FX Derivatives. Options: Brief Review

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Archibald Smith
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1 FX Derivatives 2. FX Options Options: Brief Review Terminology Major 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. The complete definition 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. 1

2 More terminology: An option is: - In-the-money (ITM) if, today, we would exercise it. For a call: X < S t (better to buy at a cheaper price than S t ) For a put: S t < X (better to sell at a higher price than S t ) - At-the-money (ATM) if, today, we would be indifferent to exercise it. For a call: X = S t (same to buy at X or S t ) For a put: S t = X (same to sell at X or S t ) In practice, you never exercise an ATM option, since there are some small brokerage costs associated with exercising an option. - Out-of-the-money (OTM) if, today, we would not exercise it. For a call: X > S t (better to buy at a cheaper price than X) For a put: S t > X (better to sell at a higher price than X) The Black-Scholes Formula Options are priced using variations of the Black-Scholes formula: C call premium e i f T N (d1) Xe N (d2) Fischer Black and Myron Scholes (1973) changed the financial world by introducing their Option Pricing Model. At the time, both were at the University of Chicago. S t i d T The model, or formula, allows an investor to determine the fair value of a financial option. Almost all financial securities have some characteristics of financial options, the model can be widely applied. 2

3 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 - Continuous pricing - Costless to short assets According to the formula, FX premiums are affected by six factors: Variable Euro Call Euro Put Amer. Call Amer. Put S t X T?? i d i f The Black Scholes does not fit the data. In general: - It overvalues deep OTM calls and undervalue deep ITM calls. - It 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) 3

4 Aside: Joke Not a joke as such, but (a true story, apparently, as told by a Finance lecturer at LSE): An economist was about to give a presentation in Washington, DC on the problems with Black-Scholes model of option pricing and was expecting no more than a dozen of government officials attending (who would bother?). To his amazement, when he arrived, the room was packed with edgy, tough-looking guys in shades. Still, after five or so minutes into the presentation all of them stood up and left without a word. The economist found out only later that his secretary ran the presentation through a spell-checker and what was "The Problem with Black-Scholes" became "The Problem with Black Schools", a rather more fascinating subject. Black-Scholes for FX options The Black-Scholes formula for currency options is given by: C call premium e i f T S t N (d1) Xe i d T N (d2) d1 = [ln(s t /X) + (i d i f )T]/( T 1/2 ), d2 = [ln(s t /X) + (i d i f.5 2 )T]/( T 1/2 ). Using the put-call parity, we calculate the put premium: P put premium C e i f T S t Xe i d T Note: S t,x,t,i d, and i f are observed. is estimated, not observed. 4

Example: Using the Black-Scholes formula to price FX options It is September, 2008. We have the following data: S t = 1.6186 USD/GBP (observed) X=1.62 USD/GBP (observed) T = 40/365 =.1096.

d1 = [ln(s t /X) +(i d i f +.5 2 )T]/( T 1/2 )= =[ln(1.6186/1.62)+(.0479.0583 +.5.08 2 )x.1096]/(.08x.1096 1/2 )= = -0.062440. d2 = [ln(s t /X) + (i d i f.5 2 )T]/( T 1/2 ). =[ln(1.6186/1.62)+(.0479.0583.5.08 2 )x.1096]/(.08x.1096 1/2 )= = -0.088923.

5 Example: Using the Black-Scholes formula to price FX options It is September, We have the following data: S t = USD/GBP (observed) X=1.62 USD/GBP (observed) T = 40/365 = (as a ratio of annual calendar) (observed) i d = (annualized) (observed) i f = (annualized) (observed) =.08. (annualized) (not observed, estimated!) (1) Calculate d1 and d2. d1 = [ln(s t /X) +(i d i f )T]/( T 1/2 )= =[ln(1.6186/1.62)+( )x.1096]/(.08x /2 )= = d2 = [ln(s t /X) + (i d i f.5 2 )T]/( T 1/2 ). =[ln(1.6186/1.62)+( )x.1096]/(.08x /2 )= = (2) Calculate N(d1) and N(d2). Now, look for the cumulative normal distribution at z = The area under the curve at z = is N(d1= ) = (= , recall z is negative!) N(d2= ) = (3) Calculate C and P. C = e x x e x = USD P = e x e x.1096 = USD

