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J.P. Morgan RiskMetricsļ methodology uses EWMA to calculate the variances and covariances of ļ¬nancial
24

asset prices, in order to compute the VaR of a ļ¬nancial portfolio.
The term 1/(1 ā’ Ī») corresponds to the asymptotic limit of the sum of the weights, i.e., the sum of the terms
25

of a geometric progression with ratio equal to Ī» (being Ī» < 1).
Portfolio Management and Information 361

Identically, the covariance corresponds to:

T
Ī»T ā’j (v2,tā’T +j ā’ v 2,tā’T ,T )(v3,tā’T +j ā’ v 3,tā’T ,T )
Cov(v2,t,T , v3,t,T ) = (1 ā’ Ī»)
Ė
j =1
(12.29)
26
Therefore, the correlation coefļ¬cient is:

T
Ī»T ā’j [(v2,tā’T +j ā’ v 2,tā’T ,T )(v3,tā’T +j ā’ v 3,tā’T ,T )]
j =1
Ļt,T =
Ė (12.30)
T T
2 2
Ī»T ā’j (v2,tā’T +j ā’ v 2,tā’T ,T ) Ī»T ā’j (v3,tā’T +j ā’ v 3,tā’T ,T )
j =1 j =1

The formula in equation (12.30) can be represented in a recursive form. In fact, the
standard deviations and the covariance can be computed recursively, in the following way:

Ļi,t,T = (1 ā’ Ī»)(vi,t ā’ v i,tā’T ,T )2 + Ī»Ļi,tā’1,T
Ė2 Ė2 (12.31)
Cov(v2,t,T , v3,t,T ) = (1 ā’ Ī»)(v2,t ā’ v 2,tā’T ,T )(v3,t ā’ v 3,tā’T ,T ) + Ī»Cov(v2,tā’1,T , v 3,tā’1,T )
Ė Ė
(12.32)
Nevertheless, EWMA measures are still past dependent. On the contrary, implied cor-
relations computed from option prices are forward looking measures that should quickly
reļ¬‚ect any perceived structural breaks in the data generating process. These correlations
may be obtained directly from volatility quotes. Both techniques will be implemented in
Section 12.6.

12.5 INDICATORS OF CREDIBILITY OF AN EXCHANGE
RATE BAND
Another relevant exercise for portfolio management when ļ¬nancial assets are denominated
in currencies managed within target zones is the assessment of the credibility of the
exchange rate policy. Several indicators have been used in the past to perform these
exercises, namely spot interest rate spreads and spot and forward exchange rates.27 More
recently, information from option prices started to be used for this purpose.
The simplest analysis using currency option prices is that consisting of deriving the
RND function and quantifying the probability of reaching the band limits. When there is
only information on at-the-money forward options, the RND may be approached using
implied volatilities to calculate the standard deviation of the distribution,28 assuming the
log-normality of the exchange rate.

26
In Campa and Chang (1996), the weights are directly attached to correlations in moving periods with a ļ¬xed
n
Ī»j ā’1 Ļtā’T ā’(j ā’1),T .
length: Ļt,T = (1 ā’ Ī»)
Ė
j =1
27
See Svensson (1991).
28
In the OTC market, the volatilities (in annual percentages) are quoted instead of the option prices.
362 Applied Quantitative Methods for Trading and Investment

The credibility of exchange rate bands can be analysed through a more rigorous method,
building indicators based on the constraints on option prices implicit in the perfect credi-
bility assumption (see Malz (1996), Campa and Chang (1996) and Campa et al. (1997)).
Consider a ļ¬‚uctuation band with upper and lower bounds denoted by S and S, respec-
tively. First, consider the extreme case of strike prices of a call-option at the upper bound
of the band or above it (X ā„ S). Under perfect credibility the call-option is worthless,
since it will never expire in-the-money. Conversely, if the strike price of a put-option is
at the lower edge of the band or below it (X ā¤ S), the put-option is also worthless, since
the probability of expiring in-the-money is nil.
It can be shown that the intrinsic value of a call-option ā“ the maximum between zero
and the option price if immediately exercised ā“ equals:29
ā—
C(X) = S exp (ā’iĻ„ Ļ„ ) ā’ X exp (ā’iĻ„ Ļ„ )
f f
(12.33)

For strike prices within the band, the maximum value of the call-option price is
given by:
C(X ā¤ S) = (S ā’ X) exp (ā’iĻ„ Ļ„ )
f
(12.34)

This value is reached only when the exchange rate is expected to get to the band ceiling
at the expiry date with certainty. This implies that the perfect credibility of the band is
rejected whenever the current value of the call-option price, with strike price X ā¤ S,
exceeds the maximum value that the option price may take according to equation (12.34),
assuming that the future exchange rate is within the band:

C(X ā¤ S) > (S ā’ X) exp (ā’iĻ„ Ļ„ )
f
(12.35)

