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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: 1.2764Symbol Bid Ask Strike price (Cents per EUR) (Cents per EUR)ECDLR 8.43 8.79 119.50ECDLV 7.52 7.85 120.50ECDLC 6.70 6.95 121.50ECDLG 5.83 6.10 122.50ECDLK 5.01 5.24 123.50XDELZ 4.61 4.85 124.00ECDLO 4.20 4.40 124.50XDELX 3.90 4.10 125.00ECDLS 3.50 3.70 125.50XDELY 3.20 3.40 126.00ECDLW 2.92 3.10 126.50XDELB 2.62 2.78 127.00ECDLD 2.36 2.50 127.50XDELF 2.08 2.22 128.00ECDLH 1.81 2.00 128.50XDELJ 1.63 1.79 129.00EPALL 1.42 1.58 129.50XDELN 1.24 1.41 130.00EPALP 1.06 1.26 130.50
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 CALLS; EXPIRY: END MARCH 2009; CENTS PER EUR
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 EUR/USD PUTS; EXPIRY: MARCH 2009; CENTS PER EUR, DEC 4, 2008
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 3.10200.00 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
X=49.50 X+c=51.00
c =1.50 ST
+
-
O
c = 1.50
Option Buyer
Option Seller
Payoff Profile of a Call Option
Elementary Option Strategies
Breakeven Spot Rate
ST : SPOT RATE AT EXPIRY
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
X-p=1.78
X=1.85
p=0.07 ST
Option Buyer
Option Seller
Payoff Profile of a Put Option
O
+
-
p=0.07
Breakeven Spot Rate atOption Expiry
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 CallsBuy Call Strike X1, Premium c1; Sell Call Strike X2, Premium c2 ;
X1 X2
Profit
ST
c1
c2
PROFIT PROFILE OF THE SPREAD STRATEGY :
Bull Spread Using Puts Buy Put Strike X1, Premium p1; Sell Put Strike X2, Premium p2
X1 X2
Profit
ST
p1
p2
Bear Spread Using Calls
X1 X2
Profit
ST
BUY CALL STRIKE X2; SELL CALL STRIKE X1
PROFIT PROFILE:
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
Payoff Profile of a Long Butterfly Spread : Payoff Profile of a Short Butterfly Spread :
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
ST
X+p+c X-p-c
X
0
-
+
X: Strike price in put and call; c : Call premium p : Put Premium ST: Spot rate at expiry
Payoff Profile of a Straddle
Elementary Option Strategies
ST
0
+
-
Payoff Profile of a Strangle
X2 X1
(X1 + p + c) (X2 – p – c)X1: Call strike
X2: Put strike
p: Put premium
c: Call premium
ST: Spot rate at Expiry
Elementary Option Strategies Strip & Strap: CALLS + PUTS
SAME STRIKE AND EXPIRY DATE
Profit
K ST
Profit
K ST
LONG STRIP LONG STRAP
LONG (1 CALL+2 PUTS) LONG (2 CALLS+1 PUT)
+
-
O
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 optionMany 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 BankSpot Reference: 1.2720 Six Month Forward: 1.2792Transaction Date: 10/11/2008 Start Date: 14/11/2008Expiry 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 of FinalTerms and Conditions
Trigger Forward : Twelve MonthsGBP import USD export hedge using currency options
Transaction between
Customer and Bank
Spot Reference 1.7379
Twelve Month Forward
At-the-money-forward (ATMF12M) : 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 PricingNotation (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 PricingNotation (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
-
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
(t) : Standard deviation of the spot exchange rate.
e(T/360)
Hi- =
HB
e(T/360)
Fi-
FB =
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
ln(SBF/XBH) - (2/2)T
d2 = -------------------------------- T in the above formula denotes the standard deviation of log-changes in the
spot rate
dze2)/z2(-
)π2
1d
- N(d) ( =
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)/ST
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 :
{1.55P(1.55)+1.60P(1.60)+1.65P(1.65)+1.70P(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
dYf(Y)b
a
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.
TTTTTT Sd)Sf(
X X- Sd)Sf(S
X = )c(Et
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
where Et(ST) is the expected value at the current
time t of the spot rate at maturity (time T).
)d 1N()S(tE = Sd)Sf(SX
TTTT
The value of d1 is given by
2
21 +
X
)S(tEln
= dTt,
Tt,
T
1
and
where d2 = d1 - σt,T
)dN( = )Sd()Sf(X
2TT
(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,
e t) - )(Tr - r(S = F fdtTt,
(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
)d2N(Xe t)(T-r- - )d1N(e t)(T-r-St = t),St(c df0
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
X]-Sd jn-ujmax[0, ) jn-p-(1pjj)!(n-j!
n!n=j
0=je Tr1 = c 0d
0
d)(u-
d - e T/n)r-r( = p
fd
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
pue )(T/n)r-r(- = p fd
p)-(1de )(T/n)r-r(- = )p-(1 fd
]e T/n)r-r(-p)-[d(1jn-
]upe T/n)r-r(-[j
e Tr- = e Tr
d jn-) jn-p-(1pjujfdfdf
d
With these modifications, equation the call value can be rewritten as
)p'n,(k, XeTr-
- )pn,(k, eTr-
S = cdf
00
where the function Ψ is the cumulative binomial probability
) jn--(1jj)!(n-j!
n!n=j
k=j = )n,(k,
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 σT if the option life is T years.
This is achieved by choosing the following values :
ln(u) = σh ln(d) = -σh
i.e set u = eσh and d = 1/u and
p = (1/2)+(μh/2σ) (1-p) = (1/2)-(μh/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 = eσh = e0.06(0.144326) = 1.0087
d = 1/u = 0.9914
p = (1/2)+(μh/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.
