CHAPTER SEVEND OPTIONS 1 • Introduction • Contract specifications • Option positions • Hedging using option contract • Strategy on currencies option • Option pricing 2 3 • A foreign c
Trang 1CHAPTER SEVEND
OPTIONS
1
• Introduction
• Contract specifications
• Option positions
• Hedging using option contract
• Strategy on currencies option
• Option pricing
2
3
• A foreign currency option is a contract giving the purchaser
of the option the right to buy or sell a given amount of currency at a fixed price per unit for a specified time period
– The most important part of clause is the “right, but not the obligation” to take an action
– Two basic types of options, calls and puts
• Call – buyer has right to purchase currency
• Put – buyer has right to sell currency
– The buyer of the option is the holder and the seller of the option is termed the writer
4
Wall Street Journal on Friday, January 31, 2007
-BRITISH POUND (CME)
62,500 pounds; cents per pound
Strike Price
Trang 26-Nov-15 5
6
• Every option has three different price elements
– The strike or exercise price is the exchange rate at which the foreign
currency can be purchased or sold
– The premium, the cost, price or value of the option itself paid at time
option is purchased
– Spot exchange rate in the market
7
• Options may also be classified as per their payouts
– At-the-money (ATM) options have an exercise price equal to the
spot rate of the underlying currency
– In-the-money (ITM) options may be profitable, excluding
premium costs, if exercised immediately
– Out-of-the-money (OTM) options would not be profitable,
excluding the premium costs, if exercised
8
FOREIGN CURRENCY OPTIONS MARKETS
• Over-the-Counter (OTC) Market – OTC options are most frequently
written by banks for US dollars against British pounds, Swiss francs, Japanese yen, Canadian dollars and the euro
– Main advantage is that they are tailored to purchaser – Counterparty risk exists
– Mostly used by individuals and banks
• Organized Exchanges – similar to the futures market, currency options are
traded on an organized exchange floor
– The Chicago Mercantile and the Philadelphia Stock Exchange serve options markets
– Clearinghouse services are provided by the Options Clearinghouse Corporation (OCC)
Trang 3• A long position in a call option
• A long position in a put option
• A short position in a call option
• A short position in a put option
The underlying assets
• Commodities
• Stock
• Foreign currency
• Index
• Futures…
10
PROFIT & LOSS FOR THE BUYER OF A CALL OPTION
Loss
Profit
(US cents/£)
+ 10
+ 5
0
- 5
- 10
160 165 170 175 180
Limited loss
Unlimited profit
Break-even price
Strike price
“Out of the money” “In the money”
“At the money”
Spot price (US cents/£)
The buyer of a call option on £, with a strike price of 170 cents/£, has a limited loss of 50 cents/£ at spot
rates less than 170 (“out of the money”), and an unlimited profit potential at spot rates above 170
cents/£ (“in the money”).
11
Loss
Profit (US cents/£)
+ 10
+ 5
0
- 5
- 10
160 165 170 175 180
Limited profit
Unlimited loss
Break-even price
Spot price (US cents/£)
The writer of a call option on £, with a strike price of 170cents/£, has a limited profit of 5 cents/£ at
spot rates less than 170, and an unlimited loss potential at spot rates above (to the right of) 175 cents/SF
Strike price
PROFIT & LOSS FOR THE WRITER OF A CALL OPTION
12
Loss
Profit (US cents/£)
+ 10
+ 5
0
- 5
- 10
160 165 170 175 180
Limited loss
Profit up
To 165
Strike price
“At the money”
Spot price (US cents/£)
The buyer of a put option on £, with a strike price of 170cents/£, has a limited loss of
5 cents/£ at spot rates greater than 170 (“out of the money”), and a profit potential at spot rates less than 170cents/£ (“in the money”) up to 165 cents
Break-even price
PROFIT & LOSS FOR THE BUYER OF A PUT OPTION
Trang 4Loss
Profit
(US cents/£)
+ 10
+ 5
0
-5
- 10
160 165 170 175 180
Loss up
To 165
Limited profit
Spot price (US cents/£)
The writer of a put option on £, with a strike price of 170 cents/£ has a limited profit of
5 cents/£ at spot rates greater than 165 and a loss potential at spot rates
less than 165 cents/£
Break-even price
PROFIT & LOSS FOR THE WRITER OF A CALL OPTION
Strike price
14
OPTION AND A STOCK
x
S T
Profit
(c)
x
S T
Profit
(d)
x
S T
(b)
x
S T
Profit
(a)
Profit patterns.