A tighter constraint on option prices in a fully credible exchange rate band can be
deļ¬ned from the convexity relationship between the option price and the strike price. The
argument can be exposed in two steps. First, each unitary increase in the strike price yields
a maximum reduction of the value of the call options expiring in-the-money that equals the
current value of that unit (when the probability of expiring in-the-money is 1).30 Second,
the higher the strike price, the less probable it is that the call-option expires in-the-money,
and hence the smaller the reduction in the call-option price for each unitary increase in
f
the strike price.31 Thus we have ā’ exp (ā’iĻ„ T ) ā¤ ā‚C(X)/ā‚X ā¤ 0 and ā‚ 2 C(X)/ā‚X2 ā„ 0.
Figure 12.7 illustrates the constraints on the behaviour of options prices in a perfectly
credible exchange rate band. Straight line (1) represents the line of maximum slope
that passes through C(S), when the unit rise in the strike price cuts the option price
by the present value of that unit. This slope is the symmetric of the discount factor
f
ā’ exp (ā’iĻ„ Ļ„ ).32 Straight line (2) joins the prices of the options with strike prices S and
fā— f
S, being C(S) and C(S) respectively equal to S exp (ā’iĻ„ Ļ„ ) ā’ S exp (ā’it Ļ„ ) and zero.33

29
Using compounded interest in continuous time. Campa and Chang (1996) consider discrete compounding.
30
This happens when the absolute value of the slope of the call-option price function is one, i.e. the cumulative
probability is zero.
31
The minimum reduction is obtained when C(X) is horizontal, i.e. when the slope of the curve is zero and
the cumulative probability is one.
The straight line crosses the point F , since its slope equals ā’C(S)/x ā’ S, with x denoting the horizontal
32
f
intercept. Therefore, equalling that ratio to the value of the slope of line ā’ exp (ā’iĻ„ Ļ„ ) yields x = F .
33
The value of C(S) is found by substituting X by S in equation (12.33).
Portfolio Management and Information 363

Option
price

C(S)

(2)

(1)

f
Slope = ā’ exp(ā’it t)

Strike
S F S
price

Figure 12.7 Option prices in a perfectly credible band (convexity test)

Since the relationship between the option price and the strike price is strictly convex,
if the exchange rate band is fully credible, the call-option price should stand in a convex
curve between (1) and (2), that contains points (S, C(S)) and (S, 0) for any strike price
ranging between S and S. Therefore, the perfect credibility of an exchange rate band
can always be rejected provided that a given point in the call-option price function (for
S ā¤ X ā¤ S) is higher than the corresponding point in line (2). This being the case, given
that the call-option price function is strictly convex, the price of a call-option with strike
price greater than S is positive, which means that the probability the exchange rate exceeds
the band upper bound is not zero.
It can be shown that the assessment of the rejection of perfect credibility corresponds
to checking whether the following convexity condition holds:34

Sā’X ā—
[S exp (ā’iĻ„ Ļ„ ) ā’ S exp (ā’iĻ„ Ļ„ )]
f f
C(X) > (12.36)
Sā’S

As mentioned in Campa and Chang (1996), condition (12.36) is more restrictive than
condition (12.35) whenever the forward rate is within the band.35 In the case of wide

34
See Campa and Chang (1996).
f
35
Multiplying and dividing the right-hand term of equation (12.36) by exp (iĻ„ Ļ„ ) yields:

Sā’X Sā’X
ā—
C(X) ā„ exp (iĻ„ Ļ„ ) exp (ā’iĻ„ Ļ„ )[S exp (ā’iĻ„ Ļ„ ) ā’ S exp (ā’iĻ„ Ļ„ )] ā” C(X) ā„ exp (ā’iĻ„ Ļ„ )(F ā’ S)
f f f f f
Sā’S Sā’S
F ā’S
ā” C(X) ā„ [(S ā’ X) exp (ā’iĻ„ Ļ„ )]
f
Sā’S

The ļ¬rst factor of the right-hand term in this inequality is the right-hand term of inequality (12.35), so (12.36)
is a more restrictive condition when the second factor is smaller than 1 (i.e., when S ā¤ F ā¤ S), as in this case
the full credibility will be rejected from a lower value.
364 Applied Quantitative Methods for Trading and Investment

exchange rate bands ā“ as happened with the ERM of the EMS before the launch of the
euro ā“ the results should be expected to point in general to the non-rejection of perfect
credibility, so the exercise provides no additional relevant information. However, more
interesting exercises can be attempted in these cases ā“ e.g. using equation (12.36) to
calculate the smaller possible bandwidth that would have permitted the non-rejection of
its perfect credibility ā“ i.e., the minimum size of a perfectly credible band.
Bearing in mind the process of monetary integration in the EU, this exercise allows us
to use option prices to identify the band in which a currency could have ļ¬‚oated without
imposing a realignment within the term to maturity of the option. With the beginning of
the EMU becoming closer, the currency would then be expected to ļ¬‚oat in a progressively
narrower interval.
Transforming (12.36) into an identity and using the principle of a bandā™s symmetry
around the central parity yields S = Sc /(1 + Ī±) and S = (1 + Ī±)Sc (where Sc is the central
parity). Substituting S and S for these expressions, the equation resulting from (12.36)
can be solved in order to Ī±, given Sc :36
ā
ā— ā—
Ī± = {[ā’SX exp (ā’iĻ„ Ļ„ ) ā’ Sc exp (ā’iĻ„ Ļ„ ) ā’ {[SX exp (ā’iĻ„ Ļ„ ) + Sc exp (ā’iĻ„ Ļ„ )]2
f f f f
2 2

