ities at any time1212
36m15m
10
48m
100–200
12m
4 – except ZAR which ismonthly
12
5 & 50
60–125dependingon contract
24m
18m
4 – except MXN, ZARand BRL which aremonthly (BRL & RUBcash-settled)
Approx. lot(contract) size(USD’ 000)
12
Max.maturity
No. maturitiesper year
Average total daily volume (USD bn)
AUD/CAD, AUD/JPY, AUD,AUD/NZD, GBP/JPY, GBP/CHF,GBP, CAD/JPY, EUR/AUD,EUR/GBP, EUR/CAD, EUR/JPY,EUR/NOK, EUR/SEK, EUR/CHF,EUR, EUR (LGE), NZD, CHF/JPY,CAD, JPY, NOK, ZAR, SEK, CHFMXN
AUD/CAD, AUD/JPY, AUD/NZD,AUD, GBP/JPY, GBP/CHF, BRL,GBP, CAD, CAD/JPY, EUR/AUD,EUR/GBP, EUR/CAD, EUR/JPY,EUR/NOK, EUR/CHF, EUR/SEK,E-mini Euro FX (E7), E-mini JapaneseYen (J7), Euro FX (EUR), JPY, MXN,NZD, NOK, RUB, CHF/JPY, ZAR,SEK, CHFBRL, BRL Mini, EUR/BRL
Currency pairstraded (vs. USDunless stated)
Note: Other exchanges exist, for example the Korean Futures Exchange, Budapest Futures Exchange.
EuronextSAFE – South African FuturesExchangeTIFFE – Tokyo InternationalFinancial Futures Exchange
MexDer – Mexican DerivativesExchangeSydney Futures ExchangePBoT – Philadelphia Board ofTrade
BM&F – Brazilian & MercantileFutures ExchangeFINEX (NY Board of Trade –Financials)
CME – Chicago MercantileExchange
Futuresexchangename
Table 3.7 Futures exchanges offering currency futures contracts
11 706
<1
<1<1
3<1
3
644
2711
8345
Averagedaily volume(USD m)
Currency Hedging
37
then they will do so without a bid/offer spread. However, the price at which they deal may benearer the bid price in the interbank market, nearer the offer price, or in the middle. Exactlywhere it trades will depend on the dynamics of the market (which way it is moving, the pressureof other buyers and sellers), the respective elasticity of the buyer and seller, and the urgencyof their need to deal.The ‘simultaneous order’ is, however, a special case of the general case of a mixed selectionof varied customer orders. In most futures markets, brokers are appointed by the exchangeas ‘ofﬁcial’ market-makers – i.e. they are allocated the task of creating liquidity continuouslythroughout the trading day. They will be prepared to accept positions (i.e. act as buyer or seller)at a price which reﬂects the risk to them of taking the position. This is exactly analogous tothe market-maker role in the interbank market. This often means that they will actively quotea bid and offer price simultaneously which is, de facto, the bid/offer spread (in the absence ofnew customer bids or offers.Currency futures market bid/offer spreads are narrow in relation to other futures markets,reﬂecting the depth and liquidity of the underlying cash market.3.1.4.4 SensitivityCurrency futures contracts are straight-line instruments with respect to the exchange rate. Withthe exception of the cash effect of margins, their economic effect is identical to that of forwardcontracts.3.1.5 Currency options3.1.5.1 NatureCurrency options are highly complex instruments, despite their seemingly simple design.I therefore make no apologies for devoting some considerable attention to this section, inparticular, their pricing.Currency options are contracts which give the option buyer the right, but not the obligation,to take delivery of one currency in exchange for another at a ﬁxed exchange rate on an agreeddate in the future. By corollary, option sellers (writers) have the obligation, but not the right,to deliver one currency in exchange for another at a ﬁxed exchange rate on an agreed date inthe future. The ﬁxed exchange rate is called the ‘strike rate’, and there are four dates relevantin each option: the deal date; the date for premium payment (2 working days later); the expirydate (2 working days prior to the maturity date); and the maturity date, when the exchange ofprincipal amounts takes place.There are a huge variety of option types now on offer, but there are two key ‘plain vanilla’option types – European-style options and American-style options. European options can onlybe exercised on the expiry date (and mature only on the maturity date), while American optionscan be exercised by the option buyer at any time between the deal date and the expiry date.An early-exercised American option will also accelerate the maturity date, which will occurtwo days after exercise. European options are dominant in the OTC market, and increasinglydominant in the traded options market (see below). The subsequent discussion will concentrateon European-style options.Options are traded at a premium, payable by the option buyer to the option seller on thegrant of the option. Currency options can be traded in the interbank market, when they aregenerally called over-the-counter (OTC), and on futures exchanges, when they are called
38
Currency Overlay
