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(Jml aHociated compallies ill



' LE 4
/3 en

where that last expression is valid in the limit as In goes to infinity, cOllesponding to continuous compounding Hence continuous compounding leads to the familiar expo~ nential growth CUlve Such a curve is shown in FigUle 2 2 for a 10% nominal interest late

We have examined how a single investment (say a bank deposit) glows over time due to intelest compounding It should be clem that exactly the same thing happens 10 debt It I bonoll' money from the biwk at an intelest rate 1 and make no payments to the bank, then my debt increases accOJding to the same formulas Specifically, if my debt is compounded monthly, then after k months my debt will have grown by a factor of [I + (I /12) l'

14 12 10



FIGURE 2.2 Expollential growth curve; cOllfinuous compoUlld growth, Under conl;nuotls compounding at 1D'X" the value of $1 doubles in abotll 7 yems In 20 yems it grows by a factor of ilbotll B

!5 ~

0 0 4 10 12 14 16 18 20 22 24

Money Markets
Although we have treated interest as a given known value, in reality thew are many different rates each day Diftetent rates apply to different cflCufllstunces, different user classes, and different period, Most .ates me established by the forces at supply and demand ill broad markets to which they apply These rates are published widely; a snmpling for one day is shown in Table 2 2 Many of these market rates are discussed

Market Interest Rates

Interest rates (August 9, 1995)
U S I rCl.lsury bUb and \lotes 1-momh bfll 6-monlh biIl

I-yeur bill
1-year note (% yield)

539 539 536 605
649 692 56875

IO-year note (r/o yield) 3D-year bOlld (% yield) Fed tu nds nUe Discount rate PIime tate Comtnercial paper Certificates of deposit I month 2 mouths I year Banker's m:ceptam:es (30 days) London late Eurodollars (I month) London Interbank offered rate (I l1Iomh) Federal Hotne Loan Mortgage Corp (Freddie Mae) (30 yeats) lIIal/l' different /'{/(e\ (/pph'
0/1 (/II\' gil't'll

875 584 5 17

528 568 575 588 794


711il' 1\' (/



Chapler 2

THE BASIC THEORY OF INTEREST mote fully in Chapters 3 and 4 Not all interest rates are broad market rales There may be pdvate rates negotiated by two private parties, Or in the context of a finn, spedal rates may be established for internal transactions or tor the purpose of evaluating projects, as discussed laler in this chapter

The theme ot the pteVfOUS section is that money invested today leads to increased value in the future as a result of intefest The fOfmulas of the previous section show how to detefminc this future value That whole set of concepts and formulas CHn be reversed in Lime to calculate the value that should be assigned flOW, in the present, to money that is to be feceived at a laler Hme This feversal is the essence of the extremely imponant concept ot present value. To introduce this concept, consider two situations: (1) you will receive $110 in I yem, (2) you receive $100 now and deposit it in a bank account fOf I year at 10% interest Cleaf"ly these situations are identical after I year-you wHl receive $110 We can restate this equivalence by saying that $110 received in I year is equivalent to the receipt of $100 now when the interest ,ate is 10% Or we say that the $110 to be received in I year has a present value of $100 In general, $1 to be received a yem in the future has a present value of $1/0 + 1), whefe , is the interest rale A similar ttansfollnation applies to future obligations such as the repayment of debt Suppose that, fOf some reason, you have an obligation to pay someone $100 in exactly I year This obligation can be regarded as a negative cash flow that occurs at the end of the year To calculate the p,esent value of this obligation, you determine how much money you would need l1n1l' in order to cover the obligation This is easy to determine It the CU[lent yearly interest rate is r, you need $100/0 + I) II that amount ot mone y is deposited in the bank now, it wHi grow to $100 at the end of the yeat You Clln then fully meet the obligation The present value of the obligation is therefore $100/(1 + 1)< The pfocess of evaluating future obligations as an equivalent present value is altefnatfvely referred to as discounting. The present value ot a tuture monetary amount is less than the face value of that amount, so the future value must be discounted to obtain the present value The tactof by which the futUfe value must be discounted is called the discount factor. The I-year discount factor is d, 1/(1 + I), where r is the I-year intetest fate So if an amount A is to be received in I year, the present value is the discounted amount d l A The tormula for presenl value depends on the inlcrest rate that is available from a bank or othef source II that source quotes rates with compounding, then such a compound interest late should be used in the calculation of present value As an example, suppose that the annual interest tate, is compounded at the end of each of III equal periods each year; and suppose that a cash payment of amount A will be received at the end 01 the "th period Then the appropriate discount

23 factor is




+ (! /111)]'

The pfesent value 01 a payment ot A to be received k periods in the future is dJ.A,

The pfcvious section studied the impact of interest on a single cash deposit 01 loan; that is, on a single cash lIow We now extend that discussion to the case where cash flows occur at several time periods, and hence constitute a cash flow stream or sequence FiIst we require a new concept

The Ideal Bank
When discussing cash lIow streams, it is useful to define the notion ot an ideal bank. An ideal bank applies the same I ate 01 interest to both deposit, and loans, and it has no service charges or transactions lees Its interest rate applies equally to any size of principal, tram I cent (or traction thel'eof) to $1 million (or even more) Furthermore, separate transactions in an account are completely additive in their effect on tuture balances Note that the definition 01 an ideal bank does 1101 imply that interest rates tor all tlansactions arc identical For example, a 2-yem celtificate of deposit (CD) might ofter tl higher rate than a I-year CD However, the 2-year CD must offer the same fate as a loan that is payable in 2 years If an ideal bank has an inter est I ate that is independent 01 the length ot time tot which it applies, and that interest is compounded according to nOfmal rules, it is said to bc a constant ideal bank. In the fest ot this chapter, we always assume that intelest fates afe indeed constant fhe constant ideal bank is tlte refcrence point used to describe the outside financial mat ket-thc public market for money

Future Value
Now we retm n to the study ot cash now SUeams Let us decide on a fixed timc cycle for compounding (for example, yeady) and let" period be the length at this cycle We assume that cash flows OCCUI at the end 01 each period (although some flows might be zero) We shall take each cash flow and deposit it in a constant ideal bank as it anives (It the flow is negative, we covel it by taking out a loan) Under the terms of a constant ideal bank, the final balance in OUt account can be found by combining the results of the individual Explicitly, consider the cash flow stleam (xo, \'f, , Xli) At the end ot 1/ periods the initial cU.lih now ro will have grown to xo( I + I yl, where, is the


Chapter 2


interesl rate pel peliod (which is the yearly rale divided by the number of pedods per received after the firsl period, will al the final time have year) The nexl cash flow, been in the account for only 11-1 periods, and hence it will have a value ot Xf (I +r )II-f Likewise, the next flow .\1 wHl coIled inlerest during 11 - 2 periods and have value X2( 1+, )n-2 The final flow XI/ wfJI not collect any interest, so will remain Xli The lotal xo(l + I)" + x,(I + 1)"-' + + x,,, value at the end of II periods is therefore FY


To summarize:
Future value oj a stream

Given a




(Xo, .rf,

1 each peliod, tbe rull11e value oj the ~/lea11l i~

+ I)" +'" (I + I )"-' +

+ x"

Example 2.1 (A short stream) Consider the cash flow stream (-2, l, I, I) when the periods are years and the interest rate is 10% The future value is FY

-2 x

(I 1)3 +

I x (l I)'


I x I I




(2, I)

This formula for future value always uses the interest rate per period and assumes that interest rates are compounded each period

Present Value
The present value of a general cash flow stream-like the future value-can also be calculated by considering each flow element sepal ately Again consider the stream (.to, tf, , .t1l ), The present value of the first elementxo is just that value itselt since no discounting is necessary The present value ot the flow .\ 1 is .\'1/( I + 1), because thal flow musl be discounled by one period (Again the inleresll ate I is the per-period late) Continuing in this way, we find that the present value of the entire stream is PY xo+X,/(I+I)+",/(I+I)'+ +X,,/(I+l)" Wesummarizelhisimporlant result as follows

Presellt value of a !;tream Gh'en Q [(1'1h /1011' \tleam (xo,.\(, late I pel peIind. the pJe\ellt \I{llue oj tl1i\ [Q'Ih /1011' \tleQ11I i\







Example 2.2 Again conside, the cush now slream (-2, I, I, I) Using an interesl rate of 10% we have PY

- 2+

II + (jI)" + (jI)"








The plesent value of a cash flow slfeam can be regufded as the pfcsenl payment amounl that is equivalent to the entire SlIeam Thus we can think of life entire stream as being replaced by a single flow at the initial time Thele is another way to inletptet the formula


pfcscnl value that is based

on ltanstortning the formula tot fUlufe value FUlUle value is the amounl ot fUlure payment that is equivalent to the cnthe stream We can think ot life stream as being

lIUnsforfned into that single cash flow at period


The present value ot this single

equivalellt flow is obtained by discounting it by (I -1- I)" That is, the plesent value and the future value are related by

In the plevious examples 101 the cash now slleam (-2, I, I, I) we have 487 FV /(1 1)3 648/1 331 487


Frequent and Continuous Compounding
Suppose that

is the nominal annual interest rate and interest is compounded at


equally spaced periods pel yeU! Suppose that cash nows oceul inititllly and at the end of each period tor a total ot 11 periods, forming a sUeHfn (Xn,Xf, ,XII) Then

according to the preceding we have


kd. [I

+ (1/111)1'


Suppose now that the nominal interest rate 1 is compounded continuously and ,1/1 (We have h kim for the stleam in the cash Hows occur at times In, If, previous pmagraph; but the more general situation is allowed here) We denote the cash now at time" by tUd In that case,

" L t(l,)e-'" k::::tl This is the continuous compounding tormula tOf pfesent value

Present Value and an Ideal Bank
We know that au ideal bank can be used to change the pattern of a cash flow slf eam
FOI example, a 10% bank can change the stream (1,0,0) into the stream (0,0, I 21)

by receiving a deposit of $1 now and payiug principal and interest of $1 21 in 2 yenrs The bank can I Hefe is an iterative technique that generates a sequence AO, AI' A2. ,Ak, ot estimates that converges to the root X > 0, solving / (X) = 0 Start with 0 close to the solution Assuming A~ has been colcl1lated, evaluate

t and define

(Ad = {{I

+ 2a2Ak + 3aJAi +
A - A _ J(Ad k + IlllnArl

k+1 -


This is Newton's method It is based on approximating the function / by a line tangent to its graph at Ab as shown in Figure 1. 4 Try the procedure on J(A) = -I + A + A1 Start with AO = I and compute four additional estimates fiGURE 2.4

Newton's method

/ /


71 1 1 1

°Exercises rollowed by


arc Inillhcnlatic;.lily more difficult thull nvcmgc


Chapter 2

5. (A prize) A major lottery advertises thilt it pays the winner $10 miiHon However, this prize money is paid at the rate of $500,000 each year (with the first payment being immediate) fot a total of 20 payments What is the present value of this prize at 10% interest? 6, (Sunk costs) A YDltng couple has tmtde u nonrefundable deposit of the first month's rent (equal to $1,000) 011 U 6-month apartment lease The next day they find a diftelcnt apartment that they Hke just as well, but its mOllthly rent is only $900 They plun to be in the apartment only 6 months Should they switch to the new aparttnent? What jf they plan to stay 1 year? Assume an interest rate of 11% 7. (Shortcut) Gavin Jones is inquisitive and tJetermined to learn both the theory and the application of investtnent theory He pressed the tree farmer for additional information and lemned that it W j = 0, 1,2, . if ,\"o < Yo and L.;I..,O ri > Let Pr{d) and P.l'(d) denote the present values of these two projects when the discount factor is d


(a) Show that there is a crossover v 0 such that Pr(c) = Py(e)

(b) For Exercise II, calculate the crossover value

14. (Depredation choice)

In the United States the accelerated cost recovery system (ACRS)

must be used for depreciation of assets placed into SCI vice afteI December 1980 In this system, assets ilre classified fnto categories specifying the effective ttlx life The classific 0 exactly when v > 0, so the condition tor acceptance of the project based on whether A > 0 coincides with the net present value criterion

(v, 0, 0,

Example 3,4 (A capital cost) The purchase of a new machine f'lf $100,000 (at time zero) is expected to generate additional revenues of $25,000 tor the next 10 years starting at year I If the discount rate is 16%, is this a profitable investment? We simply need to determine how to amortize the initial cost unifOlmly ovel 10 yems; that is, we need to find the annual payments at 16% that are equivalent to the original cost Using the annuity formula, we find that this corresponds to $20,690 per year Hence the annual worth ot the project is $25,000 $20,690 = $4,310, which is positive; thus the investment is profitable Note that if the purchase oj the machine were financed at 16% over 10 yems, the ClUIW/ yemly net cash Hows would cones pond exactly to the annual worth

Bonds lepresent by far the greatest monetulY value ot fixed~income securities and me, as a class, the most liquid oj these securities, We devote special attention to bonds, both because oj their practicnl impOitance tl i.\ tlte 1{fte 0./ inte!e!Jt between thme tintes that is c01O'htent with a given spot rate cUlve Unde! \'miom compollnding cOllFentiom the /01 ward mtelJ me Jpecijied as/allows (a) Year·ly Fm vemlv compoullding, the /onl'md rate.\ salisfv,


j > i,

+ sY


+ s,); (I + f; j )j-'


+ Il)l] Ij(j-I)

[ (I +.\.;)'


Chapter 4

(b) m periods per year F01 m peliod-pel-vem compounding, the f01I1'md zate'i WIis/v, 101 j > t, expleBed in pelioe/s,
(l + Ij/lII)j = (I + 1,/111)' (l +



Hence, (I + \}/IIl)j]i lU -') [ (1+1,/111)'


-Ill ffl.{,

(c) Continuous F01 continuoif\' Lompmmdillg, the fmwmd rate} all II alld lI'ilill'2 > and wli'l/)'



are defined 101


h:= 'i{~/2 -


'2 -',


Note again that continuous compounding produces the simplest formula As a further convention, it is useful to define spot rates, discount factors, and forward rates when one of the time points is zero, representing current time Hence we define Stu := 0 and correspondingly dto = 1, where 10 is the cunenl time (Alternatively we write So = 0 and do = I when denoting time by period integers) For forward rutes, we write similarly ffo f) = 'tfl The fOlward rates from time zero are the corresponding spot rates There are a large number of forward rates associated with a spot rate curve, In tact, it there are II periods, there are II spot rates (excluding 'to); and there are /1(/1 + 1)/2 forward rates (including lhe basic _'polrales) However, all lhese forward rates are derived from the 11 underlying spot rates The forward rates are introduced partly because they represent rates of actual transactions Forward contracts do in fact serve a very important hedging role, and their use in this manner is discussed further in Chaptel 10 They are introduced here, however, mainly because they ale important tor the full development ot the term ,'lUucture theory They me u!'ied briefly in the next section and then extensively in the section following that