ā—
ā’ 4Sc [C(X) ā’ S exp (ā’iĻ„ Ļ„ )][ā’C(X) ā’ X exp (ā’iĻ„ Ļ„ )]}]/2Sc [C(X)
f f
2

ā—
ā’ S exp (ā’iĻ„ Ļ„ )]} ā’ 1
f
(12.37)

One can also build a measure of realignment intensity (G(T )) between t and T . Intu-
itively, this measure corresponds to the weighted average of the size of the possible
realignments when the exchange rate overcomes the band ceiling:37
ā
G(T ) = (ST ā’ S)q(ST )dST (12.38)
S

Given that the upper ceiling of the band after a realignment may be higher than the
spot rate, this must be considered as a measure of the minimum intensity of realignment.
Comparing (12.38) to the call-option deļ¬nition in (12.1) and making X = S, the intensity
f
of realignment is equivalent to exp (iĻ„ Ļ„ )C(S), i.e., the capitalised value at the maturity
date of the call-option with strike corresponding to the upper bound of the exchange rate
band.38
Even when there are no options for the strike price S, it is possible to compute the mini-
mum value of a theoretical option for that strike price, based on the convexity relationship
between option prices and strike prices. A lower bound for the intensity of realignment
will then be computed according to:39

Sā’X fā—
C(S) ā„ C(X) ā’ [S exp (ā’iĻ„ Ļ„ ) ā’ S exp (ā’iĻ„ Ļ„ ) ā’ C(X)]
f
(12.39)
Xā’S

36
This corresponds to the lowest value of the solutions to the equation resulting from (12.36).
37
Therefore, this measure does not consider all possible realignments, as these may occur even when the spot
rate is inside the band.
Notice that, in the interval [S, +ā], the maximum of the function is ST ā’ X.
38
39
See Campa and Chang (1996).
Portfolio Management and Information 365
fā—
f
For at-the-money forward options (when X = S exp [(iĻ„ ā’ iĻ„ )Ļ„ ] = F ), it can be
shown that the intensity of realignment will be given as:40

Sā’S
G(T ) ā„ C(X) exp (iĻ„ Ļ„ ) Ā· +F ā’S
f
(12.40)
F ā’S

12.6 EMPIRICAL APPLICATIONS
12.6.1 The estimation of RND functions using option spreads: an application to
the US dollar/euro exchange rate
According to the evolution of the EUR/USD spot exchange rate and risk-reversals
(Figure 12.8), ļ¬ve main periods may be identiļ¬ed. The ļ¬rst one is between the euro
launch and mid-July 1999, during which the euro depreciated vis-` -vis the USD from
a
around 1.17 to 1.02 and the risk-reversals got closer to zero. The second period, until
mid-October, was marked by the euro appreciation, up to around 1.08, with the risk-
reversals becoming positive, i.e. suggesting expectations of further appreciation of the
euro vis-Ā“ -vis the USD. During the third period, the risk-reversals returned to negative
a
values with the depreciation of the euro until mid-May 2000, achieving a level around
0.895. In the fourth period, the euro recovered to a level around 0.95 at the end of June
2000. Lastly, the European currency started a downward movement until mid-October,
only interrupted after the joint exchange rate intervention on 22 September, held by the
ECB, the Federal Reserve, the Bank of Japan and the Bank of England. This intervention
moved the risk-reversals sharply upwards, again to positive values.
May-99

May-00
Aug-99
Sep-99

Aug-00
Sep-00
Nov-99
Dec-99
Feb-99

Feb-00
Mar-99

Mar-00
Jan-99

Jun-99

Jan-00

Jun-00
Apr-99

Oct-99

Apr-00

Oct-00
Jul-99

Jul-00

2.0
1.20
1.15 1.5
1.10
1.0
1.05
0.5
1.00
0.95 0.0
0.90
ā’0.5
0.85
ā’1.0
0.80
Jan-99
Feb-99
Mar-99
Apr-99
May-99
Jun-99
Jul-99
Aug-99
Sep-99
Oct-99
Nov-99
Dec-99
Jan-00
Feb-00
Mar-00
Apr-00
May-00
Jun-00
Jul-00
Aug-00
Sep-00
Oct-00

