c.h. hui - using first-passage-time density to assess realignment risk of a target zone.pdf

8/14/2019 C.H. Hui - Using first-passage-time density to assess realignment risk of a target zone.pdf

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Using first-passage-time density to assess realignment risk of a target zone

C. H. Hui*1

, C. F. Lo2,3

, T. K. Chung2

Abstract

This paper proposes a path-dependent approach for estimating realignment probabilities of

targeted exchange rates based on the first-passage-time (FPT) density instead of the

commonly used path-independent approach. We consider that path dependency is an

intrinsic characteristic of realignment risk because a realignment of an exchange rate can

occur whenever a committed band by a central bank is breached. A mean-reverting

lognormal process is considered in the FPT approach. Based on market data of the British

pound and Italian lira during the ERM crisis of 1992, the realignment probabilities of the

currencies estimated under the proposed approach show that path dependency is

quantitatively significant, compared with the path-independent approach. The first-hitting

time and the time of the maximum slope of the FPT density have forward-looking capability

of assessing realignment risk of the pound and lira target zones.

JEL Classification: F31, G13

Keywords: realignment risk, ERM crisis, first-passage-time probability

* Correspondence. E-mail: [email protected] Phone: (852) 2878 1485 Fax: (852) 2878 2485

1 Research Department, Hong Kong Monetary Authority, 55/F, Two International Finance Centre, 8, FinanceStreet, Central, Hong Kong, China.

2 Institute of Theoretical Physics and Department of Physics, The Chinese University of Hong Kong, Shatin,N.T., Hong Kong, China. E-mail: [email protected]

3 Hong Kong Institute for Monetary Research, Hong Kong Monetary Authority, 55/F, Two InternationalFinance Centre, 8, Finance Street, Central, Hong Kong, China.


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1. Introduction

Option markets have the desirable property of being forward-looking in nature and

thus are a useful source of information for gauging market sentiment about future values of

financial assets. Options, whose payoff depends on a limited range of the expected exchange

rate, offer broader information about market expectations than the forward exchange rate.

The entire risk neutral probability density function of the exchange rate can be inferred from

the volatility implied from option prices. Several studies have used option prices to estimate

realignment risk of targeted exchange rates during the Exchange Rate Mechanism (ERM)

crisis in 1992. Malz (1996), Mizrach (1996) and Sderlind (2000) estimated the realignment

risk of the British pound exchange rate based upon its risk neutral probability density

functions for the crisis period. Campa and Chang (1996) calculated a minimum realignment

size for the exchange rate using arbitrage arguments only. The estimation of the realignment

risk has become an important measure for assessing the credibility of a target zone.

The approach used by Malz (1996) to infer the probability density function is based

on the assumption that the pound exchange rate follows the Bernouilli version of a jump-

diffusion process, postulating either one jump until option maturity or no jump at all (see Ball

and Torous, 1983, 1985). Implicitly, the estimated realignment probabilities are defined as

the likelihood that the spot pound exchange rate is below the lower band (i.e., the lower

fluctuation limit) at the end of a specified time horizon. Therefore, the estimation is

independent of the path of the exchange rate within the time horizon. Such a path-

independent approach has also been used in estimating realignment risk in Campa and Chang

(1996), Mizrach (1996) and Sderlind (2000).

This paper proposes an approach for estimating probabilities of an exchange rate

hitting a band limit of a target zone based upon a first-passage-time (FPT) approach, which is

path-dependent, instead of the commonly used path-independent approach. Such a first-


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hitting event would trigger compulsory intervention by the central bank or a realignment of

the currency. The realignment probability (which represents a proxy for band credibility) can

thus be assessed by measuring the first-hitting probability. This paper demonstrates that there

is a significant difference between the probabilities measured by the path-independent

approach and by the FPT approach. It is because the risk measurement of the path-

independent approach depends on the exchange rate only at the end of some time interval, but

not on its particular path. This means that the path-independent realignment probability does

not take into account the risk of the exchange rate passing through the target zone boundary

during some time interval. It also implicitly assumes that hitting a band limit of the target

zone (or a realignment) can only occur at the end of the time horizon of risk assessment, and

that the exchange rate is otherwise free to move to any level relative to the band. The path-

independent approach therefore underestimates the realignment risk by an amount equal to

the probability of breaching the band during some time interval. This paper however

considers that path dependency is an intrinsic characteristic of realignment risk because a

realignment of an exchange rate can occur whenever a committed band by a central bank is

breached, that could be triggered by an important economic-political event. The FPT

approach explicitly recognises the consequences of a realignment whenever the exchange rate

falls below (or rises above) a band limit. Consequences like a readjustment of the band will

also move the targeted exchange rate of a currency.

