FOREIGN CURRENCY OPTIONS
“In the mid 1990s, trading in exotic options began to develop at a rapid pace. Today, dealers
routinely supply two-way bid-ask prices for a wide spectrum of exotic currency options.”1
I. Definitions
Currency Options – provide payoffs that depend on the difference between the exercise price
and the exchange rate at maturity.
Currency Futures Options – provide payoffs that depend on the difference between the
exercise price and the exchange rate futures price at maturity.
“Because exchange rates and exchange rate futures prices generally are not equal, the options
and futures-options contracts will have different values, even with identical expiration dates and
exercise prices. Today, trading volume in currency futures options dominates by far trading in
currency options.”2
Definition of Derivatives – “any apparently complex financial instrument in which someone has
suffered a loss.”3
II. Spot Foreign Exchange and Market Conventions
“There is no organisation that decides what market convention should be for any aspect of
foreign exchange trading.”4
The most important locations for foreign exchange are London, New York, and Tokyo, although
foreign exchange trading takes place in every major city around the world.
The major exchange for trading foreign currency options in the United States is the Philadelphia
Stock Exchange. (www.phlx.com)
Spot Rate (S) – a quotation for currency to be delivered in two bank business days. One
exception is the USD/CAD which is delivered in one bank business day.
Pip – the smallest unit of quotation for a currency.
Value Date – foreign exchange settlement date (cannot be a bank holiday in either currency’s
country and it must not be a bank holiday in the United States).
1
Source: Options on Foreign Exchange, 2nd Edition by David F. DeRosa.
Source: Investments, by Zvi Bodie, Alex Kane, and Alan J. Marcus (Chapter 20) p.656
3 http://www.vtmagazine.vt.edu/sum96/derivatives.html.
4 Source: http://www.londonfx.co.uk/quoting.html
2
1
Source: http://www.coppclark.com/EDF_InteriorSpread.aspx
Bid/Ask - 125.00/125.10 means the dealer is willing to sell dollars (buy yen) at 125.10 and buy
dollars (sell yen) at 125.00.
Trading Rules:
 The first currency in an exchange rate pair is the direct object of the trade.
o Example: To buy $10M euro/dollar is to buy $10M euros against dollars. If S =
1.17 then you will receive 10M euros and you will pay 1.17M dollars.
 The rule in the professional currency market is that the higher currency on the list is the
one that deals against all currencies. See Table 1 below.
Table 1: Currency Hierarchy
Symbol* Name
Conversion Factor**
EUR
Euro
(6 significant digits)
GBP
British Pound
(6 significant digits)
MTL
Maltese Lira
(6 significant digits)
AUD
Australian Dollar
(6 significant digits)
NZD
New Zealand Dollar
(6 significant digits)
FJD
Fiji Dollar
(4 significant digits)
PYG
Paraguay Guarani
(5 significant digits)
USD
United States Dollar
(6 significant digits)
JPY
Japanese Yen5
(6 significant digits)
* Source: http://www.londonfx.co.uk/quoting.html
**Source: http://coinmill.com/sources.html
5
Source: Options on Foreign Exchange, 2nd Edition by David F. DeRosa, p. 14.
2
American terms – U.S. dollars per unit of foreign currency. The pound, Australian dollar, New
Zealand dollar, and the euro are quoted using the American convention. For example: $1.7/£.
European terms – The number of units of foreign currency per dollar. The Japanese yen is
quoted using the European convention. For example ¥120/$.
Note: Most exchange-traded currency futures and options quote currencies American, even for
currencies that are quoted European in the spot market!
III. Pricing Options
C = value of a call option
P = value of a put option
S = spot rate (d/f)
K = strike price (d/f)
rf = foreign interest rate
rd = domestic interest rate
t = t0
T = t1
T – t = τ = time to expiration
Value of a call at expiration:
Value of a put at expiration:
CT = Max[0, ST – K]
PT = Max[0, K – ST]
Max value of a call is the spot price:6
Max value of a put is the strike price
C≤S
P≤K
Min value of a call option:
C ≥ S0e
Min value of a put option:
r 
P ≥ Ke d  S0e
 rf 
 Ke rd
 rf 
where, τ = T – t
Assume:
 European options
 No – arbitrage
 Price cannot be < 0 (i.e. C ≥ 0, P ≥ 0)
USD put = JPY call
USD call = JPY put
Long USD call = Long JPY put
Short USD call = Short JPY put
Long USD put = Long JPY call
Short USD put = Short JPY call
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If this were not true an arbitrage profit would exist by selling the option and buying the currency.
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Put-Call Parity
C  Kerd  P  S0e
 rf 
Portfolio 1
Long put
Short call
Total Value
Portfolio 2
Long domestic bond
Short foreign bond
ST ≤ K
ST > K
K - ST
0
K - ST
0
-(ST – K)
K - ST
K
-ST
K - ST
K
-ST
K - ST
Figure 1a:Put-Call Parity
Profit
Profit
Payoff
Call Holder
Put Writer
Combined =
Leveraged Equity
Long Call
0
0
Exchange Rate
Call Writer
Put Holder
Short Put
Exchange Rate
Exchange Rate
Figure 1b:Put-Call Parity
Long £
Buy an ATM $/£ put
Profit
Profit
0
Exchange rate, $/£
0
Exchange rate, $/£
K
Profit
Sell an ATM $/£ call
Value of the Portfolio =
Exchange rate, $/£
E
0
K
Exchange rate, $/£
0
K
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Formulas
r 
r 
C = S0e f N (d1 )  Ke d N (d2 )
P = Ker  N (d2 )  S0er  N (d1 )
f
d
where,
2 
S  
ln  0    rd  rf 