EUR / USD risk-reversals

Figure 12.8 Risk-reversals for Ī“ = 0.25

40
366 Applied Quantitative Methods for Trading and Investment
Sub Full Estimation()
ā™
ā™ Estimated Vols Macro
ā™ Macro recorded on 7-12-2002 by Jorge Barros LuĀ“s
Ä±
ā™
Row = InputBox(Prompt:="First row for the estimation (e.g. 2)",
Title:="1st Estimation Row")
Row2 = InputBox(Prompt:="Last row for the estimation (e.g. 3)",
Title:="Last Estimation Row")
NoRows = Row2 - Row
Sheets("Strike Call").Select
Range("a1", LastRow).Select
For m = 0 To NoRows
Sheets("Strike Call").Select
Range("a1").Select
ā™Estimation of the strikes corresponding to the deltas
For n = 1 To 19
SolverReset
SolverAdd CellRef:=Range(Cells(Row + m, 23 + n - 1), Cells(Row +
m, 23 + n - 1)), Relation:=1, FormulaText:=Range(Cells(Row + m, 23
+ n - 2), Cells(Row + m, 23 + n - 2)) - 0.001
SolverAdd CellRef:=Range(Cells(Row + m, 22 + n), Cells(Row + m, 22
+ n)), Relation:=3, FormulaText:="0"
Range(Cells(Row + m, 1 + n), Cells(Row + m, 1 + n)).Select
SolverOk SetCell:=Range(Cells(Row + m, 1 + n), Cells(Row + m, 1 +
n)), MaxMinVal:=2, ByChange:=Range(Cells(Row + m, 22 + n),
Cells(Row + m, 22 + n))
SolverSolve True
Next n

"For n = 21 To NoColumns = 1
SolverReset
SolverAdd CellRef:=Range(Cells(Row + m, 23 + n - 1), Cells(Row + m, 23
+ n - 1)),
Relation:=1, FormulaText:=Range(Cells(Row + m, 23 + n - 2), Cells(Row +
m, 23 + n - 2)) - 0.001
SolverAdd CellRef:=Range(Cells(Row + m, 22 + n), Cells(Row + m, 22 +
n)),
Relation:=3, FormulaText:="0"
Range(Cells(Row + m, 1 + n), Cells(Row + m, 1 + n)).Select
SolverOk SetCell:=Range(Cells(Row + m, 1 + n), Cells(Row + m, 1 + n)),
MaxMinVal:=2, ByChange:=Range(Cells(Row + m, 22 + n),
Cells(Row + m, 22 + n))
SolverSolve True
Next nā™ā™
ActiveWindow.LargeScroll ToRight:=-2
ā™Copy strikes to new sheet
Range(Cells(Row + m, 23), Cells(Row + m, 41)).Select
Selection.Copy
Figure 12.9 Visual Basic macro: the estimation of strike prices
Portfolio Management and Information 367
Sheets("Sheet3").Select
Range("A1").Select
Selection.PasteSpecial Paste:=xlAll, Operation:=xlNone,
SkipBlanks:=False, Transpose:=True
ā™Copy call option prices to new sheet
Sheets("Call").Select
ActiveWindow.LargeScroll ToRight:=-2
Range(Cells(Row + m, 2), Cells(Row + m, 20)).Select
Application.CutCopyMode = False
Selection.Copy
Sheets("Sheet3").Select
Range("B1").Select
Selection.PasteSpecial Paste:=xlValues, Operation:=xlNone, SkipBlanks:
=False, Transpose:=True
ā™Ordering by strike
Columns("A:B").Select
Application.CutCopyMode = False
OrderCustom:=1, MatchCase:=False, Orientation:=xlTopToBottom
ā™Copy of the ordered strikes
Range("A1:A19").Select
Selection.Copy
Sheets("Strikes").Select
Range(Cells(Row + m, 2), Cells(Row + m, 2)).Select
Selection.PasteSpecial Paste:=xlAll, Operation:=xlNone,
SkipBlanks:=False, Transpose:=True
ā™Copy of the ordered call-option prices
Sheets("Sheet3").Select
Range("B1:B19").Select
Application.CutCopyMode = False
Selection.Copy
Sheets("Strikes").Select
Range(Cells(Row + m, 23), Cells(Row + m, 23)).Select
Selection.PasteSpecial Paste:=xlAll, Operation:=xlNone,
SkipBlanks:=False, Transpose:=True

Figure 12.9 (continued )

The estimations of the RND functions assuming a linear combination of two log-normal
distributions were performed in the ļ¬les āOTC EUR USD.xlsā, using a Visual Basic
macro (āFull Estimationā).41 Figure 12.9 presents the block of the macro concerning the
estimation of strike prices.
The macro starts by asking the user to insert the number of the ļ¬rst and last row to
be estimated. After computing the number of rows and positioning in the ļ¬rst cell of
the sheet āStrike Callā, the strike prices corresponding to the option deltas are estimated.

