Transcript Chapter 20

Chapter 20
Option Valuation and Strategies
Portfolio 1
Buy a call option
– Write a put option (same x and t as the call
option)
What is the potential payoff of this portfolio?
–
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St < =X
Payoff of call owned
0
Payoff of put written - (X - St)
Total
St - X
St > X
St - X
0
St - X
Portfolio 2
Buy the stock
– Borrow the present value x
What is the potential payoff of this portfolio?
St <= X
St > X
Payoff of stock
St
St
Payoff of put written
-X
-X
Total
St - X
St - X
–
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Portfolio 1 and 2 have identical payoffs so
they must be worth the same amount or else
there would be an arbitrage opportunity.
 C - P = S - X / (1 + rf)t
 This is the put-call parity relationship
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Put-Call Parity Arbitrage
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Stock price = 110
Call price = $17 (T = 6 months, and X = 105)
Put Price = $? (T = 6 months, and X =$105)
rf = 5% for a six-month period
What is the equilibrium value of the put option?
Suppose the WSJ states that the above put is selling for $5,
is there an arbitrage opportunity? If so, then create a pure
arbitrage.
 The
equilibrium value of the put is
C - P = S - X / (1 + rf)t
 17 - P = 110 - 105 / 1.05
 P = $7
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If the put is selling for $5 then an arbitrage opportunity
exists.
C - P = S - X / (1 + rf)t
 17 - 5 = 110 - 105 / 1.05
 12 > 10
 Buy low sell high.
 So you would buy the right hand side of the
equation and sell the left hand side.
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Buy the Stock and borrow the present value
of the exercise price
 Sell the call and buy the put option
 To be a pure arbitrage you must show that
there was zero net investment and that there
is a guaranteed profit.
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Position
Initial CF
Buy Stock
-110
Borrow PV X
+100
Sell Call
+17
Buy Put
-5
Total
2
Cash flow in six months
St <= 105
St > 105
St
St
- 105
- 105
0
-(St - X)
105 - St
0
0
0
Black Scholes Option Pricing Model
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Vo = Ps x F (d1) - (Pe / ert ) x F (d2)
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d1 = ln (Ps / Pe) + { r + ( s2 / 2) } T
s (T1/2)
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d2 = d1 - s (T1/2)
Example of Black Scholes Model
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Stock Price
Exercise Price
Time to expiration
Standard deviation
Interest Rate (risk free)
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Calculate the intrinsic value of IBM’s call option.
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=
=
=
=
=
102 11/16
105
2 months
25%
6% annually
Solution to B-S Problem
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d1 = ln (102.69 / 105) + { .06 + ( .252 / 2) } .17
.25 (.171/2)
d1 = -.065
d2 = -.065 - .25 (.171/2)
d2 = -.1681
Vo = 102.69 x F (-.07) - (105 / e(.06)(.17) ) x F (-.17)
V0 = 102.69 x .4721 - 103.93 x .4325
V0 = $3.53
The Hedge Ratio
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The ratio of the change in the price of a call option
to the change in the price of the stock.
The hedge ratio is also called the options delta.
Is the delta positive or negative for a call option?
Is the delta positive or negative for a put option?
How do you calculate the hedge
ratio?
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The numerical value of F(d1) is the hedge ratio.
d1 = ln (Ps / Pe) + { r + ( s2 / 2) } T
s (T1/2)
Look in the cumulative normal distribution table
to find F(d1).
Example of Hedge
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In our example F(d1) = .4721, this means that the
price of the option will rise $.47 for every $1
increase in the price of the stock. Thus if the
investor owns 100 shares of stock and has written
2.12 calls, a $1 increase in the stock will generate
a $1 decrease in the option. The gain in one
position is exactly offset by the loss in the other
position.
Example of Hedge (cont.)
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Number of call options to hedge 100 shares
1 / Hedge ratio
Defines the number of call options that must be
sold for each 100 shares purchased.
Our example: 1 / .4721 = 2.118195
The hedge ratio may also be viewed as the number
of shares that must be purchased for each option
sold. In our example, the hedge ratio of .4721
implies that 47.21 shares purchased for every call
option sold is a hedged position.
Additional Option Strategies
Covered Put
 Protective Put
 Straddle
 Bull Spread
 Bear Spread
 Butterfly Spread
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Covered Put
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Short the stock and sell the put
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Profit or Loss
Price of Stock
Protective Call
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Short the stock and buy the call option
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Profit or Loss
Price of Stock
The Long Straddle
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Purchase of a put and a call with the same exercise price
and expiration date.
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Profit or Loss
Price of Stock
The Short Straddle
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Write a put and a call with the same exercise price and
expiration date.
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Profit or Loss
Price of Stock
The Bull Spread
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Purchase the call option with the lower X and sell the call
option with the higher X
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Profit or Loss
$1.5
Price of Stock
-$3.5
The Bear Spread
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Purchase the call option with the higher X and sell the call
option with the lower X
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Profit or Loss
$3.5
Price of Stock
-$1.5
The Butterfly Spread
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Involves three option at different strike prices. Example
buy two of the options with the middle strike price and sell
the options with the higher and lower strike prices
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Profit or Loss
$1
Price of Stock
-$4
The Butterfly Spread
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Involves three option at different strike prices. Example
sell two of the options with the middle strike price and buy
the options with the higher and lower strike prices
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Profit or Loss
-$4
Price of Stock
-$1