Transcript Ch08 - NTU

Chapter 8
Foreign Currency
Derivatives
The Goals of Chapter 8
• Introduce the foreign currency futures and foreign
currency options
• Study the fundamentals of their valuation and
trading strategies associated with them for
speculation
• Finally, the sensitivities of foreign currency option
values with respect to various determining factors
are analyzed
8-2
Foreign Currency Derivatives
• Financial management of the MNE in the 21st century
involves the use of financial derivatives
• Derivatives are so named because their values are
derived from underlying assets like the stock price or
the foreign currency exchange rate
• The derivatives can be used for two very distinct
management objectives:
– Speculation–use of derivative instruments to take a position
in the expectation of a profit
– Hedging–use of derivative instruments to reduce the risks
associated with the cash flows of corporate operations or
investments
• Two common foreign currency financial derivatives:
foreign currency futures and foreign currency options
– The valuation models and the use for speculative investment
8-3
Foreign Currency Derivatives
• Derivatives can be used to achieve following benefits:
1. To achieve payoffs that investors would not be able to achieve
without derivatives, or could achieve only at higher cost
2. Provide an alternative to hedge risks
• For price risks, derivatives can be used to purchase the asset at a fixed
price in the future; For future foreign cash flows, the foreign currency
derivatives can be used to minimize the corresponding volatility
3. Make underlying-asset markets more efficient
• Arbitrage transactions between derivatives and the underlying asset
could make underlying-asset markets more efficient and reduce the
volatility of returns of underlying asset
• However, the information from the derivatives market could enlarge
the fluctuation of the underlying asset price
4. Reduce tax liabilities
• Asymmetries in the tax across different countries: Derivatives can be
used to replace debts issued in tax-favored countries with debts in other
countries, e.g., through currency swaps introduced in Ch 9
5. Motivate management (to solve agency problems)
8-4
Foreign Currency Futures
8-5
Foreign Currency Futures
• A foreign currency futures contract is an agreement
for future delivery of an amount of foreign exchange at
a fixed time, place, and price
– Foreign currency futures are standard contracts traded on
an exchange, but foreign exchange forward (FX forward)
contracts are contracts traded in the over-the-counter market
– The other differences between foreign currency futures and
FX forward contracts are compared on Slide 8-9
• It is similar to futures contracts that exist for other
underlying assets, like gold, cattle, Treasury bonds, etc.
• The most important market in the world for foreign
currency futures is the CME group
– CME Group was created on July 12, 2007 from the merger
between the Chicago Mercantile Exchange (CME) and the
Chicago Board of Trade (CBOT)
8-6
Foreign Currency Futures
• Contracts of exchange-traded derivatives are standard
contracts established by the exchange on which the
derivatives are traded
• Major features of the foreign currency futures that are
standardized are as follows
– Contract size (also called the notional principal)
– Method of stating exchange rates (“American terms” are
used, i.e., the US$ price of one foreign currency)
– Maturity date (matured on the third Wednesday of Jan.,
Mar., Apr., Jun., Jul., Sept., Oct., and Dec.)