41
This macro is run directly from the Excel menu and only demands the user to copy previously the ļ¬rst row
in each sheet to the rows containing new data to be used for estimation.
368 Applied Quantitative Methods for Trading and Investment
14.0
13.5 Observed
Volatility (% / year)
13.0 Estimated
12.5
12.0
11.5
11.0
10.5
10.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Delta

Figure 12.10 Observed and estimated 1-month volatilities at 13/01/99

This step follows equations (12.3) and (12.10) and is carried out for all the call-option
deltas between 0.05 and 0.95, with an interval equal to 0.05.
Figure 12.10 illustrates the results obtained in this step for the ļ¬rst sample day, with
the larger markers representing the three volatilityā“delta pairs computed from the risk-
reversal, the strangle and the at-the-money volatility ļ¬gures, with the remaining markers
being those resulting from the curve in equation (12.10).42 Afterwards, the volatilityā“delta
function is transformed into a volatilityā“strike (smile) curve, by estimating the strikes that
minimise the squared differences to the observed volatilities. With the volatility smile and
using the pricing formula in (12.1), call-option prices are obtained.
The observed values in Figure 12.10 are computed in the āDelta Volā sheet, while the
estimated values are obtained in the āStrike Callā sheet. In this sheet, the ļ¬rst set of
columns (up to column T) contains the squared difference between the observed volatility
(in order to the delta) and the estimated volatility (in order to the strike price). The
estimated strike prices are in the second set of columns of the same sheet (between
columns V and AO). The third set of columns in the same sheet (between AQ and BJ)
contains the estimated volatility, which results from equation (12.10), being the call-option
delta obtained from inserting in equation (12.3) the estimated strike prices in the second
set of columns in the sheet āStrike Callā. Lastly, the sum of the squared residuals is
presented in column BL.
It can be seen in Figure 12.10 that the estimated volatilities are very close to the
observed ļ¬gures. This result is usually obtained only after some iterations and trials
concerning the starting values, given the non-linear features of the target function. Thus,
the choice of those starting values is crucial for the ļ¬nal result.
Afterwards, the RND is estimated, being the parameters of the distributions obtained
in order to minimise the squared difference between the estimated and the observed call-
option prices, as in equations (12.11) to (12.15).43 Figure 12.11 shows the ļ¬tting obtained
for the same ļ¬rst sample day concerning call-option prices.
42
The exercise was performed only considering call-option prices, as all relevant formulas (namely (12.5) and
(12.6)) were also derived for call options.
43
The estimated RND parameters are presented in the ļ¬le āParam.ā. Again, given the non-linear features of
the target function in the optimisation problem, the choice of the starting ļ¬gures is relevant. In the applications
presented on the CD-Rom, the estimated values for the RND parameters at the previous date were used as
starting values. In order to estimate only the RND parameters, an additional Visual Basic macro is provided
(āRND Estimationā).
Portfolio Management and Information 369

0.07

0.06
0.05

0.04
0.03

0.02
0.01

0.00
1.10 1.12 1.14 1.16 1.18 1.20 1.22 1.24 1.26
Strike price

Observed Estimated

Figure 12.11 Observed and estimated 1-month call-option prices at 13/01/99

18
16
14
12
10
8
6
4
2
0
0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40
EUR/ USD

30-07-99
13-01-99 14-07-99 13-10-99 29-10-99

Figure 12.12 1-Month EUR/USD RND functions using two log-normal mixtures (1999)

These prices are subsequently used to estimate the RND functions as illustrated in
Figures 12.12 and 12.13, employing the Solver add-in in the sheet āParam.ā, as shown in
Figure 12.14. The usual constraints are imposed, i.e., the weight parameter Īø has to be
between zero and one, while the standard deviations of the normal distributions have to
be positive.
The āPDā sheet contains the cumulative distribution functions, while the āRNDā sheet
contains the RND functions, computed simply as the arithmetic difference between con-
secutive values of the cumulative distribution function. The remaining sheets in the ļ¬les
are used to compute several statistics.44 For the quartiles, sheets āQ25ā and āQ75ā are
used to calculate the difference between the cumulative distribution function for each grid
value and the respective percentile. Conversely, the ļ¬les āq25 Minā and āq75 Minā are
used to identify the grid values with those minimum differences.

44
As Excel is being used, these computations are performed on a discrete basis. A thinner grid, in the āGridā
sheet, computed in the interval forward rate +/ā’0.05 is used to compute the mode and the median. The RND
values for this thinner grid are computed in the āF Gridā sheet.
370 Applied Quantitative Methods for Trading and Investment

14
12
10
8
6
4
2
0
0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40
EUR/ USD

17-05-00 31-05-00 30-06-00 14-09-00 29-09-00

Figure 12.13 1-Month EUR/USD RND functions using two log-normal mixtures (2000)

