-18-

From (A6) and (A10) obtain

Liw(t) IWlZ(4)] = [n+ 20°p]/ exp RFR IH? + 207p1 1/2, MRC),

oO 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 (Al10) 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

a(t) = A 6 Oa Bae sr ae

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

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

T
a(t) = LE fw(t) L2(2) 1] = fF weecr - tat,
i)

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

a(R), 1) = 6 Tp Ty ~ exp Ay KO = ir AT]

OT

pe = K(t) We = K(t) = n(T = 1)y)

o(T - ql? o(T - pt 2

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

(All)

(al2)

(A13)


-12-

_ a 2
P(T|K(t)) = x - K(t) {- (x - K(t) - n(T - 1)) }
ovPa(t ~ 13/2 exp 20(T = t)
_ x - K(t) x - K(t) - n(T - 1)
= o¢ ). (24)
o(T - 13? ovT — 1
Finally,
e(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 Cy, 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
unnormaized first moment of this p.d.f. Hence, Co/Cy = E(u(t) |u(t), T)
so that E(K(t)|K(t), T) = x - Cy/C,. Deriving the actual formulas (14) - (15)

requires the cranking out of some horrendous integrals, which we relegate

to the appendix.

III) Application

We have derived analytical expressions for f(T - tlx, 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

©

x(t) = S [x exp{t
t

T 1 T
}+ a expt} i Ge = Cy (1) /C, (1) expl- apa

- t
%2 2 %2

“£(T - t|x, K(t))dT (26)


1/2 3/

a=? 12

/2 3/2

3aii. I 2oxt
xi

2 any} eexp{-1/2e4 dae, = Qoxt “(1 - 1/T)

+o (-x*)

3aiii. The third term is

f orl - cme (any?

exp{-1/2e" ydey
-xi

- 2
and we must integrate this by parts. Set dF = (2m) 12. exp{-1/2er}de,

-1/2
and set H= €,. We know f HdF = HF - f FdH. Hence f HdF = -[(27) / 5
1/2

c cs

exp{-1/2e7 #] - f=. (2n)”

exp{-1/2es }de,
-xk 0 ay

or
J WdF = -x*$(-x*) + O(x*)
Since f Hd¥ is (A25) up to a constant we find that (A25) is

oP (L = t/ DL OG) — G(x]

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

which is (A23), is

1/2 /

of=(1 = t/t) 4/7101 ~ e/0y? oct) + 2onr/2 01 - ef 3/264)

+ oP tL = t/T) LOCK) — x*6G) 1]

(A25)

(A26)

(A27)

(A28)


(4)

(€)

(2)

(1)

Sutseyoind jo uofjdunsse ay2 st (¢) uot enby 1 .ty 2 ep esodut am AqtoTTduts

lt *

aoqg *TeIinjoOnNIAs oq 07 pounsse V1e (a «Ty <0,

©) ‘*szejoweied s3T
pue Tepow ay} Jo YOOTG BUTPTInq [TeroTAeYyaq OTseq ay ST uoTIOUNJ pueuAp
Aauow oyL *(T) JO epts puey-IyZTA oy. ‘puewep Asuow Tear Tenbe ysnu yoTYyM

Atddns Keuow ot sowop Tear ayi st (T) JO APTS puey-IjsaT syL

JaTTIee IO 3 palep saTqeTzea [Te pue
JTapouw ay2 JO ainjonzqs 24} BuTUTe UCD Yas UOoTJeWIOJUT 3 SWTZ = (3)T

({)I Uo TeuoTaTpuod (3)x Jo a8ueyd Jo ajer paqoedxe = ((3)1|(3)¥)a

*aqei a8ueyoxe = (3)x
soueqinjstp oTAseyoos = (3)A
(TeaeT) e3e1 YseteqUT = (3)T
gndqjno = (3)4

