ed in column BL.It can be seen in Figure 12.10 that the estimated volatilities are very close to theobserved ﬁgures. This result is usually obtained only after some iterations and trialsconcerning the starting values, given the non-linear features of the target function. Thus,the choice of those starting values is crucial for the ﬁnal result.Afterwards, the RND is estimated, being the parameters of the distributions obtainedin order to minimise the squared difference between the estimated and the observed calloption prices, as in equations (12.11) to (12.15).43 Figure 12.11 shows the ﬁtting obtainedfor the same ﬁrst sample day concerning call-option prices.42The exercise was performed only considering call-option prices, as all relevant formulas (namely (12.5) and(12.6)) were also derived for call options.43The estimated RND parameters are presented in the ﬁle “Param.”. Again, given the non-linear features ofthe target function in the optimisation problem, the choice of the starting ﬁgures is relevant. In the applicationspresented on the CD-Rom, the estimated values for the RND parameters at the previous date were used asstarting values. In order to estimate only the RND parameters, an additional Visual Basic macro is provided(“RND Estimation”).
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0.070.06Premium
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Figure 12.11 Observed and estimated 1-month call-option prices at 13/01/991816141210864200.70 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 1.40EUR / USD13-01-99
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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 inFigures 12.12 and 12.13, employing the Solver add-in in the sheet “Param.”, as shown inFigure 12.14. The usual constraints are imposed, i.e., the weight parameter θ has to bebetween zero and one, while the standard deviations of the normal distributions have tobe positive.The “PD” sheet contains the cumulative distribution functions, while the “RND” sheetcontains the RND functions, computed simply as the arithmetic difference between consecutive values of the cumulative distribution function. The remaining sheets in the ﬁlesare used to compute several statistics.44 For the quartiles, sheets “Q25” and “Q75” areused to calculate the difference between the cumulative distribution function for each gridvalue and the respective percentile. Conversely, the ﬁles “q25 Min” and “q75 Min” areused to identify the grid values with those minimum differences.44As Excel is being used, these computations are performed on a discrete basis. A thinner grid, in the “Grid”sheet, computed in the interval forward rate +/−0.05 is used to compute the mode and the median. The RNDvalues for this thinner grid are computed in the “F Grid” sheet.
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Figure 12.13 1-Month EUR/USD RND functions using two log-normal mixtures (2000)’Estimation of RND parametersSheets("Param.").SelectRange("a1").SelectSolverResetSolverAdd 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)).SelectSolverOk SetCell:=Range(Cells(Row + m, 53), Cells(Row + m, 53)),MaxMinVal:=2, ByChange:=Range(Cells(Row + m, 2), Cells(Row + m,6))SolverSolve TrueSheets("Strikes").SelectNext mEnd 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 functionduring the ﬁrst half of 1999 moved leftwards and the left tail increased. After the recoveryof the euro until mid-October 1999, the EUR/USD spot rate fell again and the riskreversals 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-à-vis the USD until the end of June 2000 broughta signiﬁcant increase in right tail of the RND. This movement was inverted with theeuro depreciation that occurred until mid-September. Following the joint central banks’intervention on 22 September, the RND skewness shifted again to positive values.
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16141210864200.70 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 1.40EUR / USD13-01-99
Figure 12.15
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1-Month EUR/USD RND functions using a jump distribution (1999)
141210864200.70 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 1.40EUR/USD17-05-00
Figure 12.16
31-05-00
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1-Month EUR/USD RND functions using a jump distribution (2000)
Concerning the jump distributions, the ﬁle “OTC EUR USD jump” (similar to theone previously described) is also presented on the CD-Rom. This ﬁle contains similarmacros and the results illustrated in Figures 12.15 and 12.16 are similar to those shownin Figures 12.12 and 12.13.12.6.2 Implied correlations and credibility tests from currency options data:the Portuguese escudo and the Danish crownImplied correlations from option prices were computed in order to assess the credibilityof the Portuguese and Danish exchange rate policies, respectively during the transitionuntil 1998 towards the EMU and the euro referendum in September 2000.45 In fact, theanticipation by market participants of monetary integration between two currencies shallcorrespond to a very high (in absolute value) correlation between those two currencies,on the one side, and any third currency, on the other side.45The Danish crown was kept under a target zone regime after the euro launch, integrating the ERM IIﬂuctuation band of +/−2.25% vis-à-vis the euro. From 1 January 2001 on, the Danish crown would be theonly currency in the ERM II, as the Greek drachma would be integrated in the euro.