ā™Estimation of RND parameters
Sheets("Param.").Select
Range("a1").Select
SolverReset
SolverAdd CellRef:=Range(Cells(Row + m, 2), Cells(Row + m, 2)),
Relation:=3, FormulaText:="0"
SolverAdd CellRef:=Range(Cells(Row + m, 2), Cells(Row + m, 2)),
Relation:=1, FormulaText:="1"
SolverAdd CellRef:=Range(Cells(Row + m, 4), Cells(Row + m, 4)),
Relation:=3, FormulaText:="1/1000"
SolverAdd CellRef:=Range(Cells(Row + m, 6), Cells(Row + m, 6)),
Relation:=3, FormulaText:="1/1000"
Range(Cells(Row + m, 53), Cells(Row + m, 53)).Select
SolverOk SetCell:=Range(Cells(Row + m, 53), Cells(Row + m, 53)),
MaxMinVal:=2, ByChange:=Range(Cells(Row + m, 2), Cells(Row + m,
6))
SolverSolve True
Sheets("Strikes").Select
Next m
End Sub

Figure 12.14 Visual Basic macro: the estimation of RND functions

In line with the spot exchange rate and the risk-reversal evolution, the RND function
during the ļ¬rst half of 1999 moved leftwards and the left tail increased. After the recovery
of the euro until mid-October 1999, the EUR/USD spot rate fell again and the risk-
reversals decreased, though they kept positive most of the time until mid-May 2000.
Consequently, the RND functions remained mostly upward biased or symmetrical.
Afterwards, the euro appreciation vis-` -vis the USD until the end of June 2000 brought
a
a signiļ¬cant increase in right tail of the RND. This movement was inverted with the
euro depreciation that occurred until mid-September. Following the joint central banksā™
intervention on 22 September, the RND skewness shifted again to positive values.
Portfolio Management and Information 371

16
14
12
10
8
6
4
2
0
0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40
EUR/ USD

30-07-99
13-01-99 14-07-99 13-10-99 29-10-99

Figure 12.15 1-Month EUR/USD RND functions using a jump distribution (1999)

14
12
10
8
6
4
2
0
0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40
EUR/USD

17-05-00 31-05-00 30-06-00 14-09-00 29-01-99

Figure 12.16 1-Month EUR/USD RND functions using a jump distribution (2000)

Concerning the jump distributions, the ļ¬le āOTC EUR USD jumpā (similar to the
one previously described) is also presented on the CD-Rom. This ļ¬le contains similar
macros and the results illustrated in Figures 12.15 and 12.16 are similar to those shown
in Figures 12.12 and 12.13.

12.6.2 Implied correlations and credibility tests from currency options data:
the Portuguese escudo and the Danish crown
Implied correlations from option prices were computed in order to assess the credibility
of the Portuguese and Danish exchange rate policies, respectively during the transition
until 1998 towards the EMU and the euro referendum in September 2000.45 In fact, the
anticipation by market participants of monetary integration between two currencies shall
correspond to a very high (in absolute value) correlation between those two currencies,
on the one side, and any third currency, on the other side.

45
The Danish crown was kept under a target zone regime after the euro launch, integrating the ERM II
ļ¬‚uctuation band of +/ā’2.25% vis-` -vis the euro. From 1 January 2001 on, the Danish crown would be the
a
only currency in the ERM II, as the Greek drachma would be integrated in the euro.
372 Applied Quantitative Methods for Trading and Investment

Concerning the Portuguese escudo, implied correlations were computed between the
DEM/USD and the USD/PTE, for the period between July 1996 and August 1998, con-
sidering in equation (12.20) S1 , S2 and S3 respectively as the DEM/PTE, DEM/USD
and USD/PTE.46 For the Danish crown, the implied correlation between the EUR/USD
and the USD/DKK was computed, using the exchange rates EUR/DKK, EUR/USD
and USD/DKK. The ļ¬les used, ācorrelations PTE database.xlsā and ācorrelations DKK
database.xlsā, are also included on the CD-Rom.
The ļ¬le ācorrelations PTE database.xlsā contains separate sheets for the bid and ask
at-the-money implied volatilities, though the implied correlations are computed using only
the ask quotations, in the sheet āCorr.askā. In this sheet implied correlations computed
from exchange traded options are also presented (though not computed), as well as the
spread between the Portuguese escudo and the German Mark forward overnight rate, for
settlement on 1 January 1999.47 The ļ¬le ācorrelations DKK database.xlsā contains all the
data and computations in one sheet.
Figure 12.17 shows that the implied correlation between the German Mark and the
Portuguese escudo exchange rates, vis-` -vis the US dollar, increased since the end of
a

1.00 3.0
0.99
0.98 2.5

0.97
Implied correlation

0.96
2.0
0.95
0.94
0.93 1.5
0.92
0.91 1.0
0.90
0.89
0.5
0.88
0.87
0.86 0.0
Oct-96

Apr-97

Oct-97

Apr-98
Mar-97

Mar-98
Jan-97

Jun-97

Jan-98

Jun-98
Feb-97

Feb-98
Nov-96
Dec-96

Nov-97
Dec-97

Aug-98
Jul-96

May-97

Jul-97

May-98

Jul-98
Aug-96
Sep-96

Aug-97
Sep-97

1-month 3-month 6-month

Figure 12.17 Implied and EWMA correlations between the rates of variation of the DEM/USD
and PTE/USD (3-month term) and the spread between the Portuguese and German forward
overnight interest rates (settlement on 1 January 1999)

