Amultidimensional semi-parametric upper bound for pricing a general class of complex call options

Transcription

1 Amultidimensional semi-parametric upper bound for pricing a general class of complex call options University of Mississippi University, MS Telephone (601) jdula@sunset.backbone.olemiss.edu October 1995 ABSTRACT In 1987, A.W. Lo presented a result for bounding from above the expected payoff of a call option with a single underlying risky asset. The bound is semi-parametric because it depends only on the mean and variance of the terminal distribution of the price of the risky asset. In this paper we extend this result to the multivariate case of a complex European call option on the maximum of a basket of n risky assets. This result also depends only on first and second-order information about the terminal distribution of the prices of the risky assets. Also, it is not required that the prices at maturity be independent but neither is it necessary to have information on the cross-moments; i.e., covariances. The expression for the upper bound is a closed-form formula that is very simple to evaluate.

2 Amultidimensional semi-parametric upper bound for pricing a general class of complex call options University of Mississippi, University, MS October 1995 Abstract. In 1987, A.W. Lo presented a result for bounding from above the expected payoff of a call option with a single underlying risky asset. The bound is semi-parametric because it depends only on the mean and variance of the terminal distribution of the price of the risky asset. In this paper we extend this result to the multivariate case of a complex European call option on the maximum of a basket of n risky assets. This result also depends only on first and second-order information about the terminal distribution of the prices of the risky assets. Also, it is not required that the prices at maturity be independent but neither is it necessary to have information on the cross-moments; i.e., covariances. The expression for the upper bound is a closed-form formula that is very simple to evaluate. An upper bound for the expected payoff of a call and put option with a single underlying risky asset requiring knowledge of the mean and variance of its terminal price was proposed by A.W. Lo in [1987]. Lo showed how this bound may be applied to pricing options when the mean and variance of the associated risk-neutral pricing distribution can be related to observable quantities as when the price of the asset follows a lognormal diffusion or a mixed diffusion-jump process. The upper bound is semi-parametric in the sense that no other distribution attribute besides the terminal mean and variance of the price of the risky asset are needed for its calculation. Lo considers this a remarkable property of the bound and, indeed, bounds of this type are particularly practical (witness Tchebycheff s Inequality). In this paper we present a direct extension of Lo s result for the multivariate case. It is an upper bound on the expected payoff of a call option on the maximum of a basket of n risky assets in the form of a (complex) European call option. 1 The bound, discounted at the risk-free rate, is an upper bound on the price of the option. The semi-parametric upper bound requires knowledge of the n means and variances of the terminal distribution of the prices of the underlying risky assets. As in the case of Lo s upper bound, our result remains a closed-form formula that is simple to evaluate and is remarkable in that, although it does not require an assumption of independence between terminal prices, no knowledge of covariances is necessary. The paper is organized as follows. In the first section we discuss Lo s [1987] result and how it may be interpreted geometrically as the result of a domination of the payoff function by 1

3 Page a parabola. We also look at a recent extension of this work by Grundy [1991]. We proceed with a discussion on terminology across various fields where other versions of this upper bound appear in the univariate case. We close with a discussion on what we mean when we refer to tightness and sharpness of bounds. Section presents the main result of this paper. We begin with a general discussion on the theory behind bounding convex functions such as the payoff function at hand. The multivariate upper bound appears as a proposition along with the conditions for tightness and sharpness of the bound. The complete derivation and demonstration of the result is relegated to the appendix. We close the section with a discussion on how the parameters needed for evaluating the bound are connected to the actual stochastic processes that may govern their prices through time. The performance of the bound is studied in Section 3. Here we use the exact results available from Stulz [198] for the case when n =to test the performance of the bound. We close with a summary and conclusions and we present ideas for future work. The appendix is intended to collect all the relevant mathematics required in the derivation and demonstration of the main result. 1. The univariate case. The semi-parametric bound proposed in Lo [1987] applies to the terminal payoff function of a call option of the form P(S) =max {S K;0}, (1.1) where S is the terminal price of the risky asset and K is the strike price. If the (conditional) mean µ and variance Ṽ of the price at maturity given today s price of the risky asset are known then an upper bound for the expectation of the terminal payoff of the function P is given by: ( ) 1 µ K+ (K µ) + Ṽ if K µ +Ṽ ; µ E[P(S)] ( µ( µ K)+ µ Ṽ ) / ( µ + Ṽ ) if K < µ +Ṽ µ. Note that the only knowledge about the terminal distribution required is the terminal mean µ and variance Ṽ. Therefore the bound is valid when the expectation is taken with respect to any distribution sharing these two attributes. This bound belongs to a family of bounding results for expectations of sublinear functions (i.e. positively homogeneous: graph is a v-shaped cone, see Rockafellar [1970, pg. 30])