An important prediction of the theoretical literature on targeted exchange rates is that

mean reversion of the exchange rate is expected when the central banks engage in

intramarginal intervention. Others argue that credibility of a target zone induces stability

speculation by market participants, producing forces to drag the exchange rate back to the

central parity whenever it drifts away too much from it. This mean-reverting property is

widely referred to in the literature (see for example, Krugman (1991), Svensson (1992), Rose


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and Svensson (1994), and Anthony and MacDonald (1998)). Several recent studies have

attempted to investigate empirically this theoretical prediction by examining the time-series

properties of the currencies participating in the European Monetary System (EMS) (see for

example, Ball and Roma (1993, 1994), Svensson (1993), Rose and Svensson (1994),

Nieuwland et al. (1994), Anthony and MacDonald (1999), and Kanas (1998)). While their

investigations show mixed results, the empirical results suggest that mean reversion is present.

In view of this evidence, the exchange rate within a target zone is assumed to follow a mean-

reverting lognormal process for estimating the first-hitting probabilities in this paper.1

With respect to the FPT probability density, this paper directly assesses the time

period until the exchange rate first hits a band limit of a target zone, i.e., the expectation of

escape time of the exchange rate from the target zone. The peak of the FPT density is used to

identify the first-hitting time. In addition, the time of the corresponding maximum slope of

the FPT density (i.e., the maximum rate of escape) is identified as a triggering point where

the exchange rate is expected to move towards the band limit of the target zone at the

maximum capacity. Such movements of the exchange rate may trigger intervention by the

central bank or even a realignment of the currency. These two measures are found to provide

forward-looking signals for the assessment of the credibility of a target zone, in particular

when the first-hitting (realignment) probability is close to zero and the target zone is

considered to be fully credible.

In the following section, we derive first-hitting probabilities under the mean-reverting

lognormal process using the FPT approach. In section 3, the first-hitting probabilities, first-

hitting time, and time of the maximum rate of escape of the pound and the lira during the

ERM crisis in 1992 are calculated from the FPT approach and the results are compared with

1Ball and Roma (1998) indicate that the mean-reverting behaviour of the exchange rates in the EMS may be due

to the effect of reflecting barriers where the intervention boundaries are fully credible. However, this paper usesan absorbing barrier, where the intervention boundary is not necessarily fully credible, to calculate the FPT

density.


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those obtained from the path-independent approach. The final section summarises the

findings.

2. FPT density under mean-reverting lognormal process

Regarding the derivation of the formulas for the FPT density of a mean-reverting

lognormal process, we need to solve the Kolmogorovs backward equation governing the

transition probability density ( )tSSp ,,0 connecting the current spot exchange rate (i.e., the

domestic currency value of a unit of foreign currency) S0and the rate Sat the backward time t

in the presence of two absorbing boundaries at LHS = and UHS = :

( ) ( ) ( ){ }2

2 20 0 0

2

( , , ) ( , , ) ( , , )1( ) ( ) ln ln *

2

p S S t p S S t p S S tt S t t S r t r t S

t S S

= + +

, (1)

where is the conditional mean exchange rate, is the parameter measuring the speed of

reversion to this mean, r and r* are the risk-free interest rates of the domestic and foreign

currencies respectively, and the is the volatility of the exchange rate. The FPT distribution

function (or the first-hitting probability) ( )tSPFP , is given by ( , ) 1 ( , )FPP S t P S t , where

the transition probability distribution function 0 0( , ) ( , , )U

L

H

H

P S t p S S t dS also satisfies the

backward equation with the additional boundary condition ( ) 10, =SP .