‘traded options’. The principal distinguishing features of these two types of options are thesame as the distinguishing features of interbank forward contracts versus futures contracts.The former are bilateral contracts, the latter are contracts with an exchange.Box 3.5
Currency options
Currency options are contracts which give the option buyer the right, but not the obligation,to exchange two currencies at a ﬁxed rate (strike rate) on an agreed date in the future. Theytrade at a premium which reﬂects three principal elements: the relationship between thestrike rate and the current market level (in- or out-of-the-money); the implied volatility forthe currency market; the length of time remaining until the option matures.Currency options are very different instruments from forward contracts. Forward contracts represent only a transfer of economic currency ownership; options represent a morefundamental transfer of market risk. This is because the option buyer cannot lose any moremoney than the option premium he pays the option seller. However, he can make theoretically unlimited amounts of money if the underlying currency market moves in a particulardirection. If he has bought a ‘call’ on a currency, then he will proﬁt if that currency is strongagainst the other currency in the pair; if he has bought a ‘put’, then he will proﬁt if thatcurrency is weak against the other currency in the pair.The pricing of options has been the subject of intense academic debate, and by ‘pricing’I mean the premium at which market participants will willingly buy or sell options. Modernoption pricing is based on a seminal work by Black and Scholes in 1973, although theirwork must be considered a framework rather than a solution to the problem.The currency options market is so well developed that there is, in effect, a secondarymarket in which players can pit their expertise against each other – that of currency marketvolatility. Players wish (put crudely) to sell volatility high and buy low, and it is arguablethat this expertise might be quite separate from an ability to correctly predict the directionof the foreign exchange market.There is one distinguishing feature of OTC options that sets them apart from forward contracts – one party (the option buyer) does not need to be creditworthy. This is because optionbuyers pay the premium for the option up front, and once that premium has been paid, the option buyer has no further obligations, only rights. By contrast, an option writer (option granter,option seller) needs to have a solid credit rating. The option writer receives a cash premium atthe start of the option contract, and thereafter has a potentially unlimited liability to the optionbuyer, and no upside beyond the premium already received.In traded options, the credit arrangements are very similar to those with futures: both theoption writer and the option buyer are required to post initial margins; the option buyer’s initialmargin is the option premium; the option writer’s margin is, in effect, the option premium plusan amount to reﬂect the market risk the writer is running.
3.2 OPTION PRICINGThe theory of option pricing is a huge topic, and a full review is beyond the scope of this book(and the author’s expertise!). However, the key elements in pricing theory are within the book’sscope, and I have set them out below.
Currency Hedging
39
3.2.1 First principlesThe core principle in establishing a theory for pricing options is that the theoretical ‘fair’premium should be equal to the present value of the expected payoff of the option at maturity.If options trade at fair prices, then there will be no net transfer of value long-term betweenoption sellers and buyers.A fair price is not necessarily the price at which options will trade – that depends on theavailability of risk capital willing to take the risk of selling options, and the demand for buyingoptions from customers. Since option writers tend to be professional traders, why would anoption writer wish to write an option at a theoretically correct price when there is no expectationof proﬁt, but a certainty of risk? A commercial insurer will not write insurance unless hehas an expectation that the average losses will be less than the premium plus accumulatedinterest.Other things being equal, professional option writers will not write options without theexpectation of proﬁt, and that means they will write options at higher than ‘theoretical prices’(or their view of them). However, they do need to know what the theoretical price is, and also aframework for judging the expected proﬁt and the risk. A good model can provide all of this –although it is debatable, as we shall see, whether such a model exists. There are a numberof competing theories of option pricing, but all boil down to trying to establish the expectedpayoff of an option at maturity.
