CURRENCY OPTIONS

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					CURRENCY OPTIONS
            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.
 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.
               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.
      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
     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.
               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 > X and out-of-the-money-spot
  if St < X
• A put option is said to be at-the-money-spot if
• St = X, in-the-money-spot if St < X and out-of-the-
  money-spot if St > 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.
        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 < Ft,T    (X > Ft,T)
OTMF – out of the money forward - if
     X > Ft,T   (X < Ft,T)
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
XDELY                3.20   3.40                  126.00
ECDLW                2.92   3.10                  126.50
XDELB                2.62   2.78                  127.00
ECDLD                2.36   2.50                  127.50
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
PHLX EUR/USD CALLS; EXPIRY: END MARCH 2009; CENTS PER EUR
  Symbol               Bid      Ask                Strike Price


 ECDCR                 10.60   10.95                119.50
 ECDCV                 9.95    10.25                120.50

 ECDCC                 9.20    9.55                 121.50
 ECDCG                 8.55    8.85                 122.50
 ECDCK                 7.90    8.20                 123.50
 XDECZ                 7.60    7.90                 124.00
 ECDCO                 7.30    7.60                 124.50
 XDECX                 7.00    7.30                 125.00

 ECDCS                 6.70    7.00                 125.50
 XDECY                 6.45    6.70                 126.00
 ECDCW                 6.15    6.45                 126.50
 XDECB                 5.90    6.20                 127.00
 ECDCD                 5.65    5.90                 127.50
 XDECF                 5.40    5.65                 128.00

 ECDCH                 5.15    5.45                 128.50
 XDECJ                 4.95    5.20                 129.00
 EPACL                 4.70    5.00                 129.50
 XDECN                 4.50    4.75                 130.00
 EPACP                 4.25    4.55                 130.50
 XDECR                 4.05    4.35                 131.00

 EPACT                 3.85    4.20                 131.50
 XDECV                 3.65    4.00                 132.00
PHLX EUR/USD PUTS; EXPIRY: MARCH 2009; CENTS PER EUR, DEC 4, 2008

  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
  XDEOB                 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 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
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
  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
         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
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.
Source: UBS June 6, 2006
    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
      Elementary Option Strategies
• Call Options
  –   Current spot rate, St
  –   Strike price X
  –   Call option premium c
  –   Spot rate at maturity ST
  –   Call Option Buyer’s Profit = -c for ST  X
          = ST - X - c for ST > X
  – Call Option Writer’s Profit = +c for ST  X
                                            = -(ST - X - c)
                                                for ST > X
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 Buyer’s 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
           Elementary Option Strategies
                Payoff Profile of a Call Option
       +

                         c = 1.50
       O
               c =1.50                                  ST


       -                            X=49.50 X+c=51.00

                                         Breakeven Spot Rate
                                          Option Buyer
ST : SPOT RATE AT EXPIRY
                                          Option Seller
     Elementary Option Strategies
• Put Options : Premium p
   – Put Option Buyer's Profit
   – = -p            for ST  X
     = X - ST – p for ST < X
   – Put Option Writer’s Profit
   – = +p            for ST  X
     = -(X - ST - p) for ST < X
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 Buyer’s 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
    Elementary Option Strategies
          Payoff Profile of a Put Option


+
                                          p=0.07
O
                                 p=0.07            ST
             X-p=1.78
-
                        X=1.85

    Breakeven Spot Rate at                Option Buyer
    Option Expiry
                                          Option Seller
          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
 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
Bull Spread Using Calls
Buy Call Strike X1, Premium c1;
Sell Call Strike X2, Premium c2 ;


            Profit
                        c2


                                            ST
                       X1           X2
                 c1

                                PROFIT PROFILE OF THE
                                SPREAD STRATEGY :
Bull Spread Using Puts
Buy Put Strike X1, Premium p1; Sell Put Strike X2,
Premium p2


         Profit
                                         p2

                   X1          X2       ST



                                        p1
       Bear Spread Using Calls
BUY CALL STRIKE X2; SELL CALL STRIKE X1

   PROFIT PROFILE:

