OPTIONS MARKETS
Options
• Similar to futures; however, they give the buyer (holder)
the right but not the obligation to buy/sell the underlying
asset at a specified future date and at a specified
exercise (strike) price)
• The buyer pays the price for this right (option premium),
which is determined by the supply and demand in the
market
• The seller (writer ) sells the right and receives the
premium but has the obligation to buy/sell the underlying
in the future if the option is exercised by the buyer
Options
• A call option is an option (the right but not the obligation) to buy a
certain asset by a certain date for a certain price (the strike price)
• A put option is an option (the right but not the obligation) to sell a
certain asset by a certain date for a certain price (the strike price)
• An American option can be exercised at any time during its life
• A European option can be exercised only at maturity
• Underlying: Stocks, Foreign Currency, Stock Indices, Futures
• Example LIFFE: Equity-based futures and options (FTSE 100
Index, FTSE 250 Index, FTSE Eurotop, MCSI Euro Index, MSCI
Pan-Euro Index, MSCI European); Individual equities; Commodity
futures and options (Robusta Coffee, White Sugar, Feed Wheat,
Milling Wheat, Rapeseed and Corn).
4 basic option positions
• Long call
• Short call
• Long put
• Short put
Positions at maturity
• Long call option: The buyer (holder) has the right but not the
obligation to BUY the underlying asset at a specified future date and
at a specified exercise (strike) price)
• Short call option: The seller has the obligation to deliver the
underlying asset at a specified future date and at a specified
exercise (strike) price)
• Long put option: The buyer (holder) has the right but not the
obligation to SELL the underlying asset at a specified future date
and at a specified exercise (strike) price)
• Short put option: The seller (holder) has the obligation to BUY the
underlying asset at a specified future date and at a specified
exercise (strike) price)
Actions
• The buyer of an option may:
→ Exercise the option
→ Liquidate the option
→ Let the option to expire unexercised
• The seller of an option may:
→ Wait for the option to expire
→ Liquidate the option
Closing a position
• All open positions may be closed out by doing an opposite
transaction with an option of the same series (same characteristics,
underlying, strike, maturity)
• Assume you have bought a call option on the FTSE-ASE20 that
matures in December, with a strike price of 400 index points. You
paid for the option 20 index points.
• To close the position you must sell a call option on the FTSE-ASE20
that matures in December, with a strike price of 400 points (If you
sell the option for more that 20 index points you will make a profit).
Covered and naked calls
• A covered call is a cal option where the seller already
owns the underlying and can deliver it if the buyer
exercises the option.
• A naked call is a cal option where the seller does not
already own the underlying and has to buy it from the
market in order to deliver if the buyer exercises the
option.
• Selling naked calls is a dangerous investment practice
since the losses can be very high.
Option specifications
•
•
•
•
Expiration date
Strike price
European or American
Call or Put (option class)
• E.g. a European January call on stock X
with a strike of $100.
Trading options
• Over The Counter (OTC): The major participants are
banks the make the market (market makers) big
multinationals, etc. The contracts are not standardized
and “tailor-made” for clients
• Organized exchanges: Most organized exchanges use
market makers to facilitate options trading; A market
maker quotes both bid and ask prices when requested;
The market maker does not know whether the individual
requesting the quotes wants to buy or sell
Option price = intrinsic value + time value
• Intrinsic Value: The quantity by which the current price
of the underlying is higher from the strike price
• In other words, the value of the option if it was exercised
today
• E.g. January call at 240: P = 254 pence, strike = 240
pence. Thus the intrinsic value is 14 pence
• What is the rest (25-14 = 11 pence) ?
• The rest is the time-value of the option
Intrinsic Value
• «in-the-money» options: when they have positive intrinsic value
• «at-the-money» options: price = strike price
• «out-of-the-money» options: when they have ‘negative’ intrinsic
value
• If St > E
call option is «in the money»
put option is «out of the money»
• If St < E
call option is «out of the money»
put option is «in the money»
• Εάν St = E
call & put are «at the money»
Time Value
• Time value declines as we approach maturity, until it
decays (time decay)
• The decline rate is not linear and it increases as we
approach matutrity
Margins
• Margins are required when options are sold
• For example when a naked call option is written
the margin is the greater of:
1 A total of 100% of the proceeds of the sale plus
20% of the underlying share price less the
amount (if any) by which the option is out of the
money
2 A total of 100% of the proceeds of the sale plus
10% of the underlying share price
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
8.17
Warrants
• Warrants are options that are issued (or written) by a corporation or
a financial institution
• The number of warrants outstanding is determined by the size of the
original issue & changes only when they are exercised or when they
expire
• Warrants are traded in the same way as stocks
• The issuer settles up with the holder when a warrant is exercised
• When call warrants are issued by a corporation on its own stock,
exercise will lead to new treasury stock being issued
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
8.18
Executive Stock Options
• Option issued by a company to executives
• When the option is exercised the company issues more stock
• Usually at-the-money when issued
• They become vested after a period of time (usually 1 to 4 years)
• They cannot be sold
• They often last for as long as 10 or 15 years
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
8.19
Convertible Bonds
• Convertible bonds are regular bonds that can be
exchanged for equity at certain times in the future
according to a predetermined exchange ratio
• Very often a convertible is callable
• The call provision is a way in which the issuer can force
conversion at a time earlier than the holder might
otherwise choose
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
8.20
Example: Long in a call option
• Long position on a 3-month European call option on stock ABC with
a strike price of $120, option price of $4, current stock price of $118
• Standardization: Each contract is for 100 stocks.