Introducing the new variable ( )LHSx /ln , the backward equation governing ( )tSP ,

is simplified to

( ) ( )2

2 2

2

( , ) 1 ( , ) ( ) 1 ( , )( ) ( )[ln ] * ( )

2 2L

P x t P x t t P x t t t x r t r t t

t x H x

= + +

. (2)

Without loss of generality, we assume that ( )txP , is given by


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)),()}(exp{(~

),(~

)(exp)(exp),(

3

3

ttxtcxP

txPx

txx

xtctxP

=

=

(3)

where

( ) ( )

1 2

2

1 3 0

2

2 30

30

*( ) ( ) ( )

( ) ( )( ) ( ) ln ' * ' exp{ ( )}

2

1( ) ( ) exp{2 ( )}

2

( ) ( )

t

L

t

t

x t c t c t

t tc t t r t r t c t dt

H

c t t c t dt

c t t dt

=

= +

=

=

(4)

with being a real adjustable parameter. Then it can be easily shown that ),(~

txP satisfies

the partial differential equation:

x

txPtct

x

txPtct

t

txP

=

),(~

)}(2exp{)(2

1),(~

)}(2exp{)(2

1),(~

3

2

2

2

3

2 . (5)

Next, defining )(2 tc= and { } { } ),(~

2/exp4/exp),( 2 txPxtxP = , we can cast Eq.(5) into

the canonical form of the diffusion equation

2

2),(),(

x

xPxP

=

. (6)

By direct substitution, it is straightforward to show that

=

= )(exp

)(sin

)(4exp

)(),(

0

22

1

200

LL

n

L

xnAx

L

L

L

LxP

nn (7)

is the general solution of the diffusion equation for )1()(0 0 + LLx and 0 where

0L denotes the interval of the target zone at 0=t , i.e., ( )LU HHL /ln0 = , subject to the

absorbing boundary conditions: 0),( =xP at both 0=x and )(Lx = . Here is a real

adjustable parameter and Ans are the expansion coefficients to be determined. As a result,

the function ),( txP of the corresponding boundary conditions is given by


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( )

( ) ( )

=

=

)(

)(exp

)(

)]()}(exp{[sin

)]()}(exp{[))((4

exp

)]()}(exp{[2

1exp

)(4

1exp

)(),(

20

2

22

2

3

1

2

3

2

0

3

2

2

2

0

tcLL

tcn

tcL

txtcxnA

txtcxtcL

L

txtcx

tctcL

LtxP

nn

. (8)

This function has two moving boundaries specified by

)})(exp{)]()(exp([

)})(exp{)(exp()(

321

3

tctctcH

tctxHtS

L

LL

=

=

(9)

and

2 3

1 2 0 3

1 2 3

( ) exp([ ( ) ( ( ))]exp{ ( )})

exp([ ( ) ( ) ]exp{ ( )})

exp([ ( ) ( )]exp{ ( )})

U L

L

U

S t H x t L c t c t

H c t c t L c t

H c t c t c t

= +

= +

=

, (10)

where 0L . It is obvious that the two real adjustable parameters and are

responsible for controlling the movement of the two boundaries.

The expansion coefficientsAns are determined by

4

1

2

1expsin

2 0

0

2

00

+

=

L

n xxL

xndx

LA

. (11)

The integral in Eq.(11) cannot be evaluated in closed form and numerical quadrature is

needed for the evaluation. Nevertheless, according to our calculations the numerical

quadrature can be very efficiently performed by Mathcad running on a PC with Window 98.

Furthermore, in our calculations only the first 20 expansion coefficients are usually needed to

ensure convergency of the results.

To simulate the fixed lower and upper boundaries at LHS = and UHS =

respectively, we choose the optimal values of the adjustable parameters and so as to

minimise both of the integrals


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2

0

( )ln

TL

L

S tdt

H

and2

0

( )ln

TU

U

S tdt

H

.

In other words, we try to minimise the deviations from the fixed boundaries by varying the

parametersand. Here Tdenotes the time horizon. Simple algebraic manipulations then

yield the optimal vales ofandas follows:

=

T

T

opt

dttctc

dttctctc

03

2

2

0321

)}(2exp{)(

)}(2exp{)()(

+=

T

T

optoptdttctc

dttctctc

L

03

2

2

0323

0)}(2exp{)(

)}(2exp{)()}](exp{1[

. (12)

Furthermore, we can also determine the upper and lower bounds for the exact FPT

distribution function. It is not difficult to show2that if the moving boundaries stay outside

the region bounded by the fixed boundaries, i.e., ( )U US t H

> and LL HtS < )( , for the

duration of interest, then the corresponding value will provide a lower bound for the exact

value. On the other hand, if the moving boundaries are embedded inside the bounded region,

i.e., ( )U US t H

< and LL HtS > )( , then the corresponding value will serve as an upper bound.

