Lecture multinational financial management chapter 7 ngo thi ngoc huyen

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FOREIGN CURRENCY OPTIONS CHAPTER SEVEND • 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 FOREIGN CURRENCY OPTIONS – 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 CHAPTER OVERVIEW FOREIGN CURRENCY OPTIONS MARKETS  • Introduction Table shows option prices on British pound taken from the online edition of Wall Street Journal on Friday, January 31, 2007 BRITISH POUND (CME) 62,500 pounds; cents per pound Strike Calls Price Feb Mar 162 2.36 2.94 163 1.5 2.32 164 0.86 1.7 165 0.5 1.36 166 0.26 1.02 167 0.12 0.76 • Contract specifications • Option positions • Hedging using option contract • Strategy on currencies option • Option pricing Apr 2.14 1.34 0.92 Feb 0.16 0.3 0.66 - Puts Mar 0.74 1.12 1.5 2.16 - Apr 2.02 - 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 6-Nov-15 FOREIGN CURRENCY OPTIONS 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 • 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 – Main advantage is that they are tailored to purchaser – Counterparty risk exists – Mostly used by individuals and banks – Spot exchange rate in the market • 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) PROFIT & LOSS FOR THE WRITER OF A CALL OPTION OPTION POSITIONS Profit (US cents/£) 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… + 10 +5 160 165 170 175 180 Spot price (US cents/£) Unlimited loss -5 - 10 Loss PROFIT & LOSS FOR THE BUYER OF A CALL OPTION “At the money” Strike price “Out of the money” Break-even price Limited profit Profit (US cents/£) Strike price The writer of a call option on £, with a strike price of 170cents/£, has a limited profit of cents/£ at spot rates less than 170, and an unlimited loss potential at spot rates above (to the right of) 175 cents/SF 11 PROFIT & LOSS FOR THE BUYER OF A PUT OPTION “At the money” Strike price Profit (US cents/£) “In the money” “In the money” “Out of the money” + 10 + 10 +5 +5 Profit up To 165 Unlimited profit -5 160 165 Limited loss 170 175 180 Spot price (US cents/£) 160 165 - 10 Loss 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 10 cents/£ (“in the money”) - 10 175 180 Spot price (US cents/£) Limited loss -5 Break-even price 170 Break-even price Loss The buyer of a put option on £, with a strike price of 170cents/£, has a limited loss of 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 12 PROFIT & LOSS FOR THE WRITER OF A CALL OPTION B1 BULL SPREAD CREATED USING CALL OPTIONBuying a call on a stock with a certain price and selling a call on the same stock with a higher price Profit (US cents/£) Strike price This strategy limits the investor’s upside potential as well as downside risk + 10 Break-even price +5 Profit Limited profit 160 165 170 175 Spot price (US cents/£) 180 -5 X1 X2 ST Loss up To 165 - 10 Loss The writer of a put option on £, with a strike price of 170 cents/£ has a limited profit of cents/£ at spot rates greater than 165 and a loss potential at spot rates less than 165 cents/£ 13 15 STRATEGIES INVOLVING A SINGLE OPTION AND A STOCK  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 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 Profit Profit x x ST ST X1 (a) X2 ST (b) Profit Profit x ST (c) x ST (d) 14 16 B5 BUTTERFLY SPREAD CREATED USING CALL OPTIONS- Buying one call at low price X1 and buying another call at high strike price X3 B3 BEAR SPREAD CREATED USING CALL OPTION Buying a call one exercise price and selling a call with another strike price and selling two call with a strike price X2, halfway between X1 & X3 Profit This strategy refer to an investment who fells that large stock price moves are unlikely This strategy limits the investor’s upside potential as well as downside risk Profit X1 X1 X2 X2 X3 ST ST 17 BASIC OPTION PRICING RELATIONSHIPS AT EXPIRY B4 BEAR SPREAD CREATED USING PUT OPTIONBuying a put one exercise price and selling a put with another strike price • 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] Profit X1 X2 19 ST 18 Copyright © 2014 by the McGraw-Hill Companies, Inc All rights reserved MARKET VALUE, TIME VALUE, AND INTRINSIC VALUE FOR AN AMERICAN CALL EUROPEAN OPTION