CURRENCY OPTIONS

1
CURRENCY OPTIONS
2
CURRENCY OPTIONS

• An option is a contract in which the buyer of the
  option has the right to buy or sell a specified
  quantity of an asset, at a pre-specified price,
  on or upto a specified date if he chooses to do
  so however, there is no obligation for him to do
  so
• Options are available on a large variety of
  underlying assets including common stock, stock
  indices, currencies, debt instruments, and
  commodities.
• Options are also available on financial prices
  such as interest rates.
• Options on forward and futures contracts, options
  on swaps and finally options on options are also
  traded.

3
Options on Spot, Options of Futures and Futures
Style Options

• Option on Spot Currency Right to buy or sell the
  underlying currency at a specified price no
  obligation
• Option on Currency Futures Right to establish a
  long or a short position in a currency futures
  contract at a specified price no obligation
• Futures-Style Options Represent a bet on the
  price of an option on spot foreign exchange.
  Margin payments and mark-to-market as in futures.

4
Options Terminology

• The two parties to an option contract are the
  option buyer and the option seller also called
  option writer
• Call Option A call option on currency Y against
  currency X gives the option buyer the right to
  purchase currency Y against currency X at a
  stated price Y/X, on or any time upto a stated
  date.
• Put Option A put option on currency Y gives the
  option buyer the right to sell currency Y against
  currency X at a specified price on or any time
  upto a specified date.

5
Options Terminology (contd.)

• Strike Price (also called Exercise Price) The
  price specified in the option contract at which
  the option buyer can purchase the currency (call)
  or sell the currency (put) Y against X. Maturity
  Date The date on which the option contract
  expires. Exchange traded options have
  standardized maturity dates.
• American Option An option, that can be exercised
  by the buyer on any business day from trade date
  to expiry date.
• European Option An option that can be exercised
  only on the expiry date

6
Options Terminology (contd.)

• Option Premium (Option Price, Option Value) The
  fee that the option buyer must pay the option
  writer up-front. Non-refundable.
• Intrinsic Value of the Option The intrinsic
  value of an option is the gain to the holder on
  immediate exercise. Strictly applies only to
  American options.
• Time Value of the Option The difference between
  the value of an option at any time and its
  intrinsic value at that time is called the time
  value of the option.

7
Options Terminology (contd.)

• A call option is said to be at-the-money-spot if
  Current Spot Price (St ) Strike Price (X),
• in-the-money-spot if St gt X and
  out-of-the-money-spot if St lt X
• A put option is said to be at-the-money-spot if
• St X, in-the-money-spot if St lt X and
  out-of-the-money-spot if St gt X
• In the money options have positive intrinsic
  value at-the-money and out-of-the money options
  have zero intrinsic value.
• Practitioners compare the strike price with the
  forward rate for the same expiry date.

8
Options Terminology (contd.)
Thus at time t a call (put) option expiring at
time T is ATMF at the money forward if
X Ft,T (X Ft,T) ITMF in the money
forward -if X lt Ft,T (X gt
Ft,T) OTMF out of the money forward - if
X gt Ft,T (X lt Ft,T)
9
PHLX EUR/USD CALLS EXPIRY END DECEMBER 2008
QUOTES AS ON DECEMBER 4. SPOT RATE
1.2764 Symbol Bid Ask Strike price
(Cents per
EUR) (Cents per
EUR) ECDLR 8.43 8.79 119.50 ECDLV 7.52 7.85
120.50 ECDLC 6.70 6.95 121.50 ECDLG 5.83 6
.10 122.50 ECDLK 5.01 5.24 123.50 XDELZ 4.
61 4.85 124.00 ECDLO 4.20 4.40 124.50 XDELX
3.90 4.10 125.00 ECDLS 3.50 3.70 125.50 XD
ELY 3.20 3.40 126.00 ECDLW 2.92 3.10 126.5
0 XDELB 2.62 2.78 127.00 ECDLD 2.36 2.50 1
27.50 XDELF 2.08 2.22 128.00 ECDLH 1.81 2.00
128.50 XDELJ 1.63 1.79 129.00 EPALL 1.42
1.58 129.50 XDELN 1.24 1.41 130.00 EPALP 1
.06 1.26 130.50
10
PHLX EUR/USD CALLS EXPIRY END MARCH 2009
CENTS PER EUR
11
Symbol Bid
Ask Strike Price
ECDOK 3.80 3.95 123.50 XDEOZ 3.95 4.15
124.00 ECDOO 4.15 4.30 124.50 XDEOX 4.35 4
.55 125.00 ECDOS 4.60 4.75 125.50 XDEOY
4.80 4.95 126.00 ECDOW 5.05 5.15 126.50 XD
EOB 5.25 5.40 127.00 ECDOD 5.50 5.65 127.
50 XDEOF 5.75 5.90 128.00 ECDOH 6.05 6.20
128.50 XDEOJ 6.30 6.40 129.00 EPAOL 6.55
6.70 129.50 XDEON 6.85 7.00 130.00 EPAOP
7.15 7.25 130.50
PHLX EUR/USD PUTS EXPIRY MARCH 2009 CENTS PER
EUR, DEC 4, 2008
12
PHLX USD/CHF CALLS EXPIRING IN NOVEMBER
2008 (QUOTES AS ON
SEPT 3, 2008) (STRIKES AND PREMIA QUOTED
AS CENTS PER CHF) Symbol
Bid Ask Strike Price
XDSKG 3.55 3.79
88.00 SIQKI
3.23 3.44 88.50
XDSKK 2.91
3.12 89.00 SIQKM
2.60 2.82
89.50 XDSKO 2.32
2.55 90.00 SIQKQ
2.06 2.29
90.50 XDSKS 1.82
2.06 91.00 SIQKU
1.59 1.85
91.50 XDSKW
1.41 1.66 92.00 CHF/USD
SPOT RATE 90.85
13
PHLX GBP/USD PUTS EXPIRING IN NOVEMBER 2008
(QUOTES AS ON SEPT 3, 2008)
(STRIKES AND PREMIA QUOTED AS CENTS PER GBP)
STRIKE BID
ASK 178.00
0.39 0.84
181.50 1.42
1.87 187.00
5.08 5.58
194.00 11.73
12.23 205.50
23.20
23.71 215.00
32.69 33.19 GBP/USD
CLOSING SPOT 1.8232
14
PHLX CURRENCY OPTION PRICE QUOTES (March 30,
2007, Cents per GBP) GBP/USD AMERICAN OPTIONS
(CONTRACT SIZE 31250)  Strike
CALLS PUTS
Price Apr May Jun
Apr May Jun   196.00
1.55 2.21 2.80 0.70
1.39 2.00 197.00
1.00 1.69 1.15
1.87 198.00 0.62 1.26
1.84 1.75 2.44 3.10 200.00
0.27 0.71 1.18 3.50
3.87 4.32
15
EUR/USD
EUROPEAN OPTIONS (CONTRACT
SIZE 62500) (Cents per
EUR)  Strike CALLS
PUTS Price
Apr May Jun Apr May
Jun   131.00 2.13 2.44
2.75 0.05 0.22 0.43
133.00 1.13 1.66 2.16
0.32 0.71 1.00 135.00
0.25 0.72 1.13 1.43 1.73
1.99 137.00 0.08 0.29
0.61
16
USD/YEN OPTIONS QUOTES (CME) Oct 10, 2005
Strike price CALLS
PUTS   Dec
Mar Jun Dec Mar
Jun   8600 2.77 -
- 0.33 0.58 0.76  
8700 2.02 3.15 4.29
0.57 0.84 1.02   8800
1.40 2.50 3.62 0.94 1.18
1.33   8900 0.94 -
3.04 1.48 1.64
1.73   Source Reuters/CME.
17
Source UBS June 6, 2006
18
(No Transcript)
19
Elementary Option Strategies

