A Model of Stochastic Process Switching

International Finance Discussion Papers Number 201

January 1982

A MODEL OF STOCHASTIC PROCESS SWITCHING

by

Robert P. Flood and Peter M. Garber

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an

acknowledgment by a writer that he has had access to unpublished material) should be cleared with the author or authors.

Often, a policy authority such as a central bank operates by establishing a policy rule to set the variables under its control. Such a rule iis allowed to operate freely as long as certain endogenous variables of interest to the authority remain within particular bounds; however, when those endogenous variables cross their bounds, the authority switches to a new policy rule which it had prepared to meet this contingency. Since variables such as prices are determined partly by agents’ beliefs about future events, agents’ behavior injects the probabilities that policy switches will occur at particular future times into current price determination,

In this paper we explore in a formal model the determination of a current exchange rate when future policy regime switches are possible. In order to do this we develop a new aspect of an otherwise standard exchange-rate model; this key component is the probability density function (p.d.f) for the first passage through a barrier of the endogenous variable

1/

(the exchange rate) which interests the policy authority.—’ Since analytical solutions for first passage p.d.f.'s are available for only a limited number of stochastic processes, we are restricted to these processes in formulating our exanple. However, within this class of processess, our results are generally applicable to many different kinds of macroeconomic problems.

We present our ideas in the context of a model of exchange rate determination. Our choice of a specific example is intended to add concreteness to the analysis but should not be interpreted as setting

limits on the applicability of the analysis. Indeed, the structure of

the problem at hand virtually duplicates the structures which would be

dinjny swos Aepun Jey} eTqtssod st 4T ‘iaAeMOH ‘*aqeT BYI Jes 0} SJayIEU asgueyoxe ut ATJUaIIND |suaAISIUT Jou Op sjUsMUIZAOS Jey SsuUeaU ATaeIJ

JOT OF PoMmoTTe ST SsaeTouUerTIND OM} UseMJeq |9Je1 BBuUeYoKS 9YyI IeUL

ATOXTT ST BULXTY sinjngy usymM ojey eBueyoxY JUSeaIND oy] ZuTuTWI19}0q “(1

*uoTINTOS a4ei-aZueyoxe ue

sonpoid 03 Aiessadeu sdeqjs atofew ay} Juesaad om TI uoTjOes UT ‘eTqrssod st B3UTXTJ einjnj usyM Tepow a3e1-a8ueyoxe au dr jas oM T UOTIDeS UT

"S,076T ATiee

ay] UT Sazer BBueyoxo YysTITAg pue yousryZ oyy JO sjusweAoM ay BUTApNis

0} eTqeot{dde ATie[not Ied aie sz[Nsei ino ‘ased 199}3eT BY} Jo eTdwexa ue

SY ‘wioj xeTdwod siow e aq T[TM a3e1 aZUeYOKS JUeTAIND ay} AO}J UOT NTOS

Teynoe ay. ‘snogoTeue sft enbyuyoa} uct njTos ay i y8noy} ‘uayy ‘uteqreoUN st

SUTWT O41 STTYM UMOUy ST TeAeT ay JI ‘aspeTMouy yons But oeTJeI w1z0jy

eB sounsse 9}e1 o8uRPYyoXS JUSIIND |Yy OJ UOTINTOS ay. PTIOM suOoTIeDedx~a

TvuoT eI B UT Udy} STeAST UMOUY & Je PaxXTJ 2q TTTM a}e1 VBueYoKS 9Y} oUT eAinjny e Je Jey MOUY sjuese usyM ‘Apnjs om aTdwexae opyToeds ey} Url

‘uoTje[N3e1 SuTputq Mou 9Yy} Jo ATNseI e se aB8ueyd pr[nom Awouoda ay} uT sassavo0ad

OFIseydo}JS By} JO oWlOSs Udy. BUF[F299 ay} Yee OF |1AaM SaqeI JSaT0qUT JUeAaTAI

JT ‘TeAeMOH *BUTTTe8D Jey} MOTAeq eq ATJUeTAIND Kew Saqe1 4Jsai1ajUT auwos

uo ZUTTT99 0 uoTIeTNsey e yITM Awouode ue ‘eTdwexs tog ‘suoT eTN8e1

Zutputq-uou ATjueTInD jo sjzoezja oy ButApnys 10jz [nyasn aq ptnoys, dn-qas

Ino uoTiTppe ut “yo TMs ADT[od singjny utejsaoun Asyjo Aue ATTen ATA

io aoFid p[o3 jo BZuTxTZ eainjnj vy} ‘wAroyat xe. oTqTssod ‘sto1juoo sotad

pue e8em Jo uoT}ONporjUT eTqtTssod oy} faTni a3e1 JsereqUT ue 0} UINIeI

etTqtssod s,AjTioyjne Aiejauow e se yons swetqoad B3utTApnis zo0jz ajetTAdoadde 14 49 F104 Y T 3 J ,

~3-

contigencies a government may intervene and establish a fixed rate system; this possibility will partly determine the current floating rate through its effect on expectations.

The specific example that we have in mind is that of Britain in the 1920's. The British decision to return to the gold standard at the pre-war parity of $4.86/&, was announced in the Budget Speech of April .28, 1925. and effective in the exchange market the next day (Moggeridge 1969, p. 9). However, as early as 1918 the Treasury and Ministry of Reconstruction appointed a Committee on Currency under Lord Cunliffe, which reported in 1919 "in our opinion it is imperative that after the war the conditions necessary to the maintenance of an effective gold standard should be restored without delay" (Moggeridge 1969, p. 12). Since the dollar was fixed to gold at that time, the British government was indicating that in the future it would fix the dollar-pound exchange rate at its pre-World War I level; the timing depended on achieving. purchasing power parity at the pre-war exchange rate. Adopting such a policy affects the current exchange rate. Here we present a model in which this result is explicit.

