<< ώρπ. 16(βώεψν 19)―ΈΡ≈–ΤάΆ»≈ >>
E(N ) = 1 + (T βà’ 2) βà’ arc sin(œ¹F )
2œÄ

2
A round turn is deο¬¹ned as reversing a position from one unit long to one unit short or vice versa.
Optimal Allocation of Trend-Following Rules 339

where
mβà’2 mβà’2
œ¹F = (m βà’ i βà’ 1)(m βà’ i βà’ 2) (m βà’ i βà’ 1)2 if m βâΞ 2
i=0 i=0

Subsequently, the expected transaction costs will be E(TC) = βà’cE(N ), where c is the
trading cost per round turn. Table 11.4 indicates the expected number of round turns out
of a 250-day year under the random walk assumption. The last column shows the resulting
yearly cost in percentage terms for a cost per transaction equal to c = 0.03%.
To further illustrate the relationship between moving average order and number of round
turns generated we used the spreadsheet Lequeux001.xls to calculate the number of round
turns observed for each of our currency exchange rates and plotted this alongside the ex-
ante expected theoretical number of round turns (Figure 11.5). The number of transactions

Table 11.4 Expected number of transactions and cost for simple MAs

Moving average Expected number of Expected yearly cost %,
c = 0.03%
round turnsa

βà’3.75
S(2) 125.00
βà’2.77
S(3) 92.52
βà’2.02
S(5) 67.40
βà’1.46
S(9) 48.62
βà’1.05
S(17) 34.92
βà’0.76
S(32) 25.46
βà’0.56
S(61) 18.62
βà’0.41
S(117) 13.68
a
Number of round turns assuming a year of 250 trading days.

140
USE-JPY
EUR-USD
AUD-USD
120 GBP-USD
THEORETICAL
Number of round turns per annuun

100

80

60

40

20

0
0 20 40 60 80 100
Moving average order

Figure 11.5 Observed and theoretical number of round turns as a function of the MA order
340 Applied Quantitative Methods for Trading and Investment

decreases as a function of the order of the moving average. The reader can reproduce
this chart by running the simulation and selecting βÄ€MA orderβÄù for the X variable and
βÄ€Round turnsβÄù for the Y variable in the worksheet βÄ€MAβÄù. This of course only plots the
results for the selected currency pair. To produce the comparison chart the reader must
obtain the data for each currency pair, which is stored in column AD, and generate the
plot themselves.
The formula is exact for linear rules and just an approximation for non-linear predictors.
The shorter-term moving averages generate the most trading signals. Therefore if one
takes into account transaction costs it clearly appears that for equally expected gross
returns, longer-term rules must be preferred. The number of round turns generated by
a moving average trading rule is higher under the random walk assumption than when
the underlying series exhibit positive autocorrelation and lower when exhibiting negative
autocorrelation. Indeed, the more positive the autocorrelation, the more trends and the
fewer transactions there are. However the number of transactions is higher when there
are negative autocorrelations and therefore the expected return after transactions costs
will be lower. One important implication is that negative autocorrelations will be more
difο¬¹cult to exploit than positive ones when transaction costs are taken into account. This
may well explain why there are very few active currency managers relying on contrarian3
strategies. The theoretical model can help managers who rely on linear predictors such as
simple moving averages to factor in ex-ante the cost of implementing their strategy.

11.6 THEORETICAL CORRELATION
OF LINEAR FORECASTERS
Another strong argument for using trading rules based on an autoregressive model over
a fundamental process is that it is possible to calculate ex-ante the level of correlation
between the signals generated by linear trading rules. It is therefore possible to estimate
ex-ante what diversiο¬¹cation can be provided by a set of linear trading rules. This is clearly
not possible when it comes to an exogenous process. It would indeed be very difο¬¹cult
to determine the ex-ante correlation of the returns generated by a trading strategy based
for example on interest rate movements or current account announcements. Acar and
Lequeux (1996) have shown that under the assumption of a random walk without drift
it is possible to derive the correlation of linear trading rule returns. Assuming that the
underlying time series, Xt , follows a centred identically and independently distributed
normal law, the returns R1,t and R2,t generated by simple moving averages of order m1
and m2 exhibit linear correlation, œ¹R , given by:
οΘ® οΘΙ
min(m1 ,m2 )βà’2
οΘ· οΘΚ
(m1 βà’ i βà’ 1)(m2 βà’ i βà’ 1)
οΘ· οΘΚ
οΘ· οΘΚ
2
œ¹R (m1 , m2 ) = arc sin οΘ· οΘΚ
i=0
οΘ· οΘΚ
œÄ οΘ· οΘΚ
m1 βà’2 m2 βà’2
οΘ° 2οΘ»
(m1 βà’ i βà’ 1)2 (m2 βà’ i βà’ 1)
i=0 i=0

3
Contrarian strategies are mean-reverting strategies. Therefore these strategies will be at their best when the
underlying series exhibit a high level of negative autocorrelation in their returns.
Optimal Allocation of Trend-Following Rules 341

Table 11.5 Simple moving average returns expected correlation4

œ¹ MA-2 MA-3 MA-5 MA-9 MA-17 MA-32 MA-61 MA-117

MA-2 1 0.705 0.521 0.378 0.272 0.196 0.142 0.102
MA-3 1 0.71 0.512 0.366 0.264 0.19 0.137
MA-5 1 0.705 0.501 0.361 0.26 0.187
MA-9 1 0.699 0.501 0.359 0.258
MA-17 1 0.707 0.504 0.361
MA-32 1 0.705 0.502
MA-61 1 0.704
MA-117 1

Table 11.5 utilises this equation and shows the coefο¬¹cient correlation between the
returns generated by different moving averages when applied to the same underlying
market.
Trend-following systems are positively correlated. Zero or negative correlation
obviously would require the combination of trading rules of different nature such as
trend-following and contrarian strategies. Buy and sell signals and therefore returns of
trend-following trading rules are not independent over time under the random walk
assumption. This will therefore put a ο¬‚oor on the maximum risk reduction that can be
achieved by using them.

11.7 EXPECTED VOLATILITY OF MA
Volatility of returns is an important measure of underlying risk when comparing trading
strategies. There are many fallacious statements and feelings regarding short-term versus
long-term trading strategies. This usually derives from some sort of confusion between
position-to-position and mark-to-market returns. Acar et al. (1994) have shown by using a
bootstrap approach that the volatility of the returns generated by trading rules is approxi-
mately equal to the volatility of the underlying process applied to and is thus independent
of the moving average order. To illustrate this relationship Figure 11.6 shows the volatil-
ity of the daily returns generated by moving averages of order 2 to 177 and the volatility
of USDβÄ“JPY spot daily returns over the period 15/02/1996 to 12/03/2002. The reader
can reproduce this chart by running the simulation and selecting βÄ€MA orderβÄù for the X
variable and βÄ€VolatilityβÄù for the Y variable in the worksheet βÄ€MAβÄù.
As exhibited in Figure 11.6, though the variance of the trading rule returns differs
depending on the moving average order, the overall level remains quite independent of
the time horizon. The difference in terms of risk would be of no consequence to a market
trader, the same level of riskiness as a buy and hold position in the underlying. On a
mark-to-market basis active management does not add risk relative to a passive holding
of the asset.

