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«Werner Hiirliann Allgemeine Mathematik Konzernbereich Leben WGR Paulstrasse 9 CH-8401 Winterthur - SWITZERLAND Telephone : +41 52 261 58 61 and ...»

INEQUALITIES FOR THE EXPECTED VALUE OF AN EXCHANGE

OPTION STRATEGY

Werner Hiirliann

Allgemeine Mathematik

Konzernbereich Leben WGR

Paulstrasse 9

CH-8401 Winterthur - SWITZERLAND

Telephone : +41 52 261 58 61

and

Schonholzweg 24

CH-8409 Winterthur - SWITZERLAND

Abstract.

Two exchange option strategies are considered, one of which does only depend on the

marginal distributions of the underlying sequence of random variables. Their expected total financial payoffs are compared under various assumptions, including stochastic ordering ones.

While some results are closely related to earlier prophetlgambler comparisons in gambling theory, a lot of new inequalities are presented. In particular some new sharp inequalities by known marginal means and variances are formulated.

Kewords :exchange option, bivariate dependence, Hoeffding-Frechet extremal distributions, stop-loss transform, stochastic orders

1. Introduction.

Let the random variables X,, X I,..., X, represent the unknwon future prices at future times t o, t,,.., t, of some specific goods trated in a financial market. Suppose an agent operates according to the following exchange option strategy. At present time he agrees that at each future time t,, i = 0,..., n - 1, he writes an option to exchange at time t,,, a specific good at price X, for another specific good at price X,,,, whose financial payoff at time t,,, equals (X,,, - X, )+.Neglect or suppose the X,'s are interest adjusted quantities.

Then the agent wants to know as much as possible about the total financial payoff Z:=,(X, - X,_,)+ as viewed from present time, say about the expected value E[S,] of S, = this exchange option strategy. As alternative exchange option strategy the agent could agree at present time that at each future date t,, he writes an option to exchange at time t,,, a good at the price X, for another good at the expected price p,,, = E[X,+,], whose financial payoff at time t,,, equals (p,,, - X,),. Let the total financial payoff of this alternative x:=, 01, - X,_,)+ From a statistical point of view, the expected value strategy be T, =.

E[T" ] depends only on the marginal distributions of X,,X I,...,X,, while E ~ S], depends additionally on the dependence between X,_, and X,. It is thus of general interest to study 1.

the variation of E[s,] given the value of E[T, The considered exchange option strategies have a nice interpretation in game and gambling theory, a topic developedby Dubins and Savage(1965) and for which one can recommend the recent textbook by Maitra and Sudderth(l996) for background. Let us follow Krengel and Sucheston(l987), from which we have extracted the general inequalities (2.4) and (2.12) in Theorems 2.1 and 2.2. By definition a prophet is a player with complete foresight and a gambler knows only the past and the present, but not the future. Suppose both bet on differences of consecutive non-negative random variables X, such that E[X; Ix,_,] = E[x,], where the players are supposed to multiply their stakes by uniformly bounded random variables. Then the expected payoff E[S,] - corresponds to the expected gain of the prophet while E[T,] is the optimal expected reward of the gambler. In this situation Theorem 2.1 says that the expected gain of the prophet is at most three times that of the gambler, where the constant three is optimal.

The main point, where we differ from Krengel and Sucheston(l987), is in the innovative application of their results to Finance. Furthermore, in our context, we work under less restrictive assumptions and we formulate some new properties like (2.5), (2.13) and (2.14). Potential applications lie in Asset and Liability Management, including those suggested in Hiirlimann(l997a), Section 5.

2. Ineaualities bv known means.

By knowledge of only the means, we investigate the relationship between the expected values E[S, ] and E[T, ] as defined in Section 1. For a random variable X with survival function F(x), the stop-loss transform of X is denoted by n(x) = E[(x- x),] = /:~(t)dt, where the last equality is obtained through partial integration.

Lemma 2.1.

