not themathematics by which prices are arrived at, but the assumptions underpinning the applicationof each theory. The late Professor Fischer Black himself wrote ‘I sometimes wonder whypeople still use the Black–Scholes formula, since it is based on such simple assumptions –unrealistically simple assumptions’.This is not a trivial point. In many areas of scientiﬁc enquiry, simplifying assumptions areroutinely made to limit the degrees of freedom or the complexity of the hypotheses. Howeverin ﬁnancial markets, such assumptions are not ‘simplifying assumptions’. ‘Assuming’ thatmarkets are lognormal is a very powerful statement. If it is true, then from this a large numberof signiﬁcant results ﬂow, including the maths of option pricing. ‘Assuming’ that marketsare costless to trade in continuous time is another very powerful statement, and it materiallyaffects the construction of option pricing theory. It may be clever maths to use this assumptionto ‘prove’ that there should be no risk premium in option prices, but if the assumptions aregrossly violated, is this a useful proof?Professor Black should probably not have been so surprised – or rather his surprise mighthave been redirected. The Black–Scholes model has become so popular – indeed the marketstandard – because it is elegant and logical, is easily calculable (particularly with moderncomputers), and because it validates and stands together with delta hedging and so is practicallyuseable. Everyone who buys, trades or prices options knows (or should know) that many ofthe model assumptions are violated in real markets, but they feel more able to cope with theseviolations from the safe haven of the model. Abandon the model, they argue, and they are onceagain adrift with no reference points.3.3.4.1 Implied volatilitiesThere is an interesting phenomenon which reﬂects this ‘measured acceptance’ of the basicBlack–Scholes methodology, ‘implied volatilities’. As I mentioned above, for a ‘plain vanilla’option, the only market variables needed to fully price an option are (i) its maturity, (ii) itsin- or out-of-the-moneyness, (iii) the interest rate, and (iv) the volatility. (i) to (iii) are knownvariables – they are fact. Only (iv) is not directly observable. What is observable, however, isthe option price quoted in the market. This means that (iv) can be reverse-engineered, i.e. theterms of the option pricing equation can be rearranged to calculate the volatility that wouldhave produced the option price quoted. The value so produced is called the ‘implied volatility’.Implied volatilities can be calculated for every speciﬁcation of option. In the perfect worldof the Black–Scholes assumptions, implied volatilities would not change (since Black and
Currency Hedging
47
Scholes assume that volatility is constant, and they would be independent of other variables,like in- and out-of-the-moneyness. In the world of real markets, however, implied volatilitiesare so volatile that many option traders, if asked, would claim that their principal job is to ‘tradeimplied volatilities’. Not only do they move up and down in line with the short-term volatilityexpectations of the market, but there are ‘bid’ and ‘offer’ implied volatilities (depending onwhether the option is bid or offered by the market), and changing implied volatilities dependenton the in- and out-of-the-moneyness of the option.Figure 3.5 shows the history of implied volatilities for USD/EUR, 3m at-the-money options.It illustrates the distinct lack of stability in implied volatilities.18.00%
Implied volatility
16.00%
14.00%
12.00%
10.00%
8.00%
6.00%1997
1998
1999
2000
2001
2002
Figure 3.5 History of implied volatilities – USD/EUR (formerly DEM) options; 97–02; 3m maturity;offer price
Figure 3.6 shows a snapshot of the implied volatilities for one speciﬁcation of option, butgraphed against the extent to which it is in- or out-of-the-money. The fact that it shows a strong12.00%
11.50%
11.00%
10.50%
10.00%
9.50%−10%
−8%
−6%
Out-of-the-money
−4%
−2%
0%
2%
% In-/Out-of-the-money
4%
6%
8%
10%
In-the-money
Figure 3.6 Implied volatilities for USD/EUR option versus in- or out-of-the-moneyness – euro call;6m maturity; 18 Nov 02 quotes
48
Currency Overlay
skew represents in part the scepticism of the market about the lognormality assumption. Theskew implies that either (a) extreme movements are more likely to happen than implied by thelognormal curve or (b) low time-value options (particularly very out-of-the-money) are moredifﬁcult, and more risky to hedge. Both of these possibilities are explicitly recognising failuresin the Black–Scholes assumptions, and instead of adjusting or abandoning the model, they areadjusting one of the input variables. It is a bit like aiming a gun with skewed sights left of thetarget (rather than adjusting the sights) in an attempt to hit the target!3.3.5 Practical implications of assumptions violationsOne of the key issues that arises when discussing the Black–Scholes assumptions is ‘what isthe effect on expected return’? Remember that in the pure Black–Scholes world, the