– Last trading day (the second business day prior to the
maturity date)
– Commissions (Customers pay a commission to their broker
for a round transaction, which differs from that in the
interbank market, dealers earn the bid and ask spread and do
not charge a commission)
8-7
Foreign Currency Futures
– Settlement
• Only 5% of all futures contracts are settled by the physical delivery
of foreign exchange between buyer and seller
• Most often, buyers and sellers offset their original positions prior to
the delivery date by taking an opposite position
– Collateral and margins
• To prevent the default risk, both sellers and buyers must deposit a
sum as the initial margin, which is a kind of cash collateral
• The value of the contract is marked to market and all changes in
value are paid from the margin account daily
• Marked to market means that the value of the contract is revalued
using the closing price for the day
• If the balance of the margin account falls below the maintenance
margin, the investor receives a margin call and needs to top up the
margin account to the initial margin level the next day
– Use of a clearing house as a counterparty (all contracts can be
viewed as agreements between an investor and the exchange
clearing house, rather than between two investors involved)
8-8
Exhibit 8.2 Comparisons between
Currency Futures and Forwards
8-9
Exhibit 8.1 Quotations for Currency
Futures (US$/Mexican Peso)
※ Comparing with the spot exchange rate (which is only one price), foreign currency
futures considers the maturity date as one more dimension, i.e., for a series of
maturity dates, there is one futures price for each maturity date
8-10
Payoff for Foreign Currency Futures
• Suppose the quotation of three-month futures contract
for 500,000 Mexican pesos (notional principal) is F3-mon
=$0.10958/Ps, which is called as futures price, settle
price or delivery price)
– The payoff (or the value at maturity) for the long position
Payoff at maturity = notional principal ×
(spot rate at maturity – delivery price)
– The payoff (or the value at maturity) for the short position
Payoff at maturity = – notional principal ×
(spot rate at maturity – delivery price)
– If the spot exchange rate after three months is $0.095/Ps, the
payoff for the long position is Ps500,000 × ($0.095/Ps –
$0.10958/Ps) = – $7,290, and the payoff for the short position
is $7,290
8-11
Value for Foreign Currency Futures
• Futures value (how much is the foreign currency future
worth today)
– After one month, if the two-month futures price is F2-mon, for
the long position
Futures value = notional principal × (F2-mon – delivery price)
/ (1 + rd(60/360))
– For the short position
Futures value = – notional principal × (F2-mon – delivery price)
/ (1 + rd(60/360))
where rd is the domestic interest rate, and in this case, the
domestic currency is the US$
– If the F2-mon is $0.11/Ps after one month, and rd is 6%, the
value for the long position is Ps500,000 × ($0.11/Ps –
$0.10958/Ps) / (1.01) = $207.92
8-12
Value for Foreign Currency Futures
• The delivery price is set such that the futures is worth
$0 initially
– In the beginning of the three month (the date on which the
investor enters into the futures contract), by setting the
delivery price to be the current three-month futures rate
F3-mon, the futures value is zero
$0 = notional principal × (F3-mon – delivery price)
/ (1 + rd(90/360))
• Thus, for foreign currency futures, investors cannot decide the
delivery price for themselves, and the delivery price should be the
current futures rate in the market
• In additional, since the value of a futures is zero when it is initiated,
both counterparties of a futures do not pay anything initially
※The above description about the futures price, the
futures value, and the delivery price is also applicable
to foreign exchange forward (FX forward) contracts
8-13
Foreign Currency Options
8-14
Foreign Currency Options
• A foreign currency option is a contract giving the
option purchaser (the buyer) the right, but not the
obligation, to buy or sell a given amount of foreign
exchange at a fixed per unit price for a specified time
period (until the maturity date)
• There are two basic types of options, puts and calls
– A call is an option to buy foreign currency
– A put is an option to sell foreign currency
• An American option gives the buyer the right to
exercise the option at any time between the date of
writing and the expiration or maturity date
• A European option can be exercised only on its
expiration date, not before
8-15
Foreign Currency Options
• The buyer of an option is termed the option holder,
while the seller of the option is referred to as the
option writer, issuer, or grantor
• Every option has three different price elements:
– The exercise or strike price (K) – the exchange rate at
which the foreign currency can be purchased (in call
contracts) or sold (in put contracts)
– The current underlying or spot exchange rate in the market
(S)
– The option premium (c or p) – the cost, price, or value of
the option itself (usually paid by the buyer to the seller on
the date the transaction is done)
8-16
Foreign Currency Options
• ITM (St > K) vs. ATM (St = K) vs. OTM (St < K)
– An option whose exercise price is the same as the current spot
price of the underlying currency is said to be at-the-money
(ATM)
– Excluding the cost of the premium, an option that would be
(not) profitable if exercised immediately is referred to as in-themoney (ITM) (out-of-the money (OTM))
• For the over-the-counter markets
– Foreign currency options is issued by financial institutions
– The main advantage of OTC options is that they are tailored to
the specific needs of the firm, that is, financial institutions are
willing to write or buy options that vary by amount (notional
principal), strike price, and maturity
– However, the participants in the OTC market are exposed to
counterparty risk, which is the risk that the other party in an
agreement may default on the final payment
8-17
Foreign Currency Options
• For the organized exchanges
– In 1982, the Philadelphia Stock Exchange introduced trading
in foreign currency option contracts in the U.S.