JaaeT eoftad = (3)d

Atddns Aeuow = (3)u

pue (*y'Q) ,,USsTer0s,, seqousp eTqeTiea

e JaAO (y) YSTIajse ue fswTyITAzeZoT ajouep AT[eisues siajJeT eased aMOT

“(@AI[@)pa + Oz = Gt

(x + @d = ad

ly «ly
¥ *

f(aya¢ (ay t%0 - (aslo + %
* * * * * *

o< (@d - (yu

o< «ly f(a)a + (ayo - (ay 4b + %

(a)d - (@)u

suotjenbe 3utMoT TO} aya Aq peqtiosep st [epow sul

-y-


[(a/2 - T)2]o 9 “((i/2 - 1)2]9 0
: 4 e/T u
}dxo - ¢ Gar =p wel ye J = 4,0

(atv) PL i> DF a ae =)

oz u
yo quepuedepuf iz0ojoez e Aq JTeuofjiodoad st YW aeyq pups om 6(4-)> = (2)>

3ey, Su;Aequewer pue (2 - J)U = q pue ru - xF = B YIFM A[NSeT STYyR 3utsp

10% G/+ - 129%
OW) Cae ce” GRY = eGR = =

ete eyt?
lex, az) - A oe

‘eaqe3Te Aiequowete fg
Ty 7% - X = (1)y Jo ueaw TeUCTITpUOD ayy ‘Nn Jo
uesw [BUOFITpUoD 9y ST Ty /% pue juejsuoo SurztTeusoU 9Yy3 ST T5/1 o1°H

u

(stv) smp(myy on f= 9

u

BuO

squewou 943 UT

peqsezequy ere aM “nm aeao *g*p'd e ST ({) ‘quUeIsUOD BuTZz;TEWIOU e 10z deoxy

2- Yo 2 Ao 2 2 Ao
(yT¥) “Mer apis a tg x a) Of sgigh - qo

ra

se ue} TIM 0q ued (CTY) UOFROUNZ eY42 ‘SJUEeTOTZJe0D QueysuUOD oYy2
Tre Zuyddorp pue (¢TV) O3UF n BUFINIFIASGns *(2)4 - Xa Nn 8utupTjeq

(| @ow4 ZOZ JUeJSUOD BUTZ;TeMIOU YQ BuTATIeEad *Z

+L 2000
*y¢ped TeupZzew ayy jo enTea oy3 Aq YBno1yy (ETV) ePFATP 02 peeu ayz JO
asnedeq 3Ue79zFFP 4 ITEM (L| (2)%)4 20¥ Juesuoo BuPZ;TemZou eya IYI

qdaoxe ‘| Qa ‘y¢prd [TeuofIFpuoo eyz Fo Woz |YyI OSTe ST (ETV) ‘x

-6T-


ke
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‘uin} UT
Sey 3ee1 T[FM am pue sjuaeweTe ee14, Jo wNs e ST (€ZY) UT TersequT ey.

Zepun szeyxoeIq olenbs ut W192 eyL *(ZZV) FO 3Aed ASATJ ey} Worl OTRZeIpenb

24 pepuedxe pue wi0jy TeUoTJOUNZ s,(13)6 peanztasqns sary am aroym

(ezv) Tsp(73a/T-}4Ke, (az) (79/2 229 +
I “
32 jg hlt - Day pmee + X,G/t - D1 f aqh Gir - D>I9

ST (@@V) UF Ter8equzT Asazy ouL

[ersequl ysaty ‘ee

*uin} UF 9Sey} oJeNTeAS [TTA om

pue sTeiZequr om} oOjUT ueyoIq Mou St (Z7V) JO epts puey-IystTa sy

‘el0jeq se peuTyep yx pue ¢, «ls pue z = u yata (LTV) Wory saqnsez yoTUM

(zzv) Ssp(s)¢
z 4X zo
zi hl - D2} 71,6 G/> - 1)2]9°3 + X(L/2 - T)-] J Garay say -
L./1 _ T ~aie og
3P( 3) 07), LGA - T2109 2( 2, 1st T)1}]9°3 + X(L/2 - T)] f = 49
aepe 3 MON
ene 5

-T7-


-16-

Appendix

Derivation of Cy 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
f w(t) Z(t - t,)dt,

where
—_ x - nt
=x 1
w(t, ) >—~37D $—TI 3 )
ot. ot.
1 1
and
(=e? Ree) - B= ner = &)
a(r = ty) #—— o¢ va
o(t - ty)
It is a property of the Laplace transform, L[ ], that

L[w(t)] *L[Z(t)] = uf w(t Z(t - t,)dt,] (see Simmons, pp. 407-408) .