3.2.2 Option pricing theory3.2.2.1 Coin tossingLet’s take a simple example: coin tossing. Suppose that we score 1 point for heads and −1point for tails whenever we toss a coin. The coin is true, and it is tossed once a period. The‘market price’ is the sum of the points scored. We can (I hope with only mental arithmetic)devise and calculate the fair value for a one-period call at zero strike price and at one dollar perpoint. For simplicity, assume interest rates are zero. The answer is 50c. Why? Because there isa 50% chance that the score is minus one (when the option payoff is zero), and a 50% chancethat the score is plus one (when the option payoff is $1). (50% × $0) + (50% × $1) = 50c.What about a two-period option? Again the fair value is 50c: a 25% chance of two heads(payoff = $2), a 50% chance of one head/one tail (payoff = nil), and a 25% chance of twotails (payoff = nil). What about three-period? The answer turns out to be 75c, which the readermight like to verify himself.This simple model has a number of attractive features. Firstly, the model is intuitivelyplausible – that is we believe it accurately describes a known physical process. Secondly, theunderlying distribution (i.e. heads/tails) is known with conﬁdence. Thirdly, the model makesfew assumptions, and those that it does make can be tested with some rigour and veriﬁed.This pricing model would be called a binomial model (because the underlying distribution isobjectively known to be binomial). Interestingly, an n-period price (where n is a large number,say 1000) would approximate to the Black–Scholes model price, which is derived from thenormal distribution, because the Central Limit Theorem tells us that the distribution of themean of a large binomial sample tends to normality.This forms the ideal backdrop to a reliable valuation model, a backdrop that is regrettablydiminished or absent when we move from manufactured distributions to those actually observed
40
Currency Overlay
in the ﬁnancial markets. Of course, option theorists do not necessarily regret the difﬁcultiesof modelling the real ﬁnancial world – if the models worked most of them would be out ofa job!
3.3 THE BLACK–SCHOLES MODELFor practical purposes modern option theory can be dated from a seminal paper by Blackand Scholes in 1973. In that paper the authors demonstrated for the ﬁrst time the theoreticalpossibility of risklessly hedging written options using only the underlying security, and thatthis process produced ‘costs’ exactly equal to the ‘fair’ premium of the option. The process theydescribed (although see below) was one of continuous adjustment of the underlying holdingof the security, a process we now call ‘delta hedging’.The importance of the paper was that it provided a theoretical argument for claimingthat the fair price of an option should be the expected payoff value without any additionalrisk premium, because it is risklessly replicatable using the underlying securities. Priorto this, pricing models had always tried to include a pricing of the risk taken by optionwriters.This underpinning was seen to establish the fair price, and the maths to calculate it, as alegitimate price, one which traders could have conﬁdence in. Ironically, the ‘fair price’ mathshas been almost universally accepted (although Black and Scholes did not invent this); theirinnovation, the riskless replication, was always an illusion, since it relied on assumptions aboutthe underlying markets which are clearly violated in practice. Black and Scholes knew at thetime that their riskless replication was never available to real traders – and yet this is the modelwhich has endured.Without going into too much maths, what is the Black–Scholes model?
3.3.1 Market assumptionsThe ﬁrst, and most important, aspect of the Black–Scholes model is their assumptions about thecharacteristics of the market. Note also that these were assumptions made about a single equitymarket, not the currency market. The following are the key assumptions (there are others, butthey are not key, or are not seriously violated in practice):(a) The market price√ is a lognormal random walk in continuous time;(b) Volatility (= variance) is constant;(c) There are no transaction costs, and any (fractional) amount of the option and the instrumentcan be traded in continuous time.All three of these are seriously violated in all markets, and the currency market is noexception. If (a) held, then there would be no possibility for any active management, forecastingor systematic outperformance in the currency (or any) markets – all markets would be cointossing. But the violation of (c) is the most serious, since if transactions costs are present (asthey always are), and if the market is not priced in continuous time (which it cannot be in amarket run by humans, even in the imagination), then the maths of the Black–Scholes modelbreaks down.
