Explicit Representation of Cost-Efficient Strategies

Explicit Representation of Cost-Efficient Strategies Carole Bernard

1

Phelim P. Boyle

2

May 12, 2010 Abstract This paper uses the preference free framework proposed by Dybvig (1988) and Cox and Leland (1982,2000) to analyze dynamic portfolio strategies. In general there will be a set of dynamic strategies that have the same payoff distribution. We are able to characterize a lowest cost strategy (a “cost-efficient” strategy) and to give an explicit representation of it. As an application, for any given path-dependent strategy, we show how to construct a financial derivative that dominates in the sense of first-order stochastic dominance. We provide new cost-efficient strategies with the same payoff distributions as some well-known option contracts and this enables us to compute the relative efficiency of these standard contracts. We illustrate the strong connections between costefficiency and stochastic dominance. Key-words: Stochastic Dominance, Efficiency Cost, Expected Utility, Path-Dependent Strategies.

1 University

of Waterloo, email: [email protected] Laurier University, email: [email protected]. Both authors acknowledge support from the Natural Sciences and Engineering Research Council of Canada. Thanks to Will Gornall for excellent research assistance. We thank Jean-Pierre Fouque, Mike Ludkowski, Klaus Schenk-Hopp´e, Shige Peng, Steven Vanduffel, Jiaan Yan, Jenny Young, Thaleia Zariphopoulou, Xunyu Zhou, and all other seminar participants at Heriot-Watt University, Leeds Business School, Cass Business School, Michigan University, University of Oxford, University of California at Santa Barbara, Technische Universit¨at M¨ unchen and the Chinese Academy of Sciences. 2 Wilfrid

Explicit Representation of Cost-Efficient Strategies

Abstract This paper uses the preference free framework proposed by Dybvig (1988) and Cox and Leland (1982,2000) to analyze dynamic portfolio strategies. In general there will be a set of dynamic strategies that have the same payoff distribution. We are able to characterize a lowest cost strategy (a “cost-efficient” strategy) and to give an explicit representation of it. As an application, for any given path-dependent strategy, we show how to construct a financial derivative that dominates in the sense of first-order stochastic dominance. We provide new cost-efficient strategies with the same payoff distributions as some wellknown option contracts and this enables us to compute the relative efficiency of these standard contracts. We illustrate the strong connections between cost-efficiency and stochastic dominance. Key-words: Stochastic Dominance, Efficiency Cost, Expected Utility, Path-dependent Strategies.

1

1

Introduction

This paper focuses on investors who seek a specific probability distribution of terminal wealth at a minimal cost.1 Under fairly general assumptions, we derive a formula for the unique investment portfolio that minimizes the cost of achieving a given distribution of payoff. In particular, given any specified dynamic investment strategy, that is inefficient2 , we are able to give the explicit payoff of the efficient financial derivative that strictly dominates this strategy in the sense of first-order stochastic dominance. Stochastic dominance has always played an important role in decision making under uncertainty, and numerous rules in this literature are based on first and second stochastic dominance. Importantly, stochastic dominance offers a preference-free and non-parametric alternative to expected utility theory.3 Our work is closely related to and extends the work of Dybvig(1988a), (1988b) and Cox and Leland.4 We first recall their main results. In the Black and Scholes framework, Cox and Leland (1982) provide necessary and sufficient for a dynamic investment strategy to satisfy each of the following properties: (i) the strategy is self financing, (ii) the strategy yields path-independent returns, and (iii) the strategy is consistent with the optimal behavior of some expected utility maximizer. When the market is arbitragefree and complete, Dybvig (1988a), (1988b) proves that path-dependent strategies are inefficient and cannot be optimal for agents who care only about distribution of terminal wealth. More specifically, Dybvig characterizes the efficient payoffs that rational investors with increasing preferences should choose, and gives an explicit formula for the efficiency loss of a strategy (the magnitude of the inefficiency). The present paper draws on the same basic intuition that underlies the Basak and Shapiro (2001) paper concerning an agent who maximizes expected utility but is also subject to a Value-at-Risk constraint. They note that when the agent has to satisfy the VaR constraint it may be optimal for her to incur losses in those states which are most expensive to insure. The state-price density then plays a critical role in ranking payoffs in terms of their costs. The current paper extends Dybvig’s work by providing the explicit form of the optimal path-independent strategies. We illustrate specific examples of inefficient path-dependent strategies and contrast them with their corresponding optimal strategies. Recently, Vanduffel, Chernih, Maj and Schoutens (2008) show that for general L´evy markets, path-independent payoffs are preferred by risk-averse decision makers as long 1 Goldstein, Johnson and Sharpe (2006), (2008) propose an interesting and novel approach to construct the probability distribution of wealth of a given individual at maturity. 2 We define efficiency in Section 2.2. 3 A comprehensive synthesis of first and second-order stochastic dominance can be found in Levy (1992). Recently, Kopa and Post (2009) develop a test of first degree stochastic dominance optimality, illustrated with a study of the U.S. stock market. 4 The original Cox and Leland paper was written in 1982 but not published until 2000 in Cox and Leland (2000).

2

as the arbitrage-free pricing is based on the Esscher transform. Amin and Kat (2003) in an empirical paper conclude that investing in hedge funds is inefficient. Their measure of efficiency is based on the distribution price defined by Dybvig (1988a). Dybvig’s methodology is powerful since it does not depend on preferences. These results have been obtained in a frictionless market: the inclusion of frictions such as transaction costs has been studied by several authors including (Pelsser and Vorst (1996), Jouini and Khallal (2001) among others). The results in our paper require that the market is frictionless, arbitrage free and that the agents agree on the same5 pricing operator so that effectively the market is complete. We also assume that investors have a fixed investment horizon, have lawinvariant preferences6 and prefer more to less. While some of the results in our paper are restatements of ideas that are already known it is convenient to summarize our contributions as follows. First, given any strategy (either path-dependent or path-independent) with distribution F at maturity, we are able to exhibit the unique European path-independent payoff with the same distribution as F that minimizes the cost and should thus be preferred by all investors. The cost-efficient payoff puts more weight on the least expensive states while still maintaining the same distribution F . Specifically we derive an explicit expression for the cost-efficient payoff distribution. In addition we also derive the most inefficient payoff with the same distribution as F and hence provide a lower bound and an upper bound for the cost of a given distribution of terminal wealth. Second, we provide a simpler and more general proof of the results of Cox and Leland (1982), (2000), Dybvig (1988a), (1988b), and Vanduffel et al. (2008). In particular, our proof is valid provided there exists a state-price process which is a strictly positive random variable continuously distributed on (Ω, F, P ). We do not have to specify the pricing operator. We prove the optimality of non-decreasing payoffs and derive a closedform expression for the optimal cost-efficient payoff and the efficiency loss of any strategy. Third, we characterize the necessary and sufficient conditions for a strategy to be cost-efficient. The payoff has to be a non-increasing function of the state-price process. Consequently, when the state-price process is a decreasing function of the underlying stock (as it is the case in the one-dimension Black and Scholes market for example), efficient payoffs have to be non-decreasing functions of the underlying stock price. This result is well-known in the literature: it was already noted by Dybvig (1988a), (1988b) but our conditions are more general. Fourth, we illustrate our findings by analyzing several new examples of inefficient strategies or inefficient financial derivatives. The simplest example is the put option. We also analyze several path-dependent payoffs including geometric Asian options and 5

