Tài chính doanh nghiệp - Chapter sevend: Foreign currency options

1CHAPTER SEVEND FOREIGN CURRENCY OPTIONS 1 CHAPTER OVERVIEW • Introduction • Contract specifications • Option positions • Hedging using option contract • Strategy on currencies option • Option pricing 2 3 FOREIGN CURRENCY OPTIONS • A foreign currency option is a contract giving the purchaser of the option the right to buy or sell a given amount of currency at a fixed price per unit for a specified time period – The most important part of clause is the “right, but not the obligation” to take an action – Two basic types of options, calls and puts • Call – buyer has right to purchase currency • Put – buyer has right to sell currency – The buyer of the option is the holder and the seller of the option is termed the writer 4  Table shows option prices on British pound taken from the online edition of Wall Street Journal on Friday, January 31, 2007 FOREIGN CURRENCY OPTIONS MARKETS Feb Mar Apr Feb Mar Apr 162 2.36 2.94 - 0.16 0.74 - 163 1.5 2.32 2.14 0.3 1.12 2.02 164 0.86 1.7 - 0.66 1.5 - 165 0.5 1.36 1.34 - 2.16 - 166 0.26 1.02 1 - - - 167 0.12 0.76 0.92 - - - BRITISH POUND (CME) 62,500 pounds; cents per pound Strike Price Calls Puts 26-Nov-15 5 6 FOREIGN CURRENCY OPTIONS • Every option has three different price elements – The strike or exercise price is the exchange rate at which the foreign currency can be purchased or sold – The premium, the cost, price or value of the option itself paid at time option is purchased – Spot exchange rate in the market 7 FOREIGN CURRENCY OPTIONS • Options may also be classified as per their payouts – At-the-money (ATM) options have an exercise price equal to the spot rate of the underlying currency – In-the-money (ITM) options may be profitable, excluding premium costs, if exercised immediately – Out-of-the-money (OTM) options would not be profitable, excluding the premium costs, if exercised 8 FOREIGN CURRENCY OPTIONS MARKETS • Over-the-Counter (OTC) Market – OTC options are most frequently written by banks for US dollars against British pounds, Swiss francs, Japanese yen, Canadian dollars and the euro – Main advantage is that they are tailored to purchaser – Counterparty risk exists – Mostly used by individuals and banks • Organized Exchanges – similar to the futures market, currency options are traded on an organized exchange floor – The Chicago Mercantile and the Philadelphia Stock Exchange serve options markets – Clearinghouse services are provided by the Options Clearinghouse Corporation (OCC) 39 There are four types of options positions: # • A long position in a call option • A long position in a put option • A short position in a call option • A short position in a put option The underlying assets • Commodities • Stock • Foreign currency • Index • Futures OPTION POSITIONS 10 PROFIT & LOSS FOR THE BUYER OF A CALL OPTION Loss Profit (US cents/£) + 10 + 5 0 - 5 - 10 160 165 175 180170 Limited loss Unlimited profit Break-even price Strike price “Out of the money” “In the money” “At the money” Spot price (US cents/£) The buyer of a call option on £, with a strike price of 170 cents/£, has a limited loss of 50 cents/£ at spot rates less than 170 (“out of the money”), and an unlimited profit potential at spot rates above 170 cents/£ (“in the money”). 11 Loss Profit (US cents/£) + 10 + 5 0 - 5 - 10 160 165 175 180170 Limited profit Unlimited loss Break-even price Spot price (US cents/£) The writer of a call option on £, with a strike price of 170cents/£, has a limited profit of 5 cents/£ at spot rates less than 170, and an unlimited loss potential at spot rates above (to the right of) 175 cents/SF. Strike price PROFIT & LOSS FOR THE WRITER OF A CALL OPTION 12 Loss Profit (US cents/£) + 10 + 5 0 - 5 - 10 160 165 175 180170 Limited loss Profit up To 165 Strike price “In the money” “Out of the money” “At the money” Spot price (US cents/£) The buyer of a put option on £, with a strike price of 170cents/£, has a limited loss of 5 cents/£ at spot rates greater than 170 (“out of the money”), and a profit potential at spot rates less than 170cents/£ (“in the money”) up to 165 cents. Break-even price PROFIT & LOSS FOR THE BUYER OF A PUT OPTION 413 Loss Profit (US cents/£) + 10 + 5 0 -5 - 10 160 165 175 180170 Loss up To 165 Limited profit Spot price (US cents/£) The writer of a put option on £, with a strike price of 170 cents/£ has a limited profit of 5 cents/£ at spot rates greater than 165 and a loss potential at spot rates less than 165 cents/£. Break-even price PROFIT & LOSS FOR THE WRITER OF A CALL OPTION Strike price 14 STRATEGIES INVOLVING A SINGLE OPTION AND A STOCK x ST Profit (c) x ST Profit (d) x ST (b) x ST Profit (a)  Profit patterns.  (a) Long position in a stock combined with short position in a call,  (b) Short position in a stock combined with long position in a call.  (c) Long position in a put combined with long position in a stock,  (d) Short position in a put combined with stock position in a stock 15 B1. BULL SPREAD CREATED USING CALL OPTION- Buying a call on a stock with a certain price and selling a call on the same stock with a higher price X1 X2 ST Profit This strategy limits the investor’s upside potential as well as downside risk 16 B2. BULL SPREAD CREATED USING PUT OPTION Buying a put on a stock with a certain price and selling a put on the same stock with a higher price X1 X2 ST Profit 517 B3. BEAR SPREAD CREATED USING CALL OPTION Buying a call one exercise price and selling a call with another strike price X1 X2 ST Profit This strategy limits the investor’s upside potential as well as downside risk 18 B4. BEAR SPREAD CREATED USING PUT OPTION- Buying a put one exercise price and selling a put with another strike price X1 X2 ST Profit 19 B5. BUTTERFLY SPREAD CREATED USING CALL OPTIONS- Buying one call at low price X1 and buying another call at high strike price X3 and selling two call with a strike price X2, halfway between X1 & X3 X1 X3 ST Profit X2 This strategy refer to an investment who fells that large stock price moves are unlikely BASIC OPTION PRICING RELATIONSHIPS AT EXPIRY • At expiry, an American option is worth the same as a European option with the same characteristics. • If the call is in-the-money, it is worth ST – E. • If the call is out-of-the-money, it is worthless. CaT = CeT = Max[ST – E, 0] • If the put is in-the-money, it is worth E – ST. • If the put is out-of-the-money, it is worthless. PaT = PeT = Max[E – ST, 0] Copyright © 2014 by the McGraw-Hill Companies, Inc. All rights reserved. 6MARKET VALUE, TIME VALUE, AND INTRINSIC VALUE FOR AN AMERICAN CALL E ST Profit Loss Long 1 callThe red line shows the payoff at maturity, not profit, of a call option. Note that even an out-of-the-money option has value— time value. Intrinsic value Time value In-the-moneyOut-of-the-money Copyright © 2014 by the McGraw-Hill Companies, Inc. All rights reserved. EUROPEAN OPTION PRICING RELATIONSHIPS Consider two investments: 1 Buy a European call option on the British pound futures contract. The cash flow today is –Ce. 2 Replicate the upside payoff of the call by:  Borrowing the present value of the dollar, exercise price of the call in the U.S. at i$ , the cash flow today is  Lending the present value of ST at i£, the cash flow today is E (1 + i$) ST (1 + i£) – 7-22 Copyright © 2014 by the McGraw-Hill Companies, Inc. All rights reserved. EUROPEAN OPTION PRICING RELATIONSHIPS Ce > Max ST E (1 + i£) (1 + i$) – , 0  When the option is in-the-money, both strategies have the same payoff.  When the option is out-of-the-money, it has a higher payoff than the borrowing and lending strategy.  Thus,  Using a similar portfolio to replicate the upside potential of a put, we can show that: Pe > Max STE (1 + i£)(1 + i$) – , 0 7-23 Copyright © 2014 by the McGraw-Hill Companies, Inc. All rights reserved. The Black - Scholes formula for pricing the European foreign currency call and put are where c = premium on a European call p = premium on a European put S = spot exchange rate (domestic currency/foreign currency) F = continuous compounding Forward rate E = exercise or strike price, T = time to maturity rd = domestic interest rate, rf = foreign interest rate σ = Volatility (standard deviation of percentage changes of the exchange rate) OPTION PRICING AND VALUATION )N()N( 210 dEedeSc TrTr hf    Tr T hedFdEp  )]N()N([ 12 T Tσ E F d T  2 1 2 1 ln        Tdd  12 Trr tt fheSF )(   7e-rT = continuously compounding discount factor (e=2.71828182) ln = natural logarithm operator N(x) = cumulative distribution function for the standard normal distribution, which is defined based on