If ab then one has the inequalities

–  –  –

- This follows directly f?om the stop-loss transform definition :

Proof.

and

Lemma 2.2. If E[X] = p, then the following inequalities are satisfied :

–  –  –

z 2 a,, and p, p,, hence(2.11) follows. 0 Remark 2.1. In case the X, 's are non-negative random variables, there exist an absolute upper bound for E[T,,] in terms of the means. For each i=O,...,n-I, suppose X, has 1, support [0, b, bi r p,,,. It is not difficult to see that X,,,,= Z,, where Z, is diatomic with

–  –  –

It would be helphl to know under which assumptions the ratio of E[S,] to E[T,,] can be lowered. A simple answer, which will be precised in Corollary 3.1, is as follows.

–  –  –

Then (2.9) reads which implies (2.12) by means of (2.6). Clearly (2.13) follows from Lemma 2.3, and (2.14) follows from (2.13) if one notes that by equal means the increasing concave order reduces to the concave order, which turns out to be equivalent to a reversed convex order or stop-loss order as stated in Definitions 2.1. 0 On the other side, one knows that the constant three is optimal in (2.4) and (2.5), in the sense that there exist random variables for which the bounds are nearly attained. With a slight simplification, recall the argument by Krengel and Sucheston(l987), Section 4.





–  –  –

From the mentioned result, one knows that The equality is attained provided equality is attained for each summand. The latter holds for a bivariate diatomic extremal random pair (X,,X,_,) with support

–  –  –

Furthermore the inequality is sharp and attained by diatomic random variables with the following properties. For each i=O,...,n, the support {x,,y, } of X, is given by

–  –  –

As a consequence, this best upper bound allows us to show that the constant two in Theorem 2.2 is optimal in case of equal means, and attained by diatomic random variables.

This result contrasts with the optimal constant three in Theorem 2.3,which is only nearly attained by a non-degenerate random vector.

–  –  –

Suppose further that X, = X, with probability one, then one has equality in (2.14), that is Moreover E[s,], E[T,] are maximal among all random variables with equal means and known variances.

–  –  –

4. Ineaualities bv known means. medians and variances.

Let X, Y be non-negative random variables with marginals F(x), G(x probability function H(x,y). The indicator function of a set {.} is denoted by I{.

identity one derives, by taking expectations, the expected positive difference formula where the bar denotes survival functions. By Hoeffding(l940) and Frechet(l951), the following extremal bounds hold, where H., H * are themselves distribution functions, and H(x,y) belongs to the space BD@,G) of all bivariate pairs (X,Y) with given marginals F(x), G(x). It follows that from which divers bounds for E[(X - Y),] can be obtained. In fact, the applied method allows to determine bounds for expected values of the more general type E [ ~ ( x - Y)], where f(x) is any convex non-negative function, as observed by Tchen(1980), Corollary 2.3.

As a general result, the same method allows to determine, under some regularity assumptions, bounds for expected values of the general type E[f(X,Y)], where f(x,y) is either a quasimonotone (sometimes called superadditive) or a quasi-antitone right-continuous knction (cf Lorentz(l953), Whitt(1976), Cambanis, Simons and Stout(1976), Tchen(1980), Cambanis and Simons(1982)). First, some immediate but practical results are presented. A more detailed analysis is postponed to Sections 5 and 6.

Pro~osition4.1. (Upper bounds by known medians and mean) Let X, Y be non-negative random variables with medians m x, m y, and let the mean p y of Y exists. Then one has the inequality

–  –  –

and (4.6) follows from (4.2). If X 6,, Y then (4.7) follows similarly. 0 Pro~osition4.2. (Lower bounds by known means) Let X, Y be non-negative random variables with finite means. Then one has the inequality If Y I,, X then one has the best lower bound

–  –  –

If there is an increasing uncertainty reflected by an increasing riskiness in the X,'s, for example X, 5,s X, 5,... 5, X,, then (4.6), (4.9) imply the simpler bounds where the last inequality follows by the inequality of Bowers(1969). Similarly, if there is a decreasing uncertainty reflected by a decreasing riskiness in the X,'s, for example X, t, t XI 2,..., X,, then (4.7) and (4.10) imply the bounds

5. Ineaualities by known maminals.

Let X, Y be non-negative random variables with marginals F(x), G(x), and joint probability function H(x,y). Our aim is a detailed analysis of the upper bound which follows from (4.2) and (4.4). To evaluate (5.1) two cases are distinguised.

