ńņš. 17 |

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

Ļt,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:

Ļi,t,T = (1 ā’ Ī»)(vi,t ā’ v i,tā’T ,T )2 + Ī»Ļi,tā’1,T

Ė2 Ė2 (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 Ļtā’T ā’(j ā’1),T .

length: Ļt,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

See also Campa and Chang (1996).

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

LastRow = Range("a1").End(xlToRight).Address

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

Selection.Sort Key1:=Range("A1"), Order1:=xlAscending, Header:=xlNo,

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 āOTC EUR USD.xlsā, using a Visual Basic

macro (āFull 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 āStrike 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 āDelta Volā sheet, while the

estimated values are obtained in the āStrike 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 āStrike 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 āParam.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

(āRND Estimationā).

Portfolio Management and Information 369

0.07

0.06

0.05

Premium

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 āParam.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 āPDā sheet contains the cumulative distribution functions, while the āRNDā 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 āQ25ā and āQ75ā are

used to calculate the difference between the cumulative distribution function for each grid

value and the respective percentile. Conversely, the ļ¬les āq25 Minā and āq75 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 āGridā

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 āF 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 āOTC 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, ācorrelations PTE database.xlsā and ācorrelations DKK

database.xlsā, are also included on the CD-Rom.

The ļ¬le ācorrelations 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 āCorr.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 ācorrelations 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

Interest rate spread

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

12-month EWMA O/N forward spread

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 exchange traded options

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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