Imr~ - HBS People Space Model of Warrant Pricing.pdf · lating a priori knowledge of a and . P, the...

bull Imr~ Winter 1969 Volume 10 Number 2

Avery R Johnson Organization Perception and Control in Living Systems

Paul A Samuelson A Complete Model of Warrant Pricing 17 Robert C Merton that Maximizes Utility

David T Kleinman

Robert Tannenbaum Sheldon A Davis

Marvin J Cetron Harold F Davidson

Structuring Capital Markets for 47 Efficient Use of Foreign Aid The Case of Brazil

Values Man and Organizations 67

MACRO RampD 87

Biographies 101

Book Reviews 107

Books Received Winter 1969 111

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Table of Contents 0INDUSTRIAL MANAGEMENT Massachusetts Institute REVIEW ASSOCIATION of Technology Alfred P Sloan Cambridge Massachusetts School of Management 02139

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David L Diamond Lawrence W Garrett Robert L Gipson Anthony C Taylor

Susan L Singer

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A Complete Model of Warrant Pricing that Maximizes Utility

Paul Samuelson Massachusetts Institute of Technology Robert C Merton Massachusetts Institute of Technology

This paper presents a realistic theory of warrant prices which overcomes some deficiencies or oversimplifications of an earlier model developed by Paul A Samuelson and published in the Spring 1965 issue of the IMR The analysis here is very complex and can be extended beyond warrant pricing to other types of securities Further elaboration of the theory and its application is supplied in two appendices which follow the article Good luck Ed

Introductiont

In a paper written in 1965 one of us developed a theory of rational warrant pricing) Although the model is quite complex mathematically it is open to the charge of overmiddot simplification on the grounds that it is only a firstmiddotmoment theory2 We now propose to sketch a simple model that overcomes such deficiencies In addition to its relevance to warrant pricing the indicated general theory is of interest for the analysis of other securities since it constitutes a full supply-andmiddotdemand determination of the outstandmiddot ing amounts of securities

Cash-Stock Portfolio Analysis

Consider a common stock whose current price Xt will give rise n periods later to a finite-variance multiplicative probability distribution of subsequent prices Xt+n of the form

Prob Xt+n S X IXt = Y = P(X Y n) = P(Xy n) (1)

where the price ratios Xt+nfXt = Z = ZI Zz bullbullbull Zn are assumed to be products of unimiddot formly and independently distributed distributions of the form

Prob ZIS Z = P(Z 1) and where for all integral nand m the ChapmanKolmogorov

relation P(Zn+m) = j P(Zzn)dP(zm) is satisfied This is the geometric Brownian

motion which at least asymptotically approached the familiar log-normal Ignoring for simplicity any dividends we know that a risk averter one with concave utility and diminishing marginal utility will hold such a security in preference to zero-yielding safe cash on Iy if the stock has an expected positive gain

0lt E[Z]- 1 = 1000

ZdP(Zn)- 1 = eCCQ_l that iSa gt 0 (2)

where the integral is the usual Stieltjes integral that hand les discrete probabilities and densities and a is the mean expected rate of return on the stock per unit time (We have ensured that a is constant independent of n)

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-Aid from the National Science Foundation is gratefully acknowledged tFootnotes for this article appear at the end of the article

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IMR Winter 1969

A special case would be for n 1 the following discrete distribution where gt 1

XH1 X with probability p ~ a X+l = -1X with probability 1 - P ~ 0 (3)

This simple geometric Brownian motion leads asymptotically to the log-normal distri shybution Condition (2) becomes in this special case 0 lt E[Z] 1 = px + (1 p)X-1-l If for exampleA= 11 and p= 1 P = 05 then E[Z] -1 = 12(11+ 1011) 004545 If our time units are measured in months this represents a mean gain of almost one-half a per cent per month or about 52 per cent per year a fair approximation to the recent performance of a typical common stock

To deduce what proportion cash holding will bear to the holding of such a stock we must make some definite assumption about risk aversion A fairly realistic postulate is that everyone acts now to maximize his expected utility at the end of n periods and that his utility function is strictly concave Then by portfolio analysis3 in the spirit of the classical papers of Domar-Musgrave and Markowitz (but free of their approximations) the exshypected utility is maximized when w = w where w is the fraction of wealth in the stock

Max O(w) = Max (U[(1- w) + wZ]dP(Z n) (4) w w Jo

where w = w is the root of the regular condition for an interior maximum

o 0 (w) aOOZU f [(1- wmiddot) + wZ] - U [(1 - w) + wZl dP(Z n) (5)

or

1 =it ZU [(1 - w) + wZ]d P(Zn)

fooo U[ (1- w)+ wZ]dP(Zn)

Since U is a concave function Umiddot is everywhere negative and the critical point does correspond to a definite maximum of expected utility (Warning Equations like (4) posit that no portfolio changes can be made before the n periods are up an assumption modified later)

If zero-yielding cash were dominated by a safe asset yielding an instantaneous force of interest r and hence e in n periods terms like (l-w) would be multiplied bye and (5) would become

fooo ZU [(I-wmiddot)e rn + wZ]dp(Zn) ern = lt ectn ifwgtO

foOlgt U [(1-wmiddot)e +wmiddotZjdP(Zn) (5a)

This relationship might well be called the Fundamental Equation of Optimizing Portfolio theory Its content is worth commenting on But first we can tree it from any dependence on the existence of a perfectly safe asset Re-writing (4) to involve any number m of alternative investment outlets subject to any joint probability distribution gives the multiple integral

(01) m MaxU[wh wm]= MaxJo U [kwjZJdP(ZIZmn) (4a) Wj Wj 1- 1

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Samuelson and Merton Complete Model of Warrant Pricing

19

m m Introducing the constraint Ewj I into the lagrangian expression l=O +)[1 EwJ

i=1 j=1

we derive as necessary conditions for a regular interior maximum4

for k= 1bullbullbullbullbull m

Dividing through by a normalizing factor we get the fundamental equation

1000 = 1000

ZldQ(ZIbullbullZmn) =1000

Z2dQ(ZIo Zmn) = bull ZmdQ(ZIo bull Zmn) (5b)

where m

U [Z wjZj]dP(Z bullbull Zmn) l = 1

dQ(ZIo bullbull Zm n)

The probability-cum-utility function Q(Zn) has all the properties of a probability distrishybution but it weights the probabHity of each outcome so to speak by the marginal utility of wealth in that outcome

Figure 1 illustrates the probability density of good and bad outcomes Figure 2 shows the diminishing marginal utility of money and Figure 3 plots the effectivemiddotprobability density whose integral Ioz dQ(zn) defines Q5 Conditions (5) (5a) and (5b) say in words that the effective-probability mean of every asset must be equal in every use and of course be equal to the yield of a safe asset if such an asset is held Note that 0 (0) E[l] - em= e OC

- em and this must be positive if w is to be positive Also 0 (1) 0 ZdQ(Z n) - em and this cannot be positive if the safe asset is to be held in positive amount By Kuhn-Tucker methods interior conditions of (5) could be generalshyized to the inequalities needed if borrowing or short-selling are ruled out

For the special probability process in (3) with p =lh and Bernoulli logarithmic uti lity we can show that expected utility turns out to be maximized when wealth is always divided equally between cash and the stock ie wmiddot= lh for all A

Max O(w) = Max ilog(1-w +WA)+ log(1-w + WA-1)

w w = logO + A) + logO +A-i) for aliA (6)

The llaximum condition corresponding to (5) is

0= 0 (w) = -- (- 1 +A)+ t (-1 + A-i) and (7) + A t + t A-1

wmiddot == t for all A QED

(fhe portfolio division is here so definitely simple because we have postulated the special case of an unbiased logarithmic price change coinciding with a Bernoulli logarithmic utility function otherwise changing the probability distribution and the typical persons wealth level would generally change the portfolio proportions)

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IMR Winter 1969

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3


21

Recapitulation of the 1965 Model

Under what conditions will everyone be willing to hold a warrant (giving the right to buy a share of the common stock for an exercise price of $1 per share at any time in the next n periods) and at the same time be willing to hold the stock and cash Since the warrants price will certainly move with the common rather than provide an opposing hedge against its price movements if its expected rate of return were not in excess of the safe assetmiddots yield the warrant would not get held In the 1965 paper it was arbi trarily postulated that the warrant must have a specified gain per dollar which was as great or greater than the expected return per dollar invested in the common stock Thus if we write Y~(n) for the price at time t of a warrant with n periods still to run the 1965 paper assumed for stock and warrant

E[X+TXt ] = eaT~etT (8a)

E[Yt+T(n - T)yt(n)] = efjT~ eaT if the warrantis to be held (8b)

In (8b) we recognize that after the passage of T periods of time the warrant has nmiddotT rather than n periods left to run until its exercise privilege expires It should be stressed that the warrant can be exercised any time (being of American rather than European option type) and hence in (8b) the warrant prices can never fall below their arbitrage exercise value which in appropriate units (ie defining the units of common so that the exercise price of the warrant is unity) is given by Max (OXt-1) Thus we can always convert the warrant into the common stock and sell off the stock (commissions are here neglected)

In the 1965 model the expected percentage gain fj of a warrant and the expected pershycentage gain a of a common were arbitrarily postulated as exogeneously given data instead of being deduced from knowledge of the risk aversion properties of U Postumiddot lating a priori knowledge of a and P the model was derived by beginning with the known arbitrage value of a warrant about to expire namely

(9)

Then if the warrant is to be held we can solve (8b) for Yt(l) = F1(X) from the equation

ef1 = E[Fo(XZ)F1(X)IX]

10 Fa (XZ)dP(ZI)

F1(X) (10)

In this integral and elsewhere we can write X for Xt bull If (10) is not achievable the warrant will be converted and will now be priced at its Fo(X) value Hence in every case

F1(X) = e-P10 Fo(XZ)dP(ZI) if held

= X-I ~ e-P10 Fo(XZ)dP(ZI) bull if now converted

= Max[OX-1 e-fj 10 Fo(XZ)dP(ZI)] in alt cases (lOa)

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IMR Winter 1969

Successively putting in these expressions F2 and Fl for FI and Fobullbullbullbullbull Fe+1 and Fe for FI and Fo the 1965 model deduced rational warrant price formulas Fn(X) = Fn(X) = Y(n) for any length of life and the important perpetual warrant case F (X) = F(X) can be deduced by letting n-gtoc

F (X) = e-3 10 F(XZ)dP(ZI) if X s C (a (3)

oo

= X-I e-3lo F(XZ)dP(ZI) if X C (a (3) (11)

where C(a (3) is the critical level at which the warrant will be worth more dead than alive This critical level will be defined by the above relations and will be finite if 3 gt a 6

The special case of the 1965 theory in which a=3 is particularly simple and its math ematics turns out to be relevant to the new utility theory presented here In this case where conversion is never profitable (for reasons which will be spelled out even more clearly in the present paper) the value of the warrants of any duration can be evaluated by mere quadrature as the following linear integrals show

Fn(X) = e- T1000

Fo_T(XZ)dP(ZT)

= e-laquon 1000

Fu(XZ)dP(Zn)

e- n 1x (XZ-1)dP(Zn) (12)

In concluding this recapitulation let us note that the use of short discrete periods here gives a good approximation to the mathematically difficult limiting case of continuous time in the 1965 paper and its appendix

Determining Average Stock Yield

To see how we can deduce rather than postulate in the 1965 manner the mean return that a security must provide let us first assume away the existence of a warrant and try to deduce the mean return of a common stock The answer must depend on supply and demand supply as dependent upon risk-averters willingness to part with safe cash and demand as determined by the opportunities nature affords to invest in real risky processes along a schedule of diminishing returns

To be specific suppose one can invest todays stock of real output (chocolates or dollars when chocolate always sell for $1 each) either (a) in a safe (storage-type) process-cash so to speak-that yields in the next period exactly one chocolate or (b) in a common stock which in the special case (3) gives for each chocolate invested today Achocolates tomorrow with probability p or A-1 chocolates with probability 1-p If we allocate todays stock of chocolates so as to maximize the expected utility we shall shun the risk process unless its expected yield exceeds unity For the special case7 p = I-p Y2 this will certainly be realized and as seen in the earlier discussion of (7) for al A a Bernoullimiddot utility maximizer will chooseto invest half of present resources in the safe (cash) process and half in the risky (commonmiddotstock) process


23

Now suppose that the risky process - say growing chocolate on the shady side of hills where the crop has a 5 chance of being large or small- is subject to diminishing reshyturns With the supply of hill land scarce the larger the number of chocolates planted rather than merely stored the lower the mean return per chocolate (net of any competi shytive land rents for which the limited supply of such land will be bid to at each level of total investment in risk chocolates) Although it is admittedly a special-case assumpshytion suppose that A in (3) drops toward unity as the absolute number of chocolates invested in the risky process rises but that p = 1-p = V2 throughout Then the expected yield a = eOlt -1 drops toward zero aSA drops toward one

Given the initial supply of chocolates available for safe or risk allocations the expected yield of the common stock a will be determined at the equilibrium intersection of total supply and demand in our simple case at the level determined by the A and a yields on the diminishing returns curve where exactly half of the available chocolates go into the risk process8

Determining Warrant Holdings and Prices

Using the general method outlined above we can now deduce what warrants must yield if a prescribed amount of them is to be held alongside of cash and the common stock by a maximizer of expected utility

Specifically assume that cash in an insured bank account or a safe process has a sure yield of er-l per unit time Assume that each dollar invested in the common stock has a mean ex-ante yield foooZdP(Z 1)-1 e -I per period It will be desirable now to specialize slightly our assumption of concave total utility so that the behavior of a group of investors can be treated as if it resulted from the deliberation of a single mind In order that asset totals should behave in proportions independent of the detailed allocashytions of wealth among individuals we shall assume that every person has a constant elasticity of marginal utility at every level of wealth and that the value of this constant is the same for all individuals9 Just as assuming uniform homothetic indifference curves frees demand curve analysis in non-stochastic situations from problems of disaggregation a similar trick comes in handy here

Finally we must specify how many of the warrants are to be outstanding and in need of being voluntarily held There is a presumption that to induce people to hold a larger quantity of warrants their relative yields will have to be sweetened Let the amounts of total wealth W to be invested in cash common stock and warrants be respectively Wit Wi and W3 As already seen there is no loss of generality in setting W = 1 Then subject to the constraintto WI + W2 + W3 = W = I we consider the following special case of (4a) and generalization of (4)

- 000 Fn(XZ)Max U[w W2 W3] = Max 0 U [wle rT + wZ + WI (X) ]dP(XT) (13)wJ WI n+T

where as before we assume that the decision is made for a period of length T (Setting T = 1 a small period would be typical) To explain (13) note that eU is the sure return to a dollar invested in the common stock Since we can with $1 buy IFb -f1(X) units of a warrant with n+T periods to go and since these turn out after T periods to have the

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IMR Winter 1969

random-variable price F (XZ) clearly Wa is to be multiplied by the per-dollar return F (XZ)Fn+T(X) as indicatedl As in (4a) we seek a critical point for the Lagrangian