The yield curve can be observcd, at lenst roughly, by looking at a series of bond quotes in the financial press The curve is almost never fiat but, rather, it usually slopes gIadually upward U!'i matuIity increases The spot rate CUIve has similru characteristics Typically it, too, slopes lapidly upward at shOlt maturities and continues to slope upward, but more graduaJIy as maturities lengthen It is natural to ask if there is a simple explanation tOl this typical shape Why is the CUI ve not just tIat at a common interest rate? There are three standard cxplanations (or "theories") foJ' the telln structure, each ot which plOvides some importnnt insight We outline them briefly in this section




Expectations Theory
The first explanation is that spot rales me determined by expectations ot what rales will be in the future To visualize this plOcess, suppose that, as is usually the, the spot rate curve slopes upwald, with rates increasing tOl longer maturities The 2-yeUl rate is greater than the I-year rate It is argued that this is so because the mmket (that is, the collective of all people who trade in the interest rate market) believes that the I-yeal rate will most likely go up next year (This belief may, tor example, be beCl.lUSe most people believe inflation will lise, and thus to maintain the same real rate ot interest, the nominal rtlle must increase) Thi!'i majority beliet that the interest rate will rise translates into a mmket expeL/alion An expectation is only an average guess; it is not definite information-for no one knows tor Sure what will happen next year-but people on aveIl.lge assume, accOlding to this explnnation, that the rate will increase This mgument is made mOle concrete by expressing the expectations in telms of fmward rates This more precise tormulation is the expectations hypothesis. To outline this hypothesis, consiJel the forward rate 112, which is the implied late for money loaned for I year, a year from now According to the expectations hypothesis, this forwald rate is exacl/" equal to the market expectation of what the I-year spot rate will be next yem Thus the expectation can be infelred trom existing rates Eadiel we consideled a situation where J I = 7% and 52 = 8% We tound that the implied fOlwm'd late was 112 = 9,01% Accmding to the unbiased expectations hypothesis, this value ot 90 I % is the mUlket's expected value of next yem's I-yell! spot late .'I; The Same Ulgument applies to the othel rates as well As additional spot I ates are consideled, they define cOlresponding fOlward lUtes fOl next yeal Specifically, s!, .~:!, and 5J togethel detelmine the torwUld Hltes h 2 and II J The second of these is the fOlWUld late tOl bOlIowing money for 2 yeals, stalting next yenr This Hlte is assumed to be equal to the cunent expectation of what the 2-yem spot lUte s~ will be next year In genelal, then, the CUlfcnt spot lUte Cll!ve leads to a set ot forwald lutes Ii 2, iJ.3, ,1111, which define the expected spot HUe curve ,\~, ~;, '~:I_I tOl next year The expectations are inherent in the cunent spot late structure There ale two ways ot looking at this construction One way is that the current spot rate CUI ve implies an expectation about what the spot lUte Cll! ve will be next year The othel way is to tUIn this first view mound and say that the expectation ot next year's CUlve detel111ines what the CUllent spot lUte cll!ve must be Both vicws me intellwined; expectations about future lUtes Ule paIt ot today's mmket and inlluence today's lUtes This theOlY 01 hypothesis is a nice explanation ot the spot lute cUlve, even though it has some impOltant weaknesses The primmy weakness is that, accOlding to this explanation, the market expects lates to inclease whenevel the spot late cUlve slopes upwUld; tlnd this is pIactically all the time Thus the expectations cannot be light even on avelage, since lUtes do not go up as often as ex.pectations would imply Nevellheless, the expectations explanation is plausible, although the expectations may themselves be skewed The expectations explanation ot the tel m Stl ucttHC can be leg~llded us being (loosely) based on the compmison pIinciple To sec this, considel again the 2-yem


Chapter 4

THE TERM STRUCTURE OF INTEREST RATES situation, An invcslOl can invest either in a 2-yeaI instrument or in a I-year instrument

followed by another I-year investment. The follow-on investment can also be cLlnied oullwo ways It can be llnanged cunenlly lhlOugh a fOlward contract at rate [1.2, or it can simply be "lOlled over" by reinvesting the following yeat at the then prevailing I-yelll rale A wise inveslOl would compme the two altematives. If the investor expects that next yem's I-year rule will equal the current value of 11.:1" then he or she will be indiffelCnt between these two alternatives Indeed, the fact that both are viable implies that they must seem (approximately) equal

Liquidity Preference
The liquidity prefelcnce explanation assellS that inveslOlS usually plefer shOll-telm fixed income secuIities ovel long-term secuIities, The simplest justification for this assellion is that investOls do not like to tie up capital in long-term seclJrities, since those tunds may be needed befOle the matUlity date InvestOlS pl'efel their funds to be liquid rathel than tied up, However, thc term liquiditv is used in a slightly nonstandard way in this algument Thele me large active markets tOl bonds of majOl corporations and of the TIeaSuIY, so it is easy to sell any such bonds one might hold Short-term and long-telln bonds of this type me equally liquid Liquidity is used in this explanation of the tellTI structUle shape instead to express lhe fnctthat most investors plefel shOlt-tel111 bonds to long~telm bonds The reason fOl this pletelence is that investors anticipate that they may need to sell theil bonds soon, and lhey recognize that long-term bonds ru'e mOle sensitive to interest rate changes than are shOlt-lellTI bonds Hence an investOl who may need funds in a year or so will be be leluctant to place these tunds in long-term bonds because of the relatively high near-telm risk associated with such bonds To lessen risk, such an investor prefers shOl Helm investments Hence to induce investors into long-tel m instl uments, better lutes must be offered for long bonds For this reason, according to the theory, the spot tate CUI ve I iscs

Market Segmentation
The mmket segmentation explanation of the term structure argues that the market for fixed-income securities is segmented by matUlily dates This argument assumes that investors have a good idea of the matuIHy date that they desile, based on theil projecled need fot tutUle tunds ot their fisk pleference The algument concludes that the group ot investOls competing tOl long-term bonds is different from the glOUp competing tOl shOlt-tcllTI bonds Hence there need be no relation between the pJices (defined by intelest lates) ot these two types of instftlments~ ShOlt and long Jates can move mound lather independently Taken to an extleme, this viewpoint suggests that all points on the spot rate curve are ITIUlually independent Each is detel mined by the forces ot supply and demand in its own mm ket A moderated veu;ion ot this explanation is that, although the market is basically ~egmented, individual investOl s are willing to shift segments if the rates in an adja-

4 5



cent segment are substantially mOle aUl active than those of the main target segment Adjacent rutes cannot become grossly out of line with each othel Hence the spot rate curve muSt indeed be a curve lathel than a jumble of disjointed numbels, but this CUlve can bend in various ways, depending on market forces

Certainly each ot the foregoing explanations embodies an element of t!uth The whole truth is probably some combination of them all The expectations theory is the most analytical ot the three, in the sense that it offers conclete numerical values fOJ expectations, and hence it can be tested These tests show that it works leasonably well with a deviation that seems to be explained by liquidity preference Hence expectations tempered by the I isk considerations ot liquidity preference seem to offel a good suaightfOlward explantltion

The concept ot mal ket expectations introduced in the previous section as an explanation fOl the shape of the spot I ate curve can be developed into a useful tool in its own right This tool can be used to tOlm a plaUSible fOl'eeast ot future interest tates

Spot Rate Forecasts
The basis of this method is to asSume that the expectations implied by the current spot rate curve will actually be fulfilled Under this assumption we can then predict next yem's spot late curve flom the cunent one This new CUlve implies yet anothel set of expectations fOl the following yea! It we asSUme that these, too, are fulfilled, we can predict ahead once again Going fOlwmd in this way, an entile futUle of spot late cUlves can be predicted OfcoUlse, it is undelstood that these predicted spot late cUlves are based on the assumptioll that expectations will be fulfilled (alld we lecognize that this may not happen), but once made, the assumption does plOvide a logical forecast Let us wOlk out some of the details We begin with the CUlTent spot rate CUlve _!q,,~::!, ,5/1, and we wish to estimate next yem's spot late cUlVe 5;,,\~, ,5;/_1 The ctllrent fOlward late II} can be legardcd as the expectation of what the interest late wi!! be next yem-measUled from next year's CUllent time to a time j - I yeals ahead-in othel WOlds, II j is next yem's spot late Explicitly,::!
::!Reca!! that thi~ formula tor
(!+h1)1- I (I+\'d







(4 I)

ii 1

was given in Section 43 It is derived fraIl! the relation (I + 11)1



Chapter 4

THE TERM STRUCTURE OF INTEREST RATES for I < j ::.: 11. This is the basic formula for updating a spot rate curve under the assumption that expectations are fulfilled Starting with the current curve, we obtain

an estimate of next year's curve We term this trunsfonnation expectations dynamiCS, since it gives an explicit charactetization of the dynamics of the spot rate curve based on the expectations assumption Other assumptions are certainly possible For instance, we could 0 A risI{~f['ee asset has a retUi n that is deterministic (that is, known with celtainty) and therefore has a = 0 In other words, a fisk-free asset is a pure interest-bearing instlUlTIcnt; its inclusion in a portfolio corresponds to lending or bOllowing cash at the lisk-tree rate Lending (such as the purchase of a bond) corresponds to the risk-free asset having a positive weight, whereas borrowing corresponds lo its having a negative weight The inclusion of a risk-free asset in the list of possible assets is necessUlY to obtain realism Investors invariably have the opportunity to bOrTOW or lend Fortunately, as we shall see shortly, inclusion of a risk-free assel introduces a mathematical degeneracy that greatly simplifies the shape ot the efficient frontier To explain the degeneracy condition, suppose that there is a risk-tree asset with a (determinislic) rate of relurn If Consider any other risky asset with rate of return I, having mean r and variance a:2 Note that the covariance ot these two retulns must be zem This is because the covariance is defined to be E[ (I 1')(1 J I J) Dc Now suppose that these two assets arc combined to form a portfolio using a weight ot a for the risk-free asset and i-a fOi the risky asset, with a :s i The mcan rate of return of this pOitfoiio will be al J + (I a)rc The standard deviation of the return will be /(1 - a):2a:2 = (l a)a This is because the risk-flee asset has no variance and no covaliance with the Jisky asset The only term left in the tOIlTIula is that due to the risky assct If we define, just tOi the moment, a; = 0, we see that the portlolio late of retUln has


rnean = al; standmd deviation aaJ + (I + (I

a)r ala These equalions show that bOlh the mean and the standard deviation of the portfolio vary linemly with a This means that as a valies, the point representing the portfolio tlUces out a straight line in the plane Suppose now that there are 11 risky assets wilh known mean rates of letlll n and known covariances ail In addition, there is a risk-free asset with rate ot !etUIn If The inclusion ot the risk-free asset in the list of available assets has a plOfound eitect on the shape of the feasible legion The reason 1'01 this is shown in Figure 6 l3(a) FilSt we constlUct the ordinary feasible region, defined by the II lisky asset" (This region may be either the one constructed with shOJ ting allowed or the one constructed without shOlling) This legion is shown as the dmkly shaded region in the figure Nexl,




Chapter 6


fiGURE 6.13

(a) (b) Effecl of a risk-free asset. Inclusion of il risk-free ilsset adds lines \0 the feasible region (a) If both borrowing and lending are allowed, il complete infinite triangular region is obtained (b) If only lending is allowed, the region will have a triangular front end, but will curve for larger cr

tor each assel (or porlfolio) in this I'egion we form combinalions with the risk-free assel In forming these combinalions we allow bOlTowing or lending of the risk-free asset, but only pUlchase of the risky asset These new combinalions trace oul lhe infinite straight line originating at the risk-free point, passing through the risky asset, and conlinuing indefinilely There is a line of lhis lype fOl every assel in the original feasible set The totality 01 these lines forms a tliangularly shaped feasible legion, indicated by the light shading in the figure This is a beautiful result The feasible region is an infinite lriangle whenever a risk-free assel is included in the universe of available asselS If bOlfowing of the risk-Iree asset is not allowed (no shorting 01 this asset), we can adjoin only the finile line segmenls belween the risk-free assel ,;md poinls in lhe original feasible region We cannol ex lend lhese lines further, since this would entail bonowing of the risk-flee assel The inclusion of lhese finile line segments leads to a new feasible region with a slraight-line if'Ont edge bUl a lounded top, as shown in Figure 6 13(b)

When risk-free borrowing and lending me available, the efficient set consists of a single straight line, which is the top of the triangular feasible region This line is tangent to the original feasible set of risky assets (See Figure 6 14) Tilere will be a point F in the original feasible set that is on the line segment defining the overall efficient set It is clem lhat Wl)' efficienl point (any point on the line) can be expressed as a combination of this asset and the risk-free asset We obtain different efficient points by changing the weighting between these two (including negative weights of the risk-free asset to bonow money in order to leverage the buying of the risky asset) The portfolio




One-fund lheorem When bOlh

borrowing 0

The final inequality tollows because the first bracketed term is positive and the second is zero Since fi is small this means that tan eo >- tan 8 0 Hence

the efficient trontier is lurger than it was originally


7.9 SUMMARY if everybody uses the mean-variance approach to investing, and if everybody hns the same estimates of the asset's expected leturns, variances. and covariances, then everybody must invest in the same fund F oj risky us sets and in the risk-lree asset Because F is the same for everybody, it tollows that, in equilibrium, F must correspond to the market portfoliO M-the portfolio in which each asset is weighted by its proportion

of total market capitalization This observation is the basis for the capital asset pricing model (CAPM) If the market portfolio M is the efficient portfolio of risky assets, it tollows that the efficient frontiel in the r-a diagram is 0 straight line that emanates from the lisk-free point ond passes through the point lepresenting M This line is the capitol market line Its slope is called the market pi ice of risk Any efficient portfolio must lie on this line The CAPM is delived directly from the condition that the market portfolio is 0 point on the edge oj the feasible region that is tangent to the copital market line; in other wOJds, the CAPM expresses the tangency conditions in mathematical form The CAPM I esult states that the expected rote of return of any asset i satis fies r, 'i = tJ;(rM


whelc /31 = COV{/I.//ld/a~1 is the beta of the asset The CAPM can be represented graphically as 0 secUi ity market line: the expected JUte of return of an asset is a stroight-line function oj its beta {OJ, alternatively, of its covariance with the market}; gleatel beta implies gleatel expected return Indeed, from the CAPM view it follows that the lisk of an asset is fully characterized by its beta It follows, for example, that an asset that is uncorrelated with the market (tJ = 0) will have an expected JUte of return equal to the risk-jree rate The beta of the mmket portfolio is by definition equol to I The betas of other stocks take othcr values, but the betas of most US stocks lange between 5 and 25



The beta of a pOitfolio of stocks is equal to the weighted average of the betas of the individual a"ets that make up the pOJttolio One application of CAPM is to the evaluation of mutual fund performance The Jensen index measures the histOlical deviation of a fund from the secUJity market line (This measure has dubious value tOJ funds of publicly traded stocks, however) The

Shmpe index measures the slope of the line joining the fund and the risk-tree asset on the diagram, so that this slope can be compared with the market price of lisk The CAPM can be convelted to an explicit formula fOI the pi ice of on a~set .In the simple:-;t version, this fOllTIula states that plice is obtained by discounting the expected payoff, but the interest rute used fOJ discounting must be I} + fJ(f,\/ 'I)' where f3 is the beta ot the U:-i:'iet An aitemativc form expresses the price as a discounting of the certainty equivalent ot the payoff, and in this formula the discounting is based on the I isk-lree rate I I It is important to recognize that the pricing formula oj CAPM is linear, meaning that the price of u Sum ot assets is the sum of their prices, and the pi ice ot a multiple of an asset is that same multiple oj the basic price The certainty equivalent fOl mulation of the CAPM clearly exhibits this linear property The CAPM can be used to evaluate single-peliod pJOjects within firms Managels of films should maximize the net plesent value oj the firm, as calculated using the pricing form of the CAPM formula This policy will generate the greatest wealth for existing owners and provide the maximum expansion of the elticient hontfer for all mean-variance investors


1. (Capilt\1 miJIket line)
Assume Ihat Ihe expecred rate of relUrn on the market ponlolio is 23% und the rme of return on T~bil!s (the risk~free rate) i11 7% The SIandurd deviation 01 the market is 32% Assume thutlhe murkel ponlolio is elficient

«(I) What is Ihe equution 01 Ihe capihd nJurkei line? (/;) (i) If an expected retUl n of 39)'(1 is desired, whut is the stlmdard deviation of rhis posilion? (it) II you have $1,000 to invest how should you alloc"\le il 10 achieve thl::

nbove position?
«() If you inve~a $100 in the risk-free asset und $700 in rhe market portfolio, how nlUch

money should yOIl expect 10 huve ut the end 01 Ihe year?