2 
K 
d1 
 
2 
S  
ln  0    rd  rf 

2 
K 
d2 
 d1   
 
Example:7
Currencies:
Type:
S=
K=
Rf =
Rd =
τ=
USD/GBP
GBP call (USD put)
1.6000
1.6000
11%
8%
4 months = 120/360
σ=
14.1% volatility
Steps 1 & 2 calculate d1 & d2:
0.1412 120
 1.6000  
ln 

0.08

0.11

 

360
2 
 1.6000  
d1 
0.141 120
360
0   0.02006  .33
0.141 x 0.57735
0.00669

0.081406
 0.08214

d 2  d1   
  0.08241  0.141 120
360
  0.08241  0.081406
  0.16382
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Example 14.2, p. 322 from Options, Futures, & Other Derivatives, by John C. Hull.
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Steps 3 & 4 calculate N(d1) & N(d2):
N(d1) = N(-0.0821)
= N(-0.08) - 0.21[N(-0.08) – N(-0.09)]
= 0.4681 - 0.21 x (0.4681-0.4641)
= 0.4681 - 0.0008
= 0.4673
N(d2) = N(-0.1638)
= N(-0.16) - 0.38[N(-0.16) – N(-0.17)]
= 0.4364 - 0.38 x (0.4364-0.4325)
= 0.4364 - 0.0015
= 0.4349
Step 5 calculate C:
r 
C  S0 e f N (d1 )  Ke  rd N (d 2 )
 1.6e

0.11 120
360
 0.4673  1.6e0.08120 360 0.4349
 1.6e 0.0366 0.4673  1.6e 0.266 0.4349
 0.7207  0.6776
 0.0431
or 4.3 ¢
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Figure 2: USD/JPY Options
Source: ft.com
Figure 3: USD/EUR Options
Source: ft.com
Figure 4: USD/GBP Options
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Black-Scholes-Merton8
Three Assumptions:
1. Frictionless Markets - No taxes, transaction costs, no restrictions on taking long or short
positions, and no market control.
2. Interest rates are riskless, continuously compounded, and constant.
3. Spot rates change instantaneously and they have a normal distribution.
Options Calculators
Chicago Board Options Exchange

http://www.cboe.com/LearnCenter/OptionCalculator.aspx
Currency Options Calculators
CFO.com

http://tools.cfo.com/calc/BSCurrency/input.jsp
OZForex

http://www.ozforex.com.au/cgi-bin/optionsCalc.asp
IV: LEGO® Hedging9
LEGO® hedging requires:
1. Identification of the amount of foreign currency at risk and relevant future dates.
2. The value of the FX instrument selected (forwards, options or futures) must
change (∆F) so as to offset any change in the identified exposure (∆S).

∆V = ∆S - ∆F (where the change in value of the firm (∆V) is expected to
equal zero)
Example: A U.S. based firm with domestic dollar denominated sourcing costs and euro export
earnings may want to hedge adverse movements in the value of the dollar vs. euro.
– In other words, a fear that the dollar will appreciate vs. the euro.
– The firm could purchase a series of long maturity put options on the euro.
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The Black-Scholes (1973) model was extended by Merton (1973) to include the theoretical case of an option
on shares of a stock that pays dividends continuously, Garman and Kohlhagen (1983) adapted the model to
work for European options on foreign currencies.
9 Charles W. Smithson, “A LEGO Approach to Financial Engineering: An Introduction to Forwards, Futures,
Swaps, and Options,” Midland Corporate Finance Journal vol.4 (no.4, 1987), pp. 16-28.
Michael H. Moffett, Jan Karl Karlsen (1994) “Managing Foreign Exchange Rate Economic Exposure” Journal
of International Financial Management & Accounting 5 (2), 157–175.
Joseph K. Cheung and Richard Chung (1996) “Valuation of complex financial instruments via basic
components,” Review of Quantitative Finance and Accounting, Volume 7, Number 2 / September, 1996,
p. 163-176.
http://pages.stern.nyu.edu/~adamodar/pdfiles/eqnotes/valclose.pdf
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Challenges with LEGO® Hedging
1. Even with relatively predictable cash flows total exposure may be greater than the
straight-forward currency loss.
– Example: If export sales are reduced due to FX risk, the total exposure might
include lost economies of scale or market share.
2. While the foreign currency option will protect the firm from FX liabilities, the actual
upfront cost of the option may be prohibitive.
3. Tax laws may require the company to mark-to-market options at the end of the fiscal
year.
– In the U.S. companies can apply for “Hedge Accounting” to avoid marking-tomarket a financial hedge instrument.
– Hedge Accounting means both the instrument and the underlying exposure are
treated as if they were one single account.
4. The actual exposure of a firm will likely change over time.
V. Exotic Currency Options
Exotic currency option - an option with a nonstandard feature that sets it apart from an ordinary
vanilla currency option.
Barrier option – pays (or doesn’t pay) once the spot price hit a predetermined price. Barrier
options are the most popular exotic options.
Types of Barrier Options:

Knock-out option

Double-barrier currency option
Binary option – pays a lump sum if the option is in the money at expiration.
Types of Binary Options:

One touch

Double-barrier range binary
Other Types of Options

Basket option

Compound option

Average rate currency options

Quantos options
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VI. Final Comments
All varieties of barrier options depend on knowing whether the barrier has been struck and the
industry standard is to have the option-dealing bank assume the role of the determination agent
(despite the obvious conflict of interest).
Transparency
Foreign exchange transactions are private conversations between two parties (no public record of
trades exist).
Thin Markets
Trading done during the early Monday morning hours in Australia and New Zealand might not
be counted.
Cross rates
A number of disputes have arisen asking whether a cross-rate implies a barrier event.
www.isda.org
International Swaps and Derivatives Association
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