), “A Dynamic Benchmark for Managed Currencies Funds”, European Journal of Finance, 4(4), 311–330.Levich, R. M. and L. R. Thomas (1993), “The Signiﬁcance of Technical Trading-Rule Proﬁts in theForeign Exchange Market: A Bootstrap Approach”, Journal of International Money and Finance,12, 451–474.Schulmeister, S. (1988), “Currency Speculations and Dollar Fluctuations”, Banco Nationale delLavaro, Quarterly Review, 167 (Dec), 343–366.Silber, L. W. (1994), “Technical Trading: When it Works and When it Doesn’t”, The Journal ofDerivatives, Spring, 39–44.Taylor, S. J. (1980), “Conjectured Models for Trends in Financial Prices, Tests and Forecasts”,Journal of the Royal Statistical Society, Series A, 143, 338–362.Taylor, S. J. (1986), Modelling Financial Time Series, Wiley, Chichester, UK.Taylor, S. J. (1990a), “Reward Available to Currency Futures Speculators: Compensation for Riskor Evidence of Inefﬁcient Pricing?”, Economic Record (Suppl.), 68, 105–116.Taylor, S. J. (1990b), “Proﬁtable Currency Futures Trading: A Comparison of Technical and TimeSeries Trading Rules”, in L. R. Thomas (ed), The Currency Hedging Debate, IFR Publishing,London, pp. 203–239.Taylor, S. J. (1992), “Efﬁciency of the Yen Futures Market at the Chicago Mercantile Exchange”,in B. A. Goss (ed.), Rational Expectations and Efﬁcient Future Markets, Routledge, London,pp. 109–128.Taylor, S. J. (1994), “Trading Futures Using the Channel Rule: A Study of the Predictive Powerof Technical Analysis with Currency Examples”, Journal of Futures Markets, 14(2), 215–235.Zangari, P. (1996), “An Improved Methodology for Measuring VAR”, RiskMetrics Monitor,Reuters/J.P. Morgan.
12
Portfolio Management and Information fromOver-the-Counter Currency Options∗JORGE BARROS LUÍS
ABSTRACTThis chapter looks at the informational content of prices in the currency option market.Risk reversals, strangles and at-the-money forward volatilities derived from OTC are used,along with data regarding exchange traded options. Three empirical applications of theliterature are presented. The ﬁrst one is on the EUR/USD, where option prices for severalstrikes are obtained from currency option spread prices and risk-neutral density functionsare estimated using different methods. This application is followed by the analysis ofimplied correlations and the credibility of the Portuguese exchange rate policy, during thetransition to the EMU, and of the Danish exchange rate policy around the Euro referendumin September 2000. This chapter is supported by the necessary application ﬁles, producedin Excel, to allow the reader to validate the author results and/or apply the analysis to adifferent dataset.
12.1 INTRODUCTIONPortfolio and risk management are based on models using estimates for future returns,volatilities and correlations between ﬁnancial assets. Considering the forward lookingfeatures of derivative contracts, option prices have been used intensively in order toextract information on expectations about the underlying asset prices.Compared to forward and futures contracts, option prices provide an estimate not onlyfor the expected value of the underlying asset price at the maturity date of the contract,but also for the whole density function under the assumption of risk neutrality (the riskneutral density or RND), based on the theoretical relationship developed in Breeden andLitzenberger (1978). This information is relevant for Value-at-Risk (VaR) exercises, aswell as stress tests. However, the completion of these exercises also demands correlationestimates, which can be obtained from option prices only in the case of currency options.Contrary to interest rates and stock price indexes, currency options are more heavilytraded in over-the-counter (OTC) markets.1 The information from OTC markets usually∗
This chapter contains material included in the PhD thesis of the author (Luı́s, 2001).According to BIS (2001), at the end of June 2001, the OTC market was responsible for 99.5% of the aggregatevalue of open positions in currency options.1
Applied Quantitative Methods for Trading and Investment. 2003 John Wiley & Sons, Ltd ISBN: 0-470-84885-5
Edited by C.L. Dunis, J. Laws and P. Naı̈m