expectedPV of all options is zero (and therefore it is irrelevant from a return perspective whether youare a buyer or a seller, or whether you buy or sell any particular currency against any other).In a world where these assumptions are routinely and grossly violated, what is the expectedPV of an option?For options on most underlying markets (equities, commodities), the consistent buyer ofoptions would have lost money over time. One would have thought therefore that, by analogy,the option writer would have made money. This may, however, not have been true since mostprofessional option writers will hedge their exposure using a dynamic (i.e. delta) hedge in theunderlying market. If the average options premiums charged were lower than the frictionalcosts incurred by the option writer together with the calls on the writer by the option buyers,then it is possible that both could lose money. Those option writers that write options withoutmatching them with a dynamic hedge will have a clearer run at proﬁtability, but will suffervery high risk in periods of high volatility. For most ﬁnancial houses, the risk/return trade-offof uncovered option writing is unattractive. Historical option pricing, and its effect on theproﬁtability of options, is looked at in more detail in Chapter 10.This brings me to a diversion which is brieﬂy worth a mention. Although uncommon in thewriting of currency options, it is common in the writing of equity options for an option-tradingﬁnancial house to persuade an institutional investor to write what are termed ‘covered calls’.These are call options on single equities or indices, but in the eyes of the institution they are‘covered’ by their ownership of the underlying security. If the equity in question remains stableor falls, the call will not be exercised, and they will keep the option premium. If the equityprice is strong, they will be called on the option they sold, and they will have to surrenderthe proﬁts they would otherwise have made in that period from their ownership of the equity.Their compensation will again be the option premium. As a long-term strategy this will beproﬁtable if the implied volatilities are higher than the actual volatilities, and the measurementof volatilities accurately reﬂects the market movements at the relevant horizon. I will comeback to this point in a discussion of trends in Chapter 9.As a formal risk-management exercise, ‘covered call’ writing leaves a lot to be desired. It ishighly unlikely that the investing institution will be in a position to judge whether the impliedvolatilities are attractive (and higher than the long-term historical volatility); they are highlyunlikely to be minimising the risk of option writing by matching the delta of the option withtheir holding, and they are unlikely to possess the analytical tools to determine whether writingoptions is diversifying or concentrating versus their liabilities.In theory investing institutions with foreign currency exposure could, by analogy, write‘covered call’ on foreign currencies. That they are more likely to do the opposite – hiring active currency overlay managers whose proﬁts proﬁle is option-like (from an option
Currency Hedging
49
buyers’ perspective) – is testament to the rather ill-thought-out nature of ‘covered call’ optionwriting.
3.4 CURRENCY OPTION PRICING HISTORYCurrency options have been available since November 1982. The ﬁrst options were Americanstyle options traded on the Philadelphia Stock Exchange. In the subsequent two or threeyears, a substantial over-the-counter market developed in the London interbank market andelsewhere.A full and detailed analysis of historical option premium levels is beyond the scope of thisbook, but broadly speaking, option premiums in the period 1985–1992 were generally at levelsat or below the actual average payoffs in the market, particularly for longer-maturity options.That is, over this period, passive buyers of currency options would have made average returnson their option premiums signiﬁcantly higher than the risk-free rate of interest. Of course, thepayoffs of options are highly variable, and can frequently be zero. So consistent buyers ofoptions would ideally have had pre-existing currency exposure which option payoffs would bea hedge for. This is a highly unusual observation, and will not have been generally appreciatedby the currency option community (although a number of banks entered the currency optionmarket in this period as writers, and subsequently withdrew nursing losses).In the period subsequent to 1992, it has been much less easy to make generalisations aboutaverage option premium levels. However, there remains in the currency market no obviousevidence that long-term writers of currency options are indulging in a proﬁtable exercise.3.4.1 LognormalityIn Appendix 2 we discuss lognormality in some detail, particularly in relation to the calculationof the volatility of a series of returns. Lognormality (i.e. the assumption that market prices froma lognormal random walk) represents one of the key assumptions in