Currency Hedging
Box 3.6
41
Normal and lognormal
The normal distribution is a feature of many types of random processes. Its ‘bell-shapedcurve’ represents the likelihood of outcomes ranging from people’s height to the numberof peas in a bag of frozen peas. The normal curve is deﬁned by its formula, which is quitesimple, but more importantly has taken a place at the centre of the statistical world. Thereis no ‘natural law’ which makes the normal distribution the right one for modelling the realworld, and indeed it is possible that its mathematical elegance is one of its main qualiﬁcations. However, the Central Limit Theorem” does lead directly to the normal distribution.The Central Limit Theorem says that if the average value of a sample of observations istaken (say the number of heads from tossing a coin 1000 times), then the normal distributiondescribes the likelihood of the number of heads being 450–500, 500–550, 550–600, etc. Aslong as the samples are large enough, this effect applies to a wide range of underlying distributions. Coin-tossing is ‘binomial’, the number of ﬁsh caught by ﬁsherman is Poisson, andthe length of telephone calls is exponential, but all of these will have normal distributionsof sample means for large samples.Lognormal processes are those where the randomness is multiplicative, say in the periodreturns of an asset. They are called lognormal because if returns are recorded as percentages,then ‘1+ percentage’ is the factor by which to multiply the asset to get the next periodvalue. If we take the natural log of these ‘1 + percentages’, the resulting series is normallydistributed.Appendix 2 takes a full look at the mathematical implications of this phenomenon. Itis not self-evident, however, that lognormal (i.e. a normal generator in a multiplicativeenvironment) is the ‘natural’ distribution for ﬁnancial markets. Indeed, the occurrence ofso-called ‘7-standard deviation’ events in the markets (or ‘melt-downs’) on a regular, ifinfrequent, basis does require us to abandon the normal hypothesis. 7-SD events occurunder a stable normal distribution once in every 2 billion years.
3.3.2 The modelBlack and Scholes derived their fair price not by simple calculus from the lognormal distribution(which had already been done), but by a peculiarly ‘back door’ method. They hypothesisedthat if the underlying market on which the option was written indeed satisﬁed the aboveassumptions, then it would be possible for a writer of a call option on x shares of underlyingto hold an amount of shares y(0 ≤ y ≤ x) which (measured instantaneously) would providea riskless hedge. Rather bizarrely given the subsequent development of delta hedging, theway this was actually shown in the paper was that the holder of a ﬁxed amount of underlyingshares could hedge that risk by varying the amount of options written. If the amount x wasvaried continuously (and they mean continuously, not every 1/100th of a second!), then usingcontinuous time maths (which allows calculus), they found that the formula for pricing theoption which satisﬁed the ‘no-risk’ requirement created by this continuous adjustment methodwould be, hey presto, the fair value (which equals the expected payoff value of the option).The Black–Scholes model, in its simplest form, has only three variables: the time to expiration, the market volatility, and the strike price in relation to the current market level. Forcurrency options, the interest rate is relevant only insofar as it discounts future cash ﬂows. Ifthe option premium were payable at maturity, the interest rate would fall away.
