2π

2

A round turn is de¬ned as reversing a position from one unit long to one unit short or vice versa.

Optimal Allocation of Trend-Following Rules 339

where

m’2 m’2

ρF = (m ’ i ’ 1)(m ’ i ’ 2) (m ’ i ’ 1)2 if m ≥ 2

i=0 i=0

Subsequently, the expected transaction costs will be E(TC) = ’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

’3.75

S(2) 125.00

’2.77

S(3) 92.52

’2.02

S(5) 67.40

’1.46

S(9) 48.62

’1.05

S(17) 34.92

’0.76

S(32) 25.46

’0.56

S(61) 18.62

’0.41

S(117) 13.68

a

Number of round turns assuming a year of 250 trading days.

140

USE-JPY

EUR-USD

USD-CAD

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 “MA order” for the X variable and

“Round turns” for the Y variable in the worksheet “MA”. 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

dif¬cult 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 diversi¬cation 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 dif¬cult

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, ρR , given by:

®

min(m1 ,m2 )’2

(m1 ’ i ’ 1)(m2 ’ i ’ 1)

2

ρR (m1 , m2 ) = arc sin

i=0

π

m1 ’2 m2 ’2

° 2»

(m1 ’ i ’ 1)2 (m2 ’ i ’ 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

ρ 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 coef¬cient 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 ¬‚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 USD“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 “MA order” for the X

variable and “Volatility” for the Y variable in the worksheet “MA”.

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 ρ(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 USD“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|/σ m

and the ¬rst three autocorrelation coef¬cients of the series drawn. We then regressed

the normalised drift and the ¬rst three autocorrelation coef¬cients 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

“Tools”, “Macro”, “Visual Basic Editor” and changing the line of code “For i = 1 To 10000” to “For i = 1

To 100”.

6

Figure 11.7 can be found in the “Sensitivity to Drift” workbook of Lequeux004.xls. Figure 11.8 can be found

in the “Sensitivity of Dependencies” 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

coef¬cient 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 coef¬cients of regression between returns of the moving average trading

rule and the ¬rst three autocorrelation coef¬cients of the underlying time series decrease

344 Applied Quantitative Methods for Trading and Investment

Table 11.6 Weighting of ρ 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/πσρe(’µf /2σf ) + µ(1 ’ 2 [’µf /σf ])

E(rt+1 ) =

where ρ = Corr(xt+1 , ft ), is the cumulative function of N (0,1), µ is the expected

value and σ is the standard deviation of the underlying return xt . The expected value and

standard deviation of the forecaster ft are given by µf and σf . In the case of a random

walk with drift model the above formula will simplify as E(rt+1 ) = µ(1 ’ 2 [’µf /σf ]).

In this case the expected return is a negative function of the volatility. In the special case

when there is no drift the expected pro¬t is given by:

2

E(rt+1 ) = σ Corr(xt+1 , ft )

π

In our applied example we use an adjusted form of this proposition where ρ =

3

i=1 wi Corr(xt+1 , ft ), with wi the weight attributed as a function to the sensitivity of

the forecaster to autocorrelation coef¬cients. 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 mean“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 “Regressions Data” sheet of Lequeux004.xls, A59:I62.

8

The model can be found in the “Simulation” 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 “Performance Basket” 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 ¬ve. It also provides cash ¬‚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 ¬rst, 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

MA basket

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

EUR“USD USD“JPY GBP“USD USD“CAD AUD“USD

’2.01% ’1.73%

Return 1.56% 6.06% 3.77%

Volatility 6.77% 9.01% 5.24% 3.79% 7.64%

’0.38 ’0.46

IR 0.23 0.67 0.49

’11.27% ’11.13% ’13.35% ’16.85% ’10.64%

Maximum cumulative

drawdown

’1.66 ’1.23 ’2.55 ’4.45 ’1.39

Normalised maximum

cumulative drawdown

’2.11% ’3.16% ’1.36% ’1.54% ’3.53%

Maximum daily loss

Maximum daily pro¬t 1.96% 5.64% 1.48% 0.90% 2.65%

’0.78 ’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 mean“variance allocation model