A sufficient condition for this to occur is when the market is complete. A function f defined on random variables is said to be law-invariant if for two random variables X and Y with the same distribution, then f (X) = f (Y ). 6

3

lookback options. Interestingly the corresponding cost-efficient payoff for the Asian option is a type of power call option. We also examine the Constant Proportion Portfolio Insurance (CPPI) strategy and explain why the discrete version of this strategy is inefficient whereas the strategy is cost-efficient if implemented in continuous time. Fifth, we reinforce the suboptimality of a path-dependent strategy by constructing explicitly two preferred strategies and relate these to stochastic dominance rankings. The pathdependent strategy is dominated in the sense of first-order stochastic dominance by the optimal cost-efficient strategy which is cheaper and has the same distribution. It is also dominated in the sense of second-order stochastic dominance with a strategy that has the same cost but a different distribution. In our setting, both approaches are interesting and lead to very different strategies: we compare them using a numerical example. The rest of the paper is organized as follows. Section 2 begins with a brief introduction to the framework and assumptions and then introduces a very simple binomial example. Section 3 presents our main findings on cost-efficiency and optimal cost-efficient payoffs in a preference free framework. We derive the implications for agents’ optimal investment strategies in the standard expected utility framework in Section 4. Section 5 illustrates our propositions using numerical examples based on some well known strategies and payoffs. The final section summarizes the paper. The proofs are given in the Appendix.

2

Framework and Introduction to Cost-efficiency

We start this section by describing our assumptions and the framework. Then we recall the concept of cost-efficiency and illustrate it with a simple example.

2.1

Assumptions and framework

The assumptions regarding agents’ preferences are fairly general and are similar to those used by Dybvig (1988a), (1988b). More precisely, we assume agents’ preferences depend only on the probability distribution of terminal wealth.7 Investors have a fixed investment horizon and there is no intermediate consumptions. Agents prefer more to less.8 In the expected utility framework, it means that agents’ utility functions are non-decreasing functions of wealth. 7

Such preferences are called law-invariant preferences, they include a wide range of behavioral theories, among them expected utility theory (Von Neuman Morgenstern (1947)), cumulative prospect theory (Tversky and Kahneman (1992)), and rank dependent utility theory (Quiggin (1993)). Lawinvariant utilities are also recently discussed by Carlier and Dana (2009). 8 In other words, they have increasing preferences. By “increasing preferences” it is meant that preferences satisfy first stochastic dominance. If X > Y almost surely, then the outcome X is seen as at least as good as the outcome Y by the representative investor.

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We assume that the market is free of arbitrage, perfectly liquid and frictionless (no transaction costs, no trading constraints). Under these assumptions, there exists a stateprice process (ξt )t such that (ξt St )t is a martingale for all traded assets S in this market. In an incomplete market, the state-price process is not unique. However, we assume all agents agree on the pricing operator. Therefore, the choice of the state-price process is fixed. Denote it by (ξt )t . Apart from our discrete time example in the next section we assume a continuous-time setting, and that for all non-negative t, ξt is continuously distributed. We first define the cost of a given payoff. Definition 1 The cost of a strategy (or of a financial investment contract) with terminal payoff XT is given by c(XT ) = E[ξT XT ] where the expectation is taken under the physical measure P .

2.2

Cost-efficiency

We start by defining cost-efficiency and the distributional price. These concepts were first introduced by Cox and Leland (1982) and Dybvig (1988a), (1988b). They will enable us to compare strategies while requiring fairly general assumptions on preferences. Definition 2 A strategy (or a payoff ) is cost-efficient if any other strategy that generates the same distribution costs at least as much. Given a random variable X, denote by F its cumulative distribution function (cdf) defined by F (x) = P (X 6 x) for all x ∈ R. The distributional price of this payoff is a useful concept originally proposed by Dybvig (1988a), (1988b). Definition 3 The “distributional price” of a cdf F is defined as PD (F ) =

min

{Y | Y ∼F }

{c(Y )}

(1)

where {Y | Y ∼ F } denotes the set of random variables that have the same distribution as F . Definition 4 The “efficiency loss” of a strategy with payoff XT at maturity T with cdf F is calculated as follows, c(XT ) − PD (F ). (2) Note that the cost c and the distributional price PD both depend on the state-price process ξ. However for ease of exposition, we omit the dependence on ξ in the notation. 5

2.3

Binomial example to illustrate the concept of cost-efficiency

Although the results obtained in this paper are derived in a continuous time setting with general assumptions on preferences, it is instructive to illustrate these concepts with a simple binomial model. Dybvig (1988b) illustrates his ideas using a similar example. This example shows the importance of the relationship between the risk-neutral measure used for pricing and the physical measure in determining the efficiency loss. Suppose we have a two period discrete time model in the Cox Ross Rubinstein (1979) framework. Let P denote the physical probability measure and let p = 21 denote the probability of an up jump at each node under P . Suppose the initial price of the risky asset (stock) is 16. At the end of the two periods, the stock can take three distinct values (4, 16 or 64) and the associated (physical) probabilities over the two periods are 1 1 , or 14 respectively. We have 4 2 S66 2 = 64 mmm m m mmm

P (S2 = 64) =

1 4

S66 2 = 16 mmm m m mmm

P (S2 = 16) =

1 2

S2 = 4

P (S2 = 4) =

p

S1 = 32Q QQQ1−p mm66 m QQQ m m Q(( mmm p

S0 = 16Q

QQQ1−p QQQ Q((

p

S1 = 8 Q QQQ1−p QQQ Q((

1 4

Consider a payoff X2 after 2 periods:   1 2 X2 =  3

if S2 = 64, if S2 = 16, if S2 = 4.

(3)

The expected utility of this payoff can be calculated as follows: E[U (X2 )] =

U (1) U (3) U (2) + + . 4 4 2

When the effective interest rate is 10% per period, the corresponding risk-neutral probability of an up jump is q = 52 . Thus the risk-neutral probabilities that S2 = 64, 4 12 9 S2 = 16 or S2 = 4 are respectively equal to 25 , 25 and 25 . The state-price process at time t is the ratio of probabilities under the risk-neutral measure and the physical measure

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discounted at the risk-free rate. It can be calculated as follows: ξ2 = p

ξ1 = p nn66 nnn nnn

ξ0 = 1 P PPP1−p PPP P(( ξ1 =

1 q 1+r p

h33 hhhhh h h h h hh VVVV VVV1−p VVVV V++

ξ2 = 1 1−q 1+r 1−p

p hhh33 hh h h h h hh

VVVVV 1−p VVVVV V++

ξ2 =

q2 1 (1+r)2 p2

≈ 0.5289

q(1−q) 1 (1+r)2 p(1−p)

≈ 0.7934

(1−q)2 1 (1+r)2 (1−p)2

≈ 1.1901

The time zero price of the payoff X2 is given by c(X2 )

= E[ξ2 X2 ]   1 1 1 (0.5289) 1 + (0.7934) 2 + (1.1901) 3 ≈ 1.82. = 4 2 4

Consider another payoff given by   3 2 X2⋆ =  1

if S2 = 64, if S2 = 16, if S2 = 4.