6 Empirical Check: From the WSJ quote (in USD cents): British Pound ,000 British Pounds-European Style. 161 Sep Oct Note: You can choose the volatility to match the observed premium. This is the implied volatility. Trading in Currency Options Markets for Foreign Currency options (1) Interbank (OTC) market centered in London, New York, and Tokyo. OTC options are tailor-made as to amount, maturity, and exercise price. (2) Exchange-based markets centered in Philadelphia (PHLX, now NASDAQ), NY (ISE, now Eurex) and Chicago (CME Group). - PHLX options are on spot amounts of 10,000 units of FC (MXN 100K, SEK 100K, JPY 1M). - PHLX maturities: 1, 3, 6, and 12 months. - PHLX expiration dates: March, June, Sept, Dec, plus 2 spot months. - Exercise price of an option at the PHLX or CME is stated as the price in USD cents of a unit of foreign currency. 6

7 OPTIONS PHILADELPHIA EXCHANGE Calls Puts Vol. Last Vol. Last Euro ,000 Euro-cents per unit. 132 Feb Mar Feb Mar Mar Feb Mar Size S t = USD/EUR P call = USD.019 P put = USD.0283 X=Strike T=Expiration Note on the value of Options For the same maturity (T), we should have: value of ITM options > value of ATM options > value of OTM options ITM options are more expensive, the more ITM they are. Example: Suppose S t = USD/EUR. We have two ITM Dec puts: X put = 1.36 USD/EUR X put = 1.42 USD/EUR. premium (X=1.36) = USD premium (X=1.42) = USD

8 Using Currency Options Iris Oil Inc., a Houston-based energy company, will transfer CAD 300 million to its USD account in 90 days (UP: long CAD 300M). To avoid FX risk, Iris Oil decides to use a USD/CAD option contract. Data: S t =.8451 USD/CAD Available Options for the following 90-day period: X Calls Puts.82 USD/CAD USD/CAD USD/CAD Iris Oil selects.84 USD/CAD put: Cost = CAD 300 M * USD /CAD = USD 2.04 M Iris Oil decides to use the.84 USD/CAD put => Cost of USD 2.04M. At T = t+90, there will be two situations: Option is ITM (exercised) or OTM (not exercised): If S t+90 <.84 USD/CAD If S t USD/CAD Option CF (HP): (.84 S t+90 ) CAD 300M 0 Plus (UP) S t+90 CAD 300M S t+90 CAD 300M Total USD 252M S t+90 CAD 300M Net CF in 90 days: USD 252M USD 2.04M= USD M for all S t+90 <.84 USD/CAD S t+90 CAD 300M USD 2.04M for all S t USD/CAD Worst case scenario (floor) : USD M (when put is exercised.) Remark: The final CFs depend on S t+90! 8

9 The payoff diagram shows that the FX option limits FX risk, Iris Oil has established a floor: USD M. But, FX options, unlike Futures/forwards, have an upside => At time t, the final outcome is unknown. There is still (some) uncertainty! Net Amount Received in t+90 FX Put USD M.84 S t+90 With options, there is a choice of strike prices (premiums). A feature not available in forward/futures. Suppose, Iris Oil also considers the.82 put => Cost of USD 0.63M. At T = t+90, there will be two situations: Option is ITM (exercised) or OTM (not exercised): If S t+90 <.82 USD/CAD If S t USD/CAD Option CF (HP): (.82 S t+90 ) CAD 300M 0 Plus (UP) S t+90 CAD 300M S t+90 CAD 300M Total USD 246M S t+90 CAD 300M Net CF in 90 days: USD 246M USD.63M = USD M for all S t+90 <.82 USD/CAD S t+90 CAD 300M USD.63M for all S t USD/CAD Worst case scenario (floor) : USD M (when put is exercised). 9

10 Both FX options limit Iris Oil FX risk: -X put =.84 USD/CAD => floor: USD M (cost: USD 2.04 M) -X put =.82 USD/CAD => floor: USD M (cost: USD.63M) Note: Higher premium, higher floor (better coverage). Net Amount Received in t+90 X put =.82 USD/CAD X put =.84 USD/CAD USD M USD M S t+90 (USD/CAD).8353 USD/CAD => break even S t+90 Hedging with Currency Options Hedging (insuring) with Options is Simple Situation 1: Underlying position: long in foreign currency. Hedging position: long in foreign currency puts. Situation 2: Underlying position: short in foreign currency. Hedging position: long in foreign currency calls. OP = underlying position (UP) + hedging position (HP-options) Value of OP = Value of UP + Value of HP + Transactions Costs (TC) Profit from OP = UP + HP-options + TC 10