EUR“USD USD“JPY GBP“USD USD“CAD AUD“USD

’0.47% ’0.98%

Return 10.29% 1.32% 10.73%

Volatility 9.97% 12.75% 7.58% 5.48% 11.05%

’0.05 ’0.18

IR 0.81 0.17 0.97

’16.52% ’13.54% ’16.17% ’15.34% ’18.68%

Maximum cumulative

drawdown

’1.66 ’1.06 ’2.13 ’2.80 ’1.69

Normalised maximum

cumulative drawdown

’2.67% ’3.22% ’1.49% ’1.61% ’4.72%

Maximum daily loss

Maximum daily pro¬t 2.58% 5.65% 2.15% 1.51% 2.65%

’0.19 ’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 ¬rst 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 ¬ve this would have provided

better economic value than using an equally weighted basket of trading rules. These results

are signi¬cant for active currency managers who seek to provide their investors with a

balanced risk/return pro¬le 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), “Expected Return of Technical Forecasters with an Application to Exchange Rates”,

Presentation at the International Conference on Forecasting Financial Markets: New Advances for

Exchange Rates and Stock Prices, 2“4 February 1994, London. Published in Advanced Trading

Rules, Acar & Satchell (eds), Butterworth-Heinemann, 1998.

Acar, E. and P. Lequeux (1995), “Trading Rules Pro¬ts and the Underlying Time Series Properties”,

Presentation at the First International Conference on High Frequency Data in Finance, Olsen and

Associates, Zurich, Switzerland, 29“31 March 1995. Forthcoming in P. Lequeux (ed.), Financial

Markets Tick by Tick, Wiley, London.

Acar, E. and P. Lequeux (1996), “Dynamic Strategies: A Correlation Study”, in C. Dunis (ed.),

Forecasting Financial Markets, Wiley, London, pp. 93“123.

Acar, E., P. Lequeux and C. Bertin (1994), “Tests de marche al´ atoire bas´ s sur la pro¬tabilit´ des

e e e

indicateurs techniques”, Analyse Financi` re, 4 82“86.

e

Arnott, R. D. and T. K. Pham (1993), “Tactical Currency Allocation”, Financial Analysts Journal,

Sept, 47“52.

Billingsley, R. and D. Chance (1996), “Bene¬ts and Limitations of Diversi¬cation among Com-

modity Trading Advisors”, The Journal of Portfolio Management, Fall, 65“80.

Kritzman, M. (1989), “Serial Dependence in Currency Returns: Investment Implications”, Journal

of Portfolio Management, Fall, 96“102.

LeBaron, B. (1991), “Technical Trading Rules and Regime Shifts in Foreign Exchange”, University

of Wisconsin, Social Science Research, Working Paper 9118.

LeBaron, B. (1992), “Do Moving Average Trading Rule Results Imply Nonlinearities in Foreign

Exchange Markets”, University of Wisconsin, Social Science Research, Working Paper 9222.

Lequeux, P. and E. Acar (1998), “A Dynamic Benchmark for Managed Currencies Funds”, Euro-

pean Journal of Finance, 4(4), 311“330.

Levich, R. M. and L. R. Thomas (1993), “The Signi¬cance of Technical Trading-Rule Pro¬ts in the

Foreign Exchange Market: A Bootstrap Approach”, Journal of International Money and Finance,

12, 451“474.

Schulmeister, S. (1988), “Currency Speculations and Dollar Fluctuations”, Banco Nationale del

Lavaro, Quarterly Review, 167 (Dec), 343“366.

Silber, L. W. (1994), “Technical Trading: When it Works and When it Doesn™t”, The Journal of

Derivatives, 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 Risk

or Evidence of Inef¬cient Pricing?”, Economic Record (Suppl.), 68, 105“116.

Taylor, S. J. (1990b), “Pro¬table Currency Futures Trading: A Comparison of Technical and Time-

Series Trading Rules”, in L. R. Thomas (ed), The Currency Hedging Debate, IFR Publishing,

London, pp. 203“239.

