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1、FX Derivatives,1.FX Futures and Forwards,FX RISK,Example:ABYZ,a U.S.company,imports wine from France.ABYZ has to pay EUR 5,000,000 on May 2.Today,February 2,the exchange rate is 1.15 USD/EUR.Situation:Payment due on Mayy 2:EUR 5,000,000.SSep 2=1.15 USD/EUR.Problem:St is difficult to forecast Uncerta
2、inty.Uncertainty Risk.Example:on January 2,St or 1.15 USD/EUR.,On SFeb 2,ABYZ total payment would be:EUR 5,000,000 x 1.15 USD/EUR=USD 5,750,000.On May 2 we have two potential scenarios:If the SMay 2(USD appreciates)ABYZ will pay less USD.If the SMay 2(USD depreciates)ABYZ will pay more USD.The secon
3、d scenario introduces Currency Risk.,FX Risk:Hedging Tools,Hedging Tools:-Market-tools:Futures/ForwardMoney Market(IRPT strategy)Options-Firm-tools:Pricing in domestic currencyRisk-sharingMatching Outflows and Inflows,Futures or Forward FX Contracts,Forward markets:Tailor-made contracts.Location:non
4、e.Reputation/collateral guarantees the contract.Futures markets:Standardized contracts.Location:organized exchangesClearinghouse guarantees the contract.,Comparison of Futures and Forward Contracts,FX Futures/Forwards:Basic Terminology,Short:Agreement to Sell.Long:Agreement to Buy.Contract size:numb
5、er of units of foreign currency in each contract.CME expiration dates:Mar,June,Sep,and Dec+Two nearby monthsMargin account:Amount of money you deposit with a broker to cover your possible losses involved in a futures/forward contract.Initial Margin:Initial level of margin account.Maintenance Margin:
6、Lower bound allowed for margin account.A margin account is like a checking account you have with your broker,but it is marked to market.If your contracts make(lose)money,money is added to(subtracted from)your account.,If margin account goes below maintenance level,a margin call is issue:you have to
7、add funds to restore the account to the initial level.Example:GBP/USD CME futures initial margin:USD 2,800maintenance margin:USD 2,100If losses do not exceed USD 700,no margin call will be issued.If losses accumulate to USD 850,USD 850 will be added to account.,Using FX Futures/Forwards,Iris Oil Inc
8、.,a Houston-based energy company,will transfer CAD 300 million to its USD account in 90 days.To avoid FX risk,Iris Oil decides to short a USD/CAD Forward contract.Data:St=.8451 USD/CADFt,90-day=.8493 USD/CADIn 90-days,Iris Oil will receive with certainty:(CAD 300M)x(.8493 USD/CAD)=USD 254,790,000.No
9、te:The exchange rate at t+90(St+90)is,now,irrelevant.,The payoff diagram makes it clear:Using futures/forwards can isolate a company from uncertainty.,St+90,USD 254.79M,Forward,Amount Received in t+90,Hedging with FX Futures Contracts,FX HedgerFX Hedger reduces the exposure of an underlying position
10、 to currency risk using(at least)another position(hedging position).Basic Idea of a HedgerA change in value of an underlying position is compensated with the change in value of a hedging position.Goal:Make the overall position insensitive to changes in FX rates.,Hedger has an overall portfolio(OP)co
11、mposed of(at least)2 positions:(1)Underlying position(UP)(2)Hedging position(HP)with negative correlation with UPValue of OP=Value of UP+Value of HP.Perfect hedge:The Value of the OP is insensitive to FX changes.Types of FX hedgers using futures:i.Long hedger:Underlying position:short in the foreign
12、 currency.Hedging position:long in currency futures.ii.Short hedger:Underlying position:long in the foreign currency.Hedging position:short in currency futures.Note:Hedging with futures is very simple:Take an opposite position!,Q:What is the proper size of the Hedging position?A:Basic(Naive)Approach
13、:Equal hedgeModern Approach:Optimal hedge The Basic Approach:Equal hedgeEqual hedge:Size of Underlying position=Size of Hedging position.Example:Long Hedge and Short Hedge(A)Long hedge.A U.S.investor has to pay in 90 days 2.5 million Norwegian kroners.UP:Short NOK 2.5 M.HP:Long 90 days futures for N