(a) Long position in a
stock combined with
short position in a call,
(b) Short position in a
stock combined with
long position in a call.
(c) Long position in a
put combined with
long position in a
stock,
(d) Short position in a
put combined with
stock position in a
stock
15
B1 BULL SPREAD CREATED USING CALL
OPTION-Buying a call on a stock with a certain price and selling a call on the same stock with a higher price
X1 X2 ST
Profit
This strategy limits the investor’s upside potential as well as downside risk
16
B2 BULL SPREAD CREATED USING PUT OPTIONBuying a put on a stock with a certain price and selling a put on the same stock with a higher price
X1 X2 ST Profit
Trang 5B3 BEAR SPREAD CREATED USING CALL OPTION
Buying a call one exercise price and selling a call with another strike price
Profit
This strategy limits the investor’s upside potential as well as downside risk
18
B4 BEAR SPREAD CREATED USING PUT
OPTION-Buying a put one exercise price and selling a put with another strike price
Profit
19
B5 BUTTERFLY SPREAD CREATED USING CALL
and selling two call with a strike price X 2, halfway between X 1 & X 3
S T Profit
X 2
This strategy refer to
an investment who fells that large stock price moves are unlikely
• At expiry, an American option is worth the same as a European option with the same characteristics.
• If the call is in-the-money, it is worth ST– E.
• If the call is out-of-the-money, it is worthless.
CaT= CeT= Max[ST– E, 0]
• If the put is in-the-money, it is worth E – ST.
• If the put is out-of-the-money, it is worthless.
PaT= PeT= Max[E – ST, 0]
Copyright © 2014 by the McGraw-Hill Companies,
Inc All rights reserved.
Trang 6MARKET VALUE, TIME VALUE, AND INTRINSIC
E
ST
Profit
Loss
Long 1 call
The red line shows
the payoff at
maturity, not profit,
of a call option
Note that even an
out-of-the-money
option has value—
time value
Intrinsic value
Time value
In-the-money Out-of-the-money
Copyright © 2014 by the McGraw-Hill Companies,
Inc All rights reserved.
Consider two investments:
1 Buy a European call option on the British pound futures
contract The cash flow today is –Ce.
2 Replicate the upside payoff of the call by:
Borrowing the present value of the dollar, exercise price of the
call in the U.S at i$ , the cash flow today is
Lending the present value of S T at i£, the cash flow today is
E
(1 + i$)
(1 + i£) –
7-22 Copyright © 2014 by the McGraw-Hill Companies,
Inc All rights reserved.
£) – (1 + i$) , 0
When the option is in-the-money, both strategies have the same payoff
When the option is out-of-the-money, it has a higher payoff than the borrowing and lending strategy
Thus,
Using a similar portfolio to replicate the upside potential of a put, we can show that:
£)
7-23 Copyright © 2014 by the McGraw-Hill Companies,
Inc All rights reserved.