• Assumptions
• Ignore brokerage commissions, margins etc
• Dealing with European options
• All exchange rates, strike prices, and premiums
  will be in terms of home currency per unit of
  some currency A and the option will be assumed to
  be on one unit of the currency A
• Profit profiles shown at maturity

20
Elementary Option Strategies

• Call Options
• Current spot rate, St
• Strike price X
• Call option premium c
• Spot rate at maturity ST
• Call Option Buyers Profit -c for ST ? X
• ST - X - c for ST gt X
• Call Option Writers Profit c for ST ? X
• 
  -(ST - X - c)
• for ST gt X

21
A CALL OPTION A trader buys a call option on US
dollar with a strike price of Rs.49.50 and pays a
premium of Rs.1.50. The current spot rate, St, is
Rs.48.50. His gain/loss at time T when the option
expires depends upon the value of the spot rate,
ST, at that time USD/INR ST AT EXPIRY
Option Buyers Gain()/Loss(-)
48.2500
-Rs.1.50 48.5000
-Rs.1.50
48.7500
-Rs.1.50 49.0000

-Rs.1.50 49.2500
-Rs.1.50
49.5000
-Rs.1.50 49.7500
-Rs.1.25
50.0000
-Rs.1.00 51.0000

Rs.0.00 52.0000
Rs.1.00
54.5000
Rs.3.50 56.0000
Rs.5.00
22
Elementary Option Strategies
Payoff Profile of a Call Option

c 1.50
O

ST
c 1.50
-
X49.50
Xc51.00
Breakeven Spot Rate
Option Buyer
ST SPOT RATE AT EXPIRY
Option Seller
23
Elementary Option Strategies

• Put Options Premium p
• Put Option Buyer's Profit
• -p for ST ? X
• X - ST p for ST lt X
• Put Option Writers Profit
• p for ST ? X
• -(X - ST - p) for ST lt X

24
A PUT OPTION A trader buys a put option on pound
sterling at a strike price of 1.8500, for a
premium of 0.07 per sterling. The spot rate at
the time is 1.9465. At expiry, his gains/losses
are as follows GBP/USD ST AT EXPIRY Option
Buyers Gain()/Loss(-) 1.7000
0.0800
1.7300
0.0500 1.7500
0.0300
1.7600
0.0200 1.7800
0.0000
1.7900
-0.0100 1.8300
-0.0500
1.8500
-0.0700 1.8700
-0.0700
1.9000          
-0.0700 1.9500
-0.0700
25
Elementary Option Strategies
Payoff Profile of a Put Option

p0.07
O
p0.07 ST
X-p1.78
-
X1.85
Option Buyer
Breakeven Spot Rate at Option Expiry
Option Seller
26
Elementary Option Strategies

• Spread Strategies
• Bullish Call Spread Consists of selling the call
  with the higher strike price and buying the call
  with the lower strike price
• Bearish Call spread If the investor expects the
  foreign currency to depreciate, he can adopt the
  reverse strategy viz. buy the higher strike call
  and sell the lower strike call
• Bullish Put Spread Consists of selling puts with
  higher strike and buying puts with lower strike
• Bearish Put Spread Opposite of Bullish Put
  Spread
• These strategies, involving options with same
  maturity but different strike prices are called
  Vertical or Price Spreads

27
A Bullish Call Spread The CHF/USD spot rate is
0.75. April calls with strike 0.70 are trading at
0.07 and calls with strike 0.80 at 0.005. Sell
the call with the higher strike price and buy the
call with the lower strike price. Profits at
expiration are as below   ST Gain/Loss
Gain/Loss Net on
Short on Long Gain/loss   0.6000
0.005 -0.070 -0.065
0.6500 0.005 -0.070 -0.065
0.7000 0.005 -0.070
-0.065 0.7500 0.005
-0.020 -0.015 0.7650 0.005
-0.005 0.000 0.7800
0.005 0.010 0.015 0.8000
0.005 0.030 0.035
0.8500 -0.045 0.080 0.035
0.9000 -0.095 0.130
0.035
28
Bull Spread Using CallsBuy Call Strike X1,
Premium c1 Sell Call Strike X2, Premium c2