3

expression L = 0 + )[1-~ wi1 to get the counterpart of (5b) namely 1-1

(00 Fn(XZ)Jo efT U [we rT + w2Z + W3 F+T(X) ]dP(Z T)

(14)

J w~middot) erT + WImiddot Z + Wagtt FD+I(X)1 dP(ZT)

where we have the normalizing factor

(00 F (XZ)

(15)

(16)

C = U [(1-W2middot shyo so that as in (5b)

U [(1 WI - Wamiddot) efT + WtmiddotZ + Wi ] dP(ZT) dQ (ZT) = -----------~=---

C

If the Wj were prescribed - eg as the solution to a simultaneous-equation supply and demand process that auctions off the exogeneously given supplies of common stock and warrants at the prices that will just get them held voluntarily12 - then for T = 1 (16) would become an implicit equation enabling us to solve for the unknown function FA+I(X) recursively in terms of the assumed known function F(X) Since Fo(X) is known from arbitrage-conversion considerations (16) does provide an alternative theory to the 1965 firstmiddotmoment theory

Let us now call attention to the fact that the implicit equation in (16) for F+T(X) can be enormously simplified in the special case where the number of warrants held is small Thus for wmiddot == 0 or nearly so the dependence of U [middot1 on F +TCX) becomes zero or negligible and (16) becomes a simple linear relationship for determining FD+T() reo bull cursively from F(middot) If wmiddot = 0 (15) and (16) become

erT = (OO ZU [(1 - wImiddot)efT + w2middotZ1dP(ZT) (15a)Jo c


25

erT = fo ~~(~~ U [(1 - w2middot)erT + w2middotZ]dP(ZT) (16a)

c

Our task will thus be simplified when we specify that the number of warrants to be held is small that is warrant pricing is to be determined at the critical level just necessary to induce an incipient amount of them to be voluntarily held This is an interesting case because it is also the critical level at which hedging transactions involving buying the common and selling a bit of the warrant short just become desirable)3 Most of our paper will be concerned with this interesting incipient-warrant case based on (15a) and (16a) but we will first digress briefly to show how one might deduce the quantitative level of all Wjmiddot in terms of given supplies of the various securities

Digression General Equilibrium Pricing

To illustrate how warrants would have to be priced if their exogeneously given supply is to be absorbed voluntarily by utility maximizers it suffices to consider the simplest case of one-period warrants that are available in a fixed amount V And let us assume for expositional simplicity that diminishing returns (eg in connection with the chocoshylate-growing hillsides above) operate so slowly that we can take the probability distri shybution of common-stock price changes as exogeneously given with P(ll) given and the common stocks expected yield a known parameter ea Assume that the present common stock price is known to be at the level X= x Also let the amount of the safe asset (money or near-money) be prescribed at the level M and with a prescribed safe yield er being a parameter of the problem

We can now deduce for utility maximizers the equilibrium values for the unknown number of shares of common stock held S and the unknown equilibrium pattern of warrant prices F1(x) Our equations are the balance sheet identities definitions and supply conditions

W = M 1 + Sx +VFI(x)

= M(1 +w2 + _1_-_-=-----) (17)WI WI

(18)

(19)

and also our earlier equations (15) and (16) with T = 1 and n = 0

foOD lU [WIer + Wtl + (1 shy WI shy w) F~~ ]dP(l1) er=~~--------------------------~~-------

C

(15b)

(OD Fo(xl) U [Wier + wtZ + (1 - WI shye = Jo FI(x)

c

w) Fo(xl)]dP(l1) F(x)

(16b)

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IMR Winter 1969

Equations (19) (15b) and (16b) are independent equations for the three unknowns wmiddot W2 and F1(x) Hence we do have a determinate system14 When V -gt 0 we have the simpler theory of the rest of this paper

Utility-Maximizing Warrant Pricing The Important Incipient Case

After our digression we go back to equation (16a) rearranging its factors to get for T I

F n+(X) e-r10 F(XZ)dQ(Z 1) where dQ(Z 1) is short for (20)

U [(1 - wt)er + w2Z]dP(Z1)dQ(Zl rw2) f

Jo U [(1 - w~)er + w2Z]d P(Z 1)

Here W2 is a parameter already determined from solving (15a) and indeed is precisely the sllme as the Vi determined earlier from solving equation (5a) It will be recalled that Q(Z 1) is a kind of utilmiddotprob distribution Precisely because of (15a) we know that the expected value of Z calculated not in terms of the true objective probability distribution dP(Z 1) but rather in terms of the utilmiddotprob distribution dQ(Z 1) has a yield per unit time exactly equal to that of the safe asset Rearranging (14a) we have

100010 ZdQ(ZI) = er lt eIX ZdP(ZI) (21)

Taken together with the initial condition from (9) Fo(X) = Max (OX-l) equations (20) and (21) give us linear recursion relationships to solve our problems completely proshy

vided we can be sure that they always yield Fn(X) values that definitely exceed the conversion value of Fo(X) Because of (21) we are here in a mathematical situation similar to the 1965 special case in which a = 13 and indeed no premature conversion is ever possible But of course there is this significant difference in the 1965 case d P rather than dQ is used to compute IX and 13 and to emphasize this we write a = ap = Jp for that case in the present case where dQ is used in the computation we write aQ and 3Q recognizing from (21) that aQ = r and from (20) thatJQ = r = aQ lt ap a The aQ and I3Q yields are purely hypothetical or subjective they should not be identified with the higher objective ap and (Jp yields computed with actual probability dP These are the true ex ante expected percentage yields calculated from actual dollar gains and losses they are objective in the sense that Monte Carlo experiments replicated a large number of times will within this probability model characterized by P(ZI) actually average out ex post with mean yields of ap and (Jp on the common stock and warrants respectively15

The mathematics does not care about this dP and dQ distinction The same kind of step-by-step algorithm is yielded whatever the interpretation of the probability distrimiddot bution used But this new approach does raise an awkward question In the 1965 paper it could be taken as almost selfmiddotevident that conversion can never be mandatory if both warrant and stock have the same ex ante yield In this case where the yields calculated -with dQ(ZI) are of a hypothetical kind it is desirable to provide a rigorous proof that our new theory of warrant pricing never impinges on the inequalities set by arbitrage as discussed above and in the 1965 paper


27

If we are assured of non-conversion the value of a perpetual warrant can be determined from the linear integral equation (20) For n so large that it and n+1 are indistinguishshyable we can write

and (20) becomes

F(X) = e-ro F(XZ)dQ(ZI) (22)

Substituting F(X)== X into (22) does turn out to provide a solution So too would cX but only for c = 1 can we satisfy the two-sided arbitrage conditions Xgt F(X) gt X - 1

Actually the homogeneous integral equation (22) has other solutions of the formcXn where substitution entails

cXm = e-rcXm0 ZmdQ(Z 1)

1 = e-r oZmdQ(ZI) = fgt(m) (23)

This last equation will usually be a transcendental equation for m with an infinite numshyber of complex roots of which only m 1 is relevant in view of our boundary conditions16

That our new theory leads to the perpetual warrant being priced equal to the common stock may seem paradoxical just as in the 1965 special case where ap = (Jp We shall return to this later

Explicit Solutions

In a sense our new theory is completed by the step-by-step solution of (20) In the 1965 theory however it was possible to display explicit formulas for non-converted warrants by quadrature or direct integration over the original Fo(X) function The same procedure is possible here by introducing some further generalizations of our util-prob distribution Q(ZI)

There are some by-no-means obvious complications in our new theory Given the quadrature formu la

Fl(X) = e-ro Fo(XZ)dQ(ZI) (24)

one is temped at first to write as would be possible in the 1965 case where dP reo placed dQ

F2(X) = e-2r0 F o(XZ)dQ(Z 2)

(25) or in general

F(X) e-nr0 Fo(XZ)dQ(Z n) (26)

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IMR Winter 1969

where as in (5b) we define

U [Wtmiddotern + w2middotZ1dP(Zn)d Q(Z n) = 7--------=-------------=----=--------shy

fo U [wlmiddotern + w2middotZ1dP(Zn)

But these relations are not valid They would be valid only if say in the case n = 2 we locked ourselves in at the beginning to a choice of portfolio that is frozen for both periods regard less of the fact that after one period has elapsed we have learned the outcomes of X t+1 and by (20) would want to act anew to create the proper Wimiddot proporshytions for the final period (For example suppose as in (7) we have U = 10gW and there is an equal chance of the stocks doubling or halving with h = 2 p = 12 = 1-p Suppose we put half our wealth into cash at the beginning and freeze our portfolio for two periods Then we are violating the step-by-step solution of (20) if after we have learned that the stock has doubled we do not sell-out half our gain and put it into cash for the second period)7 In summary (25) is not consistent with (24) and

F2(X) = e-rfo Ft(XZ)dQ(Z 1) (27)

If direct quadrature with Q(Zn) is not valid what is What we need are new iterated integrals Q2(Z) bullbull QIl(Z) which reflect the compound probabilities for 2 n periods ahead when the proper non-frozen portfolio changes have been made Rather than derive these by tortuous economic intuition let us give the mathematics its head and merely make successive substitutions Thus from (20) applied twice we get

FIl+2(X) = e-rfo FIl+(XZ)dQ(Z 1)

= e-rfo [e-ro FIl (XZV)dQ(VI)ldQ(ZI)

= e-2r 000 Fn[X(ZV)ld fo Q[ (~V) 1]dQ(ZI)

= e-2r000 FIl(XR)dQ2(R) (28)

where

roo RQI(R) =Jo Q (Z 1)dQ(Zl)

and where the indicated interchange in the order of integration of the double integral can be straightforwardly justified

This suggests defining the iterated integrals18 by a process which becomes quite like that of convolution when we replace our variables by their logarithms namely relations like those of Chapman-Komolgorov

QI(Z) === Q(ZI) by definition

QI(Z) = f QI (~) dQI(V) ~ Q(Z2)

-(29)


29

Then by repeated use of (28)s substitutions the results of the stepbymiddotstep solution of (20) can be written in terms of mere quadratures namely

FI(X) e-r 10 Fo(XZdQI(Z)

F2(X) = e-2r10 Fo(XZ)dQ 2(Z)

(30)

Fortunately the subjective yields aQ and (3Q calculated for the new generalized utilprob functions Qt(Z) do all equal r per unit time That is wecan prove by induction

10 ZdQI(Z) = e lt ea

1000 ZdQ2(Z) = e2

10 ZdQ(Z) = en (31)

This is an important fact needed to ensure that the solutions to our new theory never fall below the arbitrage levels at which conversion would be mandatory

Warrants Never to be Converted

It was shown in the 1965 paper that for13gta and 13 a constant the warrants would always be converted at a finite stock price level We will show that in the present model with its explicit assumption of no dividends the warrants are never converted (ie F (X) gt Fo(X) )19

Theorem If fo ZdQ(Z) e and F (X) e- fo Fo(XZ)dQ(Z)

then F(X) ~ Fo(X) == Max(O X-I)

and we are in the case where the warrants need never be converted prior to expiration

Since Fo(X) ~ X-I it is sufficient to show that

X-I s e-m10 Fo(XZ)dQ(Z) is ltgt(Xr) (32)

holds for all rgt 0 n gt 0 and Xgt O We show this as follows bull ltgt(Xr) ~ e-m10 (XZ - I)dQ(Z) because Fo(XZ) ~ XZ - 1 and dQIl(Z) ~ 0

~ xe-Iooo ZdQ(Z) em

e-rD~ X - ~ X 1 from (31) for all r ~ 0 n gt 0 and X ~ O

Therefore (32) holds and the theorem is proved

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IMR Winter 1969

Thus we have validated the step-by-step relations of (20) or the one-step quadrature formula of (30)

As an easy corollary of this theorem we do verify that longer life of a warrant can at most enhance its value ie bull F+I(X) ~ Fn(X)

For from the theorem itself FI(X) ~ Fo(X) and hence

F2(X) 10 FI(XZ)dQ(ZI) ~fo Fo(XZ)dQ(ZI) = FI(X)

And inductively if F(X) ~ FI(X) for all t ~ n it follows that

Fn+l(X) = 10 F(XZ)dQ(Zl) ~Ioltraquo Fn_I(XZ)dQ(Zl) = F(X)

If Q(ZI) gt 0 for all Zgt 0 and Q(Z 1) lt 1 for all Z lt co we can write strong inequalities Fn+l (X) gt FIl(X) gt Fn_ 1 (X) gt bullbullbull gt FI(X) gt Fo(X)

The lognormal case belongs to this class If however as in example (3) Q(ZI) = 0 for Z lt A- lt 1 and for Z gt A gt 1 Fl(X) will vanish for some of the same X values where Fo(X) vanishes Fl(X) will equal (X - 1) = Fo(X) for large enough X values

Hence our weak inequalities are needed in general However for n large enough and X fixed we can stili write the strong inequality namely FIl+l (X) gt Fn(X) for n gt n(X)

The crucial test is this If for a given X one can in T steps end up both above or below the conversion price of I then FT(X) gt Fo(X) and Fn+T(X) gt Fn(X) Also if Fn(X) gt Fo(X) for a particular X F+T(X) gt F (X) for that X

Exact Solution to the Perpetual Warrant Case

We now shall show that the stationary solution to (30) F(X) i5iE X20 is indeed the limit of the finite-duration warrant prices as n-+oo From (30)

F(X) e-m 10 Fo(XZ)dQ(Z)

== e-m fi-oltZ - I)dQn(Z)

X101 (1 - XZ)dQn(Z)] r~

e-rn [= e-m10 (XZ - I)dQ(Z) - flx o dQ(Z) (33) 10 dQ(Z)

== X - e-m + e-m 81(Xn)8(Xn) from (31)

But 181(X n)1 S 1 for i = 12 So as n -+ 00 r gt 0

F(X) = limit F(X) gt= X n-+oo

Thus the result is shown for r gt O For r - 0 the proof is similar and follows closely the proof on page 23 of the 1965 paper For r gt= 0 (30) becomes

bull

I


31

Fn(X) = i7x (Xl l)dQ(l)

= X - 1 + iJ1(X n) iJ 2(X n as before

X 101 ZdQ(Z)

1 - ollX dQn(l)

X 0 17 ZdQ(Z)limIt iJ1(Xn) = 1 - 1 n-gtoo 10 x dQ(Z)

1- 0 because Q(O+X) = 1 (34)

121for precisely the same reasons that P(O+Xoo) = Similarly

limit iJ2(Xn) = 101 x dQ(Z) = 1 because Q(O+X) 1

n-gtoo

Therefore

limit FI1 (X) = X - 1 + limit iJ)(Xn) iJ 2(Xn) n-gtoo n-gtoo

=X-1+1 X

So the result is shown for r = O

Admittedly our new theory has arrived at the same paradoxical result as the special case of the 1965 theory namely that a perpetual warrant should sell for as much as the common stock itself Such a result would seem empirically bizarre In real life pershypetual warrants generally do sell for less and since the common stock is equivalent to a perpetual right to itself at zero exercise price one would have thought it would dominate a perpetual warrant exercisible at $1 Indeed one of the purposes of the general 1965 theory was to construct a model that would keep perpetual warrants down to a price below the common