PRICING RELATIONSHIPS Profit The red line shows the payoff at maturity, not profit, of a call option Long call  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, Ce > Max Intrinsic value Note that even an out-of-the-money option has value— time value Time value  Using a similar portfolio to replicate the upside potential of a put, we can show that: ST E Pe > Max Out-of-the-money Loss ST E – ,0 (1 + i£) (1 + i$) (1 + i$) In-the-money – ST ,0 (1 + i£) E Copyright © 2014 by the McGraw-Hill Companies, Inc All rights reserved Copyright © 2014 by the McGraw-Hill Companies, Inc All rights reserved OPTION PRICING AND VALUATION EUROPEAN OPTION PRICING RELATIONSHIPS The Black - Scholes formula for pricing the European foreign currency call and put are Consider two investments: Buy a European call option on the British pound futures contract The cash flow today is –Ce Replicate the upside payoff of the call by:  c  S0e where Borrowing the present value of the dollar, exercise price of the call in the U.S at i$ , the cash flow today is ST (1 + i£) Copyright © 2014 by the McGraw-Hill Companies, Inc All rights reserved  N(d1 )  Ee rhT  N(d2 ) F  ln  T   σ 2T E  d1    T d  d1   T c = premium on a European call p = premium on a European put S = spot exchange rate (domestic currency/foreign currency) ( r  r )T F = continuous compounding Forward rate Ft  St e h f 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) Lending the present value of ST at i£, the cash flow today is –  rf T p  [E  N(d2 )  FT  N(d1)]erhT E (1 + i$)  7-23 7-22 OPTION PRICING AND VALUATION e-rT = continuously compounding discount factor (e=2.71828182…) (1+12%)1  1.12 Inputs (1  12% / 2)  1.1236 (1  12% /12)12  1.126825 (1  12% / 365)365  1.127446 12%1 e  1.1274969 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., Spot rate (DC/FC e.g USD/EUR) Strike price volatility (annualized) domestic interest rate (annualized) foreign interest rate (annualized) time to maturity in days time to maturity in years 170 Call Price = 172 10.00% Put Price = 8.00% 7.80% 90 0.25 Outputs 2.4666 4.3453 N(x) =  x - n(x)dx=  x -  x2 e dx 2 27 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) 6-Nov-15 28 Exhibit: Intrinsic Value, Time Value & Total Value for a Call Option on British Pounds with a Strike Price of $1.70/£ Option Premium (US cents/£) Valuation on first day of 90-day maturity 6.0 5.67 Total value 5.0 4.00 4.0 3.30 OPTION PRICING AND VALUATION • The total value (premium) of an option is equal to the intrinsic value plus time value • Time value captures the portion of the option value due to the volatility in the underlying asset during the option life – The time value of an option is always positive and declines with time, reaching zero on the maturity date • Intrinsic value is the financial gain if the option is exercised immediately 3.0 2.0 1.67 – On the date of maturity, an option will have a value equal to its intrinsic value (due to the zero time value at maturity) Time value Intrinsic value 1.0 0.0 1.66 1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74 Spot Exchange rate ($/£) Exhibit: Intrinsic Value, Time Value & Total Value for a Call Option on British Pounds with a Strike Price of $1.70/£ FX Call Option Value and intrinsic value • 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” Call Value (Garman-Kohlhagen modified Black-Scholes) 20.00 18.00 16.00 Total value 14.00 12.00 10.00 8.00 6.00 3.30 Intrinsic value 4.00 2.00 0.00 157.25 CURRENCY OPTION PRICING SENSITIVITY Time value 159.80 162.35 164.90 167.45 170.00 172.55 175.10 177.65 180.20 182.75 185.30 Spot exchange rate DELTA DELTA • 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 c  e -rf T N(d1 ) > S p Delta  (for puts)   e -rf T N(-d1 ) < S – 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 Delta  (for calls)  • 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 DELTA • Time to maturity sensitivity (theta): # 1.00 0.90 0.80 Delta (N(d1) 0.70 0.60 0.50 0.40 0.30 0.20 – Option values increase with the length of time to maturity c Theta θ (for calls)  0 T p Theta θ (for puts)  0 T – A trader with find longer maturity options better values, giving trader the ability to alter an option position without suffering