For our problem

_ x x - nt
wit) = “377 $ >
oT oT
and
-1/2 =
=2 K(t) = x - nt
Z(t) = —> Oar

oT

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

= -1
may be stated equivalently as finding L Tintw(r) ILIZ(x) 11, where L “[

(Al)

(a2)

(a3)

(a4)

(A5)


- 1 102, 2 x *
=[1- 7a - 1)? +o 1, 1¢(2) + oar s(1 - wach
- exp ( - aeearey - we + 0° ]0(-2) ~ oF - pict -2))) (14)
Tt a
and
c= forfa - = zd +20 - eb
T. 1 T
~ exp{(1 - ne torta - gb - 2a - De. as)
1 1
1 2 * 2
In these formulas, $(x) = —— exp {- a and @(x) = exp{- Pay.
V2n -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, f(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.


pue) pejoedxe ay} Joy (6) SATOS ued aM UOTITpUOD TeUTUIAD eB UeATD

ian lp
(6) (a) 1|()*)a at (a) 1] (wa a7 ((a)1|(@)*)4

saey om SB8uTZuerieer $(})] uo

TeuoTITpuos ae aBueyoxe pajoedxe oy. UT uofjenba TeTquestesiTp e ST STUL
(8) (a)t|@xa% + (a)T|@wa = (@)1/@»4

purty om $3 OWT] Je squase 02 BTQeTTeAR (3)I IS

UOTIEWAOJUT ay} UO TeUCTITpuod (9) JO SaPFS yI0q Jo suoTzERIedxe BuTyAeL
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qW jy Bes $,e}e1 aBueYyoKe oy} JO sWTI ey? Je MOTEq Wory xX ysnoryz
e8essed ysify e ayew 02 (2)xX = a) - (3)d qoedxe am ‘kT QUeTIND uTeIgO 07
STY2 IOJ MOT 003 ST TeAeT doFAid usfei1oz dy SNUTW TeAeT BdTAd OT ISeWOp
ey} uotqzdunsse Aq s0uTS “@d¢ — (9)d = x ueym ‘ort fx aetnoqqied
ewos 3e spToy haqtaed aamod Butseyoand usym 92e1 a8ueyoxe ay} XTF TTTM
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“(829 SON ~ (S)9 ‘O'T ‘sso00id 1eueTM e ST (Ss) pue a3e1 JJTIp YI ST U sTOyM


-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
earning assets are perfect substitutes.2/ We assume that m{t), ntt), y(t),
y(t), v(t) and a(t) are exogenous to x(t).

Combine (1) - (4) to obtain

x * kk .
m(t) - m(t) - x(t) = My - G + ay(t) - ayy(t) ~ a (R(t) |1(t))
*
+ v(t) - v(t). (5)

* kK * *
We define K(t) = Gy ~ Oy + ayy (ty - ay (t) - m(t) + m(t) + v(t) - v(t).

Hence (5) may be written as

x(t) = K(t) + aE (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
expectat:ions, 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)|I(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(0) + n t + e(t) (7)


2
(oTv) (Gp _ldee + Hl - Large, qldoe + 4] = [@)2]7

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19

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2
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soufs ‘zeyjzing *(z9¢ “d

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(9v) “(Chez + 24) - wE aro = [@)a)1

Z/t z

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*suiojsueiq aoeytdey Buzsn (Ty) 10;

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-lI-


azeyh

(€1) To/29 - x = (2 *)T]Q)wa

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T

aya wey rea-gely

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(21) = (()a *x|2 - LF

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9y} UE JUETOTJJoO0o yX 242 1OZ

at Taye = #% MnaTasqns om vey

z/T- Z/t

-€7-


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.