the probability density function for the standard normal distribution, n(x), i.e., 1 2 12 365 12% 1 (1+12%) 1.12 (1 12% / 2) 1.1236 (1 12% /12) 1.126825 (1 12% / 365) 1.127446 e 1.1274969         2x x x 2 - - 1 N(x) = n(x)dx= e dx 2     OPTION PRICING AND VALUATION OPTION PRICING AND VALUATION • The pricing of currency options depends on six parameters: – Current spot exchange rate ($1.7/￡) – Time to maturity (90 days) – Strike price ($1.72/￡) – Domestic risk free interest rate (r$ = 8%) – Foreign risk free interest rate (r￡ = 7.8%) – Volatility (10% per annum) Based on the above parameters, the call option premium is $0.0246/￡(this result is calculated based on the Black-Scholes formula in the excel file “GK” Garman Kohlhagen) 27 Inputs Outputs Spot rate (DC/FC e.g. USD/EUR) 170 Call Price = 2.4666 Strike price 172 volatility (annualized) 10.00% Put Price = 4.3453 domestic interest rate (annualized) 8.00% foreign interest rate (annualized) 7.80% time to maturity in days 90 time to maturity in years 0.25 6-Nov-15 28 8Exhibit: Intrinsic Value, Time Value & Total Value for a Call Option on British Pounds with a Strike Price of $1.70/£ 1.69 1.70 1.71 1.72 1.731.681.671.66 0.0 1.0 2.0 3.0 4.0 5.0 Spot Exchange rate ($/£) Option Premium (US cents/£) 3.30 5.67 4.00 6.0 1.74 1.67 Total value Intrinsic value Time value -- Valuation on first day of 90-day maturity -- Exhibit: Intrinsic Value, Time Value & Total Value for a Call Option on British Pounds with a Strike Price of $1.70/£ 0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.00 157.25 159.80 162.35 164.90 167.45 170.00 172.55 175.10 177.65 180.20 182.75 185.30 C al l V al u e (G ar m an -K oh lh ag en m od if ie d B la ck -S ch ol es ) Spot exchange rate FX Call Option Value and intrinsic value Time value Intrinsic value Total value 3.30 • The total value (premium) of an option is equal to the intrinsic value plus time value • Time value captures the portion of the option value due to the volatility in the underlying asset during the option life – The time value of an option is always positive and declines with time, reaching zero on the maturity date • Intrinsic value is the financial gain if the option is exercised immediately – On the date of maturity, an option will have a value equal to its intrinsic value (due to the zero time value at maturity) OPTION PRICING AND VALUATION CURRENCY OPTION PRICING SENSITIVITY • If currency options are to be used effectively, either for the purposes of speculation or risk management, the traders need to know how option values react to their various factors, including S, K, T, rf, rd, and σ • More specifically, we will study the sensitivity of option values with respect to S, K, T, rf, rd, and σ • These sensitivities are often denoted with Greek letters, so they also have the name “Greeks” or “Greek letters” 9DELTA • Spot rate sensitivity (delta): – Delta is defined as the rate of change of option price with respect to the price of the spot exchange rate. – Delta is in essence the slope of the tangent line of the option value curve with respect to the spot exchange rate – For calls, Δ is in [0, 1], and for puts, Δ is in [-1, 0] – For call (put) options, the higher (lower) the delta, the call (put) option is more in the money and thus the greater the probability of the option expiring with a positive payoff f f -r T 1 -r T 1 c Delta (for calls) e N(d ) > 0 S p Delta (for puts) e N(-d ) < 0 S            DELTA d 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 D e lt a ( N (d 1 ) Spot exchange rate DELTA • For the example, the delta of the option is 0.5, so the change of the spot exchange rate by ±$0.01/￡ will cause the change of the option value approximately by 0.5× ±$0.01 = ±$0.005. More specifically, the option value will become $0.033 ± $0.005 • Please note that the Delta estimation works well only when the change of the exchange rate S is small. (If the spot exchange rate increases by $0.1/￡, the Delta estimation predicts the option value becoming $0.083. • The larger the absolute value of Delta, the larger risk the portfolio is exposed to the exchange rate changes THETA • Time to maturity sensitivity (theta): # – Option values increase with the length of time to maturity – A trader with find longer maturity options better values, giving trader