        Profi
        t

                 X1         X2      ST
          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 :
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
Elementary Option Strategies
                 Butterfly Spread




Payoff Profile of a Long Butterfly Spread :
Payoff Profile of a Short Butterfly Spread :
    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
    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.
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
      Elementary Option Strategies
                Payoff Profile of a Straddle


            +

                           X
             0
                                         ST
            -         X-p-c X+p+c


X: Strike price in put and call; c : Call premium
p : Put Premium ST: Spot rate at expiry
          Elementary Option Strategies


                       (X2 – p – c)   (X1 + p + c)
X1: Call strike
                   +
X2: Put strike
p: Put premium 0
                                                 ST
c: Call premium    -
                              X2       X1
ST: Spot rate at
    Expiry         Payoff Profile of a Strangle
      Elementary Option Strategies
        Strip & Strap: CALLS + PUTS
      SAME STRIKE AND EXPIRY DATE

      Profit
                           Profit
+


               K    ST              K   ST
O

-       LONG STRIP            LONG STRAP
    LONG (1 CALL+2 PUTS)   LONG (2 CALLS+1 PUT)
 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.
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
              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
                   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.
                 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).
(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 it’s an American option;
under what circumstances would it be rational to exercise it
early? Assume that all future interest rates are known with
certainty.
(3) Consider a European call expiring at time T on a forward
purchase contract expiring at time T2 > 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 > T. How would you value a European option?, Ignoring
interest rate uncertainty, would an American call be
exercised early?
 Innovations with Embedded Options
• Range Forwards (Cylinder Option,
  Tunnel Option)
• Participating Forwards
• Conditional Forward (Forward
  Reversing Option)
• Break Forwards
• Many other combinations – structured
  products
             Range Forwards


Price Paid


     F2


     F1




                 F1   F2      ST
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 <  < 1
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+  max[0, (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)].
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 
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.
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 min[ST + , 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.
  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 = Spot+c(X)+p(X- )-p(X)
  Here c, p are the premiums for straight European
  call and put options
           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.
           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
2. 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.
3. 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.
            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.
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).
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.
            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?
               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
                     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.
                   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.
                    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.
                      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.
Client Summary   Trigger Forward : Twelve Months
of Final         GBP import USD export hedge using
Terms and        currency options
Conditions
Transaction      Customer and Bank
between

Spot Reference   1.7379


Twelve Month     At-the-money-forward    (ATMF12M)   :
Forward          1.7457


Amount           GBP 1 million


Style            European Strike American Barrier


Cut              Tokyo
Transaction Date   September 8, 2008


Start Date         September 12, 2008


Expiry Date        September 10, 2009


Delivery Date:     September 12, 2009


Description        If GBP 1.8578 is not seen during the life
                   of the option then Customer buys GBP
                   & sells USD at 1.7200

                   If GBP 1.8578 is seen during the life of
                   the option then Customer buys GBP &
                   sells USD at market and

                   Customer pays upfront no premium as
                   this is a zero cost strategy
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
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
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
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
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
Documentation: ISDA Master Agreement, Schedule,
Legal Opinion and Risk Disclosure Statement.
Holiday Convention: New York, Frankfurt
      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
     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 t+T
  – 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)
   Principles of Option Pricing
Notation (contd.)
– S(t) : The spot exchange rate at time t. S(t+T) 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
    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 t+T with continuous compounding