the construction of optionpricing models. It is worth noting that lognormality is not entirely consistent with returns aswe normally calculate them, unless we operate a very rigorous ‘translation’ process betweenthe ‘log world’ and the world we actually observe.The principle of lognormality is the assumption that the ‘linking’ variable, linking oneperiod’s security (or currency) value with the next, is multiplicative. This is best (and accurately)represented by converting the price into its natural log. The arithmetical differences betweenthese sequential log prices are then normally distributed. Exactly the same result can be achievedby converting the percentage returns of sequential prices into logs by R = ln(1 + r ), where ris the percentage return and R is the normally distributed variable.The following are variables in the Black–Scholes model:C = S N (d1 ) − K exp(−r t)N (d2 ),where:C = theoretical call premiumt = time until expirationr = risk-free interest rateexp(−r t) = exponential function = e−r tln(S/K ) + (r + (s 2 /2)td1 =√√ s td2 = d1 − s t
s = standard deviation of log returnsS = current stock priceK = option strike priceN = cumulative standard normal distribution
50
Currency Overlay
When working in the ‘log world’, it has to be dealt with consistently. So, for instance, is therisk-free interest rate, r , the IRR annual return (the convention used in investment management), or is it a continuous rate of return ln(1 + r )? Is the standard deviation expressed as astandard deviation of percentage returns, or of additive changes in the log price? Or exp(SD(logchanges))−1? These questions have to be carefully considered when using the Black–Scholesor any other model.
3.4.2 Monte Carlo modelsThere is one other major class of models based on numerical analysis. The principal representatives of this class are Monte Carlo models. These take a very different approach to pricing,which can crudely be characterised as ‘let’s try it and see’. (Incidentally, they are called MonteCarlo because the Principality is a major gaming centre – the epitome of testing distributionsby repeated sampling!) Generally used by banks and ﬁnancial engineers to price more intractable or complex option instruments, Monte Carlo models take an externally determinedprice distribution (e.g. lognormal, lognormal+jump, binomial or whatever), and calculate oneoutcome of the option (i.e. whether it matures in- or out-of-the-money, and if in-the-moneyhow big the payoff is) on the basis of a randomly generated price series. This process is thenrepeated (several thousands or tens of thousands of times) to generate a wide range of possibleoutcomes. The average of these outcomes (appropriately discounted if necessary) is then the‘fair value’ of the premium.Monte Carlo techniques are appealing where there are non-standard price distribution assumptions, where the option instrument is path-dependent or has other complexities, and wherethe model user wishes to ‘stress test’ the option instrument and examine the range and varietyof outcomes. The disadvantage of Monte Carlo techniques is that they are a sampling processwhose accuracy increases asymptotically with the number of simulations run. This can be avery heavy computing requirement for on-line pricing. Indeed some banks use Monte Carlo inregular batch runs, and then create ‘look-up’ tables with the results to get instant on-line quotes.
3.4.3 CostsThe premium of an option is not its ‘cost’. In the Black–Scholes model world, the optionpremium equals the discounted average payoff value, and so neither buyer nor seller expect togain on average over time. However, there has developed a particular approach to pricing thecost of an option which merits a brief discussion – that of implied volatility.If we take an accepted model for the pricing of a currency option – say the Black–Scholesmethodology with adjustments for the forward contract nature of the underlying – then withonly four input values we can calculate the fair price. These inputs are:(a) Maturity date(b) % In- or out-of-the-money. This represents the difference between the current market priceand the strike price. An in-the-money call option has a strike price lower than the currentmarket price, and vice versa. In currency options, it is usual to measure the strike rateversus the forward market rate, rather than the spot(c) Market volatility – we shall come back to this(d) Interest rate for the option maturity period
Currency Hedging
51
