, X ∗ satisfies the constraints (1). Moreover, given another payoff X subject to constraint (1), by using (3), we have, ∀ω ∈ Ω ua (X ∗ (ω)) − λ1{X ∗ (ω)≥Γ(ω)} ≤ ua (X(ω)) − λ1{X(ω)≥Γ(ω)}

(A2)

ua (X ∗ (ω)) − ua (X(ω)) ≤ λ{1{X ∗ ≥Γ(ω)} − 1{X≥Γ(ω)} }

(A3)

Thus

Therefore by condition (2), and constraints of X, we have E[ua (X ∗ )] − E[ua (X)] ≤ λ{P (X ∗ ≥ Γ) − P (X ≥ Γ)} ≤ 0 The proof of this lemma is completed. Lemma A.2 Let Γ0 > 0 a.s.. (1) and (3) of lemma A.1.   Γ0 (ω) Γ(ω) Xλ (ω) =  Γ0 (ω)

¤

The following family, indexed by λ > 0, satisfies the conditions if Γ(ω) ≤ Γ0 (ω) if Γ(ω) > Γ0 (ω) and ua (Γ(ω)) − ua (Γ0 (ω)) < λ if Γ(ω) > Γ0 (ω) and ua (Γ(ω)) − ua (Γ0 (ω)) ≥ λ

(A4)

Proof. Clearly (1) holds. Since ua (.) is non-decreasing, Γ0 (ω) solves the optimization problem (Eω ) in the region {Γ(ω) ≤ Γ0 (ω)}. Assume that Γ(ω) > Γ0 (ω). It suffices to compare ua (Γ0 (ω)), which is the minimal one among the region [Γ0 (ω), Γ(ω)), with ua (Γ(ω)) − λ, which is the minimal one over the region [Γ(ω), ∞). Thus, the lemma is proved. ¤ Proof of Proposition 2. Xλ∗ of lemma A.2 satisfies (1) and (3) of lemma A.1. Thus, we only have to prove the existence of λ > 0 such that (2) of lemma A.1 is also satisfied. 27

By assumption the distribution function of ua (Γ) − ua (Γ0 ) is continuous. Therefore the function π(x) = P (Γ > Γ0 , ua (Γ) − ua (Γ0 ) ≥ x) (A5) is continuous. Since P (Γ > Γ0 ) > 1 − α, there exists a positive constant λ∗ > 0 such that π(λ∗ ) = 1 − α. Then Xλ∗ of lemma A.2 satisfies condition (2) in lemma A.1. Then by lemma A.1, Xλ∗ solves for the optimization problem (11). ¤ Proof of Proposition 1. Proposition 1 is a particular case of prop 2. Assume that Γ0 is constant, say Γ0 = f > 0. According to assumption on the distribution of Γ, there exists a positive constant d such that P (Γ ≥ Γ0 +d) = 1−α. Then by choosing λ∗ = ua (Γ0 +d)−ua (Γ0 ) we see that Xλ∗ is the same as the optimal design as stated above in the proof of Prop 2. ¤ Proof of Proposition 3. It is similar to the proof of propostion 2 by replacing ω by τ (ω). We omit details. Proof of Proposition 4. The proof of Prop 4 (1) is similar to the proof of Prop 1. We only prove Prop 4 (2). Given a random payoff X with X > 0, a.s.,

E[ua (X)] > −∞,

P (X ≥ Γ) ≥ α.

We want to find another random payoff Y subject to the same constraints and −∞ < E[ua (Y )] < E[ua (X)]. It will then prove that there is no minimum. Lemma A.3 Given a finite random variable Z with finite expectation, and a number α ∈ (0, 1), then there exists a subset B of Ω such that 0 < P (B) ≤ 1 − α and E[Z1B ] is finite. Proof. Define Bn = {n ≤ |Z| < n + 1}, n ≥ 0. By assumption we know that lim P (Bn ) = 0, lim E[Z1Bn ] = 0

n→∞

n→∞

Then, the existence of B follows easily.