14、OK 2.5 M.(B)Short hedge.A U.S.investor has GBP 1 million invested in British gilts.UP:Long GBP 1 M.HP:Short futures for GBP 1 M.,Define:Vt:value of the portfolio of foreign assets measured in GBP at time t.Vt*:value of the portfolio of foreign assets measured in USD at time t.Example(continuation):C
15、alculating the short hedgers profits.Its September 12.An investor decides to hedge using Dec futures.Situation:UP:British bonds worth GBP 1,000,000.FSep 12,Dec=1.55 USD/GBPFutures contract size:GBP 62,500.SSep 12:1.60 USD/GBP.number of contracts=?HP:Investor sells GBP 1,000,000/(62,500 GBP/contract)
16、=16 contracts.,Example(continuation):Calculating the short hedgers profits.On October 29,we have:Sep 12Oct 29ChangeVt(GBP)1,000,0001,000,000 0Vt*(USD)1,600,0001,500,000-100,000St1.601.50 0.10Ft,T=Dec1.551.45 0.10The USD change in UP(long GBP bond position)is given by:Vt*-V0*=VtSt-V0S0=V0 x(St-S0)=US
17、D 1.5M-USD 1.6M=USD-0.1M.The USD change in HP(short GBP futures position)is given by:-V0 x(Ft,T F0,T)=Realized gain(GBP-1M)x USD/GBP(1.45-1.55)=USD 0.1M.USD Change in OP=USD Change of UP+USD Change of HP=0=This is a perfect hedge!,Note:In the previous example,we had a perfect hedge.We were lucky!VSe
18、p 12=VOct 29=GBP 1,000,000(FSep 12,Dec-SSep 12)=(FOct 29,Dec SOct 29)=USD.05An equal position hedge is not a perfect hedge if:(1)Vt changes.(2)The basis(Ft-St)changes.,Changes in Vt If Vt changes,the value of OP will also change.Example:Reconsider previous example.On October 29:FOct 29,Dec=1.45 USD/
19、GBP.SOct 29=1.50 USD/GBP.VOct 29 increases 2%.Size of UP(long):GBP 1,020,000.Size of HP(short):GBP 1,000,000.USD change in UP(long GBP bond)isVOct 29*(long)=GBP 1.02M x 1.50 USD/GBP=USD 1,530,000VSep 12*(long)=GBP 1.0M x 1.60 USD/GBP=USD 1,600,000USD Change in Vt*(long)=USD-70,000,USD Change in HP(s
20、hort GBP futures)=USD 100,000.Net change on the overall portfolio=USD-70,000+USD 100,000=USD 30,000.=Not a perfect hedge:only the principal(GPB 1 million)was hedged!,Basis ChangeDefinition:Basis=Futures price-Spot Price=Ft,T-St.Basis risk arises if Ft,T-St deviates from a constant basis per period.I
21、f there is no basis risk completely hedge the underlying position(including changes in Vt).If the basis changes equal hedge is not perfect.In general,if(Ft,T-St)(or weakens),the short hedger loses.if(Ft,T-St)(or strengthens),the short hedger wins.,Example(continuation):Now,on October 29,the market d
22、ata is:FOct 29,Dec=1.50 USD/GBP.SOct 29=1.50 USD/GBP.BasisSep 12=FSep 12,Dec-SSep 12=1.55-1.60=-.05(5 points)BasisOct 29=FOct 29,Dec-SOct 29=1.50-1.50=0.Compared to previous Example,basis increased from-5 points to 0 points.(The basis has weakened from USD.05 to USD 0.)DateLong Position(Buy)Dec Futu
23、res(Sell)September 121,600,0001,550,000October 291,500,0001,500,000Gain-100,000 50,000Note:(Ft-St)Short hedgers profits loses(USD-50,000).Equal hedge is not perfect!,Basis risk Recall IRPT.For the USD/GBP exchange rate,we have:Assume T=360.After some algebra,we have:The basis is proportional to the
24、interest rate differential.As interest rates change,the basis changes too.,Derivation of the Optimal hedge ratio Additional notation:ns:number of units of foreign currency held.nf:number of futures foreign exchange units held.(Number of contracts=nf/size of the contract)h,t:uncertain profit of the h
25、edger at time t.h=hedge ratio=(nf/ns)=number of futures per spot in UP.We want to calculate h*(optimal h).For that,we minimize the variability of h,t.h,T=ST ns+FT,T nf(Or,h,T/ns=ST+h FT,T.)We want to select h to minimize:Var(h,T/ns)=Var(ST)+h2 Var(FT,T)+2 h Covar(ST,FT,T)h*=-SF/F2.,Optimal hedge rat
26、io:h*=-SF/F2.(A covariance over a variance=A(OLS)regression estimate!)Remarks:(1)h*is the(negative)slope of an OLS regression of ST against FT:St=+Ft,T+t=h*=-b(OLS estimate of)(2)Recall IRPT:Ft,T=St(1+id)/(1+if)=St.Calculating changes:Ft,T=St=St=(1/)Ft,T Then,h*=-b estimates(1/)in the IRPT equation:
27、=1/h*.(3)Two cases regarding hedge ratios:-When Ft,T is denominated in the same currency as the asset being hedged,we can use IRPT to get the hedge ratio,h*.-When Ft,T is denominated in a different currency than the asset being hedged(cross-hedging),OLS provides an estimate of h*.,OLS Estimation of