For constant input model parameters (i.e., , , , r*and r being constant), the FPT

density (or the first-hitting probability density) can be defined by ( ) ( ),

, FP

FP

P S tp S t

t

,

whichgives a measure of the likelihood of the exchange rate hitting a band limit of a target

zone or a realignment of the currency in a specified time horizon. Based on the derived FPT

density, the expected time for the exchange rate hitting a band (i.e., first-hitting time) can be

estimated and is roughly regarded as the duration time of the exchange rate staying within the

zone. The first-hitting time 0 is defined as:

2The proof is based upon the maximum principle for parabolic partial differential equations (see the appendix

of Lo et al. (2003) for the relevant proof).


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( )

=

>= 0

,;00

t

tSpt FP , (13)

which is the time of the peak of the first-hitting probability density with respect to time.3 It is

obviously noted that longer first-hitting time implies lower likelihood of the exchange rate

hitting a band limit.

Another measure of the time 1 of the corresponding maximum slope of the FPT

density (i.e., the maximum rate of escape) is used to assess the realignment risk, which is

defined as:

( )1

,0;max

FPp S tt

t

= >

. (14)

1 can be interpreted as a measure of time that the exchange rate takes to move towards a

band limit of the target zone at the maximum capacity. Similar to the first-hitting time, a

larger value of 1 means a lower probability of the exchange rate hitting a band limit.

3. Estimating measures from FPT densityIn this section, the three measures derived previously are employed to assess the

likelihood of the exchange rates of the British pound (GBP) and the Italian lira (ITL) against

the Deutschemark (DEM) hitting the band limits.

For the pound, the model parameters used to estimate first-hitting probabilities are

those used by Malz (1996) based upon combining the information from several options with

different strike prices covering the period from 31 March 1992 to 16 September 1992. On 16

September 1992, the pound left the ERM after a period of turbulence. The first-hitting

probabilities of the pound with a one-month time horizon based on the FPT density under the

lognormal process and the mean-reverting process respectively are shown in Figure 1. They

3The definition is different from the mean first-hitting time which is in general quite large.


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are calculated for each day by using the volatility implied from at-the-money DEM/GBP

option prices, as well as market data of S, r, and r*.4 The lower band limitHLis DEM2.778

per GBP and the upper band limitHUis DEM3.122 per GBP (which put the pound into the

%6 band). For the mean-reverting process, the conditional mean exchange rate is set at

DEM2.95 which is the central parity during the period. The speed of reversion is set to be

0.478.5 The first-hitting probabilities estimated by the two path-dependent models based on

the FPT approach, where Eq.(8) is used, are compared with the realignment probabilities

based on the path-independent approach with a jump-diffusion process in Malz (1996).6 For

comparison, the realignment probabilities based on the path-independent approach with a

lognormal process are illustrated in Figure 1.

Figure 1 shows that the probabilities estimated by the four models are almost zero

during most of the time from April to early July of 1992. During mid-July and early August,

their probabilities start to increase but are below 0.3 which is relatively low. The

probabilities estimated by the path-independent approach with the jump-diffusion process are

close to those estimated by the FPT approach and by the path-independent approach with the

lognormal process.

The probabilities then rise sharply in the second half of August, peaking on

September 16. The probabilities estimated by the FPT approach with both the lognormal and

mean-reverting processes, which are above 0.5 during most of the time of the period, are

substantially higher than those (about 0.4) estimated by the path-independent approach. In

4The at-the-money option prices are provided by JPMorgan. Other market data are from Bloomberg.

5is estimated by applying the maximum likelihood technique to the mean-reverting specification of Susing394 weekly observations of the DEM/GBP exchange rate in the period from 6 March 1985 to 15 September

1992. The corresponding estimated is DEM2.8515. It is not the objective of this paper to investigate themean-reverting behaviour of the exchange rates in the ERM. The estimation is just for the purpose of numerical

illustrations of the mean-reverting model.6It is noted that the realignment probabilities estimated in Malz (1996) only consider the realignment risk at the

lower band limit. The first-hitting probabilities estimated the realignment risk at both the upper and lower bandlimits. However, the movements of the GBP exchange rate were at the side of the lower band during the study

period.