What is there to do about the paradox First one can recognize that the common stock may be paying dividends now or can be expected to pay dividends at some time in the future Therefore the analysis presented in Appendix B may be deemed appropriate and this will serve to dispel the paradox Second one might have thought that dropping the WImiddot = 0 incipient case would dispel the paradox But such a guess would seem to be erroneous since wmiddot gt 0 is compatible with having a warrant price like F00(X) = X because the variance of a perpetual warrant and the common stock are the same Finally we may dispel the paradox by accepting it as prosaic If a stocks mean gain is almost certain to rise indefinitely above the exercise price in the distant futUre and that is what a gt 0 implies why should not the $1 exercise price be deemed of -negligible percentage importance relative to the future value of the common (Recall too that the $1 is not paid now but only after an infinite time) Hence why should not the perpetual warrant sell for essentially the same price as the common And if people believe this will be the case it will be a selfmiddotfulfilling belief (If most people doubt this the person who believes in it will average a greater gain by buying warrants)

32

IMR Winter 1969

Illustrative Example

Now that the general theory is complete it is of interest to give a complete solution in the easy case of the binomial process with Bernoulli utility as was described in (3) where Xgt 1

Xt+l = XX with probability p i

Xt+1 = X-IX with probability 1 - P = i (3a)

and the Bernoulli logarithmic total utility function UW) = 10gW We further assume the yield on cash is zero (Lebull r = 0) and the mean yield of the common stock l+a is

1 + a Hgt- + X-I) (35)

The utility maximum equation corresponding to (13) for T = I is

- f F (XX) 1 -I F(X)X-I) Max U = Max l Iog[wl + wX + Wa F (X)] + lilog[wl + wX + wa F (X)]Wi Wi +1 +1 (36)

Since we already know that WI- = W2- = Ih is optimal for wa- imposed at zero from the previous analysis of (6) and (7) the firstmiddotorder conditions corresponding to equations (14) reduce to a single equation

+ iX - F(X)) + p-I- F(XX-I) 0= F+z(X) + FO+1(X)

i+igt- i+X-I (37)

Solving for the warrant prices corresponding to (20) we have

F+l(X) = (1 + X)-lF (XX) + (1 + X-l)-lF(XX-l) (38)

We have previously shown that the arbitrage conditions imposing premature conversion are not binding Therefore (38) and the initial condition

Fo(X) = Max [OX - 1] (39)

are sufficient to determine the warrant prices

The coefficients in (38) can easily be interpreted by our new notion of the util-prob function They are dQs discrete probabilities (qh q l) corresponding to the original d P discrete probabi lities (Ph P -tgt = (O being related by

ql == P1U( + X1)[P_IU(t + V-I) + P1U + IX)]

111 - H + X ] [ H +X -I ) + ( + X )]

= (1 + )1]1 for i - +1

As in the 1965 paper we convert (38) into a standard random-walk stochastic process by means of a logarithmic or exponential transformation in which X = )t k = logxX It will suffice for an example to consider only integer values of k Finally write F(X) = Ft Then (38) becomes the familiar partial difference equation22 of the classical random walk Ft+1 q IFt+1 + q 1Ft-h ql + q-l == 1 (40)

bull


k

-4 -3 -2 1 0 1 2 3 4 n

0 0 0 0 0 0 1000 2100 3300 4600

t 1 0 0 0 0476 1000 2100 3300

2 0 0363 0476 1250 2100

t 3 0363 0684 1250

4 0684

Table I

Table I illustrates in the familiar form of Pascals triangle calculation of the warrant prices for our special case The arrows in the table illustrate the step-by-step

calculations thus FI(1) = FIOI) = FOI is for X = 11 calculated as Fo = (1 11)

FlO = 0476 and Fu = Fa(11) is calculated as FI3 (1) F12 + ( 11) F02 = 1250

From Table I we calculate

ql = _1_ = 4762 q_1 = = 5238 21 21

Note that there are several re-occurring patterns within the table which are not due to the particu lar choice of X For example in the k = 0 column successive odd and even entries repeat themselves FOI = Fo2 F03 = FDbullbullbullbullbull FObull2n+1 = Fobull2n+l for all X

What is the profitability of holding the warrant as against hold ing the common or holding cash We can compute this from our table using the actual d P probabilities of (1f2 1f2) Thus the outcomes Fo(X+1) that emerge from buying Fl(l) have a mean yield of i (21) + i (0)-1 = 05 per cent per month

This turns out to be a higher actual yield than the postulated a = 04545 per cent per month of the common stock (We are here speaking of actual C(p and flp yields and not of the hypotheticalaQ = rand flQ = r yields referred to in earlier sections) One can easily verify from any other entry in the table that in every case the warrants fl yield exceeds the fixed a yield of the common Indeed from the general formulas for any and not just for = 11 one finds fl gt a Thus to find the mean yield from buying a lperiod warrant at Xt = 1 at the rational price FI (1) for any Xgt I we calculate from (38) the price FI (1)

FI(l) = (1 + X)-lFo() + (I + -1)-lFoX-1)

_x-1 -+1 +0

bull

34

IMR Winter 1969

Our mean gain per dollar b is

E [ F((Xt+21I X = IJ = (x - 1) 1 f (1) t (1 +X)1 (x---=-I)

X-l=b

X-I + X-I for Xgt I -2-gt = a from (35)

or b gt a and f3 gt lX

Is this a surprising finding When one reflects that the warrant has higher volatility than does the common it would seem intuitively reasonable that they should have to afford a higher yield than the common if they are to be held in the same portfolio Moreshyover since the degree of volatility can be expected to vary with the price of the common and the duration of the warrant there is no a priori reason to expect that the actual (l should be a constant instead it is reasonable to expect that it must be written as a function of X and n namely(l(X n)

Actually this expectation that 3(Xn) gt a which was based on our illustrative case and on a priori reasoning turns out to be true for even the most general case In the next section by means of an important lemma we shall prove the above inequality Of course in the limit when the perpetual warrant approaches the value of the common stock the divergence 3(Xn) -lX will go to zero as n-+oo

Proof of the Superiority of Yield of Warrants Over Yield of Common Stock

First we wish to state an important lemma upon which this proof and other results rest Proof of this lemma and indeed of a wider lemma of which this is a special case is releshygated to Appendix A Broadly speaking what we wish to show is that if two perfectly positively correlated securities are to be held in the same portfolio with the outcome of one being a monotone-increasing function of the other but with its possessing greater volatility in the sense of its elastiCity with respect to the other exceeding one the mean yield of the volatile security must exceed the mean yield of the less volatile one

We define the elasticity of the function ~(V) with respect to V Eit in the usual fashion as

d(log~) V~I(Y)

E~=d(logV) it(Y)

Although we work here with functions possessing a derivative this could be dispensed with and be replaced by working with finite-difference arc elasticities

Lemma (a) Let gtIT (Y) be a differentiable non-negative function whose elasticity E~ is strictly greater than one for a II V E (0 GO )

(b) Let v(V) be a positive monotone-decreasing differentiable weighting function shy(Le v(V) gt 0 v(V) lt 0) and dP(V) be a probability distribution function over nonshynegative V such that its cumulative distribution function must grow at more than one positive point (so that P(Y) takes on at least three positive values for positive Vs)


35

If 000 I(Y)v(Y)dP(Y) 000 Yv(Y)dP(Y)

Then 000 I(Y)dP(Y) gt It YdP(Y)

With this lemma we can then proceed to state and prove the following theorem

Theorem If Fu(X) is generated by the process described in equations (20) and (21) or in (29) (30) and (31) and if the actual yield (I(Xn) is defined by

efI(xn) 000 Fn(XZ)Fn+1(X)dP(ZI) then for all finite n (I(Xn) gt a

Now writing Fn(XZ)Fn+1 (X) = I(Z) we must show that I has the properties hypothe sized by part (a) of the lemma ie I 0 and pound1 gt 1 Clearly I(z) 0 and even more because Fn is an increasing function of its argument I(Z) gt 0 for all Z gt O From equation (30) and the definition of Fo(X) for all X gt 0 such that Fn(X) gt 0 we have

Fn(X)

o S Fn()lt) =

7x ZdQn(Z)

~ (XZ - l)dQu(t)

1 1 ----gtshyx dQ(Z) X

~ ZdQn(Z) (41)

So for Xgt 0 such that Fn(X) gt 0

XFn(X) gt 1 Fn(X) (42)

Therefore from (42)

Fn(XZ)X Z[----]

(XZ)F(XZ) ----gt 1

If we write v(Z) = U [1 - w2)e + W2Zj we must show that U satisfies condition (b) of the lemma Clearly by the definition of U U gt 0 and UtI lt 0 condition (b) is satisfied From (29) (30) and (31) with n = I all the conditions for the hypothesis of the lemma are satisfied

roo F (XZ) rooJo F+ (X)dQ(ZI) = e = Jo ZdQ(ZI)

1

Therefore by the lemma

roo F(XZ) roo Jo Fn+I(X) dP(ZI) gt Jo ZdP(Zl)

or

bull therefore

p(Xn) gt a

---------------~-- ---- shy

36

IMR Winter 1969

So the theorem is proved Using the Lemma as generalized in Appendix A one could give a second proof that the common itself being more volatile than the safe asset must have a greater expected yield namely a gt r as expressed earlier in equation (21)

Conclusion

This completes the theory of utilitymiddotwarranted warrant pricing We leave to another occasion the calculation by a computer of tables of values for Fn(X) based upon certain empirical assumptions about the volatility and trend of the P(X t + uXtn) process Using the general mathematical methods of the 1965 paper but with different ecoshynomic interpretations we can also prepare tables of Fn(X) for the Appendix 8 case of dividend-paying stocks_

-

Samuelson and Merton Complete Model of Warrant PriCing

37

Appendix A

The generalization and proof of the lemma to prove the theorem that t1(Xn) gt a is as follows23

Lemma let l ltgt and v be Reimann-Steiltjes integrable with respect to P where dP(Y) is a probability distribution function and v is a monotone-decreasing function on [0 QO) and v(Y) gt 0 for Y gt O Suppose

(a) there exists YE(O QO) such that l(Y) s ltgt(Y) for all Y lt Y and ltgt(Y) s l(Y) for a II Ygt Y and

(b)foryengt l(Y)v(Y)dP(Y) = fo ltgt(Y)v(Y)dP(Y)

Then foryengt f(Y)dP(Y) 2 foryengt ltgt(Y)dP(Y)

Proof

1 foY [fey) - ltgt(Y)]v(Y)dP(Y) s 0

Ii [l(Y) - ltgt(Y)]v(Y)dP(Y) 2 0 because vY) 2 0

2-foY [fey) - ltgt(Y)]v(Y)dP(Y) y [l(Y) - ltgt(Y)]vy)dP(Y) from (b)

3 let v= v(Y) gt 0

Then v(Y) 2 v for Y s Y

v(Y) s v for Y 2 V by hypothesis

4 Then

_ foY [fey) - ltgt(Y)] ild P(Y) s I [fey) - ltgt(Y)]vd P(Y) from 2 and 3

5 Therefore

10 f(Y)d P(Y) 2 fofyengt ltgt(Y)d P(Y) QED

To show the lemma stated in the text is a special case of this general lemma and to get the sharper inequality result of that lemma it is necessary to prove a corollary to the general lemma and also another lemma to the corollary (The lemma to the corollary will be referred to with a lower case I to distinguish it from the general lemma)

Corollary let 1 ltgt and dP be as in the lemma and let dP not have the property

dP = ~ suppose v(Y) is strictly monotone-decreasing and nonshyI p P ~ 0 otherwise

negative on [0 QO] Suppose

(a) there exists YE(O QO) such that fey) lt ltgt(Y) for all YE(OV) and q(Y) lt f(Y) for all YE(Vlaquoraquo and

bull(b)fo fY)v(Y)dP(Y) = foryengt qY)vY)dP(Y)

Then fo f(Y)dP(Y) gt 10 q(Y)dP(Y)

38

IMR Winter 1969

Proof

1 loy [w(Y) - p(Y)]v(Y)d P(Y) lt 0

Ii [w(Y) - p(Y)]v(Y)dP(Y) gt 0 by the property of d P and v o 2-IoY [gtJt(Y) p(Y)] v(Y)d P(Y) Ii [gtJt(Y) - p(Y)]v(Y)dP(Y) from (b)

3 Let v = v(Y) gt 0

Then v(Y) gt v Y lt Y

v(Y) lt v Y gt Y by hypothesis

4 Then (f

-)0 [HY) p(Y)]vdP(Y) ltIi [gtJt(Y) - p(y)]vdP(Y)

(Note the posited property of dP was needed for this step)

5 Therefore

10 o(Y)dP(Y) gt fo~ p(Y)dP(Y) QED

Thus the strict inequality form of the Lemma used in the text is proved

Although it is clear that the strict inequality of the corollary would not hold for the pathological dP(Y) case ruled out in the hypothesis of the corollary and of the Lemma in the text it is instructive to give an example of this case

Let d P(Y) be such that Prob Z = O = Prob Z = 3 = i (Note 0 (3) = 3 from below) and suppose that we have Bernoulli logarithmic utility Then we have 1 + a = 15 or a 5 the mean yield of the stock From the utility maximum equation for n = I

FI(X) = lFo(3X)

and by the usual recursive process we get

FI1(X) = 1Fo(3nX)

The mean warrant yield b is defined as follows

b = E[FI1(XZ)F+l(X)] - 1

= t[lDFQ(3DX 3) 1 X)] - 13 n+l

= 15 - 1 5

So b a or fJ (Xn) == a in this singular case

In retrospect the reason for fJ(Xn) = a for this type of distribution is that in it the stock and warrant are equally volatile with the chance of losing everything being the same for both stock and warrant

I I -i


39

We must show now the equivalence of the elasticity hypothesis of the Lemma in the text to the hypotheses of the general Lemma To do so we prove the following lemma to the corollary

Lemma Let 1 ltgt and dP be as in the general Lemma and in addition I and ltgt are continuous Suppose either (i) there exists an Xgt 0 such that q(V) = 0 V5 X ltgt(0) ~ 0 Eq gt Eltgt gt 0 for all Vgt X Eltgt gt 0 for all Vgt 0 and (b) holds or (ii) Eq gt Eltgt gt 0 for all Vgt 0 and (b) holds Then condition (a) of the corolshylary holds

Proof

(I) If ltgt(V) q(V) for some Ygt 0 then there does not exist Y ~ Y Ygt 0 such that ltgt(V) = 1(1)

Proof Consider any point V gt 0 where ltgt(V) = q(V) Under condition (i) Y gt X because ltgt(0) ~ 0 Eltgt gt 0 for a II V gt O Thus Y is such that E~(V) gt Eltgt(V) Le I cuts ltgt from below at Y But since Eq gt Eltgt for all Vgt X ltgt can cut I from below only once (II) There exists a Iigt 0 such that ir(V) lt ltgt(V) for all V dO Ii)