4
For instance œ¹(MA-2, MA-117) means the rule returns correlation between the simple moving average of
order 2 and the moving average of order 117 is equal to 0.102
342 Applied Quantitative Methods for Trading and Investment

12.33%
Volatility of MA returns
USD-JPY returns volatility

12.33%

12.32%
Annualised volatillity

12.32%

12.31%

12.31%

12.30%

12.30%
0 20 40 60 80 100
Moving average order

Figure 11.6 Volatility of spot USDβÄ“JPY and moving averages trading rule returns of order 2
to 117

11.8 EXPECTED RETURN OF LINEAR FORECASTERS
To depict the relationship between the returns generated by moving averages trading rules
we conducted a Monte Carlo experiment. We generated 10 000 simulated series of 400 data
points. The random variables were generated using a methodology developed by Zangari
(1996)5 where each random variable r t results from the mixture of two independent
normal distributions of mean zero: one with a probability of occurrence p = 1 and a
standard deviation of 1 and the other with a probability of success of 0.1 and a standard
deviation of 3.
r t = (0,1) + (0,3)B(0.1)

For each of the series drawn we calculated the return that would have been generated
for moving averages of order 2 to 117. We also noted the normalised drift |¬µm|/œÉ m
and the ο¬¹rst three autocorrelation coefο¬¹cients of the series drawn. We then regressed
the normalised drift and the ο¬¹rst three autocorrelation coefο¬¹cients of the underlying time
series against the computed moving average trading rule annualised returns. The results
are shown in Figures 11.7 and 11.8.6
As illustrated in Figures 11.7 and 11.8, the expected return of moving averages is a
function of both drift and price dependencies. Short-term moving average returns will be

5
The simulation can be found in the spreadsheet Lequeux002.xls. Note however that with 10 000 iterations the
simulation takes a long time to run. It is possible to change the number of iterations, to say 100, by selecting
βÄ€ToolsβÄù, βÄ€MacroβÄù, βÄ€Visual Basic EditorβÄù and changing the line of code βÄ€For i = 1 To 10000βÄù to βÄ€For i = 1
To 100βÄù.
6
Figure 11.7 can be found in the βÄ€Sensitivity to DriftβÄù workbook of Lequeux004.xls. Figure 11.8 can be found
in the βÄ€Sensitivity of DependenciesβÄù workbook of Lequeux004.xls.
Optimal Allocation of Trend-Following Rules 343
0.70

0.60

0.50
Regression coefficient

0.40

0.30

0.20

0.10

0.00
2 6 10 14 18 22 26 30 34 38 42 46 50 54 58 62 66 70 74 78 82 86 90 94 98 102106110114
Moving avarage order

Figure 11.7 Normalised drift and moving average trading rule returns

1.40
a1
a2
a3
1.20

1.00
Regression coefficient

0.80

0.60

0.40

0.20

0.00
0 20 40 60 80 100
Moving average order

Figure 11.8 Autocorrelations and moving average trading rule returns

far more sensitive to price dependencies rather than drift, whereas the return of longer-
term moving averages will be more sensitive to the exchange rate drift. This relationship
is well highlighted by Figures 11.7 and 11.8. On the one hand we note that the regression
coefο¬¹cient between the drift of the underlying time series and the delivered return of the
moving average trading rule increases with the length of the moving average. On the
other hand the coefο¬¹cients of regression between returns of the moving average trading
rule and the ο¬¹rst three autocorrelation coefο¬¹cients of the underlying time series decrease
344 Applied Quantitative Methods for Trading and Investment
Table 11.6 Weighting of œ¹ as a function of the moving average order

Weightings MA-2 MA-3 MA-5 MA-9 MA-17 MA-32 MA-61 MA-117

A(1) 1.000 0.634 0.426 0.370 0.367 0.339 0.311 0.329
A(2) 0.000 0.366 0.336 0.328 0.318 0.324 0.305 0.282
A(3) 0.000 0.000 0.238 0.302 0.316 0.337 0.384 0.390

with the length of the moving average. Research on the expected return of trend-following
strategies has been published. Acar (1994) wrote a seminal paper that proposes a closed
form solution to estimate ex-ante the return of a trend-following trading rule. The results
are interesting because they provide an unbiased framework for a better understanding of
what may drive the returns generated by linear trading rules. Assuming that the rate of
return, xt , follows a Gaussian process, Acar established the mean of the trading rule by:
2 2
2/œÄœÉœ¹e(βà’¬µf /2œÉf ) + ¬µ(1 βà’ 2 [βà’¬µf /œÉf ])
E(rt+1 ) =

where œ¹ = Corr(xt+1 , ft ), is the cumulative function of N (0,1), ¬µ is the expected
value and œÉ is the standard deviation of the underlying return xt . The expected value and
standard deviation of the forecaster ft are given by ¬µf and œÉf . In the case of a random
walk with drift model the above formula will simplify as E(rt+1 ) = ¬µ(1 βà’ 2 [βà’¬µf /œÉf ]).
In this case the expected return is a negative function of the volatility. In the special case
when there is no drift the expected proο¬¹t is given by:

2
E(rt+1 ) = œÉ Corr(xt+1 , ft )
œÄ
In our applied example we use an adjusted form of this proposition where œ¹ =
3
i=1 wi Corr(xt+1 , ft ), with wi the weight attributed as a function to the sensitivity of
the forecaster to autocorrelation coefο¬¹cients. The weights were derived from our Zangari
simulation and are shown in Table 11.6.7

11.9 AN APPLIED EXAMPLE
In the previous sections of this chapter we have highlighted that it is possible to determine
ex-ante the transaction cost, the volatility, the estimated returns as well as the correlation
between linear individual forecasters such as moving averages. This provides us with the
necessary statistical tools to establish a framework for a meanβÄ“variance allocation model
of trend-following rules. Though we could use a larger sample of linear trading rules such
as simple, weighted, exponential moving averages and momentums, we choose to focus
only on a subset of trading rules for the sake of computational simplicity. The reader
could easily translate this framework to a wider universe of trading rules if required. In
the following we try to determine ex-ante what would be the optimal weighting between
moving averages of order 2, 3, 5, 9, 32, 61 and 117 to maximise the delivered information
ratio. The model has been programmed into a spreadsheet8 to give the reader the possibility