The Black - Scholes formula for pricing the European foreign currency call and put are
where
c = premium on a European call
p = premium on a European put
S = spot exchange rate (domestic currency/foreign currency)
F = continuous compounding Forward rate
E = exercise or strike price, T = time to maturity
rd= domestic interest rate, rf= foreign interest rate
σ = Volatility (standard deviation of percentage changes of the exchange rate)
OPTION PRICING AND VALUATION
) N(
)
T r T
h
e d F d E
T
T σ E F d
T
2 1
2
1
ln
d2 d1 T
T r t t f h
e S
Trang 7e-rT= continuously compounding discount factor (e=2.71828182…)
ln = natural logarithm operator
N(x) = cumulative distribution function for the standard normal
distribution, which is defined based on the probability density
function for the standard normal distribution, n(x), i.e.,
1
2
12
365
12% 1
(1+12%) 1.12 (1 12% / 2) 1.1236 (1 12% /12) 1.126825 (1 12% / 365) 1.127446
e 1.1274969
2
x
2
-1 N(x) = n(x)dx= e dx
2
OPTION PRICING AND VALUATION
OPTION PRICING AND VALUATION
• The pricing of currency options depends on six
parameters:
– Current spot exchange rate ($1.7/￡)
– Time to maturity (90 days)
– Strike price ($1.72/￡)
– Domestic risk free interest rate (r$= 8%)
– Foreign risk free interest rate (r￡= 7.8%)
– Volatility (10% per annum)
Based on the above parameters, the call option premium is
$0.0246/￡(this result is calculated based on the Black-Scholes
formula in the excel file “GK” Garman Kohlhagen)
27
Spot rate (DC/FC e.g USD/EUR) 170 Call Price = 2.4666
Strike price 172 volatility (annualized) 10.00% Put Price = 4.3453
domestic interest rate (annualized) 8.00%
foreign interest rate (annualized) 7.80%
time to maturity in days 90 time to maturity in years 0.25
Trang 8Exhibit: Intrinsic Value, Time Value & Total Value for a Call Option
on British Pounds with a Strike Price of $1.70/£
1.69 1.70 1.71 1.72 1.73 1.68
1.67
1.66
0.0
1.0
2.0
3.0
4.0
5.0
Spot Exchange rate ($/£)
Option Premium
(US cents/£)
3.30
5.67
4.00 6.0
1.74 1.67
Total value
Intrinsic value
Time value
Valuation on first day of 90-day maturity
Exhibit: Intrinsic Value, Time Value & Total Value for a Call Option
on British Pounds with a Strike Price of $1.70/£
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
18.00
20.00
157.25 159.80 162.35 164.90 167.45 170.00 172.55 175.10 177.65 180.20 182.75 185.30
Spot exchange rate
FX Call Option Value and intrinsic value
Time value
Intrinsic value
Total value
3.30
• The total value (premium) of an option is equal to the
intrinsic value plus time value
• Time value captures the portion of the option value due to the volatility in the underlying asset during the option life
– The time value of an option is always positive and declines with time, reaching zero on the maturity date
• Intrinsic value is the financial gain if the option is exercised immediately
– On the date of maturity, an option will have a value equal to its intrinsic value (due to the zero time value at maturity)
OPTION PRICING AND VALUATION
CURRENCY OPTION PRICING SENSITIVITY
• If currency options are to be used effectively, either for the purposes of speculation or risk management, the traders need to know how option values react to their various factors, including S, K, T, rf, rd, and σ
• More specifically, we will study the sensitivity of option values with respect to S, K, T, rf, rd, and σ
• These sensitivities are often denoted with Greek letters,
so they also have the name “Greeks” or “Greek letters”
Trang 9• Spot rate sensitivity (delta):
– Delta is defined as the rate of change of option price with
respect to the price of the spot exchange rate
– Delta is in essence the slope of the tangent line of the option
value curve with respect to the spot exchange rate
– For calls, Δ is in [0, 1], and for puts, Δ is in [-1, 0]
– For call (put) options, the higher (lower) the delta, the call
(put) option is more in the money and thus the greater the
probability of the option expiring with a positive payoff
f
f
-r T 1
-r T 1
c Delta (for calls) e N(d ) > 0
S p Delta (for puts) e N(-d ) < 0
S
DELTA
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
Spot exchange rate
DELTA
• For the example, the delta of the option is 0.5, so the change of the spot exchange rate by ±$0.01/￡ will cause the change of the option value approximately by 0.5× ±$0.01 = ±$0.005 More specifically, the option value will become $0.033 ± $0.005
• Please note that the Delta estimation works well only when the change of the exchange rate S is small (If the spot exchange rate increases by
$0.1/￡, the Delta estimation predicts the option value becoming $0.083
• The larger the absolute value of Delta, the larger risk the portfolio is exposed to the exchange rate changes
THETA
• Time to maturity sensitivity (theta): # – Option values increase with the length of time to maturity
– A trader with find longer maturity options better values, giving trader the ability to alter an option position without suffering significant time value
c Theta θ (for calls) 0
T p Theta θ (for puts) 0
T
Trang 10• 90 to 89 days:
• 15 to14 days
• 5 to 4 days
• The rapid deterioration of option value in the last days prior to
expriration day
02 0 89 90
28 3 3 3
cent time
premium
05 0 14
15
32 1 37 1
cent time
premium
08 0 4
5
7093 0 7929 0
.