Proof 1) For (i) this holds trivially by setting Ii = X in view of the restrictions on ltgt and 1 2) For (ii) suppose such a Ii does not exist Then given any X gt 0 there exists a V such that Yt(O X) and q(Y) gt ltgt(V) But since Eq gt Eltgt for all Vgt 0 this implies that q(Y) gt ltgt(V) for all V gt O But this contradicts (b)

Thus [q(V) - ltgt(V)]v(V)d P(Y) lt 0 and therefore

0 [q(V) ltgt(V)]v(V)dP(V) gt 0

Thus q(V) lt ltgt(V) for some YE(O Ii)

q(Y) gt ltgt(V) for some YE(Ii 00 )

This implies since ir and ltgt are assumed continuous that there exists Ygt 0 such that ltgt(V) = q(Y) By (I) we know Yis un iq ue in (000) Therefore ir(V) lt ltgt(V) 0 lt Y 5 X and Eq gt Eltgt for V gt X so that V is such that for VX(Ooo) and q(Y) lt ltgt(V) for all V lt Yand ltgt(V) lt q(V) for all Vgt Y

QED Thus from the corollary and the lemma to the corollary and by taking ltgt(V) = V (and therefore Eltgt 1) we have proved the Lemma used in the text It was necessary in the lemma to the corollary to include the alternative hypothesis (i) because in the case where

q(l) = Fn(Xl)F+1(X)

it is possible that Fn(Xl)$5 0 for positive Xl in the neighborhood of Xl 0 in which case Eir will not be properly defined One can see that this has no effect on the Lemma because

10 II q(Y)v(V)dP(V) =k q(V)v(V)dP(V)

-

40

IMR Winter 1969

where R = YiYt(Ooo) and I(Y) gt O and similarly

1000

I(Y)dP (Y) = hl(Y)dP(Y)

Thus we could go through the entire derivation considering only YER where EI is wellmiddotdefined and then at the end substitute the integrals over all nonmiddotnegative Y

It should be emphasized that the proof of the general lemma did not even require continuity of 1 ltp and v and that the probability distribution dP can be discrete enmiddot tailing corners in the Fn(X) functions Thus it holds for quite general types of assets and probability distributions A simple extension of the corollary would prove the following general theorem of portfolio analysis

Theorem let 110 12bull r In be the set of price ratios for n perfectlymiddotcorrelated assets and let their elasticities E1lr be such that EII gt EI2gt bull gt EIo let 1 v dP be as defined in the corollary If

1000 I(Y)v(Y)dP(Y) 000 Ij(Y)v(Y)dP(Y)

for ij I n then E[1I] gt E[IlI gt gt E[wJ

Appendix B

If a common stock permanently pays no dividend the theory of the text is applicable If it does pay a dividend the nice simplifications of the 1965 nonmiddotconversion special case is lost and we are back in all the 1965 complex inequalities If we work with conmiddot tinuous rather than discrete time the complicated McKean 1965 appendix methods are needed and many unsolvable problems remain problems that can be solved to any degree of accuracy only by taking smaller and smaller discrete time intervals Here we shall sidestep all complexities stemming from continuous time and can do so with a clearer conscience since the uti lity maximization is taken always to be over some premiddot scribed finite interval (eg six months and a day to achieve capital gains tax privileges)

The simplest assumption about dividends is that the common priced at Xt will after any prescribed period say T pay a dividend proportional to its price Xt+Tbull The dividend will then be Xt+T(eOT - 1) where Ii is the force or instantaneous rate of dividend yield By convention we may set T = 1 and each common that costs us Xt today brings us

Xi+ + Xi+ (eli - 1) = XHleli

after one period (We neglect all taxation throughout despite the earlier remark about sixmiddotmonth holding periods)

Now our maximum problem becomes

- r e Ii F(XZ)Max U(Wh WI WI) = Max Jo U[wler + we Z + WI F (X)] dP(Z1)Wj Wj +1

subject to WI + WI + WI 1

bull


41

The conditions for the critical point of the Lagrangian L = U + Y (1 - 23IWj) are exactly as in (14) (15a) and (16a) except that w2eoalways appears where previously W2 alone appeared Hence the basic equations of the present theory (15a) and (16a) become

Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)

where of course dQ now involves 0 along with its other suppressed parameters Now I3Q = r as before but aQ = r - 0 lt tJQ and we are in the difficult fJ gt a area of the 1965 analysis

Now the values deduced from (43) will fall below Fo(X) conversion levels for large enough X and conversion will be mandatory Hence the recursion relation (43) above must be superseded by the inequalities

FI(X) = Max [OX - le-rlo Fo(XZ)dQ(Zl)] (45)

F2(X) = Max OX - le-rlo F1(XZ)dQ(Zl)]

Fn+l(X) = Max [OX le-r10 Fn(XZ)dQ(Zl)]

F00 (X) = F(X) = Max [OX - le-r0 F(XZ)dQ(Z 1)]

By the 1965 methods one can show that for given r fJ and 0 = fJ - a we can find conversion values (X Cll CZ coo) which are in ascending order and for which

=X - IX gt Cn

Actually for the perpetual warrant case we have the following Fredholm-like integral equation of the second kind to solve for F (X) = F(X) namely for XltC =C ()Of

ooF(X) = e-rlo F(XZ)dQ(Zl)

= e-rfoeJx F(XZ)dQ(Zl) +e7x (XZ - l)dQ(ZI)

= e-rocx F(XZ)dQ(Z 1) + 4gt(X c) where 4gt is a known function (46)

If dQ corresponds to a probability density q(Z)dZ we can tl8nsform this to bull

F(X) = e-r (c q(vX)F(v)dv + 4gt(Xc)Jo X

42

IMR Winter 1969

Suppose this is solved by any of the well-known methods for each possible c and let F(Xc) be the solution joining-up equation

Then we can solve for the unknown c ceo as the root of the

F(Xc) X-I at X = coo or (47)

F(cc) Coo -1

Thus the perpetual warrant case can be solved without going through the calculations of F(X)

Actually if the probabilities of price changes are bunched around Z = 1 with a finite range so that P(ZI) Q(ZI) = 0 for Z lt Xmiddot lt I and P(ZI) = Q(ZI) = 1 for Z gt X gt I this Fredholmmiddottype equation can be solved as a Volterramiddotlike equation which after a logarithmic transformation becomes almost of the Poisson or Wiener-Hopf type This can be seen as follows consider an X small enough so that (cXmiddotmiddot) gt X Such an X exists because Xmiddotmiddot is finite For Xs satisfying this inequa1ity we have

X F(X) e-r r F(XZ)dQ(ZI) gt X-I (48)

lXOlt

and we can now use the method of analysis shown in the section on utility-maximizing warrant pricing There is an infinite number of solutions to the homogeneous integral equation (48) of the form cXm Substituting in (48) we have

cXm = e-rcXm ZmdQ(ZI)

(49)

This is the same as the transcendental equation (23) However in this case because r = I3Q gt OtQ = r ~ m = 1 is no longer a solution The relevant real root satisfying the boundary conditions is m gt 1 giving us the power formula of the 1965 paper

F(X) = aXm = (coo _ 1) ( X ) (coo - 1) Coo

bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J

See Samuelson [5] where theorems like this one are proved without making the meanvariance approximamiddot tions of the now classical MarkowitzmiddotTobin type

Since units are arbitrary we can take any prescribed wealth level and by dimensional convention make it unity in all of our formulas This enables expressions like wW to be written simply as w where W = total wealth As will be specified later working with isoelastic marginal utility functions that are uniform for all investors will make the scale of prescribed wealth of no importance

The concavity of U is sufficient to achieve the negative semidefiniteness of the constrained quadratic forms and bordered Hessian minorsof L needed to insure thatany solutio~ to the first-order conditions does provide a global as well as local maximum Although the maximum IS unique the portfoliO proportions could take on more than one set 01 optimizing values in singular cases where the quadratic forms were semidefinite rather than definite eg where a perpetual warrant and its common stock lire perfectly linearly correlated making the choice between them indifferent and not unique This example will be presented later

At a Washington confereoce in 1953 the first author once shocked the late J M Clark by saying Although the probability of a serious 1954 recession is only onemiddotthlrd that probability should be treated as though it were twomiddotthirds This was a crude and nonmarginal use of a utilmiddotprob notion akin to dQ

61n the 1965 paper [6J pp 30-31 it was mentioned that the possibility of hedges in which the common stock is sold short in some proportion and the warrant is bought long would be likely to set limits on the discrepshyancies that in the absence of dividend payments could prevail between (J and n In a forthcoming paper Restrictions on Rational Option Pricing A Set of Arbitrage Conditions the second author develops arbi trage formulas on warrants and puts and calls which show how severely limited are such (Jn discrepancies as a result of instantaneous almost suremiddotthing arbitrage transformations

If the probability of good and bad crops were not equal or if the safe investment process had a nonmiddotzero yield the proportion of the risk asset held would be a function of the A yield factor and for utility functions other than the Bernoulli logmiddotform and a probability distribution different from the simple binomial w would be a more complicated calculable function

middotStrictly speaking a will probably be a function of time ao-I being high in the period following a generally poor crop when the )-1 yield factor rather than A has just occurred and the investable surplus is small We have here a stationary time series in which total output vibrates around an equilibrium level Spelling all this out would be another story here a will be taken as a constant

For the family

e bX e -U(X)

U(X) = a t- ___ 0 lt e r= I XU(X)= e e-l

e

The singular case where e = 1 can be found by LHopitaJs evaluation of an indeterminate form to correspond to the Bernoulli case U(X) = a + b 10gX As Arrow [1] Pratt [4J and others have shown optimal portfolio proportions are independent of the absolute size of wealth for any function that is a member of this utility

family

Actually we can free our analysis from the assumption of isoelastic marginal utility if we are willing to apply it to any single individual and determine from it the critical warrant price patterns at which he would be neither a buyer nor seller or would hold some specified proportion of his wealth in the form of warrants By pitting the algebraic excess demands of one set of individuals against the other we could determine the market clearing pattern

IOU being concave assures a maximum The problem could be formuiated with KuhnmiddotTucker inequalities to cover the nomiddotborrowing restriction w S I and the nomiddotshortmiddotselling restriction w ~ O

llThe F function in (13) is the utilitymiddotwarranted price of the warrant which is not the same as the rational warrant price of the 1965 theory discussed above even though we use the same symbol for both

bull

44

IMR Winter 1969

12This would be a generalization of the analysIs above to three rather than only two assets In the next section we digress to discuss briefly in these terms the simplest case of pricing a given supply of Imiddotperiod warrants This illustrates a general theory

IThorp and Kassoul [7J advocate hedged short sales of overpriced warrants about to expire The analysis here defines the levels at which one who holds the stock long can just benefit in the maximizing expected utility sense from shortmiddotsale hedges in the warrant

Strictly speaking F is a functIon of more than X alone it can be written as F (X rV1M) likewise the equishylibrium S is of the form G(XrVM)M where both G and F are functionals of the probabilitymiddotdistribution function P(ZI) There is a formal similarity here to the quantity theory of money and prices due of course to the homogeneity assumption made about tastes It should be fairly evident that in the same fashion by which we have here deduced the f( ) function from the known F () function one could in general deduce recursively F( ) in terms of a known f () function Similar homogeneity properties in terms of (VM) and VIM would hold finally instead of assuming completely inelastiC V supply and completely elastic common stock supply dependent on a hard parameter one could formulate a completely general equilibrium model in which r n and the probability distribution P(Zl) were all determined simultaneously

SWe will show later that 13 gt ltlp for finite-duration warrants falling toward equality as the duration time becomes perpetual

middotThe HertzmiddotHerglotzmiddotlotka methods of renewal theory are closely related once we replace X and Z by their logarithms However the fact that our dQ involves Zs on both sides of unity with positive weights introduces some new complications later without regard to formal expansions of this type we prove that f(X) ~ F(X) X for references to this literature including work by Fellner see lopez (3)

l1There is a further complication If decisions are frozen for n periods then (26) is valid superseding (24) and (20) Or put differently n of the old time periods are now equivalent to one new time period and in terms of this new time period (20) would be rewritten to have exactly the same content as (26) Now (24) or (25) would simply be irrelevant One must not suppose that this change in time units is merely a representational shift to new dimensional units as from seconds to minutes If our portfolio is to be frozen for six months that differs substantively from its being frozen for six weeks even though we may choose to write six months as twentymiddotsix weeks But now for the complication one would not expect the U(W) function relevant for a six-week frozenmiddotdecision period to be relevant for a six-month period as well Strictly speaking then in using (26) for a longmiddotfrozenmiddotperiod analysis we should require that the U(W) function which enters into dQ(Z n) be written as dependent on n or as aU[W n)ilW Two papers showing proper lifetime portfolio decisions are forthcoming P A Samuelson lifetime Portfolio Selection by DynamiC Stochastic Programming and R C Merton lifetime Portfolio Selection Under Uncertainty The ContinuousmiddotTime Case

One further remark Consider the incipientmiddotcash case where wmiddot = 0 because the common stock dominates the safe asset with a gtgt r Combining this case with our inCipient-warrant case w remains at unity in every period no matter what we learn about the outcomes within any larger period In this case the results of (20) and those of (26) are compatible and the latter does give us by mere quadrature a onemiddotstep solution to the problem The 1965 proof that F(X) ~ X as n ~ oc can then be applied directly

llf as mentioned in footnote 9 we free the analysis from the assumption of isoelastic marginal utility the definitions of (29) must be generalized to take account of the changing (Wjmiddot) optimizing decisions which will now be different depending on changing wealth levels that are passed through

lThe results of this section hold also for calls See Appendix B for the results for dividendmiddotpaying stocks

2Ofhis is the limiting case where equations (30) and (31) become identical The bordered Hessian becomes singular and w and wmiddot become indistinguishable iebullbull the warrant and the stock cease to be distinguish able assets

21See Samuelson (6) p 17 The paradox of almostmiddotcertain almostmiddottotal ruin for fairgame betters who re-bet their proceeds is involved hern Consider a hypothetical multiplicative probability prOCeSS Y - X Y = XZ Y - XZZbullbullbullbullbullbull Y ~ XZ bullbullbull Z where X is a constant and each ZI is independently distributed according to the probability distribution Prob ZI S Z - Q(Z) Then it directly follows that Prob XZZ S XZ ~ Q(Z)

and bullbullbull Prob XZ bullbullbull Z S XZ = Q(Z) Since [[Z) ~ j ZQ(Z)dZ - eO - 1 and P(Xl) and Q(Z) involve

some positive dispersion the geometric mean of dQ(Z) lies below the arithmetic mean of 1 lt [[Z] Hence