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comprises Black–Scholes implied volatilities for at-the-money forward options, as wellas prices for some option spreads, such as risk-reversals and straddles.This data provides useful information about the uncertainty, the skewness and thekurtosis of the exchange rate distribution.2 Furthermore, it allows the estimation of theRND function without knowing option prices for a wide range of strike prices (see, forexample, Malz (1997) and McCauley and Melick (1996)). Another advantage of OTCoptions is that, as they have ﬁxed terms to maturity, instead of ﬁxed maturity dates,3the RND functions don’t have to be corrected by the effect of time on the distributionparameters.4Concerning correlations between exchange rates, Campa et al. (1997) show that inforecasting correlations between the US dollar–German Mark and the US dollar–Japaneseyen exchange rates, in a period from January 1989 to May 1995, correlations implied byoption prices outperform historically based measures, namely conditional correlations, J.P.Morgan RiskMetrics correlations and a GARCH (1,1)-based correlation.As implied correlations shall be the best estimate for the future correlation between twoexchange rates, it must result from using the available information in the most efﬁcientway. Consequently, implied correlations must not be affected by the behaviour of thevariance of the exchange rates.5 OTC currency option market data also allows us toassess the credibility of target zones, by no-arbitrage tests built upon prices of options forany known strikes as in Campa and Chang (1996), and by monitoring implied correlations.This chapter contains six additional sections. In the next section, the basic issues ofcurrency option spreads valuation are introduced. The estimation of RND functions fromoption spreads is assessed in the third section. Two other informational contents of currency options are exploited in the fourth and ﬁfth sections, respectively measures ofcorrelation and arbitrage-based credibility tests.The sixth section contains the empirical applications: the ﬁrst subsection concerns theestimation of RND functions from currency option spreads, focusing on the EUR/USDexpectations between January 1999 and October 2000. As the euro is a recent currency,the limited track record regarding its path and the behaviour of the European CentralBank (ECB) reinforces the importance of analysing the market expectations on the futureevolution of its exchange rate vis-à-vis the US dollar. The analysis of the expectations onthe US dollar/euro exchange rate is also relevant given that, though the exchange rate isnot an intermediate target of the monetary policy followed by the ECB, most internationalcommodities are denominated in that currency.In the ﬁrst 22 months of its life, the euro depreciated nearly 30% vis-à-vis the US dollar.This movement was also characterised by signiﬁcant increases of the historical and impliedvolatilities of the EUR/USD exchange rate, though with much higher variability in theformer case (Figure 12.1, which may be found in the ﬁle “chart1.xls”). Two Excel ﬁlesare provided to illustrate this application, both using 1-month options, one concerning2Though one may question the leading indicator properties of the risk-reversals, as their quotes exhibit ahigh correlation with the spot rate. Using data between January 1999 and October 2000, the contemporaneouscorrelation with the EUR/USD spot rate is 0.53 and decreases when lags are considered. Dunis and Lequeux(2001) show that the risk-reversals for several currencies do not anticipate spot market movements.3These options are usually traded for maturities of 1, 3, 6 and 12 months.4Some papers have tried to correct this maturity dependence using prices of exchange traded options, namelyButler and Davies (1998), Melick and Thomas (1998) and Clews et al. (2000).5Loretan and English (2000) analyse this link between correlation and variances for the case of two stockindexes.
351
1.251.201.15
35
1.10
25
1.051.00
20
0.95
15
30
0.900.850.80
Vol. (%/year)
EUR/USD
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10
Jan-99Feb-99Mar-99Apr-99May-99Jun-99Jul-99Aug-99Sep-99Oct-99Nov-99Dec-99Jan-00Feb-00Mar-00Apr-00May-00Jun-00Jul-00Aug-00Sep-00Oct-00
5
EUR / USD
Figure 12.1
1 month ATM vol.
1 month st.-dev.