cent time
premium
Theta: Option Premium Time value Deterioration
※ The negative slope means the option value decreases with the time
approaching the expiration date
※ For the at-the-money options, the decay of option values
accelerates when the time approaches the expiration date
VEGA
• Sensitivity to volatility (Vega): # – The vega for calls and puts are the same
– Volatility is important to option value because it measures the exchange rate’s likelihood to move either into or out of the range in which the option will be exercised
– The positive value of vega implies that both call and put values rise (fall) with the increase (decrease) of σ
– The intuition for positive vega of both calls and puts is that since the options give the holder the right to fix the purchasing or the selling prices, options are more valuable in the scenario with higher volatility
f
f
-r T 1
-r T 1
c Vega ν (for calls) =Se n(d ) T 0
σ p Vega ν (for puts) =Se n(d ) T 0
σ
VEGA
• Volatility increase 1%, from 10% 11%:
• If the volatility rise, the risk of the option being exercised is increasing, the option premium would be increasing
30 0 10
11
033 0 036 0
.
%
%
.
$
$ volatility
premium
Trang 11RHO AND PHI
• Sensitivity to the domestic interest rate is termed as rho
※rd↑, domestic currency↓, foreign currency↑, because the call (put)
can fix the purchase (sale) price of the foreign currency, call↑ and put↓
• Sensitivity to the foreign interest rate is termed as phi
※rf↑, domestic currency↑ , foreign currency↓, because the call (put)
can fix the purchase (sale) price of the foreign currency, call↓ and put↑
d
d
-r T 2 d
-r T 2 d
c
R h o ρ ( f o r c a lls ) = K T e N ( d ) > 0
r p
R h o ρ ( f o r p u t s ) = K T e N ( -d ) < 0
r
f
f
- r T 1 f
- r T 1 f
c
P h i φ ( f o r c a l l s ) = S T e N ( d ) < 0
r p
P h i φ ( f o r p u t s ) = S T e N ( - d ) > 0
r
Rho
• US dollar interest rate increase 1%, from 8% 9%:
• If the US dollar interest rate increase of 1%, the ATM call option
premium increase from $0.033 to $0.035/£
2 0 0 8 0 9
033 0 035 0
.
%
%
.
$
$ rate erest int
$ US
premium
Phi
• British Pound interest rate increase 1%, from 8% 9%:
• If the £ interest rate increase of 1%, the ATM call option premium decrease from $0.033 to $0.031/£
• Phi value is -0.2
2 0 0
8 0 9
033 0 031 0
.
%
%
.
$
$ rate erest int BP
premium
Interest Differentials (rd– rf) and Call Option Premiums
※ When the interest rate differential (rd– rf) increases, the foreign currency call value indeed increases
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
ITM call (K=$1.65/￡) ATM call (K=$1.70/￡) OTM call (K=$1.75/￡)
Option premium (U.S cents/￡)
r US$ – r ￡