• We have the right to buy in 3 months 100 shares of ABC at $120
• For this right we pay today $4 per stock (i.e. $400)
• Assume that in 3 months the stock price is up by 15% approximately
at about $135. Exercise?
Example: Long in a call option
• Exercise and buy 100 shares at $120
• Sell in the market the shares for $135
• Gain: $15 per share
• Cost $4 per share
• Net profit: $11 per share
• Net profit: $1,000 (returns of aprox. 375%)
Example: Long in a call option
•
Assume that in 3 months the stock price is down by
15%, at about $100.
•
Exercise?
•
If exercised we will pay $120 for a share that is now
worth $100 and loose $2,000.
•
We do not exercise: Loss of $400
•
100% of our initial capital.
Example: Long in a call option
• Assume that in 3 months the stock price is up by 3%, at about
$121.5.
• Exercise?
• If exercised we will pay $120 for a share that is now worth $121.5
and gain $150.
• Cost: $400
• Loss: $400 - $150 = $250
• 60% of our initial capital.
Example: Long in a put option
• Assume that you buy a 2-month European put option on currency Χ
at a strike of $0.64/Χ. Each contract is standardized at 62,500Χ.
• The premium is $0.02 for each X.
• Position: In two months we have the right but NOT the obligation
• To Sell currency X
• At an exchange rate of $0.64/Χ
• We pay for this right now $1.250 ($0.02 x 62.500 Χ)
Example: Long in a put option
• Assume that in 2 months the rate is 0.58/Χ
• Profit or Loss?
• We buy spot at $0.58 for $36,250 ($0.58 x 62,500)
• Exercise the option and deliver for $40.000 ($0.64 x 62,500)
• Profit $ 3,750 ($40,000 - $36,250)
• Net Profit $2.500 ($ 3,750 - $1,250)
Example: Long in a put option
• Assume that in 2 months the rate is 0.68/Χ
• Profit or Loss?
• DO NOT Exercise the option and sell (deliver) for $0.64 a currency
that is now worth $0.68
• Loss: $1,250
• 100% of initial capital
Returns from options (ignoring the premium)
• St = price of underlying at maturity, E = strike price
• Return of long call at maturity:
= St - E
if
St > E
=0
if
St  E
• Returns of a short call at maturity:
= - (St - E)
if
St > E
=0
if
St  E
• Return of long put at maturity:
=0
if
St > E
= E - St
if
St < E
• Returns of a short put at maturity:
=0
if
St > E
= - (E - St)
if
St < E
Example of index options
• Athens Stock Exchange: Calls & Puts
• FTSE/ASE – 20, FTSE/ASE – 40
• Price in units, multiplier 5 Euro
• Table: Trading activity on January 2007 option contracts
→ underlying FTSE/ASE-20
→ 8 December 2006, time: 14.25)
→ Current FTSE/ASE-20 price: 2385 points
Option Valuation
Binomial Trees
• Consider a stock with a current price of $10
• Assume that it is known with certainty that in 3 months it
will be worth either $11 or $9
• How much will a 3-month European option should be
worth if the risk free interest rate is 8% and the strike
price is $10.5?
In 3 months
•
If p = $11 then the value of the option will be $0.5 (St - E = 11 – 10.5)
•
If p = $9 then the value of the option will be $0 (Not exercised)
•
What is the value of the call (f) today?
How to get it?
• Assume no arbitrage and create a risk free portfolio with the stock and
the option in such a way that there is no uncertainty in 3 months
• E.g. Long Δ shares, Short 1 call option
• If at maturity stock price goes from $10 to $11
→ stock value will be $11x Δ
→ option value will be $0.5
→ Portfolio Value: $(11Δ – 0.5)
• If at maturity stock price goes from $10 to $9
→ stock value will be $9 x Δ
→ option value will be $0.
→ Portfolio Value: $(9Δ - 0)
How much is Δ;
• The portfolio will be riskless only if we choose Δ in such a manner
that the final value is equal for both prospects:
11Δ – 0.5 = 9Δ  Δ = 0.25
• In other words we must buy 0.25 for every stock we sell for the
portfolio to be riskless
• If f is the value of the option today then the value of the portfolio
today will be:
10Δ – f = 10(0.25) – f
The value of the portfolio
• If at maturity stock price goes from $10 to $11
→ Portfolio Value: $(11x0.25 – 0.5) = $2.25
• If at maturity stock price goes from $10 to $9
→ Portfolio Value: $(9 x 0.25 – 0) = $2.25
• A riskless portfolio must return the risk free rate and its Present Value
will be:
PV = FV e-rt = 2.25e-(0.08)(3/12) = 2.205
• Since the portfolio value today is 10(0.25) – f
• Since the Present value of the portfolio is 2.205
• Then 10(0,25) – f = 2,205
• Solving for f : f = 0,295
Generalization
• Consider a stock with a current price of S
• Assume that it is known with certainty that in T months the price:
→ Will increase from S to Su and the call to fu
→ Will decrease from S to Sd, and the call to fd
• How much will a T-month European option should be worth if the
risk free interest rate is r% and the strike price is E?