In order to highlight the novel aspects of our study we adopt the simplest: exchange-rate model popular in the current literature. This is the monetary model of Bilson (1978), Frenkel (1978) and Mussa (1978). The model consists of semi-log linear money demand functions for the countries studied, assumptions of purchasing power parity and uncovered interest parity, and an assumption that semi-elasticities of money demand

with respect to interest rates are identical across countries.

(7)

(€)

(Z)

(T)

1 ly 2 asodut em AQTOTTduUTs Zt * ”

Jog ‘*Teinjonij4s 9q 0} pounsse aie (¢

Sutseyoind jo uoftjdunsse oyz st (¢) uot eNnby ly «0p) *siojowered sjt

pue Tepow sy} Jo YyooTq BSUTpPTIng [TertoTAeyeq JTSeq oy} ST UOT}OUNF puewesp Aauow ay] *(T) JO apts puey—-WysTI sy. ‘puewep Adsuow Teer Tenbs Jsnu YyoTYM

ATddns Aauow oTSowop Teer ay} ST (T) JO OPTS puey-IjseT sy]

JoT[T ies 10 q pelep seTqetTiea [Te pue Tepow oud JO aanjzonazjs 29} BuUTUTe UCD Jas UOTJeWAOZJUT 3 OWT = (A)I

(.)I Uo TeuoTITpuod (3)x Fo asBueyd Jo oje1 paqoedxa = ((3)1| (3) *)4

‘aqei a3ueyoxe

(a)x aoueqinistp oTAseyoojs = (9)A (TeseT) 33e1 JSetTaqUT = (9)T gndjno = (q)4

Tease, sotad = (4)d

Ajtddns Aauow = (3)w

pue (‘3°N) ,USTet0jJ,, sejJouap eTqetTiea

eB TaAO (xy) YSTASISe ue fswTYyITAeZOT vsjousp ATTeA9uas sidjJaT eseD ABMOT

*((9)1[(9)¥)a + (HE = (at

()x + @d = Cd o< Sp stp f(aya + (a)t%m - (a) dtp + Op = (a) d - (aya o< “melo f(aya + (a) Tem - (aykto + OW = (a)d - (aDU

suotjenbs ZuTMoTTOF ay} Aq peqtidsep ST Tepow auy

-y-

-5-

power parity, which is an arbitrage condition in a one-good world. Equation (4) is the condition of uncovered interest parity, which, with risk neutrality, follows from an assumption that domestic and foreign

3 ; * earning assets are perfect substitutes >" We assume that m{t), m{t), y(t), * x y(t), v(t) and v(t) are exogenous to x(t).

Combine (1) - (4) to obtain

* * kk . m(t) - m(t) - x(t) = Oy - G + a,y(t) - a y(t) - ooB (X(t) |I(t)) * + v(t) - v(t). (5)

* x * k * We define K(t) = Ay ~ A + a,y(t) - a,y(t) - m(t) + m(t) + v(t) - v(t).

Hence (5) may be written as

x(t) = K(t) + a B(X(t)|I(t)). (6)

Squation (6) is the standard sort of equation that monetary models have produced and is a structural semi-reduced form consistent with a wide variety of models. To address the problem of the future fixing of an exchange rate we must specify both the stochastic nature of the exogenous forcing function K(t) and the nature of the policy rule whereby the monetary authority decides the time for fixing the exchange rate. With rational expectations, the decision to fix the exchange rate implies a decision

to change the stochastic nature of K(t). This follows from equation (6);

when x(t) is fixed, with rational expectations, E(x(t)|1I(t)) Must be zero, hence K(t) must be fixed. For the purposes of this example we will assume that, as long as the monetary authority does not actively fix the exchange

rate, K(t) is a random walk with drift, i.e. K(t) can be written as

K(t) = K(O) + n t + e(t) (7)

pue) pajoedxe ay} 30J (6) ®ATOS ued OM UOTITpUOd TeUTUIa} eB USATD

Co fo (6) (a)1|@)*)a = + ((a)1|()wa = - = ((3)1| (4)*)4

aAeyY oM SZuUTZUeTTeDAT £(2)]I UO

TeuUOTITpuos vjzet aSueyoxe poqoedxe oy} UT uoTJenbe [eTJuetesJTp e ST STUL (g) ((a)t| @)x)a% + ((a)1/@)wa = ()1[@)*4

puTJ OM 6} BWTQ Je Squase OF sTqeTTeae (3)I 32S UOTIEWIOJUT VY} UO TeUCTITpUOD (9g) JO SapTS YyIOG Jo suoTzeIJedx~e BUT ye, *(2)a pue x uo TeUuCTITpuoD ST YoTYM *((a)4 *x|3-L)3 ‘y°p°d @ YITM Wopuetr ST sinooo a8essed 4sATJ STYI YOTYM Je VIngnyZ sy UT J Juswow ay} 62 oT Aue qv jy BOP $,9}e1 B8ueyoxe oy} JO OWT? 9Yyz 7e MOTSq WoTZ *X yznoryy a3essed ysafty e ayew OF (9)xX = (ad ~ (3)d qoedxe am *ATJUaTAIND UTeqGO OF STY JOJ MOT 002 ST [aAeT VoTAd usTer0z |UQ SnuUTU TeAeT soTAd ST IsSeUOp ayq uotzdunsse Aq aouts. ‘Gd - (4)d = xX usYyM *3°T ‘x Jeqnotqied awos 3e sptoy Ajtsed aemod ZuTseyoand uaym ajei1 aB8ueyoxe oy} XTF TTTM £1QUNOD UBTIIOJ ay} UT SseTITAOYINe ATeJeUOMU ay} Jey esoddns aM ‘pexTF 3q