4 A multidimensional semi-parametric upper bound... Page 3 which trace their origins to Scarf s article in [1958]. It is derived by showing that it corresponds to the expectation of a quadratic function that dominates the payoff function. The upper bound is possible for the special case of the payoff function P(S) since it is sublinear. Therefore, there exists a family of dominating quadratic functions that are supported by the payoff function at precisely two points. From among the elements of this family, there is one parabola which attains the minimum expectation with respect to all distributions with the same mean µ and variance Ṽ. Moreover, the two contact points between the parabola and the payoff function are the atoms of the range of a discrete random variable. It is with respect to the distribution of this two-atom random variable that the expectation of P(S) constitutes an upper bound for all distributions (see proofs in appendix in Lo [1987] and in Proposition 4 in Grundy [1991]). The occurrence of the variance in the expression makes this a second-order semi-parametric bound. Semi-parametric bounds such as the one discussed here are known by several other names. In some areas of applied probability they are called minimax bounds since they represent the smallest of all bounds guaranteed to be greater than or equal to all possible values for the actual expectation or probability. Another name is distribution-free for obvious reasons. Perhaps the most famous and useful semi-parametric, second-order, bound is the familiar Tchebycheff s Inequality (for this reason yet another name for these results is Tchebycheff-type bounds). Second-order upper bounds for the univariate case appear in other applications. Examples include: a class of univariate convex functions for which a second-order upper bound for their expectation with respect to a given discrete, two-atom, distribution can be established in Dulá [1987]; a direct application of Scarf s results for generating second-order upper bounds in stochastic optimization in Dupačová [1966]; and second-order upper bounds applied in the area of utility theory in the work by Willassen [1990]. A final point regarding the theory of bounding in probability regards the concept of tightness and sharpness of a bound. A bound is tight if it satisfies an optimization criterion usually defined by a set of analytic conditions (e.g., a mathematical program). A bound is sharp when it is actually attained by one of the distributions satisfying the conditions. As may be verified in Lo s and Grundy s proofs, the second-order semi-parametric bound on the expectation of the payoff function of a call option in the univariate case is both tight and

5 Page 4 sharp. Note that sharpness implies tightness but the converse is not necessarily true. (see Dulá [1987] for more details).. The multivariate case. The call option with payoff in (1.1) is generalized for the multidimensional case by the complex European call option on the maximum of n risky assets. We can characterize the payoff of a complex European call option on the maximum of n risky assets at maturity using the following function: P(S 1,...,S n )=max {max{(s 1 K 1 ),...,(S n K n )}; 0} ; (.1) where S 1 0,...,S n 0 are the prices of the risky assets at maturity and the K 1,...,K n the individual strike prices on the corresponding option. The random variable S =(S 1,...,S n ) has a terminal multivariate nonnegative distribution F (S) conditioned on today s prices for the assets which we do not have to know. For the purposes of our development, it is sufficient to know the expectation of the n terminal expected values, µ 1,..., µ n ; and variances Ṽ 1,...,Ṽn, of the prices of the assets. The fundamental approach to generating upper bounds on the expectation of a function consists of two steps: (i) find a dominating function over the same domain (preferably one for which it is easier to find its expectation); and (ii) evaluate the expectation of the dominating function with respect to the given distribution; this value is an upper bound on the expectation of the original function. The use of quadratic functions in this approach provides an immediate advantage; namely, there is no need for multivariate integration since the expectation of a quadratic function with respect to any distribution can be evaluated directly when the first and second-order moments of the distribution are known. Moreover, this expectation is the same for all distributions with the same first and second moments. The fact that there usually is an entire class of quadratic functions that satisfy the condition in (i) indicates that a selection criterion may be specified to identify a single element implying the necessity for some sort of optimization procedure. The result is a tight upper bound since the optimization is over a class of dominating functions. Note that when using quadratic functions to dominate the payoff function we can choose to exclude certain second-order information such as the covariances by limiting the class of dominating quadratics to those without cross terms.