Profit
c2
ST
X1
X2
c1
PROFIT PROFILE OF THE SPREAD STRATEGY
29
Bull Spread Using Puts Buy Put Strike X1,
Premium p1 Sell Put Strike X2, Premium p2
p2
p1
30
Bear Spread Using Calls
BUY CALL STRIKE X2 SELL CALL STRIKE X1
PROFIT PROFILE
Profit
X1
X2
ST
31
Elementary Option Strategies Butterfly Spreads  
This is an extension of the idea of vertical
spreads. Suppose the current spot rate NZD/USD is
0.6000. The call options with same expiry date
are available   Strike Premium
  0.58 0.07
0.62 0.03
0.66 0.01 A Butterfly
Spread is bought by buying two calls with the
middle strike price of 0.62, and writing one call
each with strike prices on either side, here,
0.58 and 0.66. The profit table is as follows
32
A BUTTERFLY SPREAD (Contd.)
ST Gain on Gain on
Gain on Net 0.62
call 0.58 call 0.66 call Gain
(long 2) (short 1) (short
1)   0.5000 -0.06
0.07 0.01 0.02
0.5200 -0.06 0.07
0.01 0.02 0.5600
-0.06 0.07 0.01
0.02 0.5800 -0.06
0.07 0.01
0.02 0.5900 -0.06
0.06 0.01 0.01
0.6000 -0.06 0.05
0.01 0.00
0.6100 -0.06 0.04
0.01 -0.01 0.6200
-0.06 0.03 0.01
-0.02 0.6400 -0.02
0.01 0.01 0.00
0.6500 0.00 0.00
0.01 0.01
0.6600 0.02 -0.01
0.01 0.02 0.6800
0.06 -0.03 -0.01
0.02
33
Elementary Option Strategies
Butterfly Spread
Payoff Profile of a Long Butterfly Spread
Payoff Profile of a Short Butterfly Spread
34
Elementary Option Strategies

• Horizontal or Time Spreads
• Horizontal spreads consist of simultaneous
  purchase and sale of two options identical in all
  respects except the expiry date
• The difference in premiums between the two
  options will be moderate at the time of
  initiation but will have widened at the time of
  expiry of the short term option provided the
  underlying exchange rate has not moved
  drastically

35
Elementary Option Strategies

• Straddles and Strangles Volatility Bets
• A long straddle consists of buying a call and a
  put both with identical strikes and maturity.
  Usually both are at-the-money-spot.
• A long strangle consists of buying an out-of-the-
  money call and an out-of-the-money put
• Both are bets that the underlying price is going
  to make a strong move up or down I.e. market is
  going to be more volatile.

36
Straddles and Strangles  A straddle consists of
buying a call and a put both with identical
strikes and maturity. As an example, suppose
sterling December call and put options with a
strike of 1.7250 are priced at 2.95 cents and
1.24 cents respectively. Profits for alternative
values of ST are   ST Gain on Call
Gain on Put Net Gain   1.6500
-2.95 6.26 3.31
1.6831 -2.95 2.95
0.00 1.7000 -2.95
1.26 -1.69 1.7250
-2.95 -1.24 -4.19
1.7669 1.24 -1.24
0.00 1.8000
4.55 -1.24 3.31
37
Elementary Option Strategies
Payoff Profile of a Straddle

X
0
ST
Xpc
X-p-c
-
X Strike price in put and call c Call premium


p Put Premium ST Spot rate at
expiry
38
Elementary Option Strategies
(X1 p c)
(X2 p c)
X1 Call strike X2 Put strike p Put premium c
Call premium ST Spot rate at Expiry

0
ST
-
X2
X1
Payoff Profile of a Strangle
39
Elementary Option Strategies Strip Strap
CALLS PUTS SAME STRIKE AND EXPIRY DATE

Profit
Profit

ST
ST
K
K
O
-
LONG STRIP
LONG STRAP
LONG (1 CALL2 PUTS)
LONG (2 CALLS1 PUT)
40
Hedging with Currency Options

• Hedge a Foreign Currency Payable with a Call.
• Hedge a Receivable with a Put Option
• Covered Call Writing. Earn a premium by writing
  a call against a receivable.
• Options are a convenient hedge for contingent
  liabilities (Note however that the risk of the
  liability materialising or not cannot be hedged
  with the option)
• Options allow hedger to bet on favorable currency
  movements with limited downside risk.

41
Over-The-Counter (OTC) Market Practices

• Like in the forex market, dealers trade directly
  with each other and through brokers
• Unless a quote for a specific option - call or
  put - is requested, the market practice is to
  quote a two way-price in terms of implied
  volatility for an At-the-Money- Forward (ATMF)
  straddle for a given period

42
Futures Options

• The underlying asset in this case is a futures
  contract
• A call option on a futures contract, if
  exercised, entitles the holder to receive a long
  position in the underlying futures contract plus
  a cash amount equal to the price of the contract
  at that time minus the exercise price
• A put option on being exercised gives the holder
  a short position in the futures contract plus
  cash equal to the exercise price minus the
  futures price