TITM a}e1 a8ueyoxe vy} useymM oF etna AdTTod e Ajtoeds 0} Aepizo UT

‘aqei a8ueyoxe

PexTJ peatsep ay} pzemMoz JFTIP TTFA (3) FeYR OS (4) xK JO Toaquos Aq

pastocexe oq ued (3)y BuTUIeACZ ssad0id |9Yy. AeAO TOI}UOD *97e1 aZueyoxXs paxTj ‘<qqtaed aem-aaid e 0} uANqer 07 TeOS JusMUIDAOS *y°Q B BUTJOeTJeI ssooo0id

eB q09Tas om ‘aTqtssod ere (3)y A0J suOoTJeOTJTONds sATJeUIEITe AUC STTUM

"(8,9 TON ~ (S)2 'a°T ‘ssoo0id ateueTM e ST (S)2 pue o2e1 JJTIp oy} ST LU oT0YyM

-9-

therefore actual) exchange rate at time t.

Suppose first that purchasing power parity at the exchange rate * occurs at time T; then the exchange rate is fixed at x for t > T and x(T) = x. Since x(T) is fixed at T, its expected rate of change conditional on fixing at T is zero at T and hence, from (6), x = K(T). That x(t) makes a first passage through x at Tis equivalent to K(t) making a first passage through x at tio!

Conditional on first passage at T, the current exchange rate (and

its current expectations) can be determined as

t -T

E(x(t)|1(t), T) = x exp { 5 2

T } +> exp cf} f E(K(t)|I(t), T) exp{- + at (10) % “7 %9

where E(K(t) |I(t), T) indicates the expected path of K(t), t < t < T, given I(t) and K(T) = x for the first time. The unconditional exchange rate is

then the integral of (11) weighted by the first passage p.d.f. x(t) = J E(x(t)|I(t), T)E(T - t|x, K(t))aT (11) t

Equation (11) is of the form of a typical solution to a rational expectations model. The problem which remains is to express the right hand side of (11) in terms of a finite number of in principle observable variables. In

linear rational expectations models this final step is often accomplished

by conjecturing that the solution is a linear function of the state

variables and then requiring the unknown coefficients in the conjectured solution to obey the model at hand. This is the method of undetermined coefficients recently popularized by Lucas (1972). Our problem, however,

is substantially more difficult because the as yet unknown non-linear

aT3ya

(€) | Th7%9 -x = (2 (IQ)

‘ST uoTze}Iedxe TeUuOTITpuod ay AOJZ eTNuAOCZ ALoOTTdxa

Ty fe T

ey wey, *———__ f= = 7 pue (3)¥ - x27 42-22 bg - 7 ay Totti - tf 2 % _ JOT “SUT JSATF SY} AOF X = (1)Y SL OWT Qe Jey UseATS pue (2)X uaaT3s

(1) JO uoTzeJDedxe ay. ST (L >1ls aS] *()1|() 4 yey. [TTeI9y “(CL *(2)1/(2)4)a JO UOTIEUTUIaJap 9Yy} UTeTdxe OM Uay, *SszUaUOdWOD sqTt ufeTdxe pue uotzeqJoedxe sTy} OJ eTNWI0J YTOTTdxe |Yy Ano |ITAIM ASATJ OM

*sassaov0id oTAseycoyS UT ssToOAZexe ue st (J *()1](2)a)4 JO UOTIBATASP ou]

q- 1) 49

Z ze ~ Diese (= He = @y- xr IO

Ga - x

(ZT) = ((3)a ‘x|3 - 1)3

ST X > (2) ueaTs ‘x ysno1ryy (1)y Jo o8essed AsATJZ ay AVAO *F°p'd sayy

“(Ege *d Saz0TAe], pue utTTaey 99S) S}xX90} piepueqis UT eTqeTTeae st 1JTIp yATM ssev0id JoueTM e jo *z*p'd a8essed 4sATy ay. A0J uot ANTOS au]

*Yees OM WIOF poonper oy} BuTpTaeTA *(TT) pue (OT) OIUT paanatTasqns aq

usy} Aew sepnjtTuseuw om} asay} 10z suotsseidxe TeoT ATeuy jg ‘()1]() ws

SUTpUyT} puodes pue ((2)¥ *x|3-1)3 uoT}ounjy AjzTsuep ay} BuTputTy jasaty ‘sdaqs

Om} UT passoid om uoTjenbsa oje1-a8ueyoxe WAi0J peonpea ay. uTeIQGO OL

(LZ *(3)T}()a)a pue ((3)y Sx] - 1)F Jo swiz0g syL (IT

*oFAeyeq JusWUIsAOZ VinjJnjZ pejoedxe pue Juszind uo puaedap TITS siejoweied esoym uoTeTe1 [ein}INIZS-uoU e 2aq [ITM Yeas aM UOT INTOS

eyL ‘seTdpoutad ysayJ Worx pajonajsuod aq YsnM UOT INTOS 9YI Jo W105

-8-

Ty T Ta 4 c, = [1 - 7 ita - 7 + ot ]6(Z) + ozn 21 - 7) 0-2)

T T T ~ exp (1 = FIG ~ 2? + 07x 10(-2) - oar fa - 4-21) (14) lo 1 , 1 and 1 Tr kok T x C, = lor p(1 - 7 *o(Z) + Z2(1 - 72) T T, 4 T - expt (1 - 714) ford - =) 74 - 2a - hed). (15) 1 1 l 2 x 2 In these formulas, $(x) = —— exp {- ae and @(x) =f exp{- thay. 27 -o ¥27

To derive formulas (13)-(15), we must find the conditional ‘density of K(t), given T, the time of first passage through X; where T>t> t. Call this density function h(K(t)|T). Then we need only multiply by K(t) and integrate to determine the first moment. We can find this density function by first determining the joint density over (K(t), T). For simplicity, let us assume that we are looking forward

from time t = 0 and that K(0) = 0. (These assumptions are relaxed in our reported results; see Appendix section 4.)