6 A multidimensional semi-parametric upper bound... Page 5 The problem of dominating a function is a problem in functional approximation and is, in general, difficult. However, the special characteristics of the payoff function of a complex European call option permit domination by a quadratic function over its entire domain. This is the case since P(S) isconvex, polyhedral, and sublinear. How this plays a role in the determination of the upper bound will be discussed in the derivation and demonstration of the main result which we present in the following proposition. The result we introduce in the proposition below is an extentsion of the work by Dulá [199] and Dulá & Murthy [199]. The paper by Dulá [199] presents a closed-form multivariate Tchebyscheff upper bound for simplicial sublinear functions where the information required is the marginal means and the total sum of marginal variances. This result was generalized to the case when individual variances are known in Dulá & Murthy [199]. However, no closed-form formula is available and the bound follows the solution of a constrained nonlinear program. In both papers, the upper bounds are derived using results from moment problem theory and semi-infinite programming. A major contribution of these two works is to demonstrate that tight and sharp upper bounds exist based on the solution to special moment problems where the usual assumption of compactness (e.g., Karr [1983] and Kemperman [1968]) of the range of the random variable is not required. The demonstration for the result in the present paper uses different arguments. Moment problem theory plays no role; rather, a more intuitive geometric arguments are used which can be easily applied to future extensions and generalizations. The literature on bounding the expectation of convex functions focuses on upper bounds because this is the difficult case, the case of a lower bound can be handled efficiently with Jensen s Inequality (a discussion on the relative merits of lower and upper bounds can be found in Kall [1988]).. Proposition. Let P(S 1,...,S n ) be the payoff function presented in (.1) with S 1,...,S n the terminal prices of the n underlying risky assets. Let µ 1,..., µ n and Ṽ1,...,Ṽn be the expectations and variances, respectively, of the terminal prices of the assets. Then, an upper bound on the expected payoff at maturity over all return distributions for the n-variate, nonnegative random variable S is given by the following inequality: E F [P] 1 n ( ) µ i K i + Ṽ +( µ Ki ), (.)

7 Page 6 This bound is valid independently of the nature of the terminal distribution of the payoff function provided the parameters exist. However, there is a region in the domain of the payoff function P for which the bound is tight and sharp. This information is given in the following table: K i Ṽi + µ i µ i The bound is tight. where i = argmin { β i + K i ; i =1,...,n} Additionally, the bound is sharp if the simplex defined by the n +1 (affinely independent) contact points contain the vector ( µ 1,..., µ n ). K i < Ṽi+ µ i µ i The bound is never tight. Notice that evaluation of the bound does not require information on cross moments of the distribution. This is due to a deliberate design choice for the family of dominating quadratics. The expression in (.) can be derived directly. However, any derivation would be incomplete if it does not provide the requisite conditions for tightness and sharpness that must accompany the introduction of a new probabilistic bound. The extensive development needed to obtain and demonstrate such conditions require an extensive development a presentation for the result in the proposition We have relegated this discussion to the appendix. We can see how this bound generalizes the result in one dimension by setting n =1to obtain E F [P] 1 ( ) µ K+ Ṽ +( µ K). The condition for this bound to be tight from the table above requires that K Ṽ + µ. The µ upper bound as presented in Lo and Grundy has a second expression corresponding to the case when K < Ṽ + µ. The absence of this second case in our proposition does not invalidate µ the bound; rather, it means that the bound is no longer tight (and sharp) if the second

