FX Futures, Options & FX Exposure

4/15/2021FX RISK & HEDGINGFX Futures, Options & FX Exposure(for private use, not to be posted/shared online) Last Class Model of FX Rate Determination not very successful. Foresting St with different approaches:- Fundamental Models- TA Models- RWM tend to do well. Main take away: Forecasting is difficult, especially in the short-run. Managing FX Risk becomes very important. Hedging Market-based Tools:- Market-tools:Futures/ForwardMoney Market (IRPT strategy)Options1

4/15/2021 Hedging with Forwards/Futures easy: Take an opposite position. Q: What is the size of the hedging position?- Equal size: Good if UP does not change in value and/or basis staysconstant.- Optimal size: Modern approach. Derivation of the Optimal hedge ratioAdditional notation:ns: Number of units of foreign currency held.nf: Number of futures foreign exchange units held. Number of contracts nf/size of the contractπh,t: Uncertain profit of the hedger at time t.h Hedge ratio (nf/ns) number of futures per spot in UP.We want to calculate h* (optimal h): We minimize the variability of πh,t.πh,T ΔST ns ΔFT,T nf(Or, πh,T / ns ΔST h ΔFT,T.)We want to select h to minimize:Var(πh,T/ns) Var(ΔST) h2 Var(ΔFT,T) 2 h Covar(ΔST,ΔFT,T) σS2 h2 σF2 2 h σSF h* -σSF/σF2.2

4/15/2021 Optimal hedge ratio : h* -σSF/σF2.(A Covariance/Variance A regression (OLS) estimate)Remarks:(1) h* is the (negative) slope of a regression: Regress ΔST against ΔFt,T .ΔST μ θ ΔFt,T εt h* -θ (OLS estimate of θ)(2) Recall IRPT:Ft,T𝑆 ΔST (1/δ) ΔFt,T1 i T/3601 i T/360𝑆 𝜹 h* -θ -1/δ(3) Two cases regarding how to compute h*:- When Ft,T is denominated in the same currency as the UP, use IRPT.- When Ft,T is denominated in a different than the UP, use regression. OLS Estimation of Optimal Hedge RatioConsider the following regression equation:ΔSt μ θ ΔFt,T εt OLS produces θ -σSF/σF2 - h* A hedge is perfect only if ΔSt & ΔFt,T are perfectly correlated: R² 1.(εt has to be always zero.) R² measures the efficiency of a hedge: The higher, the better.3

4/15/2021Example: We estimate a hedge ratio using OLS for UP GBP 1M:We use five years of monthly data for a total of 60 observations.ΔSt .001 .92 ΔFt,T,R² .95. h - θ -.92.Q: How many contracts?A: nf/size of the contract h ns / size of the contract -.92 * GBP 1M / GBP 62,500 -14.7 15 contracts sold! ¶ The high R² points out the efficiency of the hedge: Changes in futures USD/GBP prices are highly correlated with changesin USD/GBP spot prices.Note: A different interpretation of the R2: Hedging reduces the variance ofthe CF by an estimated 95%.Remark: OLS estimates of the hedge ratio are based on historical data. Thehedge we construct is for a future period. Problem! Time-varying hedge ratios: ht* -σSF,t /σF,t2There are many models that are used to forecast variances over time.Popular model: GARCH models.GARCH models: The variance changes with the arrival of news(innovations) and past variances. These models accommodate the stylizedfact that big (small) changes tend to be followed by big (small) changes.Many GARCH models and specifications. The specification depends on thenumber of lagged errors ( 2t) and lagged variances ( 2t).A GARCH(1,1) specification is a good approximation. For example, forchanges in St: 2S,t S0 S1 2S,t-1 ßS1 2S,t-1 S,t-1 forecasting error at t-1. (Under RWM, S,t-1 is the change in St-1.) 2S,t-1 variance of changes in St.4

4/15/2021 Time-varying hedge ratios: ht* -σSF,t /σF,t2GARCH models accommodate two features of financial data:- Large 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, R.To estimate a time-varying hedge ratio, we need a model for the bivariatedistribution of St and Ft,T (to get covariances). Things can get complicatedquickly.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 do not match. Another important consideration: Liquidity.5

4/15/2021 Contract Terms (Delta Hedging)Major decision: Choice of contract terms. Advantages of Short-term hedging:- Short-term Ft,T closely follows St.Recall linearized IRPT : Ft,T St [1 (id – if) * T/360]As T 0, Ft,T St(UP and HP will move closely)- Short-term Ft,T has greater trading volume (more liquid). Disadvantages of 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, 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.6

4/15/2021Three Hedging Strategies for Expected Hedge Period of 6 Months Short Term – RolloverRollover (“a roll”) occurs when a trader closes out a position in an expiringcontract (“the front month”) and simultaneously reestablishes the sameposition in a future month. A roll extends the expiration of a position.The gain or loss on the original contract will be settled by taking thedifference between the price on the day the roll is executed and theprevious day’s mark-to-market.Example: Trader is long a GBP Dec futures trading at USD 1.5530 onDec 11.A roll: Close Dec position on Dec 11 (expires Wed, Dec 16) at USD 1.5530.Open (simultaneously) Mar position at market rate, USD 1.5620.On December 10, the Dec futures was mark-to-market at USD 1.5525.Then, the long side receives USD 31.250 ( .0005 * 62,500) when the Decfutures position is closed. ¶7