The joint density function equals the conditional density function over K(t) multiplied by the marginal density function over T, £(T),

i.e.

g(K(t), T) = h(K(t)|T)£(T). (16)

The joint density also equals the conditional density over T, given K(t), which we denote by F(T|K(t)), multiplied by the marginal density over

K(t), H(K(t)), i.e.

*(Z1) wotqenba Worry) ‘ST Ty ae omra asaqy

ey. 103 x y8noiyq Zutssed s,yzed e YITM pazetoosse AYy8tem AIT[TIqeqoad suyz

. qT, -.1,0 ; Ty - 1,40 t (61) 3 = (= aa (‘a - 2)u-x - ()y¥

:se (QT) Woly peuTeqqo st 1 swt 4e (1)y Ysnoiy BSutTssed sqt

_Y3TM pelzetoosse Yy8TemM AIT[Tqeqoad TeuotjtTpuooun sy «Ty 42 xX 7® sqieqs

yqed e@ Jey} Ud=ATD *1 OWT] Je (1)y Y8norYI pue Ty ae awT} ISITJ aya Joy x | ysnozyi ssed ‘T yqed ayTT ‘yotym syqed jo AjtTuTJUT ue vie sarauy

*1 03

Aoyid x yZnoiy} pessed aaey osTe yoTYyM 4nq 1 Ae (1)y Yy8no1y ssed yoTyuM

(I e1nsTq UT T eATT) syyed [Te YyYITM peqetoosse jYysTemM ay. JOeTIqns

Istm aM £((1)4)9 Aq (21)y 02 UsATS AYSTEM AATTTqQeqoid 9yI woay ‘snyy ‘2

01 totad x MoTeq pouteweir ZuTAeYy $,(1)y UO TeuOTITpuoD (1)y Jo AITSuap

u aud SE ((1)MH ‘(9GE ‘d Sa0TAe] pue uTTszey 99S) { x4 }dxe at = (x)> az0ymr

3_0 O1uZ

1A 4 . _ > t = Cor= Gy zh axe — 7 = (Gs

LAO

(gT) Aas

‘y°p:d [euotqTpuooun ue sey (1)y ‘0 = (0O)F ONTPA ZuTIAeIS S YITM pue YJTIP YIM ssov0id AsUueTM e& ST (1) a0uTs *(@)H doTeAep aM ISATY *(9OT) worzz (Z| (2)4)4 A]TSuUsp TeUOTITpuUOD ay} sUTWIZ}e0p ued aM vay? £(] 6(2)y)3 Aatsuap qutof ay. Jonazaqsuod 03 (/T) UT suoT oUNg A}Tsusap 34}? BSN T[TM 9M SaaTAosp 0} Asea ATOATIeTOA oae Ady VoUTS ‘1 03 ZotTaid perzinoso ZurAey jou as’essed JSAT} uo TeucoTITpuod st 4TF Jnq

*‘sino00 a8essed 4SATJF YOTYM 3e 1 < J swt ey UO puadep Jou seaop ((1)y)H

(ZT) ((2)m)H((2)4{L)a = (L f(2)4a)3

K()

ar v TIME

FAGURE

* (ZT) uotjenba woilj utTese yng *((2)4|L)a4 *3°p°d ~Teuofytpuos ayy Aq

((2)x)H ATATa—Tnm yasnu am (J £(1)y)3 *F°prd autof ay. euTMAazap oF

T Tt T ryo- LAO 2 - 1A0 Ty ° 19 9 (€Z) a ee ee (3 - 1)uU-x - (2)¥ qu - x x 1 LAO LAO - = 1 rap? & = (wa (8T) wozz (22) qoOezAqns ATUO pdsU aM JUeJSUOD ZUTzT[TeUAZOU e OF dn ((1)¥)H SUTUAAaJep OF Ty - 1,9 Ty ~ 140 T3 oO Tyo 0 (ZZ) p-—__——— a (2 92/t f (2 -21ju-x - (1)y T qu - x x L . T >(TZ) JO ~3 AsAo TeisajUT ay} ST 1 3e (1)y Tenba pue 1 03 aoTad awtz awos ye x ysnoiyq ssed yIoq yoTyM syqed [Te YITM paqetoosse AYystem ARTTTqQeqoad 9yW SetoOyer9YL Ty - 1,0 Ty — 1,A0 T, 5 Tyo (12) ot i gle (72 - 2)U- ¥% - (1)¥ qu- xX — = (0 = (o)a|'a)3(x = (Ta) a] (29 ST 1 38 (1)y Y3no143 ssed pue Ty 3e aT] 3YSATJ Oy? AOZ x YyBno1ryW ssed yI0qG yotya syjed [Te Jo os ay YITM poJzetToOsse AYUsTaM AAT[Tqeqolrad ay} usy] T T T . T. 30 3_07 3 Uzso 140 — (oz) (gg 44 = 2 - jax HE = (0 = (sf '3)3 qu - x _— 2‘ qu - x) =

-12-

_ ~ 2 ; = __x - K(t) _ (« = K(t) - n@ - 1)) ¥(T|K(t)) JG RED exp{ 2o(F - 1) } _ x - K(t) x - K(t) - n(T - 1) = ee ( ). (24) o(T - 13/2 ovT - t Finally, g(K(t), T) = CH(K(t))F(T|K(t)) (25)

where C is a normalizing constant. h(K(t), T) is simply (25) divided by £(T) and evaluated at a particular value of T.