             B = e-iH(T/360)
-             H
BF(t,T) : Foreign currency price, at time t, of a pure
  discount bond that pays one unit of foreign currency
  at time t+T, with continuous compounding
             B = e-iF (T/360)
              F
- (t) : Standard deviation of the spot exchange rate.
       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(t+T), C(t+T) = max [0, S(t+T)-X]
     p(t+T), P(t+T) = max [0, X-S(t+T)]
   At any time t < T
     C(t)  max [c(t), S(t)-X] ; P(t)  max [p(t), X-S(t)]
      Principles of Option Pricing
– Consider two American options, calls or puts, which
  are identical in all respects except time to maturity.
  One matures at t+T1 while the other at t+T2 with
  T2 > 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) < C(X1, t); c(X2, t) < c(X1, t)
P(X2, t) > P(X1, t); p(X2, t) > p(X1, t))

 where X1 and X2 are strike prices with X2 > X1
 C/X , c/X < 0     P/X , p/X > 0
   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
  C[S(t),X,t,T]  c[S(t),X,t,T]
   S(t)BF(t,T) - XBH(t,T)
 C[S(t),X,t,T]
 max{[S(t) - X], [S(t)BF(t,T) - XBH(t,T)]}
    Principles of Option Pricing
– For European and American put options we have

  p[S(t),X,t,T]  XBH(t,T) - SBF(t,T)

  P[S(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

 C[S(t),X,t,T]  c[S(t),X,t,T]  BH(t,T)(Ft,T-X)

 P[S(t),X,t,T]  p[S(t),X,t,T]  BH(t,T)(X – Ft,T)
         Principles of Option Pricing
– Put-Call Parity Relationship for European Options
  p[S(t),X,t,T] = c[S(t),X,t,T]+XBH(t,T)-S(t)BF(t,T)
Using the interest parity relation: Ft,T = St(BF/BH). Thus

 p[S(t),X,t,T] = c[S(t),X,t,T]+BH(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
-p[S(t),X,t,T] = -c[S(t),X,t,T] - XBH(t,T) + Ft,T BH(t,T)
A short put = A short call + short HC bond + Long FC bond
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.
                 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                           d   1      (- z 2 / 2)
                                          N(d) =     (     )e             dz
                                                   -   2π

            ln(SBF/XBH) - (2/2)T
  d2 = --------------------------------
                     T
   in the above formula denotes the standard deviation
    of log-changes in the spot rate
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
 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
      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
    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
   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
   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
The Greeks for a call option are:
The Greeks for a put option are:
 Option Deltas and Related
Concepts: The Greeks (contd.)




          Volatility Smile
  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:
 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
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(ST=1.55)] + (1.60-1.50)[P(ST=1.60)]
+ (1.65-1.50)[P(ST=1.65)] + (1.70-1.50)[P(ST=1.70)]
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 >X          ST >X
The first sum is nothing but the mean of the
truncated distribution of ST i.e. mean of ST over
values of ST > X; the second sum is just the
cumulative probability of ST > X i.e. the probability
that the option will be in-the-money and hence will
be exercised at maturity.
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
                   b

                   f(Y)dY
                   a
The sums in (1) above are now replaced by the
corresponding integrals. The expected value of the
call at maturity is given by
                              
   Et (cT) =  ST f(ST) dST - X  f(ST) d S T
             X                 X
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.
The Black-Scholes model assumes that the spot rate S
evolves according to the following stochastic process:
          dS/S = μdt + σdz      (A.10.28)
where         dz = ε 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 σ are parameters and ε is a standard
normal random variate.
(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 σ2dt 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 = lnSt+[μ - (σ2/2)](T-t)
            σt,T = σ (T-t)
μ and σ 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
              
               ST f(ST) dST = Et (ST) N(d1)
              X

where Et(ST) is the expected value at the current
time t of the spot rate at maturity (time T).
The value of d1 is given by

                       Et (ST)    1 2
                  ln             +  t,T
          d1 =          X        2
                             t,T

and
             
              f(ST) d( ST) = N(d2)
             X
where                  d2 = d1 - σt,T
 (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, σt,T = σ (T-t)
 (3) Using the interest parity theorem, with
continuously compounded domestic and foreign
interest rates rd and rf,

               Ft,T = St e(rd - rf )(T - t)
 (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


    c0 (St , t) = St e-rf (T-t) N(d1) - Xe-rd(T-t) N(d2)
    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.
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)
σ : 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 < --------------- < u
                  (1 + rf)
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
            (1+rd)Bd + uS0Bf (1+rf) = c1u
and         (1+rd)Bd + dS0Bf (1+rf) = c1d
 Solving these, we get