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Implied correlation
Concerning the Portuguese escudo, implied correlations were computed between theDEM/USD and the USD/PTE, for the period between July 1996 and August 1998, considering in equation (12.20) S1 , S2 and S3 respectively as the DEM/PTE, DEM/USDand USD/PTE.46 For the Danish crown, the implied correlation between the EUR/USDand the USD/DKK was computed, using the exchange rates EUR/DKK, EUR/USDand USD/DKK. The ﬁles used, “correlations PTE database.xls” and “correlations DKKdatabase.xls”, are also included on the CD-Rom.The ﬁle “correlations PTE database.xls” contains separate sheets for the bid and askat-the-money implied volatilities, though the implied correlations are computed using onlythe ask quotations, in the sheet “Corr.ask”. In this sheet implied correlations computedfrom exchange traded options are also presented (though not computed), as well as thespread between the Portuguese escudo and the German Mark forward overnight rate, forsettlement on 1 January 1999.47 The ﬁle “correlations DKK database.xls” contains all thedata and computations in one sheet.Figure 12.17 shows that the implied correlation between the German Mark and thePortuguese escudo exchange rates, vis-à-vis the US dollar, increased since the end of
1-month12-month
3-monthEWMA
6-monthO/N forward spread
Figure 12.17 Implied and EWMA correlations between the rates of variation of the DEM/USDand PTE/USD (3-month term) and the spread between the Portuguese and German forwardovernight interest rates (settlement on 1 January 1999)
46Following this notation, perfect correlation between S2 and S3 corresponds to a correlation coefﬁcient equalto −1. The results will be presented in absolute values.47The forward overnight interest rates with settlement on 1 January 1999 were computed from the estimationsof daily spot curves, using the Nelson and Siegel (1987) or the Svensson (1994) models. The former methodconsists of estimating the parameters β0 , β1 , β2 and τ of the following speciﬁcation for the spot curve:
sm = β0 + (β1 + β2 )[1 − e(−m/τ ) ]/(m/τ ) − β2 [e(−m/τ ) ]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.
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the ﬁrst quarter of 1997 until May 1998, in all maturities considered.48 In March 1998,following the announcement of the 1997 ﬁscal deﬁcits of EU countries, the impliedcorrelation increased to above 0.99. On the eve of the Brussels summit of 1–3 May1998, the correlation between the exchange rate variations of DEM/USD and USD/PTEwas already between 0.99 and 1.Therefore, the results from implied correlations are consistent with the general assertionthat ﬁnancial market participants anticipated the inclusion of Portugal in the group of theeuro founding members.49The relationship between the behaviour of implied correlations and the expectations ofthe escudo’s participation in the euro area seems to be conﬁrmed by the evolution of theterm structure of interest rates. In fact, exchange rate correlations and the spread betweenthe Portuguese escudo and the German Mark forward overnight rate, for settlement on 1January 1999, show a similar trend.50In order to validate the results obtained for the implied correlations, CME data onGerman Mark/US dollar option premia were also used. For this purpose, the mismatchingbetween the structure of OTC and that of market data had to be overcome. In fact, aspreviously mentioned, whilst OTC data involves options with a constant term to maturityand a variable maturity date, the exchange traded options usually have constant maturitydate and variable term to maturity.To have comparable data, given that the database on Portuguese escudo options comprised OTC volatilities for ﬁve maturities (1, 2, 3, 6 and 12 months), the term structureof volatility was estimated from OTC data, using the method presented in Nelson andSiegel (1987) for the estimation of the term structure of interest rates.51 Compared to othermethods previously used, such as those in Xu and Taylor (1994), this method providessmooth and ﬂexible spot and forward volatility curves.52 Furthermore, it is in line withthe generally accepted assumption that expectations revert towards a long-term level (corresponding to β0 ) as the term increases, though not necessarily in a monotonic way andfrom the short-term volatility (corresponding to β0 + β1 ), as in Xu and Taylor (1994).53Concerning the term structure of volatilities, the results obtained point to the existenceof a positively sloped curve on most days included in the sample, both for DEM/PTE48These correlations are computed using ask prices for the implied volatilities. As EMU implies a structuralchange in the pattern of the correlations, the forecasting ability of implied correlations is not compared withalternative GARCH-type models.49The correlations obtained with the EWMA method revealed identical patterns of evolution, though withslightly higher values and a smoother path.50Reﬂecting 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 beforethe euro was born, on 1 January 1999. However, as announced on 3 May 1998, the current bilateral centralparities of the Exchange Rate Mechanism of the European Monetary System (ERM-EMS) would be used inthe calculation of the irrevocable conversion rates to the euro. Admitting that both the announcement of theparticipating countries and the rule of conversion to the euro were credible, the behaviour of the interest andexchange rates in the transition to the EMU became restricted by the need for full convergence, up to 31December 1998, between the short-term interest rates prevailing in the EU-11 and between the market bilateralexchange rates and the central parities. Consequently, the convergence of forward interest rates for settlementon 1 January 1999 and the correlation between variations in the EU-11 currency exchange rates vis-à-vis thirdcurrencies (say, the US dollar) should increase.51The properties of the term structure of volatility are assessed in Campa and Chang (1995).52The model used in Xu and Taylor (1994) only permits three shapes for the term structure of volatilities.53The slope of the volatility curve depends on the values of β0 and β1 , while the curvature is related to thevalues of β1 , β2 and τ . For further details, see Nelson and Siegel (1987).
