8.1 properties of stock option prices some model-independent results chapter 8
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8.1
Properties ofStock Option PricesSome 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)
c p C PVariable
S0
KTrD
+ + –+
? ? + ++ + + ++ – + –
–– – +
– + – +
8.4
American vs European Options
An American option is worth at least as much as the corresponding European option
C cP p
8.5
Pricing bounds
8.6Calls: An Arbitrage
Opportunity?
• Suppose that
c = 3 S0 = 20 T = 1 r = 10% K = 18 D = 0
• Is there an arbitrage opportunity?• (Buy the call, sell underlying, deposit PV(K)!)
8.7Puts: An Arbitrage
Opportunity?• Suppose that
p = 1 S0 = 37 T = 0.5 r =5%
K = 40 D = 0
• Is there an arbitrage opportunity?• (Buy put, buy stock, borrow
PV(K)!)
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
• For an American call option:
S0 = 100; T = 0.25; K = 60; D = 0Should 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?
An Extreme Situation
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 S D Ke
p D Ke S
rT
rT
0
0
8.16
Extensions of Put-Call Parity
• American options; D = 0
• European options; D > 0
c + D + Ke -rT = p + S0
• American options; D > 0
rTKeSPCKS 00
rTKeSPCKDS 00
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)
8.19Positions in an Option & the Underlying (Figure 9.1, page 186)
Profit
STK
Profit
ST
K
Profit
ST
K
Profit
STK
(a) (b)
(c)
(d)
8.20
Bull Spread Using Calls(Figure 9.2, page 187)
K1 K2
Profit
ST
8.21
Bull Spread Using PutsFigure 9.3, page 189
K1 K2
Profit
ST
8.22
Bear Spread Using CallsFigure 9.4, page 189
K1 K2
Profit
ST
8.23
Bear Spread Using PutsFigure 9.5, page 190
K1 K2
Profit
ST
8.24
Butterfly Spread Using CallsFigure 9.6, page 191
K1 K3
Profit
STK2
8.25
Butterfly Spread Using PutsFigure 9.7, page 192
K1 K3
Profit
STK2
8.26
Calendar Spread Using CallsFigure 9.8, page 193
Profit
ST
K
8.27
Calendar Spread Using PutsFigure 9.9, page 193
Profit
ST
K
8.28
A Straddle CombinationFigure 9.10, page 194
Profit
STK
8.29
Strip & StrapFigure 9.11, page 195
Profit
K ST
Profit
K ST
Strip Strap
8.30
A Strangle CombinationFigure 9.12, page 196
K1 K2
Profit
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.