2. (A sOlall world)

Consider a world in which Ihere ure only two risky ussets, A and B. and a Iisk-Iree asset F The two lisky asseis are in eqmd supply in the market; Ihat is, tv! ~(A + B) rhe lollowing infOlnHl,tion is known: '/, 10, l1'~ 04, a,\lI 01, l1'h = 02. tlnd r,l/ = 18





«(I) Find t\ geneHd expression (withoul substituting vulues) for l1'jl' flA ,Jnd f31l (b) According to the CAPM, wlUH are the numericul values 01 r,\ ufld r/l'1

(Bounds on reI urns) Consider a universe 01 jusl three securities They huve expected lutes of return 01 10%. 20£;h. and 10% respectively 'Two portfolios urc known to lie 011


Chapter 7 THE CAPITAL ASSET PRICING MODEL lhe minimum-variance set They {\Ie defined by the portfolio weights




"0 40


It is {llso known that the Illurkct portfolio is efficient
(a) Given this information, what nre the minimum Ilnd maximum possible values for the

(/J) Now suppose you are lOld that

expected rme 01 return on the murket ponfotio? w represents the minimum-variance portfolio Docs this change your l\ns\Vcrs 10 purl (a)?

4. (Quick CAPM derivalion) Delivc the CAPM formula for r~ -rf by using Equmion (69) in Chapler 6 lHil/J Note that

~a,,1J), =oOV(r",,,)]
Apply (69) bOlh to nS!'Iet k 1lnd to lhe ITIarket itself 5. (Uncorrelated asselS) SUPPOI-lC Lhere are 1/ mulUally uncorrelatcd assets The return on 1]sset i has variance a,2 The expecled rales of return are unspecified at Lhis point The loUJi amounl of assel i in the markel is KI We lel T = L:I::I KI and lhen set \"1 = KilT, for i = 1.1, .11 Henee the market portfolio in normalized form is x = (\"j,.t;!, ,xu) A;.;!'!ume there is a risk-free asset with Jate of return, f Find nn expression fOi f3J in terms of the XI '!'! and ai's
6. (Simp leland) In Simpleland there arc only two risky stocks. A and B, whose det1Jils are listed in Table 7 4

Details of Stocks A and B

Number of shares outstanding
Slack A Stock B

Price per share
$150 $200

Expected rate of return
15('/0 12%

Standard deviation of return

100 150


Furthelmore, lhe correl1Jlion coefficient between the returns ot stocks A and B is There is also a risk-free a. OJ This function is concave according to the preceding definition, but it is a limiting case Strictly speaking, this function represents risk aversion of zero Frequently we reSelve the phrase 1hk aWl1 ~e for the case where U is ':iofetlv cOllce/ve, which means that there is strict inequality in (9.2) whenever x =F V


Example 9.2 (A coin toss) As a specific example suppo~e thnl you face two options The first is based on a toss of a coin-heads, you win $10; tails. you win nothing The second option is that you can hove an amount M for certain Your utility function for money is .r .04x 2 Let us evaluate these two olternatives The filst has expected utility E[U(.t)] = hto 04 x lO2) + = 3 The second alternative has expected utility M 04M': If M = 5, tor example, then this value is 4, which is greater than the value of the first alternative This means that you would favOi the second alternative; that is, you would prefel to have $5 fOi .sure rather than a 50-50 chance of getting $ to or nothing We can go a step further and determine what value of M would give the same utility as the first option We solve M 04M' = 3 This gives M = $349 Hence you woutd be indifrerent bet ween getting $349 for sure and having a 50-50 chance at getting $to or 0


We can refate important properties of a utility function to its derivatives First, U (r) is increasing with respect to x if U'(r) > 0 Second. U(x) is stIictly concave with respect to .\ if U"(x) < 0 For example. consider the exponential utility function

U(r) =



We find U'(r) = _a'2e- nl < 0, so U is concave

_e- a ,

(/e- a .,

> 0, so U is increasing Also, U"(x) =

Risk Aversion Coefficients
The deglee ot risk Hvelsion exhibited by a utitity function is leiated to the magnitude of the bend in the function-the stronger the bend, the greater the risk aversion This notion can be quantified in terms 01 the second derivative 01 the utility function The degree at risk aversion is tormatly defined by the ArTow-Pratt absolute risk aversion coefficient, which is

The term U1(r) appems in the denominator to normalize the coefficient With this normalization a(r) is the same tor a11 equivalent utility functions Basicalty. the coefficient function a(r) shows how lisk t1version changes with the wealth leve! For many individuals. risk aversion dccleaSes as their wealth increases, reflecting the lact that they arc witling to take more risk when they me fInanciatly secure A,< a specific example consider again the exponential utility function U (x) _e- U .I We have U'(r) ae- a.1 and U" (l") -a:!e- u.I TherefOle a(r) = a In this case the risk Hversion coelficient is constant lor aIt r If we make the same calculation be- II . \ we find that U'(r) bCW- H •I for the equivalent utility function U (r) and U!I (r) = -lw:!e-I/·I So again a(t) = a As anothcl example, consider the logarithmic utility function U (x) = In r Herc U'(t) l/x and U"(t) = -t/r' Therefore a(t) = t/,: and in this case, risk aversion decleaseg as wealth incleases

Certainty Equivalent
Although the actual value of the cxpected utitity at a ",ndom wealth variable is meaninglcss except in comparison with that 01 anothcl alternative, there is a derived mea,stue wi!h units that do have intuitive meaning This meastrre is the certainty eqUJvulent.The certainty equivalent of 11 Hlndom weafth variable r is defined to be the amount at a certain (that is, risk-free) wealth that has a utility tevet equat to the expected utility of r In other words, the certainty equivalent C at a random wealth variable x is that value C satisfying
U(n E[U(x)]

The cCrlninty equivalent 01 a random vmiable is the same 10l atl equivalent utility Iunctions and is measured in unit.s 01 wealth
!This general t:ol\cept 01 ccrtuilHY c(luivnIL'IH iii ilHJircetly related to the concept with tlw iinme l\aO\c t1~ed in Seetiou 77


Chapter 9



FtGURE 9,3

Certainty equivatent. The certainly

equivalenl is always Jess lhan lhe expecled value for a risk-averse

inveslor Reprinted with


sion of Fidelity lnveslments

As an example, consider the coin toss example discussed eallier Our computation at the end of the example found that the certainty equivalent of the 50-50 chance of winning $!O or $0 is $.3 49 because that is the value that, if obtained with certainty, would have the same utility as the reward based on the outcome of the coin toss For a concave utility function it is always true that the ccrtninty equivalent of a random outcome x is less than or equnl to the expected value; that is, C :::::: E(x) Indeed, this inequality is another (equivalent) way to define risk aversion The certainty equivalent is illustrated in Figure 9 3 for the case of two outcomes XI and Y2. The certainty eqUivalent is found by moving horizonlalfy leftward from the point where the line between U (y,) and U (X2) intersects the vertical line drawn at E(x)

Thew are systematic procedures COl assigning an appropriate utitity function to an investor, some of which are quite elaborate We outline a few general approaches in simple foml

Direct Measurement of Utility
One way to measure an individual's utility function is to ask the individual to assign certainty equivalents to various risky alternatives One particulmly elegant way to organize this process is to select two fixed wealth values A and B as reference points probability I
A lottery is then plOposed that has outcome A with probability p and outcome B with p. For various values of p the investor is asked how much certain wealth C he or she would accept in place ot the lottery C will vary as p changes Note that the values A, B, and C arc values for total wealth, not just increments based on a bet A lottery Witll probability p has an expected value of e pA + (I p)B

However, a risk-averse investor would accept less than this amount to avoid the risk of the lollery I-Ience C < e






______L -________






Expected value





FIGURE 9.4 Experimental determination of utility function. (al For lotteries that pay either A or B and have expected value e, a pC/son is asked ro state rhe certainty equivalent C (b) Inverting this relation gives the uriliry function

The values of C reported by the investor for various p's are plotted in Figure 94(n) Tile value of C is placed above the corresponding e A curve is drawn through these points, giving a function C(e) To define a utility function from this diagram, we normalize by setting VIA) A and V(8) 8 (which is legitimate because a utility function has two degrees of scaling freedom) With this normalization, the expected utility 01 the lottery is pV(A) + (l p)V(8) pA + (1 p)8,

which is exactly the same as the expected value e TherefOic since C is defined so that V (e) is the expected utilily of the lottery, we have the relation V (C) e, Hence C V-I (e), and thus the curve defined by C(e) is the inverse of the utility function
The utility function is obtained by flipping the axes to obtain the inverse function, as shown in Figure 9 4(b) ,

Example 9.3 (The venture capitalist) Sybil, a moderately successful venture capitalist, is anxious to make her utility function explicit A consultant asks her to consider lotteries with outcomes 01 either $IM or $9M She is asked to follow the direct procedure as the probability p of receiving $IM varies For a 50--50 chance of the two outcomes, the expected value is $5M, but she assigns a certainty equivalent of $4M Other values she assigns are shown in Table 9, I e (We just read Tire utility lunction is also shown in Table 9 I, since VIC) tram the bottom lOW up to the next row to evaluate V) For example, V (4) However, the values ot C in the table arc not all whole numbers, so the table is not in the form that one would most desire A new table ot utility values could be constructed
~IJ different values of A and B are used a new utility function is obtained, whlch is cquivl\Jent to the original olle; thut is it iii just l\ J\nC1!f trl\nsfomllUion of the original onc (Sec Exercise 5 )


Chapter 9


Expected Utility Values and Certainty Equivalents p e


0 9 9

H2 784


3 66 576

4 58 484

5 5 4

7 42 324


26 196

9 I8 144

by interpolating in Table 9 I For example (although perhaps not obviously),

3 4(2 00

I 96) 256

+ 2 6(2 56
I 96



Parameter Families
Another simple method of assigning a utility function is to select a parameterized family of functions and then detellnine a sllitable set of parametci values This technique is often carrjed out by as~uming that the utility function is of the exponential form U Cr) -e- II .\' It is then only necessary to determine the parameter a, which is the risk aversion coefficient tor this utility function This parameter can be determined by evaluating a single lottcIY in certainty equivalent tClms For example, we might ask an investor how much he or she would accept in place of a loHelY that offels a 50-50 chance 01 winning $1 million or $100,000 Suppose tl1e investor felt that this was equivalent to a certain wealth of $400,000 We then set
_e--1()O DOOa

5e-! (Joo


5e- JOO,OOOa

We can solve this (by an iterative procedurc) to obtain a = 1/$623,4260 Many people pICfer to use a logarithmic or power utility function, since these functions have the property that risk aversion decreases with weaHh Indeed, for the logarithmic utility, the risk aversion coefficient is aCl) I/r, and for the power utility function V(o') VXI' the cocfficient is a(x) (I V)/x Thcrc are also good argumenls based on the theory of Chapter 15, which suggest that these are appropriate utility functions tor investors concerned with long-term growth of their wealth A compromise, or composite, approach that is commonly used is to recognize that while utility is a function of total wealth, most investment decisions involve relatively small increments to that wealth Hence if .to is the initial wealth and w is the increment. the proper function is U (-1:0 + w) This is approximated by evaluating increments directly with an exponential utility function _e- aw However, it we assume thnt the true utility function is In x, then we usc a l/.to in the exponential approximation

Example 9,4 (Curve fitting) The tabular results of Example 93 (for the ventule capitalist Sybil) can be expressed compactly by fitting a Clilve to the results If we assume a power utility function, it will have the form U (x) ax)' + c Our normal-




ization requires

a a9" +c +c

= I = 9

Thus a 8/(9)' -I) and c (9)'-9)/(9)' I) Therefore it only remains to determine y We can find the best value to fit the values matching U(C) to e in Table 9 I We find (using a spreadsheet optimizer) that, in fact, y = t provides an excellent fit Hence we set U (x) = 4ft - 3; or as an equi valent fOlm~ V (x) = f t

Questionnaire Method
The risk aversion charactclistics of an individual depend on the individual's feelings about risk, his or het cunent financial situation (such as net walth), the prospects tor financial gains 01 requirements (such as college expenses), and the individual's age One way, therefore, to attempt to deduce the appropriate risk factor and utility function for wealth increments is to administer a questionnaire such as the one .shown in Figure 95, prepared by Fidelity Invc!-ltments, Inc This gives a good qualitative evaluation, and the results can be used to assign a , 0 the tunction is striclly concave everywhere and thus exhibits risk aversion We assume that all random variables of interest lie in the feasible lange x:::: alb; is, within the meaningful I ange of the quadratic utility function


Chapter 9


This Risk Quiz" is inlendecl ilS (1 slnrlillg poinl ill ~essiQn~ bel ween il elienl nnc! ,1 fll1:I!lrl'(, (:l))I,Jf:IL'c,llnnl;h'


46 ANO HIGHER: Vou I1mh,lll!y h.we Ihe nlOOl'Y ;)mllhl' IOclm,lll()n III Inke m!.s commotlj(jl'~ Hlf~HlSk in\,I!~I_

ownl Inll hllli!cd pJflncr-

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.. hip~ . .1~ w!!11 ,1~ ~IOfk options .1m! lnVcSII1H;nl rea!c%lll' BlIlbesureloillVCr,if\·.1II(.!a'il,>omc ofl'ourpmllnhoimo ...1fl'rll1vc,>\tllCI11' [vcol'o\J IOU!J IOll'cvcrylhing and Icgr(>1 \'oUrlllg!H\~L