8 A multidimensional semi-parametric upper bound... Page 7 condition on the exercise price K from the table is satisfied. In one dimension this case corresponds to when the left contact point between the optimal dominating parabola and the payoff function is negative. In Lo s upper bound, the expression corresponding to this second case is derived by forcing the left contact point between the dominating parabola and the payoff function to be the origin. This is a situation which may not be easily extended to the multidimensional case and is left open for future work. The parameters µ 1,..., µ n and Ṽ1,...,Ṽn needed to evaluate the upper bound are the means and variances of the conditional, marginal price distributions of the individual underlying risky assets at maturity. Since no knowledge of cross-moments (e.g. covariances) is required, we may analyze individual marginal distributions to find the values for the parameters needed. For a discussion on how to obtain or estimate numerically the parameters needed to evaluate the upper bound when the governing stochastic process for each asset is a lognormal diffusion processes or mixed diffusion-jump process the reader is directed to Lo [1987] and Grundy [199]. The semi-parametric, second-order, multivariate, upper bound presented in the proposition above offers one obvious advantage; namely, it is easy and straight-forward to evaluate. For the univariate case, the upper bound offers advantages over calculating the actual price of the option. These are obvious if the distribution of the terminal price of the underlying risky asset is not fully specified. Even if it is possible, in some cases, to apply an exact pricing formula, to price the option directly is a laborious task. However, this task quickly becomes onerous as the dimensions of the random variable increases whereas the numerical requirements of the upper bound in the proposition are essentially trivial in any number of dimensions. 3. Testing and performance of the bound. A result by Stulz [198] provides an opportunity to test the performance of the bound. Stulz discussed in detail complex European call options on the maximum and minimum of two risky assets. He derived closed form expressions for the value of these complex call options directly based on the assumption of a lognormal diffusion process for each of the two underlying risky asset and a coefficient of correlation which is constant through time. The results by Stulz are generalized (and corrected) for the case of n risky assets by Johnson

9 Page 8 [1987]. The evaluation of this expression requires laborious calculations including n + 1 evaluations of the cumulative of an n-variate normal distribution. We have used the closed form expression for the bivariate case by Stulz/Johnson to test the upper bound. It is required that the value for the upper bound be discounted according to a given riskless interest rate r if the comparison with Stulz result is to be consistent. Our results are based on a rate r =.1 Figure 1 is a graph comparing the actual value and the upper bound over a wide range of the sums of variances Ṽ1 + Ṽ for the case when µ 1 = µ =1and K =1.1. The figure is selected as representative of the relation between the bound and the actual value. It can be seen that the upper bound performs well for low variances. Figure 1 shows that the performance is adversely affected as the sum of the variances increase; however, from Figure wesee that the percent difference between the bounds decreases in rate with the variance; that is, the difference does not explode as the sum of the variances increases. 4. Summary and conclusions. We have explored the geometry of the payoff function for a European call option on the maximum on n risky assets to reveal that it is convex with a conical graph. These attributes permit the design of a second-order upper bound on its expectation with respect to the distribution of the prices at maturity requiring only information on the distribution s first and second moments. The special properties of the payoff function permit a closed-form, analytic expression for the upper bound. The upper bound presented in this paper is interesting for a variety of reasons. First, duly discounted, and when the mean and variance of the terminal marginal distributions of the individual underlying risky assets are available, it is the basis for rationally the corresponding option. Also, as noted by Lo, this upper bound yields a measure of the impact of misspecifying the stochastic price process. Finally, a bound may be the only numerical value possible especially if information on the random vector of asset prices at maturity is limited to expected values and variances of prices at maturity. An idea for improving the upper bound is to introduce individual parameters for each of the n squared terms (instead of a single coefficient, as in this paper) in the definition of family of dominating quadratics. This allows the possibility to stretch and contract the

10 A multidimensional semi-parametric upper bound... Page 9 dominating quadratic independently along each axis of the space. The consequence will be abetter bound since the conditions on the quadratic are less restrictive. Another obvious idea for improving the upper bound is to incorporate information on the cross moments; i.e. covariances. Again, the additional information about the distribution permits more flexibility in the design of an optimal quadratic which dominates the payoff function since this permits the rotation of the main axes of the ellipsoidal level sets. At this point it is not clear from our investigations whether such upper bounds will have a simple, closed form, formula; rather, it appears that they may require the solution of a convex optimization problem. Finally, an important extension of our work in this paper is to derive a tight and, hopefully, sharp bound when the n + 1-st optimal contact point between the dominating quadratic and the payoff function is in the negative orthant of the space. This should provide an expression n for the bound in the case when the strike price for the option is such that K < (Ṽi+ µ ) i n. µi