– The foreign currency option contracts on an organized
exchange is standardized
– Today, two famous organized exchanges for options on the
currency in the U.S. are the Philadelphia Stock Exchange
(PHLX) and the CME group
– Exchange-traded options are settled through a clearing house,
so buyers do not deal directly with sellers
– The clearing house can be viewed as the counterparty to
every option contract, and it guarantees the fulfillment of the
option contracts, so the counterparty risk is substantially
reduced
8-18
Exhibit 8.3 Swiss Franc Option
Quotations (U.S. cents/SF)
※Comparing with the futures exchange rate, in addition to the maturity date, foreign
currency options consider one more dimension – different strike prices
※The August 58½ call option premium is 0.5 cents per franc, so the option value of one
this contract is SF62,500×$0.005/SF = $312.5
※Premiums are quoted as a domestic currency amount per unit of foreign currency (In the
OTC market, premiums are quoted as a percentage of the transaction amount)
8-19
Speculation Strategies in
Foreign Currency Markets
8-20
Foreign Currency Speculation
• Speculation is an attempt to profit by trading on
expectations about prices in the future
– Spot market
• When the speculator believes the foreign currency will
appreciate (depreciate) in value, they buy (sell) foreign currency
• If the foreign currency really appreciate (depreciate), selling
(buying back) foreign currency will make a profit
– Forward or futures market
• When the speculator believes the spot exchange rate on some
future date will be higher (lower) than today’s forward or futures
exchange rate for the same date, take the long (short) position on
foreign exchange forwards
• If the spot exchange rate on that future date is really higher
(lower) than the forward or futures exchange rate, fulfilling the
forward contract will bring profit
– Options markets
• Extensive differences in risk patterns depending on purchase or
sale of put and/or call
8-21
Option Market Speculation
• Buyer of a call:
– Consider a purchase of August call option on Swiss francs
with the strike price of 58½ ($0.5850/SF), and a premium
of $0.005/SF
– In August, at all spot rates below the strike price of 58.5,
the purchase of the option would choose not to exercise
because it would be cheaper to purchase SF on the open
market
– For all spot rates above the strike price, the option
purchaser would exercise the option, purchase SF at the
strike price and sell them into the market netting a positive
payoff (the profit is the payoff less the option premium)
– Payoff = notional principal ×
max(spot rate at maturity – strike price, 0)
8-22
Exhibit 8.4 Profit and Loss for the
Buyer of a Call Option on Swiss francs
“At the money”
Strike price
Profit
(US cents/SF)
“Out of the money”
“In the money”
+ 1.00
+ 0.50
0
- 0.50
Unlimited profit
57.5
58.0
58.5
59.0
59.5
Spot price
(US cents/SF)
Limited loss
Break-even price
- 1.00
Loss
The buyer of a call option on SF, with a strike price of 58.5 cents/SF, has a limited loss of
0.50 cents/SF at spot rates less than 58.5 (“out of the money”), and an unlimited profit
potential at spot rates above 58.5 cents/SF (“in the money”)
8-23
Option Market Speculation
• The advantage of using forwards (futures) or options
for speculation
– Comparing to investing on the spot asset, they both are
highly leveraged investments
• For forwards or futures, it is not necessary to pay anything
initially, but if the expectation of the investor is correct, the
investor can earn a profit
• For options, the buyer needs only to pay the premium
initially, i.e., 0.50 cents/SF in the above case (58.5 cents to
buy one unit of SF vs. 58.5 cents to purchase 58.5 cents/0.50
cents = 117 shares of call options)
– Comparing to forwards or futures, options are with an
additional advantage of limiting the potential losses
8-24
Option Market Speculation
• Writer of a call:
– When the buyer of an option loses, the writer gains
– Payoff = – notional principal ×
max(spot rate at maturity – strike price, 0)
– The maximum profit that the writer of the call option can
make is limited to the option premium he receives when