'J.Theamounl I havusavud 10 hilndll' emergl'fldes,suchi1sajobhlssorunexpcdedmetlic.11 expenscs,c!]naleslo:

41-45:Vllllll.Wl',ln.lbo\,I!-.1\·!!lagcloiernIlL-o-lor ll,>k.1lltlplnb,lbll'cnnlQlhllnlCoHltiW(()nWIIJ

4 Iflwercjllckingaslocklninveslin,lwoulti Inokfor companies lIlJlareinvolvetl111 t1evdop)lIglhL'llOl prmluc1snflhe folllrl', sudla~ fhe neJ(lpl'nicillin

hi Two lu ~i\ nlonlh,>' ~aL)lY II Sl'VI!l1 111lJlllh~' \l) one l'l'.1r ~ tf!OIlL'lOlwoYL'MS silfMY L'IMOIl'lh;1l1iwnYO.1r'>",


f{J\'L'ryourlossL'5 ilWL'511If'>ll1lf1hf.1\L'f(ory'lm

36-40:You h,w!;a)lJWri1f(elO!ur~l1t:('f(lrriilgme III DlsagreL' el DI,>,lgH'l' ~unl1gll' 11 1 wanl ~ntlnecdloTL't1ucelheovcralllevl'l of de hI in my person:ll financ(>s .111\gll\~ ,lIon!;ly bl ,\f:1l'!.' rI Ndlfll'r:l~\n!L'l1ll1 Ifj'>If()Il!11\'

~lc'llfy petformanec 81uL' (hfp ~Io(k~ hiflh-gIJIIL'

corpornlcilontis nlll\(/nllunil,>lnrircal C51.11L' nIL' :III pOS~IIJIt! OpllOllidlL'rb~CclU,cofr(]ur'I!l(,Ofy[)uri!1nll1lL' anti i.1nljll' l-lf("(lIll 0 As befOle, this argument can be leversed it pi > PI + P'1 Hence the pIice ot d l + d 2 must be PI + P,! In general, therefore, the price ot adl + f3d 2 mllst be equal to a P, + f3 P, T his is linear pricing:' In addition to the absence ot type A arbilIUge, the preceding argument assumes un ideal functioning ot the mmket: it aSSllmes thLlt securities can be ulbitrarily divided into two pieces, and it assumes that there Ute no transactions costs In practice these requirements are not mel perfectly, but when dealing with large number...; ot shutes of traded securilies in highly liquid malkets, they me closely met

Suppose now that there are /I securities dl, th, ,lIlI A portfolio ot these securities is represented by an n-dimensional vectol () = «()I, ()'2, ,011) The ith component (); lepresents the amount of ,e"wity i in the pOlttolio The payoff of the pOltfolio is the Hlndol11 vHtiable

lI=t();d, i=1 Under the a,sumption ot no type A alllilrnge, the price ot the pmtlolio Iineurity Thus the tOlnl plice is

e is lound by


I),p, i=1 which is a more genernl expression 01 Iinem pricing Recall that the CAPM formula in pricing lorm is linear

Type B Arbitrage
Another fOlm of arbitrage can be identified It an investment has nonpositive cost but has a positive probability 01 yielding a positive payo!! and no plObability of yielding n negative pflyoft, that investment is said to be a type B arbitrage. In other wOlds, a type B arbiuuge is a situation where an individual pays nothing (01 II negative amount) and hrls a chance ot getting something An example would be a tree 10ttelY ticket-you pay nothing for the ticket, but hHve a chance of winning a prize Clearly, such tickets me rfire in securities mmkets The two types ot arbitrage are distinguished only fOI clarity of the concepts involved In further developments we shall usually assume that neithel type A nor


Chapter 9

GENERAL PRINCIPLES type B is possible, and we shall just say that there is no arbitrage possibility. However,

we huve shown that ruling out type A is all that is needed to establish linear pricing
Ruling out type B as well allows Us to develop stronger relations, a.s shown in the next section

We at e now prepmed to put many of the earlier sections of this chapter together and consider the pOltfolio problem of un inveslor who uses an expected utility critetion to

rank alternatives If x is a random variable, we write x 2: 0 to indicate that the variable is never less than zero We write _\ > 0 to indicate that the variable is never less than zero and it is stlictly positive with some positive probability Suppose that an investor has a strictly increasing utility function U and an initial weaHh W There are 11 securities d l , {h, ,dn The investor wishes to form a portfolio to maximize tile expected utility of final wealth, say, x We let the portfolio be defined by 0 = (0"0,, , 0,,), which gives the amounts of the various securities
The investor's problem is maximize

E[U(. 0 These equations are l'el), important because they serve two loles First, and most obviously, they give enough equations to actually solve the optimal port/olio problem An example ot such II solution is given soon in Example 9 5 Second, since these equations are valid if there me no arbitrage opportunities, they provide a valuable chruacterization ot plices under the assumption of 110 arbitrage This use of the equations is explained in the next section If there is a risk-free asset with total return R, then (9A) must apply when d, = Rand P, = I Thus,
), = E[U'(t')jR


Chapter 9

Substituting this value of A in (94) yields
E[U'(x')d,] RE[U'(x')] = Pi-

Because ot fhe importance of these equations, we now highlight them:

Portfolio pricing eqllation ploblem (9 3a), lileu
E[U'(x')d i ] = ,Pi

lOI i = 1,2,

,11, w/lCle A> 0 1/ rilele i'l a I irk-jiee aHel
EfU'(x')diJ RElU'(x')] = Pi

R, rlwll

fOl i = I. 2,


Example 9.5 (A film venture) An investor is considering the possibility of investing in a venture to produce an entertainment film He has learned that such ventures Llre quile risky In this particular Case he has learned that there are essentially three possible outcomes, as shown in Table 9 2: (I) with probability 3 his investment will be multiplied by a factor of 3, (2) with probability 4 the factor will be I, and (3) with probability 3 he will lose the entire investment One of these outcomes will occur in 2 years He also has the oppo!tunity to earn 20% risk free over this peliod He wants to know whether he should invest money in the film ventule; and if so, how much? This is a .simplification ot a fairly realistic situation The expected retum is 3 x 3 + 4 x I + 3 x 0 = I 3, which is somewhat better than what can be obtained risk free How much would "Oll invest in such a venture? Think about it tor a moment The investor decides to use U (.\) = In \ as a utility function. This is an excellent general choice (as will be explained in Chaptel 15) His problem is to select amounts OJ and O of the two available seculities, the film venture and the risk-free opportunity, 2

The Film Venture

High SUccess Moder 0 because p, > 0, U'(x')' > 0, and A > 0 We also have

P showing thal the



are slale prices They are all positive


Nole that the theorem says that such positive prices exist-it does not say that they are unique If there are more slales UHm securHies, lhcle may be many different ways to assign slale prices that m'e consistent with the ptices ot the existing securities The theorem only says thal tot one ot these ways the slale prices are posilive

Example 9,8 (The plain film venture) Consider again the original Him ventute Thele are three slales, bUl only two secutities: the venture itself and the riskless security Hence slale prices are not unique We can Hnd a set of positive state prices by using (9 12) and the values of the 8;'s and A I found in Example 95 (with W = I) We have

l V1 =





128, 274


These stale prices can be used only to price combinalions of the original lWo securities They could not be applied, for example, to the purchase of residual rights To check the price of the original venture we have P = 3 x .221 + .l18 = I, as it should be

Example 9,9 (Expanded film venture) Now consider the film venture with three available securities, as discussed in Exnmple 96, which inllOduces lesidual righls Since lhere are lhree slales and lhree securilies, lhe slate prices are unique Indeed we may Hnd the state prices by setting the price of the three securities to I, obtaining lVlt I 2V/t

+ V" + I 21/1, + I 2V/,


This system has the solution vrl


Thetefore the price of a security with payoff (d t , d', d') is



+ ~d2 + t{{"

9 10



You can compUie this with the formula lor P given at the end 01 Examplc 97 It is exactly the sume

Note also that these slate prices, aHhough ditlerenl 1'10111 those 01 the preceding example, give the same values for prices 01 securities that are combinations ot just the two in the original film vcnlUie For example, the price of the basic venture ilselt is P = ~ = I


Suppose there lie positive stute prices VI" J = 1,1, , d l ) cun be round from security d = (d', d',

, S Then the price of uny


P = Ld'VI,

We now normalize these slale prices $0 that they sum to I Hence we let V/O L~-=I V'\> and lel {h = v/lll//O We can then wrile the pricing formula as

The quantities 'I" \

V/o Lq~(t

(9 13)

= 1,2, , .5, cun be thought 01 as (artineiul) probabilities, since lhey are positive and sum to I Using these as probabilities, we can write the pricing forlTIula as
P = VIoE(d) (914)


with respect to the artificial probabililies {h The value V/o has a usetul interpretation Since Vlo = L:;=I V'I, we see that

Edenotes expectntion


is the price ot the security (I, I, , I) tilal pays I in every state-a risk-tree bond By definition, its price is IjR, where R is the risk-free return Thus we can write the

pricing forlTIula as

= RE(d)


(9 15)

This equation states that the price of a securily is equal to the discounted expected value 01 its payon~ under the artificial probabilities We term this risl{-neutral pricing since it is exactly the formula that we would use iI the q\'s were real probabilities and we used a risk-neutral utility function (that is, the linear utility function) We also

refer to the q\'s as risl{-neutral probabilities. This artifice is deceptive in its simplicity; we shall find in the coming chaptels that il has proJound consequences In fact a major portion of Pmt 3 is elaboration of this simple idea Here are thlCc ways to find the risk-neutral probabilities {h:
«(/) The risk-neutral probabilities c!ln be found i'rom positive state prices by multi-

plying those prices by the risk-free rate This is how we defined the risk-neutral

probabilities at the beginning 01 this section


Chapter 9

(b) If the positive state prices were found from a portfolio problem and there is a

risk-hee asset, we can use (9 6) to define q, =

I:;=, p,U'(l")'



This formula will be useful in our later work

(c) If there are 11 slales and at leasllI independent seclllilies with known prices, and no arbitrage possibility, then the risk-neullal probabilities can be found directly by solving the system of equations i:::::: 1,2, tor ,11

the n

unknown q~ '$

Example 9,10 (The film venture) (with three securities) to be
Vf! :::::

We found the state pI ices of the full Him ventule
Vf? ::::::



0/) :::::


Multiplying tl1ese by the risk-flee rate I 2, we obtain the lisk-neutral probabilities

q, = 2,

q, = 6,

Hencc the price of a secmity with payoff (d', d', d 3 ) is




6d,+ .2d3

Here again, this pricing formula is valid only for the original secUlilies or linear combinalions of lhose securities The risk-neutral probabilities were delived explicitly to price the original securities

The Iisk-neutral pricing resull can be extended to the general situation that does not assume that there are a Hnite number of states (See Exercise 15 )

Let us review some alternative pricing methods Suppose that there is an environment of n securities for which prices are known, and then a new securily is introduced, deHned by the (random) cash How d to be obtained at the end of the period What is

the correct price of that new security? Listed here me five allernative ways we might assign it a price In each case R is the one-period risk-free return
1. Diw.:01mted eypecred value"




2. CAPM pI icing

R -I-


whelc fJ is the bela of the asset with respecllo the markel, and RA/ is the return on the market portfolio We assume that the market portlolio is equal to the Markowitz fund of risky assets

3. eel tailltv eqIli\!{IielIt lOl III oj eAPIV!

4. Log-optima/I" king

where W is the return


the log-optimal portfolio

5, RiJk-llell!Hll/JI icing,
E(d) P=R

where the expectation

E is

taken with respect to the risk-neutral probabilities,

Method I is the simplest extension ot what is true for the deter ministic case In general, however, the price delelmined this way is too Inrge (al least for assets that are positively correlated with all others) The price usually must be reduced

Method 2 reduces the answer obtained in I by increasing the denominator This method essentially increases the discount rale Method 3 reduces the answer obtained in I by decreasing the numerator, replacing il with a certainty equivalent Method 4 reduces the answer obtained in I by putting the retUln R· inside the expectation Ailhough E(I/R') = I/R. the resulting price usually will be smaller than that of method I
Method 5 reduces the answer obtained in I by changing the probabilities used to calculate the expected value

Methods 2-5 represent four different ways to modify method t to get a more appropriate result What me the differences between these four modified methods?
That is, how will the prices obtained by the ditferent formulas diller') Think about it

for it moment The answer, ot course, is thnt it the new security is a linear combination ot the original 11 securilies, all four of the modified methods give identical prices Each method is a way ot expressing linear pricing If d is not a linear combination of these II securities, the prices assigned by the different formulas may ditfer, tor these formulas are then being applied outside the domain tor which they were derived Methods 2 U11d 3 will always yield identical values Methods 3 and 4 will yield identical values if the log-optimal formula is used to calculate the risk-neutral probabilities Otherwise they will differ as well If the cash flow d is completely independent 01 the 11 original securities, then ,,/I /il'e methods, including the first, will produce the identical price (Check itr)


Chapter 9


We can obtain addilional methods by specifying other lllilily funclions in the optimal portfolio problem For the II original securilies, the price so obtained is independent 01 the ulilily function employed Howevel, the methods plescnled here seem to be the most usetul

This chapler is devoted to general theory, and hence il is somewhat more abstract than other chapters, but the tools presented are quite powerful The chapter should be leviewed unel reading Parl 3 and again after leading Parl 4 The first pan of the chapter presents the basics of expected utility theory Utility lunctions account fOl lisk avclsion in financial decision making. and plOyicie a mOle general and more useful approach than does the mean-variance tramew01k In this new applOuch, an unceltain final wealth level is evaluated by computing the expected value of the utility of the wealth One ranclom wealth level is preferred to another if the expected utility 01 the first is grealel than that of the second Often the utility function is expressed in analytic form Commonly used functions are: exponential, logarithmic. aU(t)+b power, and qlladratic A utility function U(t) can be transformed to V(x) with a > 0, and the new lunction V is equivalent to U fO! decision-making purposes It is genelally assumed that a utility function is increasing, since more wealth is preferred to less A uLilily function exhibits lisk aversion if it is concave If the utility function has derivatives nnd is both increasing and concave, then U'(x) > 0 and U"(x) < O. Conesponding to a random wealth level, thele is a l111mbcl C, called the certainty equivalent of that random wealth The certainty equivalent is the minimum (nonrandom) amount that an investol with utility function U would accept in place 01 the random wealth undel consideration The value C is defined such that U (C) is equal to the expected utility due to the random wealth level In older to use the utility function approach, an appropliate utility function must be selected One way to make this selection is to assess the certain equivalents of various lotteries, und then work backward to find the undedying utility tunction that would assign those cel tain equivalent values F/equently the utility function is assumed to be either the exponential fOlm _e- ax with {I approximately equal to the I eciprocal of total wealth, the logarithmic form In x, or a power form )I r J' with )I < I but close to 0 The parameteu; of the function are either fit to lottery responses or deduced trom the answers to a selies of questions about an investOl'S financiul situation and attitudes toward I isk The second part of the chap tel pIesents the outline of a general theory 01 linear pricing In perfect markets (without transactions costs and with the possibility of buying or selling any amount 01 each secwity), 5ecurity prices must be linear-meaning that the price of a bundle 01 seculities must equal the sum ot the prices of the component securities in the bundle-otherwise there is an arbitrage opportunity Two types of m bitrage are distinguished in the chapter: type A, which rules out the possibility ot obtaining something 1m nothing-right now; and type B, which rules out the possibility of obtaining a chance for something later-at no cost now