11 Page 10 Appendix: Demonstration and development. In this appendix we present the results that lead to the final expression for the upper bound on the expectation of the payoff function given in (.1). We begin by investigating the geometry of this function. The payoff function, P, ispiecewise linear and convex because it is the pointwise supremum of a set of linear functions in R n (see Rockafellar [1970], Thm. 5.5, p. 35). It is also sublinear in the sense that the graph of the function in R n+1 is the boundary of a polyhedral cone with vertex at the point (K 1,...,K n ) R n. Moreover, it is simplicial because its conical graph has the smallest possible number of sides for a polyhedral cone with n +1dimensions; namely, n +1. The function P is defined over n +1 regions in R n. Denote the kth unit vector in R n as e k (i.e. e k i =0;i k and e k k = 1), the vector of all 1 s in R n as u, and K =(K 1,...,K n ) R n. The n +1regions are: K +pos(u, e k ; k =1,...,n, k j), for j =1,...,n; C j = K +pos( e k ; k =1,...,n), if j = n +1. where the vector operation pos is the positive cone of the vectors in the list; that is, all linear combinations of the vectors using only nonnegative multipliers. The regions C j ; j =1,...,n+1 are simplicial cones in R n.ifthe point Ŝ is in region Cj then P(Ŝ) =Ŝj, except when the point is in region C n+1 ;inwhich case the point yields a payoff value of zero. Therefore, an equivalent and, as we will see, more convenient characterization for P is as follows: n e j i (S i K i ) if S C j ; j =1,...,n; P(S) = (A.1) 0 if S C n+1. This form for P reveals that the gradient of the function at any point in the interior of any of the regions C j ; j =1,...,n+1is e 1 if S C 1 ;.. P(S) = e n if S C n ; (A.) 0 if S C n+1.

12 A multidimensional semi-parametric upper bound... Page 11 The bound for the expectation of the terminal payoff of a complex European call option on a basket of n risky assets expressed by the inequality in (.) is an upper bound because the right-hand side of the inequality is the expectation of a special quadratic function which dominates the payoff function. This special quadratic function is a member of a class of quadratic functions specifically designed to have the property that they share the same value as the payoff function at precisely n + 1contact points. Therefore, at these points the gradients of the two functions must also be the same. All such quadratic functions necessarily dominate the payoff function and are necessarily strictly convex. Before we proceed we need to clarify that all expectations are with respect to any distribution in R n that has the same vector of means µ 1,..., µ n and variances Ṽ1,...,Ṽn as the original terminal distribution F. We will denote this class of distributions by F. Note that the random variable which induces F takes on only nonnegative values. However, we may consider F and element of F if we let the range of the random variable be the entire space and consider its density outside of the nonnegative orthant to be zero. We will address directly how the actual nonnegativity of the random variable affects our results later on. We begin our demonstration by defining analytically the class of quadratic functions which dominate the payoff function by intersecting it at precisely n +1 points; one in the interior of each of the regions C j ; j =1,...,n+1. Consider the following generic form for a quadratic function in R n : Q(S) =d 0 + n n d i (S i K i )+ δ i (S i K i ). (A.3) The quadratic functions Q(S) defined by the values for the parameters d 0,d 1,...,d n,δ 1,...,δ n for δ i 0; i are full-bodied paraboloids with ellipsoidal level sets. If, in addition, δ i > 0; i, then these are strictly convex. The use of the translated argument S i K i in the expression will facilitate the algebraic manipulations in the development. Our first task is to define values for d 0,d 1,...,d n,δ 1,...,δ n such that the resultant class of quadratic functions will satisfy two conditions: for n +1points, S 1,...,S n+1, one in each region C j of R n (i) P(S j )= Q(S j ); S j C j ; j =1,...,n+1;and (ii) P(S j )=Q(S j ); S j C j ; j =1,...,n+1