7
The results may be found in the βÄ€Regressions DataβÄù sheet of Lequeux004.xls, A59:I62.
8
The model can be found in the βÄ€SimulationβÄù sheet of the spreadsheet Lequeux005.xls. The results contained
in Tables 11.7 and 11.8 may be found in the appropriate workbooks of Lequeux003.xls. The reader can see
how they were arrived at by referring to the βÄ€Performance BasketβÄù sheet of Lequeux005.xls.
Optimal Allocation of Trend-Following Rules 345

to experiment and also investigate the effect of changing the sampling methodology to
estimate the various parameters required to estimate ex-ante returns of our set of moving
averages. Figure 11.9 details the workings of the spreadsheet.
Tables 11.7 and 11.8 show the results that were obtained when using this model and
when allocating equally between the moving averages of order 2, 3, 5, 9, 32, 61 and 117.
The returns generated by the moving average trading rule allocation model have out-
performed the equally weighted basket of moving averages in terms of return divided by
risk (IR) for four currencies out of ο¬¹ve. It also provides cash ο¬‚ow returns that are closer
to normal as denoted by the lower kurtosis of the daily returns (Tables 11.7 and 11.8).
Though these may not appear as outstanding results at ο¬¹rst, one has to take into account the
level of transaction cost incurred. Because of the daily rebalancing necessary the cost of
implementation will be far greater for the allocation model than for the equally weighted
approach. The fact that the model still manages to outperform the equally weighted basket
demonstrates somehow the higher quality of the allocation model.

Expected return,
Select currency and volatility & IR of
transaction cost for

Expected return, volatility &
transaction cost of MAs

Expected correlation
of MA returns
Solved weights
to optimise the
information ratio
Estimates of means, variances
and autocorrelation
coefficients

Autocorrelation
coefficients weightings

Solved MA weights

Figure 11.9 Simple moving averages allocation model
346 Applied Quantitative Methods for Trading and Investment
Table 11.7 Results for the equally weighted basket of moving averages

βà’2.01% βà’1.73%
Return 1.56% 6.06% 3.77%
Volatility 6.77% 9.01% 5.24% 3.79% 7.64%
βà’0.38 βà’0.46
IR 0.23 0.67 0.49
βà’11.27% βà’11.13% βà’13.35% βà’16.85% βà’10.64%
Maximum cumulative
drawdown
βà’1.66 βà’1.23 βà’2.55 βà’4.45 βà’1.39
Normalised maximum
cumulative drawdown
βà’2.11% βà’3.16% βà’1.36% βà’1.54% βà’3.53%
Maximum daily loss
Maximum daily proο¬¹t 1.96% 5.64% 1.48% 0.90% 2.65%
βà’0.78 βà’0.01
Skew 0.12 0.87 0.17
Kurtosis 3.36 12.56 3.26 5.52 5.39

Table 11.8 Results for the meanβÄ“variance allocation model

βà’0.47% βà’0.98%
Return 10.29% 1.32% 10.73%
Volatility 9.97% 12.75% 7.58% 5.48% 11.05%
βà’0.05 βà’0.18
IR 0.81 0.17 0.97
βà’16.52% βà’13.54% βà’16.17% βà’15.34% βà’18.68%
Maximum cumulative
drawdown
βà’1.66 βà’1.06 βà’2.13 βà’2.80 βà’1.69
Normalised maximum
cumulative drawdown
βà’2.67% βà’3.22% βà’1.49% βà’1.61% βà’4.72%
Maximum daily loss
Maximum daily proο¬¹t 2.58% 5.65% 2.15% 1.51% 2.65%
βà’0.19 βà’0.25
Skew 0.08 0.52 0.25
Kurtosis 0.92 4.36 0.90 1.76 2.65

11.10 FINAL REMARKS
In this chapter we ο¬¹rst highlighted some of the statistical properties of trading rules
and developed the reason for the interest in these for a market participant. In the latter
part of the chapter we went on to provide the reader with an unbiased framework for
trading rules allocation under constraint of cost and maximisation of information ratio.
The results have shown that in four currency pairs out of ο¬¹ve this would have provided
better economic value than using an equally weighted basket of trading rules. These results
are signiο¬¹cant for active currency managers who seek to provide their investors with a
balanced risk/return proο¬¹le in the currency markets whilst using trend-following models.
Though the results are encouraging, the challenge remains ahead. How to forecast the drift
and serial dependencies remains the essence in any forecasting context and remains the
key factor in bettering such a model. It may well be that the answer for forecasting those
parameters lies more within a macro-fundamental approach rather than using uniquely the
price as an information discounting process.
Optimal Allocation of Trend-Following Rules 347