11 Advantage of options over futures: Options simply expire if S t moves in a beneficial way. Price of the asymmetric advantage of options: The TC (insurance cost). Q: What is the size of the hedging position with options? A: Basic (Naive) Approach: Equal Size Modern (Dynamic) Approach: Optimal Hedge The Basic Approach: Equal Size Example: A U.S. investor is long GBP 1 million (=UP). She hedges using Dec put options with X= USD 1.60 (ATM). Underlying position: V 0 =GBP1M. S t=0 = 1.60 USD/GBP. Size of the PHLX contract: GBP 10,000. X = USD 1.60 P t=0 = premium of Dec put = USD.05. TC = Cost of Dec puts = 1 M x USD.05 = USD 50,000. Number of contracts = GBP 1 M / GBP 10,000 = 100 contracts. On December S t=t =1.50 USD/GBP < X => option is ITM (exercise put) UP = V 0 x(s t S 0 )=GBP1M*( ) USD/GBP = - USD 0.1M. HP = V 0 x(x S t )=GBP1M*( ) USD/GBP = USD 0.1M. OP = -USD 100,000 + USD 100,000 USD 50,000 = -USD 50,

12 Example: If at T, S T = 1.80 USD/GBP => option is not exercised (put is OTM). UP = GBP 1M x ( ) USD/GBP = USD 0.2M HP = 0 (No exercise) OP = USD 200,000 USD 50,000 = USD 150,000. The price of this asymmetry is the premium: USD 50,000 (a sunk cost!). Dynamic Hedging with Options (Optimal Hedge) Listed options are continually traded. Options positions are usually closed by reselling the options. Part of the initial premium (TC) is recovered. Profit from OP = UP + HP-options + TC 1 -TC 0 Hedging is based on the relationship between P t (or C t )and S t : Goal: Get P t ( C t ) and S t with similar changes. Problem: The relation between P t ( C t )and S t is non-linear. 12

13 Value of a GBP Puts in Relation to the Exchange Rate P t (USD) P t =.015 A slope = (delta) 1.60 S t (USD/GBP) P t can go up or down in response to changes in S t. Slope of the curve at A ( =delta): elasticity of premium to changes in S t Suppose, S t changes by a small amount, say Using results from calculus, we can (locally) approximate the change in P t by the derivative. P t (USD) P t P t =.015 A A Approximation error S t (USD/GBP) S t changes from 1.60 to 1.59 USD/GBP. Approximated change: P t increases by USD -.01 x (-0.5) = USD P t USD.02 13

14 AtA, S t changes from 1.60 to 1.59 USD/GBP. We approximate the change in the function by the slope (derivative): P t = [USD -.01 x (-0.5)] = USD.02 Q: What is a good hedge in point A? If GBP depreciates by USD.01, each GBP put goes up by USD.005. buy 2 GBP puts for every GBP of British gilts (2 =-1/-0.5) That is, the optimal hedge ratio is h* =-1/. (Note: negative sign to make h positive) Problem: -hedging only works for small changes of S t, where we can ignore the approximation error. Example: We are at point A (S t = 1.60, P t = USD.015). h = Hedge ratio = (-1/ )=-1/-0.5 = 2. n = 2 x 1,000,000 = 2,000,000. Number of contracts = 2,000,000/10,000 = 200. Now, at A,S t = 1.59 USD/GBP P t =.02. HP = 2,000,000 x ( ) = USD 10,000. UP = 1,000,000 x ( ) = USD -10,000. Problem: If the GBP depreciates, options protect the portfolio by its changes. 14

15 Value of GBP Puts when S t Moves P t (USD) P t =.0152 P t =.015 B ( =-0.8) A ( =-0.5) S t (USD/GBP) Suppose we move to B, with S t = 1.55 USD/GBP (a big change in S t ) The slope moves to = A new hedge ratio needs to be calculated. Example: Back to point B. Now, S t = 1.55 USD/GBP, with = New h = 1.25 (= -1/-0.8) New n = 1.25 x1m=1,250,000. New Number of contracts = 1,250,000/10,000 = 125 contracts. No over hedging: Investor sells 75 put contracts (realizes a profit). 15