13 Page 1 The first condition generates the n +1relations K, for j =1,...,n; êj S j = 1 + K, for j = n +1. (A.4) where d = (d 1,...,d n ). Notice how for each j, S j C j. Substitute the expressions for S j in (A.4) in the relations in (ii) to obtain the n +1 conditions on the parameters d 0,d 1,...,d n,δ 1,...,δ n n (d k e 1 d 0 k) =0, 4δ k k=1. n (d k e n d 0 k) =0, 4δ k k=1 (A.5) d 0 n k=1 d k 4δ k =0. This system of equalities restricts the set of values for d 0,d 1,...,d n,δ 1,...,δ n to those that define quadratic functions Q(S) which will satisfy conditions (i) and (ii) above. There are n + 1relations and n + 1variables meaning that more than one solution is possible. Note that the strict convexity of the payoff function implies that δ i > 0; i. Otherwise, we arrive at a contradiction since d i =0implies that the graph of Q would contain a line which cannot be contained in a pointed cone and d i < 0 means negative values for Q would be possible and therefore it could not dominate the (strictly convex) payoff function over an unbounded domain. 1 This result requires some algebraic manipulation. Here, as in other places in this appendix, derivations requiring extensive algebra, as well as some proofs, have been left out but are fully detailed in the Annex for the purpose of facilitating the review process. Any occurrence of the symbol followed by a number corresponds to a special discussion in the Annex. Refer to item inthe Annex for the details on how to arrive at this result.

14 A multidimensional semi-parametric upper bound... Page 13 Given that there may be a multitude of solutions to the system of equations for the variables corresponding to the parameters of Q we may apply an optimization criterion over this set. Our objective is to find the function from the family of quadratic functions represented by Q in expression (A.3), the parameters of which satisfy the relations in (A.5) such that it generates the minimum expectation, in a sense soon to be specified, with respect to any distribution in F. Recall that all distributions in this set are such that the mean is the vector ( µ 1,..., µ n ) with marginal variances Ṽ1,...,Ṽn. Define β i =E F [ (Si K i ) ] = ( Ṽ i +( µ i K i ) ). (A.6) Then βi is the marginal second moment of any distribution in F around the point K in R n. The optimization criterion we employ corresponds to finding values d 0,d 1,...,d n,δ 1,...,δ n satisfying the equalities in (A.5) such that the function n n f(d 0,d 1,...,d n,d n+1 )=d 0 + d i ( µ i K)+ δ i βi (A.7) is minimized. The choice of this objective function is a design decision. 3 The optimal solution to the optimization problem defined by this objective function subject to the n + 1 conditions in (A.5) are values d 0,d 1,...,d n,δ1,...,δ n which define a quadratic function denoted by Q. This optimal quadratic function is the one from among those with form Q as in (A.3) with a minimum expectation with respect to all distributions with mean ( µ 1,..., µ n ) and second moments around the point K. Solving the optimization problem with the constraints in (A.5) and objective function in (A.7) we get that any solution is such that d 1 = d = = d n = 1.From this we obtain that, d 0 = 1 n δ i which reduces the problem to a continuous optimization over δ 1,...,δ n. 4 Solving we obtain that δi 1 =, and replacing this value in the expression for 4β i;,...,n d 0 3 Refer to item 3 inthe Annex for the details on how to arrive at this result.

15 Page 14 we have that d 0 = 1 n β 4 i. 4 When we replace these values for d 0,d 1,...,d n,δ1,...,δ n in (A.3) we obtain the quadratic function Q (S) = 1 n β i n n (S i K i )+ 1 4β i (S i K i ). (A.8) We may verify the following regarding the quadratic function Q 5 1. The optimal contact points between Q and P are: ξ 1 β 1 + K 1 S j =. ; for j =1,...,n; and S n+1 = β 1 + K 1. ; (A.9) ξ n β n + K n β n + K n where the coefficient associated with the ith component, ξ i ; i =1,...,n, is defined as follows: { 1, if k j; ξ k = 1, if k = j At these points the conditions P(S j )= Q (S j ) and P(S j )=Q (S j ); S j C j ; j =1,...,n+1are satisfied. 6. The expectation of Q with respect to any distribution in F yields the upper bound: 7 E[Q ]= 1 n ( µ i K i + β i ) Since Q (S) P(S) for all S R n then E[Q ] E[P] with respect to any distribution in F, the upper bound is established. Note that this expression corresponds directly with the right-hand side in (.). 4 Refer to item 4 inthe Annex for the details on how to arrive at this result. 5 Refer to item 5 inthe Annex for the details on how to arrive at this result. 6 Refer to item 6 inthe Annex for the details on how to arrive at this result. 7 Refer to item 7 inthe Annex for the details on how to arrive at this result.