The expected utility E[U (X2⋆ )] is equal to the expected utility of X2 whereas its price, c(X2⋆ ), is now strictly lower:   1 1 1 ⋆ c(X2 ) = (0.5289) 3 + (0.7934) 2 + (1.1901) 1 ≈ 1.49 < c(X2 ). 4 2 4 In fact the payoff X2⋆ is the cheapest way to get the same distribution as the payoff of X2 under the physical measure. Its price c(X2⋆ ) is called the distributional price by Dybvig (1988a), (1988b). The efficiency loss of the payoff X is equal to c(X2 ) − PD = c(X2 ) − c(X2⋆ ) = 0.33.

3

Optimal Cost-Efficient Payoffs

This section contains our main results. From now on, the assumptions of Section 2.1 hold. In addition ξT is continuously distributed with cdf Fξ . Assume the random payoff variable XT has cdf F , where F is right-continuous and non-decreasing. We also assume 7

that X is nonnegative and has a finite cost. We define its inverse F −1 as follows F −1 (y) = min {x | F (x) ≥ y} . The inverse is left-continuous and non-decreasing. The next proposition gives the explicit representation for the cost-efficient payoff among a set of payoffs which have the same distribution. Proposition 1 Let F be a cdf. Define YT⋆ = F −1 (1 − Fξ (ξT )) .

(4)

Then, YT⋆ is distributed with the cdf F . Let Y be any other random variable distributed with the same cdf F . Then we can show that
c YT⋆ ≤ c(Y ),

which means Y
T⋆ minimizes the cost of achieving a payoff with distribution F . If in addition c YT⋆ < +∞, then YT⋆ is the unique9 optimal solution that satisfies PD (F ) = c(YT⋆ ).

(5)

The cheapest way to achieve a payoff with probability distribution F is YT⋆ . Note that YT⋆ is non-increasing with respect to ξT . This Proposition extends Theorem 1 of Dybvig (1988a) which is given in a complete discrete-time market model where all states are equally probable. It states that “the cheapest way to achieve a lottery assigns outcomes of the lottery to the states in reverse order of the state-price density.” We can interpret Proposition 1 as follows. Given a payoff XT distributed with cdf F , the (a.s.) unique payoff YT⋆ also distributed with the same cdf F that minimizes the cost of a strategy distributed with the same cdf F is YT⋆ = F −1 (1 − Fξ (ξT )) . It is obviously non-increasing with respect to the state-price process. One can then reduce the cost of the strategy by replacing the payoff XT by YT⋆ without changing the payoff probability distribution. Note that for any utility function U , XT and YT⋆ have the same expected utility since they have the same probability distribution (in other words, they are on a iso-utility curve, that is E[U (XT )] = E[U (YT⋆ )]). We next define the concept of path-independence in an obvious way. Definition 5 A strategy with payoff XT at time T based on the price of some security The uniqueness is only an almost surely uniqueness which means that any payoff equal to YT⋆ everywhere except on a set with probability measure zero would also satisfy the condition (5). 9

8

S is path-dependent if its payoff depends not only on the final value ST but on previous values of S as well. We note that in general path-dependent payoffs are not cost-efficient. To be costefficient, the payoff has to be of the following form YT⋆ = F −1 (1 − Fξ (ξT )) .

(6)

Therefore, the payoff has to be a function of the state-price process at time T . When the state-price process ξT can be expressed as a function of ST , YT⋆ becomes the payoff of a European derivative written on the stock price ST at time T and therefore it is path-independent. We will implement formula (6) and illustrate it with examples in Section 5. Proposition 1 leads to the following extension of a result due to Dybvig (1988a). Corollary 1 Under the same assumptions as Proposition 1, the “distributional price” can be expressed as Z 1 PD (F ) = Fξ−1 (γ)F −1 (1 − γ)dγ. (7) 0

This formula was first established by Dybvig in the case of a complete market. See Theorem 3 of Dybvig (1988a). The formula was proved in a discrete-time financial market and extended to continuous-time models using heuristic arguments. The next result provides necessary and sufficient conditions for a payoff to be cost efficient. Proposition 2 Recall that Fξ denotes the cdf of the state-price density at time T , ξT . Consider a payoff XT with finite cost which can be written as XT = h(ξT ). Then XT is cost-efficient if and only if h is non-increasing. Denote by F the cdf of XT . Moreover, if XT is cost-efficient, it satisfies XT = YT⋆ = F −1 (1 − Fξ (ξT )) a.s.

3.1

Black and Scholes market

We now give some specific results for the Black and Scholes case. Here the dynamics of the underlying stock price are given by dSt = µdt + σdWt , St 9

where Wt is a standard Brownian motion under the physical measure P , µ is the drift and σ is the volatility. The solution for the stock price is    σ2 St = S0 exp µ − t + σWt 2 The continuously compounded risk-free interest rate is r < µ. The (unique) state-price process can be computed explicitly as ξt = e−rt e− 2 ( 1

µ−r 2 t σ

W ) e−( µ−r σ ) t.

Consequently, ξT can be written as an explicit function of the stock price ST , −b ST (8) ξT = a S0       2 2 where a = exp σθ µ − σ2 T − r + θ2 T , b = σθ , θ = µ−r . Denote by Fξ the cdf σ of ξT . There is a closed-form expression for Fξ in this case in terms of the cdf of a standard normal distribution. Let M denote the mean of ln(ξT ). We can show that M = − 21 θ2 T − rT . The variance of ln(ξT ) is equal to θ2 T . Then,   ln(x) − M √ . (9) Fξ (x) = P (ξT 6 x) = Φ θ T 

Note that under these assumptions, the “cost-efficient payoff” distributed with cdf F has the closed-form expression given by (6). When a formula for F −1 is available, then the cost-efficient payoff YT⋆ can be computed explicitly using the expression for the cdf of the state-price process given by (9). Our next corollary shows that any path-dependent payoff in the Black Scholes model will be inefficient. Corollary 2 Under the assumptions of the Black and Scholes model any path-dependent payoff is inefficient. Corollary 3 Any wealth distribution (financial derivative) with a non-decreasing payoff in the underlying stock price ST is cost-efficient. Corollary 2 is a direct consequence of the uniqueness almost surely of cost-efficient strategies proved in Proposition 1. Corollary 3 follows from Proposition 2. Details can be found in the Appendix. It is possible to find the cost-efficient payoff in a Black and Scholes market with several risky assets. In this case, the state-price process depends on the value at maturity of each asset in the market in a complicated way, and the cost-efficient payoff YT⋆ can also be expressed as a function of ξT as in (6). 10

3.2

Explicit expressions for the bounds for the strategy’s cost

The next result provides an upper bound for the cost of a strategy where the payoff is distributed with cdf F . Although the lower bound is normally of more interest it is also instructive to evaluate the loss of efficiency for the most expensive strategy. Proposition 3 The strategy ZT⋆ that generates the distribution F with the highest cost solves the following optimization problem max

{Z | Z∼F }

{c(Z)} .

