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FORWARD
Payoff
prezzo sottostante
a scadenza, ST
K
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3.2
Options
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3.3
Long Call on IBMProfit from buying an IBM European call option: option
price = $5, strike price = $100, option life = 2 months
30
20
10
0-5
70 80 90 100
110 120 130
Profit ($)
Terminal
stock price ($)
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3.4
Short Call on IBMProfit from writing an IBM European call option: option
price = $5, strike price = $100, option life = 2 months
-30
-20
-10
05
70 80 90 100
110 120 130
Profit ($)
Terminal
stock price ($)
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3.5
Long Put on Exxon Profit from buying an Exxon European put option:
option price = $7, strike price = $70, option life = 3 mths
30
20
10
0
-770605040 80 90 100
Profit ($)
Terminal
stock price ($)
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3.6
Short Put on ExxonProfit from writing an Exxon European put option:
option price = $7, strike price = $70, option life = 3 mths
-30
-20
-10
7
070
605040
80 90 100
Profit ($)Terminal
stock price ($)
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3.7Payoffs from Options
X = Strike price, ST = Price of asset at maturity
Payoff Payoff
ST STX
X
Payoff Payoff
ST STX
X
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3.8
TerminologyMoneyness :
–At-the-money option
–In-the-money option
–Out-of-the-money option
• Expiration date
• Strike price
• European or American
• Call or Put (option class)
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3.9
Types of Options
• Exchange-traded options
– Stocks
– Foreign Currency
– Stock Indices
– Futures
• Warrants
• Convertible bonds
• swapoptions
• ....
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3.10
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
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3.11
Warrants(continued)
• Warrants are traded in the same way as
stocks
• When call warrants are issued by a
corporation on its own stock, exercise
will lead to new treasury stock being
issued
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3.12
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
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3.13
Executive Stock Options continued
• They become vested after a period ot
time
• They cannot be sold
• They often last for as long as 10 or 15
years
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3.14
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
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3.15
Convertible Bonds(continued)
• 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
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3.16
Exchangeable Bonds
• An exchangeable bond is a sort of
convertible bond that provides the
conversion into the shares of a
company different from the issuer
• Usually, the underlying stock is the
equity of a strategic partnership
• There can be adverse signalling
problem which are reduced with “best
of” structures
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3.17
Trading Strategies
Involving Options
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3.18
Three Alternative Strategies
• Take a position in the option & the
underlying
• Take a position in 2 or more
options of the same type (A
spread)
• Combination: Take a position in a
mixture of calls & puts (A
combination)
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3.19Positions in an Option & the
Underlying
Profit
STX
Profit
ST
X
Profit
ST
X
Profit
STX
(a)(b)
(c) (d)
cap on
long
strategy
floor on
long
strategy
floor on
short
strategy
cap on
short
strategy
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3.20
basket of options
• spread type: basket of options of the
same type (call or put)
– bull spread
– bearish spread
– butterfly spread
• combination type: basket of options of
different types
– straddles
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3.21
Bull Spread Using Calls
X1 X2
Profit
ST
initial cash outflow
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3.22
Bull Spread Using Puts
X1 X2
Profit
ST
initial cash outflow
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3.23
Bear Spread Using Calls
X1 X2
Profi
t
ST
initial cash inflow
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3.24
Bear Spread Using Puts
X1 X2
Profit
ST
initial cash inflow
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3.25
Butterfly Spread Using Calls
X1 X3
Profit
STX2
buy 2 calls and sell 2 calls
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3.26
Butterfly Spread Using Puts
X1 X3
Profit
STX2
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3.27
A Straddle Combination
Profit
STX
bottom straddle
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3.28
A 2nd Straddle Combination
top straddle
X1
Profit
ST
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3.29
A Strangle Combination
X1 X2
Profit
ST
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3.30
A Top Vertical Combination
X1 X2
Profit
ST
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4.31
Pricing:
1. binomial tree
intuition
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Intuizione per capire
K = S(0)
Call = max [ 0 ; S(T) – S(0)
]
stato «up» : Call = Su -
S
stato «down» : Call = 0
Se fossimo «risk-neutral»
Premio = E ( Call )
= p *(Su –
S) + (1-p)*0
= p *(Su –
S)
altre ipotesi statistiche:
p = 50%
Su = S(1+σ* 𝑇)
Sd = S(1-σ* 𝑇)
Binomial Tree
Premio =
0.5* S* σ* 𝑇
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4.33
Pricing:
2. Black&Scholes
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In realtà
Approssimazione binomiale: Premio = 0.5* S*
σ* 𝑇
Approssimazione Taylor 1 ordine: Premio = 0.4* S* σ* 𝑇esempio:
se T = 1 ; σ = 30%
Premio = 0.12 * S
se T = 0,25 ; σ = 30%
Premio = 0.06 * S
Black-Scholes (at-the-money): Premio =
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4.35
Pricing:
3.Monte Carlo
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An Ito Process for Stock Prices(See pages 225-6)
where m is the expected return s
is the volatility.
The discrete time equivalent is
dS Sdt Sdz m s
S S t S t m s
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Monte Carlo Simulation
• We can sample random paths for the
stock price by sampling values for
• Suppose m= 0.14, s= 0.20, and t =
0.01, then
S S S 0 0014 0 02. .
see simple_example.xls
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Monte Carlo Simulation – One Path (continued. See Table 10.1)
PeriodStock Price atStart of Period
Random
Sample for
Change in Stock
Price, S
0 20.000 0.52 0.236
1 20.236 1.44 0.611
2 20.847 -0.86 -0.329
3 20.518 1.46 0.628
4 21.146 -0.69 -0.262
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Monte Carlo SimulationWhen used to value European stock options, this
involves the following steps:
1. Simulate 1 path for the stock price in a risk neutral
world
2. Calculate the payoff from the stock option
3. Repeat steps 1 and 2 many times to get many sample
payoff
4. Calculate mean payoff
5. Discount mean payoff at risk free rate to get an
estimate of the value of the option
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A More Accurate Approach
Use
The discrete version of this is
or
e
d S dt dz
S S S t t
S S St t
ln /
ln ln( ) /
/
m s s
m s s
m s s
2
2
2
2
2
2
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Extensions
When a derivative depends on several
underlying variables we can simulate
paths for each of them in a risk-neutral
world to calculate the values for the
derivative