(A6) ¤

To prove proposition 4, we first define A = {X ≥ Γ}. Case (1). P (A) = 1. Therefore X ≥ Γ, a.s.. By using lemma A.3, we can choose a measurable set B such that 0 < P (B) ≤ 1 − α and E[ua (X)1B ] > −∞. Choose a small enough positive number γ and define Y = γ1{B} + X1{B c }

(A7)

Then Y ≥ 0 and P (Y ≥ Γ) ≥ α. Moreover, E[ua (Y )] − E[ua (X)] = ua (γ)P (B) − E[ua (X)1B ] < 0

(A8)

by choosing small enough γ > 0. Case (2). P (A) < 1. In this case P (Ac ) > 0. Since E[ua (X)] > −∞, then E[ua (X)1A ] > −∞. Define Y = γ1{Ac } + X1{A} 28

(A9)

where γ > 0 is small enough. The proof is similar to Case (1) by choosing small enough positive number γ. Then the proof of Prop 4 is finished. ¤

B

Maximum Probability Design

We now recall the result of Spivak of Cvitani´c (1999) about the maximum probability strategy (needed in section 2.2). Let x0 = 1. By Spivak and Cvitani´c (1999), the optimal terminal wealth could be written in our case as: X prob = f 1Γ≤f + Γ1{01} ˆ T (Γ−f )≤1} + f 1{ζξ

(A10)

where ζˆ will be defined below. Let C = max(Γ, f ). The no-arbitrage prices of C and f are denoted by C0 = E[ξT C] and A0 = E[ξT f ]. In particular, we assume P (Γ ≤ f ) < α and A0 ≤ x < C0 . We then define the function K0 (.) as follows: K0 (ζ) = E[ξT (C − f )1{(C−f )ζξT ≥1} ], ∀ζ > 0

(A11)

ˆ T (Γ − f ) > 1) = 1 − α. and ζˆ in ]0, ∞] by: ζˆ = inf {ζ > 0 / K0 (ζ) ≥ C0 − x}. Moreover P (ζξ The benchmark Γ can be written as: −k/b

Γ = g(ξT ) = x0 ak/b ξT

.

ˆ T g(ξT ) − ζξ ˆ T f − 1 > 0) = 1 − α. We need to investigate the Since Γ = g(ξT ), then P (ζξ property of the function K0 (ζ), which is 1−k/b n 1

K0 (ζ) = x0 ak/b E[ξT

1−k/b

x0 ak/b ζξT

o] −f ζξT ≥1

− f E[ξT 1nx

0a

o]

k/b ζξ 1−k/b −f ζξ ≥1 T T

(A12)

at the current situation. Define 1−k/b

Λ(ξT ) = x0 ak/b ζξT

− f ζξT − 1

(A13)

We now study the sign of Λ(ξT ) for a given and fixed ζ > 0. In fact, we have k −k/b Λ′ (ξT ) = x0 ak/b (1 − )ζξT − f x0 ζ b

(A14)

k/b

Let z0 := x0 af −b/k (1 − kb )b/k , which is the unique zero of Λ′ (.). Λ(ξT ) is increasing on {ξT < z} and decreasing on {ξT > z}. Note that limx↓0 Λ(x) = −1, limx↑c Λ(x) = −1, where c = af −b/k . Note that z0 < c. We now consider the two following exclusive cases to determine the sign of Λ(.). Case 1. Λ(z0 ) ≤ 0. In this case, ∀ξT > 0, Λ(ξT ) ≤ 0. The set {Λ(ξT ) ≥ 0} is empty or ˆ reduced to {z} and K0 (ζ) = 0. This is impossible because of the existence of ζ. Case 2. Λ(z0 ) > 0 . In this case, there exists aζ ∈]0, z0 [, bζ ∈]z0 , c[ such that the set {Λ(ξT ) ≥ 0} = {ξT ∈ [aζ , bζ ]}. Therefore, XTprob is determined and as shown as in the text. ¤

29

Table 3: Description of the Rebound RANGERSSM . “Rebound RANGERS” stands for Rebound Risk AdjustiNG Equity Range Securities. www.amex.com, official prospectus provided by Lehman Brothers Holdings Inc.