significant time value 0.10 0.00 d Spot exchange rate THETA • Sensitivity to volatility (Vega): # • 90 to 89 days:  premium cent 3.3  3.28 theta    0.02  time 90  89 • 15 to14 days theta   premium cent 1.37  1.32   0.05  time 15  14 • to days theta  VEGA  premium cent 0.7929  0.7093   0.08  time 54 • The rapid deterioration of option value in the last days prior to expriration day – The vega for calls and puts are the same c =Se -rf T n(d1 ) T  σ p Vega ν (for puts)  =Se-rf T n(d1 ) T  σ – 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 Vega ν (for calls)  VEGA Theta: Option Premium Time value Deterioration • Volatility increase 1%, from 10%  11%: Vega    premium $0.036  $0.033   0.30  volatility 11%  10% • If the volatility rise, the risk of the option being exercised is increasing, the option premium would be increasing ※ 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 10 Phi RHO AND PHI • Sensitivity to the domestic interest rate is termed as rho • British Pound interest rate increase 1%, from 8%  9%: c R h o ρ (fo r c a lls )  = K T e -rd T N (d ) >  rd R h o ρ (fo r p u ts )  p =  K T e -rd T N (-d ) <  rd ※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 P h i φ ( f o r c a ll s )  c =  S T e - rf T N ( d ) <  rf P h i φ (fo r p u ts )  p = S T e - rf T N ( - d ) >  rf Phi    premium $0.031  $0.033   0.2  BP int erest rate 9.0%  8.0% • If the £ interest rate increase of 1%, the ATM call option premium decrease from $0.033 to $0.031/£ • Phi value is -0.2 ※rf↑, domestic currency↑ , foreign currency↓, because the call (put) can fix the purchase (sale) price of the foreign currency, call↓ and put↑ Rho • US dollar interest rate increase 1%, from 8%  9%: Interest Differentials (rd – rf) and Call Option Premiums Option premium (U.S cents/£) 9.0  premium $0.035  $0.033 Rho     0.2  US$ int erest rate 9.0%  8.0% 8.0 7.0 6.0 ITM call (K=$1.65/£) • If the US dollar interest rate increase of 1%, the ATM call option premium increase from $0.033 to $0.035/£ 5.0 ATM call (K=$1.70/£) OTM call (K=$1.75/£) 4.0 3.0 2.0 1.0 0.0 -0.06 -0.04 -0.02 0.02 0.04 0.06 rUS$ – r£ ※ When the interest rate differential (rd – rf) increases, the foreign currency call value indeed increases 11 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 The higher (lower) the delta, the more likely the call (put) will move in-thefor a small change in the spot rate money Theta Θ Expected change in the option value For at-the-money options, premiums are for a small change in time to relatively insensitive until the final 30 expiration days Vega υ Expected change in the option value Option values rise with increases in for a small change in volatility volatility both for calls and puts Rho ρ Expected change in the option value Increases in domestic interest rates for a small change in domestic cause increasing call values and interest rate decreasing put values Phi φ Expected change in the option value Increases in foreign interest rates cause for a small change in foreign interest decreasing call values and increasing put rate values 12 [...]... Option premium (U.S cents/£) 9.0  premium $0.035  $0.033 Rho     0.2  US$ int erest rate 9.0%  8.0% 8.0 7. 0 6.0 ITM call (K=$1.65/£) • If the US dollar interest rate increase of 1%, the ATM call option premium increase from $0.033 to $0.035/£ 5.0 ATM call (K=$1 .70 /£) OTM call (K=$1 .75 /£) 4.0 3.0 2.0 1.0 0.0 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 rUS$ – r£ ※ When the interest rate differential (rd... 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 The higher ... 3.30 Intrinsic value 4.00 2.00 0.00 1 57. 25 CURRENCY OPTION PRICING SENSITIVITY Time value 159.80 162.35 164.90 1 67. 45 170 .00 172 .55 175 .10 177 .65 180.20 182 .75 185.30 Spot exchange rate DELTA DELTA... 1.66 1. 67 1.68 1.69 1 .70 1 .71 1 .72 1 .73 1 .74 Spot Exchange rate ($/£) Exhibit: Intrinsic Value, Time Value & Total Value for a Call Option on British Pounds with a Strike Price of $1 .70 /£ FX... in days time to maturity in years 170 Call Price = 172 10.00% Put Price = 8.00% 7. 80% 90 0.25 Outputs 2.4666 4.3453 N(x) =  x - n(x)dx=  x -  x2 e dx 2 27 OPTION PRICING AND VALUATION • The
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