Taylor, S. J. (1992), “Ef¬ciency of the Yen Futures Market at the Chicago Mercantile Exchange”,

in B. A. Goss (ed.), Rational Expectations and Ef¬cient Future Markets, Routledge, London,

pp. 109“128.

Taylor, S. J. (1994), “Trading Futures Using the Channel Rule: A Study of the Predictive Power

of 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 from

Over-the-Counter Currency Options—

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 ¬rst 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 ¬les, 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 ¬nancial 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

—

This chapter contains material included in the PhD thesis of the author (Lu´s, 2001).

±

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

±

™ 2003 John Wiley & Sons, Ltd ISBN: 0-470-84885-5

350 Applied Quantitative Methods for Trading and Investment

comprises Black“Scholes implied volatilities for at-the-money forward options, as well

as prices for some option spreads, such as risk-reversals and straddles.

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 ¬xed terms to maturity, instead of ¬xed maturity dates,3

the RND functions don™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 dollar“German Mark and the US dollar“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 RiskMetrics 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 ef¬cient

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 ¬fth sections, respectively measures of

correlation and arbitrage-based credibility tests.

The sixth section contains the empirical applications: the ¬rst 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 ¬rst 22 months of its life, the euro depreciated nearly 30% vis-` -vis the US dollar.

a

This movement was also characterised by signi¬cant 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 ¬le “chart1.xls”). Two Excel ¬les

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

(“OTC EUR USD”) and another related to the RND functions estimated allowing one

discrete jump to the underlying asset (“OTC EUR USD jump”).

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 ¬les “correlations PTE database” and “correla-

tions DKK database”, 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 ¬les are supplied for this application: “credibility tests DKK 1m” and “credibil-

ity tests DKK 1year” regarding the Danish crown and “credibility tests PTE 3m” con-

cerning the Portuguese escudo. Finally, the seventh section concludes.

As illustrated in Figure 12.2 (which can be found in the ¬le “credibility tests PTE 3m.

xls”), 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 ¬lled, 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 ¬le

“credibility tests DKK 1 year.xls”) 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 re¬‚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 δ = 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 Portuguˆ s do e

7

Atlˆ 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 (“BBA Currency Option Volatility Fix-

ings”). 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, ¬nancial institutions quote OTC options in implied

volatilities (vols), as annual percentages, which are translated to monetary values using

the Garman“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 (’i„ „ )[F N (d1 ) ’ XN (d2 )]

f

(12.1)

P = exp (’i„ „ )[XN (’d2 ) ’ F N (’d1 )]

f

(12.2)

√ √

where d1 = [ln(F /X) + (σ 2 /2)„ ]/σ „ , d2 = d1 ’ σ „ , 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

f—

f

rate,10 i„ is the „ -maturity domestic risk-free interest rate, i„ is the „ -maturity foreign

risk-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, instead

of monetary values. Moneyness is usually measured by the option delta (δ), which is

the ¬rst derivative of the option price in order to the underlying asset price.11 Following

equations (12.1) and (12.2), the delta values are:

‚C(X)

= exp (’i„ — „ )N (d1 )

δC = f

(12.3)

‚S

‚P (X)

= ’ exp (’i„ — „ )N (’d1 )

δP = f

(12.4)

‚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 Black“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.

f—

f

According to the covered interest rate parity, F corresponds to S exp [(i„ ’ i„ )„ ].

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 (δC ) is positive, while the delta of a put-option (δP ) 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 δ = 0.25. For simpli¬cation, the volatility of the

call-option with δ = 0.75 is regularly used as a proxy for the volatility of the put-option

with 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 = 0.25

and δ = 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 identi¬ed 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 δ = 0.25 and their prices

are de¬ned as the difference (in vols) to a reference volatility, frequently the at-the-money

f— f—

In fact, according to equations (12.3) and (12.4), δP = δC ’ exp (’i„ „ ). With δ = 0.25, δC = exp (’i„ „ ) ’

12

0.25. For short-term options, the ¬rst component of the right-hand side of the previous expression is close to

1, unless the foreign interest rate is signi¬cantly high. Thus δC = 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(σt0.25δ + σt0.75δ ) ’ atmt (12.6)

where atmt is the implied volatility of the at-the-money forward option. The market

participants consider δ = 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

about the smile curvature.