46
Following this notation, perfect correlation between S2 and S3 corresponds to a correlation coefļ¬cient equal
to ā’1. The results will be presented in absolute values.
47
The forward overnight interest rates with settlement on 1 January 1999 were computed from the estimations
of daily spot curves, using the Nelson and Siegel (1987) or the Svensson (1994) models. The former method
consists of estimating the parameters Ī²0 , Ī²1 , Ī²2 and Ļ„ of the following speciļ¬cation for the spot curve:

sm = Ī²0 + (Ī²1 + Ī²2 )[1 ā’ e(ā’m/Ļ„ ) ]/(m/Ļ„ ) ā’ Ī²2 [e(ā’m/Ļ„ ) ]

The choice of one of these models was done based on a likelihood test, given that the Nelson and Siegel (1987)
model corresponds to a restricted version of the Svensson (1994) model.
Portfolio Management and Information 373

the ļ¬rst quarter of 1997 until May 1998, in all maturities considered.48 In March 1998,
following the announcement of the 1997 ļ¬scal deļ¬cits of EU countries, the implied
correlation increased to above 0.99. On the eve of the Brussels summit of 1ā“3 May
1998, the correlation between the exchange rate variations of DEM/USD and USD/PTE
was already between 0.99 and 1.
Therefore, the results from implied correlations are consistent with the general assertion
that ļ¬nancial market participants anticipated the inclusion of Portugal in the group of the
euro founding members.49
The relationship between the behaviour of implied correlations and the expectations of
the escudoā™s participation in the euro area seems to be conļ¬rmed by the evolution of the
term structure of interest rates. In fact, exchange rate correlations and the spread between
the Portuguese escudo and the German Mark forward overnight rate, for settlement on 1
January 1999, show a similar trend.50
In order to validate the results obtained for the implied correlations, CME data on
German Mark/US dollar option premia were also used. For this purpose, the mismatching
between the structure of OTC and that of market data had to be overcome. In fact, as
previously mentioned, whilst OTC data involves options with a constant term to maturity
and a variable maturity date, the exchange traded options usually have constant maturity
date and variable term to maturity.
To have comparable data, given that the database on Portuguese escudo options com-
prised OTC volatilities for ļ¬ve maturities (1, 2, 3, 6 and 12 months), the term structure
of volatility was estimated from OTC data, using the method presented in Nelson and
Siegel (1987) for the estimation of the term structure of interest rates.51 Compared to other
methods previously used, such as those in Xu and Taylor (1994), this method provides
smooth and ļ¬‚exible spot and forward volatility curves.52 Furthermore, it is in line with
the generally accepted assumption that expectations revert towards a long-term level (cor-
responding to Ī²0 ) as the term increases, though not necessarily in a monotonic way and
from the short-term volatility (corresponding to Ī²0 + Ī²1 ), as in Xu and Taylor (1994).53
Concerning the term structure of volatilities, the results obtained point to the existence
of a positively sloped curve on most days included in the sample, both for DEM/PTE

48
These correlations are computed using ask prices for the implied volatilities. As EMU implies a structural
change in the pattern of the correlations, the forecasting ability of implied correlations is not compared with
alternative GARCH-type models.
49
The correlations obtained with the EWMA method revealed identical patterns of evolution, though with
slightly higher values and a smoother path.
50
Reļ¬‚ecting the European Union Treaty and other legal dispositions relative to the introduction of the euro,
the conversion rates to the euro of the participating currencies were only known on the last market day before
the euro was born, on 1 January 1999. However, as announced on 3 May 1998, the current bilateral central
parities of the Exchange Rate Mechanism of the European Monetary System (ERM-EMS) would be used in
the calculation of the irrevocable conversion rates to the euro. Admitting that both the announcement of the
participating countries and the rule of conversion to the euro were credible, the behaviour of the interest and
exchange rates in the transition to the EMU became restricted by the need for full convergence, up to 31
December 1998, between the short-term interest rates prevailing in the EU-11 and between the market bilateral
exchange rates and the central parities. Consequently, the convergence of forward interest rates for settlement
on 1 January 1999 and the correlation between variations in the EU-11 currency exchange rates vis-` -vis third
a
currencies (say, the US dollar) should increase.
51
The properties of the term structure of volatility are assessed in Campa and Chang (1995).
52
The model used in Xu and Taylor (1994) only permits three shapes for the term structure of volatilities.
53
The slope of the volatility curve depends on the values of Ī²0 and Ī²1 , while the curvature is related to the
values of Ī²1 , Ī²2 and Ļ„ . For further details, see Nelson and Siegel (1987).
374 Applied Quantitative Methods for Trading and Investment

1.00
0.99
0.98
0.97
0.96
0.95
0.94
0.93
0.92
0.91
0.90
0.89
0.88
0.87
0.86
Oct-96