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particular, the probabilities estimated by the FPT approach are higher than 0.7 several times

in September, while the corresponding probabilities estimated by the path-independent

approach with jump-diffusion process range only between 0.39 and 0.46. Even for the

pounds two brief respites from pressure, following the UKs announcement of plans to

borrow ECU 10 billion to defend the pound on September 3, and following the lira

devaluation on September 14, the probabilities estimated by the FPT approach with the

lognormal process are still about 0.5, while those estimated by the path-independent approach

(with the jump-diffusion process and lognormal process) are below 0.3. On 16 September

1992 when the pound left the ERM, the probabilities estimated by the FPT approach reach

almost 1, while those estimated by the path-independent approach with the jump-diffusion

are only around 0.4. The results show that the path dependency is an important characteristic

of assessing realignment risk (i.e., assessing the credibility of the target zone) during the

period of turbulence. In terms of quantitative estimation, the FPT approach gives much

clearer signals of realignment risk than those given by the path-independent approach.

It is noted that under the FPT approach the probabilities estimated with the mean-

reverting process is lower than those estimated with the lognormal process. It is because the

mean-reverting process will push the exchange rate back towards the central parity and thus

reduces the realignment risk, when the exchange rate is between the central parity and the

lower band limit during the period from 31 March to 16 September 1992. This means that

the higher the speed of reversion, the lower the probabilities will be under the mean-reverting

process.7 However, the surges of the first-hitting probabilities estimated under the mean-

reverting process after the second half of August are similar to those under the lognormal

process. The results show that the realignment risk during the period before the pound left

7A strong mean reversion could come about due to interventions in the foreign exchange market by the central

banks. Such interventions have been shown in many studies (see MacDonald (1988)). An underlying feature ofthe interventions is apparently that the authorities try to bring the exchange rate back to some normal level or

central parity. A natural way to model this is through a mean-reverting process for the exchange rate.


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the ERM is mainly attributed to the existence of the boundary which captures any

realignment within the one-month time horizon.

Based on the FPT density, the first-hitting time 0 and the time 1 of the

corresponding maximum slope of the FPT density from 2 January 1992 to 16 September

1992 are presented in Figure 2 and are compared with the one-month first-hitting

probabilities over the same period. The model parameters are the same as those used in

Figure 1.

Figure 2 shows that 0 and 1 estimated by the FPT approach move in opposite

directions compared with the movements of the first-hitting probability because the longer 0

and 1 mean a lower probability of the exchange rate hitting a band limit. However, their

movements do not change at the same time and there are time lags between them. The first-

hitting probability increases from the level of 0.02 as at 8 July to the level of 0.05 as at the

end of July. Regarding 0 and 1 , their values drop sharply before the end of June. The first-

hitting time 0 decreases from about 1 year at 26 June to 0.16 year (about 2 months) at 8 July.

Similarly, the time 1 of the corresponding maximum slope of the FPT density decreases from

0.31 year to 0.06 year over the same period. The changes in 0 and 1 indicate that the FPT

density contains forward-looking information about the likelihood of the pound hitting a band

limit.

The first-hitting probability which is obtained by integrating the FPT density from

time zero to a certain time period (say one month) may not capture the critical characteristics

of the FPT density in order to identity any changes in the likelihood of the exchange rate

hitting a band limit. On the other hand, the first-hitting time identifies the maximum of the

FPT density, and the time of the maximum slope of the FPT density identifies when the

exchange rate will move towards a band limit at the maximum capacity. As these two


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measures identify the critical quantities (i.e., the maximum measures) of the FPT density,

they are more sensitive to the changes in the information contained in the density and thus

reflect the assessment of the realignment risk (credibility) of the target zone more timely than

the first-hitting probability. Having said that, the first-hitting probabilities rise sharply in the

second half of August while the values of 0 and 1 are very small. This shows that the first-

hitting probability is a useful complementary measure for monitoring and assessing the

realignment risk during a period of turbulence.

Figure 3 shows 0 and 1 under the mean-reverting process and the lognormal

process. The values of 0 and 1 estimated with the mean-reverting process are higher than

those estimated with the lognormal process. However, when the time measures are low (i.e.,

the likelihood of the exchange rate hitting a band is high), the differences are very small. The

results are consistent with those in Figure 1 as the mean-reverting process will push the

exchange rate back towards the central parity and thus increase the first-hitting time and the

time of the maximum slope of the FPT density, when the exchange rate is between the central

parity and the lower band limit during the period. This means that the higher the speed of

reversion, the longer the time measures of 0 and 1 will be under the mean-reverting

process.