selling the call
– The amount of such a loss is unlimited and increases as the
underlying currency rises (see Exhibit 8.5)
– If the writer wrote the option naked, i.e., without owning
the currency, the writer would have to buy the currency at
the spot and take the loss of delivering at the strike price
– Even if the writer already owns the currency, the writer will
experience an opportunity loss
8-25
Exhibit 8.5 Profit and Loss for the
Writer of a Call Option on Swiss francs
“At the money”
Strike price
Profit
(US cents/SF)
+ 1.00
+ 0.50
0
- 0.50
Break-even price
Limited profit
57.5
58.0
58.5
59.0
59.5
Spot price
(US cents/SF)
Unlimited loss
- 1.00
Loss
The writer of a call option on SF, with a strike price of 58.5 cents/SF, has a limited
profit of 0.50 cents/SF at spot rates less than 58.5, and an unlimited loss potential at
spot rates above (to the right of) 59.0 cents/SF
8-26
Option Market Speculation
• Buyer of a Put:
– The buyer of a put option, however, wants to be able to sell
the underlying currency at the exercise price when the
market price of that currency drops
– If the spot price drops to $0.575/SF, the buyer of the put
will deliver francs to the writer and receive $0.585/SF
– At any exchange rate above the strike price of 58.5 cents,
the buyer of the put would not exercise the option, and
would lose only the $0.05/SF premium
– Payoff = notional principal ×
max(strike price – spot rate at maturity, 0)
– The buyer of a put (like the buyer of the call) can never
lose more than the premium paid up front
8-27
Exhibit 8.6 Profit and Loss for the
Buyer of a Put Option on Swiss francs
“At the money”
Strike price
Profit
(US cents/SF)
“In the money”
“Out of the money”
+ 1.00
+ 0.50
0
Profit up
to 58.0
57.5
58.5
59.0
59.5
Spot price
(US cents/SF)
Limited loss
- 0.50
- 1.00
58.0
Break-even
price
Loss
The buyer of a put option on SF, with a strike price of 58.5 cents/SF, has a limited loss of
0.50 cents/SF at spot rates greater than 58.5 (“out of the money”), and a profit potential
at spot rates less than 58.5 cents/SF (“in the money”) up to 58.0 cents per SF
8-28
Option Market Speculation
• Seller (writer) of a put:
– For put, if the spot price of francs drops below 58.5 cents
per franc, the option will be exercised
– If the spot price is above $0.585/SF, the option will not be
exercised and the option writer will pocket the entire
premium
– Payoff = – notional principal ×
max(strike price – spot rate at maturity, 0)
– Below the price of 58 cents per franc, the writer will lose
more than the premium received for writing the option
(falling below break-even price)
8-29
Exhibit 8.7 Profit and Loss for the
Writer of a Put Option on Swiss francs
“At the money”
Profit
(US cents/SF)
Strike price
+ 1.00
+ 0.50
Break-even
price
Limited profit
0
57.5
58.0
58.5
59.0
59.5
Spot price
(US cents/SF)
- 0.50
- 1.00
Loss up
to 58.0
Loss
The writer of a put option on SF, with a strike price of 58.5 cents/SF, has a limited profit
of 0.50 cents/SF at spot rates greater than 58.5, and a loss potential at spot rates less than
58.5 cents/SF up to 58.0 cents per SF
8-30
The Valuation of Foreign
Currency Options
8-31
Option Pricing and Valuation
• The pricing of currency options depends on six
parameters (factors):
–
–
–
–
–
–
Present spot exchange rate ($1.7/￡)
Time to maturity (90 days)
Strike price ($1.7/￡)
Domestic risk free interest rate (r$ = 8%)
Foreign risk free interest rate (r￡ = 8%)
Volatility (standard deviation of spot price percentage
changes) (10% per annum)
※ Based on the above parameters, the call option premium
is $0.033/￡(this result is calculated based on the BlackSholes formula in the excel file “Foreign Currency
Option.xlsm”)
8-32
Option Pricing and Valuation
• The Black-Sholes formulae for pricing the European
foreign currency call and put are
c=erf TSN(d1 )  erdT KN(d2 )
p=erdT KN(d2 )  erf TSN(d1 )
where
ln(S/K)  (rd  rf  σ2 /2)T
d1 
, and d 2  d1  σ T
σ T
c = premium on a European call
p = premium on a European put
S = spot exchange rate (domestic currency/foreign currency)
K = exercise or strike price, T = time to maturity
rd = domestic interest rate, rf = foreign interest rate