the ability to alter an option position without suffering significant time value c Theta θ (for calls) 0 T p Theta θ (for puts) 0 T         10 THETA • 90 to 89 days: • 15 to14 days • 5 to 4 days • The rapid deterioration of option value in the last days prior to expriration day 020 8990 28333 . ..cent time premium theta        050 1415 321371 . ..cent time premium theta        080 45 7093079290 . ..cent time premium theta        Theta: Option Premium Time value Deterioration ※ The negative slope means the option value decreases with the time approaching the expiration date ※ For the at-the-money options, the decay of option values accelerates when the time approaches the expiration date VEGA • Sensitivity to volatility (Vega): # – The vega for calls and puts are the same – Volatility is important to option value because it measures the exchange rate’s likelihood to move either into or out of the range in which the option will be exercised – The positive value of vega implies that both call and put values rise (fall) with the increase (decrease) of σ – The intuition for positive vega of both calls and puts is that since the options give the holder the right to fix the purchasing or the selling prices, options are more valuable in the scenario with higher volatility f f -r T 1 -r T 1 c Vega ν (for calls) =Se n(d ) T 0 σ p Vega ν (for puts) =Se n(d ) T 0 σ         VEGA • Volatility increase 1%, from 10%  11%: • If the volatility rise, the risk of the option being exercised is increasing, the option premium would be increasing 300 1011 03300360 . %% .$.$ volatility premium Vega        11 RHO AND PHI • Sensitivity to the domestic interest rate is termed as rho ※rd↑, domestic currency↓, foreign currency↑, because the call (put) can fix the purchase (sale) price of the foreign currency, call↑ and put↓ • Sensitivity to the foreign interest rate is termed as phi ※rf↑, domestic currency↑ , foreign currency↓, because the call (put) can fix the purchase (sale) price of the foreign currency, call↓ and put↑ d d -r T 2 d -r T 2 d c R h o ρ (fo r ca lls ) = K T e N (d ) > 0 r p R h o ρ (fo r p u ts) = K T e N (-d ) < 0 r        f f - r T 1 f - r T 1 f c P h i φ ( f o r c a l l s ) = S T e N ( d ) < 0 r p P h i φ ( f o r p u t s ) = S T e N ( -d ) > 0 r        Rho • US dollar interest rate increase 1%, from 8%  9%: • If the US dollar interest rate increase of 1%, the ATM call option premium increase from $0.033 to $0.035/£. 20 0809 03300350 . %.%. .$.$ rateerestint$US premium Rho        Phi • British Pound interest rate increase 1%, from 8%  9%: • If the £ interest rate increase of 1%, the ATM call option premium decrease from $0.033 to $0.031/£. • Phi value is -0.2 20 0809 03300310 . %.%. .$.$ rateerestintBP premium Phi        Interest Differentials (rd – rf) and Call Option Premiums ※When the interest rate differential (rd – rf) increases, the foreign currency call value indeed increases 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 ITM call (K=$1.65/￡) ATM call (K=$1.70/￡) OTM call (K=$1.75/￡) Option premium (U.S. cents/￡) rUS$ – r￡ 12 RHO AND PHI • Speculation strategy based on the expectation of the domestic interest rate – Because rd↑  c↑ and rd ↓  p↑, a trader should purchase a call (put) option on foreign currency before the domestic interest rate rises (declines). This timing will allow the trader to purchase the option before its price increases SUMMARY OF OPTION VALUE SENSITIVITY Greek Definition Interpretation Delta Δ Expected change in the option value for a small change in the spot rate The higher (lower) the delta, the more likely the call (put) will move in-the- money Theta Θ Expected change in the option value for a small change in time to expiration For at-the-money options, premiums are relatively insensitive until the final 30 days Vega υ Expected change in the option value for a small change in volatility Option values rise with increases in volatility both for calls and puts Rho ρ Expected change in the option value for a small change in domestic interest rate Increases in domestic interest rates cause increasing call values and decreasing put values Phi φ Expected change in the option value for a small change in foreign interest rate Increases in foreign interest rates cause decreasing call values and increasing put values