For assessing the realignment risk by the three measures in a narrower target zone

than that of the pound, market data of the lira from 2 January 1992 to 11 September 1992 are

used for the estimation. The lira was in the narrow %25.2 EMS band with the central

parity at ITL748.217 per DEM, which is narrower than that of the pound. On 13 September

1992, the lira left the ERM with a devaluation of about 15% afterward. The input model

parameters are market data of S, r, r* and the exchange rate volatility which is estimated

from daily exchange rate data by the exponentially weighted moving average (EWMA)


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method.8 For the mean-reverting process, the conditional mean exchange rate is set at

central parity and the speed of reversion is set to be 17.86.9

Figure 4 shows that in a one-month horizon the first-hitting probabilities under the

FPT approach with the lognormal process are consistently higher than the realignment

probabilities obtained by the path-independent approach with the lognormal process. Before

the lira left the ERM, the probabilities estimated by the FPT approach, which have been

above 0.7 since 20 August 1992, are substantially higher than those (about 0.5) estimated by

the path-independent approach. In particular, the probabilities estimated by the FPT

approach are higher than 0.8 in September, while the corresponding probabilities estimated

by the path-independent approach range only between 0.5 and 0.6. Similar to the results in

Figure 1, the realignment risk during the period before the lira left the ERM is mainly

attributed to the existence of the boundary which captures any realignment within the one-

month horizon. Because of the strong mean reversion of the lira in the narrow EMS band, the

first-hitting probabilities estimated by the FTP approach with the mean-reverting process are

much lower more than those with the lognormal process, compared with Figure 1 where the

mean reversion of the pound is relatively weak. Sometimes, they are even lower than the

realignment probabilities estimated by the time-independent approach with the lognormal

process.

8We do not have option implied volatility for the lira, i.e., the jump-diffusion process cannot be estimated. The

daily volatility of the lira based on the EWMA method is given by 2 122 )1( += ttt R , whereRtis the daily

return of the ITL/DEM exchange rate and is the decay factor which is set to be 0.94. The initial 2t is

estimated from the average of the first 74 observations of the data series starting from 2 January 1990 according

to J.P. Morgan & Co. (1995).2t is used for both the FPT approach and the path-independent approach such

that their results are comparable. Other market data are from Bloomberg.9is estimated by applying the maximum likelihood technique to the mean-reverting specification of Susing

704 daily observations of the ITL/DEM exchange rate in the period from 2 January 1990 to 11 September 1992

when the lira was in the EMS band. The corresponding estimated is ITL747.616 per DEM. It is not theobjective of this paper to investigate the mean-reverting behaviour of the exchange rates in the ERM. The

estimation is just for the purpose of numerical illustrations of the mean-reverting model.


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Figure 5 shows that in a one-month horizon the first-hitting probabilities under the

lognormal process are in general high and the corresponding values of the first-hitting time

0 and the time 1 of the corresponding maximum slope of the FPT density of the lira are low.

Their movements are more volatile compared with the corresponding measures of the pound

over the same period. The results are attributed to the narrow %25.2 band that increases

the likelihood of the lira hitting a band limit. While the surge in the first-hitting probability in

June signals an increase in the realignment risk of the lira, the measures 0 and 1 drop

almost at the same time. After the drop, 0 and 1 stayed at the level of below 0.05 year (i.e.,

less than a month) and 0.02 year respectively until 11 September. The small values of 0

mean that the lira could escape from the lower band limit within a month. During the period

between mid-July and mid-September, the first-hitting probability ranges between 0.4 and 0.9.

Compared to the first-hitting probability, the two time measures persistently give signals of

high realignment risk of the lira target zone.

Compared with Figure 5, the movements of the measures under the mean-reverting

process over time in Figure 6 are more volatile because of the more complex dynamics of the

exchange rate due to a strong mean reversion in the narrow EMS band. Nevertheless, the

first-hitting time 0 drops in mid-August before the first-hitting probability surges in early

September. This reflects that under the mean-reverting process, the first-hitting time provides

early signals for assessing realignment risk. Similar to the results in Figure 5, during the

period between mid-July and mid-September, the first-hitting probability ranges between

0.13 and 0.85, while the small values of the two time measures 0 and 1 persistently give

signals of high realignment risk of the lira target zone. The values of the first-hitting time

during this period of time indicate that the lira could escape from the lower band limit within

a month.