43
Options on Futures A call (put) on a futures
contract with strike X gives you the right to
establish a long (short) position in the futures
contract at a futures price X. If you exercise,
your position will be marked to market at the end
of the day. A September EUR futures contract on
EUR 125000 is currently trading at 1.2660 if
you exercise a call with strike 1.1950, you
become the owner of one September EUR futures
contract with a price of 1.1950. You will open a
margin account with a deposit of say 5 of the
contract value. If the settlement price is
1.2560, your margin account will be credited
with (1.2560-1.1950)(125000)
7625.
44
Futures Style Options First consider a forward
contract expiring at time T on an option with the
same expiry date. The option is on the underlying
currency. Essentially you pay the option premium
at the time of expiry. A futures style option is
like a forward-style option but with
marking-to-market. Suppose you buy a futures
style option on EUR 125000 at a price of 0.02
per EUR. You pay a margin as in futures. On the
second day the option settles at 0.03. You can
withdraw (125000)(0.03-0.02) 1250. Next day
the option settles at 0.035 and expires. You
gain a further 625 and now have to pay 0.035
premium per EUR. Ignoring time value you pay a
net amount (4375-1250-625) 2500. Whether
you exercise the option or not depends upon (ST
X).
45
(1) A European call expiring at time T on a
forward purchase contract also expiring at time
T strike price X How will it be priced relative
to a call on spot forex? If it is an American
option, under what conditions might it be
exercised early? (2) Consider a European call
expiring at time T, on a futures contract also
expiring at time T. How will it be priced
relative to an option on cash forex? Suppose its
an American option under what circumstances
would it be rational to exercise it early? Assume
that all future interest rates are known with
certainty.
46
(3) Consider a European call expiring at time T
on a forward purchase contract expiring at time
T2 gt T at strike X. If you exercise, at time T,
you will own a forward contract expiring at T2,
to buy one unit of forex at a price of X units of
HC. Ignoring interest rate uncertainty, how would
you value such an option? If the call is
American, under what conditions might it be
exercised early? (4) Suppose the option is on a
futures contract expiring at T2 gt T. How would
you value a European option?, Ignoring interest
rate uncertainty, would an American call be
exercised early?
47
Innovations with Embedded Options

• Range Forwards (Cylinder Option, Tunnel Option)
• Participating Forwards
• Conditional Forward (Forward Reversing Option)
• Break Forwards
• Many other combinations structured products

48
Range Forwards Price Paid
F2 F1
F1 F2
ST
49
A Participating Forward agreement is designed so
that the buyer can reap part of the benefit of
depreciation and the seller can reap part of the
benefit of appreciation with no up-front fee. The
contract thus guarantees a floor price to the
seller, a ceiling price to the buyer and an
opportunity of doing better than these. Consider
first the sale of a participating forward. The
seller is assured a minimum price F1 which is
less than the current outright forward rate for
the same maturity. If at maturity, the spot rate,
ST, is greater than F1, the seller gets
F1 ? (ST - F1) 0 lt ? lt 1
50
DISSECTING A PARTICIPATING FORWARD. With an
outright forward, the seller is guaranteed a
price of F (the current outright forward rate),
the present value of which is Fe-r(T-t) where r
is the risk-free interest rate, t is current time
and T is maturity date. With a participating
forward with sharing ratio ?, the seller gets
F1 ? max0, (ST - F1). A European call option
with strike of F1, maturing at T, also gives a
payoff of max 0, (ST-F1) at T. The current
value of such an option is c (St,F1,T). The
present value of the participating forward is
thus F1e-r(T-t) ?
c(St,F1,T).
51
DISSECTING A PARTICIPATING FORWARD. Since both
the outright forward and the participating
forward are costless to enter into, their values
must be identical. Thus Fe-r(T-t)
F1e-r(T-t) ? c(St,F1,T) Given St, F1 and T (and
of course an estimate of volatility), the call
value c can be determined and the above relation
can be used to determine ?, the participation
rate. The client can specify F1 the bank can
then specify ?
52
DISSECTING A PARTICIPATING FORWARD The case of
participating forward purchase can be analysed in
the same fashion. The buyer is assured of a
ceiling price of F2 which is greater than F. If
ST is below F2, buyer will have to pay F2 - ?
(F2 - ST). The buyer's cost is thus F2 - ? max
0, (F2 - ST). But max 0, (F2-ST) is the
payoff of a European put with a strike price of
F2. Thus Fe-r(T-t) F2e-r(T-t) - ?
p(St,F2,T) This relationship can be used to
determine ?, the buyer's participation rate given
F2.
53
Conditional Forward (Forward Reversing
Option) Another innovative contract is the
Forward Reversing Option. It is same as a
straight option except that the option premium is
paid in the future and is only paid if the price
of the foreign currency is below a specified
level. Thus suppose a customer is not willing to
pay more than CHF 1.7500 for a dollar. He buys a
conditional forward in which the seller quotes a
premium which is to be paid if and only if the
price of a dollar plus the premium is less than
1.7500. At maturity you pay minST ?, X where
X is the maximum price you are willing to pay, ?
is the premium for the forward reversing option
and ST is the maturity spot.
54
The payoff of this option can be replicated
by a position in which you buy the currency spot
at maturity and get the payoff from the following
portfolio Buy a European call with strike X
Plus Buy a European put with strike (X -
?) Plus Sell a European put with strike
X i.e. Forward Reversing Option Spotc(X)p(X-
?)-p(X) Here c, p are the premiums for
straight European call and put options
55
BREAKFORWARD CONTRACTS An Indian company owns a
software firm in Japan. At the current exchange
rate, one INR buys 3.5 JPY. There is some concern
in the company about Yen depreciation over the
medium term. The company has investigated selling
Yen forward for 5 years against INR. The forward
exchange rate is 3.2. The forward points are
therefore "in-favour" of the company. Rather
than lock themselves into this forward rate, the
company could elect to enter into a Break Forward
at say 3.32. While the forward rate is not as
attractive, the Break Forward gives the company
the right to terminate the contract after 3
years. This option gives the company the
flexibility to re-assess the situation in the
future. If after three years, they choose to
cancel the forward transaction, no payment is
necessary.
56

• BREAKFORWARD EXAMPLE
• USD/INR Spot 48.60 1-Year Forward 50.00
• Fixed Rate 51.00 Breakforward Rate 50.50 at
  6 months
• 1. Company agrees to buy USD at 51.00 1-year
  forward
• Company has the right but no obligation to break
  this contract at 6 months by selling USD to bank
  6-months forward at 50.50.
• The right to break may occur at multiple points.
• 6 months later suppose USD/INR spot is
  48.80, 6 months forward is 49.00, company breaks
  the original forward. Buys USD 6-months forward
  in the market. It owes the breakforward bank
  Rs.0.50, 6 months later. Its total cost is 49.50
  better than Rs.50.00, the original 1-year
  forward.