Ruling out type A arbitruge leads to line .. pI icing Ruling out both types A an B implies that the plOblem of finding the portfolio that maximizes lhe expected utility has a well-defined solution The optimal pOlttolio problem can be used to solve lealistic investment problems (such as the film venture problem) FurthennOlc, the necessmy conditions of this genelal problem can be used in a backwmd fashion to express a secUTity price as an expected value Dincrent choices of utility functions lead to different pricing formulas, but all at them are equivalent when applied to securities that are linear combinations at those considered in the original optimal portfolio problem Utility functions that lead to especially convenient pI icing equations include quadratic runctions (which lead to the CAPM lormula) and the 10gUlithmic utility function Insight and practical advantage can be derived from the use of finite state models

In these models it is useful to introduce the concept of state pi ices A set 01 positive state prices consistent with the securities under consideration exists if and only if there me no arbitrage opportunities One way to find a set of positive state prices is to solve the optimal portfolio problem The state plices are detelmined directly by the resulting optimal pOltlolio A concept of major significance is that of risk-neutral pricing By introducing mtificial probabilities, the pI icing fOlTIluia can be WIilten as P = E(d)/R, whele R, is the return of the riskless asset and E denotes expectation with respect to the mtificial (Iisk-l1eutlUl) probabilities A set of risk-neunal probabilities can be found by multiplying the state prices by the total retum R 01 the lisk-fIee asset The pI icing process can be visualized in a special space, Starting with a set of II securities defined by their (random) outcomes di, define the space S of all linear combinations of these securities A major consequence of the no-arbitrage condition is that there exists another random variable v, not necessarily in S, such that the plice at any security d in the space S is E(vd) In pmticular, fm each i, we have PI = E(vdi ), Since v is not required to be in S, there arc many choices fot it One choice is embodied in the CAPM; and in this case v is in the space S Another choice is v = l/R+, where R.t is the leturn on the log-optimal pOitiolio, and in this case v is otten not in S The optimal pOltfolio problem can be solved using othel utility functions to find othel v's If the formula P = E(vd) is applied to a secUlity d outside of S, the result will genelHlly be diffelent 10l diflerent choices 01 v II the securities are defined by a finite state model and it there ate as many (independent) securities as states, then the market is said to be complete In this case the space S contains all possible tandom vectors (in this model), and hence v must be in S as well Indeed, v is unique It may be found by solving HH optimal pOltfolio problem; all utility functioHs will plOduce the SHme v

L (Certainty cquivnlent) An investor has utility function U(x) == xl/-I for sahuy He has a new job offcr which pays $80,000 with a bonus The bonus will be $0, $10,000, $20,000, $30,000, $40,000, $50,000, or $60,000, each Wilh eqll,,1 probability What is the cCIWinty equivalent 01 lills job oftcr?


Chapter 9

2. (Wealth independence) Suppose Un investor hus exponential utility function U(t) _e- U1 aad nn initinl wealth level of W The investor is faced with an opportunity to invest un I1mOllnl wSW and obtain a random payoff x Show that his evaluation of this incremental iuvestnlcm is independent of W 3. (Risk I1version invnriance) Suppose Vex) is a utility function with Arrow-Prau risk aversion coefficient a(x) Let V (x) = c + bU(x) Wllat is rhe risk I1version coefficient of V? 4. (Relative risk aversion) The Arrow-Pran relative risk aversion coefficient is xV"( .. )

= V'(x)
U (x)

Show lhut the utility functions U (x) aversion coefllcicnts

== in y and


yx 1' hl1ve const.:ml relative risk

5. (Equivalency) Ayoung woman uses the first procedure de5cribed in Section 9 4 to deduce her utility function Vex) over the range A.::; \''::; B She USes the normalization V(A) == A, V(B) == B To check her result, she repeats the whole procedure over the range A' .:s x .:s B', where A < A' < B' < B The lesult i5 a utility function Vex), with V(A') == A', V(B') = B' [f the result5 are consistent, V and V should be equivalent; that is. V(x) = aV(.\") + b for some a> 0 nnd b Find (/ and b 6. (HARAo) defined by The HARA (fOl hyperbolic nbsolute risk nversion) cinss of utility functions is

I l' V(x)=-l'

ax -1'

+b) '

b> 0

The functions are defined for those values of x where the term in parentheses is nonnegative Show how the pmameters y, (I, and b can be chosen to obtain the follOWing special cases (or nn eqUivalent form)
(0) Lineal or risk neutral: Vex) (c) Exponential: V (x) == _e,-"y u1 = r
[Try y ==
-00 ]

(b) Qllndratic: Vex) = x - !CX 2 (d) Power: V (x)


(e) Logarirhmic: V(x) = Inx

[Try V(x) = (I

1')I-I'((,y - 1)/1')

+ d)

Show that the Arrow-Pratt ri5k aversion coefficient is of the form I/(cx

7. (The venture capitalist) A venture capitillist with a utility fUnction V (x) = .JX cnrriecl out the procedure of Example 9 3 Find an nnalyticnl expression for C as a tunction of e, and for e as a function of C Do the vnlues in Table 9 I of the example agree with these expressions? 8, (Celtnintyapproximationo) There is a Il..'>eful approximation to the certainty equivalent fhat is easy to delive A second-order expansion near = E(x) gives


V(x) '" V (X)

+ V'(X)(x -


+ !V"(X)(x - xl

Hence, ElV(x)l '" II (x)

+ !V"(x)vnr(x)



On the other hand, it we let (. denote the certainty equivalent and assume it is close to \Ve can use the ftrst~order expansion


VIc) '" V (x)
Using tilese apPloximations, show that

+ V'Ct* -


~ \: + UII(~) var(x)

9. (Quadwtic mean-v;:u-iance) An investol with unit wealth maximizes the expected value of the utility tUllction U(x) = en - b r'~)2 and obtains a lUcan-variance ctlicient pOlttolio A friend of his with wealth HI and the same utility function does fhe smne calculntion, but gets it dilfcrcnt pOI1t'oHo retuln Howevel, changing b to b' does yield the s I, this model has the property that the expected value of the price increases geometrically (that is, according to al..:) Indeed, the constant a is the growth rute factor of the model The additive model is structurally simple and easy to work with The expected value ot price grows geometrically. and all prices are nonnal random variables-, However, the model is seriously flawed because it lacks realism Normal random vmiables can take on negative vtllues, which means thut the prices in this model might be negative as well; but real stock plices are never negative Furthermore, if a :;tock were to begin at a plice of, say, $1 with a U ot, say, $ 50 and then drift upward to a price ot $100, it seems very unlikely that dIe u would remain at $ 50 It is more likely that the standard deviation would be propol tional to the price For these reasons the additive model is not a good general model of asset dynamics The model is useful for localized analyses, over short periods of time (perhaps up to a few months for common stocks), and it is a usetul building block for other models, but it cannot be used alonc as an ongoing model representing long- or intcrmediate-term fluctuations For this reason we must consider a better altelnative, which is the multiplicative model (However, OUI unden~tanding of the additive model will be important for that morc advanced model)

TIle multiplicative model has the form

+ I)


(l 14)

fot k

0, I,

, N - I Here again the quantities lI(k), k

0, 1,2,

, N - I, are

11 3



mutually independent random variables "The variable u(k) defines the ,dative change in price between times k and k + I This relative change is S(k + I)/S(k), which is independent of the overall magnitllde ot S(k) It is also independent of the units 01 price For example, if we change units from U S dollars to Get man marks, the relative price change is still/l(k) The multiplicative model takes a tamili", form it wc take the natutallogatithm 01 both sides or the equation rhis yields
In S(k

+ I) =

In S(k)

+ In!l(k)

(II 5)

for k 0, 1,2, , N - I Hence in this form the model is of the additive type with respect to the logarithm of the price. rather than the price itselt "Therefore we can use our knowledge ot the additive model to analyze the mUltiplicative model It is now natura! to specify the random disturbances directly in terms of the In /I(k)'s In particular we let w(k) = In!l(k)

for k = 0, 1,2, ,N - I, and we specify that these w(k)'s be not mal "'ndom v",iables, We assume tllat they are mutually independent and that each has expected value w(k) = v and vuriance (72 We can express the original multiplicative disturbances as (116) tor k = 0, 1,2, ,N - I Each of the variables !I(k) is said to be a lognormal random variable since its logarithm is in fact a normal random variable Notice that now there is no problem with negative values Although the normal variable w(k) may be negative, the cotresponding !I(k) given by (116) is always positive Since the random factor by which a price i, multiplied is !I(k), it tollows that prices remain positive in this model

Lognormal Prices
The successive prices of the multiplicative model can be easily found to be
S(k) /I(k I)u(k - 2)


Taking the natural logarithm ot this equation we find

In S(k)

In S(O)

+ Lin /I (i) i={) = In S(O)





The telln In 5(0) is a constant, and the w(i)'s are each normal random variables Since the sum of normal random vUliables is itself a normal random vuriahle (see Appendix A), it tollows that In S(k) is normal In other word" all prices are lognormal under the multiplicative model


Chapter 11


If each w(i) has expected value w(i) independent, then we find
E[ln S(k)J var[ln S(k)]

v and variance cr', and all are mutually

In S(O) kcr' + vk

(II 70)

Hence both the expected value and the variance increase linearly with k

Real Stock Distributions
At this point it is natural to ask how well this theoretical model fits actual stock price behavior Are real stock prices logn01J11al? The answer is that, based on an analysis of past stock price records, the price distributions of most stocks ale actunlly quite close to lognormal To verify this, we select a nominal period length of, say, I week and record the differences In S(k + 1)In S(k) for many values of k; that is, we record the weekly changes in the logarithm of

the prices tor many weeks We then construct a histogram ot these values and compare it with that of a normal distribution ot the Same variance Typically. the measured distribution is quite close to being normal. except that the observed distribution often is slightly smaller near the mean and larger at extremely large values (either positive or negative large values) This slight change in shape is picturesquely termed fat tails. (See Figure II 3 ') The observed distribution is larger in the tails than a normal

Number of samples




FIGURE 11.3 Observed distribuUon of the logarithm of return. The distribution has "fatter tails" than a normal distribution of the Si1me vi1riance

2The figure shows a liistognJnl of American Airlinl!s weekly log stock rctums for the IO-yeur period of 1982-1992 Shown ~upcrimpllSed is the nonnal distrib11tion with the same (sample) mean llnd standllrd deviation Along willi fat tails Ihere is invaril1bly a "skinny middle"

11 4



distribution This implies that large price changes tend to occur somewhat more frequently than would be predicted by a normal distribution of the same vmiance For most applications (but not an) this slight discrepancy is not impOltant

The !etum of a stock over the period between k and k+ I is S(k+ 1)IS(k), which under the multiplicative model is equal to II (k) The value of w(k) In lI(k) is therefore the logarithm of the return The mean value ot w(k) is denoted by v and the variance of w(k) by (72 Typical values of these parameters for assets such as common stocks can be intened from our knowledge of corresponding values for returns Thus for stocks, stdev [w(k)J might be typical values of 11 = E[w(k)] and u iJ= 12%,

u = 15%

when the lengtll of a period is I year If the period length is less than a year, these values scale downward;] that is, if the period length is p pmt ot a year, then

The values can be estimated trom histori.caI records in the standard tashion (but with caution as to the validity of these e~timates, as raised in Chapter 8) If we have N + I time points of data, spanning N period~, the estimate of the single-period v is -Lin - N k~O

I N-r


I I)] N N-r L[lnS(k+ I) -lnS(k)] S(k)

I In



Hence all that matters is the ratio of the last to the first price [he standard estimate of (72 is


_1_ ~ lin [5(k + I)] _vi'
N - I k~O S(k)

As with the estimation of return parameters, the error in these estimates can be chatacterized by their vari.ances For v this variance is var(il) and for


it is (assuming w(k) is normal] var(&') 2u'/(N - I)

JUsing log returns, Ih!! sCliling is I:.WlLT/V proponionul lhere is no error due to coUlpoumling (without the log) (See Exercise 2 )


with returns


Chapter 11

Hence for the values assumed earlier, namely, v ,I2 and CT 15, we find that 10 years of data is required to reduce the standard deviation of the estimatc·i of Ii to 05 (which is still a sizable traction of the true value) On the other hand, with only I year of weekly data we can obtain a fairly good estimateS of (72,

11.5 LOGNORMAL RANDOM VARIABLES is a lognormal random variable, then the vaJiable w = lnu is normal In this found that tbe prices in the multiplicative model are all lognormal random variables It is therefole usetul to study a few impoltant properties of such random variables The general shape of the probability distribution of a lognormal random variHble is shown in Figure II 4 Note that the variable is alwnys nonnegative and the distribution is somewhat skewed Suppose that w is nonnal and has expected value wand variance a 2 What is e Ul ? A quick guess might be u em, but this is wrong. the expected value of It Actually u is greater than lhis by the tactor e!(J:!; that is,


case- we


(I I 8)

This result can be intuitively undelstood by noting that as a is increased, the lognormal distribution will spread out It cannot spread downward below zero, but it can spread upwald unboundedly Hence the mean value incleases as a incleases The ext,a term is actually fairly small lor low-volatility stocks For example, 12 and a yearly a ot 15 The correction term it> consider a stock with- a yemly w


FIGURE 11A Lognormal distribution" The lognorlllal distribution is nonzero only for I( > 0

11 b



~a2 0225, which is small compared to t-he correction can be signilicant


For stocks with high volatility, however,

In Section 11 7 we will shorten the period length in a multiplicative model and take the limit as this length goes to zero This will produce a model in continuous time In preparation for that step, we introduce special random functions of time, called random walks and Wiener processes Suppose that we have N peIiods of length 6.1 We define the additive process : by

(II 10) for k = 0, 1,2, ,N This process is termed a random wall\:. In these equations E(td is a normal random variable with mean 0 and variance I-a standardized no('~ mal random variable. These random variables are mutually unconelated; that is, E[E(lj)E(ld] 0 tOl j '" k The process is started by setting z(lo) 0 Thereafter a particular realized path wanders around according to the happenstance 01 the random variables E(td [The reason tOl using -Jt;i in (II 9) will become clear shOltly] A particular path of a random walk is shown in Figure II 5 Of special interest me the difference lundom vmiables z(td - :::(lJ) 1'01 j < k We can write such a difference ns z(lk}- ;.(lj) =

L E(li)-Jt;i i=j /.:-1

This is a nOlmai random variable because it is the sum of normal random vmiables We find immediately that
E[Z(lk) - ;.(lj)]


FIGURE 11.5 Possible random walk The movements are determined by norma! r 1 is, on average, the same as z(t) but will vary from that according to a standard deviation equal to ~

11 6

FIGURE 11.6 Path of il 307

Wiener process, A

Wiener process moves continuously but is nor cJifferenti. lilt' overall rl!wlll (/fl' 1;lIIilar

11.8 ITO'S LEMMA *
We saw that the two Ito equations-tor S(I) and for In S(I)-are different, and that

the difference is not exactly what would be expected from the application of Oldinary calculus to the transformation of variables from S(t) to In S(t); an additional term is required This extra term arises because the random variables have order

v'dl, ~nd


hence their squares produce first-order, rather than second-order, efjects There is a systematic method for making such transtormations in general, and this is encapsulated

in Ito's lemma:

Ito's lemma

Suppo')e that tile 1Cmdol1l plOceH x i\ defined bv the Ito pl0ce')s

dx(t) =a(x,l)dl +b(\",t)dz


whele z is: a ')taHdmc/ WieHel pl0ceS') Suppose alw that the plOceH y(t) h defined by 1'(1) = F(." t) TIlell 1'(1) Wli\fiel lI1e 110 eqllalioll
d)'(1)= ( iho+ar+iBx,b~vhele



I a'F ')

BF dl+a;bdz

(I 122)

z i\ the wme WieHe 1 pl0cess

a~ iH

Eq (II 21)

Proof: Ordinary calculus would give a formula similar to (J I 22), but without the term with!