σ = standard deviation of percentage changes of the exchange rate 8-33
Option Pricing and Valuation
e-rT = continuously compounding discount factor
(e=2.71828182…) (1+12%)1  1.12
(1  12% / 2) 2  1.1236
(1  12% /12)12  1.126825
(1  12% / 365)365  1.127446
e12%1  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.,
x
x
1 x
2
N(x) =  n(x)dx= 
-
-
2
e
2
dx
8-34
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
– On the date of maturity, an option will have a value equal to
its intrinsic value (due to the zero time value at maturity)
8-35
Exhibit 8.8 Intrinsic Value, Time Value & Total Value for a
Call Option on British Pounds with a Strike Price of $1.70/£
Option Premium
(US cents/£)
-- Valuation on first day of 90-day maturity --
6.0
5.67
Total value
5.0
4.00
4.0
3.30
3.0
2.0
1.67
Time value
Intrinsic
value
1.0
0.0
1.66
1.67
1.68
1.69
1.70
1.71
1.72
1.73
1.74
Spot Exchange rate ($/£)
8-36
The Greeks of Foreign
Currency Options
8-37
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”
8-38
Delta
• Spot rate sensitivity (delta):
– The sensitivity of the option value to a small change in
the spot exchange rate is called the delta
c
 e -rf T N(d1 ) > 0
S
p
Delta  (for puts) 
 e -rf T N(-d1 ) < 0
S
Delta  (for calls) 
– 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
8-39
Delta
– For the example on Slide 8-32, 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. However, the
Black-Sholes formula tells us the correct option value should
be $0.1033)
– The larger the absolute value of Delta, the larger risk the
portfolio is exposed to the exchange rate changes
– Delta hedge: try to construct a portfolio with zero Delta,
such that the value of the portfolio remains the same for the
small change of the exchange rate S
8-40
Theta
• Time to maturity sensitivity (theta):
– Option values increase with the length of time to maturity
c
Theta θ (for calls) 
0
T
p
Theta θ (for puts) 
0
T
– Since options provide holders the right to fix the purchase or
the sale prices at a future time point, this right to fix prices
should be more valuable for longer time to maturity
– For a larger value of theta, a small decrease of the time to
maturity will reduce the option value substantially
– Thus, theta measures the speed of decay of the option value
– The time decay of the option value is totally from the decrease
of the time value (because the change of T will not affect the
intrinsic value of the option)
8-41
Exhibit 8.11 Theta: Time Value Decay
for ITM, ATM, and OTM Calls
※ 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
8-42
Vega
• Sensitivity to volatility (Vega):
– The vega for calls and puts are the same
c
Vega ν (for calls) 
=Se-rf T n(d1 ) T  0
σ
p
Vega ν (for puts) 
=Se-rf T n(d1 ) T  0
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
8-43
Vega
– The primary problem with volatility is that it is unobservable,
so we often use the historical exchange rate to estimate the
volatility
– Volatility is measured by the standard deviation of percentage
changes in the underlying exchange rate
– If the standard deviation of daily percentage changes in the
exchange rate is 0.007, the annual volatility is calculated as
0.007  252  0.11  11%
where 252 is the trading days in one year
– We do not use the calendar days of 365, because there are no
transactions on holidays and thus holidays do not contribute to
the annual volatility
– Another problem is that the historical volatility is not
necessarily an accurate predictor of the future volatility of the
exchange rate’s movement
8-44
Vega
• Volatility is viewed in three ways:
– Forward-looking volatility
• It is the expected volatility about a future period time over which
the option will exist
• Theoretically, we should use this volatility for pricing option,
but it is difficult to forecast the volatility of the exchange rate
about a future period of time
– Historical volatility
• It is drawn from a recent period of time
• If option traders believe that the immediate future will be the
same as the recent past, the forward-looking volatility will equal
the historical volatility
– Implied volatility
• Because volatility is the only unobservable parameter of the
option price, after all other components are accounted for, the