Notes: Public Offering Price: Interest: Issuer Rating: Maturity: Listing: Redemption Prior to Maturity:

Source:

Senior unsecured debt securities of Lehman Brothers Holdings Inc. $1, 000 per note No interest or other payments prior to maturity or earlier redemption Long-term senior debt rated A by Standard&Poor’s and A2 by Moody’s Investors Service May 20, 2007 (Expected) American Stock Exchange Annual observation dates will occur on the first, second and third anniversary of the date the notes are first offered for sale. If, on any of the annual observation dates, the ending level of the Nasdaq-100 Index is greater than or equal to the initial index level, the notes will be redeemed for an amount, per $1,000 note, equal to $1,000 plus the product of $77.50 and the number of full years that the notes were outstanding. The initial index level is $1,162.93, the closing level of the Nasdaq-100 Index on the date the notes are first offered for sale. The ending level of the Nasdaq-100 Index on any annual observation date or on the final valuation date will generally be the closing level of the Nasdaq-100 Index on that date.

Payment at Maturity:

If the notes are not redeemed prior to maturity and the ending level of the Nasdaq-100 Index on the final valuation date is greater than or equal to the initial index level, Lehman Brothers Holdings will pay to you $1,310 per $1,000 note. If the ending level of the Nasdaq-100 Index on the final valuation date is less than the initial index level, Lehman Brothers Holdings will pay to you, per $1,000 note, the lesser of: (1) $1,000, and ³ ´ (2) $1,000× Final index value + 0.2 Initial index value The final index level will be the ending level of the Nasdaq-100 Index on the final valuation date.

Offering Date:

May 15, 2003

30

Table 4: Description of the Absolute Buffer Notes. Source: www.amex.com, official prospectus provided by Lehman Brothers Holdings Inc.

Notes: Public Offering Price: Minimum Initial Investment: Interest: Upside Participation Rate: Threshold Level: Issuer Rating: Maturity: Listing: Payment at Maturity:

Senior unsecured debt securities of Lehman Brothers Holdings Inc. $1, 000 per note $10, 000 No interest or other payments prior to maturity 130% 2642.384, which is 80% of the initial index level Long-term senior debt rated A by Standard&Poor’s and A1 by Moody’s Investors Service July 29, 2008 American Stock Exchange LLC An amount, per $1, 000 note, equal to: • If the final index return is not negative, $1,000 + ($1,000 x the upside participation rate x the final index return). • If the final index return is negative and the final index level is equal to or greater than the threshold level, $1,000 + ($1,000 x the absolute value of the final index return). • If the final index return is negative and the final index level is less than the threshold level, $1,000 + ($1,000 x the final index return). The final index return will equal:

Final index level − Initial index level Initial index level The initial index level is 3302.98, which is the closing index level on the date of the attached prospectus supplement. The final index level will be the closing index level on the valuation date, which will be the third business day before the stated maturity date. The closing index level on any particular day will generally be the closing level of the Dow Jones EURO STOXX 50 Index on such day. Offering Date:

July 26, 2005

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References [1] Alexander, G., Baptista, A., 2004. “A Comparison of VaR and CVaR Constraints on Portfolio Selection with the Mean-Variance Model,” Management Science, 50, 1261-1273. [2] Allen, F., Gale, D., 1988. “Optimal Security Design,” Review of Financial Studies, 1(3), 229-263. [3] Arrow, K.J., 1963. “Uncertainty and the Welfare Economics of Medical Care,” American Economic Review, 53(5), 941-973. [4] Barrieu, P., El Karoui N., 2002. “Optimal Design of Derivatives in Illiquid Markets,” Quantitative Finance, 2, 181-188. [5] Basak, S., Shapiro, A., Tepla, L., 2006. “Risk Management with Benchmarking,” Management Science, 52, 542-557. [6] Benet B.A., Giannetti, A., Pissaris, S., 2005. “Gains from structured product markets: The case of reverse-exchangeable securities (RES)”, Journal of Banking & Finance, 30, 111-132. [7] Black, F., Perold, A., 1992. “Theory of constant proportion portfolio insurances,” Journal of Economic Dynamics & Control, 16(3-4), 403-426. [8] Blanchet-Scalliet C., El Karouri, N., Martellini, L., 2005. “Dynamic Asset Pricing Theory with Uncertain Time-Horizon,” Journal of Economic Dynamics & Control, 29, 1737-1764. [9] Bodie, Z., Crane, D.B., 1999. “The design and production of new retirement savings products”, Journal of Portfolio Management, 25, 77-82. [10] Boot, A., Thakor, A.V., 1993. “Security Design,” Journal of Finance, 48(4), 1349-1378. [11] Boyle, P.P., Tian, W., 2007. “The Design of Equity Indexed Annuities,” Insurance: Mathematics and Economics, (forthcoming). [12] Boyle, P.P., Wang, T., 2001. “Pricing of New Securities in an Incomplete Market: the Catch 22 of No-Arbitrage Pricing,”, Mathematical Finance, 11(3), 267-284. [13] Breuer, W., Perst, A., 2006. “Retail banking and behavioral financial engineering: The case of structured products”, Journal of Banking & Finance, 31, 827-844. [14] Browne, S., 2000. “Risk Constrained Dynamic Active Portfolio Management,” Management Science, 46(9), 1188-1199. 32

[15] Carlin, B.I., 2007. “Strategic Price Complexity in Retail Financial Markets”, Working Paper Available at SSRN: http://ssrn.com/abstract=949349. [16] Cox, J.C., Huang, C.F., 1989. “Optimal Consumption and Portfolio Policies when Asset Prices Follow a Diffusion Process,” Journal of Economic Theory, 49, 33-83. [17] Duffie, D., Jackson, M.O., 1989. “Optimal Innovation of Futures Contracts,” Review of Financial Studies, 2, 275-296. [18] DeMarzo, P., Duffie, D., 1999. “A Liquidity-Based Model of Security Design,” Econometrica, 67, 65- 99. [19] Detemple, J., Selden, L., 1991. “A General Equilibrium Analysis of Option and Stock Market Interactions,” International Economic Review, 32(2), 279-303. [20] Harrison, J.M., Kreps, D., 1979. “Martingales and Arbitrage in Multi-Period Security Markets,” Journal of Economic Theory, 20, 381-408. [21] Jorion, P., 2003. “ Portfolio Optimization with Constraints on Tracking Error,” Financial Analysts Journal, September, 70-82. [22] Kahneman, D., Tversky, A., 1979. “Prospect Theory: An Analysis of Decision Under Risk,” Econometrica, 47, 263-291. [23] Kulldorff, M., 1993. “Optimal Control of a Favorable Game with a Time-Limit,” SIAM Journal on Control & Optimization, 31, 52-68. [24] Madan, D., Soubra, B., 1991. “Design and marketing of financial products, Review of Financial Studies, 4(2), 361-384. [25] Merton, R.C., 1971. “Optimum Consumption and Portfolio Rules in a Continuous-Time Model,” Journal of Economic Theory, 3, 373-413. [26] Spivak, G., Cvitani´c, J., 1999. “Maximizing the Probability of a Perfect Hedge, Annals of Applied Probability, 9(4), 1303-1328. [27] Stoimenov, P.A., Wilkens, S., 2005. “Are structured products fairly priced? An analysis of the German market for equity-linked instruments”, Journal of Banking & Finance 29(12), 2971-2993. [28] Wilkens, S., Stoimenov, P.A., 2007. “The pricing of leverage products: An empirical investigation of the German market for ‘long’ and ‘short’ stock index certificates”,Journal of Banking & Finance 31, 735-750.

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