Apr-97

Oct-97
Jan-97

Jun-97

Jan-98
Mar-97

Mar-98
Feb-97

Feb-98
Nov-96
Dec-96

Nov-97
Dec-97

Apr-98
Jul-96

May-97

Jul-97
Aug-96
Sep-96

Aug-97
Sep-97
Mar.97 Jun.97
6-month 12-month
Jun.98
Sep.97 Dec.97 Mar.98

Figure 12.18 Implied correlations between the DEM and the PTE vis-` -vis the USD from OTC
a

and USD/PTE exchange rates.54 Regarding exchange rate correlations, they were slightly
below those achieved with OTC volatilities until the third quarter of 1997 (Figure 12.18).
In Figure 12.19, implied correlations between EUR/USD and USD/DKK suggest that the
market expected a very close future path for the Danish crown and the euro, in line with
the prevalent target zone regime.55
The anticipation of the Portuguese participation in the EMU is also evident in the
progressive reduction of the minimum width of the perfectly credible band vis-` -vis the
a
Deutsche mark, which corresponds to equation (12.37). According to Figure 12.20, this
variable fell sharply since January 1997, from values over 3% to less than 0.5%. This
behaviour is consistent with the shift in expectations towards a lower depreciation of the
escudo vis-` -vis the Deutsche mark, but also with the reduction of market uncertainty
a
about the future values of PTE/DEM.
Performing a similar exercise for the Danish crown, Figure 12.21 shows that the mini-
mum width of the fully credible band remained generally below the ERM bandwidth
(2.25%), which means that the necessary condition in (12.36) was fulļ¬lled. The 1-month
minimum width ļ¬‚uctuated between 0.25% and 0.5% since January 1999 and until Novem-
ber, when it decreased slightly. In the same period the 1-year minimum width fell from
1.6 to 0.6.
Conversely, between April and May 2000, implied volatilities increased, with the
increase of the interest rate spreads between the Danish crown and the euro interest rates,
as well as with the steepening of the Danish money market yield curve (Figure 12.22).

54
These results were achieved on 81% and 73% of days, concerning respectively DEM/PTE and USD/PTE
exchange rates.
55
In this case, a high correlation also corresponds to a negative coefļ¬cient, as the euro and the Danish crown
exchange rates vis-` -vis the US dollar are deļ¬ned in opposite ways.
a
Portfolio Management and Information 375

1.000

0.995

0.990

0.985

0.980

0.975

0.970
Jan-99

Jan-00
Mar-99

Mar-00
Nov-99
Sep-99

Sep-00
May-99

Jul-99

May-00

Jul-00
1 2 3 6 12

Figure 12.19 Implied correlations between the EUR and the DKK vis-` -vis the USD for several
a
terms (in months)

4.0%

3.5%

3.0%

2.5%

2.0%

1.5%

1.0%

0.5%

0.0%
Oct-96

Apr-97

Oct-97
Mar-97

Mar-98
Jan-97

Jun-97

Jan-98
Feb-97

Feb-98
Nov-96

Dec-97
Jul-96

May-97

Jul-97
Aug-96

Sep-96

Aug-97

Sep-97

Figure 12.20 Minimum bandwidth of the fully credible band: 3-month forward DEM/PTE

From May and until one month before the referendum, the slopes of the money mar-
ket yield and implied volatility curves stayed respectively around 1.1 and 1.3 percentage
points. Afterwards, the slopes decreased until negative values, which suggests that mar-
ket participants became more concerned with the short-term effects of the referendum
result.
376 Applied Quantitative Methods for Trading and Investment

2.75%

2.50%

2.25%

2.00%

1.75%

1.50%
1.25%

1.00%

0.75%

0.50%

0.25%

0.00%
Apr-99

Oct-99

Apr-00
Mar-99

Mar-00
Jan-00
Jun-99

Jun-00
Nov-99
Feb-99

Feb-00
Dec-99
Sep-99
Jan-99

Sep-00
May-99

Jul-99

May-00

Jul-00
Aug-99

Aug-00
1 month 1 year

Figure 12.21 Minimum bandwidth of the fully credible band of the EUR/DKK exchange rate

2.0

1.5

1.0
Interest rates (%)

0.5

0.0

ā’0.5

ā’1.0
Apr-99

Oct-99

Apr-00

Sep-00
Mar-99

Mar-00
Jan-00
Jun-99

Jun-00
Nov-99
Feb-99

Feb-00
Dec-99
Sep-99
May-99

Jul-99
Jan-99

May-00

Jul-00
Aug-99

Aug-00

money market volatilities

Figure 12.22 Slope of the Danish money market yield and implied volatility

Notwithstanding, some violations of the necessary condition for the target zone cred-
ibility occurred during the period studied, namely in mid-May and in the ļ¬rst half of
July 2000, when the 1-year minimum width achieved values between 2.3% and 2.5%.56

56
On these occasions, the minimum intensity of realignment computed as in equation (12.40) was 0.004.
Portfolio Management and Information 377

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