With this information, we can price an option fully. Moreover, all but one of these values aredeﬁnitive, known quantities. However one value, volatility, is not known. Its historical valuecan be estimated, there is a pool of information about the volatilities used to price other options,but it cannot be directly observed. What happens in practice is a little strange to the outsideobserver, and it is this. The market price at which options are offered or bid is recorded. Thenthe Black–Scholes calculations are ‘put into reverse’, and the level of volatility that wouldgenerate the observed premiums (all other factors remaining the same) is calculated. This canbe done for both the offer bid and the bid price of the options, and these produce a value calledthe implied volatility – bid implied volatility and offered implied volatility.Implied volatility is an excellent way of covering up the shortcomings of the market pricingmechanism, since it cheerfully reports different values for buying and selling options, differentvalues for options that are heavily in- or out-of-the-money, and so on.3.4.4 SensitivityCurrency options are not straight-line instruments. Their sensitivity to exchange rates forms acurve, the slope of which is the ‘delta’ of an option. The typical curve looks as in Figure 3.7. This‘convex curve’ characteristic of currency options gives them particular features in the hedgingof currency exposure which we shall consider in later chapters. In particular, the ‘delta’ (theslope of the option valuation) will tend towards 50% over the long-term (assuming that optionsare bought at-the-money). This means that the risk-reducing properties of an overlay portfolioconsisting of options are close to those of a passive overlay portfolio with a 50% hedge ratio.14%
% Principal – USD base
12%10%8%6%Option valuation
4%
Intrinsic value ofoption
2%0%1.30
1.40Out-of-the-money
1.50$/£ Rate
1.60
1.70
In-the-money
Figure 3.7 Option model prices % – $/£ option; £ call; strike = 1.50
3.5 INTEREST RATES AND FORWARD CURRENCY RATESOne of the distinguishing features of the currency market is its intimate relationship withthe interest rate market. In particular, forward currency rates are not, except in very special
52
Currency Overlay
circumstances, ‘forecasts’ of future exchange rates. Instead, they are a mathematical calculationbased on the currency spot rates, and the respective interest rates of the two currencies inquestion. See Box 3.7 for descriptive details.
Box 3.7
Forward currency rates
Forward currency exchange rates are not guesswork by the market-makers, but the resultof a two-way arbitrage with the respective interest rates of the two currencies in question.Let us take an example. Suppose the US dollar has interest rates at a 1-month horizon of21/8 %/21/16 %, and the UK has interest rates of 5%/415/16 %, and the spot rate is 1.5345/47.In the interest rate quotes, the ﬁrst number is the interest rate to borrow money (LIBOR)from the interbank market, the second the price to lend money (LIBID). In the exchangerate, the LHS (1.5345) is the rate to buy dollars, the RHS is the rate to sell dollars, and therate is the amount of dollars per sterling.Assuming large amounts of money can be borrowed and lent by a creditworthy bankat these rates, then these rates determine the no-arbitrage boundary rates for the forwardrates. The arbitrage works in two stages like this. Suppose a bank has £10m to depositfor one month. It can deposit in £ LIBID at 415/16 %, and have £10 041 146 in one month.Alternatively it can convert its cash into dollars, simultaneously sell the dollars 1-monthforward, and deposit the dollars at $ LIBOR. The forward rate at which this exercise isexactly cash equivalent neutral is 36.6 FX points less than the spot rate. (The main textshows the maths workings.) An FX point is 1/10 000th of a $, so if the two deals are doneat 1.5345 and 1.53104 then the £ cash at one month exactly matches. The calculation isultimately one of interest rate ratios [forward rate = spot rate × ((1 + Int$)/(1 + Int£))]rather than, as commonly supposed, interest rate differentials.If the RHS dollar swap points rise (to say −37), then UK investors will be paid to investin dollars via $ LIBID and the forward rate. If the forward swap rate falls (to say −36),then US investors will be paid to invest in pounds via £ LIBID. Forward points are subjectto bid/offer spreads (say −37/−36), and this will raise the hurdles to arbitrage. The secondlevel of arbitrage is where cash-neutral investors can ‘round trip’ proﬁtably. This meansborrowing in sterling, converting spot, lending in dollars, and selling forward (vice versa).The boundaries of this arbitrage (which has to overcome the LIBOR/LIBID spreads shown)are −37.4 and −35.8 FX swap points.
The inset box takes a particular example of 1-month forward $/£. To recap, the $/£ forwardrate is constrained by the ability of arbitrageurs to replicate a sterling one-month investmentby means of conversion into dollars, investing in $ LIBID, and selling the resulting dollarsforward to completely eliminate currency risk. We can examine the maths of this arbitrage inthe ‘perfect world’, and also in the real world of LIBID/LIBOR, spot and forward spreads.Similarly, the constraint applies in the other direction: arbitrageurs can replicate a dollar onemonth investment by means of conversion into sterling, investing in £ LIBID, and sellingthe resulting sterling forward. Finally, any market participant could borrow one currency anddeposit the other, which sets the outer bounds of the arbitrage. The boundaries of the arbitrageare fully described in Appendix 1.