How to get it?
• Assume no arbitrage and create a risk free portfolio with the stock and
the option in such a way that there is no uncertainty in T months
• E.g. Long Δ shares, Short 1 call option
• If at maturity stock price goes from S to Su
• → stock value will be Su x Δ
→ option value will be fu
→ Portfolio Value: SuΔ – fu
• If at maturity stock price goes from S to Sd
• → stock value will be Sd x Δ
→ option value will be fd
→ Portfolio Value: SdΔ – fd
How much is Δ;
• The portfolio will be riskless only if we choose Δ in such
a manner that the final value is equal for both prospects:
Su Δ – fu = Sd Δ - fd
Δ = ( fu - fd ) / ( Su - Sd )
The value of the portfolio
• A riskless portfolio must return the risk free rate and its
Present Value will be: PV = FV e-rt = (SuΔ – fu) e-rT
• Since the portfolio value today is:
SΔ – f
• The Present Value and the value today must be equal:
SΔ – f = (SuΔ – fu)e-rT

f = SΔ – (SuΔ – fu)e-rT
The value of the option
• Substitute Δ
Δ = ( fu - fd ) / ( Su - Sd )
• And solve for the option price:
f = e-rT [ p fu + ( 1-p ) fd ]
Where:
p = (erT – d ) / (u – d)
• Αντικατάσταση Δ
Δ = ( fu - fd ) / ( Su - Sd )
• Και λύση ως προς την τιμή του
δικαιώματος:
f = e-rT [ p fu + ( 1-p ) fd ]
Όπου:
p = (erT – d ) / (u – d)
In the numerical example:
• Stock
from 10 to 11
from S to Su
u = 1.1 (10 x 1.1 = 11)
• Stock
from 10 to 9
from S to Sd
d = 0.9 (10 x 0.9 = 9)
r = 8%, T = 0.25, fu = 0.5, fd = 0
• p
p
p
=
=
=
(erT – d ) / (u – d)
(e(0.08) (0.25) – 0.9 ) / (1.1 – 0.9)
0.601
• f
f
f
=
=
=
e-rT [ p fu + ( 1-p ) fd ]
e-(0.08)(0.25)[(0.601) 0.5+(1-0.601) 0]
0.295
Generalization for two steps:
• Each step will last Δt and the stock price will:
Repetitions show that:
• fu = e-rΔT [ p fuu + (1-p) fud ]
• fd = e-rΔT [ p fud + (1-p) fdd ]
• f = e-rΔT [ p fu + (1-p) fd ]
• Replace the first two in the third:
f = e-2rΔT [ p2fuu + 2p(1-p)fud + (1-p)2 fdd ]
In practice
• When we built a binomial tree we choose u and d that matches the
true volatility of the underlying (σ = standard deviation)
• u = e σ√ΔΤ
• d = e -σ√ΔΤ
• The real probability of an increase is μ = expected return)
• q = (eμΔΤ – d ) / (u – d)
• Cox, Ross, Rubinstein (1979, Journal of Financial Economics, 7)
• In practice an analyst will divide the life time of an option to steps of
duration Δt (e.g. one month = 30 steps) and in every step there will
be two possibilities (up, down)
• The analyst will end up with 31 possible final stock prices and 230
(over a billion) possible price paths
Exercise
• Stock price = $100
• In each of the following 2 6-month periods will go up or
down by 10%
• R=8%
• What is the value of a 12-month European call with E=
$100?
Solution
• fu = e-rΔT [ p fuu + (1-p) fud ]
• fd = e-rΔT [ p fud + (1-p) fdd ]
• f = e-rΔT [ p fu + (1-p) fd ]
•
f = e-2rΔT [ p2fuu + 2p(1-p)fud + (1-p)2 fdd ]
• p = (erΔT – d ) / (u – d)
Solution
u=?
d=?
100 x u = 110 → u = 110 / 100 = 1.1
100 x d = 90 → d = 90 / 100 = 0.9
p = (e0.08(6/12) – 0.9 ) / (1.,1 – 0.,9) = 0.70
f = e-2(0.08)(6/12) [0.72 (21) + 2(0.7)(0.3)0 + 0.32 0]
f = $9.61
Alternatively
• fu
• fd
=
=
=
e-rΔT [ p fuu + (1-p) fud ]
e-(0.08)(6/12) [0.7(21) + 0.3(0)] = $14.2
0
• f
f
=
=
e-(0.08)(6/12) [ 0,7 (14.2) + 0.3 (0) ]
$9.61