REFERENCES
Acar, E. (1994), βÄ€Expected Return of Technical Forecasters with an Application to Exchange RatesβÄù,
Presentation at the International Conference on Forecasting Financial Markets: New Advances for
Exchange Rates and Stock Prices, 2βÄ“4 February 1994, London. Published in Advanced Trading
Rules, Acar & Satchell (eds), Butterworth-Heinemann, 1998.
Acar, E. and P. Lequeux (1995), βÄ€Trading Rules Proο¬¹ts and the Underlying Time Series PropertiesβÄù,
Presentation at the First International Conference on High Frequency Data in Finance, Olsen and
Associates, Zurich, Switzerland, 29βÄ“31 March 1995. Forthcoming in P. Lequeux (ed.), Financial
Markets Tick by Tick, Wiley, London.
Acar, E. and P. Lequeux (1996), βÄ€Dynamic Strategies: A Correlation StudyβÄù, in C. Dunis (ed.),
Forecasting Financial Markets, Wiley, London, pp. 93βÄ“123.
Acar, E., P. Lequeux and C. Bertin (1994), βÄ€Tests de marche al¬¥ atoire bas¬¥ s sur la proο¬¹tabilit¬¥ des
e e e
indicateurs techniquesβÄù, Analyse Financi` re, 4 82βÄ“86.
e
Arnott, R. D. and T. K. Pham (1993), βÄ€Tactical Currency AllocationβÄù, Financial Analysts Journal,
Sept, 47βÄ“52.
Billingsley, R. and D. Chance (1996), βÄ€Beneο¬¹ts and Limitations of Diversiο¬¹cation among Com-
Kritzman, M. (1989), βÄ€Serial Dependence in Currency Returns: Investment ImplicationsβÄù, Journal
of Portfolio Management, Fall, 96βÄ“102.
LeBaron, B. (1991), βÄ€Technical Trading Rules and Regime Shifts in Foreign ExchangeβÄù, University
of Wisconsin, Social Science Research, Working Paper 9118.
LeBaron, B. (1992), βÄ€Do Moving Average Trading Rule Results Imply Nonlinearities in Foreign
Exchange MarketsβÄù, University of Wisconsin, Social Science Research, Working Paper 9222.
Lequeux, P. and E. Acar (1998), βÄ€A Dynamic Benchmark for Managed Currencies FundsβÄù, Euro-
pean Journal of Finance, 4(4), 311βÄ“330.
Levich, R. M. and L. R. Thomas (1993), βÄ€The Signiο¬¹cance of Technical Trading-Rule Proο¬¹ts in the
Foreign Exchange Market: A Bootstrap ApproachβÄù, Journal of International Money and Finance,
12, 451βÄ“474.
Schulmeister, S. (1988), βÄ€Currency Speculations and Dollar FluctuationsβÄù, Banco Nationale del
Lavaro, Quarterly Review, 167 (Dec), 343βÄ“366.
Silber, L. W. (1994), βÄ€Technical Trading: When it Works and When it DoesnβÄ™tβÄù, The Journal of
Derivatives, Spring, 39βÄ“44.
Taylor, S. J. (1980), βÄ€Conjectured Models for Trends in Financial Prices, Tests and ForecastsβÄù,
Journal of the Royal Statistical Society, Series A, 143, 338βÄ“362.
Taylor, S. J. (1986), Modelling Financial Time Series, Wiley, Chichester, UK.
Taylor, S. J. (1990a), βÄ€Reward Available to Currency Futures Speculators: Compensation for Risk
or Evidence of Inefο¬¹cient Pricing?βÄù, Economic Record (Suppl.), 68, 105βÄ“116.
Taylor, S. J. (1990b), βÄ€Proο¬¹table Currency Futures Trading: A Comparison of Technical and Time-
Series Trading RulesβÄù, in L. R. Thomas (ed), The Currency Hedging Debate, IFR Publishing,
London, pp. 203βÄ“239.
Taylor, S. J. (1992), βÄ€Efο¬¹ciency of the Yen Futures Market at the Chicago Mercantile ExchangeβÄù,
in B. A. Goss (ed.), Rational Expectations and Efο¬¹cient Future Markets, Routledge, London,
pp. 109βÄ“128.
Taylor, S. J. (1994), βÄ€Trading Futures Using the Channel Rule: A Study of the Predictive Power
of Technical Analysis with Currency ExamplesβÄù, Journal of Futures Markets, 14(2), 215βÄ“235.
Zangari, P. (1996), βÄ€An Improved Methodology for Measuring VARβÄù, RiskMetrics Monitor,
Reuters/J.P. Morgan.
12
Portfolio Management and Information from
Over-the-Counter Currency Optionsβà—

JORGE BARROS LU¬¥S
I

ABSTRACT
This 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 the
literature are presented. The ο¬¹rst one is on the EUR/USD, where option prices for several
strikes are obtained from currency option spread prices and risk-neutral density functions
are estimated using different methods. This application is followed by the analysis of
implied correlations and the credibility of the Portuguese exchange rate policy, during the
transition to the EMU, and of the Danish exchange rate policy around the Euro referendum
in September 2000. This chapter is supported by the necessary application ο¬¹les, produced
in Excel, to allow the reader to validate the author results and/or apply the analysis to a
different dataset.

12.1 INTRODUCTION
Portfolio and risk management are based on models using estimates for future returns,
volatilities and correlations between ο¬¹nancial assets. Considering the forward looking
features of derivative contracts, option prices have been used intensively in order to
extract information on expectations about the underlying asset prices.
Compared to forward and futures contracts, option prices provide an estimate not only
for 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 risk-
neutral density or RND), based on the theoretical relationship developed in Breeden and
Litzenberger (1978). This information is relevant for Value-at-Risk (VaR) exercises, as
well as stress tests. However, the completion of these exercises also demands correlation
estimates, 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 heavily
traded in over-the-counter (OTC) markets.1 The information from OTC markets usually

βà—
This chapter contains material included in the PhD thesis of the author (Lu¬¥s, 2001).
Ρ±
1
According to BIS (2001), at the end of June 2001, the OTC market was responsible for 99.5% of the aggregate
value of open positions in currency options.

Applied Quantitative Methods for Trading and Investment. Edited by C.L. Dunis, J. Laws and P. Na¬®m
Ρ±
ο¦™ 2003 John Wiley & Sons, Ltd ISBN: 0-470-84885-5
350 Applied Quantitative Methods for Trading and Investment

comprises BlackβÄ“Scholes implied volatilities for at-the-money forward options, as well
This data provides useful information about the uncertainty, the skewness and the
kurtosis of the exchange rate distribution.2 Furthermore, it allows the estimation of the
RND function without knowing option prices for a wide range of strike prices (see, for
example, Malz (1997) and McCauley and Melick (1996)). Another advantage of OTC
options is that, as they have ο¬¹xed terms to maturity, instead of ο¬¹xed maturity dates,3
the RND functions donβÄ™t have to be corrected by the effect of time on the distribution
parameters.4
Concerning correlations between exchange rates, Campa et al. (1997) show that in
forecasting correlations between the US dollarβÄ“German Mark and the US dollarβÄ“Japanese
yen exchange rates, in a period from January 1989 to May 1995, correlations implied by
option prices outperform historically based measures, namely conditional correlations, J.P.
Morgan RiskMetricsο¦¦ correlations and a GARCH (1,1)-based correlation.
As implied correlations shall be the best estimate for the future correlation between two
exchange rates, it must result from using the available information in the most efο¬¹cient
way. Consequently, implied correlations must not be affected by the behaviour of the
variance of the exchange rates.5 OTC currency option market data also allows us to
assess the credibility of target zones, by no-arbitrage tests built upon prices of options for
any 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 of
currency option spreads valuation are introduced. The estimation of RND functions from
option spreads is assessed in the third section. Two other informational contents of cur-
rency options are exploited in the fourth and ο¬¹fth sections, respectively measures of
correlation and arbitrage-based credibility tests.
The sixth section contains the empirical applications: the ο¬¹rst subsection concerns the
estimation of RND functions from currency option spreads, focusing on the EUR/USD
expectations 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 Central
Bank (ECB) reinforces the importance of analysing the market expectations on the future
evolution of its exchange rate vis-` -vis the US dollar. The analysis of the expectations on
a
the US dollar/euro exchange rate is also relevant given that, though the exchange rate is
not an intermediate target of the monetary policy followed by the ECB, most international
commodities are denominated in that currency.
In the ο¬¹rst 22 months of its life, the euro depreciated nearly 30% vis-` -vis the US dollar.
a
This movement was also characterised by signiο¬¹cant increases of the historical and implied
volatilities of the EUR/USD exchange rate, though with much higher variability in the
former case (Figure 12.1, which may be found in the ο¬¹le βÄ€chart1.xlsβÄù). Two Excel ο¬¹les
are provided to illustrate this application, both using 1-month options, one concerning

2
Though one may question the leading indicator properties of the risk-reversals, as their quotes exhibit a
high correlation with the spot rate. Using data between January 1999 and October 2000, the contemporaneous
correlation 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.
3
These options are usually traded for maturities of 1, 3, 6 and 12 months.
4
Some papers have tried to correct this maturity dependence using prices of exchange traded options, namely
Butler and Davies (1998), Melick and Thomas (1998) and Clews et al. (2000).
5
Loretan and English (2000) analyse this link between correlation and variances for the case of two stock
indexes.
Portfolio Management and Information 351

35
1.25
1.20
30
1.15

Vol. (%/ year)
25
1.10
EUR/ USD

1.05
20
1.00
15
0.95
0.90
10
0.85
5
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 1 month ATM vol. 1 month st.-dev.