16 Summary of Problems associated with Delta Hedging - Delta hedging only works for small changes of S t. - Δ and h change with S t n must be adjusted continually. this is expensive. In practice, use periodical revisions in the option position. Example: h changes when there is a significant swing in S t (2% or +). Between revisions, options offer usual asymmetric insurance. Hedging Strategies Hedging strategies with options can be more sophisticated: Investors can play with several exercise prices with options only. Example: Hedgers can use: - Out-of-the-money (least expensive) - At-the-money (expensive) - In-the-money options (most expensive) Same trade-off of car insurance: - Low premium (high deductible)/low floor or high cap: Cheap - High premium (low deductible)/high floor or low cap: Expensive 16

17 OPTIONS PHILADELPHIA EXCHANGE Calls Puts Vol. Last Vol. Last Euro ,000 Euro -cents per unit. 132 Feb Mar Feb Mar Mar Feb Mar Swedish Krona ,000 Swedish Krona -cents per unit. Example: It is February 2, UP = Long bond position EUR 1,000,000. HP = EUR Mar put options: X =134 and X=136. S t = USD/EUR. (A) Out-of-the-money Mar 134 put. Total cost = USD.0170 x 1,000,000 = USD 17,000 Floor = 1.34 USD/EUR x EUR 1,000,000 = USD 1,340,000. Net Floor = USD 1.34M USD.017M = USD 1.323M (B) In-the-money Mar 136 put. Total cost = USD.0283 x 1,000,000 = USD 28,300 Floor = 1.36 USD/EUR x EUR 1,000,000 = USD 1,360,000 Net Floor = USD 1.36M USD.0283M = USD M As usual with options, under both instruments there is some uncertainty about the final cash flows. 17

18 Both FX options limit FX risk: - X put =1.34 USD/EUR floor: USD 1.323M (cost: USD.017M) - X put =1.36 USD/EUR floor: USD M (cost: USD.0283M) Typical trade-off: A higher minimum (floor) amount for the UP (USD M) is achieved by paying a higher premium (USD 28,300). Net Amount Received in March if position sold X put =1.34 USD/EUR X put =1.36 USD/EUR USD M USD 1.323M S March (USD/EUR) USD/EUR => break even S March Exotic Options Exotic options: options with two or more option features. Example: a compound option (an option on an option). Two popular exotic options: knock-outs and knock-ins. Barrier Options: Knock-outs/ Knock-ins Barrier options: the payoff depends on whether S t reaches a certain level during a certain period of time. Knock-out: A standard option with an "insurance rider" in the form of a second, out-of-the-money strike price. This "out-strike" is a stop-loss order: if the out-of-the-money X is crossed by S t, the option contract ceases to exist. 18

19 Knock-ins: the option contract does not exist unless and until S t crosses the out-of-the-money "in-strike" price. Example: Knock-out FX options Consider the following European option: 1.65 USD/GBP March GBP call knock-out 1.75 USD/GBP. S t = 1.60 USD/GBP. If in March S t = 1.70 USD/GBP, the option is exercised => writer profits: USD ( ) + premium per GBP sold. If in March S t 1.75 USD/GBP, the option is cancelled => writer profits are the premium. Q: Why would anybody buy one of these exotic options? A: They are cheaper. Example (continuation): Knock-out put FX options. UP = Long bond position EUR 1,000,000. HP = Mar put option: X put =1.34 USD/EUR with X KO = 1.30 USD/EUR S t=march (in USD/EUR) Value long position S t=march 1.34 EUR 1M x S t S t=march 1.30 EUR 1M x 1.34 USD/EUR. S t=march X KO = 1.30 EUR 1M x S t. Value Long Position in March X put =1.34 USD/EUR USD 1.34M S March (USD/EUR) 19

FX Derivatives. 2. FX Options. Options: Brief Review

FX Derivatives. 2. FX Options. Options: Brief Review FX Derivatives 2. FX Options Options: Brief Review Terminology Major types of option contracts: - calls gives the holder the right to buy the underlying asset - puts gives the holder the right to sell