11 9



We shall sketch a rough plOol 01 the full formula We expand l' with respect to a chnnge L),)' In the expansion we keep terms up to first order in ~l, but since ~x is 01 order .j7;i, this means that we must expand to second order in L'.x We find l' + L'.1'

= F(.,·, t) = F(x,t)

+ a:;:-L'.x + arL'.1 + 2: ax' (L'.xl' +
BF BF Bx (aL'.I +bL'.z)+ arL'.1






I il' F

The quadratic expression in the last term must be treated in a special way When expanded, it becomes (/'(L'.I)' + 2ab L'.I L'.z + b'(L'.z)' The first two telms ot this expression are of mdel higher thnn I in ~l, so they can be dropped The term b'(L'.z)' is all that remains However, L'.z hus expected value zero and variance ~l, and hence this last tel m is of order ~l and cannot be dropped Indeed, it can be shown that, in the limit as ~l goes to zero, the term (L),z)1 is nonstochastic and is equal to ~l Substitution 01 this into the previous expansion leads to
BF BF I B'F ') aF 1'+L'.),=F(x,I)+ ( a:;:-a+ar+2:Bx,'r L'.I+ axbL'.z

Taking the limit and using l' = F (x, t) yields Ito's equation, (I I 22)

Example 11.4 (Stocl< dynamics) BlOwnian motion

Suppose that S(t) is governed by the geometric

d5 = 1"5 dl

+ a S dz

Let us use Ito's lemma to find the equation governing the process F (5(1» = In S(I) We have the identifications (/ = 1"5 and b = a 5 We also have BF las = liS and B'Flas' = _liS' Therefore according to (1122), dinS =

which agrees with our earliel lesuit

Let us consider again the binomial lattice model shown in Figure II 8 (which is identical to Figure II I) The model is analogolls to the multiplicative model discussed earlier in this chapter, since at each step the price is multiplied by a random variable


Chapter 11

FIGURE 11.8 Binomial lattice stod( modeJ At each step the stock price 5 either increases to uS or cJe~ creases to dS

Su 1 Su'd Su Su'd

Sd' Sd'


In this case, the random variable takes only the two possible values II and d We can find suitable values for If, d, and I' by matching the multiplicative model as closely as possible This is done by mUlching both the expected value of the logarithm of a price change and the variance ot the logarithm of the price change,6 To cuny out the matching, it is only necessary to ensure that the random variable SJ, which is the price after the first step, has the correct properties since the process is identical theleaftel Taking S(O) I. we find by direct calculation that E(lnSI)
I' In

var (In Sl)

p(ln II)'

+ (I + (I

1') In d

- p)(ln d)' - [pin II

+ (I

- 1') In


1'(1 - p)(inll - Ind)'

Therefore the appropriate parametci matching equations are pU +(1 p)D

(I I 23) (I I 24)

1'(1 - p)(U - D)'


where U In /I and D In d Notice that tiuce parameters are to be chosen: U, D, and p; but there are only two tequiIements Thelefore there is one degree ot freedom One way to use this -U (which is equivalent to setting d llll) In this case the treedom is to set D
('For the lattice. the probability 01 attaining the various end nodes of lhe lattice is given by the binomial dislribution Specifically. Ihe probability of reaching the value
II!_ ( ") = __k)!k! k (1/



(~) Ii (1

- P)Il-/", where

is Ihe binomial coefficient This diSlribution npproache!> (in a cenain sense) a nomal

distribUlion ror large 1/ The logllrilhm of the linal plices is 01 the form k lnu + (11- k) Ind. which is linear ill k Hence Ihe distribution or the end point prices can be considered to be nearly lognonnul

11 10 equations (I 123) and (I 124) reduce to
(2p I)U I'M




p)U J

a 2 (',J

If we square the first equation and add it to the second, we obtain
U2 a 2 M+(IIM)2

Substituting this in the first equation, we may solve for p directly, and then U In II can be determined The resulting solutions to the parameter matching equations are


:2 + r~;=;f~=7


Ja 2 M

+ (116.1)2

(I 125)

For small 6.1 (I L25) can be approximated as p II

I I " r;:-: ,.+,. (- ) ",6.1 - a

(I I 26) e-qJF:i J


These are the values presented in Section


A simple and versatile model ot asset dynamics is the binomial lattice In this model an asset's price is assumed to be multiplied either by the factor II or by the factor d, the choice being made each period accOlding to probabilities p and J-p, respectively This model is used extensively in theoretical developments and as a basis for computing solutions to investment plObJems Another broad class ot models arc those where the asset price may take on values from a continuum ot possibilities The simplest model ot this type is the additive model If the random inp!1ls of this model are normal random variables, the asset prices are also normal random variables This model has the disadvantage, however, that priceR may be negative A better model is the mUltiplicative model of the torm S(k+ I) lI(k)5(k) lithe multiplicative inputs lI(k) are lognormal, then the future prices 5(k) are also lognormal The model can be expressed in the altel11ative form as In S(k + I) - In S(k) In lI(k) By Jetting the period length tend to zelO, the multiplicative model becomes the Ito process d In S(t) v dl + a2dz(t), where z is a normalized Wiener process This special form of un Ito process is culled geometric Brownian motion This model can be expressed in the alternative (but equivalent) form d5(t) lrS(t)dl + a 2 S(t)dz(t), where J1 v + ta2


Chapter 11

MODELS OF ASSET DYNAMICS Ito processes are useful representations asset dynamics An important tool transforming such processes is Ito's lemma: If xU) satisfies illl Ito process, and vet) is defined by y(t) F(v,/), Ito's lemma specifies the process satisfied by y(t) A binomial lattice model can be considered to be an approximation to an Ito process The parameters of the lattice can be chosen so that the mean and standard deviation of the logarithm of the return agree in the two models )



1. (Stock lattice) A stock with current value S(O) 100 has an expected growlh rate of its Jogtlrithm of v 12% and a voltltility of that growth Itlte of (]" 20% Find 5uittlbJe ptlrameters of a binomitll ltlttice representing this stock with tl basic elementary period of 3 months Draw the ltlttice and enter the node values fOJ I year Whtlt tlre the probabilities of attaining the vtlrious final nodes?
2. (Time scaling) A stock price S is governed by the model In S(k + I) In S(k)

+ w(k)

where the period length is I month Let V E[w(k)] and u 2 vru[w(k)l fOJ all kNow suppose the bnsic period length is changed to I year Then the model is In5(-:,:I-,I) In S(ln

+ W(K)

where each movement in K corresponds to I yetlr What is the natural definition of W(K)? Show that ErW(K)J 121! and vru[W(K)] = 12u 2 Hence ptlrameters scale in proportion lo time 3. (Arithmetic and geometric means) Suppose that VI, V2, ,v" are positive numbels The mithmetic mean and the geometliL mean of these numbel':;; are, lespectively, and
(a) It is always true that V,I 2: VG Prove this inequality for 11 2 (b) If rl,12, ,1" are rates of return of a stock in each of 11 periods, the arilhmetic and

geometric mean rales of return me likewise and Suppose $40 is invested During the first year it increases to $60 tlnd during lhe second year it decreases to $48 Whal set undel specified terms Usually there ale a specified plice and a specified peliod of time over which the option is valid An example is the option to purchase, for a price of $200,{)OO, a certain house, say, the one you are now renting, anytime within the next yeat An option that gives the right to purchgse something is called a call option, wheleas an option that gives the Tight to sell something is called a put. Usually an option itself has a price; frequently we refel to this price as the option premium, to distinguish it from the purchase or selling price specified in the terms of the option The premium may be a small fraction at the plice of the optioned asset For example, you might pay $15,000 for the option to pUlchase the house at $200,000 It the option holdel actually does buy 01 sell the asset accOlding to the telms of the option, the option holder is said to exercise the option The original premium is not recovered in any case An option is a derivative security whose un9~Tlyi K, S K, or S < K, respectively The terminology applies at any time; but at expiration the terms describe the nature of the option value Puts have the reverse terminology, since the payofts at exelcise are positive if .s < f(

Time Value of Options
The preceding analysis tocused on the value 01 an option at expiration This value is derived trom the basic stmcture of an option However, even European options (which cannot be exercised except at expiration) have value at earlier times, :-;ince they provide the potential for future exercise Consider, for example, an option on OM stock with a strike price of $40 and 3 months to expiration Suppose the cunent price of OM

stock is $37 88 (This situation is approximately tllllt of Figure 12 I represented by the March 40 call) It is clear that there is a chance that the price of GM stock might increase to over $40 within 3 months It would then be possible to exercise the option and obtain a profit Hence this option has value even though it is currently


Chapter 12



FIGURE 12 t II the current stock price S(r) is less than the sttike price /(, we would not exercise the option, since we would lose money II, on the other hand, the stock ptice is greatel than /(. we might be tempted to exelcise HOWCVCI, il we do so we will have to pay R now to obtain the stock If we hold the option a little longel and then exelcise, we will still obtain the stock ror a pI ice of R, but we will havc earned udditional interel1t on the exercise money R -in fact, if lhc stock declines below /( in this waiting peIiod, we will nOl exeIcise und be happy that we did not do so eadiel

We now turn to the issue 01 calcululing the theoretical value 01 an 0Plion-an area of work that is called options pricing theory. Thele me sevelal applOaches to this pIOblem, bU1-lcd on dilleIem assumptions about the market, about the dynamics 01 stock pfice behaviOI, and about individual prclerences The most impOItant theoJies are based on the no mbiLIage pIinciplc, which can be applied when the dynamics at the undedying stock takc certain 1011ns The simplest 01 these theOIies is based on the binomial model at stock plice fluctuations discussed in Chapter II This the01Y is widely used in pIaclice because at its simplicily and case of calculation It is a beauliful culmination 01 lhc plinciples discussed in plCvious chapteIs The basic lheOlY at binomial oplions pricing hus been hinted at in am c;,ulicr discussions We shall develop it hCle in a self-contained munneI, bUl the readel should notice the connections to em lieI sections We shall lilst develop the. theory 101 the single-period case A single step 01 a binomial process is all lhal is used Accordingly, we suppose thal the initial price 01 a stock is S At the end 01 the peliod the price will eithel be liS with plObability p 01 dS with plobability I - P We assume II > d > 0 Also at every petiod it is possible to bOIIow OI lend at a common lisk-Iree inlerest late I We let R I + I To avoid aI bit! age OppOl tuni lies, we must have

T a see this, suppose R .2: II > (/ and 0 < p < I Then the stock peIiOlms WOJSC than the fisk-free nsset, even in the "up" bU1l1ch 01 lhe latticc Hcnce one could shOlt $1 00 of the stock and loan the plOceeds. theleby obtaining a plOfit at eithel R - II 01 R d, depending on lhe outcome state The initial cost is zero, but in cithCl cusc the pIOfil is positive, which is not possible if there ale no mbit!age oppOltunities A simii d.2: R


Chapter 12

< < , K if 5 < K

becausc the d's depend only on the sign of In(5(K) Therefore, since N(oo) N(-oo) 0, we find
CiS, T) 5- K ( 0

I and

if 5> K if 5 < K

which agrees with the known value al T.
00 Then d, 00 and e-,(7 -I) 0 Thus C(5, 00) Next let us consider T S, which agrees with the resuit derived em lie, for a perpetual call

Example 1.3,2 (A 5-month option) Let us calculate the value of the same option considered in Chapter 12, Examplc 123 That was a 5-month call option on a stock with a curtent price 01 $62 and volatility of 20% per year Thc strike price is $60 and 62, K 60, " 20, and I .10, we find the interest rate is 10% Using 5






The cOllesponding vnlues lor the cumulative normal distribution are found by the approximation in Exercise I to be


Hence the value for the cnll option is
C 62 x 739332 - 60 x 95918 x 695740 $5 798

This is close to the valuc of $5 85 found by the binomial lattice method

Although a fOlmula exists lor a call option on a non-dividend-paying slock, analogous formulas do not generally exist for other options, including an American put option The Blnck-Scholes equation, incorpOlating the corresponding boundary conditions, cannot be solved in analytic rOlln

13 4



In the binomial lattice ilamewOIk, pricing ot options and other derivatives was expressed concisely as discounted tisk-neulral valuation This concept wOlks in the Ito process framewor k as well

For the geomettic BlOwnian motion stock price proce::;s clS(t) we know from Section II 7 that E [S(t)]

(13 17)

(13 18)

In a risk-neutral setting, the price 01 the slock at time zelO is tound from its price at time f by discounting the risk-neutral expected value ilt the risk-free rale This means that there should hold 5(0) e-"E [S(t)]
S(O)e" From (1317) and (1318)

It is clear that this lor mula would hold if E[S(t)] this will be the case if we define the process dS , S df

+ 0- S di il (13 19)

where to the

f. is il standardized Wiener process, and we define E a::; expectation with respect
£ process
In other words, starling with lognormal Ito process with rate 11, we

obtain the equivalent risk-neutral process by constructing a similar process but having rate} This change of equation is analogous to having two binomial lattices for a stock process: a lattice for the real process and a lattice for the risk-neutral process In the first lattice the probabilities of moving up or down are p and I p, respectively The risk-neutral lattice has the same values as the stock prices on the nodes, but the probabilities of up and down are changed to q and I q. For the Ito process we have two processes-like two lattices Because the plObability structures are different, we use Z and i: to distinguish them Once the risk-neutral probability structure is defined, we c-an use risk-neutral valuation to value any security that is a derivative of S In particular, for a call option the pricing formula is (13.20)
This is analogous to (12 13) in Chapter 12 We know lhallhe risk-neutral disnibution 01 S( T) satisfying (13 19) is lognormal with E [In[S(J)(S(O)]1 ,J J and var [In[S(J)(S(O)11 0-' J We can use


this distribulion to find the indicated expected value in analytic form The result will be identical to the value given by lhe Black-Scholes equation for a call option price
Specifically, writing out the details of the lognormal distribution, we have