12.3 RND ESTIMATION USING OPTION SPREADS

Using risk-reversal and strangle prices with δ = 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

σt0.25δ = atmt + strt + 0.5rrt (12.7)

σt0.75δ = atmt + strt ’ 0.5rrt (12.8)

f—

From equation (12.3), it is easy to conclude that the delta of an at-the-money option is δt0.5δ = exp (’i„ )N

14

√

[σ „ /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 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 speci¬cation as

a proxy for the volatility curve. Malz (1997) uses the following quadratic polynomial

function 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 ¬t perfectly the three observed points in the delta-volatility

space (σt0.25δ , atmt and σt0.75δ ), we get:16

σtδ (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 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 ¬nal 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

ˆ ˆ

[C(Xi , „ ) ’ Ci0 ] + [P (Xi , „ ) ’ Pi0 ]2

min (12.11)

±1 ,±2 ,β1 ,β2 ,θ

i=1 i=1

s.t. β1 , β2 > 0 and 0 ¤ θ ¤ 1

which solves to provide solutions for C and P of:

∞

ˆ

C(Xi , „ ) = exp (’i„ „ ) [θ L(±1 , β1 ; ST ) + (1 ’ θ )L(±2 , β2 ; ST )](ST ’ Xi )dST

f

Xi

’ ln(Xi ) + (±1 + β1 ) ’ ln(Xi ) + ±1

2

12

= exp ±1 + β1 N ’ Xi N

f

exp (’i„ „ )θ

β1 β1

2

’ ln(Xi ) + (±2 + β2 )

2

12

= exp (’i„ „ )(1 ’ θ ) exp ±2 + β2 N

f

β2

2

’ ln(Xi ) + ±2

’Xi N (12.12)

β2

16

Replacing 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)).

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

ˆ

P (Xi , „ ) = exp (’i„ „ ) [θ L(±1 , β1 ; ST ) + (1 ’ θ)L(±2 , β2 ; ST )](Xi ’ ST )dST

f

’∞

ln(Xi ) ’ (±1 + β1 ) ln(Xi ) ’ ±1

2

12

= exp (’i„ „ )θ ’ exp ±1 + β1 N + Xi N

f

β1 β1

2

ln(Xi ) ’ (±2 + β2 ) ln(Xi ) ’ ±2

2

12

= ’ θ ) ’ exp ±2 + β2 N + Xi N

f

exp (’i„ „ )(1

β2 β2

2

(12.13)

and where L(±i , βi ; ST ) is the log-normal density function i (i = 1, 2), the parameters ±1

and ±2 are the means of the respective normal distributions, β1 and β2 are the standard

deviations of the latter and θ the weight attached to each distribution. The expressions

for ±i and βi are the following:

σi2

±i = ln Ft + µi ’ „ (12.14)

2

√

βi = σ i „ (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 dif¬culties, it offers some

advantages, as it is suf¬ciently ¬‚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 simpli¬ed 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 Black“Scholes (1973) formula given a jump and

the Black“Scholes (1973) function value with no jump.

Option price equations considering a discrete jump correspond to:

∞

— — — —

C(Xi , „ ) = exp (’i„ „ ) (»„ L(±1 , β1 ; ST ) + (1 ’ »„ )L(±2 , β2 ; ST ))(ST ’ Xi )dST

f

ˆ

Xi

— —2

’ ln(Xi ) + (±1 + β1 )

1 —2

—

= exp (’i„ „ )»„ exp ±1 + β1 ’ ln(1 + k) (1 + k)N

f

—

β1

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

—

’ ln(Xi ) + ±1 1 —2

—

’Xi N + exp (’i„ „ )(1 ’ »„ ) exp ±2 + β2

f

—

β1 2

— —2 —

’ ln(Xi ) + (±2 + β2 ) ’ ln(Xi ) + ±2

—N ’ Xi N (12.16)