To derive Cy and Cos we performed a change of variable in (25) to produce a p.d.f. over u(t) = x - K(t).— In the formula (13), C, is simply the inverse of the normalizing constant for this p.-d.f. while Cy is the unnormalized first moment of this p.d.f. Hence, co/C, = E(u(t)/u(t), T) so that E(K(t){K(t), T) =x- c./C,- Deriving the actual formulas (14) - (15) requires the cranking out of some horrendous integrals, which we relegate

to the appendix. IIL) Application We have derived analytical expressions for f(T - t|x, K(t)) and

E(K(t)|1(t), T). The next step is to substitute these results into (10) and (11) and continue the integration. However, the remaining double integral. has proven intractable to us, so we simply report our solution

for x(t) as

coe]

x(t) = Sf [x expf t

T -T, 1 — = } + ay OP)? ’ (x - Cy (1) /C, (1) exp{- att)

-£(T - t|x, K(t))dT (26)

*Aqtied azem-oad 03 uanjez ysTjtag aya

S2 pouTjep ATAeaTD se wetTqoad ZuTyoITMs ssaoo0id oOTIseyoo Is e UT A[Nser Ady 1eYyI suoteIeqTTep yons jo Teotdsye st 3E SlaAeMoH *B8UTYOITMS sseo0id oTIseyd0}s Jo JueweTe ue swatqoad SuTAseoeIOJ ,Squese o.UT Joalut Ady} Se elaeqt {ep sieyew AOTTod rsAauaym ‘psapul ‘*pesidseptm oatnb 3q Kew

SjUuseTD pue Teyverq UT petequnosue wetTqoid ay. AeYyI sn oj sulaes AT

‘uoTzenbs aje1 ssueyoOxXS ABaUT[—-UOU ANO YATM pizo0odsze UT

petstoeds useq saey p[Noys seinpssoad ATsey, jo o3ejs JSATJ oy} Si[NseA Ano 0}

SUTpAo.Dy + ‘sanpadoid soaenbs JseaT o3eqs OM] ABSUTT & pasn sjuUdUleTD) pue [eyueIrg

STeTIUSISJJTp oJeA JsoAd_UT Jo AZToUssOpus sy} AOF MOTTS OF, *BuTYOITMS

ssaooid BuTzedtotTjue useq saey Aeu sqUase usyM poTied ayA jo yxed a81eT &

osseduoous yorum *Cz76T APH OF TZ61T Atenaqey potazed ayz AaAo uoTAenbe ajei-aBueYyoxXe

WMn/SN & sJeWTIse (0861) SJUSWeTD pue TeyueT_Z ‘atTdwexe tog ~*BupTYyoITMs

ssvo0id oTAseyoojs BuTyzedtotqjue sie squase usyM potied e BuTainp spoyjow

zeautyT Trotdkéa Aq uoftienbs 3}ei-s2ueyoxs uo sjeUWTISs 07 ajZeTidoidde Jou st Tt yey} SeTTdWT IT Jey ST A[NSaAI ano jo einjzesy azeuNn,Aojun sy

*SoUuTINOI-qns uoTIe1Z94UT [eoTAeuNu

pue senbTuyoe} azeautT[uou jo uot ieutTquos e SUTSN pezeUt ssa oq atdtoutad

UT ues aAoge 9Y} WoAF ZuUTI[NSeA uoTIeNbs 93e1-a8ueYyoxs iesut[uou suy

-¢€1I-

-~14- Footnotes

1 yodels of pricing for some types of options make use of first passage probability density functions. For example, Ingersoll (1977) uses first passage profit's in studying the prices of convertible securities.

Play assuming dy = ay we are able to determine x(t) without modeling

the goods market. Alternatively we could allow a5 2 ays impose world goods market equilibrium, and produce an exchange-rate solution slightly differeit from that reported below.

i i est parity can be 3/ some empirical support for the assumption of open inter P

found in Hansen and Hodrick (1980).

4/

— In our example we are treating the U.S. as the home country and the

U.K. as the foreign country so x = 2n($4.86/b).

the nature of the exchange-rate fixing policy preciudes the existence

. of a mu.tiple solution type bubble which would cause x(t) to rise through x. However, it does not preclude the existence of negative bubbles which would prevent x(t) from passing through x from below even though K(t) passes through x. Therefore, for the stated equivalence to hold we

must explicitly rule out the existence of multiple solutions to (9) of the speculative bubble variety. Hence, the solution to (9) depends only

on market fundamentals, as the formal expressions (10) and (11) for a

solution explicitly indicate.

6/ {We are extremely grateful to J.H. Kemperman for showing us how to derive

the conditional expectation of K(t).