(10)

It can be characterized as follows ZT⋆ = F −1 (Fξ (ξT )) . For any random variable, XT , that is distributed with the cdf F , one has c(ZT⋆ ) ≥ c(XT )
If in addition c ZT⋆ < +∞, then ZT⋆ is the unique10 optimal solution of Problem (10).

We will provide several examples to illustrate this result in Section 5. The next result provides bounds for the cost of any strategy with a payoff XT .

Corollary 4 Consider a strategy with payoff XT with cdf F . The cost of this strategy satisfies Z 1 −1 Fξ−1 (v)F −1 (v)dv PD (F ) 6 c(XT ) 6 E[ξT F (Fξ (ξT ))] = 0

The lower bound was first given by Dybvig (see Theorem 3 of Dybvig (1988a)).

4

Agents’ Preferences

Cost-efficiency is a general concept that has important consequences for agents’ preferences. We first discuss its strong relationship with first-order stochastic dominance ordering. This section uses the same assumptions as Section 3. Proposition 4 Let XT be the final wealth of the investor. Assume the objective function of the agent, V (XT ), satisfies the two following properties: 10

It is only almost surely unique.

11

• The agent prefers “more to less”: this means that if α is a non-negative random variable V (XT + α) > V (XT ). • The agent has “state independent preferences”: if ZT has the same distribution as WT then: V (ZT ) = V (WT ). Then taking into account the initial cost of the payoff YT⋆ , defined by (4) in Proposition 1 is at least as good as XT to this investor. This result holds in a fairly general setting. It is not necessary to assume risk aversion as Cox and Leland (1982) and Vanduffel et al. (2008) did. Dybvig (1988a), (1988b) assumed only increasing preferences to obtain most of his results. The next result complements Proposition 4 and shows the strong connections between cost-efficiency and stochastic dominance: a connection first pointed out by Dybvig (1988a) . Proposition 5 Consider a payoff XT with cdf F , 1. Taking into account the initial cost of the payoff, the cost-efficient payoff YT⋆ of the payoff XT is a.s. equal to XT or dominates XT in the first-order stochastic dominance sense (we write ≺f sd ):

XT − c(XT )erT ≺f sd YT⋆ − PD (F )erT (11)

2. The dominance ordering given by (11) is strict unless XT is a non-increasing function of ξT .

The next corollary ties together some of our recent results and shows their inter connections. Corollary 5 The following statements are equivalent: • The payoff XT maximizes the expected utility of terminal wealth for some agent with strictly increasing Von Neumann-Morgenstern preferences over terminal wealth. • The payoff XT is cost-efficient. • The payoff XT is non-increasing in the terminal value of the state-price process. Corollary 5 is a direct consequence of Propositions 4 and 5 in the setting of expected utility theory. It shows that for an agent with preferences that correspond to a strictly increasing utility function, it is optimal to choose the cost-efficient payoff. These results were first enunciated in Theorem 1 of Dybvig (1988b) and Theorems 1 and 2 of Dybvig (1988a) and we just provide a modest generalization. 12

As another instance of this result we can cite Merton’s (1971) optimal portfolio selection in the Black and Scholes framework (where S denotes the underlying risky asset): it is an increasing function of ST (see also Cox and Huang (1989)). Since the Merton portfolio is optimal, it is not possible to construct a more preferred portfolio for an expected utility maximizer, so it has to be cost-efficient and therefore non-decreasing in ST (in light of Corollary 5). Proposition 4 and corollary 5 are true for all preferences that preserve first stochastic dominance ordering. Therefore these results are valid for expected utility maximizers, including risk-loving investors (with convex utility), risk-averse investors (with concave utility) and loss averse11 investors (with S-shaped utility)), mean-variance investors, and CPT-investors (defined as investors acting consistently with the cumulative prospect theory developed by Tversky and Kahneman (1992)).

4.1

Alternative approach to path-independent payoffs

To prove that path-dependent payoffs are not optimal for risk-averse investors in L´evy markets with Esscher pricing, Vanduffel et al. (2008) use second-order stochastic dominance and adopt a very different approach. We show that these results hold even when the pricing operator is not based on the Esscher transform. Proposition 6 Any payoff XT which cannot be expressed as a function of the stateprice process ξT at time T is strictly dominated in the sense of second-order stochastic dominance by HT⋆ = E [XT | σ(ξT )] = g(ξT ), (12) which is a function of ξT . Consequently in the Black and Scholes framework, any strictly path-dependent payoff is dominated by a path-independent payoff. The proof is given in the Appendix. Using the fact that HT⋆ dominates XT in the sense of second-order stochastic dominance, we get the following results. Corollary 6 If 1. Any risk-averse agent will prefer the payoff HT⋆ = E[XT |ξT ] to the payoff XT . 2. In a financial market, where ξT can be expressed as a function of ST , path-independent payoffs are preferred by risk-averse agents. Note that the results in the above corollary are different from our earlier results in that we now maintain the same costs but the payoff distribution differ. The payoffs HT⋆ 11

The terminology “loss-averse” refers here to the paper by Berkelaar, Kouwenberg and Post (2004).

13

and XT have the same cost but HT⋆ may not have the same distribution as XT and HT⋆ may not be cost-efficient (it will be the case only if it is a non-increasing function of ξT ). 12

Corollary 6 gives an additional way of constructing a “better” payoff. We will compare the cost-efficient payoff YT⋆ with the above conditional expectation HT⋆ in the case of a lookback option in the next section.

5

Examples

In this section, we illustrate these results using some common payoff distributions. We first investigate put options which are path-independent but inefficient. Then we examine two path-dependent payoffs: the lookback option and the geometric average call option. Finally we explain why the CPPI strategy is “theoretically” cost-efficient in continuous time but inefficient in practice. For the sake of simplicity, we assume a Black and Scholes market. We use the same notation as in Section 3.1.