16 A multidimensional semi-parametric upper bound... Page 15 The necessary Kuhn-Tucker first order optimality conditions for the optimization problem where the objective function in expression (A.7) is minimized subject to the constraints in (A.5) state that there exist multipliers π j ; j =1,...,n+1such that: 8 n+1 j=1 π j = 1; n (e k 1 d 1 ) δ 1 π i d 1 δ 1 π n+1 = ( µ 1 K 1 ) n (e k n d n ) δ n π i d n δ n π n+1 = ( µ n K n ). (A.10a) (A.10b) n (d 1 ek 1 ) 4δ 1 n (d n en i ) 4δ n π i + (d 1 ) π 4δ1 n+1 = β1 π i + (d n ) π 4δn n+1 = βn where d 0,d 1,...,d n,δ 1,...,δ n are the optimal values.. (A.10c) Let us analyze these n +1conditions on the multipliers π1,...,π n+1 and verify that our values are actually optimal. The n conditions (A.10c) are always satisfied if we replace the optimal values for d 0 and δ1,...,δ n and apply (A.10a). 8.5 The n conditions in (A.10b) can be rewritten as: ( n e i k d ) ( ) k + K δk k π i + d k + K δk k π n+1 = µ k, k =1,...,n; and, replacing the optimal values for d i,δ i ; i =1,...,n: n (ξ i β k + K k )π }{{} i +( β k + K k )π }{{} n+1 = µ i, k =1,...,n; (A.11) S i k S n+1 k where ξ i is as before. Expression (A.11) is the condition that the the vector of means µ =( µ 1,..., µ n ) can be expressed as a linear combination of the n +1 optimal contact 8 Refer to item 8 inthe Annex for the details on how to arrive at this result. 8.5 Refer to item 8.5 inthe Annex for the details on how to arrive at this result.

17 Page 16 points S j ; j =1,...,n+1(see expression (A.9)). The condition that the multipliers add up to 1 means further that this linear combination is affine; that is, the vector of means must belong to the affine hull of the n + 1optimal contact points. However, the contact points between the payoff function and any quadratic satisfying the conditions in (A.5) are necessarily affinely independent since they belong to the interior of each of the n + 1regions C j ; j =1,...,n+1inR n. Therefore, the affine hull of these n +1points is the entire space and the vectors of means is trivially in their affine hull. This along with the fact that the last condition (A.10c) isanidentity permits us to conclude that the solution we have proposed satisfies the Kuhn-Tucker conditions. Since the optimization problem of minimizing f in (A.7) subject to the constraints in (A.5) is a convex program these optimality conditions are also sufficient. Suppose that the vector of means actually belongs to the convex hull of the n +1contact points S j ; j =1,...,n +1. Then, the multipliers π j ; j =1,...,n +1 are nonnegative. Since, in addition, they are required to add to 1 by (A.10a), they constitute a proper discrete probability probability measure of a random variable the range of which is the n + 1contact points. Denote this random variable by X and its (discrete) distribution F. Due to the special structure of the contact points we may solve for π j ; j =1,...,n+1. Thus 9 π i = 1 ( 1+ µ i K i β i ) ; k =1,...,n; (A.1) π n+1 = n n µ i K i β i. This gives us the probability distribution of X. Notice that E F [S] = µ and E F [(Si K i ) ]=βi. 10 Therefore the distribution of X belongs to the class F of distributions. The 9 Refer to item 9 inthe Annex for the details on how to arrive at this result. 10 Refer to item 10 in the Annex for the details on how to arrive at this result.