*HYAIOR MON STITH-MPIDNOW 6S970N TBoTIOISTH pue suocTjeottddy yITM suoTJeNby TeTIUetessTq “7/61 6° ‘SSuowMTS

"WW ‘Butpesy ‘Aad TSay-LOSTPPY

‘Savy ssucyoxgq jo SoTWOUd DG eT (*Spe) wosuyor “H pue TeyuerZ ‘Lf

UI € “YD ,,{ZuTaeoTy peT{Torquo) Jo eutSay e aJopugq AOTTOg TeoSTty pue Azeqouoy pue squautrg jo aouerteg oy] ‘a 3ey agueyoxg ay, ‘8L61T ‘‘W ‘essny

"66-06 ‘dd ‘uoqgutysem

f*5*y-S's pue weqsks daAIaS|ay Te1epay syI JO SAoOUASADD JO prrog

‘FoUdIaJUOD VOTIbUTMISJOg SoTAg 30 SOTAWSMOUSIT Sepa Sutaqgsyog 0 ut ,,‘stsayqodAy oqey TernjeN 949 JO Zutqsaey otaqawouoog,, ‘7261 “WY “seony

“ssaig oTwepesy ‘°A°N ‘ ‘pe puz ‘Sassao01g STaseyooys Ul 9Sinoy Asity y ‘oz6T Sx0TAeL “y pue *s ‘uTTAery

*ZZE-687 ‘vy ‘SoTWOUOD, TeToUeUTY jo ~Teuanor ‘ S8TITINIVS VSTqTIAaAUOD Jo UOTIeNTeA SWTeTIJ-JUSesUTIUOD y,, ‘//6T *°Lf ‘TTOsaesuT

*19q0390 ‘¢ ‘ou ‘gg ‘Awou0dg TReOTALYTOd JO [euAnor ‘, sTsATeuy DTAJowouolg uy :seqey yods sinqng jo SI0JITpeig TeutTjdo se sajey aesueyoxY premsoy,, ‘OS6T SYOTApOH *y pue °7] ‘uasuey

"79-647 ‘dd ‘key fz ‘ou ‘oT ‘SotmdUdDF [euotyeuzsquy Jo Jeuinor ,,°S,0761 sy UT punog-zeTIog ayy :se0Tag DATIelTay pue Aosuoy ‘sajqey a8ueyoxg,, ‘O86 ‘squewat) *‘y pue “fr ‘TayuerZ

"WW fasptaqueg ‘yoieasey STwoUuODY Jo neeaing TeuotaeN ‘QG4 azedeg Zuty204 ,,*S,0L61 BUI UT Saqey afueYyoxY atqrxety,, “qOgET ‘<

"ty-sez “dd ‘key ‘7 ‘QL ‘mataAoy StwouosZ uestaeuy ,.°S,0761 aya worzZ suossay :AougXY pue sadTIg ‘sajey eBueyoxg,, ‘eQgsTt ‘

“VR SBZutTpeosy ‘AaTSay uostppy ‘saley esueyoxy Jo soTWOUGo sy (°Sspa)

uosuyor *yH pue [ayxUeTZ “fF UT T “UD ,,Se0Uaptag Teotatdwy pue sqoedsy TEUTAID0g «=FazeY OAueydXyY By 09 udtoiddy Areqouoy y,, ‘Q/6T ‘*r ‘TaxUarg

"YA ‘3utpesy ‘As Tsay-uostppy ‘Spey osuetoxy JO SoTwWOUdSY ouT {*Sp9) uosuyor *H pue Texuery *r

UT ¢ ‘YD faIeY VSuryoxg ayQ pe suojyIrzoadxg Jruotqey,, ‘g/6t ‘°r ‘uosttg

Sad39rayay

-~16-

Appendix

Derivation of Cc, and Cy

1) Solution to Integral in Text "“quation (23) Notice that the integral part of the right-hand side of equation

(23) [text] is a convolution. It is

T - Al ; w(t, )Z(t - t,)dt, (Al) where = x - nt x 1 anew emat A2 wey) 2 a7z $a 42) ot ot 1 1 and eet? xe) - ¥ - nr - &) Z(t - ty) 2—— (4 (A3) o(t - t)) It is a property of the Laplace transform, Lf ], that

T L[ w(t) ] *L{Z¢t)] = Lif w(t, )Z(t - t,)dt,] (see Simmons, pp. 407-408). For our problem

x

ee. x - NT w(t) = 372 > — TW» (A4) OT OT and -1/2 — _t K(t) - x - nt Z(t) = 3 nT ) (A5)

This property is useful to us because the problem of integrating (Al)

- -1 may be stated equivalently as finding L lintw(t) ]L[z(4) 11, where L [ ]

2°

(OTV) {(,,_[4 9% + zi -

z/t hz }dxe

Oy ee: z/t [d, 9% + gil = [(2)Z]1

ufeqqo om «*(U JO USTS BYyQ eBueYd ‘oeT) LU YITM U- ZuToeTde1 pue aa - x ya q[nsez ze ur x sutoetTdeit uayi pue x &q (gV) 3utptatp Aq (1)7 Jo wrojzsuelaq aoeTdey] oyq ureqqo aM 3eYQ SMOTTOF AT °(1)Z UE U BuTeq (1)mM ut U- pue (1)M UT Wi9eq ZuTpuodseiz109 9Y43 uF x Jo peo suUT (1)4 - x Suteq (1)Z uF ( )¢o JO AJoJZeljounu sy UL Jue RSUOD sty yatA xX/2(1)4 ST (1)Z 7eYUI BsJOU 9M (HY) YT

(6V) 3uyaedmog ‘ATJUenbesqns YyIFM YIOM TTIM om (1)Z JO w1roz |yQ ST YOTYA

10 ‘ c/T 9 (6¥) Gy aa? tae = (1)2

os (X-)> = (X)> Jey TT BDSY

*uin} mou om yoTyM oF §[(1)Z]1 BUTpuTyZ UE Tnyesn sAoad TTIM (gv)