42
Currency Overlay
The academic world regarded Black and Scholes’ proof that options could be risklesslyhedged with the underlying market instrument, and vice versa, as their major innovation. This‘discovery’ ended the speculation as to the appropriate risk premium for options, since they‘proved’ there had to be none.This innovation spawned huge interest in option pricing theory, and set Scholes (with Merton)on their way to the 1997 Nobel Prize. Unfortunately Black died in 1995, and so missed out onthe prize. It is a little ironic that the acclaim for this paper is based on the creation of risklessportfolios of options with underlying instruments. The moment there is serious violation ofBlack and Scholes’ assumptions, the riskless nature of the hedge breaks down. What mayappear to work in theory does not survive any practical tests – and under most scientiﬁcregimes this would be regarded as refutation of the theory.3.3.3 Understanding option pricingThe Black–Scholes formula is set out in the footnote below.6 I recommend that, unless youhave a strong grasp of calculus, you should leave it unread. I think the best way to understandthe pricing model is to visualise what it is trying to do.When an option matures, it will have a ‘payoff’ value equal to the curve in Figure 3.2.Ignoring the lognormal element (and forward differentials) for the moment for simplicity, thenormal distribution curve for, say, 8% annual volatility, and with the current market level at30%
% Profit – USD base
25%
20%
15%
10%
5%
0%1.10
1.20
1.30
1.40
1.50
1.60
1.70
$/£ Rate
Figure 3.2 Payoff % at maturity of $/£ option – £ call; strike = 1.506C = S N (d1 ) − K e(−r t) N (d2 )where:
C = theoretical call premiumt = time until expirationr = risk-free interest ratee = 2.7183ln(S/K ) + (r + (s 2 /2)t)d1 =√√ s td2 = d 1 − s t
s = standard deviation of log returnsln = natural logS = current stock priceK = option strike priceN = cumulative standard normal distribution
1.80
1.90
43
Probability density
Currency Hedging
1.10
1.18
1.26
1.34
1.42
1.50
1.58
1.66
1.74
1.82
1.90
$/£ Rate
Figure 3.3 Normal distribution (SD = 8%)
1.50, would be as in Figure 3.3. This tells us the ‘probability density’ of the exchange rateone year from now. ‘Probability density’ allows us to determine the chance that the exchangerate lies between any two values. I have deliberately not put values on the left-hand axis, sincewithout more information, the values of the curve do not mean a lot.From the elements in these two graphs, we can visualise that the expected payoff valueof a call option at 1.50 strike price is the sum of the likelihood of the end-year rate lying ateach interval above 1.50, and the proﬁt for the option holder at each of these intervals. Todo this accurately is a job for calculus, but we can approximate it closely by using discreteranges. Figure 3.4 shows the payoff curve superimposed on the normal distribution, and anarea, between 1.50 and 1.52, marked out.30%
% Profit – USD base
25%6.62% of areaunder curve
20%
15%
10%
5%
0%1.10
1.20
1.30
1.40
1.50
1.60
1.70
1.80
1.90
$/£ Rate
Figure 3.4 Payoff % at maturity of $/£ option – £ call; strike = 1.50
On the stated assumptions, the area under the normal curve between 1.50 and 1.52 (i.e. thelikelihood that the exchange rate will end the year lying between these values) is 6.62%. The(approximate) average proﬁt (i.e. the average of 1.50 and 1.52) to the option buyer is $0.01, or
44
Currency Overlay
0.67% (= 0.01/1.50). The expected payoff for this ‘strip’ is 6.23% × 0.67% = 0.042%. Thisprocess can be repeated for all the strips on the right-hand side of $/£ = 1.50, and the sum ofall these expected payoffs can be used as an approximation to the fair value. Table 3.8 showsthis calculation. Mathematicians will see that the strip width can be reduced and reduced,tending to zero, and that the expected probabilities can thereby be made increasingly accurate.This process, known as integration, is mathematically straightforward to compute, and theresulting formula is effectively the Black–Scholes equation. At the bottom of Table 3.8 thecalculated price is compared to the formula price (3.2% vs. 3.19%), illustrating the accuracyof the discrete approximation.Table 3.8 Option valuation – a discrete approximationOption type:Strike rate:Annual volatility:Interest rates:
Call on £ – $ base currency1.508%0% (for simplicity)
Exchange rate
Probability
Mean payoff ($)
Mean payoff (%)
Expected payoff (%)
(c)
(d)
(e) = d/((a + b)/2)
(f) = c × e
0.000%0.667%2.000%3.333%4.667%6.000%7.333%8.667%10.000%11.333%12.667%14.000%15.333%16.667%18.000%19.333%20.667%22.000%23.333%24.667%26.000%28.667%
0.000%0.044%0.129%0.203%0.262%0.301%0.320%0.321%0.305%0.277%0.241%0.202%0.163%0.127%0.096%0.070%0.049%0.034%0.022%0.014%0.009%0.012%3.20%3.19%
(a)
(b)