[[logZu - J 10gZdQ(Z) = 11 lt O By the central limit theorem applied to 10gX + ~logZI logY bullbull [(logY) = 10gX +n and [[logY) ~ - as n- so that all the probability becomes spread out t~ the left of any fixed number Z Thus Q(Z) - 1 as n - for all Z gt O (Note A fair-game (r 0) in Q-space implies a bettermiddot thanmiddotfair game (a gt 0) in Pmiddotspace from equation (21)) Warning Although Qoe(Z) becomes a logmiddotnormal distribution say L(Z11an) it is quite wrong to think that necessarily

bull


X F (X) lim e-o F(XZ)dQ(Z) n~

= ] F(X) lim e- dL(Z v~n) n--

] F(XZ)dQmiddot(Z)

Such interchanging of limits will generally not be permissible

2ZThis partial difference equation can presumably be solved by the methods of Lagrange and Laplace but there are complexities involved due to the boundary conditions of arbitrage which we do not wish to go into at this time

2lThe proofs of the general Lemma the corollary and the lemma to the corollary are by David T Scheffman PhD candidate at MIT

bull

46

IMR Winter 1969

References

111 Arrow K J Aspects of the Theory of Risk-Bearing Helsinki Yrjo Jahnssonin Saatio 1965

121 Kassouf S T Stock Price Random Walks Some Supporting Evidence Reshyview of Economics and Statistics Vol 50 (1968) pp 275-278

13] lopez A Problems in Stable Popushylation Theory Princeton Office of Popushylation Research Princeton University 1961

[4J Pratt J W Risk Aversion in the Small and in the large Econometrica Vol 32 (January 1964) pp 122-136

[5] Samuelson P A General Proof that Diversification Pays Journal of Fishynancial and Quantitative Analysis Vol 3 (1967) pp 1-13

[6] Samuelson P A Rational Theory of Warrant Pricing Industrial Manageshyment Review Vol 6 no 2 (Spring 1965) pp 13-32 Mathematical Appendix by H P McKean Jr pp 32-39

[7] Thorp E 0 and Kassouf S T Beat the Market New York Random House 1967

bull

Subscriptions

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Reprints of individual articles may be purchased (prices on request) Some back issues are available ($200 per copy)

David L Diamond Lawrence W Garrett Robert L Gipson Anthony C Taylor

Susan L Singer

-




Introductiont









0lt E[Z]- 1 = 1000



bull


18

IMR Winter 1969








or

1 =it ZU [(1 - w) + wZ]d P(Zn)








bull


19


i=1 j=1




1000 = 1000



where m










0= 0 (w) = -- (- 1 +A)+ t (-1 + A-i) and (7) + A t + t A-1



bull

20

IMR Winter 1969

bull

3


21







(9)



10 Fa (XZ)dP(ZI)

F1(X) (10)





bull

22

IMR Winter 1969



oo




Fn(X) = e- T1000

Fo_T(XZ)dP(ZT)

= e-laquon 1000

Fu(XZ)dP(Zn)

e- n 1x (XZ-1)dP(Zn) (12)






23









bull

24

IMR Winter 1969


3



(14)



(00 F (XZ)

(15)

(16)



C





25


c






= M(1 +w2 + _1_-_-=-----) (17)WI WI

(18)

(19)



C

(15b)


c


(16b)

bull

26

IMR Winter 1969






Jo U [(1 - w~)er + w2Z]d P(Z 1)







27


and (20) becomes








Explicit Solutions





F2(X) = e-2r0 F o(XZ)dQ(Z 2)

(25) or in general


28

IMR Winter 1969










= e-2r000 FIl(XR)dQ2(R) (28)

where





QI(Z) = f QI (~) dQI(V) ~ Q(Z2)

-(29)


29



F2(X) = e-2r10 Fo(XZ)dQ 2(Z)

(30)



1000 ZdQ2(Z) = e2

10 ZdQ(Z) = en (31)










~ xe-Iooo ZdQ(Z) em



30

IMR Winter 1969















X101 (1 - XZ)dQn(Z)] r~


== X - e-m + e-m 81(Xn)8(Xn) from (31)




bull

I


31



X 101 ZdQ(Z)

1 - ollX dQn(l)


1- 0 because Q(O+X) = 1 (34)



n-gtoo

Therefore


=X-1+1 X




32

IMR Winter 1969






1 + a Hgt- + X-I) (35)





i+igt- i+X-I (37)




Fo(X) = Max [OX - 1] (39)




111 - H + X ] [ H +X -I ) + ( + X )]

= (1 + )1]1 for i - +1


bull


k

-4 -3 -2 1 0 1 2 3 4 n

0 0 0 0 0 0 1000 2100 3300 4600

t 1 0 0 0 0476 1000 2100 3300

2 0 0363 0476 1250 2100

t 3 0363 0684 1250

4 0684

Table I





ql = _1_ = 4762 q_1 = = 5238 21 21




FI(l) = (1 + X)-lFo() + (I + -1)-lFoX-1)

_x-1 -+1 +0

bull

34

IMR Winter 1969


E [ F((Xt+21I X = IJ = (x - 1) 1 f (1) t (1 +X)1 (x---=-I)

X-l=b








d(log~) V~I(Y)

E~=d(logV) it(Y)





35







Fn(X)

o S Fn()lt) =

7x ZdQn(Z)

~ (XZ - l)dQu(t)


~ ZdQn(Z) (41)



Therefore from (42)

Fn(XZ)X Z[----]

(XZ)F(XZ) ----gt 1



1



or

bull therefore

p(Xn) gt a

---------------~-- ---- shy

36

IMR Winter 1969


Conclusion


-


37

Appendix A






Proof




3 let v= v(Y) gt 0



4 Then


5 Therefore









38

IMR Winter 1969

Proof



3 Let v = v(Y) gt 0



4 Then (f



5 Therefore





FI(X) = lFo(3X)


FI1(X) = 1Fo(3nX)


b = E[FI1(XZ)F+l(X)] - 1

= t[lDFQ(3DX 3) 1 X)] - 13 n+l

= 15 - 1 5



I I -i


39



Proof










q(l) = Fn(Xl)F+1(X)



-

40

IMR Winter 1969


1000







Appendix B








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull




Introductiont









0lt E[Z]- 1 = 1000



bull


18

IMR Winter 1969








or

1 =it ZU [(1 - w) + wZ]d P(Zn)








bull


19


i=1 j=1




1000 = 1000



where m










0= 0 (w) = -- (- 1 +A)+ t (-1 + A-i) and (7) + A t + t A-1



bull

20

IMR Winter 1969

bull

3


21







(9)



10 Fa (XZ)dP(ZI)

F1(X) (10)





bull

22

IMR Winter 1969



oo




Fn(X) = e- T1000

Fo_T(XZ)dP(ZT)

= e-laquon 1000

Fu(XZ)dP(Zn)

e- n 1x (XZ-1)dP(Zn) (12)






23









bull

24

IMR Winter 1969


3



(14)



(00 F (XZ)

(15)

(16)



C





25


c






= M(1 +w2 + _1_-_-=-----) (17)WI WI

(18)

(19)



C

(15b)


c


(16b)

bull

26

IMR Winter 1969






Jo U [(1 - w~)er + w2Z]d P(Z 1)







27


and (20) becomes








Explicit Solutions





F2(X) = e-2r0 F o(XZ)dQ(Z 2)

(25) or in general


28

IMR Winter 1969










= e-2r000 FIl(XR)dQ2(R) (28)

where





QI(Z) = f QI (~) dQI(V) ~ Q(Z2)

-(29)


29



F2(X) = e-2r10 Fo(XZ)dQ 2(Z)

(30)



1000 ZdQ2(Z) = e2

10 ZdQ(Z) = en (31)










~ xe-Iooo ZdQ(Z) em



30

IMR Winter 1969















X101 (1 - XZ)dQn(Z)] r~


== X - e-m + e-m 81(Xn)8(Xn) from (31)




bull

I


31



X 101 ZdQ(Z)

1 - ollX dQn(l)


1- 0 because Q(O+X) = 1 (34)



n-gtoo

Therefore


=X-1+1 X




32

IMR Winter 1969






1 + a Hgt- + X-I) (35)





i+igt- i+X-I (37)




Fo(X) = Max [OX - 1] (39)




111 - H + X ] [ H +X -I ) + ( + X )]

= (1 + )1]1 for i - +1


bull


k

-4 -3 -2 1 0 1 2 3 4 n

0 0 0 0 0 0 1000 2100 3300 4600

t 1 0 0 0 0476 1000 2100 3300

2 0 0363 0476 1250 2100

t 3 0363 0684 1250

4 0684

Table I





ql = _1_ = 4762 q_1 = = 5238 21 21




FI(l) = (1 + X)-lFo() + (I + -1)-lFoX-1)

_x-1 -+1 +0

bull

34

IMR Winter 1969


E [ F((Xt+21I X = IJ = (x - 1) 1 f (1) t (1 +X)1 (x---=-I)

X-l=b








d(log~) V~I(Y)

E~=d(logV) it(Y)





35







Fn(X)

o S Fn()lt) =

7x ZdQn(Z)

~ (XZ - l)dQu(t)


~ ZdQn(Z) (41)



Therefore from (42)

Fn(XZ)X Z[----]

(XZ)F(XZ) ----gt 1



1



or

bull therefore

p(Xn) gt a

---------------~-- ---- shy

36

IMR Winter 1969


Conclusion


-


37

Appendix A






Proof




3 let v= v(Y) gt 0



4 Then


5 Therefore









38

IMR Winter 1969

Proof



3 Let v = v(Y) gt 0



4 Then (f



5 Therefore





FI(X) = lFo(3X)


FI1(X) = 1Fo(3nX)


b = E[FI1(XZ)F+l(X)] - 1

= t[lDFQ(3DX 3) 1 X)] - 13 n+l

= 15 - 1 5



I I -i


39



Proof










q(l) = Fn(Xl)F+1(X)



-

40

IMR Winter 1969


1000







Appendix B








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull

18

IMR Winter 1969








or

1 =it ZU [(1 - w) + wZ]d P(Zn)








bull


19


i=1 j=1




1000 = 1000



where m










0= 0 (w) = -- (- 1 +A)+ t (-1 + A-i) and (7) + A t + t A-1



bull

20

IMR Winter 1969

bull

3


21







(9)



10 Fa (XZ)dP(ZI)

F1(X) (10)





bull

22

IMR Winter 1969



oo




Fn(X) = e- T1000

Fo_T(XZ)dP(ZT)

= e-laquon 1000

Fu(XZ)dP(Zn)

e- n 1x (XZ-1)dP(Zn) (12)






23









bull

24

IMR Winter 1969


3



(14)



(00 F (XZ)

(15)

(16)



C





25


c






= M(1 +w2 + _1_-_-=-----) (17)WI WI

(18)

(19)



C

(15b)


c


(16b)

bull

26

IMR Winter 1969






Jo U [(1 - w~)er + w2Z]d P(Z 1)







27


and (20) becomes








Explicit Solutions





F2(X) = e-2r0 F o(XZ)dQ(Z 2)

(25) or in general


28

IMR Winter 1969










= e-2r000 FIl(XR)dQ2(R) (28)

where





QI(Z) = f QI (~) dQI(V) ~ Q(Z2)

-(29)


29



F2(X) = e-2r10 Fo(XZ)dQ 2(Z)

(30)



1000 ZdQ2(Z) = e2

10 ZdQ(Z) = en (31)










~ xe-Iooo ZdQ(Z) em



30

IMR Winter 1969















X101 (1 - XZ)dQn(Z)] r~


== X - e-m + e-m 81(Xn)8(Xn) from (31)




bull

I


31



X 101 ZdQ(Z)

1 - ollX dQn(l)


1- 0 because Q(O+X) = 1 (34)



n-gtoo

Therefore


=X-1+1 X




32

IMR Winter 1969






1 + a Hgt- + X-I) (35)





i+igt- i+X-I (37)




Fo(X) = Max [OX - 1] (39)




111 - H + X ] [ H +X -I ) + ( + X )]

= (1 + )1]1 for i - +1


bull


k

-4 -3 -2 1 0 1 2 3 4 n

0 0 0 0 0 0 1000 2100 3300 4600

t 1 0 0 0 0476 1000 2100 3300

2 0 0363 0476 1250 2100

t 3 0363 0684 1250

4 0684

Table I





ql = _1_ = 4762 q_1 = = 5238 21 21




FI(l) = (1 + X)-lFo() + (I + -1)-lFoX-1)

_x-1 -+1 +0

bull

34

IMR Winter 1969


E [ F((Xt+21I X = IJ = (x - 1) 1 f (1) t (1 +X)1 (x---=-I)

X-l=b








d(log~) V~I(Y)

E~=d(logV) it(Y)





35







Fn(X)

o S Fn()lt) =

7x ZdQn(Z)

~ (XZ - l)dQu(t)


~ ZdQn(Z) (41)



Therefore from (42)

Fn(XZ)X Z[----]

(XZ)F(XZ) ----gt 1



1



or

bull therefore

p(Xn) gt a

---------------~-- ---- shy

36

IMR Winter 1969


Conclusion


-


37

Appendix A






Proof




3 let v= v(Y) gt 0



4 Then


5 Therefore









38

IMR Winter 1969

Proof



3 Let v = v(Y) gt 0



4 Then (f



5 Therefore





FI(X) = lFo(3X)


FI1(X) = 1Fo(3nX)


b = E[FI1(XZ)F+l(X)] - 1

= t[lDFQ(3DX 3) 1 X)] - 13 n+l

= 15 - 1 5



I I -i


39



Proof










q(l) = Fn(Xl)F+1(X)



-

40

IMR Winter 1969


1000







Appendix B








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull


19


i=1 j=1




1000 = 1000



where m










0= 0 (w) = -- (- 1 +A)+ t (-1 + A-i) and (7) + A t + t A-1



bull

20

IMR Winter 1969

bull

3


21







(9)



10 Fa (XZ)dP(ZI)

F1(X) (10)





bull

22

IMR Winter 1969



oo




Fn(X) = e- T1000

Fo_T(XZ)dP(ZT)

= e-laquon 1000

Fu(XZ)dP(Zn)

e- n 1x (XZ-1)dP(Zn) (12)






23









bull

24

IMR Winter 1969


3



(14)



(00 F (XZ)

(15)

(16)



C





25


c






= M(1 +w2 + _1_-_-=-----) (17)WI WI

(18)

(19)



C

(15b)


c


(16b)

bull

26

IMR Winter 1969






Jo U [(1 - w~)er + w2Z]d P(Z 1)







27


and (20) becomes








Explicit Solutions





F2(X) = e-2r0 F o(XZ)dQ(Z 2)

(25) or in general


28

IMR Winter 1969










= e-2r000 FIl(XR)dQ2(R) (28)

where





QI(Z) = f QI (~) dQI(V) ~ Q(Z2)

-(29)


29



F2(X) = e-2r10 Fo(XZ)dQ 2(Z)

(30)



1000 ZdQ2(Z) = e2

10 ZdQ(Z) = en (31)










~ xe-Iooo ZdQ(Z) em



30

IMR Winter 1969















X101 (1 - XZ)dQn(Z)] r~


== X - e-m + e-m 81(Xn)8(Xn) from (31)




bull

I


31



X 101 ZdQ(Z)