5.1

Put options

One of the simplest examples of a payoff distribution with a decreasing payoff in relation to stock price is the standard put option. From Corollary 2 it is not optimal for investors who prefer more to less to hold exclusively put options in their portfolio. A put option with strike K and maturity T has the following payoff XT = (K − ST )+ , and hence XT is increasing in the state-price process ξT in the Black and Scholes framework. Let us denote by F the cumulative distribution function of the payoff of the put option. The put option is therefore the (a.s.) unique payoff that has the highest possible cost and is distributed among all distributions with cdf F . This cdf, F , is  1 if x > K       σ2   K−x µ− 2 T −ln S 0 √ F (x) = P (XT 6 x) = P (ST > K − x) = Φ if 0 6 x < K σ T    0 if x < 0 It is straightforward to invert it. Define ν = Φ



2

µ− σ2



T −ln √ σ T



K S0



and consider

This is obvious since E[ξT HT⋆ ] = E[ξT XT ] because of the property of the conditional expectation. See more details in the Appendix in the proof of Proposition 6. 12

14

y ∈ (0, 1), F −1 (y) =

(

K − S0 e 0

  √ 2 µ− σ2 T −σ T Φ−1 (y)

if y ≥ ν, if y < ν.

Note that F −1 (1) = K and F −1 (0) is not well defined (it is equal to −∞). The costefficient payoff that gives the same distribution as a put option is    √  M −ln(ξ )  + 2 √ T µ− σ2 T −σ T ⋆ −1 θ T YT = F (1 − Fξ (ξT )) = K − S0 e , where Fξ is given by (9). Using (8) we obtain 

YT⋆ = K −

where c =

  σ2 2 2 µ− 2 T . S0 e

  σ2 2 2 µ− 2 T S0 e

ST

+

+   = K ST − c , ST K

The no-arbitrage price at time zero of this payoff is given by Ke−rT N (−d2 ) − S0 e2(µ−r)T N (−d1 ),

where

(13)

ln( SK0 ) + (2µ − r)T + √ d1 = σ T

σ2 T 2

,

√ d2 = d1 − σ T .

(14)

(15)

A portfolio consisting of a put option (or entirely of put options) is therefore not optimal for expected utility maximizers. It is dominated in the sense of first-order stochastic dominance by the European payoff given by YT⋆ in (13). Figure 1 displays both payoffs. Based on the parameters given in Figure 1, the cost of the put option is 5.573 while the cost of the corresponding cost-efficient payoff is 3.145. This cost-efficient payoff has the same distribution under the real probability measure as the payoff of the put option. Hence the efficiency loss for this put option is 2.428 (that is 43% of its market price). Buying a put option rather than the payoff YT⋆ represents a significant efficiency loss. This implies a rational agent would not invest her entire wealth in a put option. We can give an intuitive interpretation of the derivative payoff in formula (13). Write ST as ST (z) where z ∼ N (0, 1) and    √ σ2 ST (z) = S0 exp µ− T + σ Tz . 2 Then the payoff of the put option is [K − ST (z) ]+ and the payoff of the cost-efficient

15

100 90 80

Put Option

70 Payoff

60 50 40 Cost Efficient Payoff

30 20 10 0 0

50

100

150 200 250 300 350 Stock price at maturity ST

400

450

500

Figure 1: Payoff of a put option and its cost-efficient counterpart Parameters: σ = 20%, µ = 9%, r = 5%, S0 = 100, T = 1 year, strike K = 100. The graph shows the payoffs (as functions of the stock price ST ) of the put option and its cost-efficient counterpart that has the same payoff distribution.

strategy is 

YT⋆ = K −

  σ2 2 2 µ− 2 T S0 e

ST (z)

+

 = [K − ST (−z) ]+ .

(16)

Notice that ST (z) and ST (−z) are equal in distribution. Then, [K − ST (z)]+ has the same distribution as [K − ST (−z)]+ . This striking relationship also illustrates how the put option has a payoff that is a strictly increasing function of the state price density, while the efficient payoff is a strictly decreasing function of the state price density.

5.2

Path-dependent examples

We now consider two path-dependent payoffs: lookback options and Asian options. 5.2.1

Lookback options

Consider a lookback call option with strike K. The payoff on this option is given by  + LT = max {St } − K . 06t6T

16

Recall that we can write St = S0 e under the physical measure.

  2 µ− σ2 t+σWt

where W is a standard Brownian motion

We now derive the cost-efficient payoff for this payoff distribution. Let FL be the cdf of LT . The payoff that has the lowest cost and has the same distribution as the payoff LT is given by YT⋆ = FL−1 (1 − Fξ (ξT )) . The payoff that has the highest cost and has the same distribution as the payoff LT is given by ZT⋆ = FL−1 (Fξ (ξT )) . We show in the Appendix that the payoff YT⋆ can be written as      M − ln(a) + b ln SST0  . √ YT⋆ = FL−1 Φ  (17) θ T where a = exp

  θ σ

µ−

σ2 2



 T − r+

θ2 2

  T , b = σθ , θ =

µ−r σ

and M = − 12 θ2 T − rT .

180 160 140

Payoff

120 100 80 Y*T

60 40

Z*T

20 0 40

60

80

100 120 140 160 Stock Price at maturity ST

180

200

220

Figure 2: Payoff distributions related to lookback options Parameters σ = 20%, µ = 9%, r = 5%, S0 = 100, T = 1 year, strike K = 100. The graph shows the payoff of the cost-efficient derivative YT⋆ (as a function of the stock price ST ) that has the same distribution at time T as a lookback option. ZT⋆ shows the most expensive payoff with the same distribution as the lookback option.

Figure 2 plots the payoff distribution of the cost-efficient distribution associated with the lookback option as the blue solid line for the specified parameter values. The price of the lookback option in this case is 19.17. The price of the associated cost-efficient payoff is 18.85. Hence the efficiency loss in this case is 0.32 (which is about 1.7% of the market price). Notice that the payoff profile of the cost-efficient payoff is similar to 17

that of a standard European call. For comparison purposes, we display the profile of the payoff ZT⋆ which has the same distribution as the lookback call and has the highest initial cost (25.26). This inefficient payoff is similar to the payoff of a put option. We have discussed two methods of constructing a payoff that dominates a pathdependent payoff under our standard assumptions. The first method constructs a payoff with the same distribution and a strictly lower cost: the cost-efficient payoff dominates the lookback option in the sense of the first-order stochastic dominance. The second method is discussed in Section 4.1 in Proposition 6 and Corollary 6. There we noted that the conditional expectation with respect to the σ-algebra generated by ξT provides an alternative way of obtaining a payoff that has the same no-arbitrage price as the lookback option. In this case the constructed payoff will dominate the lookback payoff in the sense of second-order stochastic dominance. We now show how to construct this payoff that dominates the lookback in the sense of second degree stochastic dominance. We denote the conditional expectation in this case by HT⋆ , where by HT⋆ = E [LT | σ(ξT )] . In the Black and Scholes setting, this payoff is equal to H ⋆ = E [L | σ(S )] because T

T

T

there is a measurable bijection between ST and ξT . # " + max {St } − K HT⋆ = E σ(ST ) 06t6T