EUR/USD: volatility and spot rate
the RND functions estimated by a linear combination of two log-normal distributions(“OTC EUR USD”) and another related to the RND functions estimated allowing onediscrete jump to the underlying asset (“OTC EUR USD jump”).The second subsection focuses on the behaviour of the Portuguese and Danish currencies respectively during the transition until 1998 towards the Economic and Monetary Union (EMU) and the euro referendum in September 2000. In this subsection,implied correlations are computed in the ﬁles “correlations PTE database” and “correlations DKK database”, respectively for the Portuguese escudo and the Danish crown.In order to have a deeper assessment of the prospects regarding the evolution ofthe two exchange rate policies, several credibility tests developed in Section 12.5 arealso performed around the two above-mentioned episodes related to the EMU. ThreeExcel ﬁles are supplied for this application: “credibility tests DKK 1m” and “credibility tests DKK 1year” regarding the Danish crown and “credibility tests PTE 3m” concerning the Portuguese escudo. Finally, the seventh section concludes.As illustrated in Figure 12.2 (which can be found in the ﬁle “credibility tests PTE 3m.xls”), the short-term interest rate convergence of the Portuguese escudo vis-à-vis theGerman Mark increased in mid-1997. Consequently the gap between the spot and the 3month forward exchange rate started to be ﬁlled, while the exchange rate implied volatilitydiminished. This shift could have been taken by a portfolio or a market risk manager as asignal of the Portuguese participation in the third phase of the EMU on 1 January 1999.Concerning the Danish crown episode, Figure 12.3 (which can be found in the ﬁle“credibility tests DKK 1 year.xls”) shows that during the months before the Danish referendum (Thursday 28 September 2000) the interest rate spread against the Europeancurrency increased, along with the exchange rate volatility. This move could have beeninterpreted as reﬂecting an increasing probability attached to the detachment of the Danishcurrency from the euro, within the Exchange Rate Mechanism of the European MonetarySystem (ERM-EMS II).The data used in the EUR/USD application consists of a database comprising bi-monthlyquotations of the British Bankers Association (BBA), published by Reuters, between 13
3-month PTE interest rate3m.PTE / DEM impl.vol.Forward
Jan-98
3-month DEM interest rateSpot
7.60
6
7.55
5
7.50
47.4537.40
2
3-month DKK interest rate1y.DKK / EUR impl.vol.Forward
Figure 12.3
Sep-00
Jul-00
May-00
Mar-00
Jan-00
Nov-99
Sep-99
7.30Jul-99
0May-99
7.35
Mar-99
1
PTE / DEM
DEM/PTE: interest rates and implied volatility
7
Jan-99
Interest rates andimplied vol.(% / year)
Figure 12.2
Oct-98
98Jul-98
0Apr-98
100
Oct-97
2
Jul-97
102
Apr-97
4
Jan-97
104
Oct-96
6
Jul-96
106
Apr-96
8
Jan-96
108
Oct-95
10
PTE/DEM
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Jul-95
Interest rates andimplied vol. (%/year)
352
1-year Euro interest rateSpot
EUR/DKK: interest rates and implied volatility
January 1999 and 18 October 2000, for at-the-money forward volatilities and risk-reversaland strangle prices with δ = 0.25 also quoted in volatilities of the USD/euro exchangerate.6Regarding the options on the PTE exchange rates, the data used comprises volatilities(ask quotes) of OTC at-the-money forward options, disclosed by Banco Português doAtlântico,7 for the exchange rate of German Mark/escudo (DEM/PTE), German Mark/US6These quotations are published on the Reuters page BBAVOLFIX1 (“BBA Currency Option Volatility Fixings”). Forward rates were computed based on the covered interest rate parity.7Through the BPAI page of Reuters.
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dollar (DEM/USD) and US dollar/escudo (USD/PTE), between 26 July 1996 and 30 April1998. Maturities of 1, 2, 3, 6 and 12 months were considered. Call-option prices from theChicago Mercantile Exchange (CME) for the DEM/USD exchange rate were also used,with quarterly maturities between September 1996 and June 1998, from 27 July 1995 to19 September 1997.The currency option data for the Danish crown was obtained from Reuters, consisting ofimplied volatilities used in the pricing of EUR/DKK, USD/DKK and USD/EUR options(ask quotes) and spot exchange and interest rates for the EUR and the DKK, from 4January 1999 to 10 October 2000.