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6-monthSep.97
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1.000.990.980.970.960.950.940.930.920.910.900.890.880.870.86
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Figure 12.18 Implied correlations between the DEM and the PTE vis-à-vis the USD from OTCand exchange traded options
and USD/PTE exchange rates.54 Regarding exchange rate correlations, they were slightlybelow 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 themarket expected a very close future path for the Danish crown and the euro, in line withthe prevalent target zone regime.55The anticipation of the Portuguese participation in the EMU is also evident in theprogressive reduction of the minimum width of the perfectly credible band vis-à-vis theDeutsche mark, which corresponds to equation (12.37). According to Figure 12.20, thisvariable fell sharply since January 1997, from values over 3% to less than 0.5%. Thisbehaviour is consistent with the shift in expectations towards a lower depreciation of theescudo vis-à-vis the Deutsche mark, but also with the reduction of market uncertaintyabout the future values of PTE/DEM.Performing a similar exercise for the Danish crown, Figure 12.21 shows that the minimum width of the fully credible band remained generally below the ERM bandwidth(2.25%), which means that the necessary condition in (12.36) was fulﬁlled. The 1-monthminimum width ﬂuctuated between 0.25% and 0.5% since January 1999 and until November, when it decreased slightly. In the same period the 1-year minimum width fell from1.6 to 0.6.Conversely, between April and May 2000, implied volatilities increased, with theincrease 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).54These results were achieved on 81% and 73% of days, concerning respectively DEM/PTE and USD/PTEexchange rates.55In this case, a high correlation also corresponds to a negative coefﬁcient, as the euro and the Danish crownexchange rates vis-à-vis the US dollar are deﬁned in opposite ways.
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1.0000.9950.9900.9850.9800.975
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Figure 12.19 Implied correlations between the EUR and the DKK vis-à-vis the USD for severalterms (in months)
4.0%3.5%3.0%2.5%2.0%1.5%1.0%0.5%
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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 market yield and implied volatility curves stayed respectively around 1.1 and 1.3 percentagepoints. Afterwards, the slopes decreased until negative values, which suggests that market participants became more concerned with the short-term effects of the referendumresult.
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Figure 12.21 Minimum bandwidth of the fully credible band of the EUR/DKK exchange rate2.0
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Figure 12.22 Slope of the Danish money market yield and implied volatility
Notwithstanding, some violations of the necessary condition for the target zone credibility occurred during the period studied, namely in mid-May and in the ﬁrst half ofJuly 2000, when the 1-year minimum width achieved values between 2.3% and 2.5%.5656
On these occasions, the minimum intensity of realignment computed as in equation (12.40) was 0.004.
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This resulted from an interest rate and volatility increase, almost one month after theCentral Bank of Denmark repo rate increase by 0.6 percentage points (from 4.1% to4.7% on 9 June 2000), following the ECB decision to increase the reﬁ rate to 4.25%from 3.75%.After the referendum, the Central Bank of Denmark increased the repo rate to 5.6%,from 5.1%. This move was followed by a sharp decrease of the 1-year minimum widthof the perfectly credible band, from 1.9% to 1%. Then, the market seemed to expect that,notwithstanding the referendum result, the Danish monetary policy would continue to beanchored to the ECB policy, within the ERM II target zone.The violation of the full credibility condition contrasts with the evolution of theEUR/DKK exchange rate, whose spot rate kept moving in a quite narrow band, while theforward rate was always far from the band ceiling, accomplishing the Svensson (1991)simple credibility test (Figure 12.23).57The ﬁles used for the estimations illustrated in Figures 12.20–12.22 are “credibilitytests PTE 3m.xls”, “credibility tests DKK 1m.xls” and “credibility tests DKK 1y.xls”and are also included on the CD-Rom. All these ﬁles have a similar structure: the ﬁrst threecolumns contain the data on the interest rates and the spot exchange rates; the forwardexchange rates are computed in the fourth column, while the bounds of the ERM-EMS arein the two following columns; the implied (offer) volatilities and the respective call-optionprices are presented next; in the following column the minimum intensity of realignmentis computed, as in equation (12.40); the credibility tests in equations (12.34) and (12.36)are performed next; ﬁnally, the minimum width of the fully credible band in equation(12.37) is calculated.7.657.607.557.507.457.407.357.307.25
spot1-year forwardband floor
1-month forwardband ceiling
Figure 12.23 EUR/DKK Spot and forward rates
57
This test corresponds to assessing whether the forward rate exceeds the target zone limits.