Using the properties of the maximum of a Brownian bridge, one can derive the following formula (for details see the Appendix)      2  2  ST  √ S 2 1 − σ ln S +σT  ln ST +σT 2πT 0 8T σ  0 √ Φ if ST > K,  ST − K + 2 S0 σe 2 T ⋆        2 HT = (18) S S 2 √ S 2 1 ln 0 2T +T σ  ln ST +σT σ K π  8T σ √ 0  2T if S < K. Φ S σe T 0 2 2 T

The payoff HT⋆ is a function of ST . Since it is not path-dependent it can be represented on the graph. Figure 3 displays the payoffs of HT⋆ and the cost-efficient payoff YT⋆ in the case of the lookback call option. They are quite close to each other. It is also interesting to note that in contrast to Y ⋆ , the payoff H ⋆ does not depend T

T

on µ. This is a consequence of the fact that the drift parameter does not affect the distribution of a maximum of a Brownian bridge (see for instance Beghin and Orsingher (1999)). We proved that H ⋆ dominates the payoff L in the sense of the second-order stochasT

T

tic dominance (see Corollary 6). This can be easily seen from Figure 4 where the two cdf’s cross just once. The distribution of HT⋆ and of the lookback payoff have the same mean but obviously the distribution of HT⋆ has less spread. This explains why it dominates for the second-order dominance ordering.

18

Y*T

120

HT 100

Payoff

80 60 40 20 0

50

100 150 Stock Price at maturity ST

200

Figure 3: Payoffs of HT⋆ and YT⋆ Parameters: σ = 20%, µ = 9%, r = 5%, S0 = 100, T = 1 year, strike K = 100. The solid line represents the payoff HT⋆ and the dashed line represents the payoff of the cost-efficient payoff YT⋆ (as functions of the stock price ST ).

5.2.2

Geometric Asian options

We now derive the cost-efficient payoff associated with a geometric Asian call. It turns out to be a power option. First we discuss a continuously monitored geometric Asian option. Its payoff is given by  1 RT + c ln(St )dt 0 T GT = e −K where K is the strike price. The cost-efficient payoff that will give the same distribution as a continuous geometric Asian option is +  √ K 1/ 3 c,⋆ − (19) Y T = d ST d  √  2  1 1− √1 − 13 µ− σ2 T where d = S0 3 e 2 . This formula is derived in the Appendix. The payoff c,⋆ YT is a power call option which is a special case of polynomial options (see Macovschi and Quittard-Pinon (2006)). There is a simple analytical solution for its price 2

C 0 = S0 e

( √1 −1)rT +( 21 − √1 )µT − σ12T 3

3

19

N (h1 ) − Ke−rT N (h2 )

(20)

1

0.8

CDF

0.6

0.4

0.2

* T * H T

cdf of Y = cdf of lookback option X cdf of

0 0

10

20

T

30 Payoff

40

50

60

Figure 4: Cdf of HT⋆ and cdf of the payoff of a lookback option Parameters: σ = 20%, µ = 9%, r = 5%, S0 = 100, T = 1 year, strike K = 100 . The solid line represents the cdf of HT⋆ and the dashed line the cdf of the lookback call option.

where h1 =

ln

S0 K




+ ( 12 −

√1 )µT 3

σ

q

+

√r T 3

T 3

+

1 2 σ T 12

,

h2 = h1 − σ

r

T . 3

(21)

We can obtain similar results for the corresponding discretely sampled Asian option with payoff +  ! n1 n Y − K . S kT GdT =  n

k=1

In this case the efficient payoff that is distributed as the payoff GdT is given by YTd,⋆

+  √ (n+1)(2n+1)/(6n2 ) − K/α = α ST

where α=

1− S0

q

(n+1)(2n+1) 6n2

exp

n+1 − 2n

! ! p 6(n + 1)(2n + 1) (µ − σ 2 /2)T . 6n

This is also a power call option. 20

(22)

Since the price of a power call option with payoff (STα − A)+ is given by 1

C0 = S0α e(α−1)rT + 2 (α

2 −α)σ 2 T

N (h3 ) − Ae−rT N (h4 )

(23)

Sα √ )σ 2 )T ln( A0 )+(αr+(α2 − α 2 √ , h4 = h3 − |α|σ T , we can easily get the price of the where h3 = |α|σ T cost-efficient payoff in this case. We can also derive the closed-form expression of the payoff that has the highest cost and has the same distribution as the geometric Asian call option by calculating ZT⋆ = F −1 (Fξ (ξT )) . For our base case parameters the price of a geometric Asian option is 5.94. The payoff YT⋆ costs 5.77 (the lowest possible cost) and the payoff ZT⋆ costs 9.03 (the highest possible cost).

120 100

Payoff

80 60

Z*T

40 Y*T

20 0 40

60

80

100 120 140 160 180 200 220 240 260 Stock Price at maturity ST

Figure 5: Payoffs of YTd,⋆ and of ZTd,⋆ for the discrete geometric Asian option Parameters: σ = 20%, µ = 9%, r = 5%, S0 = 100, T = 1, n = 12. The solid line represents the cost-efficient payoff and the dotted line the most inefficient payoff both with the same distribution as the discrete Asian geometric option.

5.3

Constant Proportion Portfolio Insurance

The CPPI strategy was first introduced by Perold (1986), Black and Jones (1987) and Black and Perold (1992). This strategy includes as special case the buy-and-hold strategy and the stop-loss strategy. Assuming continuous monitoring, the value at time T of a CPPI strategy in the Black and Scholes framework can be written as      2 V0 − F 0 σ2 2σ VT = F T + −m T STm exp r−m r− S0m 2 2 21

where Ft stands for the floor at time t (guarantee). For example Ft = e−r(T −t) G. The parameter m stands for the multiple of the CPPI strategy (m is the percentage of the excess of the floor that is invested in the risky asset). This payoff is a European payoff increasing with respect to the underlying stock price at maturity ST . It is therefore a cost-efficient strategy (Corollary 3 page 10). The inefficiency of the CPPI strategy in practice arises from discrete monitoring. If there is discrete monitoring of the portfolio, then the final payoff of the CPPI strategy becomes path-dependent, and therefore inefficient (see Corollary 2 page 10). Numerical calculations of the inefficiency cost of the CPPI are given in Vanduffel et al. (2008) in a L´evy market when the risk-neutral pricing is based on the Esscher transform.

6

Summary and Conclusion

This paper has discussed a preference free framework for ranking different investment strategies. It draws its inspiration from the seminal papers of Dybvig and of Cox and Leland. For a given investment strategy, we derive an explicit analytical expression for the cheapest strategy that has the same payoff distribution as the given strategy. This explicit payoff is expressed in terms of the state-price density. We also derive a similar expression for the strategy that leads to the highest cost for the same payoff distribution. We discuss the inefficiency of path-dependent strategies. Our framework is used to analyze the strategies that produce some common option type payoffs. This in turn leads to some new types of relationships and insights. There are strong connections between this approach and stochastic dominance rankings. Since stochastic dominance has proved to be a valuable tool for comparing distributions in finance and economics it is hoped that some of the ideas in this paper will also prove to be useful.