57
PRICING OF BREAKFORWARDS   A simple Break Forward
(i.e. the right to cancel once only) is a FX
Forward plus a bought option. If for example
there is the right to cancel a 3 year Forward
purchase of CHF vs USD after 1 year, the Break
Forward is a 3- year FX Forward (Buying CHF) plus
a 1 year put option on 2- year Forward CHF ,
where the strike rate on the put is equal to the
rate quoted on the Break Forward. The worse than
market forward rate for the Break Forward is to
pay for the purchased put option.   Where there
is more than option to cancel, the Break Forward
is an FX Forward plus a series of Contingent
Options, i.e. the second right to cancel is
contingent upon the first right to cancel not
being exercised.  
58
In general, the "cost" of the Break Forward (the
difference between the Break Forward rate and the
market FX Forward rate) is dependent upon four
key variables  (a) Forward point volatility.
Higher volatility will lead to higher costs (b)
Number of rights to cancel. Generally, the more
rights, the higher the cost. (c) Time to first
right to cancel. Generally, the longer the
period, the higher the cost. (d) The implied
movement in forward points. The cost of the
option will clearly depend on the relationship
between the current forward points and the points
implied for the period of the option (see Implied
Forwards).
59
Like all derivatives, particularly options, the
Break Forward is priced assuming that the
counterparty acts in an economically rational
way. With simple Break Forwards (i.e. one right
to cancel), if the spot rate has fallen below the
original Break Forward rate, it is in the best
interest of the company to cancel the Break
Forward and replace it with a plain vanilla FX
Forward at the then prevailing market
rate.  Break Forwards are suitable for any
institution interested in forward foreign
exchange where there is a desire either to
protect against adverse rate movements in the
future, or where there is a business reason why
the forward contract may need to be cancelled at
some point in the future and the company wishes
to protect itself against the potential costs of
unwinding the contract.
60
Break Forward Other Versions (1) Suppose
GBP/USD spot is 1.6000 90-day Forward
1.5800 Bank offers a Floor Rate of 1.5600 and a
Break Rate of 1.60 At maturity if spot is below
1.60, you sell GBP (buy USD) at USD 1.56 per GBP
If spot is equal to or higher than 1.60 you sell
GBP at (Spot-0.04). (2) Bank offers to buy GBP
3 months forward at 1.5600 you have the right to
cancel this contract at any time upto three
months. How do you synthesize these contracts?
61
EXOTIC OPTIONS

• Barrier Options Options die or become alive
  when the underlying touches a trigger level
• Other Exotic options
• Preference Options Decide call or put later
• Asian Options
• Look-back Options Payoff based on most
  favourable rate during option life.
• Average Rate Option Payoff based on average
  value of the underlying exchange rate during
  option life
• Bermudan Options exercise at discrete points of
  time during option life. Sort of compromise
  between American and European options.
• Compound Options Option to buy an option
• Many other innovative products and structures

62
Barrier Options ? Up-and-Out or Knock-out Put
Option Consider a European put option on GBP
against USD at a strike price of USD 1.80 per
GBP. If we build into it an additional condition
that the option ceases to exist or is "knocked
out" if the spot GBP/USD goes above 2.00 at any
time during the life of the option irrespective
of what the spot rate is on the expiry date, it
becomes a Up-and-Out put or a Knock-out put. An
American firm with a GBP receivable might buy
such an option to protect the dollar value of its
asset with some side arrangement with the bank
that the moment the spot goes above 2.00, a
forward sale contract will come into effect. The
advantage of this option is its lower up- front
premium compared to a standard European put.  
63

• Barrier Options
• Up- and-In Put Option
• In the above example, a put with a strike of USD
  1.80 and a condition that the put
  becomes effective only if the spot
  rate goes above 2.00 makes it a up- and-in put.
  If the outlook for GBP is bullish in the short to
  medium run but bearish in the long run a hedger
  or trader might use such an option alternatively
  he could buy short-maturity calls and
  longer-maturity puts. The up-and-in put is a
  cheaper alternative.

64
Barrier Options ? Down-and-Out Call Option A
German firm with USD payable might buy a call on
USD with a strike price of DEM 1.60 per USD with
a knock-out at 1.55. It might have an arrangement
to buy USD forward the moment the dollar moves
below 1.55. This call would be cheaper than a
standard call with the same strike and maturity.
? Down-and-In Call Option T This opposite of a
down-and-out call. The down-and-in call comes
into existence only if the spot rate moves below
the barrier level. This option will be used when
the view is bearish in the short run but bullish
in the long run.
65
SOME STRUCTURES EUR/USD Backout Forward or
Forward Extra Transaction between Customer and
Bank Spot Reference 1.2720 Six Month Forward
1.2792 Transaction Date 10/11/2008 Start Date
14/11/2008 Expiry Date 10/5/2009 Delivery Date
14/5/2009 The Contract Works as Follows If EUR
1.2054 is not seen during the life of the option,
then if EUR trades below or upto 1.2842 on
maturity, then Customer buys EUR sells USD at
market. If EUR trades above 1.2842 on maturity,
then Customer buys EUR sells USD at 1.2842 If
EUR 1.2054 is seen during the life of the option,
then Customer buys EUR sells USD at 1.2842,
whatever be the spot rate at expiry. Customer
pays no premium upfront.
66
(No Transcript)
67
(No Transcript)
68
Client Summary of Final Terms and
Conditions  Knock-outs Range Forward Six
Months EUR import USD export hedge using currency
options Transaction between Customer and
Bank Spot Reference 1.2720  At-the-money-forward
(ATMF6M) 1.2792 Amount EUR 1 million Style
European Strike American Barrier Transaction
Date 8 SEP 2006 Start Date 12 SEP 2006 Expiry
Date 8 MAR 2007 Delivery Date 12 MAR 2007
69
If EUR/USD 1.2288 and 1.3300 are both not seen
during the life of the option then If EUR is
above 1.2288 and below or at 1.2700 on
maturity,then Customer buys EUR sells USD at
1.2700 If EUR is above 1.2700 and below or at
1.2892 on maturity,then Customer buys EUR
sells USD at market If EUR is above 1.2892 and
below 1.3300 on maturity,then Customer buys
EUR sells USD at 1.2892  
70
If EUR/USD 1.2288 is not seen during the life of
the option then If EUR is above 1.2288 and below
or at 1.2700 on maturity, then Customer buys EUR
sells USD at 1.2700 If EUR is above 1.2700
and below or at 1.2892 on maturity, then
Customer buys EUR sells USD at market If EUR is
above 1.2892 and below 1.3300 on maturity,
then Customer buys EUR sells USD at 1.2892 If
EUR is at or above 1.3300 on maturity then
Customer buys EUR sells USD at market
71
If EUR/USD 1.3300 is not seen during the life of
the option then If EUR is below or at 1.2288 on
maturity then Customer buys EUR sells USD at
market If EUR is above 1.2288 and below or at
1.2700 on maturity, then Customer buys EUR
sells USD at 1.2700 If EUR is above 1.2700 and
below or at 1.2892 on maturity,then Customer
buys EUR sells USD at market If EUR is above
1.2892 and below 1.3300 on maturity, then
Customer buys EUR sells USD at 1.2892
72
If EUR/USD 1.2288 and 1.3300 are both seen during
the life of the option then Customer buys EUR
sells USD at market Customer pays upfront no
premium as this is a zero cost strategy Documentat
ion ISDA Master Agreement, Schedule, Legal
Opinion and Risk Disclosure Statement. Holiday
Convention New York, Frankfurt
73
Option Pricing Models