This is the Black-Scholes Immuia in integral lorm

(1321 )


Chapter 13


13.5 DELTA
At ilny fixed time the value of il derivative secmity is


function of the underlying

asset's price The sensitivity of this function to chilflges in the price of the underlying asset is described by the quantity delt. (bTlf the derivatrve security's value is [(5, I),

then formally delta is

Delta is rrequently expressed in approximation form as

The delta of a call option is illustrated in Figure I J 2 It is the slope of the curve that relates the option price to the stock price Delta can be used to construct portfolios that hedge against risk As an example, suppose that an option trader believes that il certain call option is overpriced, The

trader would like to Wlite (that is, sell) the option, taking a very large (negative) position in the call option, However, doing so would expose the tradel to il great deal of price risk If the underlying slock plice should increase, the trader will lose money on the option even ir his assessment of the option value relative to its current price

is well founded The trader may not wish to speculate on the stock itself, but only to profit from his belief that the option is overpriced The trader can neutralize the effect of stock price fluctuations by offsetting the sale of options with a simultaneous purchase of the stock itselr The appropriate amount of stock to purchase is delta times the value of the options sold Then if the stock price should rise by $1, the profit on the trader's holding of stock will offset the loss on the options. The delta of a call option can be calculated from the Black-Scholes fOlmula (13 J) to be
(1 J 22)

This explicit fOlmula can be used to implement delta hedging strategies that employ call options In genelal, given u portfolio ot securities, all components of which are derivative to a common underlying asset, we can calculate the portfolio delta as the sum of the
FIGURE 13,2 Delta of iI call option, Delta measures the sensitivity of the option value to small changes in the price of the underlying se~ curity



13 5



deltas of each component of the portfolio T raden; who do not wish to speculate on the undcrlytng asset prices will form a portfolio thal is delta neutral, which means thal the overall delta is zero In the Case of the previous trader, the value 01 the portfolio was -C + 6, x S Since the delta of S is I, the overall delt" of this hedged portfolio is -6,+ 6, =0 Delta itself varies both wilh S and with L Hence a portfolio that is dclta neullal initially will not remain so~ It is necessary, therefore, to ['ebalance the potllolio by changing the proportions of its securities in ordet to matntain neutrality lilts process

constitutes a dynamic hedging strategy. In theory, rebalanctng should occur


tinuously, although in practice it is undertaken only periodically 01 when delta has materially changed from zero The amount ot rebalancing required is related to another constant termed gamma (f) Gamma is defined as


Gamma defines the curvalute of the derivative price


In Figulc 13 2 gamma is

the second derivative of the option price curve at the point under consideration Another useful number is theta (8) Theta is denned as


oj (S, t)


Theta measures the time change in the value of a derivative security Refening again to Figure I3 2, if time is increased, the option curve will shift to the right Theta measures the magnitude ot this shift These parameters are suffident to estimate the change in value of a detivmive security over small time periods, and hence they can be used to define appropriate hedging strategies In particular, using 8 f, 8S, and 81 to represent small changes in f, 5, and I, wc have

8f '" 6, 85+ as a first-order approximation to 8 f

if x (8S)'+8 x 81

Example 13.3 (Call price estimation) Consider a call option with 5 43, K 40, " 20, r i O , and a time to expiration of J 1 6 months 5 The BlackScholes fOlmula yields C $556 We can also calculate that 6, 825, f 143, and 8 -6 127 (See Exelcise 7 ) Now suppose that in two weeks the stock price increases to $44 We have 8S and 81 1(26; therelore the price of the call ar that time is approximately

C '" 5 56 + 6, x I +

if x (I)' + 8

x (I (26)


The actual value of the call at the larer date according to the Black-Scholcs formula is C = $6 23

Recall that 85 i~ proportio!1lI1 to



we mU);! indLlde thtl (05)2 term


Chapter 13


The derivation of the Black-Scholes equation shows that a derivative security can be duplicated by constructing a porttollo consisting of an aPPlOpriate combination of the underlying security and the risk-free asset We say that this portfolio replicates the derivative security The proportions of slock and the risk-tree asset in the portfolio must be adjusted continuously with time, but no additional money need be added or taken away; the portfolio is self~financing. This replication can be carried out in practice in Older to construct a synthetic derivative security using the underlying and the risk-free assets Of course, the required construction is dynamic, since the particular combination must change evelY period (or continuously in the context of the Black-Scholes framework) The process for a call option is this: At the initial time, calculate the theoretical price C Devote an amount C [0 the replicating portfolio This portfolio should have b..S invested in the stock and the remainder invested in the risk-free asset (although this will usually require bOIl mt'ing, not lending) Then both the delta and the value of the portfolio will match those of the option Indeed, the short-term behavior of the two will match A short time later, delta will be different, and the portfolio must be rebalanced However, the value of the portfolio will be approximately equal to the corresponding new value of the option, so it will be possible to continue to hold the equivalent of one option This rebalancing is repeated frequently As the expilation date of the (synthetic) option approaches, the pOI tiolio will consist mainly of stock ii the pr ice of the stock is above K; otherwise the portfolio's value will tend to zero

Example 13.4 (A replication experiment) Let us construct, experimentally, a syntiletic call option on Exxon stock with a strike pi ice of $35 and a life of 20 weeks We will replicate this option by buying Exxon stock and selling (thm is, borrowing) the risk-rree asset In Older to use real data in this experiment, we select the 20-week period from May II to September 21, 1983. The actual weekly closing prices of Exxon (with s[Ock symbol XON) are shown in the second column of Table 13.1. The measmed sigma corresponding to this period is a ;::;:: 18% on an annual basis, so we shall use that value to calculate the theoretical values of call prices and delta We assume an intelest rate of 10% Let us walk across the first lOW or the table There are 20 weeks lemaining in the life of the option The initial stock price is $3550 The third column shows that the initial value of the call (as calculated by the Black-Scholes formula) is $262 Likewise the initial value of delta is 70 I To construct the leplicating portfolio we devote a value 01 $2 62 to h, matching the initial value of the call This is shown in the column marked "Portfolio value" However, this portlolio consists of two parts, indicated in the next two columns The amount devoted to Exxon stock is $2489, which is delta times the current stock value The remainder $262 $2489;::;:: -$2227 is devoted to the risk-free asset In other words we borTOW $2227, add $262, and use the total of $24 89 [0 buy Exxon stock


TABLE 13 1


An Experiment in Option RepHcation


XON 3550 3463 3375 3475 3375 33 00 3388 3450 3375 3475 3438 35 13 36 00 37 00 3688 3875 3788 38 00 3863 3850 3750



remaining price price Delta
10 19 18 17 16 15 14 13 12 II 10 9 R 7 6 5 4 3 262 196 140 I 89 125 085 I 17 142 096 140 I 10 144 194 265 244 4 10 3 17 321 376 357 250 701 615 515 618 498 397 494 565 456 583

262 196 I 39 I 87

Stocir portfolio

Bond portfolio
-2227 -1932 -1598 -1959 -1558 -1228 -1560 -18 07 -1443 -1889 -1681 -2043 -2475 -29 II -2912 -3384 -3323 -3403 -3479 -349.3 rCl'liwted h\'

624 743 860 858 979 961 980 998 1000

81 I 14 141 96 I 38 113 149 2 00 269 253 4 08 3 16 322 376 357 250

1737 2147 1679 13 09 1674 1948 1539 2027 1794
21 92

2674 3180 31 65 3792 3639 3725 3856 3850

A call


XON I\·it/l I'trike ,nicc .i5 mill 20 wah' to expiration

blll'illg XON Hod mul sel/ing tlte rilk-ftee (/He! at /0% 7/11: IJOltjo/io iI' 1IIljll.ltel/ weft I\'eek (/u"ol'llillg to tlte I'll/lie oj Ilelta a/ IluII time. IVIII:'" tilt: l'oll1#lity ii' .Iet (//18% (the lIcllla/I'oblt! III/rlllg f/tat peliod) the fJOItJo/iO I'O/IIL C/iH"e/1' /lit/Idle I tlU! B!l/(k-SdlOlel NI/tll! of tlte ((III

Now wnlk across the second lOW, which is calculated in a Slightly ditfelent way The first four entries show that there are 19 weeks remaining, the new stock price is $3463, the cOlTesponding Black-Scholes option price is $1 96, and delta is now 615. The next entty, "Portfolio value," is obtained by updating from the row above it The eallier stock pUlchase of $24 89 is now worth (3463/35.50) x $24.89 = $24.28 The debt 01 $2227 is now a debt of (I + 0 10/52)$2227 = $22 31 The new value of the portfolio we constructed last week is thelefore now $2428 $2231 = $196 (adjusting lor the lOund-of! elIor in the table) This new value does not exactly nglee with the cunent call value (although in this case it happens to agree within the two decimal places shown) We do not add or subtract from the value However, we now rebalance tlle portfolio by allocating to the stock $21 28 (which is delta

times the slock price) and borrowing $1932 so that the net portfolio value remnins at $196

Succeeding rows are calculated in the same fashion At each step, the updated por tfolio value may not exactly match the current value of the call, but it tends to be velY close, os is seen by scanning down the table and compnring the call and portfolio values The maximum difference is II cents At the end of the ::W weeks it happens


Chapter 13


in this case that the pOitfolio value is exactly equal (to within a fraction of a cent) to the value of the call The results depend on the assumed value of volatility The choice of a = 18% represents the actual volatility over the 20-week period, and this choice leads to good resuits, Study of a longer pcriod of Exxon stock data before the date of this option indicates that volatility is more typically 20% If this value were used to construct Table 13 I, the resulting final portfolio value would be $2,66 rather than $250 If a = 15% were used, the nn.1 pOitfolio value would be $227 The degree of match would also be affected by transactions costs The experiment with an Exxon call assumed that transactions costs were zero and that stock could be purchased in any fractional al110unt In practice these assumptions me not satisfied exactly But lor large volumes, as might be typical of institutional dealings, the departure from these assumptions is small enough so that replication is in fact practical

EX'h1ple 13.5 (Portfolio insur.nce) Many institutions with large portfolios of equities (stocks) are interested in insuring against the lisk of a major mmket downturn They could protect the value of uleir portfolio if they could buy a put, giving them the right to sell their portfolio at a specified exercise pi ice K Puts are available for the major indices, such as the S&P 500, and hence one way to obtain piOlection is to buy index puts However, a p:.uticular portfolio may not match an index closely, and hence the plOtection would be imperrect Anothel approach is to construct a synthetic put using the actual stocks in the portfolio and the risk-tree asset Since puts have negative deltas, construction of a put requires a short position in stock and a long position in the risk-free asset Hence some of the portfolio would be sold and later bought back if the market moves upward This strategy has the disadvanttlgc of disrupting the portfolio and incurring trading costs A third IIpproach is to construct a synthetic put lIsing futures on the stocks held in the portfolio instead of using the stocks themselves To implement this strategy, one would calculate the total vallie of the puts required and go long delta times this amount of futures (Since ;:, < 0, we would aClllally shOi t futures) The difference between the value of stock shorted and the value oj a put is placed in the risk-tree aSseL The positions must be adjusted periodically as delta changes, just as in the previous example This method, termed portfolio insurance, was quite popular with investment institutions (such as pension runds) 1'01 a shOll time until the US stock market fell substantially in October 1987, and it was not possible to sell lutmes in the quantities called for by the hedging rule, resulting in loss of protection and actual losses in portfolio value

The theory presented in this chapter can be transformed into computational methods in several ways Some of these methods me brieRy outlined in this section




Monte Carlo Simulation
Monte Cario simulation is one of the most powerful and most easily implemented methods tOi the calculation of option values However, the procedure is essentially only useful 1'01 European-style options, where no decisions are made until expiration~ Suppose thal there is a derivative secUlity that has payoff at the terminal time J of I (S( J) and suppose the slock pi ice S(t) is governed by geomeuic Brownian mOlion according to


= /1Sd! +o-Sd~

where z is n stnndmdized Wienel process The basis for the Monte Carlo method is the Iisk-neunal pricing formula, which Slales that the initial price of the dedvative secudty should be

= e- d Ell (5(J)]

To evaluate the right-hand side by Monte Carlo simulation, the stochastic stock dynnmic equation in a risk-flee wodd dS = ISd! +o-Sdz

is simulated over the time intel val [0, J] by dividing the entire time period into sevell.ll periods of length b..! The simulation equation is

where E (td is chosen by a random number generator that produces numbers according to a normal distribution having zero mean and variance b..! (Or the multiplicdtive version of Seclion II 7 can be u,ed) After each Simulation, the value I (S(J» is calculated An estimate P of the true theoretical price of the delivative security is found hom the for mula



where the avelage is taken over all simulation trials A disadvantage of this method is that suitable acculacy may require a very large numbel of simulation trials In geneiUl, the expected enor decreases with the number of trials II by the facto! 11.Jli; so one more digit 01 Hccuracy requires 100 times as many trials Often tens of thousands of trials are required to obtain two-place accuracy

Example 13.6 (The S·mon!h call) Simulation is unnecessary fOi a call option ,ince better methods are available, but this example, which WHS solved earlier in Example 132, provides a simple illustlation 01 the method 1'01 lhis call S(O) = $62, K = $60, 0- = 20%, and I = 12% The time lo malulily is 5 months To cany Ollt the simulation the 5-month period was divided into 80 equal small time intervals The stock dynamics weiC modeled as
S(t + t>t)

= S(t) + 15 (t)t>! + o-S(t)E(t)~


Chapter 13

Running average value
5 H5


Monle Carlo t.!valualion of a il call. The value of

c[lll is estimated as the

discounted ilvemge of final p[lyoff when sim¥ ulillions ,He governed by the risk~neulml process The method is easy implement but