22

Appendix Proof of Proposition 1 page 8. Recall that the distributional price is defined in definition 3 page 5 by the optimization problem (1). It can be reformulated as follows min E [ξT Y ] Y

subject to ∀x ∈ R, P (Y ≤ x) = F (x)

(24)

The objective is to minimize the cost of a payoff Y such Y has the same distribution as F (that is the cdf of Y is F ). This specific optimization problem has been solved by Jin and Zhou (2008) in the case when F (0) = 0 [see Theorem B.1 (ii)] in a different context. Jin and Zhou show that for any feasible solution Y , that is for any random variable that is distributed with the cdf F , one has
E ξT YT⋆ ≤ E (ξT Y ) , (25)
where YT⋆ is given by (4). If in addition E ξT YT⋆ < +∞, that is c(YT⋆ ) < +∞, then YT⋆ is the unique optimal solution almost surely. In our notation, (25) can also be written as c(YT⋆ ) 6 c(Y ). In a note entitled “Erratum to “Behavioral Portfolio Selection in Continuous time”” available at SSRN, Jin and Zhou explain in Footnote 1 that the assumption “F(0)=0” is not needed. Thus Proposition 1 is proved.  Proof of Corollary 1 page 9. Using the expression for YT⋆ given in Proposition 1, the distributional price can thus be written as   PD (F ) = E ξT F −1 (1 − Fξ (ξT )) . Let fξ be the density of ξT . We have PD (F ) =

Z

+∞

uF −1 (1 − Fξ (u)) fξ (u)du.

0

We then change the variable, by Fξ (u) = v which is possible because Fξ is strictly increasing over the support R∗+ of ξT since ξT is atomless. One has fξ (u)du = dv. Therefore, Z 1

PD (F ) =

0

Fξ−1 (v)F −1 (1 − v) dv.



Proof of Proposition 2 page 9. Denote by F¯ξ = 1−Fξ the survival function of ξT . Assume XT is cost-efficient and denote its cdf by F . Then Proposition 1 (the a.s.
uniqueness of the cost-efficient payoff) and formula (6) prove that XT = F −1 F¯ξ (ξT ) a.s. and therefore that XT = h(ξT ) a.s. 23

where h is decreasing. Conversely, consider h (where h is a decreasing function) and a payoff XT = h(ξT ). Denote its cdf by F . Then, F (x) = P (XT ≤ x) = P (ξT ≥ h−1 (x)) = F¯ξ (h−1 (x)) Then F −1 = h ◦ F¯ξ−1 . Thanks to Proposition 1, YT⋆ = F −1 (F¯ξ (ξT )) is cost-efficient. But YT⋆ = F −1 (F¯ξ (ξT )) = h(ξt ) = XT . This is possible because F¯ξ and h are invertible. F¯ξ is perfectly invertible since ξT is a positive r.v. that admits no atom. Since h is decreasing it is invertible. The proof is complete.  Proof of Corollary 2 page 10. Consider a path-dependent payoff XT . Let F be its cdf. From Proposition 1, the only cost-efficient payoff with cdf F is YT⋆ , given by (6). In the Black and Scholes model, ξT is directly written as a function of the stock price ST (see expression (8)). Therefore any cost-efficient payoff with cdf F is a function of ST and is therefore not path-dependent.  Proof of Corollary 3 page 10. Consider a European derivative with payoff XT = h(ST ) where h is non-decreasing. In the Black and Scholes market, we are able to express ST as a decreasing function of ξT and the result follows from Proposition 2. Indeed ξT can be written explicitly as a function of the stock price ST (see expression (8)). Consequently, ST = S0



a ξT

 1b

      2 2 where a = exp σθ µ − σ2 T − r + θ2 T , b = σθ , θ = ξT is clearly a non-increasing function of ST .

(26) µ−r . σ

When µ > r, b > 0 and 

Proof of Proposition 3 page 11. The proof also follows from the result of Jin and Zhou (2008). Problem (10) can be formulated as max E [ξT Z] Z

subject to ∀x ∈ R, P (Z ≤ x) = F (x)

Define ZT⋆ = F −1 (Fξ (ξT )) then Jin and Zhou (2008) show that ZT⋆ solves the optimiza
tion problem, and that if in addition E ξT ZT⋆ < +∞, then ZT⋆ is the almost surely unique optimal solution. 

24

Proof of Corollary 4 page 11. A formula for the lower bound has been established in Corollary 1, see expression (7). For the upper bound, it follows readily from Proposition 3. Z +∞ ⋆ −1 uF −1 (Fξ (u)) fξ (u)du E[ξT ZT ] = E[ξT F (Fξ (ξT ))] = 0

where fξ is the density of ξT . The change of variable v = Fξ (u) yields Z

1 0

Fξ−1 (v)F −1 (v)dv 

Proof of Proposition 4 page 11. The payoffs YT⋆ and XT have the same distribution, therefore V (YT⋆ ) = V (XT ). Let c(XT ) be the cost of the payoff XT and c(YT⋆ ) be the cost of the payoff YT⋆ . Then c(XT ) = c(YT⋆ ) + α where α > 0. Therefore V (YT⋆ − c(YT⋆ )erT ) > V (XT − c(XT )erT ) where r is the continuously compounded interest rate and T is the maturity of the contract.  Proof of Proposition 5 page 12. Consider x ∈ R, P XT − c(XT )erT 6 x




= P XT 6 x + c(XT )erT




> P XT 6 x + PD (F )erT




because PD (F ) 6 c(XT ). However,

P XT 6 x + PD (F )erT = P F −1 (1 − Fξ (ξT )) 6 x + PD (F )erT
= P F −1 (1 − Fξ (ξT )) − PD (F )erT 6 x
= P Y ⋆ − P (F )erT 6 x T

D

The only inequality is an equality for all x when PD (F ) = c(XT ). By the uniqueness (almost surely) of the payoff that minimizes the cost, one has XT = YT⋆ a.s.. In other words, XT is non-increasing in ξT (which is also enough to characterize cost-efficiency (Proposition 2)).  Proof of Proposition 6 page 13. Consider XT a payoff that is not necessarily a function of the state-price process. Define HT⋆ as follows HT⋆ = E [XT | σ(ξT )] = g(ξT ),

25

where σ(ξT ) is the σ-algebra generated by ξT and g is a Borel function. Obviously c(HT⋆ ) = c(XT ) because of the properties of conditional expectations,   c(HT⋆ ) = E ξT HT⋆ = E [E [ξT XT | σ(ξT )]] = E [ξT XT ] = c(XT ).