18 A multidimensional semi-parametric upper bound... Page 17 expectation of the payoff function with respect to the distribution of X is: 11 n+1 E F (P) = P(S j )π j = 1 j=1 n (β i + µ i K i ) (A.13) which is exactly the value of E F [Q ]. With this result it is established that if the vector of means is in the convex hull of the n +1 optimal contact points S j ; j =1,...,n+1, then there exists a (discrete) distribution in F with respect to which the upper bound is attained (i.e., sharpness). The supports of this distribution are the optimal contact points with probability mass the corresponding values of the Kuhn-Tucker multipliers. The upper bound we have developed is demonstrably tight with respect to all distributions in F. However, the terminal prices of the assets cannot be negative and therefore the range of the random variable S is the nonnegative orthant of R n. Imposing a nonnegativity condition on the random variable does not affect the tightness of our upper bound if the resultant optimal dominating quadratic function Q happens to intersect the payoff function P at n + 1contact points which are in the nonnegative orthant. In this case the nonnegativity condition is redundant. To see this note that we would arrive at a contradiction if the optimal dominating quadratic function, when the nonnegativity condition is imposed, is not Q since then this other quadratic function would have a smaller expectation with respect to any distribution with the given first and second-order attributes defined over the entire space. The nature of the contact points makes it possible to determine when they are nonnegative. Notice that all the components of the optimal contact points S j are nonnegative whenever min i { β i + K i } 0(see expression (A.9)). Replacing the β i with the value in (.3) we have that this condition is equivalent to 1 K i Ṽi + µ i (A.14) µ i where i = argmin { β i + K i ; i =1,...,n}. This condition is verified expo facto and it depends intrinsically on the nature of the data of the problem. If this condition is not 11 Refer to item 11 in the Annex for the details on how to arrive at this result. 1 Refer to item 1 in the Annex for the details on how to arrive at this result.

19 Page 18 satisfied then the bound is no longer tight when nonnegativity of the random variable is an explicit condition. 5 The bound remains valid nonetheless since the quadratic Q in (A.8) still dominates P everywhere on R n.atight upper bound under nonnegativity for the case when the condition in (A.14) is not satisfied appears possible and it is the topic for future research. Finally, if in addition to satisfying the condition in (A.14) for tightness under nonnegativity, the vector of means belongs to the convex hull of the optimal contact points S j ; j = 1,...,n+ 1, then the bound is sharp attaining its minimum value at the random variable with range the n +1 optimal contact points and with measures given by the values of π j ; j =1,...,n+1in expression (A.1).

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21 Page 0 Scarf, H., [1958], A min-max solution of an inventory problem, in Studies in the Mathematical Theory of Inventory and Production, K.J. Arrow, S. Karlin, and H. Scarf, eds., pp , Stanford University Press, Stanford, CA. Stulz, R. [198], Options on the minimum or the maximum of two risky assets: analysis and applications, Journal of Financial Economics, Vol. 10, pp Willassen, Y., [1990], Best upper and lower Tchebycheff bounds on expected utility, Review of Economic Studies, Vol. 57, no. 3, pp

22 A multidimensional semi-parametric upper bound... Page 1 Footnotes. 1 For adiscussion on applications of European call options on the maximum of n risky assets the reader is directed to the paper by R. Stulz [198, especially Section 4]. Here it is stated that A wide variety of contingent claims of interest to financial economists have apayoff function which includes the payoff function of a (European) put or call option on the minimum of maximum of two risky assets. Although the presentation in the paper is intended to illustrate applications for the case when the number of underlying risky assets is n =,itisclear that the models can be generalized to any number of risky assets. The payoff function is not differentiable at the points on the boundary of any of the cones C j. However, our demonstrations and discussions will be limited only to points in the interiors of these cones. Therefore, we henceforth exclude boundary points from our discussion. 3 Other functions are certainly possible, however, it turns out this function is particularly well-suited for our purposes especially since it results in the upper bound that generalizes Lo s result directly. If we limit ourselves to second-order information, we may try excluding the mean (as in Grundy [1991] for the univariate case), or incorporate information about the covariances. Note that in the case of covariances the expression for the family of feasible quadratic functions will include cross terms. It is not known what other choices for objective function yield a final closed-form formula for the upper bound. 4 Notice the strict convexity of the resultant n-dimensional multivariate problem when δ i > 0; i implying that the original optimization problem was a convex program with a linear objective function. 5 although it remains tight for the unconstrained problem.