28、the Optimal Hedge RatioConsider the following regression equation:St=+Ft,T+t.=OLS produces b=SF/F2=-h*A hedge is perfect only if St and Ft,T are perfectly correlated:to have a perfect hedge we need t to be always zero.The R of regression measures the efficiency of a hedge.,Example:We estimate a hedg
29、e ratio using OLS:We use five years of monthly data for a total of 60 observations.St=.001+.92 Ft,T,R2=.95.h=-.92.nf/size of the contract=h ns/size of the contract=-.92 x 1,000,000/62,500=-14.7 15 contracts sold!The high R2 points out the efficiency of the hedge:Changes in futures USD/GBP prices are
30、 highly correlated with changes in USD/GBP spot prices.Note:A different interpretation of the R2:Hedging reduces the variance of the CF by an estimated 95%.Remark:OLS estimates of the hedge ratio are based on historical data.The hedge we construct is for a future period.Problem!,Time-varying hedge r
31、atios:ht*=-SF,t/F,t2GARCH models:The variance changes with the arrival of news.A GARCH(1,1)specification is a good approximation:2S,t=S0+S1 2S,t-1+S1 2S,t-1S,t-1=forecasting error at t-1.(Under RWM,S,t-1 is the change in St-1.)Recall that GARCH models accommodate two features of financial data:-Larg
32、e changes tend to be followed by large changes of either sign.-Distribution is leptokurtic i.e.,fatter tails than a normal.Statistical packages that estimate GARCH models:SAS,E-views.To estimate a time-varying hedge ratio,you need a model for the bivariate distribution of St and Ft,T(to get covarian
33、ces).,Hedging Strategies,Three problems associated with hedging in the futures market:-Contract size is fixed.-Expiration dates are also fixed.-Choice of underlying assets in the futures market is limited.Imperfect hedges:-Delta-hedge when the maturities do not match-Cross-hedge when the currencies
34、do not match.Another important consideration:liquidity.,Contract Terms(Delta Hedging)Major decision:choice of contract terms.Advantages of a short-term hedging:-Short-term Ft,T closely follows St.Recall linearized IRPT:Ft,T St 1+(id-if)xT/360As T 0,Ft,T St(UP and HP will move closely)-Short-term Ft,
35、T has greater trading volume(more liquid).Disadvantages of a short-term hedging:-Short-term hedges need to be rolled over:cost!,Contract Terms(Delta Hedging)Short-term hedges are usually done with short-term contracts.Longer-term hedges are done using three basic contract terms:-Short-term contracts
36、,which must be rolled over at maturity;-Contracts with a matching maturity(usually done with a forward);-Longer-term contracts with a maturity beyond the hedging period.,Three Hedging Strategies for an Expected Hedge Period of 6 Months,Different Currencies(Cross-Hedging)Q:Under what circumstances do
37、 investors use cross-hedging?An investor may prefer a cross-hedge if:(1)There is no available contract for the currency she wishes to hedge.Futures contracts are actively traded for the major currencies(at the CME:GBP,JPY,EUR,CHF,MXN,CAD,BRR).Example:Want to hedge a NOK position using CME futures:yo
38、u must cross-hedge.(2)Cheaper and easier to use a different contract.Banks offer forward contracts for many currencies.These contracts might not be liquid(and expensive!).Empirical results:(i)Optimal same-currency-hedge ratios are very effective.(ii)Optimal cross-hedge ratios are quite unstable.,Exa
39、mple:Calculation of Cross-hedge ratios.Situation:-Veron SA,a U.S.firm,has to pay HUF 10 million in 180 days.-No futures contract on the HUF.-Liquid contracts on currencies highly correlated to the HUF.Solution:Cross-hedge using the EUR and the GBP.Calculation of the appropriate OLS hedge ratios.Depe
40、ndent variable:USD/HUF changes Independent variables:USD/EUR 6-mo.futures changes USD/GBP 6-mo.futures changes Exchange rates:.0043 EUR/HUF.0020 GBP/HUFSUSD/HUF=+.84 FUSD/EUR+0.76 FUSD/GBP,R2=0.81.The number of contracts bought by Veron SA is given by:EUR:(-10,000,000 x.0043/125,000)x-0.84=2.89 3 contracts.GBP:(-10,000,000 x.0020/62,500)x-0.76=3.20 3 contracts.,