fromto01.5050.00%01.501.526.62%0.011.521.546.44%0.031.541.566.09%0.051.561.585.60%0.071.581.605.02%0.091.601.624.37%0.111.621.643.70%0.131.641.663.05%0.151.661.682.44%0.171.681.701.90%0.191.701.721.44%0.211.721.741.06%0.231.741.760.76%0.251.761.780.53%0.271.781.800.36%0.291.801.820.24%0.311.821.840.15%0.331.841.860.10%0.351.861.880.06%0.371.881.900.03%0.391.90upwards0.04%0.43Expected payoff (sum of expected payoffs by step)Black–Scholes model price (for comparison)
3.3.3.1 The GreeksOption theory has spawned a series of Greek letters to represent various rates of change (orelasticities) in an option’s price. The main ones are:
Currency Hedging
45
r Delta (, δ) – The change in the option value as a percentage of the change in the underlyinginstrument price. An option with a delta of 0.5 will change in value by 0.5% (of the principal)for every 1% move in the underlying marketr Gamma (, γ ) – The rate of change of the delta with respect to the pricer Theta (, θ ) – The rate of change of the option price with respect to timer Vega (bizarrely, not a greek letter) – The rate of change of the option price with respect tochanging (implied) volatilityr Rho (P, ρ) – The rate of change of the option price with respect to the interest rateWhen option traders trade options, these ‘Greek’ values are the key ones for them in constructing hedges for their portfolios. By adding second- and third-derivatives (in the mathematicalcalculus sense), one could go on adding these letters for ever, so the ones above are just themost popular.3.3.3.2 The portfolio insurance storyThis surfeit of maths brings us to an interesting point about ﬁnancial theory in general, andoption pricing in particular. Financial theory starts by modelling an imaginary world where‘everything is perfect’ (i.e. Black and Scholes’ assumptions are not violated). As long as thisworld is maintained in the imagination, then a huge and complex set of theories can be created,all of which ﬁt the world neatly. Actual markets, however, are not like physical laws. Theynot only do not conform at all closely to the assumptions of the imaginary world, but they arealso reﬂexive. By reﬂexive, I mean that they change their nature depending on the theoriesabout them that have common currency, and also by what other participants think yet otherparticipants are doing.While this is instinctively understood by market participants, there are numerous exampleswhere this phenomenon has fundamentally undermined even the least heroic assumptions onwhich these mathematical models are based. Let’s pick an example: the stock market crashof 1987. For several years prior to October 1987, US fund managers had sold programmes of‘portfolio insurance’ to the major US institutional investors. These were based precisely on theBlack–Scholes notion that by varying the underlying instrument, an investor could ‘risklessly’create the payoff of an option – in this case a put option on the US equity market, the S&P500. Generally, these programmes set the notional strike price somewhat out of the money, sothat deltas were very low at the initiation of these schemes. (Remember that deltas representthe proportion of the underlying put that is held in a short futures position.) There had been asubsequent rising market, and so, despite some resetting upwards of strike prices, deltas werestill very low. The providers of these programmes knew the processes they employed werenot riskless because they could not trade costlessly in continuous time. However, they thoughttheir proprietary ways of minimising the risks would work well enough. They could not havebeen more wrong.In October 1987, an extraneous external factor caused a downturn in the S&P. This triggeredsome selling of the market, using S&P500 futures as the chosen instrument. In the absence ofother buyers, and no new good news, this caused the equity market to fall (changes in stockfutures market levels affect the underlying securities market by arbitrage). This fall in theequity market triggered more selling of the futures, as the deltas on the (notional) put optionsrose. This triggered yet more selling of the futures market, which increased deltas and alsogammas (gamma is the rate of change of delta as the option goes towards the money).
46
Currency Overlay
These circular waves of selling and sell orders overwhelmed the market. There were inadequate buyers, the futures market continuously hit its ‘limit’ (when trading is suspended),and the underlying equity market was pulled down by arbitrageurs (probably the only futuresbuyers), who then realised that they could not sell the underlying securities because buyershad panicked and ﬂed there too.When the waves of selling ﬁnally subsided, the customers of portfolio insurance had notonly failed to be effectively protected against these market falls, but where trades had been executed (many programmes were abandoned), they were at extremely disadvantageous prices –undermining the rationale of the programmes. The whole concept was consigned to the rubbishbin with little sophisticated reﬂection, and the world moved on.3.3.4 The role of assumptions in option pricingThis example brings us to the key point on option pricing. The meat of option pricing is