1 - ollX dQn(l)


1- 0 because Q(O+X) = 1 (34)



n-gtoo

Therefore


=X-1+1 X




32

IMR Winter 1969






1 + a Hgt- + X-I) (35)





i+igt- i+X-I (37)




Fo(X) = Max [OX - 1] (39)




111 - H + X ] [ H +X -I ) + ( + X )]

= (1 + )1]1 for i - +1


bull


k

-4 -3 -2 1 0 1 2 3 4 n

0 0 0 0 0 0 1000 2100 3300 4600

t 1 0 0 0 0476 1000 2100 3300

2 0 0363 0476 1250 2100

t 3 0363 0684 1250

4 0684

Table I





ql = _1_ = 4762 q_1 = = 5238 21 21




FI(l) = (1 + X)-lFo() + (I + -1)-lFoX-1)

_x-1 -+1 +0

bull

34

IMR Winter 1969


E [ F((Xt+21I X = IJ = (x - 1) 1 f (1) t (1 +X)1 (x---=-I)

X-l=b








d(log~) V~I(Y)

E~=d(logV) it(Y)





35







Fn(X)

o S Fn()lt) =

7x ZdQn(Z)

~ (XZ - l)dQu(t)


~ ZdQn(Z) (41)



Therefore from (42)

Fn(XZ)X Z[----]

(XZ)F(XZ) ----gt 1



1



or

bull therefore

p(Xn) gt a

---------------~-- ---- shy

36

IMR Winter 1969


Conclusion


-


37

Appendix A






Proof




3 let v= v(Y) gt 0



4 Then


5 Therefore









38

IMR Winter 1969

Proof



3 Let v = v(Y) gt 0



4 Then (f



5 Therefore





FI(X) = lFo(3X)


FI1(X) = 1Fo(3nX)


b = E[FI1(XZ)F+l(X)] - 1

= t[lDFQ(3DX 3) 1 X)] - 13 n+l

= 15 - 1 5



I I -i


39



Proof










q(l) = Fn(Xl)F+1(X)



-

40

IMR Winter 1969


1000







Appendix B








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull

20

IMR Winter 1969

bull

3


21







(9)



10 Fa (XZ)dP(ZI)

F1(X) (10)





bull

22

IMR Winter 1969



oo




Fn(X) = e- T1000

Fo_T(XZ)dP(ZT)

= e-laquon 1000

Fu(XZ)dP(Zn)

e- n 1x (XZ-1)dP(Zn) (12)






23









bull

24

IMR Winter 1969


3



(14)



(00 F (XZ)

(15)

(16)



C





25


c






= M(1 +w2 + _1_-_-=-----) (17)WI WI

(18)

(19)



C

(15b)


c


(16b)

bull

26

IMR Winter 1969






Jo U [(1 - w~)er + w2Z]d P(Z 1)







27


and (20) becomes








Explicit Solutions





F2(X) = e-2r0 F o(XZ)dQ(Z 2)

(25) or in general


28

IMR Winter 1969










= e-2r000 FIl(XR)dQ2(R) (28)

where





QI(Z) = f QI (~) dQI(V) ~ Q(Z2)

-(29)


29



F2(X) = e-2r10 Fo(XZ)dQ 2(Z)

(30)



1000 ZdQ2(Z) = e2

10 ZdQ(Z) = en (31)










~ xe-Iooo ZdQ(Z) em



30

IMR Winter 1969















X101 (1 - XZ)dQn(Z)] r~


== X - e-m + e-m 81(Xn)8(Xn) from (31)




bull

I


31



X 101 ZdQ(Z)

1 - ollX dQn(l)


1- 0 because Q(O+X) = 1 (34)



n-gtoo

Therefore


=X-1+1 X




32

IMR Winter 1969






1 + a Hgt- + X-I) (35)





i+igt- i+X-I (37)




Fo(X) = Max [OX - 1] (39)




111 - H + X ] [ H +X -I ) + ( + X )]

= (1 + )1]1 for i - +1


bull


k

-4 -3 -2 1 0 1 2 3 4 n

0 0 0 0 0 0 1000 2100 3300 4600

t 1 0 0 0 0476 1000 2100 3300

2 0 0363 0476 1250 2100

t 3 0363 0684 1250

4 0684

Table I





ql = _1_ = 4762 q_1 = = 5238 21 21




FI(l) = (1 + X)-lFo() + (I + -1)-lFoX-1)

_x-1 -+1 +0

bull

34

IMR Winter 1969


E [ F((Xt+21I X = IJ = (x - 1) 1 f (1) t (1 +X)1 (x---=-I)

X-l=b








d(log~) V~I(Y)

E~=d(logV) it(Y)





35







Fn(X)

o S Fn()lt) =

7x ZdQn(Z)

~ (XZ - l)dQu(t)


~ ZdQn(Z) (41)



Therefore from (42)

Fn(XZ)X Z[----]

(XZ)F(XZ) ----gt 1



1



or

bull therefore

p(Xn) gt a

---------------~-- ---- shy

36

IMR Winter 1969


Conclusion


-


37

Appendix A






Proof




3 let v= v(Y) gt 0



4 Then


5 Therefore









38

IMR Winter 1969

Proof



3 Let v = v(Y) gt 0



4 Then (f



5 Therefore





FI(X) = lFo(3X)


FI1(X) = 1Fo(3nX)


b = E[FI1(XZ)F+l(X)] - 1

= t[lDFQ(3DX 3) 1 X)] - 13 n+l

= 15 - 1 5



I I -i


39



Proof










q(l) = Fn(Xl)F+1(X)



-

40

IMR Winter 1969


1000







Appendix B








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull


21







(9)



10 Fa (XZ)dP(ZI)

F1(X) (10)





bull

22

IMR Winter 1969



oo




Fn(X) = e- T1000

Fo_T(XZ)dP(ZT)

= e-laquon 1000

Fu(XZ)dP(Zn)

e- n 1x (XZ-1)dP(Zn) (12)






23









bull

24

IMR Winter 1969


3



(14)



(00 F (XZ)

(15)

(16)



C





25


c






= M(1 +w2 + _1_-_-=-----) (17)WI WI

(18)

(19)



C

(15b)


c


(16b)

bull

26

IMR Winter 1969






Jo U [(1 - w~)er + w2Z]d P(Z 1)







27


and (20) becomes








Explicit Solutions





F2(X) = e-2r0 F o(XZ)dQ(Z 2)

(25) or in general


28

IMR Winter 1969










= e-2r000 FIl(XR)dQ2(R) (28)

where





QI(Z) = f QI (~) dQI(V) ~ Q(Z2)

-(29)


29



F2(X) = e-2r10 Fo(XZ)dQ 2(Z)

(30)



1000 ZdQ2(Z) = e2

10 ZdQ(Z) = en (31)










~ xe-Iooo ZdQ(Z) em



30

IMR Winter 1969















X101 (1 - XZ)dQn(Z)] r~


== X - e-m + e-m 81(Xn)8(Xn) from (31)




bull

I


31



X 101 ZdQ(Z)

1 - ollX dQn(l)


1- 0 because Q(O+X) = 1 (34)



n-gtoo

Therefore


=X-1+1 X




32

IMR Winter 1969






1 + a Hgt- + X-I) (35)





i+igt- i+X-I (37)




Fo(X) = Max [OX - 1] (39)




111 - H + X ] [ H +X -I ) + ( + X )]

= (1 + )1]1 for i - +1


bull


k

-4 -3 -2 1 0 1 2 3 4 n

0 0 0 0 0 0 1000 2100 3300 4600

t 1 0 0 0 0476 1000 2100 3300

2 0 0363 0476 1250 2100

t 3 0363 0684 1250

4 0684

Table I





ql = _1_ = 4762 q_1 = = 5238 21 21




FI(l) = (1 + X)-lFo() + (I + -1)-lFoX-1)

_x-1 -+1 +0

bull

34

IMR Winter 1969


E [ F((Xt+21I X = IJ = (x - 1) 1 f (1) t (1 +X)1 (x---=-I)

X-l=b








d(log~) V~I(Y)

E~=d(logV) it(Y)





35







Fn(X)

o S Fn()lt) =

7x ZdQn(Z)

~ (XZ - l)dQu(t)


~ ZdQn(Z) (41)



Therefore from (42)

Fn(XZ)X Z[----]

(XZ)F(XZ) ----gt 1



1



or

bull therefore

p(Xn) gt a

---------------~-- ---- shy

36

IMR Winter 1969


Conclusion


-


37

Appendix A






Proof




3 let v= v(Y) gt 0



4 Then


5 Therefore









38

IMR Winter 1969

Proof



3 Let v = v(Y) gt 0



4 Then (f



5 Therefore





FI(X) = lFo(3X)


FI1(X) = 1Fo(3nX)


b = E[FI1(XZ)F+l(X)] - 1

= t[lDFQ(3DX 3) 1 X)] - 13 n+l

= 15 - 1 5



I I -i


39



Proof










q(l) = Fn(Xl)F+1(X)



-

40

IMR Winter 1969


1000







Appendix B








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull

22

IMR Winter 1969



oo




Fn(X) = e- T1000

Fo_T(XZ)dP(ZT)

= e-laquon 1000

Fu(XZ)dP(Zn)

e- n 1x (XZ-1)dP(Zn) (12)






23









bull

24

IMR Winter 1969


3



(14)



(00 F (XZ)

(15)

(16)



C





25


c






= M(1 +w2 + _1_-_-=-----) (17)WI WI

(18)

(19)



C

(15b)


c


(16b)

bull

26

IMR Winter 1969






Jo U [(1 - w~)er + w2Z]d P(Z 1)







27


and (20) becomes








Explicit Solutions





F2(X) = e-2r0 F o(XZ)dQ(Z 2)

(25) or in general


28

IMR Winter 1969










= e-2r000 FIl(XR)dQ2(R) (28)

where





QI(Z) = f QI (~) dQI(V) ~ Q(Z2)

-(29)


29



F2(X) = e-2r10 Fo(XZ)dQ 2(Z)

(30)



1000 ZdQ2(Z) = e2

10 ZdQ(Z) = en (31)










~ xe-Iooo ZdQ(Z) em



30

IMR Winter 1969















X101 (1 - XZ)dQn(Z)] r~


== X - e-m + e-m 81(Xn)8(Xn) from (31)




bull

I


31



X 101 ZdQ(Z)

1 - ollX dQn(l)


1- 0 because Q(O+X) = 1 (34)



n-gtoo

Therefore


=X-1+1 X




32

IMR Winter 1969






1 + a Hgt- + X-I) (35)





i+igt- i+X-I (37)




Fo(X) = Max [OX - 1] (39)




111 - H + X ] [ H +X -I ) + ( + X )]

= (1 + )1]1 for i - +1


bull


k

-4 -3 -2 1 0 1 2 3 4 n

0 0 0 0 0 0 1000 2100 3300 4600

t 1 0 0 0 0476 1000 2100 3300

2 0 0363 0476 1250 2100

t 3 0363 0684 1250

4 0684

Table I





ql = _1_ = 4762 q_1 = = 5238 21 21




FI(l) = (1 + X)-lFo() + (I + -1)-lFoX-1)

_x-1 -+1 +0

bull

34

IMR Winter 1969


E [ F((Xt+21I X = IJ = (x - 1) 1 f (1) t (1 +X)1 (x---=-I)

X-l=b








d(log~) V~I(Y)

E~=d(logV) it(Y)





35







Fn(X)

o S Fn()lt) =

7x ZdQn(Z)

~ (XZ - l)dQu(t)


~ ZdQn(Z) (41)



Therefore from (42)

Fn(XZ)X Z[----]

(XZ)F(XZ) ----gt 1



1



or

bull therefore

p(Xn) gt a

---------------~-- ---- shy

36

IMR Winter 1969


Conclusion


-


37

Appendix A






Proof




3 let v= v(Y) gt 0



4 Then


5 Therefore









38

IMR Winter 1969

Proof



3 Let v = v(Y) gt 0



4 Then (f



5 Therefore





FI(X) = lFo(3X)


FI1(X) = 1Fo(3nX)


b = E[FI1(XZ)F+l(X)] - 1

= t[lDFQ(3DX 3) 1 X)] - 13 n+l

= 15 - 1 5



I I -i


39



Proof










q(l) = Fn(Xl)F+1(X)



-

40

IMR Winter 1969


1000







Appendix B








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull


23









bull

24

IMR Winter 1969


3



(14)



(00 F (XZ)

(15)

(16)



C





25


c






= M(1 +w2 + _1_-_-=-----) (17)WI WI

(18)

(19)



C

(15b)


c


(16b)

bull

26

IMR Winter 1969






Jo U [(1 - w~)er + w2Z]d P(Z 1)







27


and (20) becomes








Explicit Solutions





F2(X) = e-2r0 F o(XZ)dQ(Z 2)

(25) or in general


28

IMR Winter 1969










= e-2r000 FIl(XR)dQ2(R) (28)

where





QI(Z) = f QI (~) dQI(V) ~ Q(Z2)

-(29)


29



F2(X) = e-2r10 Fo(XZ)dQ 2(Z)

(30)



1000 ZdQ2(Z) = e2

10 ZdQ(Z) = en (31)










~ xe-Iooo ZdQ(Z) em



30

IMR Winter 1969















X101 (1 - XZ)dQn(Z)] r~


== X - e-m + e-m 81(Xn)8(Xn) from (31)




bull

I


31



X 101 ZdQ(Z)

1 - ollX dQn(l)


1- 0 because Q(O+X) = 1 (34)



n-gtoo

Therefore


=X-1+1 X




32

IMR Winter 1969






1 + a Hgt- + X-I) (35)





i+igt- i+X-I (37)




Fo(X) = Max [OX - 1] (39)




111 - H + X ] [ H +X -I ) + ( + X )]

= (1 + )1]1 for i - +1


bull


k

-4 -3 -2 1 0 1 2 3 4 n

0 0 0 0 0 0 1000 2100 3300 4600

t 1 0 0 0 0476 1000 2100 3300

2 0 0363 0476 1250 2100

t 3 0363 0684 1250

4 0684

Table I





ql = _1_ = 4762 q_1 = = 5238 21 21




FI(l) = (1 + X)-lFo() + (I + -1)-lFoX-1)

_x-1 -+1 +0

bull

34

IMR Winter 1969


E [ F((Xt+21I X = IJ = (x - 1) 1 f (1) t (1 +X)1 (x---=-I)

X-l=b








d(log~) V~I(Y)

E~=d(logV) it(Y)





35







Fn(X)

o S Fn()lt) =

7x ZdQn(Z)

~ (XZ - l)dQu(t)


~ ZdQn(Z) (41)



Therefore from (42)

Fn(XZ)X Z[----]