12.2 THE VALUATION OF CURRENCY OPTIONS SPREADSAccording to market conventions, ﬁnancial institutions quote OTC options in impliedvolatilities (vols), as annual percentages, which are translated to monetary values usingthe Garman–Kohlhagen (1983) valuation formula. When the strike price corresponds tothe forward rate (at-the-money forward options), the formula respectively for call andput-option prices is:8C = exp (−iτf τ )[F N (d1 ) − XN (d2 )]
(12.1)
P = exp (−iτf τ )[XN (−d2 ) − F N (−d1 )]
(12.2)
√√where d1 = [ln(F /X) + (σ 2 /2)τ ]/σ τ , d2 = d1 − σ τ , F is the forward exchangerate,9 X is the strike price, N (di ) (i = 1, 2) represents the value of the cumulativeprobability function of the standardised normal distribution for di , S is the spot exchangeff∗rate,10 iτ is the τ -maturity domestic risk-free interest rate, iτ is the τ -maturity foreignrisk-free interest rate and σ 2 is the instantaneous variance of the exchange rate.Strike prices are usually denominated in the moneyness degree of the option, insteadof monetary values. Moneyness is usually measured by the option delta (δ), which isthe ﬁrst derivative of the option price in order to the underlying asset price.11 Followingequations (12.1) and (12.2), the delta values are:∂C(X)= exp (−iτf ∗ τ )N (d1 )∂S∂P (X)= − exp (−iτf ∗ τ )N (−d1 )δP =∂SδC =
(12.3)(12.4)
In the OTC currency option market, option spreads are traded along with option contracts.Among these spreads, risk-reversals and strangles are the most commonly traded. Riskreversals are composed of buying a call-option (long call) and selling a put-option (shortput), with each option being equally out-of-the-money, i.e., they have the same moneyness.8This formula was originally presented in Garman and Kohlhagen (1983) and is based on the assumption ofexchange rate log-normality. It is basically an adaptation of the Black–Scholes (1973) formulas, assuming thatthe exchange rate may be taken as an asset paying a continuous dividend yield equal to the foreign interest rate.ff∗9According to the covered interest rate parity, F corresponds to S exp [(iτ − iτ )τ ].10Quoted as the price of the foreign currency in domestic currency units.11Given that the pay-off of a call-option increases when the underlying asset price increases and the oppositehappens to put-options, the delta of a call-option (δC ) is positive, while the delta of a put-option (δP ) is negative.
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Pay-off
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XP
F
XC
Strike price
Figure 12.4
Risk-reversal pay-off
Therefore, the forward price will be lower than the strike price of the call-option (XC )and higher than the strike price of the put-option, corresponding to XP (see Figure 12.4).Risk-reversals are usually traded for δ = 0.25. For simpliﬁcation, the volatility of thecall-option with δ = 0.75 is regularly used as a proxy for the volatility of the put-optionwith δ = 0.25.12 Consequently, the price of a risk-reversal (in vols) is:rrt = σt0.25δ − σt0.75δ
(12.5)
with σ 0.25δ and σ 0.75δ representing the implied volatilities of the call-options with δ = 0.25and δ = 0.75, respectively. The price of a risk-reversal may be taken as a skewness indicator, being positive when the probability attached to a given increase of the underlyingasset price is higher than the probability of a similar decrease.Strangles (usually also identiﬁed as a bottom vertical combination, due to the graphicalrepresentation of its pay-off) are option portfolios including the acquisition of a calloption and a put-option with different strike prices but with the same moneyness. Bothoptions being out-of-the-money, the strike price of the call-option is higher than that ofthe put-option, as in the case of the risk-reversals (see Figure 12.5).13These option spreads are also usually traded for options with δ = 0.25 and their pricesare deﬁned as the difference (in vols) to a reference volatility, frequently the at-the-moneyf∗
f∗
In fact, according to equations (12.3) and (12.4), δP = δC − exp (−iτ τ ). With δ = 0.25, δC = exp (−iτ τ ) −0.25. For short-term options, the ﬁrst component of the right-hand side of the previous expression is close to1, unless the foreign interest rate is signiﬁcantly high. Thus δC = 0.75.13The difference is that the put-option is bought in the strangle case, instead of being sold as happens with therisk-reversal.12
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Pay-off
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XP
F
XC
Strike price
Figure 12.5 Strangle pay-off
forward, as follows:strt = 0.5(σt0.25δ + σt0.75δ ) − atmt
(12.6)
where atmt is the implied volatility of the at-the-money forward option. The marketparticipants consider δ = 0.5 as a proxy for the delta of the at-the-money forward option.14If the implied volatility is similar for the strike prices of the options included in thestrangle, the average of those option volatilities must be close to the implied volatility ofthe at-the-money forward option and the strangle price will be around zero. Therefore, thestrangle price may be considered as a kurtosis indicator, given that it provides informationabout the smile curvature.
