Transcript Properties of Stock Options

8.1
Properties of
Stock Option Prices
Some model-independent results
Chapter 8
8.2
Notation
• c : European call
•
•
•
•
•
option price
p : European put
option price
S0 : Stock price today
K : Strike price
T : Life of option
: Volatility of stock
price
• C : American Call option
•
•
•
•
price
P : American Put option
price
ST :Stock price at option
maturity
D : Present value of
dividends during option’s
life
r : Risk-free rate for
maturity T with cont comp
8.3
Effect of Variables on Option
Pricing (Table 8.1, page 168)
Variable
S0
K
T

r
D
c
+
–
?
+
+
–
p
–
+?
+
–
+
C
+
–
+
+
+
–
P
–
+
+
+
–
+
8.4
American vs European Options
An American option is worth
at least as much as the
corresponding European
option
Cc
Pp
8.5
Pricing bounds
Calls: An Arbitrage
Opportunity?
• Suppose that
c=3
T=1
K = 18
S0 = 20
r = 10%
D=0
• Is there an arbitrage opportunity?
• (Buy the call, sell underlying, deposit
PV(K)!)
8.6
Puts: An Arbitrage
Opportunity?
• Suppose that
p =1
T = 0.5
K = 40
S0 = 37
r =5%
D =0
• Is there an arbitrage
opportunity?
• (Buy put, buy stock, borrow
PV(K)!)
8.7
8.8
Put-Call Parity; No Dividends
(Equation 8.3, page 174)
• Consider the following 2 portfolios:
– Portfolio A: European call on a stock + PV of the
strike price in cash
– Portfolio C: European put on the stock + the stock
• Both are worth MAX(ST , K ) at the maturity of the
options
• They must therefore be worth the same today
– This means that
c + Ke -rT = p + S0
8.9
Arbitrage Opportunities
• Suppose that
c =3
S0 = 31
T = 0.25
r = 10%
K =30
D=0
• What are the arbitrage possibilities and
how are they exploited when
p = 2.25 ?
• p=1?
8.10
Early Exercise
• Usually there is some chance that an
American option will be exercised
early
• An exception is an American call on a
non-dividend paying stock
• This should never be exercised early
8.11
An Extreme Situation
• For an American call option:
S0 = 100; T = 0.25; K = 60; D = 0
Should you exercise immediately?
• What should you do if
1 You feel confident about the performance of
the stock for the next 3 months?
2 You do not feel that the stock is worth holding
for the next 3 months?
8.12
Reasons For Not Exercising a
Call Early, Case 1)
(No Dividends )
• No income is sacrificed by holding the option
• We delay paying the strike price
• Holding the call provides insurance against
stock price falling below strike price
• Everything else is the same
• => wait, don’t exercise
8.13
Case 2)
• The investor should rather sell
the option
• He will receive more than the
intrinsic value since other
investors prefer to hold the
stock. Otherwise price could not
be S0.
8.14
Should Puts Be Exercised
Early ?
Are there any advantages to
exercising an American put
when
S0 = 60; T = 0.25; r=10%
K = 100; D = 0
8.15
The Impact of Dividends on
Lower Bounds to Option Prices
(Equations 8.5 and 8.6, page 179)
c  S0  D  Ke
p  D  Ke
 rT
 rT
 S0
8.16
Extensions of Put-Call Parity
• American options; D = 0
S0  K  C  P  S0  Ke rT
• European options; D > 0
c + D + Ke -rT = p + S0
• American options; D > 0
S0  D  K  C  P  S0  Ke rT
8.17
Trading Strategies
Involving Options
Note: Net profit diagrams
Chapter 9
8.18
Three Alternative Strategies
• Take a position in the option and
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)
Positions in an Option & the
Underlying (Figure 9.1, page 186)
Profit
Profit
K
K
ST
ST
(a)
(b)
Profit
Profit
K
ST
(c)
K
(d)
ST
8.19
8.20
Bull Spread Using Calls
(Figure 9.2, page 187)
Profit
ST
K1
K2
8.21
Bull Spread Using Puts
Figure 9.3, page 189
Profit
K1
K2
ST
8.22
Bear Spread Using Calls
Figure 9.4, page 189
Profi
t
K1
K2
ST
8.23
Bear Spread Using Puts
Figure 9.5, page 190
Profit
K1
K2
ST
8.24
Butterfly Spread Using Calls
Figure 9.6, page 191
Profit
K1
K2
K3
ST
8.25
Butterfly Spread Using Puts
Figure 9.7, page 192
Profit
K1
K2
K3
ST
8.26
Calendar Spread Using Calls
Figure 9.8, page 193
Profit
ST
K
8.27
Calendar Spread Using Puts
Figure 9.9, page 193
Profit
ST
K
8.28
A Straddle Combination
Figure 9.10, page 194
Profit
K
ST
8.29
Strip & Strap
Figure 9.11, page 195
Profit
Profit
K
Strip
ST
K
Strap
ST
8.30
A Strangle Combination
Figure 9.12, page 196
Profit
K1
K2
ST
8.31
Warning!!
• Next week (but only Tuesday) our use of
quantitative methods will accelerate. Chapter
10 onwards.
• We need models for uncertainty to price
options.
• We will start using stochastic variables and
processes.
• The fall break is a good opportunity to
refresh/review your stat and prob knowledge.