Currency Hedging
53
3.6 CURRENCY SURPRISE3.6.1 What is currency surprise?Currency surprise (see Box 2.2) is an expression ﬁrst coined relatively recently, but now incommon use in the currency overlay industry.7 It represents a measure for the unanticipateddeviation of the spot rate from the rates ‘predicted’ by past forward rates.One of the difﬁculties that investors have faced as they internationalised their portfolios wasto get adequate information about what their currency returns were. Custodians reporting oninternational portfolios quite naturally used their standard ‘mark-to-market’ technology forvaluation. This was based on two valuation elements – the market valuation expressed in local(i.e. foreign) currency, and the spot rate to convert the local currency to GBP (or whateverthe base currency was). This naturally threw up an attribution split – the local market returnsand the currency returns (i.e. spot-to-spot). Unfortunately, this split has systematically biasedthe currency returns reported from those that are investable – because spot currency is notR) issued ainvestable. The Association for Investment Management and Research (AIMR8paper on benchmarks in 1998 which discusses this issue clearly and in some detail.These investable elements are (a) currency return = currency surprise and (b) foreign assetreturns = hedged asset returns expressed in the home currency (note not ‘local’ returns).3.6.2 Currency surprise calculationCurrency surprise (Ct ) for a foreign currency expressed as a percentage in period t is:(Foreign currency spot ratet ) − (Foreign currency forward ratet−1 )Ct =× 100%Foreign currency spot ratet−1Assume for this example that the investor has a sterling base currency, so:Foreign currency spot ratet−1= GBP/foreign currency at the end of period tForeign currency forward ratet−1 = GBP/foreign currency forward rate at the end ofperiod t − 1 for delivery at the end of period tMost exchange rates are quoted foreign currency/GBP, so for currency surprise (expressed inthis way) their reciprocals are used.The denominator is the spot rate at the end of the prior period. While there is no fundamentalnecessity for this to be the case, there is a strong case that this is the best approach. This isbecause currency surprise has been created as a concept because it is an investible element in aportfolio. Investible means, as we have seen, also dis-investible. The mechanics of disinvestingin currency surprise for those investors with international assets requires currency hedging. Itis computationally convenient if currency surprise and ‘contribution from hedging’ are exactlythe same values, but with opposite signs. It also means that the ‘contribution from hedging’ canbe combined with other asset returns within periods by addition. We have already discussedabove the attraction of having currency hedging returns additive to unhedged returns of theunderlying asset, and this requires the spot rate at the end of the prior period as the denominator.Hence it is adopted here for currency surprise.7I ﬁrst came across the expression coined by Grant W. Gardner in a 1994 Russell working paper. I immediately liked its descriptivepower, so enthusiastically adopted it. Its extensive use in the AIMR paper cited below is largely as a result of my intervention.8AIMR Benchmarks and Performance Attribution Subcommittee Report, John C. Stannard et al., August 1998. This paper canbe viewed on AIMR’s website at www.aimr.com/standards/pps/benchmark.html., and is reproduced here as Appendix 3.
54
Currency Overlay
3.6.3 Why not spot returns?Many investors measure their currency returns by reference to the movements in the spotexchange rates. If this is wrong, why is it wrong?It is wrong because spot rates are ‘uninvestible’. Let’s take the example of a UK institutionalinvestor buying $10m worth of US equities at the end of period 0. The values in Table 3.9apply for the subsequent period (I ignore bid/offer spreads and commissions for simplicity).Table 3.9 US equity returns for UK investorEndEquityperiod value ($m)(a)012345Cum.
10101110.511.511
Equity returns
Spot rate
(b) = (at /at−1 ) −1
(c)
0.00%10.00%−4.55%9.52%−4.35%10.00%
Spot returns
Combined equity &spot returns
(d) = (ct−1 /ct ) −1 (e) = [(at /ct ) /(at−1 /ct−1 )] −1
1.50001.40001.30001.35001.40001.4200
7.14%7.69%−3.70%−3.57%−1.41%5.63%
7.14%18.46%−8.08%5.61%−5.70%16.20%
Note: The algebra for column (c) is inverted because the exchange rate is expressed as dollar/sterling, i.e. the priceof sterling in dollar terms. In this context we need to calculate the price of the dollar in sterling terms, i.e. use thereciprocal.