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4. Conclusion

This paper proposes a path-dependent approach based on the FPT density for

estimating realignment probabilities of currencies in target zones and considers that path

dependency is an intrinsic characteristic of realignment risk. The path-independent approach

adapted by Malz (1996) ignores the possibility of realignment prior to the end of a specified

time horizon. In essence, this perspective assumes that the realignment risk remains zero

regardless of the changes in the exchange rate during a time horizon. However, if the

exchange rate breaches a pre-specified band limit (i.e., a boundary), realignment can occur.

Based on the ERM crisis in 1992, the realignment probabilities of the pound and lira

estimated under the FPT approach show that boundaries are quantitatively significant,

compared with the path-independent approach. A central bank which adopts a targeted or

managed-floating exchange rate regime could interpret implied first-hitting probabilities

based on the FPT approach as an indicator to assess the realignment risk.

The two other proposed measures of realignment risk, the first-hitting time and the

time of the maximum slope of the FPT density, are found to have forward-looking capability

for assessing realignment risk of the pound and lira target zones. These two measures signal

the changes in the realignment risk in the pound target zone when the first-hitting probability

is close to zero and the target zone is believed to be fully credible. Their values had been

persistently low before the lira left the ERM that demonstrates their capability for assessing

realignment risk of the lira.

Acknowledgements

The authors gratefully acknowledge useful comments from Hans Genberg and assistance

from Chi-sang Tam and Laurence Fung. The conclusions herein do not represent the views

of the Hong Kong Monetary Authority.


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0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

31/03/92 16/04/92 02/05/92 18/05/92 03/06/92 19/06/92 05/07/92 21/07/92 06/08/92 22/08/92 07/09/92

Date

Probability

Path-independent approach (jump diffusion)

Malz (1996)

FPT approach (lognormal)

FPT approach (mean reversion)

Path-independent approach

(lognormal)

Figure 1. First-hitting probability based on FPT approach and realignment probability based

on path-independent approach of GBP in next month from 31 March 1992 to 16 September

1992.


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0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

02/01/92 06/02/92 12/03/92 16/04/92 27/05/92 01/07/92 05/08/92 10/09/92

Date

Probability

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Time(year)

First-hitting probability

(LHS)Time of maximum slope of FPT density 1(RHS)

First-hitting time 0(RHS)

Figure 2. First-hitting probability in next month, first-hitting time and time of maximum

slope of FPT density of GBP under mean-reverting process from 2 January 1992 to 16

September 1992.


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0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

02/01/92 06/02/92 12/03/92 16/04/92 27/05/92 01/07/92 05/08/92 10/09/92

Date

Time(year)

Time of maximum slope of FPT density 1(mean reversion)

Time of maximum slope of FPT density 1(lognormal)

First-hitting time 0(lognormal)

First-hitting time 0(mean reversion)

Figure 3. First-hitting time and time of maximum slope of FPT density of GBP under mean-

reverting process and lognormal process from 2 January 1992 to 16 September 1992.


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0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

02/01/1992 06/02/1992 12/03/1992 16/04/1992 21/05/1992 25/06/1992 30/07/1992 03/09/1992

Date

Probability

FPT approach

(mean reversion)FPT approach

(lognormal)

Path-independent

approach(lognormal)

Figure 4. First-hitting probability based on FPT approach and realignment probability based

on path-independent approach of ITL in next month from 2 January 1992 to 11 September

1992.


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0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

02/01/1992 06/02/1992 12/03/1992 16/04/1992 21/05/1992 25/06/1992 30/07/1992 03/09/1992

Date

Probability

0.00

0.05

0.10

0.15

0.20

0.25

Time(year)

Time of maximum

slope of FPT

density 1(RHS)


(LHS)

First-hitting time 0(RHS)


slope of FPT density of ITL under lognormal process from 2 January 1992 to 11 September

1992.


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0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

02/01/1992 06/02/1992 12/03/1992 16/04/1992 21/05/1992 25/06/1992 30/07/1992 03/09/1992

Date

Probability

0.00

0.05

0.10

0.15

0.20

0.25

Time(year)

Time of maximum

slope of FPT

density 1(RHS)


(LHS)First-hitting time 0

(RHS)


slope of FPT density of ITL under mean-reverting process from 2 January 1992 to 11

September 1992.

c.h. hui - using first-passage-time density to assess realignment risk of a target zone.pdf

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