Figure 12.1 EUR/USD: volatility and spot rate

the RND functions estimated by a linear combination of two log-normal distributions
(βÄ€OTC EUR USDβÄù) and another related to the RND functions estimated allowing one
The second subsection focuses on the behaviour of the Portuguese and Danish cur-
rencies respectively during the transition until 1998 towards the Economic and Mon-
etary Union (EMU) and the euro referendum in September 2000. In this subsection,
implied correlations are computed in the ο¬¹les βÄ€correlations PTE databaseβÄù and βÄ€correla-
tions DKK databaseβÄù, respectively for the Portuguese escudo and the Danish crown.
In order to have a deeper assessment of the prospects regarding the evolution of
the two exchange rate policies, several credibility tests developed in Section 12.5 are
also performed around the two above-mentioned episodes related to the EMU. Three
Excel ο¬¹les are supplied for this application: βÄ€credibility tests DKK 1mβÄù and βÄ€credibil-
ity tests DKK 1yearβÄù regarding the Danish crown and βÄ€credibility tests PTE 3mβÄù con-
cerning the Portuguese escudo. Finally, the seventh section concludes.
As illustrated in Figure 12.2 (which can be found in the ο¬¹le βÄ€credibility tests PTE 3m.
xlsβÄù), the short-term interest rate convergence of the Portuguese escudo vis-` -vis the
a
German Mark increased in mid-1997. Consequently the gap between the spot and the 3-
month forward exchange rate started to be ο¬¹lled, while the exchange rate implied volatility
diminished. This shift could have been taken by a portfolio or a market risk manager as a
signal 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 ο¬¹le
βÄ€credibility tests DKK 1 year.xlsβÄù) shows that during the months before the Danish ref-
erendum (Thursday 28 September 2000) the interest rate spread against the European
currency increased, along with the exchange rate volatility. This move could have been
interpreted as reο¬‚ecting an increasing probability attached to the detachment of the Danish
currency from the euro, within the Exchange Rate Mechanism of the European Monetary
System (ERM-EMS II).
The data used in the EUR/USD application consists of a database comprising bi-monthly
quotations of the British Bankers Association (BBA), published by Reuters, between 13
352 Applied Quantitative Methods for Trading and Investment

10 108
implied vol. (%/ year)
8 106
Interest rates and

PTE /DEM
6 104

4 102

2 100

0 98
Jul-95

Oct-95

Jan-96

Apr-96

Jul-96

Oct-96

Jan-97

Apr-97

Jul-97

Oct-97

Jan-98

Apr-98

Jul-98

Oct-98
3-month PTE interest rate 3-month DEM interest rate
3m.PTE / DEM impl.vol.
Spot
Forward

Figure 12.2 DEM/PTE: interest rates and implied volatility

7 7.60
6 7.55
implied vol.(% /year)
Interest rates and

5
7.50

PTE / DEM
4
7.45
3
7.40
2
7.35
1
0 7.30
Jan-99

Mar-99

May-99

Jul-99

Sep-99

Nov-99

Jan-00

Mar-00

May-00

Jul-00

Sep-00

3-month DKK interest rate 1-year Euro interest rate
1y.DKK / EUR impl.vol. Spot
Forward

Figure 12.3 EUR/DKK: interest rates and implied volatility

January 1999 and 18 October 2000, for at-the-money forward volatilities and risk-reversal
and strangle prices with Έ¥ = 0.25 also quoted in volatilities of the USD/euro exchange
rate.6
Regarding the options on the PTE exchange rates, the data used comprises volatilities
(ask quotes) of OTC at-the-money forward options, disclosed by Banco PortuguΥÜ s do e
7
AtlΥÜ ntico, for the exchange rate of German Mark/escudo (DEM/PTE), German Mark/US
a

6
These quotations are published on the Reuters page BBAVOLFIX1 (βÄ€BBA Currency Option Volatility Fix-
ingsβÄù). Forward rates were computed based on the covered interest rate parity.
7
Through the BPAI page of Reuters.
Portfolio Management and Information 353

dollar (DEM/USD) and US dollar/escudo (USD/PTE), between 26 July 1996 and 30 April
1998. Maturities of 1, 2, 3, 6 and 12 months were considered. Call-option prices from the
Chicago 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 to
19 September 1997.
The currency option data for the Danish crown was obtained from Reuters, consisting of
implied 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 4
January 1999 to 10 October 2000.

12.2 THE VALUATION OF CURRENCY OPTIONS SPREADS
According to market conventions, ο¬¹nancial institutions quote OTC options in implied
volatilities (vols), as annual percentages, which are translated to monetary values using
the GarmanβÄ“Kohlhagen (1983) valuation formula. When the strike price corresponds to
the forward rate (at-the-money forward options), the formula respectively for call and
put-option prices is:8

C = exp (βà’iœ„ œ„ )[F N (d1 ) βà’ XN (d2 )]
f
(12.1)
P = exp (βà’iœ„ œ„ )[XN (βà’d2 ) βà’ F N (βà’d1 )]
f
(12.2)
βàö βàö
where d1 = [ln(F /X) + (œÉ 2 /2)œ„ ]/œÉ œ„ , d2 = d1 βà’ œÉ œ„ , F is the forward exchange
rate,9 X is the strike price, N (di ) (i = 1, 2) represents the value of the cumulative
probability function of the standardised normal distribution for di , S is the spot exchange
fβà—
f
rate,10 iœ„ is the œ„ -maturity domestic risk-free interest rate, iœ„ is the œ„ -maturity foreign
risk-free interest rate and œÉ 2 is the instantaneous variance of the exchange rate.
Strike prices are usually denominated in the moneyness degree of the option, instead
of monetary values. Moneyness is usually measured by the option delta (Έ¥), which is
the ο¬¹rst derivative of the option price in order to the underlying asset price.11 Following
equations (12.1) and (12.2), the delta values are:

βà‚C(X)
= exp (βà’iœ„ βà— œ„ )N (d1 )
Έ¥C = f
(12.3)
βà‚S
βà‚P (X)
= βà’ exp (βà’iœ„ βà— œ„ )N (βà’d1 )
Έ¥P = f
(12.4)
βà‚S
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. Risk
reversals are composed of buying a call-option (long call) and selling a put-option (short
put), with each option being equally out-of-the-money, i.e., they have the same moneyness.