–  –  –

which implies that It is remarkable that both upper bounds coincide.

Theorem 5.1.

(Minimax property o the Hoeffdg-Frechet upper bound) Let X, Y be nonf negative random variables with distribution~ F(x), G(x), and suppose the means p,, p, of X, Y exists. Then one has

–  –  –

The following result has been shown Theorem 6.1. Let X, Y be non-negative random variables with finite means P,, F,, and standard deviations o x, o,. Then the distribution-free upper bound for the expected positive 1, difference E[(x - Y), given by the Hoeffding-Frechet upper bound (6.3), is determined in

tabular form as follows :

–  –  –

Remark 6.1.

As shown by the author(l997b), generalized versions of the minimax Theorem

5.1 as well as of the Theorem 6.1 can be formulated for random variables X, Y with arbitrary ranges. The similar proofs are a bit more technical and require the distinction between several more cases and subcases. For pedagogical reasons, only the simplest extension of the bivariate inequality of Bowers in Hiirlimann(l993) has been presented here.

References.

Black, F. and M. Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy 81, 637-59. Reprinted in Luskin(1988).

Bowers, N.L. (1969). An upper bound for the net stop-loss premium. Transactions of the Society of Actuaries MX, 21 1-16.

Cambanis, S., Simons, G. and W. Stout (1976). Inequalities for E[~(x,Y)] when the marginals are fixed. Zeitschrift fir Wahrscheinlichkeitstheorie und verwandte Gebietd Probability Theory and related Fields 36, 285-94.

Cambanis, S. and G. Simons (1982). Probability and expectation inequalities. Zeitschrift f i r Wahrscheinlichkeitstheorie und verwandte Gebietd Probability Theory and related Fields 59, 1-25.

Cox, J.C., Ingersoll,.and S. Ross (1985). A theory of the term structure of interest rates.

I Econometrica 53,385-467.

Dubins, L.E. andL.J;Savage (1965). Inequalities for Stochastic Processes (How to Gamble if you must). McGraw-Hill, New York. 2nd edition (1976), Dover, New York.

Frichet, M. (1951). Sur les tableaux de correlation dont les marges sont donnbs. Ann. Univ.

Lyion Section A 14, 53-77.

Hoeffding, W. (1940). Massstabinvariante Korrelationstheorie. Schriften des Math. Instituts und des Instituts fiir Angewandte Mathematik der Universitat Berlin 5, 179-233.

Hiirlimann, W. (1993). Bivariate distributions with diatomic conditionals and stop-loss transform of random sums. Statistics and Probability Letters 24, 329-35.

Hiirlimann, W. (1995). A stop-loss ordered extremal distribution and some of its applications.

XXVI. ASTIN Colloquium, Leuven.

Hurlimann, W. (1997a). A unified approach to some loss variance bounds. Appears in Bulletin of the Swiss Association of Actuaries.

Hiirlimann, W. (1997b). Bounds for expected positive differences by means of Hoeffding Frechet extremal distributions. XXVIII. ASTIN Colloquium, Cairns, Australia.

Kaas, R., Heerwaarden, Van A.E. and Goovaerts, MJ.(1994). Ordering of Actuarial Risks.

CAIRE Education Series 1, Brussels.

Krengel, U. and L. Sucheston (1987). Prophet compared to gambler : an inequality for transforms of processes. The Annals of Probability 15(4), 1593-99.

Lorentz, G.C. (1953). An inequality for rearrangements. American Mathematical Monthly 60, 176-79.

Luskin, D.L. (1988) (Editor). Portfolio Insurance : a guide to Dynamic Hedging. John Wiley.

Maitra, A.P. and W.D. Sudderh (1996). Discrete Gambling and Stochastic Games.

Applications of Mathematics, Stochastic Modelling and Applied Probability, vol. 32. Springer.

Shake4 M. and J.G. Shanthikumar (1994). Stochastic Orders and Their Applications.

Academic Press, New York.

Tchen, A.H. (1980). Inequalities for distributions with given marginals. Annals of Probability 8, 814-27.

Vasicek, O.A. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics 5, 177-88.

Whitt, W. (1976). Bivariate distributions with given marginals. Annals of Statistics 4, 1280-89.





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