(XZ)F(XZ) ----gt 1



1



or

bull therefore

p(Xn) gt a

---------------~-- ---- shy

36

IMR Winter 1969


Conclusion


-


37

Appendix A






Proof




3 let v= v(Y) gt 0



4 Then


5 Therefore









38

IMR Winter 1969

Proof



3 Let v = v(Y) gt 0



4 Then (f



5 Therefore





FI(X) = lFo(3X)


FI1(X) = 1Fo(3nX)


b = E[FI1(XZ)F+l(X)] - 1

= t[lDFQ(3DX 3) 1 X)] - 13 n+l

= 15 - 1 5



I I -i


39



Proof










q(l) = Fn(Xl)F+1(X)



-

40

IMR Winter 1969


1000







Appendix B








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull

24

IMR Winter 1969


3



(14)



(00 F (XZ)

(15)

(16)



C





25


c






= M(1 +w2 + _1_-_-=-----) (17)WI WI

(18)

(19)



C

(15b)


c


(16b)

bull

26

IMR Winter 1969






Jo U [(1 - w~)er + w2Z]d P(Z 1)







27


and (20) becomes








Explicit Solutions





F2(X) = e-2r0 F o(XZ)dQ(Z 2)

(25) or in general


28

IMR Winter 1969










= e-2r000 FIl(XR)dQ2(R) (28)

where





QI(Z) = f QI (~) dQI(V) ~ Q(Z2)

-(29)


29



F2(X) = e-2r10 Fo(XZ)dQ 2(Z)

(30)



1000 ZdQ2(Z) = e2

10 ZdQ(Z) = en (31)










~ xe-Iooo ZdQ(Z) em



30

IMR Winter 1969















X101 (1 - XZ)dQn(Z)] r~


== X - e-m + e-m 81(Xn)8(Xn) from (31)




bull

I


31



X 101 ZdQ(Z)

1 - ollX dQn(l)


1- 0 because Q(O+X) = 1 (34)



n-gtoo

Therefore


=X-1+1 X




32

IMR Winter 1969






1 + a Hgt- + X-I) (35)





i+igt- i+X-I (37)




Fo(X) = Max [OX - 1] (39)




111 - H + X ] [ H +X -I ) + ( + X )]

= (1 + )1]1 for i - +1


bull


k

-4 -3 -2 1 0 1 2 3 4 n

0 0 0 0 0 0 1000 2100 3300 4600

t 1 0 0 0 0476 1000 2100 3300

2 0 0363 0476 1250 2100

t 3 0363 0684 1250

4 0684

Table I





ql = _1_ = 4762 q_1 = = 5238 21 21




FI(l) = (1 + X)-lFo() + (I + -1)-lFoX-1)

_x-1 -+1 +0

bull

34

IMR Winter 1969


E [ F((Xt+21I X = IJ = (x - 1) 1 f (1) t (1 +X)1 (x---=-I)

X-l=b








d(log~) V~I(Y)

E~=d(logV) it(Y)





35







Fn(X)

o S Fn()lt) =

7x ZdQn(Z)

~ (XZ - l)dQu(t)


~ ZdQn(Z) (41)



Therefore from (42)

Fn(XZ)X Z[----]

(XZ)F(XZ) ----gt 1



1



or

bull therefore

p(Xn) gt a

---------------~-- ---- shy

36

IMR Winter 1969


Conclusion


-


37

Appendix A






Proof




3 let v= v(Y) gt 0



4 Then


5 Therefore









38

IMR Winter 1969

Proof



3 Let v = v(Y) gt 0



4 Then (f



5 Therefore





FI(X) = lFo(3X)


FI1(X) = 1Fo(3nX)


b = E[FI1(XZ)F+l(X)] - 1

= t[lDFQ(3DX 3) 1 X)] - 13 n+l

= 15 - 1 5



I I -i


39



Proof










q(l) = Fn(Xl)F+1(X)



-

40

IMR Winter 1969


1000







Appendix B








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull


25


c






= M(1 +w2 + _1_-_-=-----) (17)WI WI

(18)

(19)



C

(15b)


c


(16b)

bull

26

IMR Winter 1969






Jo U [(1 - w~)er + w2Z]d P(Z 1)







27


and (20) becomes








Explicit Solutions





F2(X) = e-2r0 F o(XZ)dQ(Z 2)

(25) or in general


28

IMR Winter 1969










= e-2r000 FIl(XR)dQ2(R) (28)

where





QI(Z) = f QI (~) dQI(V) ~ Q(Z2)

-(29)


29



F2(X) = e-2r10 Fo(XZ)dQ 2(Z)

(30)



1000 ZdQ2(Z) = e2

10 ZdQ(Z) = en (31)










~ xe-Iooo ZdQ(Z) em



30

IMR Winter 1969















X101 (1 - XZ)dQn(Z)] r~


== X - e-m + e-m 81(Xn)8(Xn) from (31)




bull

I


31



X 101 ZdQ(Z)

1 - ollX dQn(l)


1- 0 because Q(O+X) = 1 (34)



n-gtoo

Therefore


=X-1+1 X




32

IMR Winter 1969






1 + a Hgt- + X-I) (35)





i+igt- i+X-I (37)




Fo(X) = Max [OX - 1] (39)




111 - H + X ] [ H +X -I ) + ( + X )]

= (1 + )1]1 for i - +1


bull


k

-4 -3 -2 1 0 1 2 3 4 n

0 0 0 0 0 0 1000 2100 3300 4600

t 1 0 0 0 0476 1000 2100 3300

2 0 0363 0476 1250 2100

t 3 0363 0684 1250

4 0684

Table I





ql = _1_ = 4762 q_1 = = 5238 21 21




FI(l) = (1 + X)-lFo() + (I + -1)-lFoX-1)

_x-1 -+1 +0

bull

34

IMR Winter 1969


E [ F((Xt+21I X = IJ = (x - 1) 1 f (1) t (1 +X)1 (x---=-I)

X-l=b








d(log~) V~I(Y)

E~=d(logV) it(Y)





35







Fn(X)

o S Fn()lt) =

7x ZdQn(Z)

~ (XZ - l)dQu(t)


~ ZdQn(Z) (41)



Therefore from (42)

Fn(XZ)X Z[----]

(XZ)F(XZ) ----gt 1



1



or

bull therefore

p(Xn) gt a

---------------~-- ---- shy

36

IMR Winter 1969


Conclusion


-


37

Appendix A






Proof




3 let v= v(Y) gt 0



4 Then


5 Therefore









38

IMR Winter 1969

Proof



3 Let v = v(Y) gt 0



4 Then (f



5 Therefore





FI(X) = lFo(3X)


FI1(X) = 1Fo(3nX)


b = E[FI1(XZ)F+l(X)] - 1

= t[lDFQ(3DX 3) 1 X)] - 13 n+l

= 15 - 1 5



I I -i


39



Proof










q(l) = Fn(Xl)F+1(X)



-

40

IMR Winter 1969


1000







Appendix B








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull

26

IMR Winter 1969






Jo U [(1 - w~)er + w2Z]d P(Z 1)







27


and (20) becomes








Explicit Solutions





F2(X) = e-2r0 F o(XZ)dQ(Z 2)

(25) or in general


28

IMR Winter 1969










= e-2r000 FIl(XR)dQ2(R) (28)

where





QI(Z) = f QI (~) dQI(V) ~ Q(Z2)

-(29)


29



F2(X) = e-2r10 Fo(XZ)dQ 2(Z)

(30)



1000 ZdQ2(Z) = e2

10 ZdQ(Z) = en (31)










~ xe-Iooo ZdQ(Z) em



30

IMR Winter 1969















X101 (1 - XZ)dQn(Z)] r~


== X - e-m + e-m 81(Xn)8(Xn) from (31)




bull

I


31



X 101 ZdQ(Z)

1 - ollX dQn(l)


1- 0 because Q(O+X) = 1 (34)



n-gtoo

Therefore


=X-1+1 X




32

IMR Winter 1969






1 + a Hgt- + X-I) (35)





i+igt- i+X-I (37)




Fo(X) = Max [OX - 1] (39)




111 - H + X ] [ H +X -I ) + ( + X )]

= (1 + )1]1 for i - +1


bull


k

-4 -3 -2 1 0 1 2 3 4 n

0 0 0 0 0 0 1000 2100 3300 4600

t 1 0 0 0 0476 1000 2100 3300

2 0 0363 0476 1250 2100

t 3 0363 0684 1250

4 0684

Table I





ql = _1_ = 4762 q_1 = = 5238 21 21




FI(l) = (1 + X)-lFo() + (I + -1)-lFoX-1)

_x-1 -+1 +0

bull

34

IMR Winter 1969


E [ F((Xt+21I X = IJ = (x - 1) 1 f (1) t (1 +X)1 (x---=-I)

X-l=b








d(log~) V~I(Y)

E~=d(logV) it(Y)





35







Fn(X)

o S Fn()lt) =

7x ZdQn(Z)

~ (XZ - l)dQu(t)


~ ZdQn(Z) (41)



Therefore from (42)

Fn(XZ)X Z[----]

(XZ)F(XZ) ----gt 1



1



or

bull therefore

p(Xn) gt a

---------------~-- ---- shy

36

IMR Winter 1969


Conclusion


-


37

Appendix A






Proof




3 let v= v(Y) gt 0



4 Then


5 Therefore









38

IMR Winter 1969

Proof



3 Let v = v(Y) gt 0



4 Then (f



5 Therefore





FI(X) = lFo(3X)


FI1(X) = 1Fo(3nX)


b = E[FI1(XZ)F+l(X)] - 1

= t[lDFQ(3DX 3) 1 X)] - 13 n+l

= 15 - 1 5



I I -i


39



Proof










q(l) = Fn(Xl)F+1(X)



-

40

IMR Winter 1969


1000







Appendix B








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull


27


and (20) becomes








Explicit Solutions





F2(X) = e-2r0 F o(XZ)dQ(Z 2)

(25) or in general


28

IMR Winter 1969










= e-2r000 FIl(XR)dQ2(R) (28)

where





QI(Z) = f QI (~) dQI(V) ~ Q(Z2)

-(29)


29



F2(X) = e-2r10 Fo(XZ)dQ 2(Z)

(30)



1000 ZdQ2(Z) = e2

10 ZdQ(Z) = en (31)










~ xe-Iooo ZdQ(Z) em



30

IMR Winter 1969















X101 (1 - XZ)dQn(Z)] r~


== X - e-m + e-m 81(Xn)8(Xn) from (31)




bull

I


31



X 101 ZdQ(Z)

1 - ollX dQn(l)


1- 0 because Q(O+X) = 1 (34)



n-gtoo

Therefore


=X-1+1 X




32

IMR Winter 1969






1 + a Hgt- + X-I) (35)





i+igt- i+X-I (37)




Fo(X) = Max [OX - 1] (39)




111 - H + X ] [ H +X -I ) + ( + X )]

= (1 + )1]1 for i - +1


bull


k

-4 -3 -2 1 0 1 2 3 4 n

0 0 0 0 0 0 1000 2100 3300 4600

t 1 0 0 0 0476 1000 2100 3300

2 0 0363 0476 1250 2100

t 3 0363 0684 1250

4 0684

Table I





ql = _1_ = 4762 q_1 = = 5238 21 21




FI(l) = (1 + X)-lFo() + (I + -1)-lFoX-1)

_x-1 -+1 +0

bull

34

IMR Winter 1969


E [ F((Xt+21I X = IJ = (x - 1) 1 f (1) t (1 +X)1 (x---=-I)

X-l=b








d(log~) V~I(Y)

E~=d(logV) it(Y)





35







Fn(X)

o S Fn()lt) =

7x ZdQn(Z)

~ (XZ - l)dQu(t)


~ ZdQn(Z) (41)



Therefore from (42)

Fn(XZ)X Z[----]

(XZ)F(XZ) ----gt 1



1



or

bull therefore

p(Xn) gt a

---------------~-- ---- shy

36

IMR Winter 1969


Conclusion


-


37

Appendix A






Proof




3 let v= v(Y) gt 0



4 Then


5 Therefore









38

IMR Winter 1969

Proof



3 Let v = v(Y) gt 0



4 Then (f



5 Therefore





FI(X) = lFo(3X)


FI1(X) = 1Fo(3nX)


b = E[FI1(XZ)F+l(X)] - 1

= t[lDFQ(3DX 3) 1 X)] - 13 n+l

= 15 - 1 5



I I -i


39



Proof










q(l) = Fn(Xl)F+1(X)



-

40

IMR Winter 1969


1000







Appendix B








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull

28

IMR Winter 1969










= e-2r000 FIl(XR)dQ2(R) (28)

where





QI(Z) = f QI (~) dQI(V) ~ Q(Z2)

-(29)


29



F2(X) = e-2r10 Fo(XZ)dQ 2(Z)

(30)



1000 ZdQ2(Z) = e2

10 ZdQ(Z) = en (31)










~ xe-Iooo ZdQ(Z) em



30

IMR Winter 1969















X101 (1 - XZ)dQn(Z)] r~


== X - e-m + e-m 81(Xn)8(Xn) from (31)




bull

I


31



X 101 ZdQ(Z)

1 - ollX dQn(l)


1- 0 because Q(O+X) = 1 (34)



n-gtoo

Therefore


=X-1+1 X




32

IMR Winter 1969






1 + a Hgt- + X-I) (35)





i+igt- i+X-I (37)




Fo(X) = Max [OX - 1] (39)




111 - H + X ] [ H +X -I ) + ( + X )]

= (1 + )1]1 for i - +1


bull


k

-4 -3 -2 1 0 1 2 3 4 n

0 0 0 0 0 0 1000 2100 3300 4600

t 1 0 0 0 0476 1000 2100 3300

2 0 0363 0476 1250 2100

t 3 0363 0684 1250

4 0684

Table I





ql = _1_ = 4762 q_1 = = 5238 21 21




FI(l) = (1 + X)-lFo() + (I + -1)-lFoX-1)

_x-1 -+1 +0

bull

34

IMR Winter 1969


E [ F((Xt+21I X = IJ = (x - 1) 1 f (1) t (1 +X)1 (x---=-I)

X-l=b








d(log~) V~I(Y)

E~=d(logV) it(Y)





35







Fn(X)

o S Fn()lt) =

7x ZdQn(Z)

~ (XZ - l)dQu(t)


~ ZdQn(Z) (41)



Therefore from (42)

Fn(XZ)X Z[----]

(XZ)F(XZ) ----gt 1



1



or

bull therefore

p(Xn) gt a

---------------~-- ---- shy

36

IMR Winter 1969


Conclusion


-


37

Appendix A






Proof




3 let v= v(Y) gt 0



4 Then


5 Therefore









38

IMR Winter 1969

Proof



3 Let v = v(Y) gt 0



4 Then (f



5 Therefore





FI(X) = lFo(3X)


FI1(X) = 1Fo(3nX)


b = E[FI1(XZ)F+l(X)] - 1

= t[lDFQ(3DX 3) 1 X)] - 13 n+l

= 15 - 1 5



I I -i


39



Proof










q(l) = Fn(Xl)F+1(X)