8
This formula was originally presented in Garman and Kohlhagen (1983) and is based on the assumption of
exchange rate log-normality. It is basically an adaptation of the BlackβÄ“Scholes (1973) formulas, assuming that
the exchange rate may be taken as an asset paying a continuous dividend yield equal to the foreign interest rate.
fβà—
f
According to the covered interest rate parity, F corresponds to S exp [(iœ„ βà’ iœ„ )œ„ ].
9
10
Quoted as the price of the foreign currency in domestic currency units.
11
Given that the pay-off of a call-option increases when the underlying asset price increases and the opposite
happens to put-options, the delta of a call-option (Έ¥C ) is positive, while the delta of a put-option (Έ¥P ) is negative.
354 Applied Quantitative Methods for Trading and Investment

Pay-off

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 Έ¥ = 0.25. For simpliο¬¹cation, the volatility of the
call-option with Έ¥ = 0.75 is regularly used as a proxy for the volatility of the put-option
with Έ¥ = 0.25.12 Consequently, the price of a risk-reversal (in vols) is:

rrt = œÉt0.25Έ¥ βà’ œÉt0.75Έ¥ (12.5)

with œÉ 0.25Έ¥ and œÉ 0.75Έ¥ representing the implied volatilities of the call-options with Έ¥ = 0.25
and Έ¥ = 0.75, respectively. The price of a risk-reversal may be taken as a skewness indi-
cator, being positive when the probability attached to a given increase of the underlying
asset price is higher than the probability of a similar decrease.
Strangles (usually also identiο¬¹ed as a bottom vertical combination, due to the graphical
representation of its pay-off) are option portfolios including the acquisition of a call-
option and a put-option with different strike prices but with the same moneyness. Both
options being out-of-the-money, the strike price of the call-option is higher than that of
the put-option, as in the case of the risk-reversals (see Figure 12.5).13
These option spreads are also usually traded for options with Έ¥ = 0.25 and their prices
are deο¬¹ned as the difference (in vols) to a reference volatility, frequently the at-the-money

fβà— fβà—
In fact, according to equations (12.3) and (12.4), Έ¥P = Έ¥C βà’ exp (βà’iœ„ œ„ ). With Έ¥ = 0.25, Έ¥C = exp (βà’iœ„ œ„ ) βà’
12

0.25. For short-term options, the ο¬¹rst component of the right-hand side of the previous expression is close to
1, unless the foreign interest rate is signiο¬¹cantly high. Thus Έ¥C = 0.75.
13
The difference is that the put-option is bought in the strangle case, instead of being sold as happens with the
risk-reversal.
Portfolio Management and Information 355

Pay-off

XP F XC
Strike price

Figure 12.5 Strangle pay-off

forward, as follows:
strt = 0.5(œÉt0.25Έ¥ + œÉt0.75Έ¥ ) βà’ atmt (12.6)

where atmt is the implied volatility of the at-the-money forward option. The market
participants consider Έ¥ = 0.5 as a proxy for the delta of the at-the-money forward option.14
If the implied volatility is similar for the strike prices of the options included in the
strangle, the average of those option volatilities must be close to the implied volatility of
the at-the-money forward option and the strangle price will be around zero. Therefore, the
strangle price may be considered as a kurtosis indicator, given that it provides information

12.3 RND ESTIMATION USING OPTION SPREADS
Using risk-reversal and strangle prices with Έ¥ = 0.25 and the at-the-money forward
volatility, the volatilities for 0.25 and 0.75 deltas may be computed following
equations (12.5) and (12.6):15

œÉt0.25Έ¥ = atmt + strt + 0.5rrt (12.7)
œÉt0.75Έ¥ = atmt + strt βà’ 0.5rrt (12.8)

fβà—
From equation (12.3), it is easy to conclude that the delta of an at-the-money option is Έ¥t0.5Έ¥ = exp (βà’iœ„ )N
14
βàö
[œÉ œ„ /2]. For short-term options and for the usual volatility values (in the range 10βÄ“40% per year), Έ¥t0.5Έ¥ βàΦ 0.5,
=
given that the discount factor is close to one and the value at which the cumulative normal distribution function
is computed is also close to zero. This implies that the normal distribution value will be around 0.5.
15
See, e.g., Malz (1996, 1997) or McCauley and Melick (1996).
356 Applied Quantitative Methods for Trading and Investment

Given that only three values in the delta-volatility space are available, the RND esti-
mation is not possible. Consequently, one needs to assume a functional speciο¬¹cation as
a proxy for the volatility curve. Malz (1997) uses the following quadratic polynomial
function resulting from a second-order Taylor expansion for Έ¥ = 0.5:

œÉtΈ¥ (Έ¥) = Έ≤0 atmt + Έ≤1 rrt (Έ¥ βà’ 0.5) + Έ≤2 strt (Έ¥ βà’ 0.5)2 (12.9)

Restricting the curve to ο¬¹t perfectly the three observed points in the delta-volatility
space (œÉt0.25Έ¥ , atmt and œÉt0.75Έ¥ ), we get:16

œÉtΈ¥ (Έ¥) = atmt βà’ 2rrt (Έ¥ βà’ 0.5) + 16strt (Έ¥ βà’ 0.5)2 (12.10)

Thus, knowing only the at-the-money forward volatility and the risk-reversal and
strangle 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 strike
price (volatility smile) is obtained. Next, the option prices are computed from those
volatilities, using equations (12.1) and/or (12.2), allowing the RND estimation.
The ο¬¹nal step in the estimation procedure will be to extract the RND from the option
prices. Different techniques for the estimation of the RND functions from European option
prices are found in the relevant literature.17 Among these, one of the most popular tech-
niques has been the linear combination of two log-normal distributions.18 It consists of
solving the following optimisation problem:

N N
2
ΥÜ ΥÜ
[C(Xi , œ„ ) βà’ Ci0 ] + [P (Xi , œ„ ) βà’ Pi0 ]2
min (12.11)
Έ±1 ,Έ±2 ,Έ≤1 ,Έ≤2 ,ΈΗ
i=1 i=1

s.t. Έ≤1 , Έ≤2 > 0 and 0 βâ¤ ΈΗ βâ¤ 1

which solves to provide solutions for C and P of:
βàû
ΥÜ
C(Xi , œ„ ) = exp (βà’iœ„ œ„ ) [ΈΗ L(Έ±1 , Έ≤1 ; ST ) + (1 βà’ ΈΗ )L(Έ±2 , Έ≤2 ; ST )](ST βà’ Xi )dST
f
Xi