12.3 RND ESTIMATION USING OPTION SPREADSUsing risk-reversal and strangle prices with δ = 0.25 and the at-the-money forwardvolatility, the volatilities for 0.25 and 0.75 deltas may be computed followingequations (12.5) and (12.6):15σt0.25δ = atmt + strt + 0.5rrt
(12.7)
σt0.75δ = atmt + strt − 0.5rrt
(12.8)f∗
14Fromequation (12.3), it is easy to conclude that the delta of an at-the-money option is δt0.5δ = exp (−iτ )N√[σ τ /2]. For short-term options and for the usual volatility values (in the range 10–40% per year), δt0.5δ ∼= 0.5,given that the discount factor is close to one and the value at which the cumulative normal distribution functionis computed is also close to zero. This implies that the normal distribution value will be around 0.5.15See, e.g., Malz (1996, 1997) or McCauley and Melick (1996).
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Given that only three values in the delta-volatility space are available, the RND estimation is not possible. Consequently, one needs to assume a functional speciﬁcation asa proxy for the volatility curve. Malz (1997) uses the following quadratic polynomialfunction resulting from a second-order Taylor expansion for δ = 0.5:σtδ (δ) = β0 atmt + β1 rrt (δ − 0.5) + β2 strt (δ − 0.5)2
(12.9)
Restricting the curve to ﬁt perfectly the three observed points in the delta-volatilityspace (σt0.25δ , atmt and σt0.75δ ), we get:16σtδ (δ) = atmt − 2rrt (δ − 0.5) + 16strt (δ − 0.5)2
(12.10)
Thus, knowing only the at-the-money forward volatility and the risk-reversal andstrangle prices, a curve in delta-volatility space is obtained. Substituting equation (12.3)and/or equation (12.4) in equation (12.10), the relationship between volatility and strikeprice (volatility smile) is obtained. Next, the option prices are computed from thosevolatilities, using equations (12.1) and/or (12.2), allowing the RND estimation.The ﬁnal step in the estimation procedure will be to extract the RND from the optionprices. Different techniques for the estimation of the RND functions from European optionprices are found in the relevant literature.17 Among these, one of the most popular techniques has been the linear combination of two log-normal distributions.18 It consists ofsolving the following optimisation problem:min
N
α1 ,α2 ,β1 ,β2 ,θ
2[Ĉ(Xi , τ ) − Ci0 ] +
i=1
N
[P̂ (Xi , τ ) − Pi0 ]2
(12.11)
i=1
s.t. β1 , β2 > 0 and 0 ≤ θ ≤ 1which solves to provide solutions for C and P of:Ĉ(Xi , τ ) = exp (−iτf τ )
 ∞
[θ L(α1 , β1 ; ST ) + (1 − θ )L(α2 , β2 ; ST )](ST − Xi )dST
Xi
 1− ln(Xi ) + (α1 + β12 )− ln(Xi ) + α1− Xi N= exp (−iτf τ )θ exp α1 + β12 N2β1β1 − ln(Xi ) + (α2 + β22 )1= exp (−iτf τ )(1 − θ ) exp α2 + β22 N2β2− ln(Xi ) + α2(12.12)−Xi Nβ216Replacing δ in (12.9) by 0.25, 0.5 and 0.75, respectively, and using simultaneously equations (12.5) and(12.6) (see Malz (1997)).17See for instance Abken (1995), Bahra (1996), Deutsche Bundesbank (1995), Malz (1996) or Söderlind andSvensson (1997).18This technique is due to Ritchey (1990) and Melick and Thomas (1997).