— —

β2 β2

Xi

— — — —

P (Xi , „ ) = exp (’i„ „ ) (»„ L(±1 , β1 ; ST ) + (1 ’ »„ )L(±2 , β2 ; ST ))(Xi ’ ST )dST

f

ˆ

’∞

— —2

ln(Xi ) ’ (±1 + β1 )

1 —2

—

= ’ exp + β1 ’ ln(1 + k) (1 + k)N

f

exp (’i„ „ )»„ ±1 —

β1

2

—

ln(Xi ) ’ ±1 1 —2

—

+Xi N + exp (’i„ „ )(1 ’ »„ ) ’ exp ±2 + β2

f

—

β1 2

— —2 —

ln(Xi ) ’ (±2 + β2 ) ln(Xi ) ’ ±2

—N + Xi N (12.17)

— —

β2 β2

The parameters » and k are respectively the probability and the magnitude of a jump.

The parameters ±i— and βi— correspond to:

σ2

±i— = ln Ft + ln(1 + k) ’ »k + „ (12.18)

2

√

βi— = σ „ (12.19)

Input Taylor Strike prices Melick & Output

Thomas21

expansion and premiums

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 brie¬‚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 ’ si,t’1 by vi,t , we have:

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

Let σi,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:

σ1,t,T = σ2,t,T + σ3,t,T + 2 Cov(v2,t,T , v3,t,T )

2 2 2

(12.23)

As Cov(v2,t,T , v3,t,T ) = ρt,T σ2,t,T σ3,t,T , where ρt,T is the correlation coef¬cient between

v2,t and v3,t , solving equation (12.23) in order to ρt,T we get:

σ1,t,T ’ σ2,t,T ’ σ3,t,T

2 2 2

ρt,T = (12.24)

2σ2,t,T σ3,t,T

In equation (12.24), the correlation coef¬cient 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

σ1,t,T ’ σ2,t,T ’ σ3,t,T

ˆ2 ˆ2 ˆ2

ρt,T =

ˆ (12.25)

ˆ ˆ

2σ2,t,T σ3,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 σ 2 (T ’ t).

23

360 Applied Quantitative Methods for Trading and Investment

There are several ways to estimate the correlation coef¬cient 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 ’ T days:

T

[(v2,t’T +j ’ v 2,t’T ,T )(v3,t’T +j ’ v 3,t’T ,T )]

ˆ

Cov(v2,t’T ,T , v3,t’T ,T )

j =0

ρt,T =

ˆ =

ˆ ˆ

σ2,t’T ,T σ3,t’T ,T

T T

(v2,t’T +j ’ v 2,t’T ,T ) (v3,t’T +j ’ v 3,t’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

offers some advantages over the traditional equally weighted moving averages, namely

because volatilities and correlations react promptly to shocks and have less memory, given

the higher weight attached to recent data.

˜

Using EWMA, the weighted returns (v) are obtained by pre-multiplying the matrix of

returns (v) by a diagonal matrix of weights (»), as follows:

v = »v

˜ (12.27)

with

® ®

···

1 0 0 0 0 ··· ···

√ vt,1 vt,2 vt,3 vt,k

··· vt’1,1 · · · vt’1,k

···

» 0 0 0

vt’1,2 vt’1,3

√

vt’2,1 · · · vt’2,k

···

···

vt’2,2 vt’2,3

»2 0 0

.

»= and v = . . . . . .

..

. . . . .

. .

. . . . .

0

. .

. . . .

.

.. °. .»

. . . .

.

° »

. . . . . . .

.

√

··· · · · vt’T ,k

vt’T ,1 vt’T ,2 vt’T ,3

»T ’1

where 0 ¤ » ¤ 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

»T ’j (vi,t’T +j ’ v i,t’T ,T )2

σi,t,T =

ˆ (1 ’ ») (12.28)

j =1

with i = 2, 3.