KQ)

v TIWE
A

F\Gore


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T
3o- 1,0 a - 10
(6T) Fz =A yo
(ta = ayu-x- (yx

= (= (Fayx] (M9

:se (QT) Wo1z peuteqqo st 1 owt] Je (1)y Ysno1ryy BuTssed sqT

YITM peqetoosse 3YyZTem AITTTqeqord TeuoTyTpuooun 9yz Ty qe x 38 sqieqs

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(I 2an8tq UT T ayxTT) syyed TTe yITM peqepoosse jYyZTem ayI qoeazqqns

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aya St ((1)y)H ‘(9gE ‘d ‘10TAe] pue utTiey 92S) { (XH }dxo =u = (X)> azeym

alo Olu

270 _yg f° }dxa = ((4)W9

4 .
(8T) Gre @jyx T rer - @)y) & T

*y*p'd TeuotTafpuooun ue sey (1)y ‘0 = (0)X
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“sinoo0 e8essed ysAT} YOTYM We 1 < J SWF 9Yz UO puedep jou seop ((1)y)H

(LT) (@)MH((2)a[Da = (1 6 (2)y)3

-0T-


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 T is equivalent to K(t) making a first passage
through x at 72!

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

its current expectations) can be determined as

T
E(x(t) x(t), T) = x exp (22) ++ exp (£3 F BcK(r)|1(t), 1) expl- Lar
*2 %2 2 %2

where E(K(t)|I(t), T) indicates the expected path of K(t), t< 1<T, given
I(t) and K(I) = x for the first time. The unconditional exchange rate is

then the integral of (11) weighted by the first passage p.d.f.

c

x(t) = J E(x(t)|I(t), T)£(T - t]x, K(t))aT
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

(10)

(11)


-20-

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

perform a change of variables using the following definitions

-Uus- (= t/TD)x
1 a (a18)

€
ot = t/t) }t/?

-ut+ (1 - t/T)x
€, = . (A19)
ESE ROTT Fa
2 lta - nyt 2

We have

che f(a -t/De + e6 fe - e/ joe - ey?

oe (ede,

= exp Qh =4/D 20 TG. = «/D¥ + cate - 1/11", ofc ~ v/0)1"/? .o(eq)den (420)
o x*

where

ke XL - pil?
xk 5 L/2
ot

Ww
Recall that. 6(w) = (21) 7!/exp{-1/2w"} and define o(w) = f4(u)du so

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

1/2 3/2 -1/2 2

2
cy = ot “(1 - t/T) >“ O(x*) + (277) ot(1 - t/T)exp{-1/2x*" }

. exp = T/T) 2K yp _ Get? ( 3/2

-1/2 2
Oo
2

1 - t/T)~°O(-x*) + (27)

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


-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/h 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.


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=24-

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

Since we are interested only in the ratio C3/cy we can remove

1
all coefficients common to the terms in cy and C). Since ott - t/T)?

is common to both cy and cy it is not included in the values which we

report for C

1 and Cy in the text.

p a and Z* in

place of t, T, x and x*, respectively, which we have used in the appendix.

Notice also that the text uses for notation Ty T

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,
T)=T-t in the text notation is substituted for T in the notation of

the appendix. In the appendix K(0) 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

1- T/T,

Zz —————, we substitute Z*
oO a

for x*, This produces the formulas for Cy and Cc, in the text.

of the appendix. Finally, letting 2*


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 is 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 ordei: 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
(the exchange rate) which interests the policy authority .2/ 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 example. However, within this class of processess, our results are
generally applicable to many different kinds of macroeconomic problems.

Ve 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


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-TI-


-14-

Footnotes
Vayodeis of pricing for some types of options make use of first passage
probability density functicns. For example, Ingersoll (1977) uses first

passage profit's in studying the prices of convertible securities.

2/ ay assuming Oy = a, we are able to determine x(t) without modeling

*
the goods market. Alternatively we could allow Gy 2 Ap» impose world
goods market equilibrium, and produce an exchange-rate solution slightly

differeat from that reported below.

i st parity can be
3/ some empirical support for the assumption of open intere: Pp y

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).

4

The nature of the exchange-rate fixing policy precludes the existence
; of a mutiple solution type bubble which would cause x(t) to rise through
xX. 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.

fli are extremely grateful to J.H. Kemperman for showing us how to derive

the conditional expectation of K(t).