Portfolio Management and Information Xi[θ L(α1 , β1 ; ST ) + (1 − θ)L(α2 , β2 ; ST )](Xi − ST )dSTPˆ (Xi , τ ) = exp (−iτf τ )
357
−∞
 ln(Xi ) − (α1 + β12 )1ln(Xi ) − α1+ Xi N= exp (−iτf τ )θ − exp α1 + β12 N2β1β1= exp (−iτf τ )(1 − θ )
 ln(Xi ) − (α2 + β22 )1 2ln(Xi ) − α2+ Xi N− exp α2 + β2 N2β2β2(12.13)
and where L(αi , βi ; ST ) is the log-normal density function i (i = 1, 2), the parameters α1and α2 are the means of the respective normal distributions, β1 and β2 are the standarddeviations of the latter and θ the weight attached to each distribution. The expressionsfor αi and βi are the following:
σ2αi = ln Ft + µi − i2√βi = σ i τ
τ
(12.14)(12.15)
where µ is the drift of the exchange rate return. Though this method imposes somestructure on the density function and raises some empirical difﬁculties, it offers someadvantages, as it is sufﬁciently ﬂexible and fast. Therefore it will be used in the followingsections.19Alternatively, it will be considered that the exchange rate follows a stochastic processcharacterised by a mixture of a geometric Brownian motion and a jump process. FollowingBall and Torous (1983, 1985) and Malz (1996, 1997), when no more than one jump isexpected during the period under analysis, the Poisson jump model presented by Merton(1976) and Bates (1991) may be simpliﬁed into a Bernoulli model for the jump component.20 Therefore, an option price with an underlying asset following such a process is a weighted average of the Black–Scholes (1973) formula given a jump andthe Black–Scholes (1973) function value with no jump.Option price equations considering a discrete jump correspond to:Ĉ(Xi , τ ) = exp (−iτf τ )
 ∞Xi
= exp (−iτf τ )λτ
(λτ L(α1∗ , β1∗ ; ST ) + (1 − λτ )L(α2∗ , β2∗ ; ST ))(ST − Xi )dST
1 ∗2− ln(Xi ) + (α1∗ + β1∗2 )∗exp α1 + β1 − ln(1 + k) (1 + k)N2β1∗
19For instance, Bliss and Panigirtzoglou (2000) conclude that a smile interpolation method dominates thelog-normal mixture technique in what concerns the stability of the results vis-à-vis measurement errors inoption prices.20As the US dollar/euro exchange rate is under analysis, the assumption of existing no more than one jumpduring the lifetime of the option is reasonable.
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Applied Quantitative Methods for Trading and Investment1 ∗2− ln(Xi ) + α1∗f∗+ exp (−iτ τ )(1 − λτ ) exp α2 + β2−Xi Nβ1∗2− ln(Xi ) + α2∗− ln(Xi ) + (α2∗ + β2∗2 )− Xi N(12.16)×N∗β2β2∗ XiP̂ (Xi , τ ) = exp (−iτf τ )(λτ L(α1∗ , β1∗ ; ST ) + (1 − λτ )L(α2∗ , β2∗ ; ST ))(Xi − ST )dST
−∞
ln(Xi ) − (α1∗ + β1∗2 )1 ∗2∗− exp α1 + β1 − ln(1 + k) (1 + k)N2β1∗1 ∗2ln(Xi ) − α1∗f∗+exp(−iβτ)(1−λτ)−expα++Xi Nτ2β1∗2 2ln(Xi ) − α2∗ln(Xi ) − (α2∗ + β2∗2 )+X(12.17)N×Niβ2∗β2∗
= exp (−iτf τ )λτ
The parameters λ and k are respectively the probability and the magnitude of a jump.The parameters αi∗ and βi∗ correspond to:αi∗ = ln Ft + ln(1 + k) −√βi∗ = σ τInput
Tay