2° (sv) "EC, 14202 + yu] - uy dee 492 + 2U)x = [(2)2]7

sAeY eM (ZOy¥ *d ‘suoumTS eas) (LV) ® /(@)a8)te- = [(2)42)1

aoutTs ‘zayqing ‘(z79¢ ‘“d

‘zoTAey pue ufzTeyY ves) wrojsuerq soe Tde] oy jo AJeqjewered syq st d sizaym

Zz”

[doz + ju) - upSydxe = [(apa)t

(9¥) | LC jhe,

ajou 0} ST daqs AsATJ ano *suiojsueij ooetdey] Butsn (Ty) 103

uofpssaidxs ot4ATeue Ano doTaaep TT#4 eM ‘*WIOJSuUeI] sdRTdeY ssiTVAUT 34} ST

-lT-

-18—

From (A6) and (A10) obtain

Liw(t)]L(2¢n)] = tn? + 20°%p] exp eEt RO? + 207)? + MO} cant) fo} [o}

(All) is the Laplace transform of the integral we seek so we are now looking for the inverse Laplace transform of (All).

Notice that if in (A10) we replace K(t) with K(t) - x then we will produce the expression on the right hand side of (All) multiplied by the

factor exp {-2nx/o-}. Thus, we create the function

~1/2 — a(t) = exp{2nx/o”} + — ¢&O = fe = (a12) (o} OT

and by construction we know L[q(t) ]

L[w(t) JL[Z(t)}. Hence

a

At) = LUELLWO)IL(Z(2) T= fF we dar = tty,

so q(t) is the analytic integral we have sought.

We were attempting to solve the integral in text equation (23) so that we could obtain an analytic expression for g(K(t), T). To obtain this expression we now use (Al2) in text equation (23) and we use equations

(23) and (24) in (25) yielding

-1/2 _ _ ox g(K(t), Th = C= 3 pEOamy _ exp CTF} 5 EO a NT) y OT oO oT Ge = K(t),_ 1, ® = R(t) - n(T = 1) | ag rrr (A13) o(T - rl? o(T - x)?

C is a normelizing constant. Given T, the first time K( ) passes through

[(u/2 - T)2]© 9 as - T)2]9 0

6 4 idxa - (fb a

(zTv) mpl ¢ “(iyi = 1) + a Gaza =! ¢ x(/i - 1) = he eAL f = 40 0} u jo juepuedepur aojoez e Aq Teuopyzodoad st "5 2ey. Pury om 6(4-)> = (1) qey SUTAequewer pue (2 - LU = q pue iu - xF = B YAFM ATNser sty sutsp

L627 (L/2 - T)2 92 (L - 1)09 10 a Se dxo = C/T C/T (9TV) a + TaET) “BGT - b= aye! (42) ¢ qc n OG a?

‘eaqe3te Arejueuetea Ag Ty 7% - X = (1)y Jo ueaw TeUOCTRTpUOD ey, ‘Nn Jo

ues TBUOTITPpUOS 9y2 ST Ty 7/9 pue quejsuod B8uTzZ~T[ewAOU 94 ST Ty /t a129H

0 u (STV) “mp(n)y A f= 9

Squowow ey. UT

peqyserequT o1e eM “Nn I9AO ‘y°p'd e@ sf (2) ‘queqsuod ZupzftTeurzau e 103 deoxy

t1=- [AO 1 Ao 2° 1 Ao (TV) (NUN = yap aw) hg ee Pg OP? Grae dP

Se usq}TIM oq ued (ETY) UOFZOUNZ 9YyQ ‘SQUeTOTJJe0d JuBysUOD eYy

Ire Bupddoap pue (ETV) OFUF N BuTanzFAsqns *(1)y - Xan SUTUT JOG (| (2)4)4 1OJ Quejsuod BUTZT[TewAou oyQ BuTATieqd °Z *] IeAo *y°p'd TeupsZiew oy jo entea oyy Aq Ysnory (ELV) ePFATP 02 pesu oyq jo asneoved JUeAsJITP 94 TTF! (1| (4) 4 AOJZ QueYsuUOD BuPZT[euzou sy} 7eUI

qdeoxe *(1|(2)¥)y '¥'prd TeuofITpuod aya Jo WIOZ eyy osTe ST (ETV) ‘x

~20-

To find the normalizing constant we set n = 1 in (A17) and we

perform a change of variables using the following definitions

~u- (1 - t/T)x

€ oft(1 - t/t) ]t/?

(A18)

-ut (1 - t/T)x

(A19) ott(1 - t/t) y2/?

2

We have

foe)

che S 1G - c/n tee [r - c/ty Tote - 2/1/74 (e, Ve,

-x*

exp (k= 4/D 2m ra - c/T)x + e,oft(l c/T)}1/2Joft1 - 1/1) 1*/? .o (ede (420) fo} x

where

_— x(1 - 2/t)1/2

ort/2

-1/2 2 . _ Recall that o(w) = (27) exp{-1/2w'} and define 4(w) = JS(u)du so

d(w) + o(-w) = 1. (A20) reduces to

ct ot. = aft) lacey + (any o?e (A - 1/T)exp{-L/2x**}

p(l_ = 1/1) 2nx 1/ /

Cc

-1/2 2 oO

- exp H-oxt 24 - t/T)? 2 6 (x) + (27)

-t(1 - 1/T)exp {-1/2x*" }] (A21)

x-

¥* (y2¥) | (¥)0,% (1/2 - 1) = "ep(7sz/T-}dx9, | (42) (4/2 - 1) fF

*uin} UT aSeyj Jeet] TTT“ om pue sjusweTe ve14} Jo wns e ST (€ZV) UT TeasaquUT 243