• Origins in similar models for pricing options on
  common stock the most famous among them being the
  Black-Scholes option pricing model
• The central idea in all these models is Risk
  Neutral Valuation
• The theoretical models typically assume
  frictionless markets

74
Principles of Option Pricing

• Notation
• t The current time
• T Number of days from t to expiry of the option
  i.e. the option expires at time tT
• C(t) Value measured in HC, at time t, of an
  American call option on one unit of spot foreign
  currency
• P(t) Value in HC, at time t, of an American put
  option on one unit of foreign currency
• c(t), p(t) Values of European call and put
  options in HC
• Exchange rates, strike prices stated as units of
  HC (home currency) per unit of FC (foreign
  currency)

75
Principles of Option Pricing

• Notation (contd.)
• S(t) The spot exchange rate at time t. S(tT)
  is thus the spot rate at option maturity. The
  spot rate is in terms of units of HC per unit of
  FC
• X The exercise or strike price, units of HC per
  unit of FC
• iH Domestic risk-free, continuously compounded
  annual money market interest rate. It is assumed
  to be constant
• iF Foreign risk-free, continuously compounded
  annual money market interest rate, assumed to be
  constant

76
Principles of Option Pricing

• Notation (contd.)
• - BH(t,T) Home currency price, at time t, of a
  pure discount bond that pays one unit of home
  currency at time tT with continuous compounding
• -
• BF(t,T) Foreign currency price, at time t, of a
  pure discount bond that pays one unit of foreign
  currency at time tT, with continuous compounding
  
• ?(t) Standard deviation of the spot exchange
  rate.

77
Principles of Option Pricing

• Determinants of option values
• S(t), X, T, iH, iF, ?
• Basic principles of option valuation
• Option values can never be negative. At any time
  t, c(t), C(t), p(t), P(t) ? 0
• ct, Ct ? St and pt, Pt ? X
• On exercise date, the option value must equal the
  greater of zero and the intrinsic value of the
  option
• c(tT), C(tT) max 0, S(tT)-X
• p(tT), P(tT) max 0, X-S(tT)
• At any time t lt T
• C(t) ? max c(t), S(t)-X P(t) ? max
  p(t), X-S(t)

78
Principles of Option Pricing

• Consider two American options, calls or puts,
  which are identical in all respects except time
  to maturity. One matures at tT1 while the other
  at tT2 with
• T2 gt T1. Let C1, C2 and P1, P2 denote the
  premiums.
• Then
• C2(t) ? C1(t) P2(t) ? P1(t) for all t ?
  T1
• ?C/?T ? 0 ?P/?T ? 0
• Two options differing only in strike price
• C(X2, t) lt C(X1, t) c(X2, t) lt c(X1, t)
• P(X2, t) gt P(X1, t) p(X2, t) gt p(X1, t))
• where X1 and X2 are strike prices with X2 gt X1
  
• ?C/?X , ?c/?X lt 0 ?P/?X , ?p/?X gt 0

79
Principles of Option Pricing

• At any time t we must have
• c(S,X,T) XBH(t,T) ? S(t)BF(t,T) or,
• c(S,X,T) ? S(t)BF(t,T) - XBH(t,T)
• and therefore
• CS(t),X,t,T ? cS(t),X,t,T
• ? S(t)BF(t,T) - XBH(t,T)
• CS(t),X,t,T
• ? maxS(t) - X, S(t)BF(t,T) - XBH(t,T)

80
Principles of Option Pricing

• For European and American put options we have
• pS(t),X,t,T ? XBH(t,T) - SBF(t,T)
• PS(t),X,t,T
• ? max X-S(t), XBH(t,T)-SBF(t,T)
• Since SBF/BH Ft,T T-day Forward Rate at t
• CS(t),X,t,T ? cS(t),X,t,T ?
  BH(t,T)(Ft,T-X)
• PS(t),X,t,T ? pS(t),X,t,T ? BH(t,T)(X
  Ft,T)