4/15/2021 Long Term – Close FX FuturesA hedger decides to go for a longer maturity than the date of the UP (FCreceivables/payables). The hedger closes the hedging position by taking anopposite forward position with the exact remaining maturity.Example: It is June 2020. Six month ago, Goyco Corporation, sold a oneyear JPY forward contract at FDec19,Dec20 .0105 USD/JPY.Now, a 6-month forward contract trades at FJune,Dec20 .0102 USD/JPY.Goyco closes its short Dec position by buying JPY forward at FJune,Dec20.The CFs occur at expiration. That is, in Dec 2020, Goyco Corp. receives:F0,one-year - Ft 6-mo,6-mo 12.5M * (.0105 - .01025) USD 3,125.Assume iUSD 6%. Then, today's value of the forward contract is:F0,one-year - F6-mo,6-mo USD 3,125 USD 3,034. ¶[1 iUSD * (180/360)][1 .06 * (180/360)] Different Currencies (Cross-Hedging)Q: Under what circumstances do investors use cross-hedging? An investor may prefer a cross-hedge if:(1) 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 HUF position using CME futures: you must cross-hedge.(2) Cheaper and easier to use a different contract.Banks offer forward contracts for non-major currencies. These contractsmay not be liquid (and expensive!). Empirical results:(i) Optimal same-currency-hedge ratios are very effective.(ii) Optimal cross-hedge ratios are quite unstable.8

4/15/2021Example: Calculation of Cross-hedge ratios.Situation:- A U.S. firm has to pay HUF 10M 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.Dependent variable:USD/HUF changes ( SUSD/HUF )Independent variables:USD/EUR 6-mo. futures changes ( FUSD/EUR)USD/GBP 6-mo. futures changes ( FUSD/GBP)Exchange rates:.0043 EUR/HUF.0020 GBP/HUF SUSD/HUF α .84 FUSD/EUR 0.76 FUSD/GBP,R2 0.81.Number of contracts bought is given by:EUR: (-10M * .0043 /125,000) * -0.84 2.89 3 contracts.GBP: (-10M * .0020 /62,500) * -0.76 3.20 3 contracts. ¶Options: Brief ReviewTerminologyMajor types of option contracts:- calls give the holder the right to buy the underlying asset- puts give the holder the right to sell the underlying asset.Terms of an option must specify:- Exercise or strike price (X): Price at which the right is "exercised."- Expiration date (T): Date when the right expires.- Type. When the option can be exercised: Anytime (American)At expiration (European).The right to buy/sell an asset has a price: The premium (X), paid upfront.9

4/15/2021More terminology: An option is:- In-the-money (ITM) if, today, we would exercise it.For a call: X StFor a put: St X- At-the-money (ATM) if, today, we would be indifferent to exercise it.For a call: X StFor a put: St XIn practice, you never exercise an ATM option, since there are (small)costs associated with exercising an option.- Out-of-the-money (OTM) if, today, we would not exercise it.For a call: X StFor a put: St XThe Black-Scholes Formula Options are priced based on the Black-Scholes formula. For a call optionon a stock, whose price is St:𝐶𝑆 𝑁 𝑑1𝑋𝑒𝑁 𝑑2where T: time to maturity, X: strike price, σ: stock price volatility and𝑁 𝑑𝑒𝑑𝑥d1 [ln(St/X) (i 2/2) (T – t)]/( 𝑇d2 [ln(St/X) (i – 2/2)(T – t)]/( 𝑇𝑡),𝑡)) d1 – 𝑇𝑡. Black and Scholes (1973) changed the financial world by introducing theirOption Pricing Model. Many applications.10

4/15/2021The Black-Scholes Formula Almost all financial securities have some characteristics of financialoptions, the Black-Scholes model can be widely applied. The variation for a call FX option is given by:𝐶𝑒𝑆 𝑁 𝑑1𝑋𝑒𝑁 𝑑2 The Black-Scholes formula is derived from a set of assumptions:- Risk-neutrality.- Perfect markets (no transactions costs, divisibility, etc.).- Log-normal distribution with constant moments.- Constant risk-free rate.- Costless to short assets.- Continuous pricing. According to the formula, FX premiums are affected by six factors:VariableEuro CallEuro PutAmer. CallAmer. PutSt X T? id if11

4/15/2021 The Black–Scholes model does not fit the data. In general:- Overvalues deep OTM calls & undervalue deep ITM calls.- Misprices options that involve high-dividend stocks. The Black-Scholes formula is taken as a useful approximation. Limitations of the Black-Scholes Model- Trading is not cost-less: Liquidity risk (difficult to hedge)- No continuous trading: Gap risk (can be hedged)- Log-normal distribution: Not realistic (& cause of next limitations).- Underestimation of extreme moves: Left tail risk (can be hedged)- Constant moments: Volatility risk (can be hedged)Black-Scholes for FX options The Black-Scholes formula for currency options is given by:𝐶call premium𝑒𝑆 𝑁 d1𝑋𝑒𝑁 d2d1 [ln(St/X) (id – if .5 2) T]/( T1/2),d2 [ln(St/X) (id – if – .5 2) T]/( T1/2).Using put-call parity, we calculate the put premium:𝑃put premium𝐶𝑒𝑆𝑋𝑒Note: St, X, T, id, and if are observed. is estimated, not observed.12