580 575 570 565



large number of lri[lls for reason-

able accuracy

Number of trials
5,000 10,000 is,OOG 20,000 25,000

where application to oplions valuation is treated in [6,7J A textbook Irclltmenl of genera! finile~differl.:nce methods i~ l8J Application to options valuation is di~cussed in 19, !OJ For a discussion of exotic options see II I, 12J The idea 01 Exerci~e 4 is in [13J Black, F, and M Schole~ (1973), ""fhe Pricing of Options and Corporate Liabilities," Joulllal oj Pohtnal EUJIlol1lv, 81, 637-654 Merton, R C (1990), COllfinIfO/t\>~71me FilwlH.e, Blackwell, Cambridge, MA CharcnteliHics and Ri.\k oj 5umdmdi;:.ed Optiom" American Stock Exchange, New York; Chicago Board Options Exchange, Chicago; New YOlk Stock Exchllnge, New York; Pa~ cHic Stock Exchange, San Francisco; Philadelphia Stock Exchange, Philadelphia, Febru~ my 1994 Leland, H E (1980), "Who Should Buy Portfolio Insurance," IO/fT/lal oj Finmne, 35, 581-594 Rubinstein, M , and H E Leland (198 I), "Replicating Options with Positions in Stock and Cash," FinalH.ial ;1flaly\'!\ JOIllllal, 37, luly/Augusl, 63-72 Boyle, P P (1977), "Options: A Monte Carlo ApprolIch," 101ff1lal oj Fillcmc.ial EWlIomiu, 4,121-Jl8 HuH, 1 C, lInd A White (1988), "The Use 01 tbe Control Variate Technique in Option Pricing," JOIflllal oj FillaIH.ial alld Qualltitatil'e A,Wl\·\i\, 23, 237-251 MitcbeH, A, tlnd 0 Grilfiths (1980), Jhe Finite Di!!elenu: Method ill Pmtial D(UerentiCiI Equatiom, Wiley, New York Brennan, M ,and E S Schwartz (1977), "The Valuation 01 American Put Options," hml/lCIl (~I FinClw.e, 32, 449-462 Courtadl1n, G (1982), "A More Accurate Finite DilJcrence Approximation 1m the Valuation ot Options," 10fffflal oj Fiawnial and Quat/titative ;1n(lI\'.\1\" 17, 697-705 Rubinstein, M (1991), "Pay Now, Choose LmeI," Rhk (Februmy) (Also see simiiaI articles on otheI exotic options by the same amhOf in subsequent issues 01 Rhk ) HuH,l C (1993), Optiom', FUlfue\ aud Otllet Detil'{ftil'c 5euuitic\', 2nd ed, Prentice f-Iall, EnglewOOd eliJt.-. NJ Brenner, M , and M G Subwhmanyam (1994), "A Simple Approf

The variable Rl.'f is the amount to which $l loaned at time f will grow at time s if it earns interest at the prevailing short rate each period from f to s The quantity R,\ is, of course, random If f is fixed, then at time j its specific value depends on the node at s It is a conceptually attractive quantity, as we shall see, but it is computationally unattractive It is unattractive because for 5 f > I its description can require a full tree representation, even if the underlying short rate process is defined on a lattice, because the overall return between two periods is path dependent (See Exercise 2 )

We now tum to one of the main themes emphasized throughout the book: risk-neutIal pricing We assume throughout this section that short-term fisk-free bOlTowing exists for all periods, as described in the previous section Hence there is a short rate defined for every node in the tree Assume again that there are 11 assets defined on the underlying state process graph From these assets, new assets can be constructed by using trading strategies We say that risk-neutral probabilities exist if a set of risk-neutral probabilities can be assigned to the arcs of the graph such that the price of any asset or any trading policy satisfies



= Rr r+1 Er(Sr+1

+ or+d

(16 I)

for every f = 0, 1,2, ,J I and whele Er denotes expectation at time f with respect to the risk-neutral probabilities This definition applies only one period at a time, and it is explessed in a backward fashion It gives Sr as a function of the reachable values of Sr+1 and Or+1 We cannot assume that risk-neutral probabilities exist for the particular set of assets in OUI collection After all, the actual prices of the assets may not be related in a systematic fashion However, as one might suspect, we can guarantee the existence of risk-neutral probabilities when the prices of the original assets are consistent in a way that makes aIbitrage impossible This is the content of lhe following theorem, which follows immediately from our earliel result in Chapter 9 on risk-neutral pI icing because the risk-neutral pricing formula (16 I) is a single-period formula


Chapter 16



Existence of risk-Ileutral probabilities Suppm'e a ~ef of 11 assets h; defined all a stafe Suppme that flOlIl fhe~e ane'';, 'iIrOif-fel1ll r;'Ik-free bOl1owil1g is pOHible at every time f Then fhele are risk-l1ellfw/ probabilities 51iCIt fltat tire price5 of trading 'l(rofegie'l with le'lpecf fa flte'ie onefs ale given by the rh-k-Ilellfral pricing formula

S, = _1_E,(S,+r +8,+r) Ru -H {f alU/ onlv

if 110 arbitrage

i'i ponible

Proof: We already h,lVe all the elements It is clear that risk-neutral pricing implies that no arbitrage is possible This was shown in Section 14"3 for a short rate lattice, and the proof carries over almost exactly It lemains to be shown that if no arbitrage is possible, then there are riskneutral probabilities Howevel, if no arbitrage is possible over the T periods, certainly no arbitrage is possible over the single period at f, starting at a given node It was shown in Chapter 9 that this implies that risk-neutral probabilities exist for the arcs emanating from that node Since this is true for all nodes at all times t, we obtain a full set of risk-neutral probabilities H
The !isk-neutral pricing formula (16 I) can be wlitten in a nonrecursive form as

where now Ef denotes expectation of all future quantities starting at the known state at time f, This formula expresses Sf as a discounted risk-neutral evaluation of the entire remaining cash flow stream It has the nice interpretation of generalizing the familiar present value formula used ror detenninistic cash flow streams However, this form is not convenient for calculation because the quantity Rfl generally requires a full tree replesentation (See Exercise 2) There are cases where the result simplifies, of course, such as when interest rates are deterministic The preceding result is just a slight generalization of concepts developed in earlier chapters We have already seen many examples of the application of the riskneutral pricing equation Binomial option pricing was the simplest and earliest example More complex examples, involving interest rate derivatives, were discussed in Chapter 14 We will look at additional examples in this chapter that exploit the general formula, but first we need a bit more theory

According to the definition, risk-neutral probabilities exist if there is no opportunity for atbitrage among the available assets, The theorem does not say that these plObabilities are unique, and, in general, they are not




If the assets span the degrees 01 freedom in the underlying graph, as is the case of two assets on a binomial lattice, then the 1 isk-neutlul prices are unique If they do not span, as in the case of two assets on a trinomial lattice, there will be additional degrees of freedom, and the risk-neutral probabilities are not unique When there are extla degrees of freedom, a specific set of tisk-neutIal probabilities can be defined by introducing a utility lunction U, measuring the utility of tlle final wealth level, and finding the trading policy tllat maximizes the expected value of U (X r) This optimal tIading policy will imply a set of lisk-neutral plices in a manner similar to that fot the single-period case discussed in Chaptel 9 We shall limit our consideration to utility functions thut have a separation property (as was done in Chapter IS) To review, suppose that we begin with a wealth u x Xo, where is level Xu Aftel the filSl petiod, OUI wealth will be X t a random return factor that depends on the hading policy vruiables at period zero x x x x X O, If we Continuing in this fashion we see that X I :=



ago afl


select U(X,) = In then U(X,) = Inag" + Ina~' + + Ina~!::: + In Xo Hence we maximize EoIU(Xr)] by maximizing Edln(a;")] for eneh t, where E, denotes expected value as seen at time f This maximization is equivalent to maximization of E, [In(a;)' X,)] = E, [U(X,+,)] with respect to e, This is the separation property Maximization of the expected final utility is obtained by maximizing the same utility function at each step of the process The separation property holds for the logarithm, and it also holds for the power utility function U (X 1) = (I /y)X ~ When the separation property holds, the multiperiod case reduces to a series of single-period plOblems, all having the same form of utility function This greatly simplifies the necessary calculations (although most of the general conclusions hold lor other utility functions)


The Single-Period Problem
Recall that there are 11 assets The single-period problem at time f, and at a specific node at that time, is to select amounts ej 101 i = I, 2, ,11 of the 11 assets, lorming a portfolio We wish to maximize the expected utility of the value of this portfolio at t + I subject to the conditiou that the total cost of the portfolio at time t is I Hence we seek ej' s to solve maximize


subject to

" Le:s; = i=1 I


" Le;(S:+1 + 8:+ 1) = Xt+1


The expectation is taken with respect to the actual probabilities 01 sllccessor , PK nodes It there ale K such nodes, we denote these probabilities by PI, P'2, Given amounts i = 1,2, ,11, the value 01 next-period wealth X t +1 depends on



Chapter 16


the partie-ulal' successor node k that occurs, The objective function can be written as PkU(XH-,h, where U(X'-I-,h denotes the value of U(X,+,) at node k


Using the results of Chapter 9, a set olrisk-neutral probabilities ean be found from the solution Specifieally, the risk-neutral probabilities are p,U'(X:+,h (166)

where X:+, is the optimal (landom) value of next-period wealth If the utility function U is increasing, U'( X:+,h will be positive, and honee all the q,'s will be positive

These tisk-ncuttal probabilities can be used to price any asset using the general tormula

which takes the specific form

If this method is used to find a set of risk-neutIal probabilities when there arc more states than basic assets, the risk-neutral probabilities will depend on the choiee of utility function The Vat fations in the risk-neutral probabilities will not affect the prices of the original assets, but will lead to variations in the prices assigned to other (new) assets

The price assigned to a new asset this way is such that an individual with the given utility lunction will not choose to include that asset in the optimal portfolio (either long or short)

Example 16.1 (Log-optimal pricing of an option) The optimal pricing method provides the toundation fOi a new latlic-e procedure for pIking a call option Suppose that we plan to use moderately large period lengths in our lattice, but to maintain accuracy we decide to use a multinomial (rathel than binomial) lattice We assign
(real) probabilities to the arcs of this lattice to closely mateh the actual characteristics of the stock

In this situation, !isk-neutral probabilities are not uniquely specified, but we can infer one set of such probabilities by using a utility runction, say, the logarithmic utility function U (X) = In X Once the risk-neutral probabilities are found, we can

price the call option by the usual backward computational plocess
What does the resulting price assigned to the call option represent? It is the price of the call that would cause someone with a logarithmic utility to be indifferent about including it in his or her portfolio Specifically, this person could first form




a log-optimal portfolio (rebalanced every period) of the stock and the risk-free asset Then if the call were offered at the derived price, this person would find that inclusion of the call, either short or long, would not increase utility Hence it would not be added to the portfolio In other words, the utility-based price is the price that leads to a zero level of demand

Example 16,2 (A 5-month call) As a specific example let us consider the 5-month call option studied in Example 123 The underlying stock had S(O) $62, J1 12, and a = 20 The risk-free rate is 1 = 10% per annum, and the strike plice of the option is K = $60. We use a trinomial lattice with I-month periods To match the parameters of the stock, we decide on the trinomial parameters 1/ = I I, d = 1/1/, and the middle branch has a multiplicative factor of I To find the real probabilities we must solve the equations that correspond to: (I) having the probabilities sum to I, (2) matching the mean, and (3) matching the variance These equations, first given in Section 137, are



2 1/ pr

+ + +



+ + +

1'3 = I ~P3 = /1 I:1f 'J d-p3 = ,,-6.1 +(1 +J16.I)-


They have solution Pr 228, 1'2 = 632, and 1'3 140 Now that the lattice parameters are fixed, we must solve one step of the Iogoptimal portfolio problem Hence we solve the problem n,;x Elln[O'R + (I - 0') Roll where R is the random return of the stock Dvel one pedod and Ro is the fisk-flee return Written out in detail this is



max (prln[O'I/ + (I - O')RoJ + 1'2 In[O' + (1 - O')RoJ + 1'3 In[O'd + (I - 0') Roll
This has optimal solution a:= 505 The corresponding risk-neutral plObubililies are then readily found from (16.6) to be


= 0'1/+(1Pr-O')Ro c

(167) (168)

'I' =

= ad +

1'3 c (1-0')//0


where c is the normalizing constant When normalized the values me q] = 218, q2 = 635, and q3 = .I 48 With these values in hand it is possible to proceed through the lattice in the normal backward solution method The results are shown in Figure l62 The price obtained is $5 8059, which is very close to the the Black-Scholes value of $5.80


Chapter 16

9985 9077 8252 7502 68 20 62 00 5636 51 24 4658 4235 3850 3985 3077
2252 FIGURE 16.2 Log·optimal priCing of a 5-month call option using a trinomial lattice,
The upper lattice contains the possible stock

Slock PI ice laUice 68 20 62 00

62 00

9077 8252 8252 75 02 7502 7502 68 20 68 20 68 20 62 00 62 00 62 00 5636 56~ 5636 51 24 51 24 51 24 46 58 46 58 42 35

prices. The lower l[lllke is found by


valuation using inferred probabilities

Log pricing lattice 10 43 520 192 1651 985 456 143 26 2351 1601 927 385 92 09 00


31 27 2302 1552 870 303 43 00 00 00

1502 820 200 00 00 00 00 00

The stmting point for genelal investment analysis as presented in this chapter is a glaph that represents a family of asset processes How can we construct such a graph

to embody the characteristics of each asset and the relations between assets? Clearly, this construction may be quite complex This section shows how a graph for two lisky assets can be constructed by combining the sepruate replesentations tor each asset Specifically, two binomial lattices are combined to ploduce a double lattice that faithfully represents both assets, Suppose that we have two assets A and B, each represented by a binomial lattice Each has up and down factors and plObabilities, but movements in the two may be conelated A replesentation of one time period is shown in Figure 16,3 The combination of these two lattices is really a lattice with fOlir branches at each time step It is most convenient to use double indexing for this new combined lattice; call the nodes II, 12, 21, and 22 The first index refers to the first lattice and the second

FlGURE 16 3

One step of two separate lattices Their

movemenls may be correlated

12 22





FIGURE 16,4 Nodes of the combination, There are four possible successor nodes from the centrell node


to the second We define the twnsition probabilities as PI!, PI']., P21. and 1122, respectively A pictUle of the combined lattice is shown in Figme 164 Here lhe cenler node is the node at an initial time, and the four outer nodes are the four possible successors Suppose the lattice for stock A has node factors u A and ciA with probabilities p~ and p~, respectively; and the lattice fOI stock B has node factors u B and d U with probabilities p? and p~ If the covariance of the logarithm ot the two retUln factors aM is known, we may select the probabilities of the double latlice to satisfy'


+ PI2 = p~ + P22 = p~\

PI! +P2I =

pi' u'u" + (PI2 + (P22 p~p~) U'D" p~ p~) D'D"

(PI! - p~p?)

+ (P2I where U A = Inu', D' A special case is of the two asset returns are PI] p~pll, Pl2 =

p~ pn D'U"



= Ind', U" = Inu", and D" = Ind" when the covariance is zero, corresponding to independence In that case it follows thai the appwpriate lattice probabilities p~p~, P2! = p~pT. and P22 = p~p~

Example 16.3 (Two nice stocI 0 The computing procedure uses certainty equivalenls rather than expected values let us briefly review the certainty equivalent concept Suppose lhat an investor has a utility function U Suppose that X is a random variable describing the investor's wealth at the terminal point. Then the expected utility of thrs wealth is E[U(X)] The certainty equivalent is the (nonrandom) amount x such that UCX) = E[U(X)]. We often write CE( X) for the certainty equivalent of X As a specific case suppose that U (X) -e-" < and suppose that the random variable X has two possible outcomes X I and X2 occurring with probabilities PI and p" respectively The expected utility is
E[U(X)] prU(Xr)+ p,U(X,) -pr e -"

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