81
Principles of Option Pricing

• Put-Call Parity Relationship for European Options
  
• pS(t),X,t,T cS(t),X,t,TXBH(t,T)-S(t)BF(
  t,T)
• Using the interest parity relation Ft,T
  St(BF/BH). Thus
• pS(t),X,t,T cS(t),X,t,TBH(t,T)(X-Ft,T)
• This can be manipulated in several ways
• p c BH (t,T)Ft,T BH (t,T)X
• Long put Short call Long FC bond Long HC
  bond
• -pS(t),X,t,T -cS(t),X,t,T - XBH(t,T) Ft,T
  BH(t,T)
• A short put A short call short HC bond Long
  FC bond

82
(No Transcript)
83
(No Transcript)
84
Garman and Kohlhagen (1983) Option Pricing
Formula In the interbank foreign exchange
market, options are not quoted with prices. They
are quoted indirectly with implied volatilities.
The convention for converting volatilities to
prices is the Garman-Kohlhagen (1983) option
pricing formula. Mathematically, the formula is
identical to Merton's (1973) formula for options
on dividend-paying stocks. Only in place of the
stock's dividend yield, substitute the foreign
currency's continuously compounded risk-free
rate. Like the Merton formula, the
Garman-Kohlhagen formula applies only to European
options. Generally, OTC currency options are
European.
85
Option Pricing

• European Call Option Formula
• c(t) S(t)BF(t,T)N(d1) - XBH(t,T)N(d2)
  (10.24)
• ln(SBF/XBH) (?2/2)T
• d1 --------------------------------
• ??T
• ln(SBF/XBH) - (?2/2)T
• d2 --------------------------------
• ??T
• ? in the above formula denotes the standard
  deviation of log-changes in the spot rate

86
Option Pricing Models (contd.)

• c(t) BH(t,T) Ft,TN(d1) - XN(d2) (10.25)
• ln(Ft,T/X) (?2/2)T
• d1 ----------------------------
• ??T
• ln(Ft,T/X) - (?2/2)T
• d2 ----------------------------
• ??T

87
Option Pricing Models (contd.)

• European Put Option Value
• p(t) XBH(t,T)N(D1) - S(t)BF(t,T)N(D2)
• BH(t,T)XN(D1) - Ft,TN(D2)
• where, D1 -d2 and D2 -d1

88
Option Deltas and Related Concepts The Greeks

• The Delta of an Option
• ? ?c/?S for a European call option
• ?p/?S for a European put option
• Having taken a position in a European option,
  long or short, what position in the underlying
  currency will produce a portfolio whose value is
  invariant with respect to small changes in the
  spot rate

89
Option Deltas and Related Concepts The Greeks
(contd.)

• The Elasticity of an option is defined as the
  ratio of the proportionate change in its value to
  the proportionate change in the underlying spot
  rate. For a European call, elasticity would be
  (?c/c)/(?S/S)
• The Gamma of an option
• ? ?2c/?S2 for a European call
• ? BFN?(d1)/S??T

90
Option Deltas and Related Concepts The Greeks
(contd.)

• A hedge which is delta neutral as well as gamma
  neutral will provide protection against larger
  movements in the spot rate between readjustments
• The Theta of an Option
• ?c/?t for a European call

91
Option Deltas and Related Concepts The Greeks
(contd.)

• The Lambda of an Option
• Rate of change of its value with respect to the
  volatility of the underlying asset price
• Concept of implied volatility
• Compute the value of ? which, when input into the
  model, will yield a model option value equal to
  the observed market price
• Volatility smile depicted in figure below

92
The Greeks for a call option are
93
The Greeks for a put option are
94
Option Deltas and Related Concepts The Greeks
(contd.)
Volatility Smile
95
AN INTUITIVE APPROACH TO THE BLACK-SCHOLES
MODEL We have seen that for a European option,
the current value is given by the expected value
at maturity discounted at the risk-free rate of
interest to the current time. Consider a call
option on 1 US dollar at a strike price of CHF
1.50. The current spot rate is CHF 1.5000 and the
option expires 90 days from now. The risk-free
interest rate is 6 p.a. The USD/CHF spot rate at
maturity has the following discrete distribution
96
ST
Probability P(ST) 1.30
0.05 1.35
0.08 1.40 0.10
1.45 0.20
1.50 0.20 1.55
0.20 1.60
0.07 1.65 0.05
1.70 0.05
97
The value of the call option for any ST is max
0, ST-X where X is the strike price. For the
above distribution, the expected value of the
call at maturity is given by (1.55-1.50)P(ST1.55
) (1.60-1.50)P(ST1.60)
(1.65-1.50)P(ST1.65) (1.70-1.50)P(ST1.70)
This can be broken down into two parts
1.55?P(1.55)1.60?P(1.60)1.65?P(1.65)1.70?P(1
.70) -(1.50)P(1.55)P(1.60)P(1.65)P(1.70) In
general this can be written as ? STP(ST)
- (X)? P(ST) (1) ST gtX
ST gtX
98
The first sum is nothing but the mean of the
truncated distribution of ST i.e. mean of ST over
values of ST gt X the second sum is just the
cumulative probability of ST gt X i.e. the
probability that the option will be in-the-money
and hence will be exercised at maturity.
99
Valuation for Continuous Distributions Now let us
apply these ideas when the spot rate at maturity,
ST, has a continuous probability distribution
with density function f(ST). Further, ST can take
any non-negative value. If a random variable Y
has a continuous distribution with density
function f(Y), the probability that a ? Y ? b for
given constants a and b is given by