Ze2uN sjayxoeIq aAleNdDS UT WIe2 GUT *(ZZV) JO JAed ASATJ ay Wor ITJeApenb

ey pepuedxe pue wiojy [TeUuoTJoOUN; s,('3)6 peanitjasqns sary oM areyM

(€Z¥) Psp 792/T-}4x9, (uz) (73 (L/2 - T)259 +

xX

T 92 /¢ (bl? - D2 ree + % (i/2 - 1] f z/qi t/2 ~ T)i]9

ST (@@V) UP Ter8eqUT AsaTZ eu]

Teasaqul ASATA ‘ee

*uin} UT oSey} aqenyTeAs [TTM om

pue sTeiZeqUL om] OUT usyoIq MoU ST (ZZV) JO epts puey-jystT1z sy

*glojeq se peuTjep yx pue 5 «Ts pue Z = U YFTM (/TV) Wory sqypNsea yoTym

(zzv) | Zap(4sy¢ Z yh Zz zh se ~ Dale tz ph G/s ~ Da}e"s + Xe ~ Do] s Gaga = ty - T,./T. ere: aye 3p( 2)07 ZA - T)2]9 p02 qh tL T)2j39°3 + X(L/2 - T)] f = 40 Tesseavy C., aus Bupatser ce

22-

Oo 2 _ —1 3 3aii. f 2oxr/ 2 - cv)?! 2 (28) M2. exp{-1/2e, de, = 2oxt 124 - t/T) [2 -xi*t - (-x*) (A25) 3aiii. The third term is fot - t/T) e421) exp {-1/2e" Ide, (A26) -x* . . -1/2 2 and we must: integrate this by parts. Set dF = (27) e,exp{-1/2e) Ide, -1/2 and set H = Ey: We know f HdF = HF - f FdH. Hence f HdF = -[(27) / € exp{-1/2e7}] - sf -. (any 72/? expf-1/2e2 }de 1 ] 1 x* —~x* or f HdF = -x*$(-x*) + O(x*) Since f Hd¥ is (A25) up to a constant we find that (A25) is on (L = t/DLO(xt) - kb (-xk)] (A27)

Summarizing, the first integral on the right hand side of (A22),

which is (A23), is

o{r(1 - ty} ta - 2/t)? x 0 (x*) + ooxtt! 24 - 2/7)! 24 (=)

+o t(L = T/T) LOC) - xb (x*) 1] (A28)

(1G) | a/t - D 2 /q3*? ~ Ce¥-)O250 + (4X) 0, X(L/2 - T)I{2_0xUZ(L/2 - 1)}dxe-

CH)007 G/ = Dez -px t+ (sd ,0 + RU/- DG - v,

(ZEV)

(TEV)

(O€V)

(67V)

c

te = 20

PUTZ eM (67V) Pue (ZEV) BuTUTquoD

[Get JO /2 = T)2 0+ (ex) O71 (L/2 ~ D2 ee (4X-) 0 9X A(2/2 - D127 ,Gh - De pqrPl _Pxue(L/2 ~ T)}dxe- SATS 07 poBuertesz oq Aew (TEV) ut suze] LCC)? ¥X + (%X-)0) (2/2 - T)1,9 + (-) 07 (2 - D2) oz~ (xX) OX 2(L/2 - D172 /,4/2 - Dost 20{ 7_OxUT(L/2 - T)}dxosqTenbo [e130 UT Ppuoses ey (gzVv) pue (czV) ‘(¥7V) UE SaTNsea ey (OEY) UF SuTanaTasqns Za Sayyed Z ” sP("S) #179 (Z/2 - T)4,0 + 89), a (L/2 - TX 7 > OX G/2 - 1] s ay Gl - T) 2 qth lO xUe (L/2 - [)}dxe- st (7zy) ur [Tea8equT puosas sUL TeaZequy puovas ‘q¢E "(€ZV) «0F UoTsseidxa ano st sty pue [(¥¥-) 97 CL/2 - Dek + GeX)e{1 ot QX(L/2 - DI 2/.(4/4 - D2)

UFeqIGO 8M (87V) JO WIe} ASeT

9y} UL JUSTOTJJoeo0d yx OY 10zZ

7 ——=-> = ¥X ONTISqns om uEYyM c/T- 2/T

o (i/2 - T)x

-24-

4, Alterations needed to Produce the Form reported in the Text

Since we are interested only in the ratio c,/C} we can remove

1 1 all coefficients common to the terms in Cy and Cy. Since ot (1 - t/T)?

is common to both Cy and C, it is not included in the values which we

report for Cc, and Cy in the text.

L? T,> Z) and Z* in

Notice also that the text uses for notation Tt place of t, T, x and x*, respectively, which we have used in the appendix. Recall that for simplicity we assumed that the time at which this forecast is made is time zero for the derivations in the appendix. The time for which the forecast is made is called t in the appendix. In the text the time at which the forecast is made is called t; the time for which the forecast is made is called t. Hence the variable Ty = t- t, in the text notation, is substituted for t, in the notation of the appendix. Similarly, qT, = T - t in the text notation is substituted for T in the notation of the appendix. In the appendix K(O) is set at zero; in the text K(t), the value of K( ) at the time at which the forecast is made, need not be

zero. Hence, we subtract K(t) from x to derive a barrier equivalent to

x in the text. Defining Z = x - K(t) we substitute Z for x in the notation

zt 7 7y/Ty of the appendix. Finally, letting Z* = Soe substitute Z* 1 for x*. This produces the formulas for C, and C, in the text.

2 1