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12.7 CONCLUSIONSIn the OTC currency option market, ﬁnancial institutions quote regularly at-the-moneyforward volatilities, as well as risk-reversals and strangles, which provide informationrespectively on the skewness and kurtosis of the distribution of the underlying asset. Theinformational content of these prices is supposedly higher than that from exchange tradedoption prices concerning currency options, as in this case the over-the-counter marketinvolves a signiﬁcantly higher transactions volume. Besides, using only this information,it is possible to estimate the whole RND function, as well as to test the credibility of atarget zone.Currency option prices also provide relevant information on implied correlationsbetween exchange rates, that can be necessary for portfolio management or riskassessments. Compared to the usual conditional correlations, the former offer theadvantage of being forward looking, allowing one to obtain correlation measures frommarket data in periods in which regime shifts are expected. Considering the EMU, impliedcorrelations permit market expectations to be assessed on the degree of convergencebetween potential participating currencies.Three empirical applications of the literature were presented: ﬁrstly on the EUR/USD,followed by the analysis of implied correlations and the credibility of the Portugueseexchange rate policy, during the transition to the EMU, and of the Danish exchange ratepolicy around the euro referendum in September 2000.As in many other cases found in the literature, the euro depreciation vis-à-vis theUS dollar has occurred against the UIP predictions. Simultaneously, market expectationsabout the future behaviour of the exchange rate also changed markedly, which is reﬂectedby OTC quotes for EUR/USD options. The results obtained conﬁrm the general assertionthat the expectations on the euro evolution deteriorated from January 1999 to October2000, though some improvements happened after the joint exchange rate intervention inSeptember 2000.Regarding the credibility of the Portuguese exchange rate policy, the results are consistent with the idea that the probability of the escudo’s participation in the euro area fromthe beginning increased persistently over the course of 1997. Additionally, the marketattributed a high degree of credibility to the bilateral central parity announced at the 1–3May 1998 European summit.On the eve of the summit, market participants were expecting a perfect correlationbetween the movements of the Deutsche Mark and the Portuguese escudo, vis-à-vis theUS dollar. Furthermore, the central moments of the PTE/DEM exchange rate RND function converged to the central parity and the uncertainty regarding the future values ofPTE/DEM decreased strongly.Though the Danish crown and the euro kept almost perfectly correlated during the ﬁrstthree quarters of 2000, the credibility of the ERM target zone was challenged by optionmarket participants before the referendum, mainly in the beginning of July. This changein market expectations followed the widening of the gap between the key interest rates ofthe Danish crown and the euro. After the referendum, though the integration in the eurowas rejected, the credibility of the Danish monetary policy was enhanced, reﬂected in thedecrease of the minimum width of the fully credible band.
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REFERENCESAbken, P. A. (1995), “Using Eurodollar Futures and Options: Gauging the Market’s View of InterestRate Movements”, Federal Reserve Bank of Atlanta Economic Review, Mar/Apr, 10–30.Bahra, B. (1996), “Implied Risk-Neutral Probability Density Functions from Option Prices: Theoryand Application”, Bank of England Economic Bulletin, Aug, 299–309.Ball, C. and W. N. Torous (1983), “A Simpliﬁed Jump Process for Common Stock Returns”, Journal of Financial and Quantitative Analysis, 18, March, 53–65.Ball, C. and W. N. Torous (1985), “On Jumps in Common Stock Prices and Their Impact on CallOption Pricing”, Journal of Finance, XL(1), March, 155–173.Bates, D. S. (1991), “The Crash of ’87: Was It Expected? The Evidence from Options Markets”,Journal of Finance, XLVI(3), July, 1009–1044.BIS (2001), “The Global OTC Derivatives Market at end-June 2001: Second Part of the TriennialCentral Bank Survey of Foreign Exchange and Derivatives Market Activity”.Black, F. and M. Scholes (1973), “The Pricing of Options and Corporate Liabilities”, Journal ofPolitical Economy, 81 (May/Jun), 637–654.Bliss, R. R. and N. Panigirtzoglou (2000), “Testing the Stability of Implied Probability DensityFunctions”, Bank of England Working Paper, No. 114.Breeden, D. T. and R. H. Litzenberger (1978), “Prices of State-Contingent Claims Implicit inOption Prices”, Journal of Business, 51(