OPTIONS
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Call Option
Put Option
Option premium
Exercise (striking) price
Expiration date
In, out-of, at-the-money options
American vs European Options
1
Option Valuation
• Valuation of a call option at Expiration =
max{P-X, 0}
Vc
P
X
Valuation of a put option at expiration:
max{X - P, 0}
Vp
P
X
2
Option Valuation (Cont’d)
Binominal Call Pricing (one period)
70
40%
P0 = 50
45
-10%
70 - 50 =20
V0 = ?
0
70 - 45 25 5
Hedge Ratio =
= =
20 - 0 20 4
HR: number of calls sold for each stock bought
Buy 1 shr of stock, sell 1.25 calls
If P1=$45, portfolio value = $45
If P1=$70, portfolio value = 70 - 20(1.25)=45
Return = 45/(50-1.25Vc)-1 = 0.10
Vc = $7.27
3
Option Valuation (Cont’d
Binominal Call Pricing (two periods)
P2=98.00
V2=48.00
P1=70.00
V1=24.55
P2=63.00
V2=13.00
P0=50.00
V0=11.60
P2=63.00
V2=13.00
P1=45.00
V1=4.73
P2=40.50
V2=0
4
Option Valuation (Cont’d
At T=1, If P1 = $70.00
HR = (98.00 - 63.00)/(48.00 - 13.00) = 1
Buy 1 stock, sell 1 call
If P2 = 98.00 Port. Value = 98 - 48 = 50
P2 = 63.00 Port. Value = 63 - 13 = 50
1+Return = 50/(70 - V1) = 1.1
V1 = $24.55
At T=1, If P1 = $40.50
HR = (63.00 - 40.50)/(13.00) = 1.73
Buy 1 stock, sell 1.73 call
If P2 = 63.00 Port. Value = 63 - 1.73x13 = 40.50
P2 = 40.50 Port. Value = 40.50 - 0 = 40.50
1+Return = 40.50/(40.50 - 1.73V1) = 1.1
V1 = $4.73
5
Option Valuation (Cont’d
At T=0
HR = (70.00 - 45.00) / (24.55 - 4.73)= 1.26
Buy 1 stock, sell 1.26 call
If P1 = 70.00 Port. value = 70 - 1.26x24.55 =39.07
P1 = 45.00 Port. Value = 45 - 1.26x4.73 = 39.07
Return = 39.07 / (50 - 1.26V0) = 1.1
V0 = $11.60
6
Black and Scholes OPM
X
VC  P0 N (d1 )  rt N (d 2 )
e
d1 and d2 are deviations from the expected
value of a unit normal distribution. N(d) is
the probability of getting a value below d.
ln 
d1  
P0
  [ R  (1 / 2) 2 ]t
f
X 
 t
d 2  d1   t
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Black and Scholes Eg.
P0= $50.00
X = $50.00 Rf =10% =0.60
d1 ={ ln(50/50) + [0.10+ (1/2)0.602 ]1} / 0.60
= 0.28 / 0.60 = 0.4667
d2 = 0.4667 - 0.60 = -0.1333
N(0.4667) = 0.6796
N(-0.1333) = 0.4470
Vc = 50 (0.6796) - 50 e-0.10 (0.4470)
= $13.76
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Put-Call Parity
Buy a share at P, sell a call, buy a put at the
same exercise price (X) as call.
Stock
call
put
Portfolio
Value of Portfolio if
P<X
P>X
P
P
0
X-P
X-P
0
X
X
Therefore the value of the portfolio today must
be equal to the PV of X:
P + Vp -VC = X/(1 +Rf)
or
Vp = Vc + X/(1 +Rf) - P
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Option Investment Strategies
Writing covered calls - buy stock, write cals
Synthetic long: Buy call, sell put
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Option Investment Strategies
Straddle: simultaneously buying puts and calls
with the same X and t on the same underlying
asset
Long Straddle
Short Straddle
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Option’s Delta, Gamma, and
Theta
Delta: Rate of change in position value in
response to a change in the value of the
underlying asset.
Gamma: Rate of change in delta in response
to change in the value of the underlying asset.
Theta: Change in position value as time to
expiration gets closer (other things being the
same)
delta zero; gamma +
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Portfolio Insurance
Investing in a portfolio of stocks and a put
option on the portfolio simultaneously.
The problem is when you cannot find a put
option on your portfolio.
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Portfolio Insurance Cont’d
Alternatively one can combine stock portfolio
with the risk free asset to have the same
portfolio insurance, using OPM:
N(d1) = slope of the call option value. It gives the
fall in position value for a decline of $1 in stock
value.
For portfolio insurance, invest 1 -N(d1) in t-bills,
and N(d1) in the risky portfolio.
Potential problem
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