                           (c1u – c1d )
                    Bf = ------------------
                          S0(u-d)(1+rf)

                         (uc1d – dc1u )
                   Bd = -------------------
                           (u-d)(1+rd)

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
 Substituting for Bf and Bd

                  pc1u + (1-p)c1d
           c0 = ----------------------
                      (1 + rd )
Where
                {[(1+rd)/(1+rf)] – d}
           p = ----------------------------
                         (u –d)

From the restrictions on u and d cited above it is easy to see
that Bd < 0 and Bf > 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 < p < 1 and can be interpreted as a probability.
 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 = max[0, (1.07)(45) - 46.50] = 1.65
       or c1d = max[0, (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)(1+rd)         (1.07-0.85)(1.06)

                       = -Rs.6.0141 (A loan)
               (c1u – c1d )              1.65
       Bf = ------------------- = ---------------------- = $0.1602
              S0(u-d)(1+rf)         (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?
 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.
          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 of exchange rate values
while panel (b) call option values. one down move etc.
Similar notation appalies to call option values. Thus c1u
denotes value of the call at time node 1 after one up move.
             A Two Period Binomial Tree


                 S2UU                     c2UU

       S1U                          c1U

S0              S2UD = S2DU    c0         c2UD = c2DU

       S1D                          c1D

                 S2DD                     c2DD



     SPOT RATE TREE             CALL VALUE TREE
Now consider call values at the end of period 1. They are
denoted c1u and c1d as before. We now apply the one period
analysis at the end of period 1. From c1u, the option value can
go to c2u2 or to c2ud. To find c1u, we must construct a portfolio
which will pay c2u2 if the spot rate moves up or c2ud if it
moves down. Following this procedure as in the 1-period case
we will obtain

                    pc2u2 + (1-p)c2ud
            c1u = -----------------------
                       (1 + 0.5rd )
                    pc2du + (1-p)c2d2
            c1d = ----------------------
                       (1 + 0.5rd )
 Note that the discounting factor has changed from (1+rd) to
(1+0.5rd) to account for the fact that rd is the interest rate per
annum whereas now each of our "periods" is half-year. You
can easily see that the definition of p has to be changed to :



                   {[(1+0.5rd)/(1+0.5rf)] – d}
             p = --------------------------------------
                           (u –d)

Now work back to period 0. The current value of the call is
given

        pc1u + (1-p)c1d           p2c2u2 + 2p(1-p)c2ud + (1-p)2c2d2
 c0 = ---------------------- = -------------------------------------------
          (1 + 0.5rd )                     (1 + 0.5rd)2
 The numerator of the last expression consists of the sum of
possible call values at maturity, each weighted by the probability
of its occurrence. Thus the call will have value c2u2 if the spot rate
goes up in each of the two periods. The probability of this is p2.
Similarly, probability of both down moves is p2. The other two
possibilities are an up move followed by down and a down move
followed by up. The probability of both is p(1-p).
Thus the current value of the call is its expected value at maturity
discounted at the risk-free interest rate.
The same logic can be extended to an option with “n periods” to
expiry. In fact, as we increase the number of nodes in the binomial
tree i.e. reduce the length of the unit period the accuracy of the
computational procedure improves.
Now consider n periods to maturity each of length T/n. Let
cnu(j)d(n-j) denote the value of the call at maturity if there have
been j up-movements and (n-j) down-movements in the spot
rate. The probability of this is given by
                            [n!/j!(n-j)!]pj(1-p)n-j
The payoff from the call is
               cnu(j)d(n-j) = max[0, (ujdn-jS0-X)]
Hence the current value of the call option is given by