implied volatility is the volatility implied by the market price
• More specifically, the implied volatility is derived based on
matching the theoretical and market option value
8-45
Vega
• Implied volatility reflect the consensus of option traders about
the expected volatility for a future period
• The following table shows the implied volatilities of the foreign
exchange rates with different time to maturity
• Option volatilities vary considerably across currencies, and the
relationship between volatility and maturity is not monotonic
8-46
Vega
• All currency pairs have historical series that contribute
to the formation of the expectations of option writers
about volatility
• In the end, the truly talented option writers are those
with the intuition and insight to price the future
effectively
• Speculation strategy based on the expectation of future
volatility
– Traders who believe that volatilities will fall significantly in
the near-term will sell options now, hoping to buy them back
for a lower price and thus make a profit, because the
immediate volatility fall will cause option values to decrease
8-47
Rho and Phi
• Sensitivity to the domestic interest rate is termed as
rho
c
Rho ρ (for calls) 
=KTe-rd T N(d 2 ) > 0
rd
p
Rho ρ (for puts) 
=  KTe-rd T N(-d 2 ) < 0
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
Phi φ (for calls) 
c
=  STe-rf T N(d1 ) < 0
rf
Phi φ (for puts) 
p
=STe-rf T N(-d1 ) > 0
rf
※rf↑, domestic currency↑ , foreign currency↓, because the call (put)
can fix the purchase (sale) price of the foreign currency, call↓ and put↑
8-48
Rho and Phi
• For calls, since ρ > 0 (rd↑  c↑) and φ < 0 (rf↑  c↓),
the option values increase as the interest rate differential
(rd – rf) increases
• For puts, since ρ < 0 (rd↑  p↓) and φ > 0 (rf↑  p↑),
the option values decrease as the interest rate differential
(rd – rf) increases
• According to the IRP, the forward rate is at a higher
premium if the interest rate differential (rd – rf) increase
Fndomestic dollars / foreign dollar
n 

1   rd 

360 

domestic dollars / one foreign dollar
S

n 

1   rf 

360 

• Thus we can conclude that when the forward rate is at a
higher premium, the foreign currency call value
increases and the foreign currency put value decreases
8-49
Exhibit 8.13 Interest Differentials (rd –
rf) and Call Option Premiums
※ When the interest rate differential (rd – rf) increases, the foreign currency
call value indeed increases
※ Note that Exhibit 8.13 in the text book is wrong, and the correct one is
shown above and in the excel file “Foreign Currency Option.xlsm”
8-50
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
8-51
Sensitivity to the strike price
• The sixth and final element that is important to option
valuation is the selection of the strike price
• The sensitivity to the strike price for calls and puts
c
For calls,
=  e -rd T N(d 2 ) < 0, so K  c 
K
p -rd T
For puts,
=e N(-d 2 ) > 0, so K  p 
K
• Investors prefer calls (puts) with lower (higher) K, but
these options are more expensive
• A firm must make a choice as per the strike price it
wishes to use in constructing an option
• Consideration must be given to the tradeoff between
favorable strike prices and costs of premiums
8-52
Exhibit 8.15 Summary of Option Value
Sensitivity
Greek
Definition
Interpretation
Delta Δ
Expected change in the option The higher (lower) the delta, the
value for a small change in
more likely the call (put) will
the spot rate
move in-the-money
Theta Θ
Expected change in the option For at-the-money options,
value for a small change in
premiums are relatively
time to expiration
insensitive until the final 30 days
Vega υ
Expected change in the option Option values rise with increases
value for a small change in
in volatility both for calls and
volatility
puts
Rho ρ
Expected change in the option Increases in domestic interest
value for a small change in
rates cause increasing call values
domestic interest rate
and decreasing put values
Phi φ
Expected change in the option Increases in foreign interest rates
value for a small change in
cause decreasing call values and
foreign interest rate
increasing put values
8-53