-

40

IMR Winter 1969


1000







Appendix B








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull


29



F2(X) = e-2r10 Fo(XZ)dQ 2(Z)

(30)



1000 ZdQ2(Z) = e2

10 ZdQ(Z) = en (31)










~ xe-Iooo ZdQ(Z) em



30

IMR Winter 1969















X101 (1 - XZ)dQn(Z)] r~


== X - e-m + e-m 81(Xn)8(Xn) from (31)




bull

I


31



X 101 ZdQ(Z)

1 - ollX dQn(l)


1- 0 because Q(O+X) = 1 (34)



n-gtoo

Therefore


=X-1+1 X




32

IMR Winter 1969






1 + a Hgt- + X-I) (35)





i+igt- i+X-I (37)




Fo(X) = Max [OX - 1] (39)




111 - H + X ] [ H +X -I ) + ( + X )]

= (1 + )1]1 for i - +1


bull


k

-4 -3 -2 1 0 1 2 3 4 n

0 0 0 0 0 0 1000 2100 3300 4600

t 1 0 0 0 0476 1000 2100 3300

2 0 0363 0476 1250 2100

t 3 0363 0684 1250

4 0684

Table I





ql = _1_ = 4762 q_1 = = 5238 21 21




FI(l) = (1 + X)-lFo() + (I + -1)-lFoX-1)

_x-1 -+1 +0

bull

34

IMR Winter 1969


E [ F((Xt+21I X = IJ = (x - 1) 1 f (1) t (1 +X)1 (x---=-I)

X-l=b








d(log~) V~I(Y)

E~=d(logV) it(Y)





35







Fn(X)

o S Fn()lt) =

7x ZdQn(Z)

~ (XZ - l)dQu(t)


~ ZdQn(Z) (41)



Therefore from (42)

Fn(XZ)X Z[----]

(XZ)F(XZ) ----gt 1



1



or

bull therefore

p(Xn) gt a

---------------~-- ---- shy

36

IMR Winter 1969


Conclusion


-


37

Appendix A






Proof




3 let v= v(Y) gt 0



4 Then


5 Therefore









38

IMR Winter 1969

Proof



3 Let v = v(Y) gt 0



4 Then (f



5 Therefore





FI(X) = lFo(3X)


FI1(X) = 1Fo(3nX)


b = E[FI1(XZ)F+l(X)] - 1

= t[lDFQ(3DX 3) 1 X)] - 13 n+l

= 15 - 1 5



I I -i


39



Proof










q(l) = Fn(Xl)F+1(X)



-

40

IMR Winter 1969


1000







Appendix B








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull

30

IMR Winter 1969















X101 (1 - XZ)dQn(Z)] r~


== X - e-m + e-m 81(Xn)8(Xn) from (31)




bull

I


31



X 101 ZdQ(Z)

1 - ollX dQn(l)


1- 0 because Q(O+X) = 1 (34)



n-gtoo

Therefore


=X-1+1 X




32

IMR Winter 1969






1 + a Hgt- + X-I) (35)





i+igt- i+X-I (37)




Fo(X) = Max [OX - 1] (39)




111 - H + X ] [ H +X -I ) + ( + X )]

= (1 + )1]1 for i - +1


bull


k

-4 -3 -2 1 0 1 2 3 4 n

0 0 0 0 0 0 1000 2100 3300 4600

t 1 0 0 0 0476 1000 2100 3300

2 0 0363 0476 1250 2100

t 3 0363 0684 1250

4 0684

Table I





ql = _1_ = 4762 q_1 = = 5238 21 21




FI(l) = (1 + X)-lFo() + (I + -1)-lFoX-1)

_x-1 -+1 +0

bull

34

IMR Winter 1969


E [ F((Xt+21I X = IJ = (x - 1) 1 f (1) t (1 +X)1 (x---=-I)

X-l=b








d(log~) V~I(Y)

E~=d(logV) it(Y)





35







Fn(X)

o S Fn()lt) =

7x ZdQn(Z)

~ (XZ - l)dQu(t)


~ ZdQn(Z) (41)



Therefore from (42)

Fn(XZ)X Z[----]

(XZ)F(XZ) ----gt 1



1



or

bull therefore

p(Xn) gt a

---------------~-- ---- shy

36

IMR Winter 1969


Conclusion


-


37

Appendix A






Proof




3 let v= v(Y) gt 0



4 Then


5 Therefore









38

IMR Winter 1969

Proof



3 Let v = v(Y) gt 0



4 Then (f



5 Therefore





FI(X) = lFo(3X)


FI1(X) = 1Fo(3nX)


b = E[FI1(XZ)F+l(X)] - 1

= t[lDFQ(3DX 3) 1 X)] - 13 n+l

= 15 - 1 5



I I -i


39



Proof










q(l) = Fn(Xl)F+1(X)



-

40

IMR Winter 1969


1000







Appendix B








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull


31



X 101 ZdQ(Z)

1 - ollX dQn(l)


1- 0 because Q(O+X) = 1 (34)



n-gtoo

Therefore


=X-1+1 X




32

IMR Winter 1969






1 + a Hgt- + X-I) (35)





i+igt- i+X-I (37)




Fo(X) = Max [OX - 1] (39)




111 - H + X ] [ H +X -I ) + ( + X )]

= (1 + )1]1 for i - +1


bull


k

-4 -3 -2 1 0 1 2 3 4 n

0 0 0 0 0 0 1000 2100 3300 4600

t 1 0 0 0 0476 1000 2100 3300

2 0 0363 0476 1250 2100

t 3 0363 0684 1250

4 0684

Table I





ql = _1_ = 4762 q_1 = = 5238 21 21




FI(l) = (1 + X)-lFo() + (I + -1)-lFoX-1)

_x-1 -+1 +0

bull

34

IMR Winter 1969


E [ F((Xt+21I X = IJ = (x - 1) 1 f (1) t (1 +X)1 (x---=-I)

X-l=b








d(log~) V~I(Y)

E~=d(logV) it(Y)





35







Fn(X)

o S Fn()lt) =

7x ZdQn(Z)

~ (XZ - l)dQu(t)


~ ZdQn(Z) (41)



Therefore from (42)

Fn(XZ)X Z[----]

(XZ)F(XZ) ----gt 1



1



or

bull therefore

p(Xn) gt a

---------------~-- ---- shy

36

IMR Winter 1969


Conclusion


-


37

Appendix A






Proof




3 let v= v(Y) gt 0



4 Then


5 Therefore









38

IMR Winter 1969

Proof



3 Let v = v(Y) gt 0



4 Then (f



5 Therefore





FI(X) = lFo(3X)


FI1(X) = 1Fo(3nX)


b = E[FI1(XZ)F+l(X)] - 1

= t[lDFQ(3DX 3) 1 X)] - 13 n+l

= 15 - 1 5



I I -i


39



Proof










q(l) = Fn(Xl)F+1(X)



-

40

IMR Winter 1969


1000







Appendix B








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull

32

IMR Winter 1969






1 + a Hgt- + X-I) (35)





i+igt- i+X-I (37)




Fo(X) = Max [OX - 1] (39)




111 - H + X ] [ H +X -I ) + ( + X )]

= (1 + )1]1 for i - +1


bull


k

-4 -3 -2 1 0 1 2 3 4 n

0 0 0 0 0 0 1000 2100 3300 4600

t 1 0 0 0 0476 1000 2100 3300

2 0 0363 0476 1250 2100

t 3 0363 0684 1250

4 0684

Table I





ql = _1_ = 4762 q_1 = = 5238 21 21




FI(l) = (1 + X)-lFo() + (I + -1)-lFoX-1)

_x-1 -+1 +0

bull

34

IMR Winter 1969


E [ F((Xt+21I X = IJ = (x - 1) 1 f (1) t (1 +X)1 (x---=-I)

X-l=b








d(log~) V~I(Y)

E~=d(logV) it(Y)





35







Fn(X)

o S Fn()lt) =

7x ZdQn(Z)

~ (XZ - l)dQu(t)


~ ZdQn(Z) (41)



Therefore from (42)

Fn(XZ)X Z[----]

(XZ)F(XZ) ----gt 1



1



or

bull therefore

p(Xn) gt a

---------------~-- ---- shy

36

IMR Winter 1969


Conclusion


-


37

Appendix A






Proof




3 let v= v(Y) gt 0



4 Then


5 Therefore









38

IMR Winter 1969

Proof



3 Let v = v(Y) gt 0



4 Then (f



5 Therefore





FI(X) = lFo(3X)


FI1(X) = 1Fo(3nX)


b = E[FI1(XZ)F+l(X)] - 1

= t[lDFQ(3DX 3) 1 X)] - 13 n+l

= 15 - 1 5



I I -i


39



Proof










q(l) = Fn(Xl)F+1(X)



-

40

IMR Winter 1969


1000







Appendix B








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull


k

-4 -3 -2 1 0 1 2 3 4 n

0 0 0 0 0 0 1000 2100 3300 4600

t 1 0 0 0 0476 1000 2100 3300

2 0 0363 0476 1250 2100

t 3 0363 0684 1250

4 0684

Table I





ql = _1_ = 4762 q_1 = = 5238 21 21




FI(l) = (1 + X)-lFo() + (I + -1)-lFoX-1)

_x-1 -+1 +0

bull

34

IMR Winter 1969


E [ F((Xt+21I X = IJ = (x - 1) 1 f (1) t (1 +X)1 (x---=-I)

X-l=b








d(log~) V~I(Y)

E~=d(logV) it(Y)





35







Fn(X)

o S Fn()lt) =

7x ZdQn(Z)

~ (XZ - l)dQu(t)


~ ZdQn(Z) (41)



Therefore from (42)

Fn(XZ)X Z[----]

(XZ)F(XZ) ----gt 1



1



or

bull therefore

p(Xn) gt a

---------------~-- ---- shy

36

IMR Winter 1969


Conclusion


-


37

Appendix A






Proof




3 let v= v(Y) gt 0



4 Then


5 Therefore









38

IMR Winter 1969

Proof



3 Let v = v(Y) gt 0



4 Then (f



5 Therefore





FI(X) = lFo(3X)


FI1(X) = 1Fo(3nX)


b = E[FI1(XZ)F+l(X)] - 1

= t[lDFQ(3DX 3) 1 X)] - 13 n+l

= 15 - 1 5



I I -i


39



Proof










q(l) = Fn(Xl)F+1(X)



-

40

IMR Winter 1969


1000







Appendix B








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull

34

IMR Winter 1969


E [ F((Xt+21I X = IJ = (x - 1) 1 f (1) t (1 +X)1 (x---=-I)

X-l=b








d(log~) V~I(Y)

E~=d(logV) it(Y)





35







Fn(X)

o S Fn()lt) =

7x ZdQn(Z)

~ (XZ - l)dQu(t)


~ ZdQn(Z) (41)



Therefore from (42)

Fn(XZ)X Z[----]

(XZ)F(XZ) ----gt 1



1



or

bull therefore

p(Xn) gt a

---------------~-- ---- shy

36

IMR Winter 1969


Conclusion


-


37

Appendix A






Proof




3 let v= v(Y) gt 0



4 Then


5 Therefore









38

IMR Winter 1969

Proof



3 Let v = v(Y) gt 0



4 Then (f



5 Therefore





FI(X) = lFo(3X)


FI1(X) = 1Fo(3nX)


b = E[FI1(XZ)F+l(X)] - 1

= t[lDFQ(3DX 3) 1 X)] - 13 n+l

= 15 - 1 5



I I -i


39



Proof










q(l) = Fn(Xl)F+1(X)



-

40

IMR Winter 1969


1000







Appendix B








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull


35







Fn(X)

o S Fn()lt) =

7x ZdQn(Z)

~ (XZ - l)dQu(t)


~ ZdQn(Z) (41)



Therefore from (42)

Fn(XZ)X Z[----]

(XZ)F(XZ) ----gt 1



1



or

bull therefore

p(Xn) gt a

---------------~-- ---- shy

36

IMR Winter 1969


Conclusion


-


37

Appendix A






Proof




3 let v= v(Y) gt 0



4 Then


5 Therefore









38

IMR Winter 1969

Proof



3 Let v = v(Y) gt 0



4 Then (f



5 Therefore





FI(X) = lFo(3X)


FI1(X) = 1Fo(3nX)


b = E[FI1(XZ)F+l(X)] - 1

= t[lDFQ(3DX 3) 1 X)] - 13 n+l

= 15 - 1 5



I I -i


39



Proof










q(l) = Fn(Xl)F+1(X)



-

40

IMR Winter 1969


1000







Appendix B








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull

36

IMR Winter 1969


Conclusion


-


37

Appendix A






Proof




3 let v= v(Y) gt 0



4 Then


5 Therefore









38

IMR Winter 1969

Proof



3 Let v = v(Y) gt 0



4 Then (f



5 Therefore





FI(X) = lFo(3X)


FI1(X) = 1Fo(3nX)


b = E[FI1(XZ)F+l(X)] - 1

= t[lDFQ(3DX 3) 1 X)] - 13 n+l

= 15 - 1 5



I I -i


39



Proof










q(l) = Fn(Xl)F+1(X)



-

40

IMR Winter 1969


1000







Appendix B








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull


37

Appendix A






Proof




3 let v= v(Y) gt 0



4 Then


5 Therefore









38

IMR Winter 1969

Proof



3 Let v = v(Y) gt 0



4 Then (f



5 Therefore





FI(X) = lFo(3X)


FI1(X) = 1Fo(3nX)


b = E[FI1(XZ)F+l(X)] - 1

= t[lDFQ(3DX 3) 1 X)] - 13 n+l

= 15 - 1 5



I I -i


39



Proof










q(l) = Fn(Xl)F+1(X)



-

40

IMR Winter 1969


1000







Appendix B








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull

38

IMR Winter 1969

Proof



3 Let v = v(Y) gt 0



4 Then (f



5 Therefore





FI(X) = lFo(3X)


FI1(X) = 1Fo(3nX)


b = E[FI1(XZ)F+l(X)] - 1

= t[lDFQ(3DX 3) 1 X)] - 13 n+l

= 15 - 1 5



I I -i


39



Proof










q(l) = Fn(Xl)F+1(X)



-

40

IMR Winter 1969


1000







Appendix B








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull


39



Proof










q(l) = Fn(Xl)F+1(X)



-

40

IMR Winter 1969


1000







Appendix B








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull

40

IMR Winter 1969


1000







Appendix B








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull


41


Fn+1(X) e- r10 Fn(XZ)dQ(Z 1) (43)

10 ZdQ(Z 1) = er-o (44)








=X - IX gt Cn







42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull

42

IMR Winter 1969




F(cc) Coo -1




lXOlt



(49)



bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull


43

Footnotes

See Samuelson [6J

2See Kassouf [2J








For the family

e bX e -U(X)


e


family




bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull

44

IMR Winter 1969















bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull




] F(XZ)dQmiddot(Z)




bull

46

IMR Winter 1969

References








bull

46

IMR Winter 1969

References








bull

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