βà’ ln(Xi ) + (Έ±1 + Έ≤1 ) βà’ ln(Xi ) + Έ±1
2
12
= exp Έ±1 + Έ≤1 N βà’ Xi N
f
exp (βà’iœ„ œ„ )ΈΗ
Έ≤1 Έ≤1
2
βà’ ln(Xi ) + (Έ±2 + Έ≤2 )
2
12
= exp (βà’iœ„ œ„ )(1 βà’ ΈΗ ) exp Έ±2 + Έ≤2 N
f
Έ≤2
2
βà’ ln(Xi ) + Έ±2
βà’Xi N (12.12)
Έ≤2

16
Replacing Έ¥ in (12.9) by 0.25, 0.5 and 0.75, respectively, and using simultaneously equations (12.5) and
(12.6) (see Malz (1997)).
17
See for instance Abken (1995), Bahra (1996), Deutsche Bundesbank (1995), Malz (1996) or S¬® derlind and
o
Svensson (1997).
18
This technique is due to Ritchey (1990) and Melick and Thomas (1997).
Portfolio Management and Information 357
Xi
ΥÜ
P (Xi , œ„ ) = exp (βà’iœ„ œ„ ) [ΈΗ L(Έ±1 , Έ≤1 ; ST ) + (1 βà’ ΈΗ)L(Έ±2 , Έ≤2 ; ST )](Xi βà’ ST )dST
f
βà’βàû

ln(Xi ) βà’ (Έ±1 + Έ≤1 ) ln(Xi ) βà’ Έ±1
2
12
= exp (βà’iœ„ œ„ )ΈΗ βà’ exp Έ±1 + Έ≤1 N + Xi N
f
Έ≤1 Έ≤1
2

ln(Xi ) βà’ (Έ±2 + Έ≤2 ) ln(Xi ) βà’ Έ±2
2
12
= βà’ ΈΗ ) βà’ exp Έ±2 + Έ≤2 N + Xi N
f
exp (βà’iœ„ œ„ )(1
Έ≤2 Έ≤2
2
(12.13)
and where L(Έ±i , Έ≤i ; ST ) is the log-normal density function i (i = 1, 2), the parameters Έ±1
and Έ±2 are the means of the respective normal distributions, Έ≤1 and Έ≤2 are the standard
deviations of the latter and ΈΗ the weight attached to each distribution. The expressions
for Έ±i and Έ≤i are the following:

œÉi2
Έ±i = ln Ft + ¬µi βà’ œ„ (12.14)
2
βàö
Έ≤i = œÉ i œ„ (12.15)

where ¬µ is the drift of the exchange rate return. Though this method imposes some
structure on the density function and raises some empirical difο¬¹culties, it offers some
advantages, as it is sufο¬¹ciently ο¬‚exible and fast. Therefore it will be used in the following
sections.19
Alternatively, it will be considered that the exchange rate follows a stochastic process
characterised by a mixture of a geometric Brownian motion and a jump process. Following
Ball and Torous (1983, 1985) and Malz (1996, 1997), when no more than one jump is
expected during the period under analysis, the Poisson jump model presented by Merton
(1976) and Bates (1991) may be simpliο¬¹ed into a Bernoulli model for the jump com-
ponent.20 Therefore, an option price with an underlying asset following such a pro-
cess is a weighted average of the BlackβÄ“Scholes (1973) formula given a jump and
the BlackβÄ“Scholes (1973) function value with no jump.
Option price equations considering a discrete jump correspond to:

βàû
βà— βà— βà— βà—
C(Xi , œ„ ) = exp (βà’iœ„ œ„ ) (Έ»œ„ L(Έ±1 , Έ≤1 ; ST ) + (1 βà’ Έ»œ„ )L(Έ±2 , Έ≤2 ; ST ))(ST βà’ Xi )dST
f
ΥÜ
Xi

βà— βà—2
βà’ ln(Xi ) + (Έ±1 + Έ≤1 )
1 βà—2
βà—
= exp (βà’iœ„ œ„ )Έ»œ„ exp Έ±1 + Έ≤1 βà’ ln(1 + k) (1 + k)N
f
βà—
Έ≤1
2

19
For instance, Bliss and Panigirtzoglou (2000) conclude that a smile interpolation method dominates the
log-normal mixture technique in what concerns the stability of the results vis-` -vis measurement errors in
a
option prices.
20
As the US dollar/euro exchange rate is under analysis, the assumption of existing no more than one jump
during the lifetime of the option is reasonable.
358 Applied Quantitative Methods for Trading and Investment
βà—
βà’ ln(Xi ) + Έ±1 1 βà—2
βà—
βà’Xi N + exp (βà’iœ„ œ„ )(1 βà’ Έ»œ„ ) exp Έ±2 + Έ≤2
f
βà—
Έ≤1 2
βà— βà—2 βà—
βà’ ln(Xi ) + (Έ±2 + Έ≤2 ) βà’ ln(Xi ) + Έ±2
Ο—N βà’ Xi N (12.16)
βà— βà—
Έ≤2 Έ≤2
Xi
βà— βà— βà— βà—
P (Xi , œ„ ) = exp (βà’iœ„ œ„ ) (Έ»œ„ L(Έ±1 , Έ≤1 ; ST ) + (1 βà’ Έ»œ„ )L(Έ±2 , Έ≤2 ; ST ))(Xi βà’ ST )dST
f
ΥÜ
βà’βàû

βà— βà—2
ln(Xi ) βà’ (Έ±1 + Έ≤1 )
1 βà—2
βà—
= βà’ exp + Έ≤1 βà’ ln(1 + k) (1 + k)N
f
exp (βà’iœ„ œ„ )Έ»œ„ Έ±1 βà—
Έ≤1
2
βà—
ln(Xi ) βà’ Έ±1 1 βà—2
βà—
+Xi N + exp (βà’iœ„ œ„ )(1 βà’ Έ»œ„ ) βà’ exp Έ±2 + Έ≤2
f
βà—
Έ≤1 2
βà— βà—2 βà—
ln(Xi ) βà’ (Έ±2 + Έ≤2 ) ln(Xi ) βà’ Έ±2
Ο—N + Xi N (12.17)
βà— βà—
Έ≤2 Έ≤2

The parameters Έ» and k are respectively the probability and the magnitude of a jump.
The parameters Έ±iβà— and Έ≤iβà— correspond to:

œÉ2
Έ±iβà— = ln Ft + ln(1 + k) βà’ Έ»k + œ„ (12.18)
2
βàö
Έ≤iβà— = œÉ œ„ (12.19)

Input Taylor Strike prices Melick & Output
Thomas21

ATM
Volatility

Risk
Reversal

Strangle

OTC option Garman & Linear combination RND
volatilities Kohlhagen of 2 log-normal
(observed) (market convention) distributions

Figure 12.6 Estimation procedure of RND functions from option spreads
Source: Adapted from McCauley and Melick (1996).