100
The sums in (1) above are now replaced by the
corresponding integrals. The expected value of
the call at maturity is given by The first
integral is the mean of the truncated
distribution of ST, over values of ST ? X while
the second integral is just the probability that
the option will be exercised at maturity i.e.
probability that ST ? X.
101
The Black-Scholes model assumes that the spot
rate S evolves according to the following
stochastic process dS/S µdt
sdz (A.10.28) where dz e
?dt This is known as "geometric Brownian
motion". It is a special case of a more general
class of processes known as "Ito processes". Here
µ and s are parameters and e is a standard normal
random variate.
102
(dS/S) which is instantaneous proportional
exchange rate return (over the time interval dt)
is a random normal variable with expected value
µdt and variance s2dt and (dS/S) has a normal
distribution. This means that the distribution
of the spot rate is lognormal i.e. the natural
logarithm of the spot rate is normally
distributed. If St is the current spot rate and
ST is the spot rate at time T then
lnST N(?t,T, ?t,T) where µt,T
lnStµ - (s2/2)(T-t)
st,T s ?(T-t)
103
µ and s are the mean and standard deviation of
the expected proportionate change in exchange
rate per unit time. Now using the properties of
the lognormal distribution and after some algebra
it can be shown that where Et(ST) is the
expected value at the current time t of the spot
rate at maturity (time T).
104
The value of d1? is given by
and where d2? d1? - st,T
105
(1) Under the assumption that perfectly
riskless portfolios can be constructed which
replicate option payoff, Et(ST) can be replaced
by Ft,T the forward rate at time t for a contract
maturing at T (2) As seen above, st,T s
?(T-t) (3) Using the interest parity theorem,
with continuously compounded domestic and foreign
interest rates rd and rf,
106
(4) Finally, the current value of the call
option is given by its expected value at maturity
discounted at the domestic riskless interest rate
rd over the time interval (T-t). The discount
factor is e-rd(T-t). Making these substitutions
leads to the Black-Scholes call option pricing
formula
107
THE BINOMIAL OPTION PRICING MODEL Model
Assumptions Apart from the assumption about spot
exchange rate movements discussed below, the
model makes the following additional assumptions
1. Foreign and domestic interest rates are
constant. 2. No taxes, transaction
costs, margin requirements and
restrictions on short sales. 3. Required hedging
instruments are readily available.
108
A Single Period Option A European call option on
US dollar with a single period (say an year) to
maturity. The following notation is defined
S0 The current USD/INR spot rate X The
strike price in a European call option on one
USD. T Time to maturity in years (taken to
be one year but can be any length)
rd Continuously compounded domestic risk
free interest rate (i.e INR interest
rate) rf Continuously compounded foreign
risk free interest rate (USD interest
rate)
109
s Volatility i.e. standard deviation of the
spot rate on an annual basis u The
multiplicative factor by which the spot rate will
increase at the end of the year if USD
appreciates d The multiplicative factor by
which the spot rate will decline at the end of
the year if USD depreciates. We are assuming that
at option maturity, the spot rate will be either
uS0 or dS0. The following condition must be
imposed on u and d
(1 rd) d lt ---------------
lt u (1 rf)

110
Now consider a European call option on one CHF.
Let its current value be denoted c0. At expiry,
the option will be worth c1u max 0, uS0-X
or c1d max 0, dS0-X depending upon whether
an upward or a downward movement has taken place
in the exchange rate. Let us construct a
portfolio the payoff from which will be identical
to the payoff from the call option at option
expiry. The portfolio consists of Bd units of
domestic currency (in this case rupees), invested
in domestic risk-free bonds and Bf units of
foreign currency invested in foreign risk-free
bonds. Bd and Bf must be chosen such that
(1rd)Bd uS0Bf (1rf) c1u and
(1rd)Bd dS0Bf (1rf) c1d
111
Solving these, we get
(c1u c1d )
Bf ------------------
S0(u-d)(1rf)
(uc1d
dc1u ) Bd
-------------------
(u-d)(1rd) Since the portfolio
and the call option have identical payoffs at the
end of the period, they must have identical
values at the beginning of the period. Hence
c0 Bd S0Bf

112
Substituting for Bf and Bd
pc1u (1-p)c1d c0
----------------------
(1 rd ) Where
(1rd)/(1rf) d p
----------------------------
(u d) From the restrictions
on u and d cited above it is easy to see that Bd
lt 0 and Bf gt 0. Thus a portfolio consisting of
rupee borrowing and a USD deposit is equivalent
to a long position in a call on one USD. Also, p,
defined above satisfies 0 lt p lt 1 and can be
interpreted as a probability.
113
Let us take a numerical example. Consider the
following data pertaining to a call option on one
USD S0 45.00 rd 6 rf 4 u
1.07 d 0.85 X 46.50 T 1 year. At
the end of the year the option will be worth
either c1u max0, (1.07)(45) - 46.50
1.65 or c1d max0, (0.85)(45) 46.50 0
From the definitions of equivalent
portfolio can be computed as (uc
(uc1d dc1u )
-(0.85)(1.65)
Bd ------------------- ----------------------

(u-d)(1rd) (1.07-0.85)(1.06)

-Rs.6.0141 (A loan)
114
(c1u c1d )
1.65 Bf -------------------
---------------------- 0.1602
S0(u-d)(1rf) (45)(0.22)(1.04) To
acquire and deposit this amount of USD the
investor requires Rs.(45)(0.1602) Rs.7.209. Of
this she should borrow Rs.6.014 as seen above
thus requiring own cash outlay of Rs.1.195. What
will be her portfolio worth at the end of the
year?
115
The USD deposit will grow to (0.1602)(1.04)
0.1666. The rupee loan repayment will require
Rs.(6.014)(1.06) Rs.6.375. Thus the (Rupee loan
Dollar deposit ) portfolio will have a value of
Rs. -6.375 (0.1666)(45)(1.07) Rs.1.65
if the USD/INR exchange rate goes up by a factor
of 1.07 Rs.-6.375(0.1666)(45)(0.85) 0 if
the exchange rate goes down by a factor of
0.85. Thus the portfolio payoff is identical to
the call option payoff under any state of
nature. Hence the present value of the portfolio
must equal the present value of the call option.
Thus the 1-year European call on 1.00 with a
strike price of Rs.46.50 should be valued at
Rs.1.195.
116
Extension to a Multi-Period Option Now suppose
the interval T which was assumed to be one year
is divided into n "periods" each of length T/n.
During each period, the spot rate either moves up
by a factor u or moves down by a factor d. A
clear idea of how the binomial model works in a
recursive fashion can be obtained by considering
n 2. Panel (a) of the figure below shows the
tree diagram