The result that the costs of the two payoffs are equal corresponds to Theorem 3 of Vanduffel et al. (2008). However our proof does not rely on any properties of L´evy processes or the Esscher transform. Finally applying Jensen’s inequality to any concave function φ leads to φ (E [XT | σ(ξT )]) > E [φ(XT ) | σ(ξT )] , and after taking the expectation, E[φ(HT⋆ )] > E[φ(XT )]. Therefore HT⋆ and XT have the same cost, same expectation but HT⋆ dominates XT in the second-order stochastic dominance.  Proof for the cost-efficient payoff (17) distributed as the lookback option Consider a lookback call option with strike K with payoff  + LT = max {St } − K . 06t6T



2

µ− σ2



t+σW

t where W is a standard Brownian motion under the physical One has St = S0 e measure. Let FL be the cdf of LT . Let x ∈ R+ ,   FL (x) = P (LT ≤ x) = P max {St } 6 K + x .

06t6T

But P (max06t6T {St } 6 K + x) is equal to       σ2 K +x µ− P max t + σWt 6 ln . 06t6T 2 S0 Thus, when x > S0 − K, 

FL (x) = Φ 

ln



K+x S0



 − µ− √ σ T

σ2 2

  2       2 µ− σ2 K+x   T − ln − µ− 2 σ S0 − K + x √ Φ S0 σ T

σ2 2

  T 

where Φ denotes the cdf of N (0, 1). This formula can be found for instance in Dana and Jeanblanc (2003), page 255. This function is invertible for any y ∈ [0, 1) since it is continuous and strictly increasing from R∗+ to (0, 1), (it seems difficult to obtain a 26

nice closed-form expression for the inverse). Define M and V , M = − 12 θ2 T − rT and V = θ2 T where θ = µ−r , then, σ   ln(x) − M √ . Fξ (x) = Φ θ T The payoff that has the lowest cost and is distributed such as the lookback call option is given by YT⋆ = FL−1 (1 − Fξ (ξT )) . The payoff that has the highest cost and is distributed such as the lookback call option is given by ZT⋆ = FL−1 (Fξ (ξT )) . Replacing  −b where Fξ by its expression and ξT by ξT = a SST0       2 2 , the payoff of YT⋆ can be written a = exp σθ µ − σ2 T − r + θ2 T , b = σθ , θ = µ−r σ as      M − ln(a) + b ln SST0  , √ YT⋆ = FL−1 Φ  θ T which is well-defined when S0 > K.

Proof of Formula (18). Define



    1 σ2 mT = max µ− t + Wt . t∈[0,T ] σ 2

From Beghin and Orsingher (1999), equation (3.3) page 162, for x > η       1 σ2 2x(x − η) Q mT > x µ− T + WT = η = exp − , σ 2 T

and this one for all x 6 η. The density of mT conditional on the n probability   is equal to o 1 σ2 event σ µ − 2 T + WT = η is given by 

2x(x − η) exp − T



4x − 2η 1{x>η} . T

Note that max St = S0 eσmT .

t∈[0,T ]

Therefore, when s > K, after some tedious calculations, # "     +  σ2 1 s + 1 σmT µ− T + WT = ln max {St } − K − K) E ST = s = E (S0 e 06t6T σ 2 σ S0 √   (2η+σT )2 −2η + σT 2πT 8T √ Φ = s−K + S0 σe 2 2 T 27

where η = E

"

1 σ

ln

  s . When s < K, the result is slightly different, S0

# √ +    2 2 1 2T π ln Ss +σT 8T σ 0 max {St } − K Φ S0 σe ST = s = 06t6T 2

2 σ

ln

S0 s K2




√ 2 T

+ Tσ

!

Then

HT⋆ =

    2  2  ST  S 2 1 − σ ln S +σT ln ST +σT 2πT 0 8T σ √ 0 Φ S0 σe 2 2 T        2 S S 2 √ S 2 1 ln 0 2T +T σ ln ST +σT σ K 2T π 8T σ √ 0 Φ S σe 0 2 2 T

    ST − K +   

√

if ST > K if ST < K 

Proof of Formula (19) for the continuous geometric Asian option RT From Kemna and Vorst (1990), T1 0 ln(St )dt is normally distributed (under the physical 2 measure) with mean 21 (µ − σ2 )T + ln(S0 ) and variance σ 2 T /3. We can easily calculate the distribution of GcT ,  if x < 0  0   K+x  1  σ2   c R ln S − 2 µ− 2 T T F (x) = P (GT 6 x) = 1 0 √T  P (e T 0 ln(St )dt 6 K + x) = Φ if x > 0 σ

We can easily invert it. Define ν = Φ

F −1 (y) =

(

S0 e 0

1 2



 2

µ− σ2

ln





K S0



   2 − 21 µ− σ2 T σ

T +σ

√T

√T

3

3

Φ−1 (y)

−K

3

and consider y ∈ (0, 1), if y ≥ ν, if y < ν.

The cost-efficient payoff that will give the same distribution as a continuous geometric Asian option is +    √ T  M −ln(ξT )  σ2 1 √ µ− T +σ c,⋆ −1 2 3 θ T −K , (27) YT = F (1 − Fξ (ξT )) = S0 e 2 where Fξ is given by (9). Using (8), one obtains (19) YTc,⋆ where d =

1− √1 S0 3 e



1 − 2

√ 1  3

2

µ− σ2



T

=d



√ 1/ 3 ST

.

K − d

+ 

28

Proof of Formula (22) for the discrete geometric Asian option Let K be the strike. Consider an Asian option with payoff

GdT

=



Qn

k=1

S kT n

Under Black-Scholes assumptions, the following random variables are iid    ST /n S2T /n ST σ2 T 2 T , ,σ , ..., ∼ LN µ− S0 ST /n ST (n−1)/n 2 n n

 n1

−K

+

.

Denote by Υ=

Y

SiT /n .

i=1...n

Then, Υ

1/n

So taking

∼ LN



n+1 (n + 1)(2n + 1) 2 (µ − σ 2 /2)T, σ T 2n 6n2



GdT = (S0 Υ1/n − K)+ , we get  0   x < 0, FL (x) = 2 /2n)T (n+1)/2 Φ ln(x/S√0 +K)−(µ−σ x ≥ 0, 2 2 σ T (n+1)(2n+1)/6n

and FL−1 (y) =

S0 exp

√ n+1 (µ − σ 2 /2)T + φ−1 (y) σ 2 T 2n

r

(n + 1)(2n + 1) 6n2

!

−K

!+

.

We can write a r.v. YTd,⋆ with identical distribution as an increasing function of ST , and determine a formula for the distributional price.  +   q    √ ln(ST /S0 )−(µ−σ 2 /2)T (n+1)(2n+1) d,⋆ n+1 2 −1 2 √ φ −K YT = S0 exp 2n (µ − σ /2)T + φ σ T 6n2 σ T + +  √  √ (n+1)(2n+1)/6n2 (n+1)(2n+1)/6n2 α−K − K/α = α ST = ST where α = exp

n+1 − 2n

! p 6(n + 1)(2n + 1) (µ − σ 2 /2)T + 6n

1−

r

(n + 1)(2n + 1) 6n2

!

!

ln(S0 ) . 

29

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