21
Using a mixture of two log-normal distributions with no jumps or with up to one jump, following Merton
(1976), Bates (1991) or Malz (1996, 1997).
Portfolio Management and Information 359

The whole estimation procedure, including both RND estimation techniques, can then
be brieο¬‚y presented using Figure 12.6, based on that presented in McCauley and Melick
(1996).

12.4 MEASURES OF CORRELATION AND OPTION PRICES
The implied volatilities of options on cross exchange rates provide enough information to
compute implied correlations between the exchange rates involved. Assuming no arbitrage
opportunities, the exchange rate between currencies X and Y at time t, denoted by S1,t ,
may be written as:
S1,t = S2,t S3,t (12.20)

where S2,t and S3,t are the exchange rates, respectively, between X and a third currency
Z and between Z and Y .
Let si,t = ln(Si,t ), with i = 1, 2, 3. Thus:

s1,t = s2,t + s3,t (12.21)

Denoting the daily exchange rate variation si,t βà’ si,tβà’1 by vi,t , we have:

v1,t = v2,t + v3,t (12.22)

Let œÉi,t,T (with i = 1, 2, 3) be the standard deviation of daily returns over a period of
time from t to t + T and let Cov(v2,t,T , v3,t,T ) be the covariance between v2,t and v3,t
over the same period. The variance of v1,t from t to t + T is given by:

œÉ1,t,T = œÉ2,t,T + œÉ3,t,T + 2 Cov(v2,t,T , v3,t,T )
2 2 2
(12.23)

As Cov(v2,t,T , v3,t,T ) = œ¹t,T œÉ2,t,T œÉ3,t,T , where œ¹t,T is the correlation coefο¬¹cient between
v2,t and v3,t , solving equation (12.23) in order to œ¹t,T we get:

œÉ1,t,T βà’ œÉ2,t,T βà’ œÉ3,t,T
2 2 2
œ¹t,T = (12.24)
2œÉ2,t,T œÉ3,t,T

In equation (12.24), the correlation coefο¬¹cient between the daily returns of the exchange
rates of two currencies vis-` -vis a third currency may be obtained from the variance of
a
the daily returns of the exchange rates between the three currencies.22 Accordingly, when
t is the current time, it is possible to estimate at t the correlation between future daily
returns of two exchange rates, using forecasts of the variances of the daily returns of the
exchange rates between the three currencies:23

œÉ1,t,T βà’ œÉ2,t,T βà’ œÉ3,t,T
ΥÜ2 ΥÜ2 ΥÜ2
œ¹t,T =
ΥÜ (12.25)
ΥÜ ΥÜ
2œÉ2,t,T œÉ3,t,T
22
Notice that it is irrelevant how exchange rates are expressed, since the variance of the growth rate of a
variable is equal to the variance of the growth rate of its inverse.
The variances used are œÉ 2 (T βà’ t).
23
360 Applied Quantitative Methods for Trading and Investment

There are several ways to estimate the correlation coefο¬¹cient in equation (12.25), from
the information available at time t. The simplest way is to compute the historical corre-
lation over a window of t βà’ T days:
T
[(v2,tβà’T +j βà’ v 2,tβà’T ,T )(v3,tβà’T +j βà’ v 3,tβà’T ,T )]
ΥÜ
Cov(v2,tβà’T ,T , v3,tβà’T ,T )
j =0
œ¹t,T =
ΥÜ =
ΥÜ ΥÜ
œÉ2,tβà’T ,T œÉ3,tβà’T ,T
T T
(v2,tβà’T +j βà’ v 2,tβà’T ,T ) (v3,tβà’T +j βà’ v 3,tβà’T ,T )2
2

j =0 j =0
(12.26)
where v i corresponds to the average rates of return.
Instead of a single past correlation, a moving average of several past correlations can be
used as a forecast of future correlation, attaching equal or different weights to each past
correlation. Equally weighted moving averages have the disadvantage of taking longer to
reveal the impact of a shock to the market and to dissipate that impact.
One of the most currently used weighting techniques is the exponentially weighted
moving averages (EWMA), where higher weights are attached to most recent observations
in the computation of the standard deviations and the covariance.24 This methodology
because volatilities and correlations react promptly to shocks and have less memory, given
the higher weight attached to recent data.
Υ€
Using EWMA, the weighted returns (v) are obtained by pre-multiplying the matrix of
returns (v) by a diagonal matrix of weights (Έ»), as follows:
v = Έ»v
Υ€ (12.27)
with
οΘ® οΘΙ οΘ® οΘΙ
¬·¬·¬·
1 0 0 0 0 ¬·¬·¬· ¬·¬·¬·
βàö vt,1 vt,2 vt,3 vt,k
οΘ· οΘΚ
¬·¬·¬· οΘ· vtβà’1,1 ¬· ¬· ¬· vtβà’1,k οΘΚ
¬·¬·¬·
Έ» 0 0 0
οΘ· οΘΚ vtβà’1,2 vtβà’1,3
βàö οΘ· οΘΚ
οΘ· οΘΚ οΘ· vtβà’2,1 ¬· ¬· ¬· vtβà’2,k οΘΚ
¬·¬·¬·
¬·¬·¬·
οΘ· οΘΚ vtβà’2,2 vtβà’2,3
Έ»2 0 0 οΘ· οΘΚ
οΘ· οΘΚ οΘ· .οΘΚ
Έ»=οΘ· οΘΚ and v = οΘ· . . . . . .οΘΚ
..
οΘ· οΘΚ . . . . .
οΘ·. .οΘΚ
. . . . .
οΘ· οΘΚ
0
οΘ· οΘΚ
οΘ· οΘΚ οΘ·. .οΘΚ
. . . .
οΘ· οΘΚ
.
.. οΘ°. .οΘ»
. . . .
.
οΘ° οΘ»
. . . . . . .
.
βàö
¬·¬·¬· ¬· ¬· ¬· vtβà’T ,k
vtβà’T ,1 vtβà’T ,2 vtβà’T ,3
Έ»T βà’1

where 0 βâ¤ Έ» βâ¤ 1.
Each column of v corresponds to the returns of each asset price included in the portfolio,
while each row corresponds to the time at which the return occurred. Consequently,
standard deviations are calculated as:25
T
Έ»T βà’j (vi,tβà’T +j βà’ v i,tβà’T ,T )2
œÉi,t,T =
ΥÜ (1 βà’ Έ») (12.28)
j =1

with i = 2, 3.

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