It is very common for returns to be presented to investors in this way, and it is temptingfor investors to argue, faced with this information, that their being exposed to the US dollar,which was strong over this period, yielded 5.63% cumulatively over the period, and thereforewas proﬁtable. But it is only possible to say whether an investment or exposure is proﬁtableor not by reference to the alternative – in this case eliminating all exposure to the US dollar.Exposure to the dollar can only be eliminated by selling US dollars forward, let us say at aone-period horizon.We have to examine the forward rates applying at each period (Table 3.10) to assess theimpact of eliminating exposure using forward contracts.Table 3.10 Example spot & forward ratesEnd period
Spot rate
Forward rate
012345
1.50001.40001.30001.35001.40001.4200
1.47001.37001.27001.32001.37001.3900
We can calculate the proﬁt or loss, both in cash and as a percentage of the initial amounthedged, from settling the one-period forward currency contracts necessary to fully hedge theequity exposure. Table 3.11 illustrates this.If we add the % proﬁt and loss each period to the combined equity and spot (or unhedged)returns, we get the hedged returns as given in Table 3.12 (we can add the percentage returns
Currency Hedging
55
Table 3.11 Currency hedge proﬁt & lossEndperiod
012345Cum.
Equityvalue ($m)
Spot rate
Forward rate
Currency hedgeproﬁt/loss (£m)
Currency hedgeproﬁt/loss (%)
(a)
(b)
(c)
(d) = (at−1 /ct−1 ) − (at−1 /bt )
(e) = d/(at−1 /bt−1 )
10101110.511.511
1.50001.40001.30001.35001.40001.4200
1.47001.37001.27001.32001.37001.3900
−0.34−0.390.510.450.300.53
−5.10%−5.50%6.07%5.84%3.60%4.30%∗
∗ See Section 3.6.4.
Table 3.12 Hedged return calculationEndperiod
Equityvalue ($m)
Spotrate
Equityvalue (£m)
012345Cum.
10101110.511.511
1.50001.40001.30001.35001.40001.4200
6.677.148.467.788.217.75
Unhedged equityreturn (%)
Currency hedgeproﬁt/loss (%)
Hedged equityreturn (%)
7.14%18.46%−8.08%5.61%−5.70%16.20%
−5.10%−5.50%6.07%5.84%3.60%4.30%
2.04%12.96%−2.01%11.46%−2.10%23.24%
each period because they occur contemporaneously, and have the same denominator). Thecumulative 23.24% is the return we get from taking US equity exposure only, and no currencyrisk. This mechanism, using forward contracts to sell currency forward (or the equivalent), isthe only way to eliminate currency risk.This allows us to complete the circle. Currency surprise is the difference between the hedgedequity return and the unhedged equity return, and will therefore in this case be the negative ofthe currency hedge proﬁt and loss %. This gives us Table 3.13, with the equity and currencyTable 3.13 Correct attribution of equity & currency returns
End period012345Cum.
Unhedged equity return (%)
Currency surprise
Hedged equity return (%)
(a)
(b)
(c) = a − b
7.14%18.46%−8.08%5.61%−5.70%16.20%
5.10%5.50%−6.07%−5.84%−3.60%−5.46%
2.04%12.96%−2.01%11.46%−2.10%23.24%
56
Currency Overlay
components correctly divided into their investible parts. So we discover that, far from beinga strong currency, which contributed to investment returns, the dollar has in fact been a weakcurrency, which has detracted from investment returns.
Box 3.8
Local currency returns
It is universal in the casual reporting of international equity market behaviour to use thecurrency of the target country. This is always implicit, rather than explicit, because equitymarkets are expressed in index numbers, not currency prices. So when a commentatorsays that the ‘Nikkei is down 0.6% to 10 100’, the value and percentage change is alwaysexpressed in yen. However, for a non-Japanese investor, yen returns are unattainable. Theonly returns that are obtainable are (a) yen returns converted to the