       1 j=n n!
                         p j (1 - p )n- j max[0, u j dn- j S0 - X]
      rdT j0 j! (n- j)!
c0 =
     e     =
                   p= e(rd-rf ) T/n - d
                           (u- d)
This can be simplified further by recognizing that for
sufficiently low values of j, say j < k,
               max[0, ujdn-jS0-X] = 0
i.e. if the number of upward movements in the spot rate is
smaller than some critical value k, the option will finish out-of-
the-money and will be worthless at maturity. Hence the
summation in the above equation need go only from j=k to j=n.
Further, define p as

                       p = ue-(-rf )(T/n)
                               rd      p

                  (1 - p) = de-(rd-rf )(T/n) (1 - p)
       j        n- j
  u j p (1 - p ) dn- j = -rf T      -(rd-rf )T/n ]j[d(1 - p) e-(rd-rf )T/n ]n- j
                        e      [upe
          e rd T
 With these modifications, equation the call value can be
 rewritten as

              -rf T                 -rdT
     c0 = S0 e      (k, n, p) - Xe      (k, n, p' )

where the function Ψ is the cumulative binomial probability


                        j=n     n!
           (k, n,  ) =               j (1 -  )n- j
                        j=k j! (n- j)!


 As the number of periods n increases and approaches infinity,
 the binomial model approaches the Black-Scholes model.
   Choosing the Parameters for the Binomial Model
• The implementation of the binomial lattice model discussed
above requires that we specify the parameters u, d, the
probability p and the number of intervals into which the period
till option expiration is to be divided.
• This choice cannot be arbitrary because these values imply an
expected rate of return per unit time and its variance; these
implied values must agree with values estimated from actual
data. How to choose values for u, d and p?
• Suppose the expected per annum change in the spot rate is
estimated to be μ and its standard deviation to be σ. We must
choose values for u, d and p such that over the life of the option,
the expected (log) change in exchange rate is μT and its
standard deviation is σT if the option life is T years.
This is achieved by choosing the following values :
        ln(u) = σh         ln(d) = -σh
        i.e set u = eσh and d = 1/u and
        p = (1/2)+(μh/2σ)          (1-p) = (1/2)-(μh/2σ)

Suppose the estimated proportionate change in exchange rate,
μ, is 15% per annum, its standard deviation, σ, is 6% and
option life is 6 months. We decide to divide the six month
period into 24 weekly periods so that h=0.02083 (=0.5/24). The
appropriate values for u,d and p are
  u = eσh = e0.06(0.144326) = 1.0087
  d = 1/u = 0.9914
  p = (1/2)+(μh/2σ) = (1/2)+[0.15(0.144326)]/0.12 = 0.68
The Black-Scholes model for currency options, derived
by Garman and Kohlhagen(1983) also assumes constant
domestic and foreign interest rates and leads to the
following valuation equation for a European call:
     c(St , t) = St e-r(f)(T-t) N(d1) – X e-r(d)(T-t) N(d2)
The current time is t, the option matures at time T, time
to maturity is (T-t) measured in years, St is the current
spot rate, X is the strike rate and r(f) and r(d) are foreign
and home currency T-year risk free interest rates. N(.)
denotes the cumulative normal distribution function and
the parameters d1 and d2 are defined as follows:
             ln(St/X) + [ rd - rf + (2/2)](T-t)
    d1 = -------------------------------------------------
                         (T – t)

    d2 = d1 - σ (T-t)
    Empirical Studies of Option
         Pricing Models
• There is substantial evidence of pricing
  biases in case of the Black-Scholes as well
  as alternative models
• Recent research has focussed on relaxing
  some of the restrictive assumptions of the
  Black-Scholes model
       Currency Options in India
• Starting in January 1994, the RBI permitted
  Indian banks to write "cross-currency" options
  i.e. options between two foreign currencies
  including barrier options and other
  innovations. They were required to cover
  themselves on a back-to-back basis.
• Currency options between the Indian rupee
  and foreign currencies were launched w.e.f.
  July 7, 2003.
• Corporates can buy options only for hedging
  underlying exposure. They cannot write
  options; more correctly they cannot be net
  receiver of premium in any structured deal.

				
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